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Investigation of the force between two non-commutative U(2) monopoles Chu, Karene Ka Yin 2004

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Investigation of the Force between Two Non-Commutative U(2) Monopoles by K A R E N E K A Y I N C H U B.Sc, The University of Br i t ish Columbia, 2001 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 2004 ©Karene K a Y i n Chu , 2004 JUBCL THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF G R A D U A T E STUDIES Library Authorization In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. \x.fyE-ENE K A - YIN £ 4 W Name of Author (please print) Date (dd?mm/yyyy) Title of Thesis: [ VWf h " t k 0 f ^ WfW^^ Two' Degree: Year: 2 Department of The University of British Columbia Vancouver, BC Canada grad.ubc.ca/forms/?formlD=THS page 1 of 1 last updated: 20-Jul-04 Abstract The force between two widely separated 't Hooft-Polyakov monopoles involves an ordinary Coulomb force as well as an attractive force wi th the same magni-tude mediated by a scalar field. Manton arrived at this fact using an ansatz he discovered for a weakly accelerating monopole [1]. We study Manton's method, eliminate its ambiguities, interpret the ansatz as the external force law for a monopole, and compare it wi th another approach that uses the stress-energy tensor [2]. We find that the force between two monopoles in non-commutative spacetime does not alter from that in commutative spacetime to first order in the non-commutative parameter, 6, both by extending Manton's method and by finding the total energy of the system. We investigate Manton's method at 0(92) but find it not very promising . We understand that the non-commutativi ty starts to affect dynamics only at 0{62). i i A c k n o w l e d g e m e n t I thank my supervisor, Moshe Rozal i , for giving me this very interesting topic, supporting me for longer than we thought this would take, being always available and approachable for me to talk to, always communicative, responsive and kind-hearted, for offering his insights on the problem from which I learn, being flexible about where I need to be, being caring about my progress and my future, (the Rajaraman book on solitons,) and finally for rushing to read this thesis for me. I thank Mark van Raamdonk for being my reader, for being concerned about me during and after the electromagetism class. I thank Kr is ten Schleich and Tom Matt ison, my undergraduate thesis super-visor, for taking an interest in my affairs and for looking out for my future. I thank al l my great friends for al l their k ind words, for l istening to me, for believing in me and encouragine me in times of difficulty. On top of that, I thank K i m for lett ing me stay at her place and using everything of hers, Henry for supporting me when unfairness happened in the string theory course, Michael for tel l ing me I need to do things according to my own standards, Somayeh for believing entirely in me, Jen de Benedictis for being on my side and encouraging me, Tara, Christ ine for motivating me and thinking for me, and Oswald for supporting me and making his computer available to me. I also thank my friends here for always being there to party and have fun with me. Final ly, I thank my father for giving me the value that education is important i i i and for lett ing me make my decision about where to go to school, and my mother for always bringing me good food and being concerned about my l iv ing conditions. Of course, the list does not stop here. iv C o n t e n t s Abstract ii Acknowledgement iv 1 Introduction 1 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory 3 2.1 Monopoles in Maxwel l Electromagnetism 4 2.2 Monopoles in SU(2) Yang-Mi l ls Theory 5 2.2.1 The Act ion and the Equations of Mot ion 6 2.2.2 The Asymptot ic Condi t ion and the Factorized Equations of Mot ion 7' 2.2.3 Monopole Solut ion-both charges 12 2.2.4 Topological Nature and Quantizat ion of Charge 14 2.2.5 Solution at the Core of the Monopole and the B P S L im i t . 15 3 Manton's Method to Find Force Between Two Commutative Monopoles 18 3.1 Manton's Ansatz for a Single Accelerating Monopole 20 3.1.1 Derivation of the Accelerated Equat ion of Mot ion and M a n -. ton's First Order Ansatz 20 3.1.2 The Solution of a Single Accelerating Monopole 25 3.1.3 The First Order Ansatz as the External Force Law . . . . 36 3.2 Manton's Method to Determine the Acceleration between Two Monopoles ! 37 3.2.1 The Globa l Solutions and the Matching Procedure 38 3.2.2 Clarif ications and Comments on Manton's Method . . . . 45 3.2.3 Limitat ions of the Manton's Method 52 3.3 F ind ing the Force through Calculat ing the Momentum F lux . . : 59 3.3.1 Stress-Energy Tensor and Reduction to the Electr ic problems 60 3.3.2 Manton's O(e0) Global Solution as the Static Solution . . 63 3.4 Conclusion for the Commutat ive Problem 66 4 Background: Non-Commutative U(N) Gauge Theory 69 4.1 Operator Formal ism to Star product Formal ism 69 4.2 The Act ion of Non-commutive U(N) Gauge Theory • . 75 4.2.1 Gauge Invariant Quantit ies 79 4.2.2 Broken Lorentz and Rotat ional In variance 81 4.3 The Equations of Mot ion 82 4.3.1 Expansion of the Equat ion of Mot ion 83 5 First Order Force between Two Non-commutative Monopoles 90 5.1 Non-Commutat ive Monopoles 90 5.2 Force Correction from the Stress-Energy Tensor 92 5.2.1 Non-Conservation of the Stress-Energy Tensor 92 5.2.2 O (8) correction to the Force Between Two Non-Commutat ive Solitons . 97 5.3 Force correction Using the Manton Method 100 vi 5.3.1 First Order Ansatz for Single Accelerating Non-Commutat ive Monopole 100 5.3.2 0(0) Force Correction 103 6 P r e l i m i n a r y i n v e s t i g a t i o n o f t h e M a h t o n M e t h o d t o O(02) 1 0 9 6.1 The SU(2) Component of the First Order Ansatz . . . . . . . . . 110 6.2 Calculat ion of the Loca l U ( l ) Solution I l l 6.2.1 U ( l ) static Monopole Solution '. I l l 6.2.2 U ( l ) perturbed monopole 116 6.3 Problems to be Solved: the Globa l Solutions 128 6.4 Conclusion for the Non-Commutat ive Problem 135 B i b l i o g r a p h y 1 4 0 vi i Chapter 1 I n t r o d u c t i o n Magnetic monopoles are interesting because they are solutions to al l grand unified theories yet they have not been found in nature [3]. Non-commutative geometry is interesting because it seems to always come up when we look for quantum gravity solutions [4]. Non-commutative Yang-Mi l ls theory, for instance, describes a l imi t ing case of string theory [5] and is also the only gauge theory in which gauge transformations include translations and therefore can serve as a toy model for quantum gravity [6]. As well, it is the only known generalization of ordinary Yang-Mi l ls theory that preserves maximal supersymmetry [7]. In this project, we were interested in magnetic monopoles in a non-commutative Yang-Mi l ls theory. In particular, we wanted to find the force between two monopoles separated widely by a distance, s, in the U(2) perturbatively non-commutative gauge theory with a scalar field in flat space-time. We noticed that Manton has found the force to order between two monopoles in the SU(2) gauge theory in commutative space-time. He first discovered how the solution near one monopole changes under a weak acceleration, then uses the structure of the equations of motion to arrive at a solution for the region in between the two monopoles, and finally determines the force by equating the local and global solution where both are valid [1]. We were interested in extending this method for our problem. We achieved the following: 1 Chapter 1 Introduction 1. We studied the Manton method in detail. We demonstrated the ambiguities of the method , and proposed the principle wi th which to find the correct solution. Using our understanding, we reinterpreted the method as the application of a constant external force law on either monopole. We then investigated the scope of the method by carrying out the method at dipole order and found that it works only for the lowest order force. We then studied Goldberg's way [2] to find the force between two monopoles using the stress-energy tensor on a static solution of the system. We found that its statement about the contributions to the force agrees wi th the force law above. We discussed the possibil i ty of using Manton's global solution as the static solution to conclude that there is no higher order forces between two monopoles using Manton's global solution as the static solution. 2. For the non-commutative U(2) theory, we derived the analogue of M a n -ton's ansatz for a single accelerating monopole. We showed using both the stress-energy tensor and the Manton method that the force between two non-commutative monopoles remains the same as that between two com-mutative monopoles to first order in the non-commutative pararrlter 9. We started to investigate the Manton method at second order in 6. We found that we can calculate the local accelerating monopole solution without dif-ficulty with the help of the symmetry of the theory [8] [9], and showed a sample calculation. We studied how the non-commutativi ty hinders us from finding the global solutions in the same way Manton did. This thesis was written in essentially the order described above. To make the report easier to follow, we chose to explain the background theories at dif-ferent parts rather than all in the beginning. We included al l calculations in the main text instead of appendices but made sure that before each long algebraic calculation a summary was given. 2 Chapter 2 B a c k g r o u n d : S i n g l e M o n o p o l e i n C o m m u t a t i v e A c c e l e r a t e d Y a n g - M i l l s T h e o r y Magnetic monopoles are classical solutions to field theories whose magnetic field far away from its center looks as though there is a magnetic charge at the center, that is, f B —y — in the asymptotic region. They have not been detected in nature yet, but is predicted by al l theories in which an internal symmetry group spontaneously breaks down to the U ( l ) group of Maxwel l Electromagnetism [3]. In these grand unified theories, monopoles typ-ical ly have such big masses that they are not likely to be produced by supernovae or current accelerators, but rather would have been produced copiously shortly after the B ig Bang and would have hardly annihilated [10]. Their absence then is quite puzzl ing and should inform us about the very early universe. This is one of the main reasons why we study monopoles. 3 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory 2.1 Monopoles in Maxwell Electromagnet ism Maxwel l electromagnetism is built without magnetic charges. The divergence of the magnetic field being zero in this theory allows us to write it in the form of the curl of a vector potential field and to bui ld the field tensor = d^A" - dvA» for the Lagrangian formalism. The only way to bui ld a "monopole" solution is to approximate a magnetic source by the end of a infinitely long and thin solenoid. In this case, the vector potential would be undefined where the solenoid is. Expl ic i t ly , if we choose coordinates such that the solenoid is placed at the negative z-axis, then a vector potential (in spherical components) whose curl gives the monopole magnetic field would be Ar = 0, A9 = 0, A* = g (1 - cos0) where g is the apparent magnetic charge. We can see that on the negative z-axis, where 6 = IT, the vector potential does not make sense since it points in all directions curl ing about the z-axis. This half-line where the vector potential is ill-defined is known as the Dirac String and is unavoidable however the vector potential is chosen. The half-line singularity is only a mathematical defect if it cannot be detected by experiment. This is true classically but not quantum mechanically. If we per-form a double slit diffraction experiment in which charged particles pass through two slits on a screen and are to be detected on a second screen, we would find that the interference pattern detected in the case where no solenoid is present in between the two different paths of the particles is different from the inference pattern when a solenoid is present [3]. This is because the probabil i ty density which determines the interference pattern is the square of the total wave function: P = |*! + *2|2 4 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory where and ^ 2 are the wave functions of particles passing through the two different slits. The presence of a solenoid in between the two paths would con-tr ibute to a phase difference between \I>i and \&2, and the probabil i ty density would become: Psolenoid = | * i + e i e ( 4 ^ * 2 | 2 where e is the electric charge and ing is the magnetic flux through the solenoid. Dirac's statement is that the two interference patterns would be the same if the phase difference contributed by the solenoid is 2imi, which translates to the following relation between the electric and magnetic charge [11]: N q = —where N is an integer y 2e Other arguments would show that with the above relation, the solenoid could not be detected by any other conceivable experiments [3]. Therefore, the monopole is a genuine monopole which is not distinguishable experimentally from a monopole created by a single magnetic charge. The theory of monopole is thus started. 2.2 Monopoles in SU(2) Yang-Mills Theory Monopoles takes a more elegant presence in Yang-Mi l ls Theory, which is essen-t ia l ly a generalization of the classical field theory of electromagnetism wi th U ( l ) gauge symmetry to one with a larger gauge group SU(2). We wi l l now see how U ( l ) electromagnetism can be embedded in this "bigger" theory, which can be seen as a prototype for grand unified theories, specifically, an SU(2) Yang-Mi l ls theory with a scalar field in a Mexican-hat potential in four-dimenional Minkowski space-time, and how monopoles exist as non-singular classical solutions to it. 5 c Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory 2.2.1 The Action and the Equations of Motion The action of such theory looks like: S = ^ ^ d z 4 T ^ [ - C 7 ^ ) G ^ ^ where G " " = 0 M " - dvA" - ie[A», Av]\ (2.1) D»(j) = 0"0 - i e ^ " , 0] fo r / i = 0 ,1 ,2 ,3 (2.2) where the fields are in the adjoint representation of SU(2), i.e., 0 and A^ are 2 x 2 Hermit ian matrices and so can be written as linear combinations of the three Pau l i matices: 0 = 0 a y ; A' = A{^ for a = 1,2,3 Since the Pau l i matrices satisfy the following identities: 0- acr b i crc 1 1 «7C = -eQfcc — o a h — ; I r — = 0 2 2 2 2 2 2 2 we can treat them as the basis of the vector space R 3 and represent the fields as vectors: 0 = (0 i , 02i ^3) ! A 1 = (A\, A%2, Al3) where i is the spatial index with the vector cross product corresponding to the commutator of the matrices and the dot product to the trace of products of matrices: - i [ ^ , 0 ] |->• A " x 0 TriG^G^) |-> G^-Gfu, The action in this vector notation becomes S = \ j d x A [-G^ix) • G^(x) + 2D"0(:r) • D M 0 ( x ) - A (0(x) • 0(x) - c 2)] where G " " = 9 " A " - duAfi + e A " x A„; • D " 0 = d"0 + e A " x 0 6 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory Vary ing the action with respect to the scalar field (j) gives the first equation of motion: r D " D „ 0 = - A (0 • cj) - c 2 ) <j> (2.3) Varying with respect to the gauge field A " gives the second: D M G ^ = -eD u(j) x </> (2.4) In this vector notation, the infinitesimal gauge transformations of the gauge field look like 6A» = e x A " + d"e where e is the infinitesimal gauge parameter, but those for the the scalar field and the field strength are simply infinitesimal rotations in R 3 8<f) = e x cj) 8GT = f x G " " Gauge invariant quantities are then invariants of this rotation, length of the vector fields which rotate in this internal R 3 space under a gauge transformation. 2 . 2 . 2 The Asymptotic Condition and the Factorized Equa-tions of Motion We are looking for a finite energy configuration of fields that would give rise to some U ( l ) monopole magnetic field in some asymptotic region. Therefore, we need to impose some conditions on the fields such that the total energy is finite, define what it means to be in the asymptotic region, and also find a way to embed the U ( l ) electric and magnetic fields in this SU(2) theory such that the U ( l ) fields satisfy the Maxwel l equations. 7 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory The C o n d i t i o n s at r —> oo The energy of a classical solution is given by the Hami l tonian which is related to the action in the usual way: H = \fdx4 [G^ix) • G^x) + 2D*V(a;) • D ^ x ) + A {<f>(x) • 4>{x) - c 2)] This is only finite if each term vanishes at infinity. Ignoring the first term for now, the last two terms vanishing at infinity implies the following boundary conditions for the scalar field and the gauge field: \<j>\ —> c as r oo |DM0| ->• 0 as r ^ oo If we write <j> as the product of its magnitude and a unit vector field (in SU(2) gauge space): 4> — h(x)(f)(x) where |</>(z)|2 = 1 the conditions above become conditions on h(x) and 0(x): DM</> = 0 as r -> oo d^h — 0 as r oo h —>• c as r —¥ oo The first two follow from the fact that DM</> can be separated into two perpen-dicular components and each needs to vanish: D"0(x) = (d»h)<t> + KDfl4>(x) = (5^)0 + h (8*4 + e A " x 4>) but £"(<£ • 0) == 0"(1) = 0 implies 0 - 5 ^ = 0 and ( A " x j>) • j> = 0 therefore DM0 _L 4> 8 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory The Asymptotic Condition We want to define an asymptotic region between infinity and the core of the monopole where the above conditions may not all be satisfied but where the embedded U ( l ) magnetic field defined later on would satisfy the vacuum Maxwel l equations. If we treat the right hand side of the equation of motion (Eq 2.4) as some "matter" current [2] jn x ^ (2.5) then the matter current vanishes when = 0 (2.6) This is the asymptotic condit ion that we wi l l use in the next chapter [1]. Note that unlike at r —> oo, d^h = 0 is not imposed in the asymptotic region. Factorization of Equations of Motion We wi l l now see how this condition gives rise to the definition of the U ( l ) field strength tensor that wi l l define the magnetic field for the monopole. The condit ion is true if the following relation between the gauge field and (j) is satisfied: A" = -d"4> x 4> + A"0 e because the second term of the covariant derivative of <p: A" x 4> = (-6*4 x 4> + \»$) x 4> = (d"4> • dl) 4> - {4> • 4>)d"4> = e would cancel with its first term. W i t h this relation, the-field strength tensor G ^ " can be factorized into a unit vector field, d)(x), and the magnitude of the field strength, a scalar in the gauge 9 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory space, which also varies over space-time. Expl ic i t ly , G " " = d^Au - dvA» + e A " x A„; where Au - d u A" = -dv4> x d»4> + [d^W - a"A") 4> + {\ vd»4> ~ A " ^ ) e A" x A" = -(d*4> x 0) x x $) + x 0) x (A">) + (A"0) x (du4> x e = [d'V • {du4> x $)] 4 - [A"3"^ - A " ^ ] = - [ ( ^ x 9 > ) - ^ - [ A " ^ - A " ^ ] But we already know that for any fi and u JL <j> and dv4> ± <f> such that x 9 " ^ || ^ which means e e Therefore, the SU(2) field strengh points in the direction of <j>: G"" = — e x 0"0) -j> 4> + [ ^ A " - 9"A"-] We define /^ " (x ) to be the length of the SU(2) vector G ^ [1]: 1 / G^-4> = (d^cj) x d v(j)) • (j) + d^X - d ' A " and note that it is a gauge invariant quantity. Now, the equation of motion and the Bianchi Identity for G^" imply the free Maxwel l equations for 1. Equat ion of motion: D ^ G " " = -eD"<j> x <f> {dpf") 4> + / " " D " ^ = -e ({dvh) 4> + KD"<j>} x h<f> =^d»r = 0 (2.7) 10 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory 2. Bianchi Identity: D ^ e ^ G ^ = 0 d^fap = Q (2.8) We have embedded U ( l ) electromagnetism in this theory with f^u being the Maxwel l field strength tensor. If <f> points in only one direction, for instance <f> = (0, 0, h), then the field strength takes the usual form: Unl ike ordinary electromagnetism, the field strength contains also the term in-volving <f>, and this wi l l allow the monopole solution to be non-singular by giving rise to a topological charge. We wi l l discuss this in section 2.2.4 Notice that the equations for f^u are decoupled from h. It is the other equation of motion E q 2.3 that factorizes in the asymptotic region to give the equation of motion for h: D„D "0 = -A(|0| 2 - c2)4> = ([d^h)<fi + h(d fi(f>+ A M x </>)^  = {d^h) 4> + (&Mh)T>ll4> = {d^h)i =>d^h = -\{h2-c2) (2.9) Although h and seem to be independent of each other in the asymptotic region, they are not at the core of a monopole where the equations of motion are not decoupled. The relation they need to satisfy in the core is given through a first order ansatz in section 3.1.1. Another important property of these asymptotic equations is their l inearity in the U ( l ) fields / ' " ' and h. In the next chapter (section 3.2.1), we wi l l explain 11 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory how Manton relies on this fact to find the solutions for the region between two monopoles. 2.2.3 Monopole Solution—both charges Now that we have a U ( l ) field strength that satisfies the Maxwel l equations, we can define a U ( l ) magnetic field in the usual way: B'=l-e^kp\l\l) Since / J ' f c can be written in terms of only </> and A' without involving h, re-str ict ing Bl to the monopole drop-off in the asymptotic region gives a condition for 4> and A' decoupled from h. Since the monopole magnetic field satisfies the free Maxwel l equations which come from the equation of motion and the Bianchi Identity, any 0 and A' that produces the monopole field is automatical ly a solution to the equations of motion. Since the SU(2) scalar h is related to the magnetic field B% through the first order ansatz mentioned above which can be evaluated in the asymptotic reion also, the asymptotic profile of Bl in fact gives a condit ion on the scalar h. We wi l l discuss this in section 3.1.1. To solve for </>, we first try to find a relationship between the direction 4> is pointing at and the magnetic field B%. It turns out that there is a solution for the choice A' = 0 (which Manton referred to as a gauge choice but is incorrect). For each gauge index (d=l ,2 or 3), the gradient in real space of 4>d is perpendicular to 12 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory the magnetic field Bl: (Below we write the gauge indices explicit ly as subscripts.) (d^d) B{ = Fk&h ^-Yeeabcdj4>bdkdlc^ .., . _ . . , „ 0 for d = b, b = c or c = d, by antisymmetry of e*7'* but eX3k 8f(j>dai<f>b8rk<f>c = I ~ , - -A I ± • ( d J 0 x d * 0 J for d / 6 ^ c and ( d J 0 x d f c 0 ) || 0 , d > J_ 0 imply d * 0 • ( d J 0 x d f e 0 ) therefore B* = 0 (2 This means that al l the components of 0 is constant along the field lines of B. For a single monopole then, 0 is constant along the radial direction and so depends only on the spherical coordinate angles, 9 and x, where 9 is the angle a vector makes wi th the H-z-axis and x is the azimuthal angle. The solution for a single charge monopole can be very simple in some fixed gauge: 0 a = ± 6 1 -r and we can check that its magnetic field is indeed the monopole field: _—eijkeabc (^ - — ^ (±—) 2e o c 1 r r 3 / \ r r 3 / V r / 2e a V r 3 / x{ = =p—- for 0 = ±f er3 Accordingly, the gauge field is: A ^ = -eabcdl4>b4>c e -eabc((±Si-)(±^)-(±°^-) (±^)) e \ r r rA r ) = ~eac ^ for 0 = ±f a c 9 13 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory Now, for the monopole with a single negative charge, there is another solution. Instead of reflecting the solution for the positive charge monopole (0® = —f) about the origin in the SU(2) space (such that 0 e = + f ) , we can reflect < f^fi about only one plane to obtain </>e; for instance, 1 4>ei ^ -<t>m \ 0©3 ) Since (f>e has only one component with a relative minus sign, B l, given by E q 5.1, would be negative. This is the solution that comes about when we generalize the solution to multiple charges. 2.2.4 Topological Nature and Quantization of Charge We now try to generalize the above solution to higher charge single monopoles. First , the divergence of B does not depend on the term wi th A f c , and is actually zero everywhere except at the origin: d'B 1 = e ijk (^^eabcd j b^dk4>cd i^>a + 2d id ja k^ = e^ k(-^eabcd b^dk4>cd a^ = 0 by similar arguments as in E q 2.10 B y the divergence theorem, then, the magnetic flux through a surface enclosing the monopole core depends also only on <j>. Now, for a monopole with a single positive charge, <j> maps the 2-sphere in physical space, which is parametrized by 9 and x, to a 2-sphere in gauge space once. Since <j) depends only on the angles and not on r, we can generalize to higher charges by choosing one angle, for instance X, and defining 0 such that it has mapped a section of the 2-sphere in real space (described by x = 0 to some xo) to an entire 2-sphere in gauge space before x 14 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory reaches 2TT, such that 4> = (sin # cos Nx, sin # cos Nx,cos6). For x > 2n/N, the mapping of <f> to the 2-sphere starts anew. Now, for <j> to be single-valued, the values of (f> at x = 0 and x = 27r have to be the same, and so N needs to be an integer, whether positive or negative. We can then check that this (j) gives the magnetic flux for N integral magnetic charges: 4nN L Bldal = Surface Here, the Dirac quantization condit ion, g oc is recovered. N is called the winding number or the topological charge of the solution. Notice that the negatively charged monopoles are <f> wi th —N. This means that looking from above the x-y plane, if the vector 0® (at al l z values) rotates counterclockwise as x increases, the vector 0 e (at al l z values) would rotate clockwise as x increases. A n important point is that there is no smooth gauge transformation that would take the solution from one N to another. Two solutions with different iV's are said to be in different homotopy sectors. It is because the magnetic charge in this theory occurs as a topological charge that the situation of the Dirac string in the Maxwel l theory can be avoided. 