ON A 1+1 - DIMENSIONAL INTERACTING WITH S O L I TON-FERMI ON SYSTEM SUPERSYMMETRY by WERNER H. Diplom-Physiker, Technische A T H E S I S SUBMITTED THE KEIL Universitat Clausthal, IN PARTIAL FULFILLMENT R E Q U I R E M E N T FOR MASTER OF THE DEGREE OF SCIENCE in THE FACULTY OF GRADUATE Department We accept to THE this thesis the required UNIVERSITY Werner Physics as conforming standard OF B R I T I S H Apri1, © of STUDIES COLUMBIA 1985 H. K e i l , 1985 OF 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of requirements f o r an advanced degree a t the the University o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it f r e e l y a v a i l a b l e f o r reference and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f department or by h i s o r her r e p r e s e n t a t i v e s . my It i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my permission. Physics Department o f The U n i v e r s i t y of B r i t i s h Columbia 1956 Main M a l l Vancouver, Canada V6T 1Y3 Date 26. 7. 85 written i i Abstract A space supersymmetric and soliton the one time sector "method of of dimension the evaluated are that the boson to to and the for constrained equations fermion in a admit zero-energy of the theory. solutions theory spatial are broken include i s doubly by translations, supersymmetry. of developed and way to a using equations is of expansion. solutions for the Hamiltonian motion We find for both The fermion consequences supersymmetry charges broken. connected soliton. the nontrivial The degenerate. t o be the has topological i s spontaneously found the the field. supersymmetry state The perturbative soliton ground construct constraints the one generalization canonical systems. the modified to with c o o r d i n a t e s and order in transformation leads i n the of supersymmetry the variables presence be using a canonical is quantized first field The the theory field in collective solved theory A "collective" method quantum system We theories. set Dirac's is investigated. coordinates" previously bosonic transformed soliton-fermion collective purely of interacting It and algebra as follows a that Finally, the with symmetries boson the has result the zero-energy zero-mode zero-mode for of the corresponds is associated with to iii Table of C o n t e n t s Abstract i i Acknowledgement iv INTRODUCTION 1. Q U A N T I Z A T I O N 1 BY 1.1 Quantization 1.2 Collective 2. Q U A N T I Z A T I O N around 3. a classical static 6 field 6 coordinates OF S O L I TON-FERMI ON A Collective 1+1-DIMENSIONAL INTERACTING 2.2 Construction 2.3 The 2.4 The H i l b e r t 13 coordinates for a soliton-fermion system of the Hamiltonian equations SUPERSYMMETRY 10 SYSTEM 2.1 field C O L L E C T I V E - C O O R D I N A T E METHODS space FOR A 23 i n g°-order f o r the s o l i t o n 1+1-DIMENSIONAL 31 sector 38 SOLITON-FERMION THEORY 3.1 The 40 supersymmetry generators f o r the soliton-fermion theory 3.2 15 Spontaneous sector Bibliography 40 supersymmetry breaking i n the soliton 47 53 iv Acknowledgement I would Semenoff, work. like for His contributed to thank his help patient and my research guidance explanations essentially gratefully acknowledge University of British to the and throughout numerous progress financial Columbia supervisor, of assistance Summer the Dr. Gordon course suggestions this in Graduate work. form of I of have also a Fellowship. W. this 1 INTRODUCTION It has been classical mean known field has systems nontrivial the Skyrme one fractional model these properties time dimension Although this may seem to physical s i g n i f i c a n c e , there Like 1-dimensional 1+1-dimensional without the presenting question field of theory, i s of Despite the are no the many gauge the for shown are such as have best are s e e m s .to be reasonable these closely various related Examples interacting therein) attracted a good toy reasons provides a and the for in assumption any studying i t . nontrivial in point the for higher mechanics, structure Particularly for models basic more for every we will features complicated dimensions. especially in attention. i n quantum crucial valid theories without difficulties. importance remain much model, oscillator theories, the admit that fields. simplifications certain to to we considerable which references technical major systems, a at theory expected solutions years have which ([7]-[9]). renormalizability, a consider. theory too be harmonic field ten by interacting soliton-fermion one the of and baryons theories solutions, which numbers ([2]-[6] for nonlinear theories studies features quantum systems last field and a space unusual certain finite-energy the Recent topological soliton Among independent During that soliton in quantized many soliton-fermion time possess solutions. exhibit include long ([1]). arisen soliton-like to time equations interest a theories nontrivial field for This properties . of 2 that depend In only on addition, physical systems. that dynamics the polymer i s given the global these theories Several of by structure authors electrons a are in of known the to theory. describe ([ 4 ] , [ 1 0 ] - [ 1 4 ]) a linearly 1+1-dimensional actual have shown conjugated diatomic soliton-fermion Hamiltonian. Summarizing 1+1-dimensional methods, is now as this theories well well as a as model discussion, a "proving for supersymmetric. many can choose the Supersymmetric unusual properties, of divergences are now to play a combines this unusual believed topology by (solitons) several the trivial of that topology new of ideas matter and systems, major ([15]-[20]) a l t e r s the as at in first with a interest. discuss conventional known the description They A nontrivial for is be symmetry physics. This how to to order. particle symmetry who are systems boson-fermion least role i s c e r t a i n l y of these theories such to for field confirmed presence of fields with topology. One fact authors soliton for condensed action field cancellation that usefulness ground" certain and model the established. Furthermore,we possess brief the problems standard - of theories perturbation nontrivial boundary containing theory cannot conditions on solitons take the the is the field solutions of the 3 field equations these systems ranging new on open Jackiw, developed a way they c-number spaces for formalism soliton fields a to the method, thirty the years path theories many a u t h o r s the present integral The a