2.2.5 Solution at the Core of the Monopole and the BPS Limit We have only looked at the solution in the asymptotic region so far, since it already captures the most important aspects of the monopole solution and is what wi l l be discussed in the rest of this report. 15 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory The explicit solution at the core of a single charge monopole was found by Bogonolmy, Prasad and Sommerfield [12] [13] in what is known as the B P S l imit fo l lowing' t Hooft and Polyakov's ansatz [14] [15]. The outline is as follow. Instead of using an asymptotic condition such that the SU(2) field strength tensor G M " would factor under it, 't Hooft defined the U ( l ) electromagnetic field strength everywhere as the gauge-invariant expression [16]: 1 tHooft (D^d) x D"0 ) • <f>\ + G^-</> Whi le in the asymptotic region, the first term vanishes because D M 0 = 0 and the second term factors as before, in the core, after being expanded, this U ( l ) field tensor st i l l takes the form we had before in the asymptotic region: 1 tHooft (d"d) x du(j)) • (j) + 0 "A" - 9 "A" with A*1 = A " • do. Now, without the asymptotic condit ion, the Bianchi Identity of G M " does not imply the Bianchi Identity for f ^ 0 0 f t D u ^ rather Fortunately, the right hand side is identically zero because of its form as argued in E q 2.10 and would integrate to a non-zero magnetic flux over a surface enclosing the monopole. The other two Maxwel l equations d f " - 0 uf*J tHooft —  u follow from the equation of motion of C " . 't Hooft and Polyakov proposed an ansatz such that f!*tHooft would give a single charge monopole magnetic field far away from the monopole core: 4>a = —h(r) (2.11) r K = *ajPyW(r) (2.12) 16 Chapter 2 Background: Single Monopole in Commutative Accelerated Yang-Mills Theory where h(r) and W(r) satisfy the following boundary conditions: 1. h(r) —> c as r —> oo for the'energy reason discussed before; 2. W(r) ^ as r -» oo such that A 1 = (1/e) 9*0 x 0 as r —>• oo, which is the asymptotic relationship between the gauge field and <j> derived before. In the asymptotic region, the 't Hooft-Polyakov solution would reduce to the solution discussed in the previous sections. The BPS L i m i t For the equations of motion to be satisfied, h(r) and W(r) need to satisfy a set of coupled "non-autonomous" differential equations [16] and has been solved only in the l imit where the amplitude of the Mexican-hat potential [A(|</>|2 — c2)] goes to zero, i.e., A -> 0, while the value of \<f>\ at which this potential is min imum is retained such that \(j>\ st i l l needs to approach c at infinity. This is known as the B P S l imit [12]. The ful l solution of the single charge monopole in this l imit is the ansatz in E q 2.11 and E q 2.12 with h and W solved: h(r) t anh (ce r ) er W(r) = — - C er s i nh (ce r ) Note that these, functions are smooth and do not diverge at the origin. 17 Chapter 3 M a n t o n ' s M e t h o d t o F i n d F o r c e B e t w e e n T w o C o m m u t a t i v e M o n o p o l e s In electromagnetism, the force acting on an electrically charged particle by the electric and magnetic field is given by the Lorentz force law F = q(E + v x B) which is the equation of motion derived from varying the particle action Sparticle = f [~ \f" U» ~ -M") dx" + / V^O% where J M is a conserved current, i.e., d^J^ — 0, with its t ime component being the electric charge density and the spatial components being the electric current densities flowing in the three different spatial directions. The magnetic monopole, on the other hand, is not a point particle but field configurations that extend over space, and there is no separate "particle" action for its dynamics. How do we find the force acting on it then? How do we find the force between two opposite charge monopoles and that between two same charge monopoles? 18 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Canonical ly, we can find the force on an enclosed region, in which a monopole can situate, by calculating the momentum flux through its boundary surface us-ing the stress-energy tensor. However, Manton arrived at the correct answers by his own method, and we outline it below: Suppose two monopoles with same or opposite charges separated by a large distance, s, accelerate with a small acceleration, e2a, from rest due to the force each experiences. For this instant, Manton assumes the fields of the monopoles to be rigidly accelerating in opposite directions, and using this assumption simpl i -fies the time-dependent equations of motion to equations that involve only spatial derivatives and terms with e2a. He then discovers a first order ansatz for each of the different charge monopoles to solve the modified equations up to 0(e2). Now, in the asymptotic region defined by-Eq 2.6, the ansatz and its derivative for each monopole become equations for h, and (/>; recall that h, and <p determines the full solution since (j) — h(j) and A1 = d%4> x 4> after having chosen \l — 0. Manton then cleverly chooses a gauge in which the ansatzes are linear in terms one of the components of 0, and in which 0 for different charge monopoles have the same dependence on \t such that a solution for \I> in the region between two monopoles would imply a solution for (f> in the same region as well. He solves for h and * for each monopole with its own charges and direction of acceleration for both O(e0) and 0{e2). F inal ly, he uses the linearity in h and ^ of the ansatz and builds the "global" solutions, hgi0bai, and ^global, for the region between the monopoles by adding the h and ^ functions from the different monopoles, but with the freedom of adding any homogeneous solutions. He requires that to 0(e2), these global functions reduce to the solutions for each ansatz at each monopole, and deter-mines the assumed acceleration e2a in the matching process. 19 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles In this chapter, we study Manton's method in details, correct and clarify a few of his statements. We interpret the method as the application of an external force law on each monopole and find the l imitat ions of this force law. We also study the canonical method mentioned above and look at statements made in the Manton method from that viewpoint. 3.1 Manton's Ansatz for a Single Accelerating Monopole 3.1.1 Derivation of the Accelerated Equation of Motion and Manton's First Order Ansatz Manton starts by deriving the modification to the static equations of motion for the instant, t = 0, when a monopole starts to move. He assumes that the monopole accelerates "a l i tt le bit" r igidly from rest such that the scalar field and the gauge field only have time-dependence in terms of the Taylor-expanded spatial coordinates: <K*V) = <f>( xi-\ *2) ; A V ) = A j (*' - \ ( e V ) e where e2a% is the small acceleration. The time derivatives of these fields become non-zero: d°d}-e2aitdi(j) ; d°Aj = -e2aH diAj (3.1) Manton also makes the other term in the covariant t ime derivative depend on time in the same way. To accomplish that, he chooses a gauge in which A 0 = 0 in the instantaneous rest frame of monopole such that at a small t ime t, A 0 in the non-accelerated lab frame would be obtained by a Lorentz boost with the 20 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles relative velocity v = —e2at: A 0 = -e2aitAi Combining the two terms, he writes the covariant t ime derivative of 4> and of G j 0 in terms of the covariant spatial derivatives: D °0 = -e2aH {dl<p + eAi x 0) = -e2aH D > G>° = -e2aH (VA1 - dlAi + eA^' x A 4 ) = -e2aH Gji Then, using these, he manipulates the equations of motion. He applies another covariant t ime derivative on these quantities, but keeps terms up to only 0(e2): D0D°</> = e V ( D V ) + (e2aHAj) x ( - e V t D V ) = e V ( D V ) + C(e 4)(3.3) B0G0j = -e2alGji + (e2aHAj) x (-e2aHGij) = -e2aiGji +C>(e4)(3.4) and he substitutes these in the equations of motion. The one equation of motion involving only d), in terms of the acceleration e2a\ looks like = D ^ D ' + e V ^ = A ( | 0 | 2 - c 2 ) (3.5) The time component of the other equation of motion becomes DjGj0 = - e D V x d> -e2aH BjGji = ^e2aH D J > x 0 (3.6) Factoring out e2alt, this equation reduces to simply one of the static equations of motion and is to be satisfied by the O(e0) solution: The spatial component of this second equation of motion can be written with the covariant t ime derivative replaced with terms with the acceleration as well: BiGij + D 0 G 0 j = (D* + e2al)Gij = -eB j(f> x 4> (3.7) Note that 21 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles 1. to O(e0), al l three equations, E q 3.5, 3.7, and 3.6, reduce to the static equations of motion, so the O(e0) solution is also also the static monopole solution; 2. there are no explicit time derivatives in al l three equations, and since the spatial derivatives of the fields with the accelerated coordinate de-pendence equal those of the fields with the static coordinate dependence, d% (cj)(x -4- l /2e 2 at 2 ) ) = <9J(</!>(:?)), changing the argument of the fields from the spatial coordinates to the accelerated coordinates is consistent with these equations. A t t = 0, the accelerated and the non-accelerated coordi-nate are the same, (x + l/2e2at2) = x, so we can now look for the solutions that have x as the argument. The First Order Ansatz Manton discovers a first order ansatz that solves the "perturbed" equations of mot ion(Eq 3.5, 3.7) in the B P S l imit (section 2.2.5): Gij = ±eijk(Dk + e2ak)(j> (3.8) where the different signs correspond to 1 the different charge of the monopoles, as explained below. We check that it indeed solves the perturbed equations of motion. First , we substitute the ansatz in E q . 3.5: D ' ( D V ) + e V ( D V ) = Bi(±hijkGjk-e2ai(f)) + e2ai(Bi(f)) = ± - € y * D ' G J ' * = 0 2 and see that it is satisfied using the Bianchi Identity. Second, we substitute the 22 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles ansatz in E q . 3.7, and find that this second equation is also satisfied: (D* + e2ai) Gij = T>i[±eijk(Dk + eV)0] + f?ai{±ei'kDk<f>) + 0(e4) = ±eijkDiBk(f> + 0(e4) = ±eijk [didkd> + eAi x dkd> + di (eAk x <f>) + e2Ai x (Ak x = ±eijk [e(dr' Ak) x 0) + e ^ A * • (j))] Now - € i i k G i k x d) = eijk(diAk) x i> + ^eijk{eAi x Ak) x 0 = eijk [(diAk) xcj) + eAk(Ai • d>)] so ( D ' + e2ai)Gii = ±\eijkeGik x <f>. = ± [Te(D J - + 62aj)d> x 0] = - e D J > x 0 The First Order Ansatz in the Asymptotic Region Recal l from the last chapter that we write <f> = h$ and that the asymptotic condit ion D f c 0 = 0 allows the gauge field to be determined by 4> only if ^ = 0, so solving for h and 4> wi l l give the complete solution in the asymptotic region. Recal l also that in this region, the U ( l ) magnetic field B is given in terms of 0 through the asymptotic condit ion, and the monopole requirement that B -> r/r2 has sufficed to give <j> to C(e°). Manton's ansatz then provides the relation between B and h up to C(e2), and so allows us to first solve for the O(e0) h which is part of the static solution. It also allows us to solve for the C(e2) corrections to both (j) and h for the accelerating monopole. Manton's ansatz factorizes and reduces to the following relation between B and h in the asymptotic region: Bk4l=\eijkGijd) = ±\(dkh)4> + hDkd] + e2akh4> => Bk = ± (dkh + e2akh) asymptotical ly (3.9) 23 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles As expected, this ansatz is consistent with the original equation of motion (Eq 2.9): d^h = dk(dkh + e2akh) = dkBk = 0 Now, the ansatz needs different relative signs for the opposite charge monopoles. This is because if oppositely charged monopoles are described by the same ansatz, h wi l l switch sign as the charge and therefore Bk is switched; but we already know that under the charge inversion, one of the components of .</> also switches sign; this means that both © and 0 monopoles wi l l have the same solution <f> = h<f> if both satisfy the same ansatz. To avoid this ambiguity, we need the different ansatzes for different charge monopoles. We choose the sign convention that Bk = +(dkh + e2akh) for the © monopole. Consequently, h for both © and © monopoles are the same to O(e0): fc<°> = - - + c r k such that ± dkhw = ± ^ 3 = Bke{Jl and h{0) -)• c as r -> oo Secondly, we want to derive the equation for <f> for the accelerating monopole which preferably would not depend on the unknown hM2\ Expanding E q 3.9,in orders of e: £*(»>(,£)+ fl*(e2)(0) = ±\dk(hW + h^) + e2akhM we see that applying the curl operator to both sides would get r id of the term with h^: eijkdj t-BhW + B k ( e  ^ = ± £ i j k ^JQk ^(0) +  ^ + e2ak ^(0) j = ± e i j k (£20fc) Bj(t Substi tut ing back the zeroth order magnetic field, we obtain an equation for decoupled from f x e2a V x 5 e / e < 0 ) = ± — ^ — (3.10) 24 Chapter 3 Manton's' Method to Find Force Between Two Commutative Monopoles where e 2 a points in the direction towards which the monopole in question is accelerating. We can see that when the acceleration is zero, the right hand side of the equation vanishes and the equation turns back into the static equation. When the acceleration is non-zero, the right hand side can be interpreted as the time derivative of the electric field as in the Maxwel l equation, just that it is very small . 3.1.2 The Solution of a Single Accelerating Monopole Solution of 4> In this section, we present the O(e 0) (f> solution in a different gauge and solve E q 3.10 for 0(e 2) correction to 4>. To simplify the problem, Manton chooses a gauge such that the solution would preserve the symmetry about the axis of separation of the monopoles. In the last chapter, we have defined 0 for opposite charges according to which direction <f> rotates as the azimuthal angle x increases. For a system of two opposite charge monopoles on the z-axis, this choice of angle to define the monopole charge would break the convenient axial symmetry of the solution, since (f) near each monopole would be winding in different directions about the z-axis. Recal l that gauge transformations of (f> are simply its rotations in the internal R 3 space, so we can choose another angle to define the winding. Manton chooses 9 such that for a single monopole, if we look down on a plane defined by a constant x, for instance the x-z plane, the oppositely charged monpoles would respectively have (f> rotating clockwise and counterclockwise as 9 increases. This way, axial symmetry for the two-monopole system can be preserved provided that the corrections of (f) due to the acceleration also exhibit axial symmetry. We wi l l see in section 3.2.1 how this gauge is crucial for solving for the two monopole system. 25 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles M a g n e t i c f i e ld i n t e r m s o f \I> a n d T The next step is to write E q 3.10 in terms of the two degrees of freedom of 0 . Suppose due to the acceleration, 0 depends on the angles x a n d 9 differently than for a single static monopole, and is written in the form / y/1 ~ *(J)»T(X) X i = (3.11) J such that | 0 | = 1 is st i l l true. Note that for a single static monopole, $f(9) = =F cos 9 and T(x) = cos x-To see what E q 3.10 means for \l>(0) and T(x), we first write the magnetic field expl ici t ly for each gauge index and apply the real space gradient operator, denoted V s , in spherical coordinates: B €abc2 0 '* <Pb x , V . (3.12) where the explicit lower index is the gauge index and the gradient of the compo-nents of 0 are as follow: V s 0 i V , 0 2 —— T V l - * 2 dr r + 1 T 9 -\& Q^f V i - * 2 d r + V i - * 2 r s i n f l V V l - T 2 #X r V l - * 2 #0 ^ [1 a* r + [r V l - #2 5cJ - T r sin 0 \ ax V T ^ P 0 V ^ 3 = ^ r + ^ * 26 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Figure 3.1: Two monopoles with different charges accelerating in opposite direc-tions Then, we obtain B explicit ly in terms of ^{9) and X ( x ) : Vl - V28% r sin 9 dr - T dT 9 r2 sin 9 89 - T dT (Vi - #2 x) Vi - * 2 or Vi - * 2 a* ax rs in# 5r 8\ ^ ^ r 2 s i n# 89 8\ f) \ (Vl-# 2 V l - T 2 ) a* 8T (-0) + r s i n 0 dr [y/i - T 2 dx\ 1 ^ - 1 - 1 0 + a# r 2 sin 9 89 - 1 ax a* - l ax r s i n f l 5r V l - X 2 8\ r 2 sin 0 50 V l - X 2 8\ We are now ready to solve E q 3.10 for \P and X . S o l v i n g for ^ a n d X Suppose the 0 monopole is situated at — | on the z-axis and accelerating with (e 2a) z while the © monopole is situated at + § on the z-axis and accelerating with (—e 2a) z, then the equations for them are exactly 27 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles the same except in terms of coordinates with different origins: e CLQ = e2a(fi cos #1 — 9\ sin #1) V x B f l = + e 2qxri © < a® = — a e = e 2a(—r 2 cos 92 + 02 sin 92) v x i = - e 2 ^ x r " 2 The coordinates subscripted 1 has its origin at the center of the monopole on the negative z-axis and the ones subscripted 2 at the center of the one on the positive z-axis. Consequently, the equations for *(#) and T(x) are the same for both monopoles as well: 8 1 dV\ 8 1 a * +r dr \sm6 dr J 89 \r2sin9 89 1 8V 8 / -1 8T 8T VT^dx +9 r2 s in 2 9 dr 8x V V l - T 2 8x 1 a* 8 ( -1 8T r3 s in 2 9 89 8x\ Vl - T 2 5x e 2 a sin 0 = X (3.13) We wi l l first show that T(x) remains unchanged from the static solution even when the monopole is accelerating. First , the inhomonogeneous term on the R H S has only a x component with coefficient that does not depend on the angle X', therefore, the particular solution needs to give a x component on the L H S that is also independent of X- Thus, for the particular solution, the possible x dependence needs to be removed: - 1 8T = constant VT-r^dx This is solved by T(x) = cos(Nx) but as explained above, N = 1 for both © and © monopoles for the chosen gauge. This means the particular solution of T has no 0(e2) correction. Also, this solution renders the f and 9 components of E q 3.13 zero as needed, regardless of what the particular solution of the other function, *(#), would be. 28 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles For the homogeneous solution of T ( x ) , assuming that ^{9) depends on 9 and perhaps r, we deduce the following equation for T ( x ) from the 9 and f components of E q 3.13: dX {VT^dx) which implies that the term inside the bracket, say (J), is linear: (I) = Ax + B. We already know that the O(e0) T ( x ) solution gives (I) = constant, and that the magnetic field is proportional to (/) from E q 3.13, which means (/) = Ax would make B discontinuous at x = 0; therefore, the only admissible solution for T ( x ) is st i l l just the static solution T ( x ) = cosx- We conclude that T ( x ) is not affected by the acceleration of the monopole. We now simplify E q 3.13 to an equation for ^ (9) only by putt ing in (J) = 1: - r ^ r w - [-zTzr + TT^W) = e a r sin 9 (3.14) dr2 \d92 sin# 89 J v ' and proceed to solve for \I>. This equation can also be writ ten in the following form which manifests its l inearity in \I>(#): - V x (x x V * ) = x e 2 a r sin 9 (3.15) The equation's l inearity in \& is crucial for Manton to bui ld the global two monopole solution as wi l l be discussed in section 3.2.1, P a r t i c u l a r s o l u t i o n o f \1/ Since the first term in E q 3.14 involves the second derivative in r of \&, if * oc r, then the first term vanishes. Now, if the 9 dependence is ^ j^-, then d 2 ( s i n 2 0 /2 ) cos# d (s in 2 0/2 ) n n , „ . 2 n o9l sin 9 o9 Combining, the correction of ^ due to the inhomogeneous term is = \ t 2 a r s in 2 9 (3.16) 29 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles which gives the magnetic field correction e 2 acos# ~e 2asin# B. part +r- 9-r 2r Note that since Bpart is proportional to e2a/r, it would be of 0(e3) when far away from the monopole 1/r is comparable to e. H o m o g e n e o u s s o l u t i o n o f * We use separation of variables to find the, ho-mogeneous solution of \I>: then Let = R(r)Q{6) r2R" 9 " cos 0 0 ' R e sin e e = A = const 1. For the 9 dependence, e»-^ |e ' -Ae = o sm9 0 Propose that Q ~ cos f c 9 sin1 9, then (-2kl -l-X) cosk 9 sin' 9 + k{k - 1) c o s * - 2 9 sin'+ 2 9 + 1(1-2) cosk+2 9 sin1'2 9 Each term vanishing gives the conditions for k, 1, and A: 0 for k = 0, / = 0 - 2 for k = 0, / = 2 Jfc = 0, 1 ; I = 0, 2 ; A = -2kl - I = < - 6 for k = 1, I = 2 2. For the radial function, r2R" - XR Let R(r) = rn, 0 where - A = 0, 2, 6 this means n(n — 1) = —A 0,1 for - A = 0 n = < - 1 , 2 for - A = 2 - 2 , 3 for - A = 6 30 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Ar + B „2 Combining the angular and the radial parts: for - A = 0 (Cr2 + f ) s in 2 9 for - A = 2 [ (Er3 + £ ) cost fs in 2 0 for - A = 6 where A to F are constants of 0(e2). Plugging these into the relation between B and * (Eq 3.13), we obtain the following magnetic field correction: B -9 r sin 6' C(rcos9- 9sin9) . +D + for * ~ Ar + B E +F for * ~ (Cr2 + f ) s in 2 9 for * ~ (Er3 + cos 0 s in 2 9 r(2r - 3r s in 2 9) - 9Sr sin 9 cos 9 f±(2 - 3s in 2 0) + 9^ sin 9 cos 9 However, only of one these magnetic fields is admissible and relevant. The first of these terms diverges at 9 = 0, so it is not allowed. The term with coefficient E is proportional to e2ar which becomes of 0(e1) when 1/r is comparable to e. This order was not mentioned by Manton and is t r iv ia l as wi l l be shown in section 3.2.2. The terms with coefficients D and F are of 0(e3) for 1/r ~ e and is irrelevant in the determination of the acceleration e2a as wi l l be shown in section 3.2.1. The only term left is B ~ C(f cos9 — 9sin9) = C i , which comes from the following ^ j , ^ : hom © °1 2 2 - 2 / 3 e arf sin oh 2 1 (e2) _ ^2 2 / iom © = — e a r , sin 0 2 (3.17) where — o\t2a and cr 2e 2a are simply Manton's names for the coefficient C for the different monopoles. We have found the homogeneus solution of the magnetic field from ^ h o ^ ; however, if we were not interested in \P, we could have noticed that the homoge-nous solution of B in E q 3.13 simply satisfies the vacuum Maxwel l equations, and 31 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles could have concluded that the solution in terms of a scalar potential U such that B = VC7, is simply any linear combination of the multipole expansion terms with cyl indrical symmetry: U B Pi(cos8) where A, B are constants and P;(cos 0) is the Legendre Polynomia l of cos 9 of order I. The homogenous magnetic field we obtained from ^ h o ^ simply corresponds to the term with the lowest I which would give rise to a non-zero magnetic field and has no 1/r dependence: U = ArPi(cos9). The complete solution for (j) for either monopole accelerating in its own direc-t ion is then f ( * ( 0 ) + + c o s * \ J with the corresponding , ^pjt, and ^oL- Note that 0 f ° r both monopoles to depend on ^ in the same way for the chosen gauge. In section 3.2.1, we wi l l see how Manton needs this to bui ld the global two monopole solution. Solution for h Solving for 4> has given us both the particular and homogeneous solutions for the magnetic field due to the acceleration. Using these and the first order ansatz E q 3.9, we can easily solve for h to 0(e2). For the 0 monopole, Bci = -{Vshe + e2ah) and from the last section BQ fi 2 cos flx - i 2 5- + rie a 0 i -e a er{ eri 2 erx 1 o sin 0i 2 -G\e a 32 Chapter 3 Manton's, Method to Find Force Between Two Commutative Monopoles The equation for h is then V / i e = rx— {-Ox dri ri d9i 2r i V / Solving for the f icomponent of the equation, 1 he = c + (<JI — c)e o r i cos 9\ + f ($i) r I The #i component of the equation then determines such that he = c — - + (oi — c)e 2 ar x cos 6\ 4- ^ e 2 a cos #x + &i r 2 For the © monopole, the (9(e0) magnetic field has a different sign from the © case, and the magnetic field correction obtained from also has a different sign because of the definition of the unknown coefficient cr2: f2 o c o s 2^ J 1 2 s i n #2 2 -5© = H—~ + r 2 e a v2-e a hu 2e a r% r2 2 r 2 The first order ansatz has a different relative sign between B and h and the direction of the acceleration is reversed: (V s h© - e 2ah) Thus, the equation and solution for he are f 2 . t 1 2 s i n ( - 2 + 0 2 Z C a r 2 2 r 2 V,/ i© = i l + ^ e 2 ^ ^ ^ 2 - + ( a 2 - c ) e 2 ( f 2 c o s 0 2 - 02sin.02) =>• fr© = c + (cr 2+c)e 2ar 2 cos 0 2 — - e 2 a cos #2 + fc2 r 2 2 We now have the full solution in the asymptotic region for a single © monopole accelerating in the +z direction and a © in the —z direction. 33 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Comparison with Fields of Accelerating Electric Charge The magnetic field obtained above for an accelerating monopole is not analogous to the electric field for an accelerating point charge in normal electromagnetism. We wi l l describe how and briefly why these fields are different, but show also that far away from the monopole where r ~ s, the term in the magnetic field that is relevant to how Manton obtains the acceleration, the O(e0) term, is actually equal, up to 0(1/s2), to the Coulomb term in the electric field of an accelerating electric particle. The differences between the fields result from the different ways we solve the two problems. For the magnetic monopole, we propose a time dependence for the solution, check that it is legitimate by evaluating the time derivatives of the fields with such time dependence in the equations of motion, and then solve these "half-stat ic" equations, since they do not have t ime derivatives anymore, for the magnetic field, both the 1/r 2 and 1/r terms. For the electric point charge, we simply solve the time dependent equations and let the t ime dependence of the fields come out of solving the equations: E, electric (x, t) = e n 7 2 1 ( / 3 x n ) 3 2\ 2 e + -c x {(n-p) x / ? } ( l - J3 • nj3 r2 ret where P(t) = f 0 (* ) , r(t) = \x(t) - x0(t)\ and n(t) = x(t) - xQ(t) r(t) The main difference between the fields is that the 1/r radiat ion fields above for the accelerating electric charge are in terms of quantities related to the path of the charge that are to be evaluated at an earlier t ime t0 defined by: \x - x0(t0) c (t — to) where Xo(to) is the path of the electric charge but the fields obtained by Manton for the accelerating monopole at t = 0 depends on the motion of the monopole at the same instant. 34 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles The magnetic field for the accelerating monopole being not time-retarded causes a violation of special relativity: even if the monopole starts to accelerate only at t = 0, the 1/r term of its magnetic field, which is the analog of the radiation of the accelerating electric charge, takes no t ime to reach the other monopole. This is consistent with the fact that the assumption of the fields to be rigidly accelerating over al l space also violates special relativity. However, the ~ 1/r magnetic field for the accelerating magnetic monopole does not part ic i-pate in the determination of the force between the monopoles (as described in section 3.2.1) and so we can ignore this problem. Note that even if we take away the time-retardation of the radiat ion of the accelerating electric charge and chooses the charge to be constantly accelerating, 0/c = e2at, the radiation terms of the magnetic monopole st i l l has a different functional form. This is because the E term in the equation, V x B — E, for the magnetic monopole problem comes directly from Manton's assumed time dependence of the fields, while the B term in the equation, V x E = B, for the electric charge problem both affects and is affected by the radiation term in the electric field. On the other hand, the static 1/ r 2 term in the magnetic field of the acceler-ating monopole, when compared to the analogous electric field, lacks the factors that depend on the velocity of the particle. However, since Manton's fields de-scribe the instant when the monopole has zero velocity, the factors become zero, and so the 1/r 2 fields for the electric charge and the monopoles are exactly analo-gous and are simply the respective static Coulomb fields. The facts that only the 1/r 2 term of the magnetic monopole field is relevant in the determination of the acceleration and that this term is the same as the static monopole field are what make Manton's method gives the same result as the stress-energy tensor method described in section 3.3. 35 7 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles 3.1.3 The First Order Ansatz as the External Force Law We wi l l now extract physical information from Manton's first order ansatz for the weakly accelerating monopole. Manton argues that the first order ansatz implies the Lorentz Force Law for a single monopole, but this is only half of the story: the first order ansatz informs us about the contribution of forces that can act on the monopole. He argues that for a single accelerating monopole,since 1' B Vh + e2a I c to G(e2) and Vh needs to vanish at infinity for the monopole to have finite energy, B must be e2ac at infinity. He claims that the Lorentz Force Law directly follows since c is the ratio between the mass and the charge of a single charge monopole: -> 9 , m(e2a) . „ . B = e2ac = ± — (3.19) 9 I do not agree with the reason for Vh = 0 or that the Lorentz Force law necessarily holds at infinity. Rather, for the monopole to have finite energy, both Vh and the magnetic field B need to drop to zero at infinity. Therefore, what Manton really assumes when he allows B to be non-zero at infinity but not Vh is that the uniform "external" field that is left over even at infinity is comprised of only the magnetic field. If we choose the external field to include a gradient field of h, then these two types of field both contribute to the forces acting on a monopole and together satisfy the force law: \ , rn(e2a) (Bext + Vhext) = ±— For example, let us look at the solution to the first order ansatz in the asymptotic region near a 0 monopole: -< h ~ 2 c o s z 1 2 s m ^l 2- ___ _ o _ 2^1 Be = T: + r\e a 0 i - e a 0\e a = —Vhe — eac + ea—. • erf eri 2 eri r\ 36 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles At infinity where all the terms with 7*1 in the denominator vanish, the undeter-mined external magnetic field reduces to — a\e2a, not e2ac, and Vh reduces to e 2a(c — o i ) . Choosing the value of 0\ would determine if the Lorentz force law is followed: if we force Vh to go to zero at infinity, such that o\ = c, then the Lorentz Force Law (Eq 3.19) is satisfied; otherwise, if o\ / c and Vh ^ 0 at infinity, the Lorentz Force Law is incorrect. The interpretation of the first order ansatz as an external force law is valid not only at infinity, but also in the asymptotic region, since in this region, although the terms with 1/r" has not dropped to zero, the ansatz st i l l independently relates the constant magnetic and Vh fields to the constant term e2ac. 3.2 Manton's Method to Determine the Accel-eration between Two Monopoles We now consider a system with two widely separated monopoles accelerating in opposite directions. We know that at the core of each monopole, the first order ansatz needs to be satisfied such that the unfactorized non-linear equations of motion there are satisfied. In the asymptotic region close to the core of each monopole then, the solution is simply the asymptotic l imit of the first order ansatz and we call this the " local" solution. Now, for the region between the monopoles, Manton discovers that he can easily obtain a global solution' by "almost" superimposing the local solutions. He finds the acceleration by requiring this global solution to become the local solutions in regions close to the cores of the monopoles. However, although Manton's result is correct, his method in fact does not give an unique answer. We wi l l show two examples of global solutions which are buil t in the same manner that Manton's is built but which conclude a different 37 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles acceleration between the monopoles. We then propose a rule to bui ld the global solution such that it gives the right conclusion, and notice that under this rule Manton's procedure can be interpreted as the application of the external force law E q 3.20, and the undetermined terms in each local solution as perturbative external fields produced by the opposite monopole. We then do an exerices to find out if the method can be used to give a higher order (0(1/(separation)3) or above) force between the monopoles, conclude that it cannot, and examine the difference between the method at higher order and at the order for which it works. 3.2.1 The Global Solutions and the Matching Procedure Manton's Way of Building the Global Solution Manton discovers an easy way to bui ld the global solution in between two monopoles. First , recall from section 2.2.2 that in the region between two monopoles, the magnetic field B satisfies the vacuum Maxwel l equations and h satisfies the Laplace equation. Both equations are linear differential equations. Secondly, concentrate on the equation for B. Recal l from section 3.1.2 that B depends on 4> (Eq 3.12) and 0 is given in terms of the function ^(9) and T(x) (Eq 3.11). In the gauge that Manton has chosen, the solutions for T(x) of the ansatzes for both © and © monopoles are the same, T = cosx , and given this, 4> depends on ^(9) the same way for both monopoles and the magnetic field B depends l inearly on \P(0) the same way for both monopoles. Thus, we can write down the following function (f)giobai which has Tgi0bai = Y e / e = cos% and depends 38 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles on global as the 0 for either monopole depends on \I>: 4>global ( global ™SX ^ (3.20) y/1- ^global s i n X y ^global J such that the global magnetic field Bgiobai again depends l inearly on ^gi0bai through its dependence on 4>gi0bai- Then, the Maxwel l equation for Bgiobai would translate into a linear differential equation for ^global-Now, if we solve for ^global, then (f)giobai wi l l be automatical ly determined by E q 3.20 and wi l l satisfy the equations of motion, and the global gauge field wi l l in turn be determined in terms of (f)gi0bai- Therefore, for the asymptotic region between the two monopoles, solving for hgiobai and ^gi0bai wi l l give the ful l solution. Final ly, since the equations for hgi0bai and ^gi0bai in the region between the monopoles are both linear, the solution of hgi0bai and bgiobai can simply be the sum of the local he, h® and local * e , * ® functions, which satisfy the equations of motion by satisfying the respective first order linear ansatzes. Here, the sum of solutions means only the sum up to constants and homogeneous solutions of the local ansatzes so there are choices to make for the global solutions. Manton also requires the global solutions to 1. be symmetric under monopole exchange; 2. satisfy the appropriate boundary conditions at infinity; . 3. reduce to the local solutions, hQ, h& and * e , near each monopole. Note that if the global function bgiobai reduces to the local * f f i and * e near the different monopoles, then ^global wi l l also automatical ly reduce to the local 0s. Manton claims that in the process of matching the global solutions that sat-isfy the above requirements to the local solutions, the acceleration between the 39 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles P e - " " -© z Figure 3.2: the two monopole system, the distances r i , r 2 and the angles 0\, 62-monopoles is determined. He obtains the correct acceleration, and we wi l l recount his choice of global functions in this section, but we wi l l also see in next section that the acceleration is actually not uniquely determined. The global solutions would also need to satisfy boundary conditions at infin-ity and be symmetric under monopole exchange. Final ly, requiring the global functions to reduce to the respective local solution near each monopole, M a n -ton claimed, would determine the acceleration of the monopoles uniquely. We can already see, however, that we have freedom to add constants or homoge-neous solutions of the ansatz to the global solutions, and wi l l discuss this in the section 3.2.2. Here, we recount Manton choices of global solutions and how he determined the acceleration. The notation used is such that ^ e / l o m denotes the homogeneous solution of h to the © ansatz which is also of 0(e2). Matching \? In choosing ^global, Manton must have noticed that when expanded near the opposite monopole, the O(e0) solution of either local ^ function would give rise (e2) to a function which is proportional to the homogeneous solution ^ h o ^ for the opposite monopole. 40 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Expl ic i t ly , wi th r i , r 2, 6\, 92 defined in figure 3.2, near the © monopole, 9i is smal l , and M 4 0 ) - l = 008(9,) - 1 « i ( 0 ! ) 2 ^ s i n 2 ^ . Then using the sine law and that r\ is approximately the separation distance, s, in this region, (0) „ l r 2 s i n 2 ( 0 2 ) ^ l r 2 s i n 2 f l 2 Similarly, near the © monopole, (IT — 92) is small and r 2 ~ s, so ^ o ) - l = - c o s 0 2 - l = c o s ( 7 r - 0 2 ) - l » _ l ^ i 2 s m 2 0 i + c ) ( j _ ) o c ^ e ^ Therefore, if ^gi0bai + 1 is the sum of the local ^ functions without the homo-geneous part, i.e., vT/' _ vr,(0) , VT/(0) _ i _,_ \r/(e2) , VT/Ce2) * global ^ 0 ' J - ' ^ e pari i *ffi part = cos #i — cos #2 — 1 + ^e2ari s in 2 9\ + ^e2ar2 s in 2 #2 then ^piofcaf would reduce to the local ^ near each monopole up to 0(e2) provided (e2) that the coefficient of the local is matched with the coefficient of the term from the expansion of \Er(°) of the opposite monopole. Thus, near the © monopole, *global —> cos 91 - ^o^ar2 s in 2 9X + ^ e 2 a r i s in 2 9, + Q(e3) = #e •f 2 1 it a\e a = — S2 Note that the radiation term from the opposite monopole, p a r t , is omitted here because it is of an irrelevant order in this geometric l imit : 1 2 -2/. 1 2 r 2 s in 2 (9 i 3 *ffi part = ^ e °r2 S m #2 ~ ^ ° ~ 6 Using exactly analogous arguments, for ^gi0bai to reduce to \& e near the © monopole a2e2a = s 41 2 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Note that *global is also symmetric under monopole exchange and remains less than 1 to O(e0) due to the added constant —1 such that 0 is a real unit vector; thus, it satisfies all of Manton's requirements for the global solutions. We have now two equations involving the acceleration and two unknown co-efnents o\ and a2. One more equation of these quantities would determine the acceleration. Matching h Manton chooses the following rather adhoc looking hgi0bai, which includes only one of the local homogeneous solutions of h but with a term proport ional to 1 /s 2 put in by hand: hgiobai - [KQ + n® - cj —2 V \ hQ p a r t + PARTJ + ne h o m + const 1 1 \ r2 cos 0 2 , ( 1 2 1 2 = c h — e a cos 0i — — e a cos 0 2 eri er2 J es2 \2e 2e +e2a ^— + cj r2 cos 0 2 + const such that near the © monopole, the term r 2 c o s 0 2 / s 2 cancels wi th the term from the expansion of h?Q near this monopole, which equals 1 _ 1 | r 2 cos 0 2 | 0 ( 1 er i es es 2 \ s 3 7^2 cos ^ 2 since r i = .' s\l 1 H 1—|- by the cosine law and the term e 2 acos0 i /2e reduces to 0(e2) to simply a constant near the © monopole and can be absorbed by the constant in the global function. Thus, hgiobai has been constructed to match / i e near the © monopole: 1 1 er2 2e * V e This hgi0bai satisfies al l of Manton's requirement for a global solution since the added term is a homogeneous solution of the equation of motion and the terms . 4 2 hgiobai —> c ^ - e 2 a c o s 0 2 + e2a (— + c\ r 2 cos 0 2 + const = hQ Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles (o2/e + c)r2 cos 0 2 could also be written as. (o2/e + c)ri cos 6^ +const so that hgi0bai is st i l l symmetric under monopole exchange. The acceleration is determined by the condition given when hg[0bai is matched with he near the 0 monopole. Near the © monopole, the term that cancels the expansion term of h^Q near the © monopole does not cancel but adds up with the expansion from hJ$ since 1 1 r i c o s 0 i . 1 . = h 0(—J b e c a u s e er2 es s2 sA and with r 2 cos 62 wi th ri cos 0i — s, hgi0bai reduces to: 1 2 r i c o s 0 i 1 , „ i (°2 \ ' „ haiobai —> c — z h — e a cos 0i + e a I h c m cos 0i + const er\ es1 2e V e / This only equals he if al l the terms proportional to r-y cos 0i together form the local homogeneous h, hQ hom- This implies the condit ion: 2 e 2 a ( ? + c ) - ^ = e 2 a (v- c ) Then, substituting the values of a i e 2 a and o2e2a from before, Manton obtains twice the Coulomb attractive acceleration for a pair of opposite charge monopoles: 2 e2a = ecs2 Force between two same charge monopoles Manton finds the force between two same charge monopoles by the same proce-dures. Suppose we switch the monopole on the +z-axis in the previous case to a © monopole, so that both monopole 1 (on -z-axis) and monopole 2 (oh +z-axis) are ©. Then all we need to change in the steps above are,the local functions near the new monopole 2, and the global functions accordingly. Since monopole 2 has the same charge as monopole T, but accelerates in the opposite direction, the local functions, call them 2 and hQ 2 , are the same 43 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles as the ones at monopole 1 except with the sign of e2a changed.and in its own coordinates: 2 — * e 1 l hom ^ ^ 9 1 part J cos 02 + ^-o2e2arl s in 2 92 — ^-e2ar2 s in 2 92 Notice that the term involving a2 has the same sign as in * ® in the previous opposite charge monopole system. Similarly, i he 2 = c (o2 — c)e2ar2 cos 92 e 2 acos 92 + k2 . e r 2 2 Now, as before, the requirement of the global function ^gi0bai to reduce to the local functions near each monopole would give the unknown coeffients, o~\ and cr2, and the acceleration, e2a, in terms of the monopole separation s. Aga in , the particular solutions of * do not participate in the matching. Therefore, we only need to take note that in global function for the same charge system , * Q ^ 2 has a different sign from in the opposite charge system, and that the homogeneous terms, the ones with coefficients <7i and a2, remain as before, to conclude that the matching procedure would give the same expression as before for o2e2a, and one with opposite sign from before for o\e2a. More explicit ly, ^global ee = cos 0i + cos 0 2 ± 1 + ¥ Q \ P A R T + P A R T where the second term is now + cos 0 2 for the © monopole 2 and after expansion near the first monopole gives 2 1 2 1 (Tie a = - — ; o2e a = —-sz sz Now, when we construct the global h function in the same manner as before, we find that the only change is the sign change of the term involving 02: 1 1 r2 cos 0 2 1 o 1 o hgiobai = c h — e a cos 9X - — e a cos 0 2 er i er2 esz 2e 2e +e2a ^ + cj r 2 cos 02 4- const 44 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles where the O(e0) local h function for the new 0 monopole, (c— l / e r 2 ) , is the same as the one for the © monopole in the previous case. The matching procedure is exactly the same and gives es1 e Therefore, the force between two same charge monopoles vanishes. 3.2.2 Clarifications and Comments on Manton's Method Other Consistent Solutions that Give Different Acceleration We wi l l now look at how the flexibil ity in choosing the global solution even ac-cording to Manton's requirements allows for global solutions that lead to different conclusions for the acceleration. We demonstrate this by the following two ex-amples. Example I The first example results in a zero acceleration between two monopoles with different charges. We choose the global function ^gi0bai to include the ho-2 2 mogenous solutions of the ansatzes, ^ e e h o m and ^§hom-*global = cos 0i - cos 02 - 1 + ^ e 2 a r i s in 2 0X + ^ e 2 a r 2 s in 2 02 Z Z 1 2 2 * 2 / i "1 2 2 - 2 / 1 — -(Tie a r x sin #1 + -cr 2e a r 2 sin tV2 z z This satisfies the equation of motion and are symmetric under monopole ex-change. Recal l that when expanded near the first monopole, . . l r 2 s i n 2 0! - C O S 0 2 - 1 « - - g 2 . Now, from figure 3.2, r 2 s in0 2 = r 1 s in0 1 => ^-o2e2a ( r 2 s in 2 02) = ^ a 2 e 2 a ( r 2 s in 2 0X) z z 45 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Then near the © monopole, if this term with <T2 cancels the expansion term from the © monopole, ^global would reduce to the local function * e , and the condition on 02 would be 2 1 o-2e a = — s2 Similarly, near teh © monopole, the homogeneous solution \J>^ 'hom can cancel wi th the term from expanding * Q ^ = cos 6\: _ 1 9 o o n l r 2 02 1 2 2 - 2 / 1 cos 6\ — 1 — -o ie ar{ sin #i « — - - o ^ e a r ^ sin 0 2 = 0 Z Z S Z =4> a 2 e a = - —. s2 What we have chosen here is that the homogeneous solution for each monopole is used to cancel the effect of the O(e0) solution of the same monopole near the other monopole. Now, we can again bui ld the global function h to include the homogeneous solutions of h: hgiobai = c - — + -e2a cos 61 - -e2a cos 92 1 1 1  n 1 h - e a cos 6\ < rx r2 2  1 2 (oi — c)e2ari cos d\ + [o2 + c)e2ar2 cos 92 + k\ + k2 and use them along with the terms +ce2ar cos 6 to cancel wi th the terms from the expansion of —1/r near both monopoles. That is, near the © monopole, ( , \ 2 a ncosOi (o2 + c)e ar\ cos Vi — = 0 and since o2e2a = from before, e2ac = 0 In the same manner, the conditions near the © monopole also result in a zero acceleration: / \ 2 n r2cos02 2 (0-1 — c)ear2 cos d2 -\——-— = 0 = > e ac = 0 sz 46 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Example II Our second example does not yield information about the acceler-ation. If we choose the global function ^giobai to include a term "put in by hand" similar to the one in Manton's hgi0bar-^global = COS Qx - COS 92 + ^ ^ l s i n 2 #1 + \ ^ a T ^ ^ ^ 1 2 2-2/> i 1 rl S i n 2 1^ --ole2arlsiVL261-l + --^—2 Z z s then it is st i l l symmetric under monopole exchange because r\ s in 2 6\- = r\ s in 2 0 2-Near either monopole, the added term would cancel the O (j?) contribution from the expansion of the corresponding cos 6, and therefore the only condit ion needed for global to reduce to the local functions \I>e and \ P e is c i = -0-2 Consequently, if we choose hgi0bai to be Manton's hgiobai, which gives also only one condit ion between <7i, 02 and the acceleration, there is not enough constraints to determine e 2ac. Thus, it is not true that coming up with symmetric solutions for h and \I> in the region between the monopoles and matching them to the corresponding local functions in regions close to the monopoles would give a unique correct answer for the acceleration. Another Requirement for the Global Solutions and the Matching Prin-ciple Let me now propose that Manton's method provides the correct answer only when it obeys the following exchange principle, which is much like what we use in electromagnetism to determine the fields for a system with two widely separated sources. What I called the exchange principle is the assumption that the ambiguity of the local solution near one monopole is due to the presence of the other monopole. 47 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles In other words, in the matching process, the local homogeneous solution of * and h near one monopole should be "produced" by the expansion of the \T/ and h solution of the other monopole, and accordingly, the global functions should not include any of the homogeneous solutions. Manton's ^global is the one prescribed by this principle but his hgi0bai is not. We now show how the matching of hgiobai would be done under this principle. First , similar to how Manton bui ld ^gi0bai, w e bui ld hgi0bai for two opposite charge monopoles by adding the C(e°) solutions of h and the 0 (e 2 ) particular solutions of h: 1 1 1 2 n 1 2 n hgiobai = c - — - — + - e a c o s 0 ! - - e a c o s 0 2 r i r2 z z -e2acr1 cos 6\ + e2acr2 cos 82 + kgiobai Now, unlike the situation for * , one term in the particular solution of h for each monopole has the same functional form, ~ r cos 8, as the term from the expansion of the solution of the other monopole. This term in the particular solution is of 0(e2) and so cannot be neglected in the matching process. The other term, ~ e2a cos 8, coes not participate in the matching as argued before. Therefore, near the 0 monopole, , v 1 / I , n c o s M 1 2 1 2 hgiobai —> V<? —s2—) 2 6 a c o s ^ ~ 2 C ' -e2acr1 cos #i + e 2 a c ( - 5 + rt cos ^ i) 4- kgiobai and for hgi0bai to reduce to hQ: 1 1 2 h - e a + kgi0bai = « i S Z 1 \ 2 — + e2ac ) ( r i cos^ i ) = axe2a [r\ cos^ i ) => e2ac = — s2 ) s2 Similarly, the l imit near the © monopole gives 1 I 1 r2cos82\ 1 2 f\ 1 2 t-\\ hgiobai —> c — - -e a cos ^ 2 + ^ a(l) r2 \ s s / z z +e2acr2 cos 82 - e2ac(s + r2 cos 82) + kgiobai 48 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles which gives the matching conditions: 1 1 2 , h - e a + kgiobai = k2 = h S u ^+-^ - e2acj ( r 2 cos# 2 ) = o2e2a (r 2 cos 92) = ^ e2ac = For two same charge monopoles, the matching condit ion near monopole 1 does not change except that the value of o\ has been determined to be different by the matching the ^ functions: — \ + t2ac = G\e2a but o~\e2a = —-s2 s2 which implies e2ac = 0. The matching condition near monopole 2is simply the negative of the one from the first monopole and so gives the same conclusion. Notice for both systems, we obtain two conditions from matching h that agree with each other while Manton obtains one only. Why ea is trivial It is very clear under this matching principle why the 0(e) acceleration vanishes. Since the O(e0) \I> solutions, ~ cos#, do not expand to give any order 1/s terms, the terms with o\ and o2 cannot be not "produced," i.e., o\ta = o~2ea = 0. Then for hgi0bai, the local part icular solution for either monopole, ~ eacr cos 8, which is to combine with the expansion from the O(e0) h solution of the other monopole to produce the local homogeneous term, the term wi th the corresponding o, has to be zero, because both the expansion term and the o term are zero. Interpretation of the Matching Process W i t h the exchange principle, we could have found the correct acceleration by matching the magnetic and Wh fields instead of h and ^ . In this case, we would only need to show that the global solution (f> exists but not solve for it explicit ly. 49 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Also, we can interpret this matching process as the application of the respective external force laws (Eq 3.20) on the monopoles. First , according to the exchange principle and due to the linear dependence of the magnetic field B o n $ , the global magnetic field is the superposition of the O(e0) Coulomb fields and the 0(e2) particular solution to the first order ansatzes and does not include the undetermined local homogeneous solutions: R - J-l - J . I i , D(*2) , o(e2) ^global — 2 • 2 ^Qpart ' ^(Bpart rl  r2 Note that this global magnetic field has similar contributions from individual monopoles as the electric field does for two separated electric charges except for the differences discussed in section 3.1.2; however, unlike in the two electric charge system, the superposition of fields here is only valid in the asymptotic region. Similarly, the global Vh field is the superposition of the Vh fields from the different monopoles, which can be easily written in terms of the local magnetic fields from the first order ansatzes: Vh global ->-|+«-HK)j + f 2 s=>(e2) n_ ( 1 "2 + Kpart + e a I c - -r2 \ r2 (3.21) That the constant terms e2ac contributed by the different monopoles cancel each other wi l l be important for our interpretation of the matching process. This can-cellation is due to the monopoles accelerating in opposite directions and happens regardless of the charges of the monopoles. Now, in the process of matching these global fields to the local fields near each monopole, the matching of the constant vectors is what gives the information about the acceleration. The local magnetic field at each monopole contains only one constant term, the homogeneous solution to the ansatz, ±ae2a, which is to be equated to the 50 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles expansion of the field from the other monopole under the exchange principle. To 0(e2), only the expansion of the static Coulomb field from the other monopole contributes. For instance, near the © monopole, the undetermined constant, —oi€2d, is matched to the following: On the other hand, the local Vh expressions contain the constant terms <7e2a=F e 2ac. For each monopole, this constant term, under the exchange principle, is to be given rise only by the expansion of the Vh field from the other monopole because the constants that appeared in the global expression (Eq 3.21) cancelled each other. Aga in , to 0(e2), only the expansion of the C(e°) field of the other monopole contributes, and this, depending on the charge of that other monopole, is simply plus or minus the contribution of the magnetic field from that monopole. For the © monopole, then, the constant terms (oie2a — e2ac) in the local Vh is equated to the far-field l imit of V/i®^ = +B^\ We can now see that matching the global fields to the local ones under the exchange principle implies that the constant part of the first order ansatz at each monopole relates only the "external" fields produced by the other monopole to its acceleration. Thus, matching with the exchange principle and using the different first order ansatzes to determine the accelerations is like applying external force laws to the monopoles: Two © monopoles: ± (e2a)— = —Bext — Vhext 9 = —Bext — (—Bext) = 0 ©/©monopo les : ±e2a— = TBext-Vhext 9 -, - 2 = TBext - (±Bext) = T2Bext = z — s where the upper signs are for the monopole on the negative z-axis and the lower signs for the other one. 51 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles This way of finding the acceleration between two monopoles, then, has become similar to the way of finding the lowest order force between two widely separated local and spherically-symmetric electric sources by the mult ipole expansion in normal electromagnetism. There are a few differences: 1. the external force law (Eq 3.20) used for the monopole pair problem, unlike the Lorentz Force Law used in the electric problem, involves an extra Vh force which is attractive regardless of the charges of the monopoles; 2. while in the electric problem, the mass of each electric source is free to vary with its total charge and so its acceleration under the external electric field from the other source varies accordingly, the mass of the monopoles is determined solely by the charge and the parameter c, and consequently, the acceleration of the monopoles is fixed once the external fields are known; 3. in the electric problem, the Lorentz Force Law can be applied at each point in either of the local charge distributions to give the induced multipole moments, but the external force law (Eq 3.20) for the magnetic monopoles is not to be applied pointwise (there is no pointl ike magnetic sources to be acted on either) and does not allow us to find the deformation of the non-pointlike monopole under the influence of the external field. 