set of freedom". extended, added. using time index therein). that the approaches framework quantization. system to theory as a can the of include the consistent classical then that To in a variables This to "collective quantize method as very theory of with be quantized via Dirac developed more developed the particle to be special problems soliton system "internal as the quantum c a l c u l a t i o n s these by care in is for i s favored transformations viewed which systems Faddeev method ordering v a r i a b l e s " and can by i t requires canonical for constrained elegant but operator transformation practical within techniques dependent classical theory field to ( [ 2 ] , [ 2 6 ] - [ 2 8 ]) viewed of developed, methods references quantum - been about ([29],[30]). Physically this For the set integral due field system new ([31]). case, canonical new in ([23 ] , [ 3 2 ] , [ 3 3 ] ) , for have a l . fashion constrained ago et canonical a l t e r n a t i v e approach Feynman gauge scalar and the This An a transformed - and Tomboulis solution constraints. than information techniques ([21]~[25] straightforward canonical obtain semiclassical approximation method Hamiltonian To different Goldstone, in a classical into account. several from theorems problem - the in path ([26]). will lead degrees of soliton being contributions to an are "quantum f l u c t u a t i o n s " 4 have to be expanded emphasized, however, approximation constrained In be given to extend described to include does supersymmetry We will In be "collective scalar the method full quantum we of the series. It involves no theory for should be semiclassical a classical with a 1+1-dimensional supersymmetry. Special problems: of method we have just and the soliton affect the theory. as follows: chapter reference discuss quantization coordinate cft-l'theory, two fermions, the first to will system presence proceed the following power this soliton-fermion will how a paper how (b) suitable that gives present interacting (a) but a system. the emphasis in give a brief quantization [26] the we and [28]. simplest review of the method" by This illustrated theory is that Jackiw admits et a l . with a soliton solutions. case In the of a transform second chapter 1+1-dimensional the old field constraints and canonical. Subsequently in the new show of the will that new generalize this method interacting soliton-fermion operators coordinates. interpretation we this we The to a set of transformation evaluate the result will variables as new is to the system. variables We with indeed energy-momentum confirm "collective tensor the coordinates". 5 Using the Hamiltonian equations and in a fermion existence general of a first operators solution. of t h e H i l b e r t third and last a the ground doublet. a spatial t h e boson due t o t h e A preliminary results are given with are quite f o r the special of a whereas t o an case we will These to the which have central t o be charges "N=i" supersymmetry. show the c l a s s i c a l fermion supersymmetric corresponds of our theory the bosonic transformation. the charges" algebra. down state theory addresses or " c e n t r a l Finally the existence transformation, Our of the soliton supersymmetry supersymmetric implies chapter. chapter i n t h e supersymmetry consequence that field system. "topological" t h e N=1 the structure f o r a theory the calculations existence of space the second The presence in find soliton supersymmetric degenerate We of the c l a s s i c a l properties. break approximation. and s o l v e a zero-mode although included construct contain concludes The order we both discussion solitons formalism will consist As a of a non-perturbatively that soliton zero-mode zero-mode solution via a always supersymmetry i s associated with 6 1. 1.1 a method field molecular as theoretic quantum quantum equations the full quantum operator The full quantum theory with a Consider c6(x t) r i n two this field equations for classical approximations around scalar The These order field c a n be viewed t o the "Born-Oppenheimer" method i n mechanics. operators. static METHODS i n the f o l l o w i n g section analog zeroth expansion COLLECTIVE-COORDINATE a classical described differential than as BY Quantization around The as QUANTIZATION c-number classical i s obtained classical an field. rather c a n be expectation classical by are treated fields solutions t o t h e vacuum describing a theory of motion viewed value extended operator power object. series We will illustrate for a real scalar this ( [26 ] , [ 2 8 ] ) . the Lagrangian density field dimensions d) where U(<p) power series parameter use from i s a polynomial expansion g like now on we U((p) = ^ i n <p. assume 2 i s the well U(q<p). known Since that The U we are interested d e p e n d s on standard a e x a m p l e we (2) The field equation for this Lagrangian i s given by in a coupling c^-self-interaction of will 7 (3) The t- solutions independent), dependent. For solutions, A found: constant <p 0 For on are which = the we ±jjj. to (3) static us can be classified ( t - independent) only the first two and and as constant space- (x- and especially time- the static is easily important. solution </> 0 to We will return static case the impose the (3) with potential to this solution field equation (2) later reduces on. to following conditions (asymptotic non-triviality) 0 0 Then (3) is integrated to or + where the prime and will + denote from c o r> if. now on (4) differentiation with 8 respect to the explicit argument. The remains constant (at i s set least on These to the zero, classical static An explicit The s o l u t i o n s are 0 (x) = L with x a 0 real the finite ±™- integration is ensured level). example the tanh parameter i f the physical consistency s o l u t i o n s with solitons. static so energy finite energy i s provided well-known by are usually our ^-interaction. "kinks" m(x-x ) (5) 0 denoting called the center of the soliton. We i note that will be for the We since 0(g~ of V= 1 ) U as well solutions. will construct now solutions. The operator field I <p 0 quantum of (*.t) 0(g _ 1 Hilbert ) - r\ . and the values of techniques. This part $ quantum the (4). the The solution same around constant is true our solution <j> . 0 (x.t) usual be classical as c o n t r i b u t i o n of 0(g ),n>O. The n Fockspace from multiparticle can our theory consider generated of i n g, equation quantum i s the bosonic Expectation sector". *I i s expanded space describe we ) _ 1 to the First { | 0 > , | k > . . . | k , . . . k „ > . . .