3.2.3 Limitations of the Manton's Method Apar t from not being applicable as a local force law, Manton's ansatz also does not help us determine the force between two opposite charge monopoles above the lowest order. We wi l l first show that the ansatz can actually be extended to the first order above lowest order in e but then the matching procedure that works for the lowest order breaks down despite of the valid ansatz. Through this process, We wi l l understand better how the matching process works for the 52 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles lowest order. Manton's method, however, does work at al l orders of e for the same charge monopole pair. We wi l l argue this at the end of this section. Extension of First Order Ansatz to 0(e3) Because the 0(e1) acceleration is zero between two monopoles, Manton's ansatz for the accelerating monopole can easily be shown to work for an assumed acceleration of one higher order in e, e3a'. We can simply replace e2a by (e2a + e3a!) in every step of the derivation (sec-t ion 3.1.1) for the 0(e2) acceleration, find that each step remains valid because any terms with explicit t ime dependence (Eq 3.3, E q 3.4) that would ruin the derivation are of the order of the square of the first non-tr ivial order, i.e., C ( (e 2 ) 2 ) , and arrive at the following extended ansatz: 1' B = ± Vh + (e2a + e3a') c This says that any^0(e 3 ) constant external B and Vh fields would contribute to an 0(e3) constant force on the monopole, m(e3a'), on top of the 0(e2) force. Repeating Manton's Method at 0(e3) In electromagnetism, the first order force above the Coulomb order on a local charge distr ibut ion with a constant dipole density involves the gradient of the external electric field, F i / S 3 ~ p • VE. The ansatz derived above involves only uniform external fields to 0(e3) and already signals that it may not work in a system where the gradient of the external fields is not uniform. Here, we show explicit ly how matching the local and global solutions of the opposite charge monopole pair up to 0(e3) results in a questionable conclusion for e3a' as well as an inconsistency. We wi l l then see that Manton's procedure works at 0(e2) by "rescuing" the same inconsistent situation had we used only the static ansatz to bui ld the global solution. F i rs t , the equations for the local ^ functions near both monopoles in the 0 / © 53 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles system includes the new acceleration: VxB($) = x (e2a + e3a') sin# (3.22) Solving, the local * ' s near both monopoles contain a part icular solution for that new term as well as the homogeneous solutions to this order: cos #1 + - e 2 a r i s in 2 6\ — -o\e2ar\ s in 2 9\ Z Z + - e 3 a ' r i s in 2 9X - ]-o[e3 a'r\ s in 2 91 - pie3a'r\ s in 2 6\ cos 6\ 2 2 — cos 92 + ^-e2ar2 s in 2 92 + \^a2e2ar\ s in 2 92 z z + ]re3a'r2 s in 2 92 + \o-'2e3a'rl s in 2 92 + p2e3dr\ s in 2 92 cos 92 z z We write the global solution without the undetermined terms as prescribed by the exchange principle: *global = cos 6\ - cos 92 - 1 + ^e2ar1 s in 2 0X + \e2ar2 s in 2 #2 . ^ z + -e3a'ri s in 2 6>i + -e3a'r2 s in 2 6>2 Again , to match the global solution with the local solution near each monopole, we expand the terms "belonging" to the other monopole in the global solution to 0(e3) and equate the resulting terms with the ambiguities in the local solutions. This t ime, the expansion of the O(e0) static parts of ^global, cos 9i cos 92 1 - -l r | s i n 2 ^ 2 _ r\ cos 92 s in 2 92 2 s2 + + O —7 near © monopole 1 rl s in 2 9i r\ cos 6\ s in 2 $ 1 1 - - + O I — near © monopole gives 0(e3) terms that can "produce" the terms with coefficients pi>2 in the op-posite local solutions provided that -pie a = —- ; p2e a = —, (3.23) 54 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles whereas the expansion of the 0(e2) particular solutions are proportional to the 0(e3) terms with coefficient u'12: 1 2 2 1 2 r\ s in 2 0 2 , -e ari sin 0i —> -e a near © monopole 2 2 s 1 2 • 2 1 2 r i s i n 2 -e ar2 sin 0 2 — > -e a near © monopole 2 2 s and therefore the matching conditions are ——e a = — e a = (3-24) 2 2 s 2 2 s v 1 However, we are matching terms analogous to the radiation terms from an ac-celerating electric charge (section 3.1.2) to the local unknowns, and if, without the assumption that the fields of the monopole accelerate r igidly everywhere, these terms were retarded in time as the radiation in electromagnetism, then at the instant when the monopoles start to accelerate, their effect would not have reached the opposite monopoles to produce the undetermined homogeneous so-lutions there. Thus, the above condition seems to violate special relativity and is questionable. We wil l discover yet a more blatant break-down at 0(e3) of this method in the following. To solve for the local h, we first find the magnetic field with the addit ional 0(e3) terms near each monopole from the local * functions: B A = r i 2 cos#! « 1 , s i n^ i 2 _ •-z 4- rie a 0 i -e a o^ e a r{ ri 2 r i 3 , cos 0i ~ 1 3 , sin 0i + f i e V - 0 i - e V n 2 n -o[e3a' - p ie 3 a ' ^fi(2r-i - 3r x s in 2 0i) - 0 i3 r i s in0 i cos0 i ) -» [ f t „ ( Bffi = f 2 2 COS 0 2 a 1 2 S i n 0 2 . 2-—~ + r 2 e a 0 2 - e a - Yo2e a r2 r2 2 r 2 +a'2e3a' + p 2 e 3 a ' ( f 2 ( 2 r 2 - 3 r 2 s in 2 0 2) - 0 2 3r 2 sin 0 2 cos 0 2 ) „ , ,COS02 - 1 , ,sin 0 2 + r 2 e 3 a' 0 2 - e V r 2 2 r2 Note that as before (section 3.2.2), matching B instead of \P gives the same equations for the unknown parameters but involves approximating the unit vector 55 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles f i in terms of the unit vectors f 2 and 0 2 near the second monopole and vice versa near the first one. For example, near the © monopole, in cyl indrical coordinates, r2 r i sin 9i \ ( 1 1r\ cos 9\ -z H p ) I — + „ 1 „ n s in0 ! ^2ri cos0i -z— + p z while the undetermined terms with p\ in the local © solution is proport ional to the 0(s~3) vector in the above expansion: P i e 3 a ' (ri(2ri - 3 r i s in 2 9X) - 0 i3 r i sin 9\ cos 9^j — p ie 3 a ' ( - p r\ sin 9i + z2rx cos t The matching of these then gives the same condition for p i as before (Eq 3.23). As well, notice that the local 0(e3) homogeneous B fields diverge at infinity, but since they are to be evaluated only near the monopoles, they are admissible. We proceed to solve for h near both monopoles using the ansatz, which relates h to B: he = c—— + (<7i — c)e2ari cos0i 4- ^ e 2 a c o s 0 i + ki [ r i 2 1 3 +(o[ - c)e3a'r1 cos 9X + - e V cos 0X + P ie 3 a ' ( r 2 - -r\ s in 2 9X) Z z r i i he = c h (cr2 + c)e 2 ar 2 cos 0 2 — - e 2 a cos 0 2 + k2 [ r2 2 1 3 + (o'2 + c)e 3 a' r 2 cos 0 2 - - e 3 a ' cos 92 + p 2 e 3 a ' ( r 2 - - r 2 s i n 2 0 2) Aga in , we expand the static solutions to 0(e3) for the matching: 1 r 2 c o s 0 2 A 3 r | c o s 2 0 2 3 r f c o s 0 2 5 r | c o s 3 0 2 / 1 7 S^  ^ 3 + 2 3^ + 2 ^ 2 S1 + 1 r i cos 0X r 2 3 r 2 cos 2 9X 3 r 3 cos 0X 5 r 3 cos 3 0X ^ / 1 I I r~2 y s + s 1 2s 3 ' 2 s 3 2 s 4 ' 2 s- \ s Predictably, this expansion does not give a 0(j§) term proportional to r c o s 0 , which is what it gives at the lower order O(j^); hence the matching conditions for o[ and a'2 are simply o[e3a' e a c ; cr2e a -e3a'c 56 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles and both imply the C(e 3 ) acceleration is given in terms of the 0(e2) one: , , e 2 a . 2 e V c = = — — s s6c This result, however, is based on the questionable conditions, E q 3.24. On the other hand, the expansions of h above can give rise to the terms wi th p near the monopoles provided that Pie°a = p2e° a = —-This is in contradiction with the conditions obtained from matching * (Eq 3.23). T h e S c o p e o f t h e E x t e r n a l F o r c e L a w What this contradiction says is more transparent when we look at it in terms of the gauge invariant fields, B and Vh. First , we write down the first order ansatz accurate to C (e 3 ) but this t ime include also the set of magnetic multipole moments, rhg n , which are the homoge-neous solutions for the perturbed equation E q 3.22, and the mult ipole moments of Vh, rnvh, n , which are determined by the ansatz in terms of the magnetic moments: (e 2a + e 3a') c - (e 2a + e 3a') J = [BstaUc ± Vhstati^j + ^Brad ± Vhradj n Now, both the static part of the fields and the analog of the radiation fields, which are the particular solutions of E q 3.22, are total ly determined and do not possibly lead to any contradiction. It is the fact that the undetermined multipole moments of both B and Vh are to be matched under the exchange principle to the respective external fields that causes the contradiction: since the L H S of the ansatz contains no terms proportional to any moments above the lowest order of the multipole expansion, if the higher multipole moments of the external B and Vh fields when equated with rhg n and rh^h,n respectively do not combine 57 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles according to the R H S to vanish, then the ansatz has become false. For example, in the two different charge monopole system, although the ansatz is derivable for C ( e 3 ) , the 0(e3) external fields for each monopole in the © / © system add up instead of cancel because of the difference in charge: 0 = m g 2 ± m V / i , 2 near © / © monopole —* = Bext ± V/lext = Bext ± (±-3ext) = 2Bext 7^ 0 This failure of the ansatz means that unless there exists a solution other than Manton's ansatz for weakly rigidly accelerating monopoles for which there is no inconsistencies at 0(e3) when the two non-uniform external fields do not combine to zero, the assumption that the fields up to C(e 3 ) are r igidly accelerating under non-zero non-uniform total external field is incorrect. This is reasonable if the monopole were to behave similarly to a finite size bal l of electric charge with spherically symmetric charge density under a non-uniform field: the bal l would deform instead of accelerate rigidly. The contradiction, however, does not imply any values for the 0(s~3) acceler-ation; in particular, it does not imply that the 0(s~3) force between two opposite charge monopoles is non-zero. We can now also see that for the static solution of two opposite charge monopoles, for which neither monopole is accelerating, the static ansatz, which does not include the constant term e2ac, would be satistfied near each monopole, and the above contradiction for higher multipole moments would appear even for the lowest moment, the constant. The accelerated ansatz allows the two monopole solution to be consistent to 0(e2) by providing a "way out" for the lowest moment. Final ly, note that at C ( e 4 ) , if the acceleration at 0(e2) is non-zero, the ansatz is not satisfied even if there exists only uniform external fields at 0 (e 4 ) . 58 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles On the other hand, for the two same charge monopole system, the contradic-t ion above does not occur because near each monopole, the external B and Vh moments arising from the fields of the other monopole already have the relation-ship required by the ansatz: rhgn = T " W M and Bext = T^hext near .©/© monopole Also, since the 0(e2) has been determined to be zero, the 0(e2) "radiat ion" terms that could potentially give a non-zero result for the acceleration at 0(e3) vanish, and so e 3 a' = 0. Now, since e 2 a = 0, the derivation of the ansatz is val id for 0 (e 4 ) , and again, the matching at this order does not involve any inconsistencies and the lower order "radiat ion" terms being zero would lead to the 0(e4) acceleration being zero. We can do this at al l orders of e and conclude that the acceleration of monopoles in a two same charge monopole system vanishes to al l orders of e, i.e., the vanishing force between two same charge monopoles in the B P S l imit is an exact result. 3.3 Finding the Force through Calculating the Momentum Flux We look at another way to find the force between two monopoles proposed by Goldberg et al [2], which gives the result as Manton's procedure, and reinforce our interpretation of the first order ansatz as the uniform external force law. We also discuss the possibil i ty of using Manton's two-monopole global solution without the 0(e2) terms as the static solution in Goldberg's method and the possibil i ty of concluding that the only force between two monopoles is the 0(l/s2) force. In the process, we understand better what is essential in Manton's method. Goldberg et al [2] find the force between two monopoles by calculating the 59 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles rate of change of momentum of either monopole in a static two-monopole con-figuration. The momentum of the monopole here means the momentum of the fields wi th in a bal l that encloses al l the "matter fields" J" (as defined before in E q 2.5) of the monopole; the surface of the bal l , then, has to be in the asymptotic region where J " = 0. The momentum current is the spatial component of the stress-energy ten-sor, which is the Noether current obtained from translational symmetry, and is conserved: = 0, therefore, the rate of change of each space component of the momentum inside the bal l equals its current, p 3 , integrated over the surface of the bal l : r &pj r Force3 = / ~^-dV = / V • p> dV Jball °T Jbaii = [ p> dA where p j = Tij and Pj = Toj JSball This integral is by definition the force on the enclosed monopole and what we need to evaluate. 3.3.1 Stress-Energy Tensor and Reduction to the Electric problems First , since the boundary of the balls is in the asymptotic region , the magnetic field on it is given, without X1 being set to zero, by: P r o b l e m for U Now, we already know that satisfies the vacuum Maxwel l equations, i.e., V x B = 0, in the asymptotic region; therefore, we can write B% 60 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles as the gradient of a scalar potential U: B = W The divergence of B being zero implies that U satisfies the Laplace equation: V2U = 0 and the flux conditions on B for each monopole implies the flux conditions on U: f B-da = ± 1 j VU • da = ± 1 J Sball J Shall Final ly, along with the requirement that U approaches a constant at infinity since the magnetic tends to zero there, the problem for U up to the monopole order is exactly analogous to the problem for the electric potential, V, for two separated local electric charge distributions with the same or opposite total charges. S t ress E n e r g y T e n s o r i n t e r m s o f sca la r p o t e n t i a l s We can write the stress energy tensor in terms of U and h in the asymptotic region: T " " = Tr -g^F^Fpx- FwFvp + \g»vDP4>Dp§ - D^^D^ = \<r (/pA0 • W - r i • / > ) + {^gr&hi • dPw - &>w • srhfi) = (^g^&UdkU - S^&UdkU + VU^U^ + (^gtMUdphdph-d»hd,/hJ Notice that for a static configuration of electric charges for which the magnetic field vanishes, the stress-energy tensor in terms of the electric potential, V, is: rpiiV ^ ui/ rOi f fflOfV fllifV 1 electric ~ ^<J J JOi J Jo J J i = •-\g,"'dhVdkV + &lV&'V-5ffidkVdkV 61 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles We can compare the first bracket in T^v wi th T^ectic for different values of p and v, and see that the dependence of the former on U is exactly the same as the dependence of the latter on V. Since the equations for U are also the same as those for V in the analogous electric problem, the bracket involving U in T^u would give the same force law up to the monopole order for the two magnetic monopole system as the force law for two ordinary Maxwel l electric monopoles. The higher moments are not determined because we have only the flux conditions on U and not a charge distr ibution for the magnetic sources. On the other hand, the second bracket in T^v involving h depends on h just as — T 7 " ' depends on V except for the irrelevant case pv = 00. We wi l l now show that the problem for h can also be reduced to a static electric problem. t h e p r o b l e m for h We know that in the asymptotic region, the static first order ansatz can be factorized: B = ±Vh for © / 0 monopole; (3.26) the equation of motion DxD{d) = 0 reduces to the Laplace equation for h: V2h = 0 and unlike for [7,the flux conditions for h at both monopole are the same, due to the change of sign in E q 3.26 when the monopole charge is changed: / Vh-da= j ±B - da = ± . ± 1 = 1 (3.27) JSball JSball Therefore, the problem for h, for both same and opposite charge monopole pairs, is analogous to the electric potential problem for two separate local electric sources with the same total charges. The terms in the stress-energy tensor involving h would then give the force opposite to that between two same electric charges, i.e., Coulomb attraction, for both the same charge and opposite charge monopole pairs. 62 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Adding the force contribution from both U and h, we obtain twice the Coulomb attraction between two opposite charge monopoles, and zero force between two same charge monopoles. This is the same statement as the one given by our inter-pretation of Manton's ansatz as the external force law, that for the two opposite charge monopole system, the external forces on each monopole add up while for the same charge monopole pair, the external forces on each monopole cancel. 3.3.2 Manton's O(e0) Global Solution as the Static Solu-tion In order for the above result to be val id, we need to show there exists a static solution in terms of h, <f> and X1) that would give the required potential U. We already know that h has a solution since it simply satisfies the Laplace equation with boundary conditions; hence we need to show only that there are 4> a n d A fields that would give a magnetic field B that satisfies Maxwel l equations in the region between the monopoles as well as the flux conditions (Eq 3.26) at the monopoles, or equivalently, fields that give the potential U that satisfies the Laplace equation and the proper flux conditions. Whi le Goldberg shows the existence of the static solution by solving the second order static equations from scratch,' we already use Manton's O(e0) global solution as the static solution: *global = COS 0X - COS 02 - 1 ] hgiobai = C - - — J Xglobal = 0 and (f>global = hgiohai 4>global{^ global) ', -^global = ~ ^%4> global x 4>global In this solution, however, higher multipole fields of O(e0) that may be needed to solve the equations of motion to higher order in e are omitted through Manton's choice that B depends only on 0, i..e., A = 0. This means that this solution, as a static solution for Goldberg's method, also does not determine the higher order 63 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles force. We wi l l first show that with the presence of these fields in Manton's global solution, the discussion in this chapter remains val id, and then briefly look at how these higher order fields could possibly be zero. Effects of multipole terms in static solution on Manton's method The inclusion of any higher order fields in the O(e0) global solution would not alter Manton's matching procedure or conclusion. First , in terms of the gauge invariant B and Vh fields, even if there is a higher order field in the global B and Vh solutions, the external force law to 0(e2) would st i l l use only the Coulomb terms in the global solution and the acceleration e 2 a would st i l l be determined to be the same. Now, in terms of the h, and A fields, the argument is more complicated. Having a higher order field from each monopole means that A from each monopole is not zero, because A vanishing implies that B depends only on the unit vector </>,' but B being gauge invariant means that however </> rotates in the SU(2) gauge space, provided that the change is continuous, B remains invariant, and so changing </> (continuously) cannot add a higher order contribution to B. Thus, the curl of A term in E q 3.25 needs to be non-zero O(e0) to give rise to any higher multipole fields which, just as the Coulomb fields, are of O(e0) in the global solution. W i t h A ^ 0, the local ansatz for an accelerating monopole is modified to: B0{V) + V x A = Vh + e2ah (3.28) where B is st i l l l inear in ^ but now also linear in A. Since the terms with A cannot contribute without singularities (the Dirac string) to the monopole term of B, the Coulomb order fields in the global B field st i l l depends only on thus, the 0(e2) homogeneous solutions of ^ near each monopole, which is to be determined by the lowest order term in the expansion of the \I>(0) solution from the other monopole, is st i l l matched as before without any influence from A. On 64 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles the other hand, the matching of A is not described by Manton's method because near each monopole, the external fields to be matched, which corresponds to the far field limit of the magnetic dipole or above fields from the other monopole, is of 0(1/s3) and above. As for h, the added magnetic multipole fields do imply more terms for the O(e0) solution of h, but the matching of the local and global h up to 0(e2) does not involve these extra terms. In conclusion, the global O(e0) magnetic field containing higher multipole fields from each monopole, does not interfere with the procedure discussed in previous sections to obtain the 0(1/s2) force between monopoles. P o s s i b i l i t i e s f o r t h e g e n e r a l i z e d a n s a t z The higher order fields can be non-zero or zero depending on how Manton's ansatz for a single accelerating monopole generalizes to higher order in e. For instance, at 0(e3), Manton's first order ansatz no longer holds true, and it is possible that the correct relation between the 0(e3) fields has a consistent solution only in the presence of some external dipole fields. Then, we would need to include dipole contributions in the 0(e°) global magnetic field, and there would be an 0(1/s3) force between the monopoles due to the coupling between the monopole charge and the added dipole field, just as the force equals q • Emon(Q) + E(iip(0) for an electric charge in the presence of an external electric field having both Coulomb and dipole contributions. However, it is also possible that extra degrees of freedom exist in the 0(e3) ansatz and no field needs to be added to the global magnetic field, just as Manton's 0(e2) ansatz contains the degree of freedom, e2a, which is determined by and does not impose any condition on the already determined 0(e°) global solution. Thus, if we can argue that the higher order local equations does not require higher order external fields, we can use Manton's global solution in Goldberg et al's method to conclude that the higher order forces are zero. 65 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles 3.4 Conclusion for the Commutative Problem Manton's first success is his idea of solving the time-dependent equations of mo-t ion for the instant when a monopole accelerates perturbatively from rest such that the time dependence of the solution can be specified and the time-dependent equations can be modified accordingly, and his discovery of the first order ansatz for this scenario. We interpret his ansatz in its factorized form in the asymptotic region as the lowest order external force law that says that both the magnetic field and the field Vh contribute to the force on a monopole. Manton's second success is his choice of gauge and his discovery that in this gauge the magnetic field B can be written as a linear function of one of the components, \I>, of 0, which is defined by <j> = h(f), and the solutions of \T/ and h determine the ful l solution, <f> and A^. Then, because the first order accelerated ansatzes as well as the equations of motion in the region between the monopoles are linear in B (and so in * ) and h, the solutions of both * and h in the middle region are simply the solutions to the sum of the accelerated ansatzes, and the magnetic and Vh fields are in turn simply superpositions up to homogenous solutions of the first order ansatz of those produced by both monopoles. Manton claims that requiring the global solutions of h and * to reduce to the local ones near each monopole determines the acceleration between the monopoles. On the other hand, we explore the ambiguities of the global solutions and find that they lead to ambiguity of the conclusion for the acceleration. We propose el iminating these ambiguities by a simple exchange priniple, which says that the global solutions should not include any homogeneous solutions to the accelerated ansatzes and that the homogeneous solutions at each monopole are to be deter-mined by the far field l imit of the solutions from the opposite monopole. This again suggests the interpretation of the Manton's ansatz as an external uniform force law at each monopole, with the external fields Bext and Vhext being simply 66 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles the lowest order term in the multipole expansion of the fields from the opposite monopole. We then discover that although Manton's ansatz is derivable for the next order in the small parameter e quantifying the monopole acceleration, the ansatz cannot be satisfied near the monopoles within the two opposite charge monopole system. We speculate that this implies that the monopoles in such a system deform and is not accelerating rigidly at this order. We show that the acceleration between two same charge monopoles vanish to al l orders of e. Goldberg et al arrives at the same conclusion for the force on a monopole in a two monopole system by calculating the momentum flux through a surface that encloses that monopole in a static two monopole configuration. More explicit ly, they solve the static equations of motion for the two monopole system, substitute this static solution into the stress-energy tensor, and integrate the momentum currents given by the tensor over surface enclosing the monopole. Their success is in noticing that in the asymptotic region, the stress-energy tensoris composed of two pieces, each depends on a scalar potential as the U ( l ) electromagnetic stress-energy tensor depends on the electric scalar potential when only static electric fields are present; and in solving for the static two monopole configuration in terms of these two scalar potentials, U (B = VU), and h ((f) = h(f>). Goldberg's approach gives another perspective on the monopole force problem. Fi rs t , the stress-energy tensor written in terms of the scalar potentials shows clearly the force contributions on a monopole in this theory and reinforces our interpretation of Manton's ansatz as the external force law. Secondly, Goldberg et al's assumption that the static solution needs to sat-isfy only the flux condit ion, that the integral of the divergence of the magnetic field over a volume enclosing the monopole be proportional to the charge of the monopole, coincides with the ambiguity of the higher order multipole fields in 6 7 Chapter 3 Manton's Method to Find Force Between Two Commutative Monopoles Manton's O(e0) global solution, such that both Goldberg et al's and Manton's methods yield the same undetermined result for the higher order (0(1/s3) and above) force between two monopoles, although Manton's method of using an ansatz seems to allow us to guess better at what happens at the next order of e. We are interested in finding out if it is possible to argue without solving completely for the higher order ansatz that the higher order forces between two opposite monopoles vanish, or otherwise if we can find the force to the dipole order by proposing a specific t ime dependence which accounts for the deformation of the monopole to this order and then modifying and solving the time-dependent equations of motion. 6 8 Chapter 4 Background: Non-Commutative U(N) Gauge Theory In order to reach our goal of applying Manton's method on monopoles in non-commutative flat space, we need to know the formalism and the classical equations of motion of the non-commutative gauge theory. We do not cover the quantum aspects in this introduction [17] [18]. 4.1 Operator Formalism to Star product For-malism Non-commutative geometry on flat space-time can be described by coordinates that are not numbers but operators whose commutation relation is given by the non-commutative parameter 9^u: [x^r] = id*" where 6^ is antisymmetric and constant under Lorentz transformation. The imaginary "z" is there because xu is Hermit ian and the commutato of Hermit ian operator are ant i-Hermit ian. 69 Chapter 4 Background: Non-Commutative U(N) Gauge Theory We wi l l consider only spatial non-commutativity, not space-time non-commutativity, which poses more complexities. We choose coordinates such that the first two coordinates do not commute: ( 0 9 0 ^  - 0 0 0 0 0 0 pi \ (4.1) To write down an action for a field theory in non-commutative geometry, we need to first define the derivative and the integral for the non-commutative coordinates. We want these linear operators to retain the properties they have in commutative geometry [7]: di(fg) = (b\f)g + f(b\g) j Tr d j = 0 for /V 0 , / J Tr[ / ,$] = 0 0 at infinity (4.2) The following choice of derivative for the non-commutative coordinates, i , j ,=l, or 2) satisfies al l of the above dj. = [d,/] = [ - t ( 0 ^ , / l where 9i3(9~1)ji — 1; and the integral is uniquely determined by the rules above [7]. Mixing of gauge space and real space The integral is written as f Tr because in non-commutative U(N) gauge theories, the notions of integrating over real space and tracing over the gauge indices cannot be separated. First , note that functions of the non-commutative coordinates being operators does not prevent the incorporation of gauge symmetries into theories, although it can affect what gauge groups are allowed (this wi l l be discussed in the next section). Secondly, in 70 Chapter 4 Background: Non-Commutative U(N) Gauge Theory a U(N) gauge theory in non-commutative spatial background, an operator field, f, in the adjoint representation transforms formally as in the commutative theory (this wi l l be derived in the another formalism in section 4.2): f — > U f T j t ; UTj t = i except that the "unitary" matrices are now unitary operator matrices and the mult ipl icat ion is an operator matr ix mult ipl icat ion. Note that because f and U are operators and do not commute even if f and TJ are not matrices, the transformation above is not t r iv ia l even if the gauge group is U ( l ) . Final ly, a special property of the non-commutative gauge theory is that the set of translations in the non-commutative directions are also gauge transformations. W i t h the above derivative, an infinitesimal translation 5f becomes a commutator: tf^x' + a' => fix') /(£') + a'idJix')) = /(fj-i^cr1)^',/(£*)] the exponential form of which is f(xi + ai) = e - W - 1 ) ^ * " / ( £ ) g ^ - 1 ) ^ (4.3) It remains to show that U = e~l(e 1 ) J ; q ' i J is unitary in the operator sense: (JTJ\ _ g - i C t f - 1 ) , - ^ ^ ' eH0-1)ikakxl _ e±[(9-1)jiaixJ,(e-1)lkakx>] This means that when we move in real space, we also move in the gauge space. Therefore integrating over real space is not orthogonal to tracing over the gauge indices: J and Tr are to be used together. B a s i s for t h e n o n - c o m m u t a t i v e space Since the commutation relation be-tween x1 and x2 is analogous to the commutator of x\pl in quantum mechanics, 71 Chapter 4 Background: Non-Commutative U(N) Gauge Theory we can introduce creation and annihi lat ion operators just as in quantum mechan-ics then: Since the derivative operator can be written in terms of the coordinate oper-ators, they can also be written in terms of c and c\. We can then describe functions of x1 and x2 in terms of matr ix elements < m\f(xl,x2)\n > where \n > for n = 0 to oo is the basis of the standard annihi lat ion and creation operator Fock space: c 'c \n >= n \n > ; < m\n >= 8mn Integrating over the two non-commutative direction then amounts to Tracing over these states. D e r i v a t i o n o f t h e S t a r P r o d u c t There is a way to write down non-commutative geometry without involving operators: Bayen et al [19] introduced a map between the operator-valued functions f(x) and number-valued functions f(x) such that the operator product f(x)g(x) would map to a product, called the star product, f(x) * g(x), which reduces to the ordinary pointwise product f(x)g(x) when the non-commutative parameter 9 goes to zero. We wi l l need two maps. The first map f(k)[f] is defined as a formal fourrier transform of the operator function f(x to commutative momentum space: ^(x1 + ix2) (4.4) c t n >= \/n + 1 In + 1 > c = y/n | n — 1 > The second map f(x)[f] is the formal inverse fourrier transform back to coordinate 72 Chapter 4 Background: Non-Commutative U(N) Gauge Theory operator space, whose definition needs a prescription for the ordering of the non-commutative operators x1, x2. We wi l l the standard Weyl-order definition: /»[/] = ^jd2kf{k)eik^ A n alternative formula may help to show what Weyl ordering is: 1=0 ' ' Once we have mapped the operator function to momentum space, we can inverse Fourrier transform it to ordinary coordinate space. Thus, an operator 0(x) would map to 0(x) as follow: 0(x)[6] = j d2k e~ikixi j Tr eik'&i6{x) Now, the operator product, 0{x) = f(x)g(x), can be written in terms of f(k')[f] and <?(&")[$]:' 6{x) = f{x)g{x) ' • = (-= J d2k' f(k') eik^ ) J d2k" g(k") eik"^ = _L J d2k' j d2k" (/(fc,)p(fc")ei(*,+fc")'*<+^ ifc^ ''^ )^ = i_y d 2 f c / y d 2 f c W ( /( f c ') 5( f c») e i(*'+fc")i* i- i^*{*^-and maps to 0(x)[0] = j d2k eik'x j Tr e~ikS:d{x) = - ± - j d2k' j d2k" e<k'+k''^e-^kU{k')g{k'') iff'i d d = e * ^^f(yl)g(z%=z=x This 0(x) = f,(x) * is the Moyal or star product of the functions f(x) and g(x). 73 Chapter 4 Background: Non- Commutative U(N) Gauge Theory ^ Let us check that the commutator relation between the non-commutative coordinates st i l l holds: [x\ x3} ' -» x1 *x3 - x3 * x% = i9lj and that the derivative operator in the operator formalism maps to ordinary-derivatives in the star product formalism: 'dj =[-i{B~1)ijX1,f(x)] -> -i{9~1)ijiOmndmx3dnf = (9 1)ij9jndnf = dif The properties of the derivative and integral in E q 4.2 can also be checked to remain true. The star product has the following properties: 1. associative: (/ * g) * h = / * (g * h) 2. non-commutative: / * g ^ g * f 3. non-local: it involves al l order derivatives of both functions f and g 4. J Tr f * g = / Tr fg since the higher order terms of the star product expansion can al l be written as total derivatives. This implies that functions can cycle inside the integral: fTrf*g*h = J Tr h * / * g The advantage of the star product formalism over the operator one is obvi-ous in theories with perturbatively non-commutative background. In such the-ories, expanding the star product to the leading orders in 9 wi l l allow us to capture the main features of the theories and to distinguish effects due to the non-commutativity. We wi l l use the star product formalism from now on. 74 Chapter 4 Background: Non-Commutative U(N) Gauge Theory 4.2 The Action of Non-commutive U(N) Gauge Theory We can bui ld the action for a commutative U(N) gauge theory with a scalar field with min imal coupling by simply requiring it to contain terms quadratic in the first derivative of each of the scalar and gauge fields, and to be invariant under Lorentz and gauge transformations. We wi l l first outline this and then argue that we can bui ld the non-commutative action in the same way. We first write al l fields as Lorentz tensors, such that contracting the space-time indices of these fields wi l l easily yield a Lorentz scalar in the end. Secondly, if the scalar field in the adjoint representation of the given U(N) gauge group and transforms as 4> -+ u<t>u\ UU] = 1 we can bui ld a first derivative of this scalar field that transforms in the same way, namely the covariant derivative D^d) (as in E q 2.2), such that the term quadratic in this derivative would transform like the term quadratic in the field itself, which is needed if the scalar field has a non-zero mass. The operation that would make the mass term gauge invariant would then also make the kinetic term gauge invariant, the operation being to take the Trace of the matrices in question. Now, the covariant derivative calls for a gauge field (explicit ly shown in the next section) that transforms like > —> UA^lP +iU{d^) and we again want a first derivative of this gauge field such that it transforms also like the scalar field, and such that the square of this derivative can be made gauge invariant by the same operation that made the other terms gauge invariant. We arrive at the normal expression for the field strength F^v (as in E q 2.1). Final ly, 75 Chapter 4 Background: Non-Commutative U(N) Gauge Theory we assemble the integrand of the action by summing the Lorentz and gauge invariant "squares" of D^fi and F^v. We can bui ld the action for a non-commutative' U(N) gauge theory with a scalar field in the adjoint representation by the same steps, except that the Trace operator is to be used with integration over the non-commutative directions (Sec 4.1) to make the action gauge-invariant. The step are the same because al l the manipulations above do not depend on what type of non-commuting product acts between the matrices, be it the ordinary matr ix product or the star ma-tr ix product, as long as it is st i l l associative and satisfies the normal axios for a product, such as 1 <g) f = f. In particular, both the covariant derivative and the field strength in a non-commutative gauge field come about in the same way and have the same form as those in the (non-Abelian) commutative theory. As an examply, we wi l l show explicit ly the derivation of the non-commutative covariant derivative. Derivation of noncommutative covariant derivative Given that the scalar field transforms in the adjoint representation as follow: cj) —y U*d>*U\ U*U* = 1 the covariant derivative is buil t such that it transforms similarly, D»4> —v U*D"(f>*U\ U*U] = l and the following quadratic expression to be used in the action also transforms similarly: Now, the space-time part ial derivative of the scalar field does not satisfy this 76 Chapter 4 Background: Non-Commutative U(N) Gauge Theory requirement: d>*(f> —> &i(U*<l)*U*) = U*(d"d)) *[/f + [(d^U) * <f> * rf + U* 4> (d" * t/ f)] (4.5) but is to be included in the covariant derivative. The combination, (dfi+T(AfJ,))d), however, wi l l transform as required if its extra term, T(A^), gauge transforms to give both the term U * T * W and terms to cancel the last two terms in the transformation of d^d) in E q 4.5, which can be rewritten as [-U * (d^U1) *(U'*<f>* U^) + (U* d> *£/f) *U * ( < W f ) ] (4.6) using *-unitary of U as well as the product rule for the derivative of star products: (b^U) *U] = d"{U*U]) - U* ( c W f ) = -U* {d»U]) If the extra term T(A^) consists of the *-commutator of the scalar field and a field that transforms as follow: A^ —> U * A*1 + iU * (d»tf) then the terms produced by the gauge transformation of T(Ati) wi l l cancel with the terms in line 4.6. The covariant derivative therefore looks exactly the same as the famil iar one in commutative gauge theories: D^d> = d^d> - i[A* d> - <p * A] The Action Since the non-commutative action is buil t in the same way the commutative action is buil t , it is simply the commutative action with ordinary matr ix products replaced with star products: SNC = \ J dx4Tr [F^{x) * F„v{x) + 2D»d>{x) * D^(x) - A(0(x) * <f>(x) - c 2 ) 2 ] 77 Chapter 4 Background: Non-Commutative U(N) Gauge Theory where F " " = 3 M " - WA" - ie(A" * Av - Au * A " ) ; = d M 0 - ie(i4" * ^  - 0 * A* 1) for t^, i / = 0,1,2,3 where the potential term is present such that in the l imit 9 —> 0, this theory reduces to the commutative theory we studied before. As before, we wi l l only consider the theory in the B P S l im i t - A —>• 0 but (4>(x) * <f>(x) — c 2 ) = 0 at inf inity-such that the last term in the action vanishes and does not contribute to the derivation of the equations of motion. S U ( N ) no t a l l o w e d Al though the form of the gauge transformation is the same for non-commutative and (non-Abelian) commutative gauge theories, the *-gauge transformation does not allow SU(N) to be the gauge group of a non-commutative theory. The following decomposition of the infinitesimal U(2) *-gauge transformation clearly demonstrates this and the argument can be easily generalized for U(N) *-gauge transformations. Let the infinitesimal *-unitary matr ix be U = 1 - i(a0t0 + aata); a = 1,2, 3 where to is the identity generator, t 0 the generators for SU(2), and cto, cta are infinitesimal gauge parameters. Then, infinitesimally, the scalar field transforms as follow: <f> —> (f> ~ i[(oi0to + aata) * (</>0to + (f)ata) - {<f>ot0 + (f>ata) * (a0t0 + aata) := (f>-i [(a0t0 + aata), (</>oto + <pata)}t Before expanding this, note that the *-commutator of two SU(2) fields produce 78 Chapter 4 Background: Non-Commutative U(N) Gauge Theory a term that is not proportional to any of the SU(2) generators: [Aa(x) ta,Bb(x) t6], = Aa(x) * Bb(x)-Qea6ctc + ^6abt0^j - Bb(x) * Aa(x) Qe 6 a c t c + ^ a t t 0 ^ (Aa * Bb + Bb * Aa) {Aa*Bb-Bb*Aa) — 2~ eabc^c H 2~ ^afeto The *-gauge transformation -above then expands to l i i i J . J . i / a 6 * </>c + 0C * a 6 \ (p —> 0oto + (pata + eabc t a [<*0, <Pa)* + [aa, <p0] \ , . ( K , 0o]* + [aa, (pa}^ * « *a - M ~ * A n important difference between this gauge transformation and an ordinary U(2) gauge transformation is that even when the infinitesimal form of the *-unitary matr ix U involves only the SU(2) generators, i.e., a0 = 0, the transfor-mation would st i l l "create" a term that is proportional to the identity generator, which is not in the SU(2) space. In other words, SU(2) is not a close group under the *-gauge transformation and cannot be the gauge group for non-commutative gauge theories. Our problem wi l l be set in a U(2) non-commutative theory. 4.2.1 Gauge Invariant Quantities We already know that the gauge space and real physical space are not orthogonal in non-commutative gauge theories; we now check that simply taking the Trace of (without the integrating over space) any operator 0(x) that transforms like the adjoint scalar field indeed does not make it gauge invariant: TrO(x) —> Tr \U(x) * 0{x) *U*(x)} ^ Tr 0(x) sinceTr [A *' B] ^ Tr [B * A] The integrand of the action, for instance, is not gauge invariant, unlike in the commutative theory. 79 Chapter 4 Background: Non-Commutative U(N) Gauge Theory This means that if we do not want to study only quantities integrated over the non-commutative directions, we need to find a way to construct semi-local gauge-invariant'operators. Gross et al [20] constructed gauge invairant operators in .momentum space by attaching open Wi lson lines to adjoint operators and then *-fourrier-transforming the combination. F i rs t , the Wi lson line is defined as the *-path-ordered exponential of the integral of the gauge field along a curve C starting at x: W(x,C) = P *exp (ie j dX ^  A^x" + s" (A))^ W i t h the same argument as in the commutative theory, this Wi lson line *-gauge transforms as follow: W(x,C) —• U(x)*W(x,C)*U](x + l) Now, W(x + I) is simply W(x) translated and can be writ ten as the *-gauge transformed U(x): tf(x + l) = eikx * U\x) *e~ikx . where the non-zero components of the momentum k^ is given by equation 4.3: kj = -(r1)^ kjP' (4.7) The combination W(x,C) * elkx then transforms as the adjoint sclar: W(x,C) *eikx U{x)*W(x,C)*[uKx+J)*eikx] = U(x) * W(x, C) * [eikx * tf(x)] Therefore, for each operator O(x) in the adjoint representation, we can define a corresponding operator 0(k) in momentum space by first attaching a Wi lson line to it and then *-Fourrier transforming: d(k) = J dx4TrO(x)*W(x,C)*eikx 80 Chapter 4 Background: Non-Commutative U(N) Gauge Theory and it wi l l be gauge invariant: 0{k) ' J dx4TrU(x)*0(x)*W(x,C)*eikx*U](x) = 0(k) provided that the open Wi lson line extend a vector ll(k) (Eq 4.7) from its starting point x. Note that for an operator at momentum k, the Wi lson line extends in a direction transverse to the momentum and to the commutative direction. A lso, in the commutative l imit , / ' reduces to 0 and the operator 0(k) reduces to the ordinary Fourier transform of the orginal operator 0(x). F inal ly, for operators at large k, the Wi lson line is long and dominates such that al l operators at large momentum would exhibit similar large k behaviour. [20] 4.2.2 Broken Lorentz and Rotational Invariance Since the non-commutative tensor 9^v is the same in any inert ial frame, i.e. does not Lorentz Transform, the star products of two Lorentz tensors, and therefore the action, are only invariant under boosts in the commutative direction. The following simple example of the star product of two Lorentz scalars illustrates this: oo ( i _ 4 _ QH" -4-1" f*9 — > ( / * < ? ) = 2^ ~\ f(A V M A V)W=y' n=0 ^ ( H & A y l e»> [ ( A - y £])" n\ - - • x = y n=0 = f*g only if A ' „ 0"" ( A " 1 ) / = 9<» where A is the 4 x 4 linear Lorentz transformation matr ix. If 9^v is defined such that its only non-zero coponents are 912 = [-921 = 9, the last condit ion is satisfied only if A (A ^ 1) has non-tr ivial entries only in the 0 or 3 (time- or z-) components and therefore represents a boost in the commutative z-direction. 81 Chapter 4 Background: Non-Commutative U(N) Gauge Theory 4.3 The Equations of Motion The equations of motion in the non-commutative theories can be obtained by the normal variation procedure due to the properties of the star product. We avoid varying the action twice, separately with respect to the gauge field and the scalar field, by rewriting the action. We write the scalar field as a extra space-time component of the gauge field and name the new 5-dimensionsal gauge field A1*, and require that all of its components are constant along the added spacial direction [21]: A4 — d) ; 9 4 A ' " = 0 The covariant derivative of the scalar field can then be written as the fourth spatial component of the new field strength, F ^ , defined by D»*d> = <9M4 - d4Afi -.te [A",4 4], = F>fi4 The action, with its potential term which does not affect the equations of motion in the BPS limit omitted, simplifies to: SNC = \fdx4Tv [F'^(x) * F'^x)] for y.,v = 0,1, 2, 3, 4 Now, we vary the action with respect to the new gauge field: 5S = \ j Tr [8F,»V*F'IXV + F,»V*8F'IW} where 8F'^ = &i(6Av) - du{8A't) + [(5A»), A \ + [A^ {8AV)\ We assume the field strength drop to zero sufficiently fast at infinity; therefore, the factors inside the J Tr can be cycled (by property 4 of the star product section 4.1). Integrating by parts, the variation becomes SS = \ j Tr [ 5 F ' ^ * F ; j = - I Tr 82 5A^(d^F^+[A^F^}) Chapter 4 Background: Non-Commutative U(N) Gauge Theory Now, the star product between 5A" and the other factor can be replaced by the ordinary local product (again by property 4 of the star product). So, for arbitrary 5AV, the variation of the action vanishes only when Dli*F'liV-=0 (4.8) or equivalently in terms of the original scalar and gauge fields: for v = 4 : * (Dp * d>) = 0; (4.9) for v = 0 ,1, 2, 3 : * F " " = ie [{Dv * d>) , <j>]m (4.10) For the U(2) non-commutative theory, each space-time component of the equation of motion has four components, one for each generator of the non-commutative U(2) gauge group. We wi l l call the equation for t 0 , the identity generator, the U ( l ) sector and the ones for t a , the Pau l i matices, the SU(2) sector. 4.3.1 Expansion of the Equation of Motion We wi l l be studying the problem of the force between two non-commutative monopoles in a perturbatively non-commutative theory. Therefore 8 « 1 and we can expand each sector of the equation of motion in the small parameter 8 and study the equation order by order. U ( l ) O{80) F i rs t , we expand the U ( l ) component of E q 4.8 to 0{92): dv{sr/ti - = e- [dv [A^4r}+d„ {A>, A1:}+eabc [A':AB\ A':} 92 + j [{A';,{A^,A'»}} + {A'-{A:»,A':}}} where {f,g} = (d^^g) - (d2g)(d1f) 83 Chapter 4 Background: Non-Commutative U(N) Gauge Theory and terms with {/, g} originate from the O(0) terms in the expansion of the *-commutator of f and g and therefore are antisymmetric under exchange of f and g-To O(0°), the R H S of E q 4.11 is irrelevant, and expanding the gauge field A'* 4 in orders of 0: j±» = -+ + Aw2) we find that the equation for is total ly decoupled from the SU(2) sector and is simply a sourceless U ( l ) electromagnetism equation (in 5 dimensions): DV ^ A ^ 0 ) - 9 M ' ; ( 0 ) ) = a , F ^ ( 0 ) = o . As in normal electromagnetism, the gauge field A 0 ^ obviously has some gauge freedom, but here, we also have the freedom to choose the value of the gauge invariant quantity FQUfi^ without affecting the O(60) SU(2) sector. To show the gauge freedom at different orders of 0 of the gauge field and fields that transform in the adjoint, we expand the infinitesimal *-gauge transformation (Eq 4.7) of these fields to 0(d2), wri t t ing the extra terms present only in the transformation of the gauge field in square brackets: for U = 1 - i (aQ0) + a{06) + af^ t 0 - i (aa + + af >) t „ ; a = 1,2, 3 84 Chapter 4 Background: Non-Commutative U(N) Gauge Theory f -to/ 0 ( 0 ) " t0d^ (0) A9) + ' t a ^{^\^} + d-{a^,f^}+eabMC - [ta0"a?