} elements 0(g due constant classical with i s of 1 a states evaluated theory vacuum will by be state with standard called |0> momenta whose k . perturbative the "vacuum 9 To construct include "quantum general state The soliton Jackiw matrix the the theory, takes the "soliton vacuum of that g given by so to that a | p , k k „ > . m a these systematic on. The soliton B assumptions is possible. the p, have < p , 1 , . . . l | . , # | k , . . . k > - 0. that later of mentioned, soliton i s however, there we would sector the different left We part into lead expansion will of to a of return the account as like $ = t o expand <p ti + two the to theory is called the field analogous to the which b) how impose solution new set of set of theory 0tl example of solutions (1) the the boost <p i s n o t ti <t>l ~ theory parametrized i s invariant i s not, but d e p e n d i n g on <j>^ s h a l l under transforms parameter. the field be expanded translational covariance to these field our since by x . 0 Lorentz into Thus we problems around The the straightforward As solution solutions to so whole while a) parameter a static with not invariant. exists generally, boosts, are especially momentum i s now stable: shown structure with we sector. translationally More |p> solutions sector". i n the This shown have static space t o be existence already operator states" i n powers space f o r the Hilbert ([28]) elements As soliton i s assumed Hilbert that theory i n our et a l . consistent a problems variables dependence of 0.. i s given which enable and on by a transformation us to control the to a has 10 1 .2 Collective We start level. with the field now I In impose f field - ~ U ( $ ) variables are with the {*,n,X,P} t o keep by - 0 t o a s e t o f new defining (« - X(t)) , XID £ ,t) (6a) , £(t) = Jdx*'tf' 2 t h e number of independent variables constant we the constraints we modify a r e now the form A Given i J dx £ <j> J i x TT ^ d dealing with our Poisson ([29],[30]). in $ 2 ^ = 4> t o g e t h e r { TT(-.t) . F ( j . - t ) J transformation r Since • I cononical momentum n - a canonical = Jdx^' order - c-number brackets («.t) - 0 2 density The s t a n d a r d <t> a n d t h e c o n j u g a t e perform M <t>^. $ ( * . t ) ; £ Lyt)\ variables where our Lagrangian solutions Poisson I We the discussion at the c l a s s i c a l Consider static usual coordinates brackets = 0 (6c) -. 0 a constrained according (6d) system to the Dirac we have t o procedure a s e t o f c o n s t r a i n t s on t h e f i e l d variables 11 = h {<p,ir) K we impose for the f,g. these on Furthermore we this a l l the from our standard can simply by the system variable, variables now brackets procedure our f o r any case we two brackets functions obtain vanish. defines to a set construct the the equations them We here. turns be the a of canonical new transformation "collective constraints. and of {f,g} Dirac (7b) calculation can coordinate The for energy-momentum of motion s u b s t i t u t i n g (6a)-(7b) reproduce the s u b s t i t u t i n g the bracket procedure variable with and by 1 described generators Since = The We Poisson remaining coordinates" system take {X,P} and our conventional Adopting a=1 i • • • t N 0 out to into results note be are however, just P. for the our old rather that Thus tensor, X, p h y s i c a l l y i n t e r p r e t e d as new ([26]). lengthy its Lorentz variables theory the the we total will momentum conjugate the center-of-mass new collective soliton. quantization of our is straightforward. theory We in the simply follow the not canonical of 12 quantization Hilbert see space, from field scheme: quantum been care To maintain change theory general precisely by part o f by the c o n s i s t e n c y we will c-number translational to and part in a As desired result. i n t o d u c t i o n of a had operators t o commutators. our the s t a t i c The a r e now # cl we plus noninvariance "position omit a detailed are going o f an an has operator" i n t r o d u c e c o n s t r a i n t s and the commutation d i s c u s s i o n of to extend interacting can The relations in a way. s i n c e we case variables correspond t h e c a n o n i c a l momenta nontrivial We gives are given additional taken field the brackets (6a-f) t h i s operators the these the p u r e l y methods fermion-boson bosonic t o the theory. more X. 13 2. Q U A N T I Z A T I O N OF A 1+1-DIMENSIONAL INTERACTING SOLITON-FERMION SYSTEM. We by will <(> a real 2-spinor boson We not however, but imposes that condition. results only technical brackets for Dirac will fermions will classical we fermion remain and l e a v e s now chapter part examine a n d some how 1 with of the theory the Majorana o f them are i n polymers. degree First remains method, anticommutators. simplest (8). I t does that our familiar The of freedom i n possible spinor. to generalize our r e s u l t s to the Lagrangian 0(g~ ) i n . the following . the charge us w i t h . t h e charge chapter.. i n t h e sense standard fractionization removes the of our q u a n t i z a t i o n with unchanged of charge £ provides the are interested t o be c o m p a t i b l e fermions (a complex i n the previous restrictions have described t r a n s p o s i t i o n ) a n d V(</>) of a Majorana the generality c o n d i t i o n merely previous discussed the choice affect the context We matrix I t i s s t r a i g h t f o r w a r d to repeat construction Dirac term 2-spinor T denoting T theory c o n d i t i o n <// = C i / / , of the Lagrangian, anticommutator Majorana and interaction note \p a M a j o r a n a field, field density t o the Majorana matrix supersymmetry in Lagrangian scalar subject conjugation The the 1+1-dimensional the supersymmetric with the consider we note unchanged from the that the i f we take 14 the V interaction i s at least fulfills this operator field In fermion The so of 0(g°). condition. the lowest - to between 0(g). the unchanged. On to between quantum since the the the We and can shift </>'-model, and for (8) leads set the that example to the only introduce level the operator the field 0(q°) will to the V'(0 t L occur theory. will suffices to 0(g°). collective the occur be zero. be at 1 ) least remain are Interactions in higher orders. is trivial the soliton due position t fermions ). In V'(0 ) the to in general classical to i n 0(g~ motion It by - <9(g°)-term i s seen potential effects up of field Furthermore fermions free t V to classical 0(g°)-approximation the V'(<£ ). spinor 0(g°). fermions nontrivial boson of quantum a can fermions equations sector operators the the and to we Grassmann c-number bosons vacuum now a least first potential quantum 1 bosons static however, background Lagrangian 0(g" ), as at i t corresponds sector, The classical the coupled In standard couples be interaction Thus Our q<p s o proportional to • 0 . order, viewed equation \p h a s V(<p) equations V - V(*) Y field = V" (</>) £ X Dirac that term our to soliton consider coordinate solution # . X 15 It is known should to possess interpreted is be m e n t i o n e d fermion transformations part we 2.1 Collective static, equation discuss we collective on t h e s e of the theory. when As based We that ( [ 2 ] , [20]) partner" will c a n be analogously to this coordinates s o l i t o n - f e r m i o n system. energy before, the Lagrangian solutions {4> ( x - x (L 0 <p . 0 e R} to the theory. (8) admits )| x ti possibility p r o p e r t i e s of our for a a set of to the classical of motion with <L (* • * • ) • so that the equation (we have The quantum standard with arbitrarily , ^ (* • t 00 ) • 0 of motion fixed i s integrated to our sign theory described canonical variables the following equal-time i [ T ( « ) , £ (j)] f o rV ) . by t h e L a g r a n g i a n commutation • S(»-j) (8) and c a n o n i c a l modes return which of the s o l i t o n coordinates fermion the Lagrangian the supersymmetry discussed finite point zero-modes as the "supersymmetry possible to define boson at this (8) h a s t h e relations ( 9 a ) It 16 • i § c o , n ^ l • [ IT c o , * ( ) ] 3 • [ iW The anticommutators Majorana fermions, T h u s we a r e the , VCjU The on that our in Chapter charge is, The which gives t, a constrained conjugation condition for . <0 the » - r reads components - by the matrix by Majorana to from the the ( 9 b ) fact that constraint procedure, are $ = t \p . J s y s t e m a n d we w i l l Dirac V>'s impose already 1. { given arise subject theory l f z' is f fermions dealing with constraints mentioned for • [ irw. * t y 1 • 0 0 • f. C, d e t e r m i n e d by the condition 17 On thec l a s s i c a l instead level of theusual - we h a v e Poisson { * ( « ) , < ( • > } t o take brackets Dirac brackets {,} ( [ 2 9 ] , [ 3 0 ] ) : h ( » ) . f ( i " ) j " (HO.^j)} where so that and o u r new b r a c k e t s a r e g i v e n by Ux-j) analogously { ^ . n ^ l , . - sc-j) The theory Dirac i sq u a n t i z e d brackets compatibility From set this by p o s t u l a t i n g t h e c o r r e s p o n d e n c e and anticommutators of anticommutators set of "collective" which between ensures the and constraints. of canonical variables and " i n t e r n a l " we t r a n s f o r m variables t o a new {$,n,#,#^,X,P} b y 18 def ining £ ( x . t ) - T T ( y . t ) , f ^ ^ = t ( x . x(t)) T T ( x - • i ( x - / ( t ) , t ; (10a) (10b) X ( t ) , t j (X-X(t)) (t) (10c) with and the [ equal-time commutation T ( o , t(3)j . .[ i w . • [ I ( « k n3)] { HO. * tyj f relations are |(,)] • [ I w . ^f(j)j • S('-j) (1 1a) (11c) (1 1d) 19 i [ ? ( 0 , X ( t ) j . 1 f o r *,n,i $* . t J The constraints as are taken to §(„,!) be • 0 ^(,)TT(*.t) « (12) before. A Expressions a formal power guarantees and will remains is leaves the the used show compatible will is obvious; We obvious drop but the an will to the our original of standard integral sign the and [ TT(0, £ The and that our whenever (12) is, for the the is notation integration c a l c u l a t i o n only (^)J (10) It verification simplify spatial conserved zero-modes. by as (12) is formalism, operators denotes explicit variables defined To understood constraint unchanged. lengthy. in always unphysical canonical cases [ H0.1U})] The transformation somewhat are independent commutators the i n X. eliminate arguments give <t> ( x - X ( t ) , t ) expansion later with form number that straightforward we the series that be to of meaning /dx. two less 20 [ £M),{ |4f, j PJ I'i} f ' L Using X our = 0 relations (11 a - f ) Aini) ( Using the formal power V • T 1 we t « T 1 7 obtain ['£(,.*) , ? J series f o r \}/ t h e c o m m u t a t o r evaluated n >o n CD A n' >o (-0 f * 5 can be 21 HE z A 1 • J 0 IT Rewriting the fermion condition gives commutator with the help fty of the Majorana | f (3), f (*-*)] 07 / 6t 1 LJf- . ( jT [ , TT(V.tr 0 . P , A A , 22 [ TT ( x - X ) , * J (j-'x)] (x-X) Again using t h e power A 15 P. [ ?. K C-x) 1 • ; ^ we obtain („.x) the commutator [ f ( 0 , which f o r our commutator I (.-x) ] • i i'(,.x) Furthermore I'd) J series Ic,-x)] simplifies 4 ' 1 I z , $(«j-X')J - the terms (y-X) 4* (* - x) reduces i ' ( 0 • c-o ( U y x - 0 - i to o to i ' o o f' ( jf) t r 23 The c a l c u l a t i o n s f o r the other and need n o t be Thus of we quantum We 2.2 of 3 0 We (a) = proceed canonical transformation to.a set to construct the Hamiltonian and the motion. (8) f t ' straightforward variables. of the energy-momentum Lagrangian are here. a consistent collective Construction The where have c a n now equations reproduced commutators 3 i s given 1 = Hamiltonian tensor by f x* obtain f o r the energy density density, T , , v = 0 0,1 f o r our 24 (b) f o r t h e momentum where We has been 0 I write T0I To T symmetrized • $'TT + ; f ( TT construct the Hamiltonian substitute The most difficult transformation the bulk ir(«.t) part (10b).for measure and this notation -if' - i (14) T oo i n o u r new variables we i s the evaluation n i s highly will of ^ J j x T T nonlinear. again drop To (i ^ since reduce arguments and the - i ( ^(-/w) ( ? « . ( r ; . ;*•*•) , n2 A define • ir(»-x(t),t) A - cf f' ) H = Jdx o f o u r c a l c u l a t i o n s we integration IT _ (10) i n t o ( 1 3 ) . the With in $ (a) a n d (b) as • : simply density given by j + ^ a; T • T a ^ + c, ^ f ) 25 IT Using - IE X the basic commutation >. we i The relation r e w r i t e II2 - - - - x i n terms i of commutators ( T <j>' ( £ + 0 t IT L, £ { expression for I • ( TT ft, • CA. i - : i s more and anticommutators: [ complicated: 1 , 1 CI a a <£ a + cx, <f>i 4 i a c • so (15) [M*'<•«•*>] * i r i ^hr = ^ 1 <x a L * x J• ^cl ^ A. CX r\, a a, 6i r + 4 CD n that ( <f>* Z t l + J c ~ l • ce' ^ * r~ aa + • * _ A <1 0 + a aa" - <- a hi hi ^ 26 2 + and we The first can be A have the four A rk ' <p„ A 4 terms ' ~a- ^ Z 1 are using I = AT + following preliminary evaluated f L X T * in their expression final form. for The n2: next two terms (15): - o• r co ^ "• AT J ' - A ^ ' 4 " ou- A• I *. a. and ^ a AT furthermore, L 1 " A * T A T 1 ' 1 * O a " a. X (AT)* ' AT AT + A* I (AT)* (AT) 4 ^* W 4. 