>] + t a e a 6 c f a » / c + / ' + 2 " f c / c - ^ { { a i ° \ / i ° ) } } j - [ t a c ^ a f +t 0 ^ ({<*?>,/?>}+{««/?>}+{«?<•,/<•>}+KU(O)}) (4.11) Now, the second line of this transformation shows that no *-gauge transforma-t ion can alter the U ( l ) zeroth order adoint fields, F ^ ^ and (f)Q°\ whereas the U ( l ) zeroth order gauge field has the gauge freedom, d^a0°\ where a 0 ° ' is a free infinitesimal parameter. We wi l l refer back to this equation when we discuss the gauge freedom of the higher order fields. For we non-commutative monopoles, we choose F ^ ^ and <j)^ to vanish and the gauge in which vanishes. 85 Chapter 4 Background: Non-Commutative U(N) Gauge Theory + 4 + 4 + 4 + 4 SU(2) O(80) Next, we expand the SU(2) component of the equation of motion: dv (P-A? - d»X:) + eabc [dv [A^A>) + Al (d»Xc» - d»A'?) A-A' A'V A'^ _ A'^A' A'" +1 [{X, (FAX - rXf)} + {AI, (d»A': - STA^) }" +^abc [{Al, AfX?) + Aihv ({4", Xf) + { ^ , A ^ } ) " "2[{^K,^}} + {<,,{^ }^}] |^oi" {AT; •^•a"}} {A)I/> {Ar\ A)"}}] [ { { 4 „ ( ^ - ^ ) } } ] ^ [e a 6 c ( - 2 9 , { { ^ , ^ } } + { { (FA{ - d»A'f) ,Al}})\ (4.12) where {{/,<?}} = (d21f){d2g) + {d2lg){d22f)-2(d1d2f)(dld2g) is symmetric under exchange of f and g and terms with these double brackets originate from the mathcalO(92) terms in expansion of the *-anti-commutators. A t zeroth order in 9, this equation has an irrelevant R H S and is total ly de-coupled from the U ( l ) sector. In terms of the original fields, A^ and </>, the equation is simply the SU(2) sector of the E q 4.9 and E q 4.10 with the *-product replaced by the ordinary product, which are obviously the equations of motion in the commutative SU(2) theory. Thus, the 0(9°) SU(2) field, A f ( 0 ) , is simply the solution to the commutative theory. According to the first line of E q 4.11, its gauge freedom at this order is also exactly the same as in the commutative case. U ( l ) 0{9) We can use the choice for the U ( l ) 0(0°) fields, A0Ko) = 0, to simplify the U ( l ) equation, E q 4.11. In fact, since the terms on the R H S of this 86 Chapter 4 Background: Non-Commutative U(N) Gauge Theory equation are quadratic in the U ( l ) fields AQ, they wi l l contain at least one factor of the vanishing AQ^ at 0(62) when the 9 expansion of A£ is put in. Thus, the U ( l ) equation accurate up to 0(82) simplifies to: d^dr-Af-PA^) = °- [du{Aa^A:} + eabc{A:Ab',A^}' + 1 [ { ^ ( 0 ^ - 0 ^ ) } ] (4.13) Note that the first order U ( l ) field, A0^9\ is determined independently of the first order SU(1) fields and depends solely on the zeroth order SU(2) fields, A'j*; and that the second order U ( l ) field, A0 , is determined only by the zeroth and first order SU(2) fields. In general, the U ( l ) component of the fields of any order in 6 is determined independently of the SU(2) components at the same order, and is determined only by the lower order U ( l ) and SU(2) fields, which would have been determined already by lower order equations. This is because expanding the equation of motion, E q 4.8, to an arbitrary order in 9 only adds more terms with explicit 6 dependence to E q 4.13 but does not change its property that the terms that involve the highest order fields depend only on the U ( l ) component of the fields. Another property of the U ( l ) equation is that to al l orders of 9, it takes the form of the ordinary Maxwel l equations, A$ ' being the Maxwel l gauge field, wi th a non-localized source comprising of the terms on the R H S , which involve lower order (lower than nth) fields and spread out over space-time. We now look at the gauge freedom of the U ( l ) fields at this order. According to the fourth line of E q 4.11, the transformation of the 0(9) U ( l ) component of the field strength, the scalar field and fields that transform like them is governed only by the zeroth order gauge parameters a0°^ and aa°\ This means that these U ( l ) fields (FQ"^ etc.) have no gauge freedom at 0(9) if we have completely fixed the gauge for the zeroth order fields. In general, at an arbitrary order n of 9, the U ( l ) component of fields that transform like the field strength has no 87 Chapter 4 Background: Non-Commutative U(N) Gauge Theory gauge freedom that is not already determined by the lower order fields. This is because the infinitesimal transformation of these U ( l ) fields involve only *-commutators (Eq 4.7), the expansion of which already has an explicit factor of 9, and so no gauge parameter to 0(9n) can be involved. On the other hand, the U ( l ) component of the gauge field, however, has a new gauge freedom at each order of 9 parametrized by a0° :^ it transforms with the extra term —daQd \ In contrast, according to the third, fifth and sixth line of E q 4.11, the SU(2) components of the field strength and the scalar field, and the SU(2) component of the gauge field, do have a new gauge freedom that depends on a new gauge / o n \ parameter aa ' at each order n of 9. A lso, the terms in these lines arising from the star product expansion renders the gauge transformation of the SU(2) component of fields which transform like the field strength not simply a rotation in the SU(2) space, and therefore the magnitude of such SU(2) vectors are not gauge invariant unlike in the commutative theory. S U ( 2 ) 0(9) We use the choice that the zeroth order U ( l ) fields vanish again to simplify the SU(2) equation (Eq 4.12). To 0(9), since al l the terms on the R H S of this equation depend on the zeroth order U ( l ) fields Aa^°\ they vanish, and the equation does not differ from the zeroth order SU(2) equation. This implies that both A f ^ ° ' and (Aa^ + Aa^) solve the same equation, and so are related by 0(9) symmetry transformations of the theory. Now, the transformed solution, (Aa^ + Aa^), is physically different from the original solution, Aa^°\ only if the symmetry transformation is not a symmetry of the solution. However, these symmetry transformations actually simply change some choices we have had when solving for the zeroth order solution Aa , and so the transformed solution Aai 4- Aa ' could really have been the zeroth order solution we have chosen. Therefore, we can choose Aa = 0 without any loss of information of the solution. For instance, for a monopole solution A^°\ we have the freedom 88 Chapter 4 Background: Non-Commutative U(N) Gauge Theory to choose its in i t ia l 4-position on the coordinate system, a first order correction A a m W h i c h is an O(e) translation would simply change that choice, but that choice is arbitrary to begin with. Also, as in the U ( l ) sector, the determination of the SU(2) fields, Aa^e ^ to each order n of 9 is decoupled from the determination of the U ( l ) field, A0, to the same order. U ( l ) O(02) Now, using the result from above that the first order SU(2) fields can be set to zero, we can further simplify the O(02) U ( l ) equation (Eq 4.13): dv [dvAf2) - d»A»{d2)) = 0 (4.14) Interestingly, this has the same form as the zeroth order U ( l ) equation and is again total ly decoupled from the SU(2) sector. Note however that this simplif i-cation is not a regular occurrence in even orders of 9 and happens here only due to the tr iv ial i ty of the first order SU(2) solution. Bo th the th i rd and fourth order U ( l ) fields depend on both the lower order SU(1) and U ( l ) fields. S U ( 2 ) 0{92) F inal ly, the 0{92) SU(2) fields needs to satify the non-tr ivial E q 4.12 and depend on both the lower order SU(2) and U ( l ) fields. 89 Chapter 5 First Order Force between Two Non-commutative Monopoles The force between two non-commutative monopoles-does not alter from the force between two commutative monopoles to first order in the non-commutative pa-rameter 9. In fact, the effect of the non-commutativity in the dynamics is not seen to this order. We wi l l show this both by the stress-energy tensor as well as by a slight extension of the Manton method. 5.1 Non-Commutative Monopoles Magnetic monopoles in the commutative theory, as discussed in chapter 2 and 3, are defined by the asymptotic behavior of the U ( l ) magnetic field embedded in the SU(2) field strength tensor. In a non-commutative theory with small 9, the field strength is dominated by the lowest order term, i.e. simply the field strength of the commutative theory; therefore, the same embedded U ( l ) magnetic field can be used to define the non-commutative monopole. To 0(92) , the magnetic field 90 Chapter 5 First Order Force between Two Non-commutative Monopoles is: 2 + 2 = e^^Aka~l-([Ai,Ak]+zeabcAi*Akc)ya +e^k^dUk-l-[Ai,Aka\yo l^F!Lmutauve + & + \ { ^ a ( 0 ) , AT } ) to \e-k (o {A^\Ar} + 2d3AkP + eabc ^ ( 0 ) + A ^ A ^ -[jeabc{{A^°\AT}}y. + O(0*) (5.1) where several terms have vanished because A0^ = Aa^ = Alff ^ = 0 as discussed in section 4.3. Note that whereas the zeroth order commutative field strength is in the SU(2) sector and can be factorized, F 3 c k m m u t a t i v e = (fjk4>a) t a , to give an embedded U ( l ) field strength which satisfies the Maxwel l equations, the O(0) correction to the field strength is in the the U ( l ) sector, and the O(02) correction (as in hte last two lines of E q 5.1), although in the SU(2) sector, cannot be factorized into a Maxwel l U ( l ) field strength and a unit vector field. Thus, the higher order corrections to the field strength cannot be easily described by corrections to the embedded U ( l ) field strength j1*" which we used to define the commutative monopole. The definition of the non-commutative monopoles in terms of the O(e0) em-bedded magnetic field and the fact that to 0(0°) the equation of motions are the same as those in the commutative SU(2) theory imply that any system of non-commutative monopoles is simply the solution of the analogous commutative SU(2) system (with t r iv ia l O(0W) U ( l ) fields) plus 0(0) and above corrections for both the U ( l ) and SU(2) fields, which satisfy the equations of motion ex-panded to higher order. We wi l l only need the tr iv ia l 0(0) SU(2) correction in 91 Chapter 5 First Order Force between Two Non-commutative Monopoles this chapter. 5.2 Force Correction from the Stress-Energy Ten-sor We have established two facts: that to O(90), a system of two non-commutative monopoles equals the solution of two commutative monopoles, and that for any classical solution to the non-commutative theory, the 0{6) SU(2) sector can be chosen to be zero as reasoned in section 4.3. These, along with the statement in this section that the O{0) correction to the non-commutative stress-energy tensor depends only on the 0{9) SU(2) fields, and therefore vanishes, determines the O{0) correction to the force between two non-commutative monopoles to be zero. We wi l l also generalize that the forces within any system of non-commutative solitons [22] are the same, to 0(9), as those within the system of commutative solitons to which the non-commutative ones reduce at zeroth order. 5.2.1 Non-Conservation of the Stress-Energy Tensor There are more than one definitions of the stress-energy tensor in non-commutative gauge theories. For example, Yukawa and Ooguri obtain one by computing disk amplitudes in string theory in a large NS-NS two-form background field and tak-ing the Seiberg-Witten l imit [23] [24] [5]. This tensor is locally kinematical ly conserved, gauge invariant and vanishes as 9 —>• 0. To arrive at our statement about the 0(9) correction to the force between monopoles, we wi l l use the tensor obtained from the Noether procedure, because it seems more intrinsic to the theory. We wi l l find that this tensor, interestingly, has very different properties than the one mentioned above: it is not locally conserved, not gauge-invariant, and reduces to the tensor for the commutative 92 Chapter 5 First Order Force between Two Non-commutative Monopoles theory at the lowest order. We derive the energy and momentum currents from translational invariance using Noether's theorem. The resulting current is neither locally conservative nor locally covariantly conservative. To obtain covariant conservation for the stress-energy tensor, * T^U = 0, we need to add a term which equals zero to the variation of the action, 5SNC, and carry out the derivation which is slightly different from the commutative case as in the usual way. We note that the final tensor and its conservation equation is not gauge invariant.The detailed deriva-t ion follows. D e r i v a t i o n o f t h e S t r e s s - E n e r g y T e n s o r We then add a term which equals zero to the "conservation" equation, (the algebra is a l i tt le dissimilar from the commutative case), to obtain a covariant conservation for the stress-energy ten-sor, DP * T^U = 0. We also note that the tensor is not gauge invariant and that its conservation equation is also only gauge covariant. The detailed derivation follows. For simplicity, we switch back to the notation that the space-time indices go from 0 to 4, with AA = 0 and = 0 such that The transformation of the gauge field due to the translation has an ordering ambiguity and needs to be symmetrized so that in the operator formalism the transformation would be Weyl-ordered: S x' x" + c" (x) 1 , „ ... 93 Chapter 5 First Order Force between Two Non-commutative Monopoles The variation in the action is then 5S = ^ j Tr F " v * SFpV dx4 We substitute 5A11 in this expression: SS = dx4Tr F»v * {{dpdpAu) * e" + {dpAv) * ( ^ e p ) 4- ( d p A „ * e») * A„ + A„ * ( 0 P A , * e + i y d x 4 T r F"" * [ep * (<9pdpA,) + (dllep) * (5 ,4,) 4- (C * dpAj *AV + A„* [ep * dpA and rearrange the terms: 5S = ^- j o?x4Tr F"" * { ( 3 ^ 4 , 4- d P 4 * 4 , 4- A M * d p 4 ) * e p 4- dpAp * [e", 4 ] J + ^  y d^ 4 TV {e> * ( d p d p 4 4- <9P4 * 4 4- 4 * d p 4 ) 4- [Ap, e\ * d p 4 } * F"" +^Jdx4Tx {{dpAv)*F'u' + F>u'*(dpAl,)}*dl>ep = X- j da;4TV ^ p (F"" * F„„) * e" - 0„ ( d p 4 * F " v 4- F"" * dpAv) * e' 4-^  y ^ 4 T V (<9P4 * F"" 4- F"" * d p 4 ) * [Ap, e% We add f d ^ [du (F^ * Ap) 4- dv (Ap * F'*")] to the integrand, which vanishes because of antisymmetry of the / i , v indices in F^ and the symmetry in dp du. Upon expansion, the first term in the expression is 0„ dv (F"" * Ap) = {dvF^ * Ap 4- F"" * d„Ap) = d, {-[Au,F»%*Ap + F^*dvAp) . . where the equation of motion * F^v = 0 has been used in the last step. These added terms combine with the terms in integrand (denoted as s/2 in the 94 Chapter 5 First Order Force between Two Non-commutative Monopoles following) to form more combinations of Fpv: s = ^ ( F ^ * F p „ ) * e' [(dvAp - dpAv + [ 4 , APl) * F " v - 4 * F " " * Ap + F " " * Ap * 4 ] * +aM [ ( F " " * ( c U p - dpAu) + [ 4 , APl) - A V * A P * F " v + 4 * F " " * Au] + {dpAu * F " " + F " " * 0 , 4 , ) * e ' ] , = Q5p(^ *^ ) + a , ( F ^ * F w + F w * F H ) *ep +0 M ( - 4 * F " " * Ap + 4 * F " " * Av + F " " * Ap * Av - Au * Ap* F " " ) * + ( 0 , 4 * F " " + F " " * 0 , 4 ) * [ A , e l , Final ly, the last two lines of the integrand regroup into [A„, F»v * Fpv + Fpv* F»% + (1/2) [Ap, F " " * F ^ ] , in the following manner: Inside the integral, the factor Ap at the end of the last line can be "cycled" to the front and so the last line can be rewritten as [ 4 , ( - 0 , 4 , ) * + F " " * ( - 0 , 4 , ) ] , * e» (5.2) For the second line, we expand the derivative. The derivative on the factor Ap in all four terms gives the following commutator bracket [Ati,(d1/Ap)*F^ + F^*(d1/Ap)l*ep; (5.3) the derivative on F'*" gives ( 4 * [ 4 . *AP-A"* [ 4 , F""] J * 4 * e" (5.4) - ( [ 4 , F H , * 4 * ^ + ^ * ^ P * K . ^ ] J * e p (5.5) which, after the interchange of some of the fj,, u indices, equal the following: [ 4 , ( F ^ * [ 4 , 4 1 * + [4, . APl * F"" ) ] , * e' (5.6) ( - F " " * 4 * 4 * 4 - 4 * 4 * 4 * F " " ) * e"; (5.7) 95 Chapter 5 First Order Force between Two Non-commutative Monopoles and the derivative on Av gives [Ap, dpAvd»Av - dpA^AX * e" (5.8) + {Ap*[A»,.Avl*dlxAv-dllAv*[A>l,A,'l*Ap)*ep (5.9) + ([A", A"], * Ap * dpAv - dpAv * Ap * [A\ AX) * ep. (5.10) The terms 5.2, 5.3, and 5.6 combine to [A^ -F^ * Fpv - Fpv * F»\*ep. The remaining terms, 5.4, 5.5, 5.7, 5.8, 5.9, and 5.10, combine to (1/2) [Ap, F^ * F^]^* ep. Therefore, since ep (x) is arbitrary, when we take away the star product be-tween ep (x) and the other factors in the integrand, the covariant conservation law is obtained: DP*T^U = 0 (5.11) \F* * Fa0 - \F»P * F\ - \. where = +g^-Fa^  0 - "p  vp -Fvp * F»p (5.12) The difference in this derivation from that in the commutative theory, in which the tensor is gauge invariant as well as locally conserved, VTm = 0 (5.13) where = Tr {^sTF^F^ - F»PF^ , is that in the commutative theory, the matrices in the integrand can cycled under the trace operator without involving the translation non-matrix parameter ep (x), but in the non-commutative theory, because ep (x) is a function of x and because it is related to the other factor by the star product, any cycling involves al l factors inside the integral, including ep (x) . 1 Note that T^m and therefore E q 5.13 is invariant under the commutative gauge transformation whereas the non-commutative tensor T^v and its equation are not invariant under the non-commutative gauge transformation. 96 Chapter 5 First Order Force between Two Non-commutative Monopoles 5.2.2 O (6) correction to the Force Between Two Non-Commutative Solitons Since the stress-energy tensor is not locally conserved and not gauge invariant, it does not allow us to find the force between two non-commutative monopoles by calculating the momentum current flux through a surface enclosing a monopole as in the commutative theory (Sec 3.3): we need to either solve the problem of non-conservation or extract information from only the conserved and gauge invariant total energy and momenta. Because of the form of the stress-energy tensor, it is easy to arrive at a statement about the force at 0(9) between solitons, but much more non-tr ivial to obtain one at 0(92). Since our goal for this project is to extend Manton's method to the non-commutative theory, we have not pursued this method furthur at the non-tr ivial second order. C o n s e r v a t i o n o f g l o b a l ene rgy a n d m o m e n t u m To show the conservation of global energy and momentum, we first integrate the covariant. conservation equation over al l space: where the second term vanishes since TIU = 0 at infinity for finite energy-momentum solitons, i.e., there is no current flowing in or out of the boundary of space, the energy and momentum charges, T0l/, integrated over all space is conserved in time: and find that / Tr d^T^dx3 — 0 because the star-commutator vanishes inside the space integral. Rewri t ing in components, 0 0 97 Chapter 5 First Order Force between Two Non-commutative Monopoles T h e fo rce i n t e r m s o f t o t a l ene rgy The force between two non-commutative solitons, or monopoles, is the rate of change of the total energy of the system with respect to the separation distance s, dE(s)/ds, where E(s) is the total energy for the solitons at separation s, E(s) = f Tr T00(s)dx3. Expanding E(s) , we obtain the 0(9) correction to the force along the axis of separation of the monopoles in terms of T 0 0 : Force = ^-(E^(s) + E^(s) + E^\s)) OS dx3 d_ ds . / Tr [T°°W (s + As) - T°°W (s)] dx3 j iv [T 0 °(°)(S) + T ° °W ( s ) + T°°^(s) + l im As->0 As / T r \T00^(S + AS)-T00^(S) dx3 As->0 ' As = Forcecommutative + Forced + Forced N o 0(9) c o r r e c t i o n to t o t a l e n e r g y To O(0), the argument is simple: The force correction involves the difference of T°°^(s) and T°°^(s + As), which are the time-component of the 0(9) stress-energy tensor for two solitons at separa-tions s and s + A s respectively. For both separations, the 0(9) stress-energy tensor depends only on the 0(9) correction to the SU(2) components of the field strength, FaU^d\ which in turn depends only on the 0(9) correction to the SU(2) components of the gauge field, which can be chosen to vanish as argued in sec-t ion 4.3. Thus, the 0(9) force correction is zero. 98 Chapter 5 First Order Force between Two Non-commutative Monopoles Expl ic i t ly , the stress-energy tensor to O(02) is: where al l the O(90) FQV^ terms are omitted since the U ( l ) fields are al l set to zero to this order). The 0(9) correction is on the second line. These terms do not include derivative terms of the form {/, g}, which comes from commutators of star products because of the symmetricness of the tensor, and they vanish when p»v{9) v a n i s n (Note that any correction to the U ( l ) part of the field strength begins to contribute only at 0(92).) Now, recall from E q 5.1 that the first order correction to the field strength for any system is only in the U ( l ) sector, i.e., F£"^ = 0, because the SU(2) 0(9) fields vanish by the equation of motion provided the zeroth order U ( l ) fields are chosen to be zero. This means to 0(9), for any solution of a perturbative non-commutative U(N) gauge theory, the stress-energy tensor is the same as that for the commutative SU(N) solution to which the non-commutative solution reduces at zeroth order. In particular then, for the cases in which two solitons [22] [25] are at distances s and s + As apart, the 0{9) corrections of the total energy, T°°W(s) and T°°W(s + As), are zero; therefore, the force correction to this order vanishes. +i 0 /«V A ( * a ) F ( 0 ) - W * 2 ) Fv <°> - FVP^ F " W ° P^a r a r pa r a r pa + ^9 b o *p\o ~ Y o b Po ~ Y o * Po + J 9 ^ { { F ^ \ F J 0 ) } } - £ { { F ^ , F ^ } } - £ { { F 1 » > F ; ( » > } } (5.14) 99 Chapter 5 First Order Force between Two Non-commutative Monopoles This fact does not depend on the geometry of the system; for instance, it is true even if the solitons are separated by only a small distance or in the non-commutative direction. It also does not capture any effect of the non-commutative geometry. t h e P r o b l e m i n 0(92) To calculate the force between two non-commutative monopoles to O(02) by calculating T00^(s), we need the solution of the fields it depends on: the 0(9) U ( l ) and 0(92) SU(2) fields for a two monopole system. We do not pursue this path. 5.3 Force correction Using the Manton Method We now start investigating how the Manton method can be extended in the non-commutative theory. We wi l l derive the first order ansatz for a single accelerating non-commutative monopole and check that the 0(9) correction to the force be-tween two non-commutative monopoles vanishes by this method. 5.3 .1 First Order Ansatz for Single Accelerating Non-Commutative Monopole In the non-commutative theory, Manton's first order ansatz wi th the product re-placed by the star product solves the equations of motion under the assumption that the monopole is accelerated globally in the commutative direction. The ar-gument also follows Manton's but with the product replaced by the star product. F i rs t , as in the commutative case, we describe a non-commutative monopole accelerating rigidly from rest by a small amount by putt ing in the specific ac-cording time dependence in the solution: 4>{xv) = - \e2aH2) • Aj{xv) = Ai{xi - \e2aH2) Z Z 100 Chapter 5 First Order Force between Two Non-commutative Monopoles where e2dl is the small acceleration of the non-commutative monopole and is in the commutative direction (discussed below). Aga in , we choose the gauge in which the time component of the gauge field A° vanishes in the instantaneous rest frame of the monopole such that a Lorentz boost along the direction of motion back to the non-accelerating " lab" frame yields the following A0 in the " lab" frame: A0 = -JaHA* Since only Lorentz boost in the commutative direction is a symmetry of the action, and the derivation of the first order ansatz relies on the above expression for A0, this method only works for acceleration in the commutative direction. Then, the part ial t ime derivative of the fields and the form of the time com-ponent of the gauge field allow the covariant t ime derivative of fj> to be written in terms of the covariant spatial deriviatve of <f> and the t ime component of the field strength tensor to in terms of its spatial components: D°*(f>= -e2ait(di(p-ie [4, 0] J =e2aitDi*<p Gj0 = -e2aH (VA1 - frA* - ie [A*, A\) = -e2aH Gji Replacing ordinary products by star products in each step in the same deriva-t ion in the commutative theory (section 3.1.1), the time dependent equations of motion for the instant when the monopole starts accelerating from rest, as in the commutative case, can be written to 0(e2) wi th terms depending on the acceler-ation in place of terms with any time derivatives or explicit t ime dependence: Dt * (Di + eV) * (f) = 0; (5.15) (e2ait) Dj * Gji = (e2ait) ie [(£>' * (f>) * 0 - 0 * (D{ * </>)] ; (5.16) [Di + e2ai]*Gij = ie [(Dj * 0) * <f> - 0 * {Dj * 0)] (5.17) where again the second equation does not give any information about the 0(e2) solution and is automatical ly satisfied by the static monopole solution. 101 Chapter 5 First Order Force between Two Non-commutative Monopoles Since the star product does not involve time derivatives, these rewritten equa-tions of motion do not involve time derivatives; thus, the form of t ime dependence introduced in the argument of the fields are allowed as in the commutative tease. These equations of motions can be satisfied by an analogous ansatz to the one proposed by Manton for the perturbed commutative monopole: Gij = ±eiik(Dk + e2ak)*d, (5.18) where besides the fields, the acceleration is also expanded in orders of 0: e V = eV<°> + eVW + e V ^ Note that E q ?? is *-gauge covariant, so the uniform acceleration ak, which does not *-gauge transform, can be determined in any gauge chosen. This ansatz satisfies the equation of motion E q 5.15 because the non-commutative Bianchi Identity: ejkiDi*Gjk = 0 depends only on the symmetry of the indices and holds independently of what kind of product acting on the factors. The ansatz also satisfies the equation of motion E q 5.17. The proofs are exactly analogous to the ones shown in sec-t ion 3.1.1 wi th ordinary products replaced with star products and the internal vector cross product replaced by the star commutator times (—i): We now examine the U ( l ) and SU(2) sector of the ansatz to O(02). U ( l ) c o m p o n e n t o f t h e ansa t z The component proport ional to t 0 of 5.18 is as follow: = F ( ^ + ^ ( W , ^ o ] , + [ 4 . 