4. ' CL 27 Together with the next two terms i n our H -expression 2 we a, To complete h b r . * J our a l i ^ ' * ] ' d ( ?• 0/ so n computation that the - X 7 of n2 the power series i are reduced expansion r ±^2 ^0 • I the commutator becomes 0 and since evaluate - z f ' f ))TT to I! Using to w h e r e M*f commutators i t remains rt . t > have 28 we have Ii A IT * S r i L AT Summing M. form only o n x-X integration dx T n terms gives the f i n a l , for II : 2 (ATr (*-X(i)j (A* O of t h i s that « CL J result - * a A (ATJ . into hr: ' up a l l t h e s e purpose n -1 1 t The 00 rather permits a n d commutes variable dx TT 4 1, lengthy calculaton integration with X. over Thus, we x. was Each to transform factor can t r a n s l a t e x t o x + X and use our c o n s t r a i n t s dx 7T <jf> a + a. n2 depends our (12): 29 I A£ / , i • The we h a v e used remaining Thus Hamiltonian ,l * I C W ) * the c l a s s i c a l soliton mass M 0 is evaluated i s given Since we 4 want consider We ( M by the orders want only - *• fy M ~g~ , S/M ~g h e n c e f )) de) a) of g orders $,\p,n,\p 0 £ p e r t u r b a t i v e expansions we h a v e 2 # y ( v c ^ t i ) 4 0 M Z * us that terms a r e e a s i l y the t o t a l H the fact I / / 4 where A i n g f o r our q u a n t i t i e s involved. up t o 0 ( g ° ) ,P,X^g , and ^ - g " 0 (M +£)- ~ 1 0 l e t f o r our operators, 1 as before. g " , n>2. Thus we that i s , have 30 Furthermore we c a n e x p a n d Hamiltonian H decomposes H = M 0 + H + 0 V and V 2 around . Then o u r as follows H x where (17) with H describes exactly 0 of V 2 this " ( ^ ) This chapter: can x just free momentum Substituting P that "position that i s already <p = c o n s t . , 0 standard at the beginning dependent background potential. at the 0(g°) l e v e l the on t h e o t h e r hand, (17) field. i n higher order i n g. perturbation theory Its effects with the H . 0 operator P =Jdx T 0 1 i s easily (14) and p r o c e e d i n g evaluated. as before = P the total mass" nontrivial. and boson interactions (10) i n t o with discussed in a static soliton Hamiltonian we statement sector be i n c l u d e d u s i n g The so a describes unperturbed system of our theory t h e vacuum describes system our e a r l i e r sector For H a boson and a fermion confirms soliton the situation we obtain (18) momentum of the system i s given by t h e 31 collective We momentum note commutes take P 2.3 The with P. t o be a field Given function i n agreement our Hamiltonian We the pure boson (16) i s i n d e p e n d e n t and P case. o f X, simultaneously c-number. equations f of the f i e l d particular with can d i a g o n a l i z e H i n q°-order the Hamiltonian Hamiltonian In that P, H 0 the evolution equation variables i s given by f o r any the canonical formalism: we (1 ) obtain A A. (19) r A IT (2) A and + A hence and 32 or (20) (3) A L H.. A. A | j A . A I Using the r e p r e s e n t a t i o n matrices 7 °=a , 2 7 *«/a 5 Jf 7 = - a , , f o r the D i r a c we can reduce the (pseudo) s c a l a r s t o . A ^ A A t t -A t 2 - ' S the anticommutation. r e l a t i o n s ( 8 ) we o b t a i n Using A A , A A i A A V y* v . ^ ] r " -f - which y i e l d s , i A | for A , the fermionic - - v(fc) i equation , 1 i l of motion 33 which is written in matrix form as u X • V'(* ) u or «0 ( ^ i - i f ( o • v'(<U)) V Since of our motion (4) The Hamiltonian are 0 does not i n v o l v e P o r X, t h e i r (22) X = P = 0 (20) equations shows that our Majorana We equations trivial a n d (21) are familiar i n 1+1-dimensional fractionalization to H (21b) •o x t (21a) method i s indeed i nthe context o f charge ([3],[34]). systems quite general and not This restricted fermions. will complete constructing the an e x p l i c i t discussion solution o f our t o (20) quantum s y s t e m by a n d (21). Using the "ansatz" A A $ (x.-t) (*.t) v to separate the (" time • 1 'tut * < / > CO t c KOt . dependence V*"(0 ) we o b t a i n fo>) ' » h > ) <23) •«r " W (24) We note that the fermionic solutions can be e x p r e s s e d i nterms o f 34 the ( 2 1 bosonic one: ) 0 v'UJJ ^ • « £ -3* If »e d e f i n e the first ^ equation i , ^ turns . into y' ( < ) J ) t h e boson (- V * v"ifc) • v'( so for $ equation i 2 i •«£ that (25) f 00 - $ provides a solution boson equation we c a n e x p a n d annihilation f o r (21). (20). The f i e l d the f i e l d operators. Thus equations operators We i tsuffices t o solve the a r e now l i n e a r , i n terms of c r e a t i o n o b t a i n f o r t h e boson and operator *(-*) - L L (r,,wr% . r w - 4 ; ; with £n a s o l u t i o n runs over spectrum. both to equation the discrete The ( n o r m a l i z e d ) ft. * (20) and w k «f k and the.continuous zero-energy 4 • hence 1 (26) • part T n e s u m of the solution (27) £ 35 is excluded hence for from we h a v e properly operator with 0tL' of theorthogonality relation (12), t h ecompleteness normalized £ k r e q u i r e s somewhat zero-energy t $ because . relation The expansion more care. Equation (24) admits t h e solution W " "I* ( " H f V(*J N a normalization constant. = V(0cl) f o r t h e fermion - a n d t h e change ) (o) Using (28a) equation of variables d0 c L (4)- = 0 'dx tL this can integrated t o (28b) To e l i m i n a t e t h e zero-mode case. First f(*.t) with we r e p l a c e analogously transformation . X (x-X(M) a + i* new a n t i c o m m u t a t i o n Xf f relations equation ?(«-*W.O constraint \ and we p r o c e e d • o t o t h e boson (10c) by (29) 36 [ f.w. v^i • K - i.