0 « ] j ) + e ^ 4 + I e ^ ( [ 4 , 4 ] t + [ 4 , ^ ] J = eV0 o (5.19) 102 Chapter 5 First Order Force between Two Non-commutative Monopoles Interestingly, if we expand the R H S to an arbitrary order n of 9, because the U ( l ) O(90) fields are chosen to be zero, 0Q 0 ' = 0, the term that de-pends on the G(9n) correction to the acceleration, e2a^e"' vanishes. This means that at any order n, no matter what e2a'(e") is, the corrections to the U ( l ) fields are the same, i.e., the 0(9n) U ( l ) fields takes no part in determining the 0(9n) correction to the acceleration. In particular, the 0(9X) correction to the acceler-ation does not depend on the U ( l ) sector at al l and the 0(92) correction to the acceleration depends on the U ( l ) fields only up to ~0(91). These are the same as the statements obtained in section 5.2 by inspecting the stress-energy tensor. 5.3.2 O{0) Force Correction We now know that only the SU(2) component of the first order ansatz (Eq 5.18), expanded as follow: T (W - \ ( [Al 4>a] , + [4. M , + itabc [Al * 0c + 0c * A\] ) ) +ei>kd>Aka - - ^ k ([Al Ak], + [A{, Ak}^ + 2zeabcA{ * Ak) = ± e V 0 a ~ (5.20) can contain information about the 0(9) correction to the acceleration. To 0(9), since the 0(9°) U ( l ) fields vanish, this equation is simply the ansatz for the commutative theory with an extra 0(9) modif ication to the assumed acceleration: +-{Dl<f)a + \eijkFik = ± ( e V + 6 V W ) 0 a where the fields are expanded in orders of the different small parameters e and 9: 0a = 0 i O ) + *P + *?> + 0 f '>. A\ = 40) + Af} + Af) + Af^ 103 Chapter 5 First Order Force between Two Non-commutative Monopoles Without the acceleration, E q 6.1, is a linear fluctuation equation for <f>a and A\, just like the second order equation of motion as discussed in section 4.3: it follows that 4>ae^ = Aa^ = 0, but (pa6e ' and Aade ' are st i l l unknown. A s y m p t o t i c c o n d i t i o n t o 0{9) The asymptotic condit ion of the commutative theory can be extended to first order in 9 as well. We can define the matter field J p to be what the covariant derivative of the field strength * F^v equals to in E q 4.10 such that at O{90), J» reduces to the matter field (Eq 2.6) defined for the commutative case: r = ie [ D " < M * ( 5 - 2 1 ) = eta(eabcD^c 4>b) - e t 0 9 { D ^ \ ^ } + O(92) We see that to first order in 9, the SU(2) component of J M depends on the SU(2) fields as the commutative matter field does. (We now switch to the vector notation for the SU(2) components of the fields as in section 2.2.1.) Thus, if we again write (j> in terms of a magnitude field h times a unit vector field (p: <P = (hW + h^ + hW) (0(o> + ^ a > + #*a>) where <j> depends as before on a function that captures its dependence on the angle it makes with the z-axis (the commutative direction): <P ( y/l - [tf (0)(°) + y(9)(<2) + V(9)&2)}2 cos(x) ^ yjl - [tf (0)(°) + (0)(£2) + V(9)^2)}2 sin(x) ^ (0)(°> 4- * ( 0 ) ( f 2 ) + ${9)^ j such that (j> remains of unit length even with the 0(9e2) correction, we would obtain, by the same reasons as in the commutative case, the same asymptotic condition: = 0 ==> D M 0 x (j) = 0 = > = 0; 104 Chapter 5 First Order Force between Two Non-commutative Monopoles the same relation between the gauge field and d>: e and the same factorization of the SU(2) field strength into its magnitude J1*" and the unit vector field d>: as in the commutative case, except now all the fields include 0(9e2) corrections on top of the 0(e2) corrections. Thus, the non-commutative first order ansatz For a system of two opposite charge monopoles separated in the commutative direction by a large distance s (fig 3.2) (s much bigger than the characteristic ra-dius of the monopoles) accelerating from rest towards each other, we can solve the above equation with the respective signs and accelerations for the local magnetic / field B (or the local \T/) to 0(9e2) near each monopole: F" " = = — [(0"0 x d'cp) • $\ 4> + [d'1*' - d'aX/j] 4> (Eq 5.18) can also be factorized in the asymptotic region: Bk = ± [dkh+{e2ak + e2ak^) h] The equation for \I> (9) would be modified to V x B = - V x (x x V*) = ±x ( e V ' + e V ) sing ' e R(°) + R(<2) + R ( e 2 ) +B{6e2) +B[6e2) - ° 0 ^ part ^  DQ hom ' -°e part ' he (5.22) B( R W + R ^2 ) + R(<2) +R(^ 2 ) + R^2> ^ f f i i - ° e part ' - ° e hom ' part > - " ® he (5.23) 105 Chapter 5 First Order Force between Two Non-commutative Monopoles The ansatz for each monopole then gives Vh near each monopole: Vhe _ "R(°) + B(e2) + B, R(^ 2) ^ R(^2) 0 hom '  J J Q part - o'^aW _ ( e 2 5 ( O ) c + e2£(0) c ) + ; + [40) + Bppart + B\ 6 2a(°) + e2a^ Vh, + ( e 2 5 (0) c + e 2 ^ ) c ) e2a(°) + e 2 a' + 3 + o'2e2aW Now, as in the commutative theory, since the solution of the system is in terms of the functions ^ and h, and the equations for these functions are linear, we can write down the global solution for ^ and h simply by adding the solutions near the monopoles but also applying the exchange principle that the undetermined homogeneous terms be determined by the expansion of the fields of the opposite monopole and not appear in the global solution: The construction of these global solutions implies that the global magnetic and Vh fields are also the sum of the fields near each monopole without the homoge-neous terms: We now determine the 0(9) constant homogeneous terms and the acceleration by requiring these global fields to reduce to the ones near the monopoles (Eq.5.22,. E q 5.23). As in the commutative case, this amounts to equating the undetermined terms at one monopole to terms from the multipole expansion of the static B and Vh fields of the other monopole. *global = ( * 0 - * 0 horn) + ( * © - * G horn) + Const hgiobai = ( ^ e — hom) + (h$ — h e hom) + const (BQ — BQ hom) + ( - B © — - B e horn) (Vhe~ Vhe hom) + (Vh® - Vh® hom) 106 Chapter 5 First Order Force between Two Non-commutative Monopoles j Specifically, near the 0 monopole, we need to expand the terms in the global 1 magnetic field that originates from the opposite © monopole and is of a higher order than 0(e2). Now, although the particular solutions, B®PART and BQpJrt) are of at least 0(e3) when expanded, they should be time-retarded as argued in section 3.1.2 and should not affect the © monopole at the ini t ia l instant. Also, there is no relationship between the different small parameters 9 and e and we cannot compare the order 0(9e2) wi th C ( e 3 ) . Therefore, at first order in 9, the only term to expand would be the 0(9) correction of the magnetic field from< the © monopole, which is zero. This, then, determines the unknown 0(9e2) approximately uniform magnetic field near the © monopole, — a[e2aS9\ to be zero. For Vhgiobai, because the monopoles are accelerating in opposite directions, the terms (e2a^ + e2a^)c from both monopole cancel and the expansion of the O(9e0) Vh term of the © monopole, which is zero, is to give rise to o[e2a^ (o[ —c) near the © monopole. Expl ic i t ly , Vhglobai = - ( B Q 0 ) + BQ part) + (B®0) + PART) _ o ( * 2 ) , 5(0e2) - " e part ' iJ© part_ + ( e 2 a ( ° ) + e 2 a W ) ( - - - ) where the first line are the 0(0°) terms, and the last line would be irrelevant since it is of 0(e3) when expanded near either monopole. Aga in , the radiative terms BQ PART and B E PART do not participate in the matching. Near © monopole then, to C ( e 2 ) , the condition for the global Vh field to reduce to Vhe is o[e2d^-e2d^c = 0 We already know o[e2a^ = 0, therefore e2a^c = 0. 107 Chapter 5 First Order Force between Two Non-commutative Monopoles As in the commutative case, this matching procedure can be interpreted'as applying a force law, which is the constant part of the factorzied first order ansatz (Eq 5.22), to the monopoles: Bill = -VhieJt-e^% where the external fields near one monopole are the far field l imi t of the fields produced by the other monopole (the exchange principle), except in this case, the force law is accurate up to O(0e2). Since both monopoles produce no non-vanishing 0(0) fields, the 0(0) external fields on both monopoles are zero, and the force correction to this order is zero. 108 Chapter 6 Preliminary investigation of the Manton Method to O(02) As shown before by the non-tr ivial O(02) corrections to the stress-energy tensor (Eq 5.14), the non-local property of the star product starts to affect the dynamics of a non-commutative gauge theory at 0(92). In fact, these effects render most of the simplifications in and the interpretation of the commutative calculation (ch 3))not val id here. Our objective, then, is to employ only the general scheme of finding the local solution near an accelerating monopole and the global solution valid in between the two monopoles, and equating these solutions in a region both decribe, in order to determine the force between two monopoles up to 0(92). This chapter reports on our prel iminary efforts towards this goal. We wi l l look at the non-commutative first order ansatz to show which local fields need to be calculated, show a sample calculation of such field, and wi l l conclude by discussing the difficulties in finding the global solutions. 109 Chapter 6 Preliminary investigation of the Manton Method to 0(92) 6.1 The SU(2) Component of the First Order Ansatz Because the O(02) U ( l ) sector of the first order ansatz for an accelerating non-commutative monopole (Eq 5.19) has no dependence on the 0(92) correction to the acceleration e2a^\ the local 0{62) SU(2) fields are the ones to be matched with the corresponding global solution to determine this acceleration correction and they satisfy the SU(2) component of the first order ansatz expanded to 0(92): ± (0Vfl + eQbc40c) + e^k&Aka + l-e^keabcA{Akc = ±(e2ak^ + e2ak^)d>a 4 + {A0,<f>a}) - ? c « * {Al Aka} + 0(9S) (6.1) where the fields are expanded up to O(02e2) wi th vanishing 0(9) terms: Ak = Ak(o) + Ak{S) + Ak(6e2) + Ak(e*) + Ak(e*e*) 0 = 0(o) + ^ ) + ^ ) + ^ ) + Note that in this equation, 1. the 0{92) and 0(92e2) fields are on only the L H S ; 2. the last two lines are non-zero only at 0{92) or above because the 0(9°) U ( l ) fields are chosen to vanish; 3. at 0(92), there is no dependence on the 0(9e2) fields, because the terms wi th these fields also involve the SU(2) 0(9) fields which vanish as explained in section 4.3; 4. there is dependence on al l the lower order -0(e 2 ) ) , 0(0)) and 0(9e2)-\J(l) fields and on the O(90) and 0(92) SU(2) fields. 110 Chapter 6 Preliminary investigation of the Manton Method to O(02) Therefore, before we look for the local O{02) SU(2) solution, we need to first solve for the local 0{9) U ( l ) fields. 6.2 Calculation of the Local U(l) Solution These local fields may also help in finding the 0(9) global solution, as the local fields do in the commutative case, and this global solution may in turn be needed in the determination of the global 0(92) SU(2) solution, to which the local SU(2) solution is to be matched. However, we have found no way in bui lding the global solution from the local ones and wi l l discuss the problem of doing it in sec 6.3. The local 0{9) U ( l ) fields involves both the static solution and its 0(9e2) correction which is due to the O(90) acceleration of the monopole. The static solution has been solved [26] [8]. We follow method used by Hata et al [8] [9] and calculate the 0(9e2) correction. 6.2.1 U(l) static Monopole Solution The U ( l ) component of the equation for a static © / © monopole is obtained by expanding the ansatz (Eq 5.19) to O(9e0): € ^ A k ^ + \e^{A^,A^} = T ^ ) + 1 - { A i } ° \ ^ (6.2) (Note that the U ( l ) ansatz can see the acceleration indirectly through its depen-dence on the SU(2) fields.) 0{9) U(l) solution for the © monopole We first look at the © case first and wi l l argue that the solution for the © monopole is the same. The equation above is a first order linear differential equation wi th inhomonogenous terms depending on the O(90) SU(2) fields, whose exact solutions in terms of the 111 Chapter 6 Preliminary investigation of the Manton Method to 0(82) W(r) and F(r) functions (Eq 2.11 and E q 2.12) wi l l be used following Hata et al [8]. Also, we now generalize the non-commutativity parameter 9 back to the original form 9^ such that symmetry in the choice of the commutative direction can help in determining the form of the solution. The inhomogeneous terms are explici t ly {AaAa} = 9efde[eaicXfW{r))ds[^F{r)) r \ ' f* f* f 7* / \ f f* y 7* 7* WF ( WF WF' WF = €ief6ef + eiefXe8fcXc ^ ^ ^~ + 2 — /. W F „ 1 9 / WF\ = e i e f 9 e f — + eiefxe9fcxc-—^-—j (6.3) and €ijk { 4 ' 4 } = eijkeefde{eajc^W{r)^df(^akd^W(r)j n ( W XCTTrlXe xeTTr\ ( W Xd-,jr,Xf X / T T A = eijk9ef ( e a je 1- eajc—W — - eajcxc—W I \eakf— + eakd—W — - eajdxd—W I WW „ ( W ' W nWW\ = eief9ef — + eiefxe9fcxc ^ - 2 - ^ - + 2— j ' ww a i d ( ww\ l a A . = e i e f 9 e f — + eiefxe9fcxc-—^ — .(6.4) These respect two symmetries: 1. A generalized rotation that rotates both the coordinates xk and the non-commutative vector parameter 9k, which is denned as 9k = ^e%^k9^k and points in the commutative direction; 2. space inversion The generalized rotation symmetry is intuitive. Suppose 9k points in the +z direction. If we rotate the coordinate system by an angle a about the +y axis, 112 Chapter 6 Preliminary investigation of the Manton Method to 0(92) and at the same time we rotate the vector 6k in the same way, 9k would st i l l point in the +z direction after the rotation. In fact, this is a symmetry of the equations of motion, and so the solution to it should be tensors wi th respect to this rotation. To 0(6)), the only independent tensors are rank 0: 9ixi rank 1: 9k,(8ixi)xk,eiik9ixi since others can be written as linear combinations of these using the following identity: CbjkSrc = ecjk$rb + tbckdjr + C&jc^ fcr (6-5) Since the inhomogeneous terms (Eq 6.4 and E q 6.3) satisfy space inversion sym-metry, the terms dl4>a and eljkd:>Ak also need to be unchanged under spatial inversion. Now, the derivative operator changes sign when space is inversed, so the fields also need to change sign, i.e., be odd under this discrete symmetry. Then, for the 0(0) U ( l ) static equation, expanded below: M^+M" = ( - ^ - ^ ) 1 9 ' W W F W °r dr V 4 r 2 2r 2 (6.6) The part icular solution is given in terms of the odd tensor structures: Af] = A{r)eijk9jxk = A(r)9ijxj . = B ( r ) 2 0 V = B[r)eiheihxi such that their derivatives in E q 6.2 are even tensor structures: d{(f)0e) = B(r)eijk9jk + eljk9jkxlB'(r)-€ijkdJAH0) = €ijkgkjA(r) + €HkdkbxbA,(rj xJ 113 Chapter 6 Preliminary investigation of the Manton Method to O{02) We rewrite the tensor structure el^k9^kxlx% in terms of the ones in the inho-mogeneous terms: to obtain the following ordinary differential equations in r for the coefficients of both tensor structures: r 2 \ r 2 / 4 \ r2 J The particular solution of these equations are simple: ™^ n A , s *LFW 1WW 5(r) = 0 ; i4(r) = - - ; r + i - ^ - . We now show that the homogeneous solution is tr iv ia l . Assume Ahom and Bhom to be polynomials in r: A(r) = Anrn ; B(r) = Bnrn such that eijk0jk . _ A n + B n + n B n = 0 for al l n -eijkxj9kbxb : An + 2Bn = 0 'for al l n^O: Combining these: for al l n ^ 0: Bn{2 + 1 + n) = 0 Then B n o m is only non-zero for n Bn= { 5 _ 3 for n = —3 BQ for n = 0 0 otherwise —3 or 0 and Aaom is given in terms of it: - 2 5 _ 3 for n = - 3 A , 0 for n = 0 otherwise 114 Chapter 6 Preliminary investigation of the Manton Method to 0(82) But the homogeneous solution with these coefficients is not admissble: the terms with J5_ 3 is proportional to r~2 and blows up at the origin; while those with Bo become infinite as r —> oo. So the U ( l ) static 0 monopole is simply the part icular solution: A, - & x [2 r2 + A r2 ) 0 A s i d e Recal l from section 4.3.1 that the 0(6) U ( l ) field strength has no gauge freedom apart from that for the O(90) SU(2) fields. This implies the combination dJAm _ dkAm a l s o h a s no gauge freedom. We can then interpret the solution as a Maxwel l gauge field that defines a "magnetic" field BlQ = epsilonljkdj and note that this magnetic field has a dipole term and a 1/r 4 term that does not have the quadrupole angular dependence, which is due to the non-localized source for this "Maxwel l " gauge field. (9(0)'U(1) so lu t i ons for t h e © m o n o p o l e For the © monopole, we avoid solving the equation again by wri t ing the inhomo-geneous terms in terms of the ones for the © monopole. We already know how 0(°) and A l (°) in the asymptotic region change when the monopole changes sign: 1 0ei ^ 0e = 92 0 V e x 0 e = ( (d^e2)(-4>e3) - (-#0 e 3)0e2 ^ (= #003)091 - (d^eiX-^es) ^ (#001)002 - (#0 e 2)0ei / -A -A: 91 02 \ ^ 9 3 J 115 Chapter 6 Preliminary investigation of the Manton Method to O(02) This leads to the sign change of one of the inhomogeneous terms: | -<4©a i </4a } defde •^ei ^ © 2 ^ © 3 S°01 \ ( _ A M \ ^01 - A m ^ © 2 \ ^ © ^ J X ©1 _ / 1 0 2 fc(0) + {4»°U*<0)* © Q J \ ^©V / In the asymptotic region then, the equation for the © monopole is the same as for the © monopole except the term +dl(f)0e^ has changed sign: However, 0 Q Q was zero, therefore the particular solution for the © monopole is just the same as the © one in the asymptotic region. Now, because of the similari ty of the equations for the two different monopoles, we can deduce that the solutions are also the same at the monopole core, i.e.: 1FW 1WW\ AW = 9ijxj <t>%\ = o 2 r 2 4 r 2 j The homogeneous solution vanishes for the same reasons as before. Therefore, although the oppositely charged monopoles have different 0(0°) SU(2) fields, they have the same 0(0) U ( l ) corrections. 6.2.2 U(l) perturbed monopole We now calculate by the same procedure as above how the U ( l ) fields change locally when a monopole is accelerating. As in the commutative case, we include 116 Chapter 6 Preliminary investigation of the Manton Method to 0(82) the homogeneous solution with undetermined coefficients, for potential ly absorb-ing expanded terms from the global solution such that the global solution would reduce to the local one near each monopole. The equation to be solved now takes into account the accelerating 0(9°) SU(2) fields as well: ^ by E q 6.8, E q 6.9, and E q 6.13. The following section is the explicit calculation of the O(0e2) corrections, <f>al result is given in E q 6.19 and E q 6.20. C a l c u l a t i o n o f t h e I n h o m o g e n e o u s t e r m s As in the static case, we can use the invariance under the generalized rotation, in which both the coordinates and the non-commutative vector 6% is rotated, to find the form of the solutions. We do this by wri t ing both the inhomogeneous terms and the solution in tensor structures built from 9k and xk. We wi l l now show the explicit calculation of the inhomogenous terms, which involves mostly tensor mult ipl icat ion, derivative and rewriting. The results are shown in E q 6.11, E q 6.12, E q 6.15, E q 6.16, E q 6.17, and E q 6.18. N o t a t i o n We wi l l use the following notation: where 4>a ^ and Aa are given in terms of the commutative solutions and and Aa6e \ to the U ( l ) fields for an accelerating non-commutative monopole. The 117 Chapter 6 Preliminary investigation of the Manton Method to O(02) C a l c u l a t i o n o f We wi l l again first calculate this term for the 0 monopole and obtain the result for the © monopole by sign change arguments. This term is explicit ly the contraction of the first derivatives of the commu-tative SU(2) solution withe non-commutative parameter: where Af> = eaic-2 ; 4>f] = ^ ( 0 ) Sk + Sh ^ The field </>££ ^ is st i l l needs to be written the tensor structure form as follow. For both © and © monopole, 8(j) = v & s i n * V 6V J where \ P e = cos# for the © one and according to E q 3.16 and E q 3.17, 2r 2r Then, 5(f> can be written as the following using 9^/9: 8(j> = e2ac e2aoi 2r 2r ^ — xz ^ -xy \x2 + y2 J ^abcObkxkxc— I -z~ar - -e'aai 1 2 l 2 - ^ 2 ~ M ? ) U2) such that one of the terms in <% ' looks like this: he 6(j)ea = -^eabc9bkxkx, 1 (1 2 1 2 2 Q~uu,-u.~.~c^ I ^ € a r - 2 e a a r 1 c r (6.8) The other term involves 6he, which can be easily written in tensor form: 1 „ 1 / 1 $he (pea — -eabc9bkxkxc 9 rl -e a — e a(a — c)r + ^abcObc^ Q e 2 Q r 2 + ~ C) r 3) + k ^ f (6-9) 118 Chapter 6 Preliminary investigation of the Manton Method to 0{92) ( 2 \ Next, we need to calculate the derivatives of both terms in <f>a £ ; . For the first term: df(h8(pa) = ^ (eabcebfxc + eabf9bkxk) ^ Q e 2 ° r ~ ^aar2} ( c - -+ ^eabcdbkxkxc^ Q e 2 a - e2aar2^j °y - ^ + ^abc9bkXkxc (\e2ar - ^aar2] ( + ^Xf 0 " " " " " " ^ \ 2 ~ 2 J V r4 ' r5 We eliminate the tensor structures that are not l inearly independent of the others using the following identities, which are variations of the identity 6.5: tabcQbfXc — ZfpqQpaxq + 2Cfab6bkXk ^abcObkXkxcxf = -efbc8ckxkxbxa + eabf8bkxkr2 such that df{h54>a) = i (efpqepaxq) Q e 2 f l r - \e2aar2^j (c ~ + pfabebkxk^ Q e 2 a r - ^ W 2 ^ - ^ 1 . o / l e 2 a c l e 2 a a i e 2 a \ + , e/a6< W \ - - — + — ) 1 / l e 2 a c 1 e2aoi e2a\ pfbcObkXkxcxa [ + + 2 — " -Similarly, we take the derivative of the other term in d>a ^ and use the above identities, df(5h<j)a) = ^[efpqepaxa + 2efabebkxk} (+^7^ ~ —^——^ 1 r - 21 (e2° e2a(ai — c) + Q[-ZfbcVckXkXbXa + eabfObkxkr \ I — + ——— r a i / e 2 a ( ° i ~ c ) + L e/pg0pa^g + 2efabVbkXk\ I . + k l \ r r3 119 Chapter 6 Preliminary investigation of the Manton Method to O(02) Final ly, we add these two terms and obtain the following for the derivative of l e ft T ( 1r2nrr r 1 F 2 A T R I I 3 e2ac e2a\ gtfpqVpa-Lq y 2 C a u 1° 2 r ' 2 r r2 J ~f g €fpq9pqXa 1 efa£i 2 r 1 t2ac\ 2 r ) df{64>) = + \ e f a p e p q x q ( - e 2 a o l C + \ ^ + \Sf) -\-lft r f) r r 1. (—l<?a°\ 4 . IZLQC _ n(2a\ ~ g^fpqJjpuqr-LvLar2 \ 2 r ' 2 r r2 ) The derivative of the field that is present even when the monopole is not accelerating is much more readily obtained: cal l the.bracket A cal l the bracket B call the bracket C call the bracket D 2xcxe We can now combine the two derivatives to find the inhomogeneous term itself: 9ef deAf^ d'tp = \9e} eaie efpg9paxg $ ~fg@ef ^aie ^ fpqOpqXa ^ ~fg9ef £aie €fap9pq%q ^ ~^~Q@ef ^aie ^ fpqXp9grXrXa ^4 2 A ~Q9ef ( ^ o i c - ^ c ^ ' e ) €fpq@paXq ^4 2 B ~Q9ef (^otc^'c^'e) €fpq@pqXa ^ 2 C ~Q^ef (.^aicXcXe) efap9pqXq — 2 D ~Q^ef ( ^ a i c ^ ' c ^ ' e ) £ fpqXpOgrXrXa 2 , s ( 5a-f XaXf\ gVef [CaicXcXe) «1 I 5 ~ r 7 I (6.10) We wi l l label the left column I and the right one II and refer to the terms by its coefficient and the column label (e.g. the very first term is A l ) . This expression looks long but in fact many of its terms vanish identically: 1. B l and BII vanish since 9ef tfpqdpq = 2{Cbef9b0f) = 0 120 Chapter 6 Preliminary investigation of the Manton Method to 0{92) 2. CI I vanishes since ®ef ( ^ a i c ^ c ^ e ) €fap@pq%q — 9efXfXe9iqXq 9eiXcXe9CqXq 0 3. DII vanishes since ®ef i^aicXcXejXa 6jpqXpO*qrXr — 0 We can disregard the terms k l and k l l which are proport ional to 1/ r 3 because it is subleading to the other terms in the far field l imit . The non-vanishing tensor structures are not independent of each other, but can be rewritten using identities derived from E q 6.5 in terms of the three inde-pendent ones,(9x)9i, 92Xi, and (9x)2Xi. The details are as follow. The terms in column one are proportional to eaie9ef which satisfies the identity: £aie9ef = Caieeefb6b = 5af9i — Sfi9a Using this, we rewrite A l , C I and D l respectively as A l : -9ef eaie efpq9paxq — = -{{-29qxq)9i - eipq9paxq9a)^ 1 A -{-29x9i - €ipq (ebpa0b) Xq9a) — U T 1 A -{-29x9i - (9iXa9a - 9qxq9i))^ U T 2 _ . A = — 9 x 9 i - ; 9 rl 1 C 1 C CI . ~Q^e^ ^aie £fap9pqXq ~^ — ~Q^aap0i9pqXq e{ap9a9pqXq) — 1 C ^( €iap@a{.£pqb9b)%q) ~~J g\ -tup-u\~pqu- ui~qi ^2 1 C = ^{-9a9aXi + 9a9iXa)— 9 rl = -(9x9i-92xi) % 9 rz D l . ~Q9ef ^a'e ^fpq^-'p9qr'^r'^a ^4 — ~9^" 9^9* (^ '^) y 4 ' 121 Chapter 6 Preliminary investigation of the Manton Method to O(02) Similarly, we can rewrite A l l : A l l : 2 a — Uef €aicXcXe ejpqUpaXq — 2(r29x6i-(ex)2xi) 4 The inhomogeneous terms for the © monopole is simply the sum of the above: 1 n n ( e2ao\c „e2ac 1 . . 2 /e2ao\c 1 e2ao\ 1 e2ac + 26{) X i ~ 2~r^~ ~ 2~r*~ + 7TQ(0x)2x 1 /e2ao\C 3 e2ao\ 9 e2ac + (6.11) 20K'~' ~ ' r 2 \ r2 ' 2 r3 2 r3 For the © monopole, the O(60e2) SU(2) fields are different and al l of the expre-sions h!^54>9, (Sh®)^ and AA°Q cannot be written in terms of tensor structures contructed using 6lj and the coordinates. This is because these vectors have only their the first two components different from those for the © monopole, which are the tensors. To il lustrate, since SP® = — cos 6 while 64>Q oc \ S<1> ^ smx oc yz ^ —xz ^ \x2 + y2 J -yz \x2 + y2 J and (She)4>e oc xz yz \ —xz -yz \ \x2 + y2 ) while (8he)<f>Q oc \x* + y' / Lucki ly, as in the static case, since the O(e09°) gauge field, also has a relative sign change from the © monopole field in the first two components, and the entire term, that concerns the dot product of the (SU2) vectors dd>a ^ and dAa°\ and 122 Chapter 6 Preliminary investigation of the Manton Method to O(02) sti l l be can be sti l l be written in tensor form: e2aa2c e2ac + 2 1 / M I ( e2aa2c le2aa2 le2ac ;(*) x* — z r - + 20 v ' % \ r2 ' 2 r 3 2 r 3 1 . . 2 If e2aa2c 3 e 2 a 0 2 9 '26^ ' XiV2 V r 2 2~r^~ ~ 2 r 3 Calculation for {^(ea),40)} The second inhomogeneous is calculated in the same way but involves the 0(e2) correction to.the SU(2) gauge field: where Ala^2) = eabcdl54>b4>c + eabcd%(f>b5(i>c ^(o) _ _a f o r Q monopole r We first need to write A^e ^ in terms of the independent tensor structures A a e 2 ) = pabc{€bpq6pixq + ebpi9pqxq) Q e 2 ° J - \e2(X(J^j ~ l i f t \ (A 2 J _ \ "I" Q £abc\€bpqVprxrxq )%i I ^ *^ ,^3 ) 1 \ Xc r 1 (5ib XbXi\ f n , (I 2 1 1 2 + gtabc I — ) (ecpq9prxrxq) I - e a - - - e aa i = T;(ZA&Z& + 2xa9ibxb + r 2 0 a i ) ( - ^ -e 2 a-^ - + ^-e 2 acr i - ] , (6.13) 9 \ I rz 2 ,r J We then take the derivatives of the this and the scalar field dy^ and mult iply 123 Chapter 6 Preliminary investigation of the Manton Method to 0(92) these to obtain 9ef deAf2) d ^ 0 ) : ^K^) = ^{SieQabXb + OaeXi + 2{6ae8ibXb + Xa9ie)) {^\^''a^ + ^ l " ) 1 . . . / 2 1 1 2 1 + 7 ( x A 6 a ; 6 3 ; e + 2xa9ibxbxe) e a— - -e aax — 9 \ rq 2 . r6 1n (l 2 1 +^9aixe - e a a i -0 V 2 r 0 e / d e A ^ df(f>W = (9if0ebxb + 9ef9feXi + 29ef9iexf) [ ~ f a ^ + l-e2aox 1 , 1 1 , ! ) ( £ _ ' r j \ r i +9efXi9fbxbxe ( e 2 a ^ - \e2aox-^ ( 2. „ s / 1 2 1 1 2 ! \ +(9efxaxf9aeXi + 2r'9ef9iexf) I >--e a — 4- - e ac r x - I Final ly, we use the identity: 0ifdfbxb = {gx)0i-e2xi (6.14) to obtain the result for the © in terms of the three independent tensors: e2ao\C 1 , M 9 ( e2aoxc l e 2 a o x + 2 J W * ~ 2 — We use the sign change argument again to obtain the following expression for 124 Chapter 6 Preliminary investigation of the Manton Method to O(02) the © monopole: e2aa2c e2aooc 1 e2aa2 7"2 2 r 3 It is important that the leading order terms (~ 1/s) in the expansion of this inhomogeneous term cancels with the leading order terms in ^A0^°\d>0€ ^ j . This causes the U ( l ) 0(9) solution to not change its leading order assymptotic behaviour, i.e., the dipole term in the U ( l ) static solution is not corrected when the monopole starts accelerating. Result for e^k { A ^ 2 \ Ak{0)} Similar calculation as above gives the final inhomogeneous term for the © monopole: 20 V - \ 2 r3 +S<fe>,*M3 l^ ( 6 - 1 7 ) For the © monopole, the expression is the same with o~\ replaced by a2: 2 X"» • °® 1 ~ 20 ' \ 2 r> ) The particular solution Adding up the inhomogeneous terms and substituting in the values 0\e2a = 1/s2, a2e2a = —1/s2 and e2ac = 2/s2, we obtain the equations for the U ( l ) corrections 125 Chapter 6 Preliminary investigation of the Manton Method to O(02) near each monopole in the two monopole system in fig 3.2: 0v-> " V 2 V 2 s 2 r 3 where the signs on top are for the solution near the © monopole, the one below for the © Now, the solution contains a factor e 2 a ^ such that it is of O(0e2), and since e2a^ is odd under space inversions, the other factor of the solution, the tensor structure part, needs to be even such that the solution would be odd, as required by the equation. The proposed form of solution is then: Afe) = A(r) e2a^ ±(9x)9ijXj 4>W = M r ) e ^ ^ 2 9 2 + Mr)e2a^(9x)2 and the equations for the coefficient of the different tensors are: \{9x)9i-. ±20 2 - rA' - 3A = T ? — {9x)2Xi: ±rcj)'2 + rAl = ± — -r29 2s2r6 The particular solutions for the © / © monopole then has the following coeffi-cients: Mr) = ~ ; h(r) = ; A(r) = ± ^ (where the top sign is for ©) The 0(9) U ( l ) fields without the undetermined homogeneous terms near each monopole in a system of two non-commutative monopoles are then: <k = (\$ + + sWMi ( ± - p ) = ° + e{-sk) + l^{-sk) (6 20) 126 Chapter 6 Preliminary investigation of the Manton Method to 0(62) Notice that the U( l ) d> field becomes non-zero when the monopole starts to accelerate; it vanishes for the single static monopole. Also, the particular solution does not change the asymptotic behaviour of the U(l ) fields. In the region near the axis of the commutative direction (such that the vector from the monopole center to a point in this region makes a small angle with the axis of the commutative direction), as r becomes comparable to s, the static term of the gauge field is of the order s~3, but the particular term contributes only at the next order, s - 4 . This means that the particular solution may not be relevant in the deciding the acceleration, as in the commutative case. The homogeneous solution As in the commutative case, the undetermined homogeneous term is included to take into account the presence of the opposite monopole in the global solution. We have not found the global solution and do not exactly know which of the homogeneous solutions are relevant in the determination of the acceleration. But all of these solutions are well-known, so we will list them below. Removing the inhomogeneous terms from the 0(9) U( l ) component of the first order ansatz (Eq 5.19) gives the Laplace equation for the U( l ) component of <j>: With only the cylindrical symmetry, the solution is given by any linear com-bination of some polynomial of r multiplied by a Legendre Polynomial of cos 9 of some order /, P/(cos#): ± = eijk d ± d i d 1 ^ = = o 127 Chapter 6 Preliminary investigation of the Manton Method to 0(82) For each order 1, the <p0e ' solution can be written in terms of the tensor structures, with the vector 6k being the axis of the cylindrical symmetry. Then, the homogeneous solution of the gauge field AQ€6^ can be written in tensor form interms of . The important point is that AQ ^ and d>0 e^ are related differently for different charge monopoles, and this may help in giving enough matching conditions to determine the acceleration. 6.3 Problems to be Solved: the Global Solu-tions (Notation: in this section, Oidn) represents both orders 0{6n) and O(0ne2)) To determine the acceleration between two monopoles using Manton's idea, we still need the O{02) local SU(2) solution (solution to Eq 6.1), and the global SU(2) solution, which depends on the global 0{d) U( l ) solution. The local SU(2) solution can be obtained by a calculation similar to the one for the local U(l ) solution with no furthur difficulty. Instead of proceeding with the calculation, we now discuss the predictable problems in finding both the global 0(0) U( l ) and the global O{02) SU(2) solutions. U(l) sector We immediately notice that the 0(9) U( l ) equation is linear in cfi^ and A Q ^ and an equation for the linear combinations (0 O e~ 0o©) a n a - (^oe^ + which involve the local fields at both monopoles can be obtained easily by summing the local equations: W (Kf + A™) + #(« - *jg) = {A%\A™} - \i» {A%\J& ~ 2 I ' 1 2 I ^ a<® ' 1 However, these linear combinations cannot be the global U( l ) solutions. While global U(l ) soluttions need to satisfy the second order differential equations of 128 Chapter 6 Preliminary investigation of the Manton Method to 0(92) motion (Eq 4.8) which depends on the O(90) global SU(2) fields, the above linear combinations depend on some combination which is dictated by the star product of the local O(90) SU(2) fields and cannot be easily rewritten in terms of the 0(9°) global SU(2) fields. This non-abil i ty to find the O(90) U ( l ) global solution by superimposing the local equations is due to the dependence of the U ( l ) equation on the non-linear SU(2) fields. We note as well that the tensor structure method used for finding the local solutions cannot be used for any equation involving the O(90) global SU(2) fields simply because these fields cannot be put into the form of a tensor structure. For example, the first two components of 4>^\i0bai contains a square root of the sum of two terms both wi th coordinate dependence: Unl ike the local solutons, since the terms inside the square root, and ^ , are both large and depend on different coordinates, the square root cannot be ex-panded as tensor structures involving polynomials of the coordinates. SU(2) sector Suppose we solved the U ( l ) component of the second order dif-ferential equation of motion for the global 0(9) U ( l ) fields. Can we do anything other than solving the SU(2) second order equation of mot ion (Eq 4.12) to find the global 0{92) SU(2) solution? We have already shown that the SU(2) com-ponent of the first order ansatz is non-linear, and so simply adding the SU(2) components of the ansatzes for the two monopoles wi l l not give an equation for the global SU(2) fields. Can we extend Manton's way of finding the global solu-t ion through factorizing the SU(2) field strength and isolating parameters that satisfy linear equations such that the global solution of these parameters can be a global 129 Chapter 6 Preliminary investigation of the Manton Method to 0(92) found by superposition? The following crude investigation shows that factorizing the SU(2) field strength to 0(92) does not give a linear equation for the O(02) correction to the parameter ^ (Eq 3.11) and that as in the U ( l ) sector, combi-nations of the ansatzes as candidates for the global equation most l ikely does not involve the lower order global fields and so is most l ikely inconsistent with the global second order equation of motion which does. F i rs t , in this sector, the star product modifies the SU(2) component of both the field strength tensor and the asymptotic condit ion from the commutative case. This rendors Manton's way of finding the global solution not valid at 0{92). As shown in E q 5.1, the SU(2) component of the field strength up to 0(92) includes the "normal" commutative dependence on the "ful l " gauge field ( A M = + + A ^ 2 ) ) , . b u t also extra terms originating from the expansion of the star product that depend only on the lower order fields. Furthurmore, the relation between the gauge field A1* and the scalar field '<f> in the asymptotic region is also changed from the commutative case because the asymptotic condit ion, that the matter field J M (Eq 5.21) vanishes, also has extra terms due to the star product: = etaeabc ^ f \ D ^ ) + ^ - e t a 0 ( { D ^ ( ^ ^ } 4 - { D ^ , 4 O ) } ) where al l fields have corrections due to the acceleration when the monopole is accelerated, and terms with the O(90) U ( l ) fields and the O{0) SU(2) fields have been omitted. According to this expression, when we factorize the SU(2) components (in vector form) of </>: 0 = (/»(»)+/»(*')) ( ^ ( » ) + ^ ) ) ; \$® + fi*)\,= 1 and write the zeroth order term of the SU(2) gauge field using the 0(0°) asymp-totic condit ion, A"(°> = d ^ ( 0 ) x the relation between the ful l SU(2) gauge 130 Chapter 6 Preliminary investigation of the Manton Method to 0(92) field Ala and the the ful l SU(2) scalar field <j>, on top of the terms which are just the linear fluctuation of the relation = 0 x D M 0 that is satisfied in the com-mutative case (written in the first two lines of the next expression), involves an extra combination of many different vectors given in terms of </>(°) and its part ial 131 Chapter 6 Preliminary investigation of the Manton Method to 0(82) spatial derivatives: JU^ ) = (>)2|^(o)|2^  A ^ 2 ) - ( > > 2 A ^ 2 ) • 0<°) + ( ^ 2 ) ^ x ^ 2 ' + ( ^ o ) ^ 2 ) | 0 ( o ) | 2 - ^ o ) 2 0 ( ° ) - ^ 2 ) ) ^ > x 3 ^ ( ° ' above is the linear fluctuation of (J> x D " 0 ) + 62 (2d1h^di2&th^ - 2d2h^dl8^ ) x 0(°) + 02 [ - d2h^dxd2dtih^ + d^h^d^h^] MM x 0(°). + e2 (2d1h^d2d^h^ - 2d22h^d1d^ ) x a 2 0 ( ° ) + #2 j _ d l h W d l d 2 d ^ h ( 0 ) + dld2hWdld»hW] 0(0) x ^ 0 ( 0 ) + 62'(h.w%d>ihW - d2h^d»h^) d24>^ x 0(0) + e2 (-h^d\d^h^ + a^w^^o)) a20(o) x 0(0) + 02 ((4 + [i])a1/i(0)a2a^w - (4 + [i])d 2^ 0 )cWh ( 0 )) x + 02 (-2d2h^d»hW + 2hWd2&thW) x d24>{0) + e2 (2dlh^dph^ - 2h^d1d»hW) dx^ x d2j>{0) + e2 [d^h^d^hW - h^d1o2dfih^} didtft® x 0(°> + #2 [ - s ^ W 0 * + h^d2d»h^\ dt4>w x a ^ 0 * + d2[+d1hW&*hW-hWd1d*hW] 3 , 8 2 ^ x 8 2 ^ + 6 {drd^o ]d2h^ - 8 2 8 ^ 8 ^ } 0<°> + ^  1^ 0(0) x 0(0)^(0)0(0)^^^(0)1 0(0) _ ^2 | l a 2 | a M 0 ( o ) x 0(0)^ (0)0(0)1^ 5^ (0)1 0(o) + 6 {d^824>t] - h^d2d^{0e)} 8 ^ + ^ {1^ { ^ ( 0 ) x 0 ( O ),^ ( O )^ ( O )} ( )^ ( O )} d i 0 ( O ) + e [d^dJP - h^d.d^} d2^ + e2^d1{d^^ x ^ ° \ h ^ } ^ h ^ d24>^ 132 = 0 Chapter 6 Preliminary investigation of the Manton Method to O(02) Here, the dependence of the O(02) fields are in the first two lines only. E l iminat ing A ^ 9 ^ from the field strength with the above relation, we find that the field strength is not simply the "original" 0(9°) dependence on the entire </> = (^°) + 0(fl2) plus corrections from the star product that are independent of <^2)), i.e.: but that even the terms with <f>^ are not the 0{92) linear fluctuation of the original commutative form and has dependence on unless we could use some constraint to eliminate it as d^(j) • (f> = 0 does in commutative case: = C — t a uvem^ + C o r r - ( ^ ) , ^ 2 ) , # , A f ' , ^ ) , ^ ) ) + Corn{^\h^\$\Af)) (6.21) F ^ ' s change in the dependence on <f> renders the most important simplication in the calculation in the commutative theory not valid here,the simplication being the factorization of the SU(2) field strength and therefore of the first order ansatz into a magnitude and the unit vector (j), such that the curl of the factorized ansatz (Eq 3.13) gives a linear differential equation (Eq 3.14) for one component of 4> and the factorized ansatz itself gives a linear differential equation (Eq 3.18) of h, such that the global solutions could be obtained simply by solving the sum of these linear equations for the two monopoles. The global solution in the non-commutative theory to 0(92) is not found in this "l inear" way. F a c t o r i z a t i o n o f t h e f i e l d s t r e n g t h t o O{02) We look at what would happen if we sti l l factorize the non-commutative SU(2) as an attempt to obtain the global equations for the correction parameters,' ty(9)(d2\ T(x)^ 2 \ the degrees of freedom of ^  (Eq 3.11), and hS°2\ There is advantage if we succeed: the global solution in the factorized form might allow us to use the SU(2) component of the 133 Chapter 6 Preliminary investigation of the Manton Method to 0{92) factorized ansatz as a uniform external force law to determine the acceleration: where Dh is the magnitude of the SU(2) vector D 0 up to O{02). Note, however, that this force law is not *-gauge invariant since the length of the SU(2) com-ponents of a field is not a *-gauge invariant quantity. We proceed to investigate inspite of this problem. The SU(2) field strength would factorize as follow: = \/(CmmutauJ2 + \Corr%, h^))\2 + \CorrJ2 (4> + v<^) and is not proportional to 0 anymore. The magnitude part has the following problmes: 1. it depends on i^ differently than fcommutative a n ( ^ s o IS m o s t l ikely not linear in ^ f^; this ruins the property that the U ( l ) embedded field strength is linear in ^ as in commutative case; 2. it most l ikely involves more than one of the correction parameters (not just but maybe also h or T) unlike in the commutative case and so applying the curl to it would not give a decoupled equation for any of the parameters Therefore, the factorization of the field strength does not help in finding the global SU(2) solutions in Manton's way. Final ly, any global equations obtained from combining the factorized local ansatz st i l l involves only the local lower order fields, 4>^e and h^jQ, which are hardly l ikely to combine into the lower order global fields 4>g°iobal that appears in the second order equation of motion, which is definitely a global equation. In conclusion, we have not found any equation for the 0{9) global U ( l ) and the 0(92) global SU(2) fields that is simpler than the second order equations of motion, which is what the first order ansatz tries to simplify in the first place. 134 Chapter 6 Preliminary investigation of the Manton Method to 0(92) 6.4 Conclusion for the Non-Commutative Prob-lem We consider the perturbatively non-commutative U(2) gauge theory with a scalar field in the adjoint representation, in which the space-time non-commutative pa-rameter 9 defined by [xl, x2] = iQ is small. We employ the star product formalism such that the equations of motion can be expanded in 9 and reduce to those in the commutative theory in the l imit 9 —> 0. The U(2) gauge group means that al l fields have a component proportional to the identity matr ix, called the U ( l ) fields below, and three components proportional to the Pau l i matrices, called the SU(2) fields below. The U(2) non-commutative monopole is defined to be the SU(2) commutative monopole (with tr iv ia l O(90) U ( l ) fields) wi th corrections of 0(9) and higher. Our original goal is to find the 0(9) term in the force be-tween two non-commutative monopoles, but finding that it is t r iv ia l , we start investigating the problem at the 0(92). We show that the 0(9) force correction is zero in the following two ways. 1. We first derive the non-locally conserved stress-energy tensor for the theory to show that at 0(9), the tensor depends only on the SU(2) components of the 0(9) corrections to the gauge field and scalar fields, both of which can be set to zero because the time dependent equations of motion at this order are simply the linear fluctuation of the equations of the commutative theory. This means that to this order, the total energy of any solution of the non-commutative theory does not change from the total energy of the O(90) solution to which the non-commutaive solution reduce when 9 — 0. Since the force between two non-commutative solitons separated by any distance s in any direction can be defined as the derivative of the total energy of the system with repect to the the separation distance, it is unchanged to 135 Chapter 6 Preliminary investigation of the Manton Method to 0(92) this order from the force between two commutative solitons in the same configuration. The force between two non-commutative monopoles is simply a particular case of this. 2. We derive the first order ansatz for a single non-commutative monopole weakly and rigidly accelerating in the commutative direction by replacing the ordinary product with the star product in Manton's derivation for the commutative ansatz and by assuming an addit ional 0(9e2) correction to the acceleration. To 0(9), only the SU(2) components of the non-commutative ansatz are relevant to the determination of the force between the monopoles. We expand both the fields and the star products in this SU(2) ansatz in orders of 9, and find that to first order, it depends on the SU(2) components of the gauge field and scalar field just as the commutative ansatz does on the commutative SU(2) fields, but has a modified acceleration which includes the extra 0(9e2) correction. Because of the definition of the star product, the SU(2) fields to 0{9) (only) can st i l l be written as vectors in the SU(2) subspace of the U(2) gauge group of the theory. When we factorize the SU(2) scalar field into its magnitude h and the unit vector </>, we find that the asymptotic condit ion and the SU(2) field strength have the same dependence on </> as in the commutative case, except that 4> has 0(9) corrections. Therefore the magnetic field obtained from the SU(2) field strength st i l l depends l inearly on the third component of </>, which also has 0(9) corrections. We can then factorize the SU(2) ansatz and find that it remains linear in \I>, but is in terms of the modified acceleration with the 0(9e2) correction. For the two monopole system, we can bui ld the global SU(2) solutions again by adding the solutions of the local ansatzes, and note that both the local 136 Chapter 6 Preliminary investigation of the Manton Method to O(02) and global solutions now include 0(0e2) terms unlike in the commutative case. Then, by the exchange principle, because there is no O(0e°) B and Vh fields from either monopole to expand near the other monopole, the external fields near each monopole is zero , and the SU(2) ansatz as a uniform force law gives zero for the 0(9e2) acceleration. We proceed to look for the 0(92) force between two non-commutative monopoles by using Manton's general idea of solving for the local and global solutions of the system and equating them near each monopole. We can easily solve the non-commutative first, order ansatz for the local so-lutions up to 0(92) by first assuming the solutions to be linear combinations of tensor structures which are products of the coordinates, xl, and a more general non-commutative parameter, 6*i, defined by [a^o;-7] = iOli [8] [9], and corresponds to different angular dependence; and then reducing the ansatz to one ordinary differential equation for each tensor structure and solving these. Knowing the 0(92) SU(2) solutions can be obtained similarly, we show ex-pl ic i t ly only the calculation for the local 0(9) and 0(9e2) U ( l ) solutions. The results are as follow: 1. The 0(0) 'U( l ) solution [8] consists of a vanishing U ( l ) 0(9) scalar field and a U ( l ) 0(9) gauge field that has a dipole potential as well a non-qradruple but ~ 1/ r 3 term. We find it interesting that a "magnetic" field defined as the curl of the \](\) 0(0) gauge field has no gauge freedom inherited from the star product gauge transformation once the gauge is fixed at the lower order, 0(0°). 2. For the accelerating monopole, the 0(9) U ( l ) fields are independent of the O(0e2) acceleration (although this has been determined tobe zero above) but have O (0/(s2)) corrections due to the 0(0°) acceleration that do not alter 137 Chapter 6 Preliminary investigation of the Manton Method to 0(62) the asymptotic behavior of the U ( l ) gauge field. The 0,(6/(s2r)) correction to the U ( l ) scalar field,however, is its only non-vanishing behavior at 0(9). We have not found a way to bui ld the global solutions from the local solutions. In particular, we show how the O(02) expansion of the star product prevents us from bui lding the O(02) SU(2) global solution using Manton's simplications. We first expand the star product in the non-commutative field strength and find that its SU(2) components have extra terms (compared to the commutative field strength) that depend only on the 0(0) and 0(0°) fields. If we then use the star-product extension of the asymptotic condit ion used in the commutative case to obtain a relation between the gauge field and the scalar field to O(02), we wi l l find that the part of the SU(2) field strength that involves the O(02) fields is no longer a topological term in (j) as in the commutative case, and involves also the O(02) h field. Then, if we define a magnetic field using the magnitude of the SU(2) field strength such that it reduces at 0(0°) to the magnetic field Manton uses in the commutative case, we wi l l find that it does not depend linearly on \I> (the third component of (f>) as it does in the commutative case. Moreover, even without only the O(02) part of the asymptotic condit ion, we can see that to O(02), this " mag-netic field" is not gauge-invariant, and does not even satisfy a linear equation, not to say the Maxwel l 's equations; thus, the superposition of magnetic field in the region between the monopoles, and the determination of the external fields near each monopole by multipole-expanding the fields from the opposite monopoles (that work in the commutative theory) are no longer val id to O(02). W i t h the asymptotic condit ion extended to O(02), we can see also that there are no linear decoupled equations for ^ and h, and so unlike in the commutative case, the solution in the region between monopole is not easily found by superimposing the local \I> and h from the different monopoles. 138 Chapter 6 Preliminary investigation of the Manton Method to 0(82) We conclude that Manton's approach to find the commutative global solution does not work at O{0 2. We think of three routes to proceed to look for the O(0 2 force: 1. We can use the second order differential equations of motion as the equations for the global solution, and look for behaviours of the global solution that would give rise adn can be matched to the homogeneous terms in the local solutions near the monopoles. We can then use the matching conditions, if there are enough, to determine the force. 2. We can try to find the difference in total energy of the monopole pairs separated by distance s and s + 5s to 0(62) using the stress-energy tensor. This involves again the solution to the second order equations of motions, but hopefully obtaining the difference between the total energies do not require solving the equations of motion entirely. 3. We can look for a way to keep track of some sort of flow of the covariantly conserved energy-momentum currents [27] and then find the flux of the momentum currents into a region enclosing a monopole. None of these routes seems more promising than the other two. 139 B i b l i o g r a p h y [1] N . Manton, Nuclear Physics B 126, 525 (1977).. [2] J .N.Goldberg, P.S.Jang, S.Y.Park , and K . C . W a l i , Physics Review D 18, 542 (1978). • ; [3] S. Coleman, The Magnetic Monopole Fifty Years Later, Harvard University, 1981. [4] J . Kowalsk i -Gl ikman, Towards Quantum Gravity: Proceedings of the Xxxv International Winter School on Theoretical Physics, Springer-Verlag Telos, 2000. [5] N . Seiberg and E. Wi t ten, Journal of High Energy Physics 1999, 032 (1999). [6] D. J . Gross and N. A . Nekrasov, Journal of High Energy Physics 2000, 021 (2000). [7] M . Douglas and N. Nekrasov, Reviews of Modern Physics 73, 977 (2001). [8] K. Hashimoto, H. Hata, and S. Mor iyama, Journal of High Energy Physics 12, 021 (1999). [9] S. Goto and H. Hata, Physics Review D 62, 085022 (2000). [10] J . Preski l l , Physics Review Letters 43 (1979). 140 [11] P. Dirac, Procedures Royal Society A 133, 60 (1931). [12] M . Prasad and C. Sommerfield, Physics Review Letters 35, 760 (1975). [13] E . Bogomolny, Soviet Journal of Nuclear Physics 24, 801 (1976). [14] G . 't Hooft, Nuclear Physics B 79, 276 (1974). [15] A . Polyakov, Journal of Experimental and Theoretical Physics Letters 20, 194 (1974). [16] Rajaraman, Solitons and Instantons: An Introduction to Solitons and In-stantons in Quantum Field Theory, Elsevier Science L t d , 1982. [17] S. Minwal la , M : Van Raamsdonk, and N. Seiberg, Journals of High Energy Physics 2000, 020 (2000). [18] M . Van Raasdonk and N. Seiberg, Journals of High Energy Physics 2000, 035 (2000). [19] F. Bayen, M . Flato, C . Fronsdal, A . Lichnerowicz, and D. Sternheimer, A n -nals of Physics 111, 61 (1978). [20] D. Gross, A . Hashimoto, and N. Itzhaki, Advances in Theoretical and Ma th -ematical Physics 4, 893 (2000). [21] E . Corr igan and P. Goddard, Annals of Physics 154, 253 (1984). [22] R. Gopakumar, S. Minwal la, and A . Strominger, The Journal of High Energy Physics 2000, 020 (2000). [23] Y . Okawa and H. Ooguri , Nuclear Physics B 599, 55 (2001). [24] Y . Okawa and H. Ooguri , hep-th/0103124. (2001). 141 [25] D. J . Gross and N. A . Nekrasov, Journal of High Energy Physics 2001 , 044 (2001). [26] D. Bak, Physics Letters B 471 , 149 (1999). [27] M . Abou-Zeid and H. Dorn, hep-th/0104244 (2001). 142 

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