^co<t,)) <30b> (30c) i M •{ It •{ ^ . M • i s straightforward to verify canonical, that \p a n d i//t a i s a Hilbert zero-mode, i// i s now of solutions operators i s , i t preserves of \ that space orthogonal (21) w i t h • this ( 3 0 d ) transformation i s the Majorana operator o anticommutators f o r a s s o c i a t e d with the t o \j/ a n d c a n b e e x p a n d e d 0 fermion creation and i n terms annihilation , ^ (31) k with The a commutation For Using (for simplicity been evaluated ( * ) • - t o (24) g i v e n relations our standard analytically. ! solution we f o rf . j ^"-model taken ( 2 5 ) a n d <»>„ •-/ k * * ' . are fixed the f i e l d the c l a s s i c a l have by equations solution x =0,g=1) o by ( 3 0 ) . c a n be solved ( 5 ) , <f> = m - t a n h the solution mx o f (23) has ([13]) : — *x - ) i k ' , Ic * T R 37 and equation (24) Equation (26) in f o r our 0(g°) Having a comment We have and a seen of viewed the be as that (31) the full boson Lagrangian (8) with potential that, the in first zero-mode order appears soliton. The interpreted supersymmetric as generated by and theory solutions fermion c o n s t r u c t i o n of i n our infinitesimally in a spinor the zero-modes the corresponding give completed the fermionic existence show and thus on yields the seems to field operator be order. perturbation theory, as a bosonic direct operators in a consequence "translation bosonic of mode" c a n translated soliton. Later theory fermionic zero-mode an the infinitesimal we theory complete we have to the d i s c u s s i o n of find the Hilbert the soliton space sector f o r our of our operators. be will supersymmetry transformat ion. To the can 38 2.4 for The H i l b e r t space f o r the soliton The space f o r t h e vacuum a free {|k,...k q. Hilbert n field ;q,...q >} m generated fermion In by theory. from (k),f (q). the s o l i t o n sector we ([28]). space be state These states states energy k; a n d by t h e b o s o n - a n d this the s o l i t o n mechanical o p e r a t o r s P,X. a n d momentum space w i t h momenta to enlarge t h e quantum f o r the center-of-mass sumultaneously t |P> t o t a k e form states |0 >»|0^> have i s t h e s t a n d a r d one of the Fock formultiparticle operators b a s e t of e x t r a sector It consists a vacuum sector They product into space account representation are required to eigenstates (32) This of i s always X. We note remains static state. Thus the product |0 > » | 0 4 k The usually As >»|0 possible furthermore (X=0), space 4 {| P 4 our Hamiltonian that s o we the Hilbert in first s e t P =0, space order 0(g°) the soliton E(0)=M 4 o f o r the soliton ; k , ... k ;q, . . . q > ] w i t h r w (16) i s independent f o r the ground sector vacuum i s g i v e n by state >. problem of possible associated discussed since with d e g e n a r a c i e s of t h e ground the existence i n the previous chapter of t h e f e r m i o n i c the f i r s t - o r d e r state i s zero-mode. field 39 equation (24) admits a zero-energy s o l u t i o n so that the fermion A operator \p has to be expanded as V ( * . t ) - t OO £ The » ?(».t) . zero-mode operator a does not have an i n t e r p r e t a t i o n as a c r e a t i o n operator but i s only r e q u i r e d t o provide a r e p r e s e n t a t i o n of the a l g e b r a In a theory with D i r a c fermions these r e l a t i o n s are s u b s t i t u t e d by the standard anticommutator and the zero-mode operator commute with the Hamiltonian. The r e p r e s e n t a t i o n of a and a* i s then given by a doubly degenerate oJ I + | 0. > •- I 0 > a + T h i s i s not v a l i d l0 > + ^ V e x i s t e n c e of a zero-energy and no d e f i n i t e statement of The Hamiltonian (16) and the r e s u l t i n g crossterms w i l l not The c o n v e n t i o n a l view that the mode s i g n a l s degeneracy does not hold can be made a t t h i s p o i n t . s o l u t i o n to t h i s problem i s given by the supersymmetry the theory. possess -- 1 0 . > A v a n i s h under commutation with a. The |0+> ([3]) with * O f o r the present case. A ^ contains b i l i n e a r s ground s t a t e a, 1 0. > - 0 0 > will The supersymmetry o p e r a t o r s w i l l turn out to j u s t the r i g h t p r o p e r t i e s to p r o v i d e the expected degeneracies f o r the s o l i t o n states. We w i l l q u e s t i o n s at the end of the f o l l o w i n g c h a p t e r . r e t u r n t o these 40 3. SUPERSYMMETRY One fact of that the whose supersymmetry to published I t has mass Bogomolny soliton a set of shown, at the saturated the this exist field theory first is breaking have order, Here of of a been we for that that are no bound c o r r e c t i o n s and ([20]). conserved c o r r e c t i o n s to Bogomolny in the theories several papers the is generators quantum quantum least quantum spontaneous supersymmetry recall algebra For years s a t u r a t i o n of remains the few concerning r e c e i v e s no on form p r o p e r t i e s and last theory THEORY the the the the will supersymmetry in the sector. The We the been bound concentrate 3.1 and charges our i . e . there Supersymmetric the SOLITON-FERMION f e a t u r e s of supersymmetry, unusual ([15]-[20]) energy. soliton remarkable algebra. During mass 1+1-DIMENSIONAL corresponding possess exception. soliton most A i t possesses currents known FOR generators briefly the for definition the of soliton-fermion theory. an (N=l) supersymmetry ([35]): given action operators anticommutation Q S 0^ the , supersymmetry Q r * i.^f)^ algebra , ^,o,i i s generated together with the relations o r (33a) (33b) I) - o (33c) by a 41 with P t h e energy-momentum e For topologically solitons, ([16]). Lagrangian possesses generates to include "topological ) 2(*,t corresponding to , currents ( jf<fr t ; V ( o » i t scomponents Q. ^ r algebra generate . Q,J [ Q«. T j ) T (34) "supercharge" a supersymmetry { c o n f i g u r a t i o n s , e.g. d e n s i t y (8) the corresponding i.e. field S = JdS The a c t i o n conserved j r and nontrivial ( 3 3 b ) h a s t o be m o d i f i e d charges" our operator. Q with |j« j ( * , t ) * topological a supersymmetry • -if, t \ • [ Q . . ?, ] * » (Q° ) 6 i ; charges algebra T, with y r T <35b) -o where T c a n be i n t e g r a t e d t o z e r o trivial, 0 (x=t») To using i s , <pcl(x=t<x> that = 0 t verify f,F: <p i s a s y m p t o t i c a l l y 0 has t o i n c l u d e the s u r f a c e ( 3 5 ) we A ) = <f> . A s o l i t o n will the following identity B,b a n d i f the f i e l d evaluate field term however, T. the anticommutator f o r two b o s o n with and fermion directly operators 42 {Fb,bf} The = Ff[B,b] + {F,f}bB (37) generators a can be v i e w e d a s c o m p o n e n t s Q. i. L d* J ' Now a p p l y The Kij'f.* ZL commutators (a) the following matrix *f A' - i s r v( )) r 5>pj L |. r to the anticommutator are evaluated t o be o * I,. V product y « |i« E (37) of S [f U ) > ( j ) ] - i l r $,„ [ v(4>), ity] 43 (b) SO • that the boson product H L can be reduced to y Ux-O [ ( *-yV *<yX*)) (y"f - / V V O ) ) IK T(«) -i; ^ * - v'it) Ho) J J and for the fermion part fj, u..ij • - * f p7;.^ ,t r r 1 f * v(+) Thus we have as a preliminary result . v'(*) f' r f r . f , f -** r;. r \ 44 V ( * Y < r , . / ' ( • ) O V ) ,( / » , . * - V'( As the last ordering Formally, That we and We choose term arising however, we our have for the there terms a l l these indeed familiar e v a l u a t e the obtain indicates from S(o) S.^ <t>) like are 7 our will of particular our 7 0 matrix terms \ . m H;' + (b) + \ , 1 , the for operator x-*y. cancel. equality representation elements with \p (x)i//(y)-» ^ 6 ( 0 ) terms found problems • = can a , (ML,. be 7 2 = 1 easily = {Q. / 03, ,Q.}. seen 7 5 = " 45 (c) * *(o) f 3L (d) ^7) • (ye^) , o - •x'x which gives the final expression \ 0 / f o r the v'u) (^7 - + + 1 , Ho) i anticommutators ) - U7'*^7) 46 whereas the right hand - X side - I 1 « = o , fl - (37) gives «\ 1 1 eo « . ft of e q u a t i o n 72 f 0| r 6 r" 7 and of the the other In Q's supersymmetry on the the two next relation relations section s o l i t o n ground is we (35b) i s proved. The s -r verification straightforward. will state. investigate the a c t i o n of the 47 3.2 Spontaneous Having our action supersymmetry established we will due to the discussion of supersymmetry broken [35] and [36]. now the We will i n the e x i s t e n c e on show presence breaking that of the N=1 symmetry central can their be sector supersymmetry is charges. breaking modify an soliton spontaneously A general found arguments for in references f o r our present case. First we derive three supersymmetry taking the relation trace equations (35a). g i v e s the from our Multipling well known by Furthermore we deduce • CL* - Q,4 It i s obvious (39) implies H which H ground will state both now |0>. and - ) (38) (35a) (39) T (40) equation the sides Q4 (38) quantum that H Bogomolny is positive. bound on Equation H > T i s intended We - from on 0 H * T from then Q directly 7 formula H" • i ( {Q..<M t - -M ) : i + central (41) to hold examine i n the these Obviously we weak sense. operator have the equations f o r our alternative soliton Q-|0>*0, 48 which means Q-|0>=0, It that supersymmetry i n which case i s obvious i s spontaneously supersymmetry from equations I t i s n o t so easy unbroken o r , e q u i v a l e n t l y , whether to decide, the c l a s s i c a l central charge Hamiltonian H|P=0> T acts = M |P=0> o question whether ([l5]-[20]). level = j * ^ A t l (cf. 0(g-') f ' whether t h e Bogomolny the solution reduces on a p u r e soliton state result h a s been I t h a s been hold discussed found corrections, so t h a t the combination result already on t h e d e n s i t y this will Imbimbo remain Their fermions i n lowest zero-mode breaking that Q bound remains 0 ( 4 1 ) on H argument i s based order appears order the other modes would to zero, authors identical remains in quantum zero. qualitatively i n perturbation zero. The more theory fermion t othe Goldstone to renormalize a process that of Goldstone 2.3) c o r r e s p o n d s have This f o r H and T argued To g e n e r a t e The orders i n order on t h e a b s e n c e (cf. section exactly i s saturated. by s e v e r a l level have with t o higher H-T ,the 1 a |P=0> receive perturbation corrections j <*•* <t> = 0 i s easy: the to first to a l l orders o f t h e Ch-symmetry. higher unlikely. a n d Mukhi valid ([19]). H and T that theory ([19],[20]). both will perturbation holds to M ( 3 2 ) ) and t h e bound this perturbation theory of however, i s always saturated. On the i s preserved. (38) and (40) t h a t broken. is broken, or that modes, t h e masses seems very 49 We can breaking by H' = reformulate i n the H spectrum Hilbert now soliton. - T. by a supersymmetric rewritten field the N=1 due the presence can section not Hamiltonian of affect system. the This with the more defined energy dynamics new equations H' or the Hamiltonian general (39) and results (40) of with the the of ( 4 3 a i s broken central the charge supersymmetry state down can ground be which can state broken. to an the In "N=i" other symmetry to be possible state generators according ground Q that we started and Q, both 0 (35c). taken as Q, does not a degenarate with t + , Q, |0.> - | 0<> (44) a, l o 4 > 4 the in ) T. d i s c u s s i o n of Hamiltonian ground f o r the spontaneously soliton-sector The the |0 > T remains a, fo > ~ I0_> Thus has (42) H' W • complete 2.4. annihilate doublet shift symmetry * H supersymmetry now degeneracies commute a i n agreement 0.,-symmetry the words, in new to and spontaneous the i s obviously conserved 0 We our of as Q -symmetry to does theories, Q* whereas and energy Qo' The Consider s t r u c t u r e of vacuum problem corresponds constant space vanishing This the spontaneous - T i os > breaking of supersymmetry provides the 50 degeneracy of with fermions Dirac We will connection fermion the conclude zero-modes the boson classical have to be as corresponding the consequence on the was set by quantum of the consider fields our analogous of are theory. not as field theories mode zero-mode infinitesimal to the the the part 0 (x-X) t l quantum and 4> (x-x ). t l invariant X x they which 0 is the be Since 0 is soliton 0 ct infinitesimal viewed as a symmetry. supersymmetry conserved transformation charges Q 0 and ([35],[37]) 5 (45a) (45b) with e an anticommuting transformations - with Majorana time 2-spinor. dependent then the <f>lL v i a a n can and far, on parameter translational corresponding So solutions Furthermore the boson soliton the variable translation on and based The translation boson note translationally operator. broken been classical the the the has a theory. interpreted collective Hence from solitons supersymmetric of solutions a between formalism zero-energy translation. Now a labelled considered induces in a operator of familiar d i s c u s s i o n with earlier symmetry representative the this coordinate translational state ([3],[34]). mentioned collective of ground If we t-parameter apply - to these the 51 classical static we t h e new obtain solutions solutions (46a) 6i// o(/ i s - up (28), now to a transforming by an ([38]). "supertranslated" transformation the f i e l d s constraint This system collective for both bosonic c a n now 6 - the zero-energy B ) mode supersymmetry quantize these s o l u t i o n s as b e f o r e (*) by treating coordinate £ Ut) + V 7 quantized • 0 the and (47) ( of B a a k l i n i fermionic c a n o n i c a l l y as o r by ([31]). 4 8 ) . either coordinate systems 4 to the c o l l e c t i v e and constant E.(t) a s a c o l l e c t i v e ) <U i n t h e work ( a position generalized ta 6(*) • SY («) tCO • c a n be constrained found • (*.t) with We parameter 4 infinitesimal classical i UO T- (o) normalization generated transformation ta t h e Feynman A detailed ([39],[40]) method parameters are path integral d i s c u s s i o n can who, coordinate f o r the be in addition, t o the case present. where 52 To of zero-energy which are summarize are broken generated the fermion by by mode. to In the quantization the theory, gives a be of the 0cL, as that of the existence the soliton. Lagrangian The zero-modes symmetry in particular, a translation transformation parameters collective for associated with coordinates for the these the theory. i t has been information in particular, fermion symmetries supersymmetry used shown infinitesimal a d d i t i o n , the can Conversely, an have to presence soliton and transformations of the we i s due applying boson-mode zero-modes results, solutions transformation for the presence of "hidden" supersymmetry. that argued about in a zero-modes that the theory implies ([34]). the existence symmetry with the of properties Dirac fermions existence of a of a the 53 BIBLIOGRAPHY 1] A.C. S c o t t , F . Y . F . (1973) 1443. 2] J . Goldstone 3] R. 4] J . Goldstone 5] M.B. 6] A.J. Niemi and 7] T. Skyrme, Proc. RoySoc.,London 8] E. Witten, Nucl. Phys. 9] R. Mackenzie and 10 M. Rice 1 1 W. S u , (1979) 12 R. Jackiw and J . R. 1 3 R. Jackiw and G. 14 A.J. 15 A. D'Adda and P. Di Vecchia, 16 E. Witten and D. Olive, 17 J.F. Schonfeld, 18 R.K. Kaul 19 C. Imbimbo 20 H. Yamagishi, 21 R. D a s h e n , 4114. 22 N. 23 J.L. 24 G. Jackiw and and and R. C. and F. and G. G. Christ and B. and Gervais Semenoff, Phys. Hasslacher and H. B. Lee, 127. Heeger, B161 Phys. B190 (1981) Rev. Lett. 50 Lett. Lett. 78B 73B Rev. D12 P h y s . Rev. and H. 253. 439. 2077. 162. 97. (1983) 357. 84/04. April Neveu, 42 (1983) (1978) (1978) preprint 1158, (1983) 51 131B Phys. 1455. Nucl. Phys. L e t t . A. 2260. Lett. 125. and Matsumoto (1982) (1979) CTP Sakita, (1984) 369. 121. Phys. Rev. Phys. LPTENS preprint T.D. (1961) Phys. L e t t . Mukhi, MIT A260 Phys. Rev. Rajaraman, S. (1984) 49 986. (1983) 135B Lett. Phys. (1981) 132B D30 61 1486. Lett. Rev. IEEE 3398. Lett. Lett. Phys. A. Semenoff, Rev. 433. Schrieffer, Nucl. R. and and (1975) (1976) (1983) Rev. i n Proc. D11 D13 Phys. Phys. B223 Semenoff, G. Rev. Semenoff, Phys. Rev. Phys. Wilczek, E. Mele, and Phys. Semenoff, F. McLaughlin Phys. Wilczek, J.R. S c h r i e f f e r 1698. Niemi D.W. Jackiw, Rebbi, and Paranjape Chu 1984. Phys. (1975) D11 Umezawa, Rev. D10 (1974) 1606. (1975) J . Math. 2943. Phys. 21 54 (1981) 2208. [25] A . J . Niemi a n d G. S e m e n o f f , [26] E . Tomboulis, Phys. [27] E . T o m b o u l i s a n d G. Woo, A n n . P h y s . [28] R. J a c k i w , [29] P.A.M. D i r a c , [30] P.A.M. D i r a c , L e c t u r e s o n Q u a n t u m U n i v e r s i t y , New Y o r k , 1 9 6 4 . [31] L.D. Faddeev, [32] J.L. Gervais [33] A. Hosoya [34] G. S e m e n o f f , H. M a t s u m o t o (1982) 1054. [35] J . Wess a n d J . B a g g e r , S u p e r s y m m e t r y a n d S u p e r g r a v i t y , P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , 1983. [36] E. Witten, [37] A. S a l a m [38] P. R o s s i , [39] N. B a a k l i n i , Nucl. [40] N. B a a k l i n i , J . P h y s . A11 R e v . D12 R e v . Mod. P h y s . Theor. Math. a n d A. N e v e u , a n d K. K i k k a w a , Phys. (1975) 1678. 98 ( 1 9 7 6 ) 1. 2 (1950) 129. Phys. Mechanics, 1 (1970) Yeshiva 1. P h y s . R e p . 23C ( 1 9 7 6 ) 2 3 7 . Nucl. Phys. B101 ( 1 9 7 5 ) 2 7 1 . a n d H. Umezawa, Phys. R e v . D25 B188 (1981) 513. and J . Strathdee, Phys. L e t t . (1984) 809. 49 ( 1 9 7 7 ) 6 8 1 . Can. J . Math. Nucl. P h y s . R e v . D30 Nucl. Phys. B76 ( 1 9 7 4 ) 4 7 7 . 71B ( 1 9 7 7 ) 145. Phys. B134 (1978) 169. (1978) 2083.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- On a 1 + 1 - dimensional interacting soliton-fermion...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
On a 1 + 1 - dimensional interacting soliton-fermion system with supersymmetry Keil, Werner H. 1985
pdf
Page Metadata
Item Metadata
Title | On a 1 + 1 - dimensional interacting soliton-fermion system with supersymmetry |
Creator |
Keil, Werner H. |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | A supersymmetric interacting soliton-fermion system in one space and one time dimension is investigated. We construct the soliton sector of the quantum theory using a generalization of the "method of collective coordinates" previously developed for purely bosonic theories. A canonical transformation leads to a set of "collective" field variables with constraints and the transformed theory is quantized in the canonical way using Dirac's method for constrained systems. The Hamiltonian is evaluated in collective coordinates and the equations of motion are solved to first order in a perturbative expansion. We find that the field equations admit zero-energy solutions for both the boson and the fermion field. The presence of the soliton has nontrivial consequences for the supersymmetry of the theory. The supersymmetry algebra has to be modified to include topological charges and as a result supersymmetry is spontaneously broken. It follows that the ground state is doubly degenerate. Finally, the zero-energy solutions are found to be connected with the symmetries of the theory broken by the soliton. The boson zero-mode corresponds to spatial translations, the fermion zero-mode is associated with the supersymmetry |
Subject |
Solitons Fermions |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-19 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0084999 |
URI | http://hdl.handle.net/2429/24825 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1985_A6_7 K43.pdf [ 2.32MB ]
- Metadata
- JSON: 831-1.0084999.json
- JSON-LD: 831-1.0084999-ld.json
- RDF/XML (Pretty): 831-1.0084999-rdf.xml
- RDF/JSON: 831-1.0084999-rdf.json
- Turtle: 831-1.0084999-turtle.txt
- N-Triples: 831-1.0084999-rdf-ntriples.txt
- Original Record: 831-1.0084999-source.json
- Full Text
- 831-1.0084999-fulltext.txt
- Citation
- 831-1.0084999.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0084999/manifest