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Restricting the form of interaction Hamiltonians for systems of fermions, antifermions, and bosons Asman, Charles Phillip 1999

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RESTRICTING T H E F O R M OF INTERACTION HAMILTONIANS FOR SYSTEMS OF FERMIONS, ANTIFERMIONS A N D BOSONS By Charles Phillip Asman B . Sc. Simon Fraser University, 1997 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E FACULTY OF G R A D U A T E STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA September 1999 © Charles Phillip Asman, 1999 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. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B . C . , Canada V 6 T 1Z1 Date: A b s t r a c t In this thesis, we define two general interaction Hamiltonians, one for describing systems of interacting nucleons, antinucleons and pions, the other for describing systems of interacting electrons positrons and photons. We then demand that the Hamiltonians should each remain invariant under the action of a certain class of transformations. By imposing this requirement, the allowed forms of the Hamiltonians are greatly restricted. ii Table of Contents Abstract ii List of Figures vii Acknowledgments viii 1 Introduction 1 1 Nucleons, Antinucleons and Pions 3 2 Introduction 4 3 Fundamental Dynamical Variables and Their Properties 5 3.1 Fundamental Dynamical Variables 5 3.1.1 Nucleon Creators and Annihilators 5 3.1.2 Antinucleon Creators and Annihilators 6 3.1.3 Pion Creators and Annihilators 6 3.2 Commutation Relations 7 3.3 Transformation Properties 7 3.3.1 Spatial Translations 8 3.3.2 Spatial Rotations 8 3.3.3 Isospin Rotations 9 3.3.4 Space Inversion 9 3.3.5 Time Reversal 10 3.3.6 Charge Conjugation 11 3.3.7 Charge Conjugation and Space Inversion 11 3.3.8 Charge Conjugation, Space Inversion and Time Reversal 11 ii i 4 A Trilinear Nucleon-Antinucleon-Pion Interaction 13 4.1 Defining the Total Hamiltonian, H 13 4.2 Diagrams of the Interaction Hamiltonian, Hi 14 4.3 Restricting the Form of Hx Under Hi, TZ, V, T and C 15 4.3.1 Restricting the Form of Hi Under Hi, Tl, V and T 17 4.3.2 Restricting the Form of # / Under Hu K, V and T 19 4.3.3 Restricting the Form of Hi Under Hi, H, V and T 20 4.3.4 Imposing Charge Conjugation Invariance on Hi 22 4.4 Restricting the Form of Hi Under 7li,7l,T and CV 23 4.5 Restricting the Form of Hi Under TZi, Tl and CVT 24 II Electrons, Positrons and Photons 25 5 Introduction 26 6 Fundamental Dynamical Variables and Their Properties 27 6.1 Fundamental Dynamical Variables 27 6.1.1 Electron Creators and Annihilators 27 6.1.2 Positron Creators and Annihilators 28 6.1.3 Photon Creators and Annihilators 28 6.2 Commutation Relations 28 6.3 Transformation Properties 29 6.3.1 Spatial Translations 29 6.3.2 Spatial Rotations 30 6.3.3 Space Inversion 30 6.3.4 Time Reversal 31 6.3.5 Charge Conjugation 31 6.3.6 Charge Conjugation, Space Inversion and Time Reversal 32 7 A Trilinear Electron-Positron-Photon Interaction 33 7.1 Defining the Total Hamiltonian, % 33 7.2 Diagrams of the Interaction Hamiltonian, Hi 34 iv 7.3 Restricting the Form of H i Under Tl, V, T and C 36 7.3.1 Restricting the Form of Ui Under Tl, V and T 36 7.3.2 Restricting the Form of 77/ Under Tl, V and T 38 7.3.3 Restricting the Form of 77/ Under Tl, V and 7~ 39 7.3.4 Imposing Charge Conjugation Invariance on H i 40 7.4 Restricting the Form of Ui Under Tl and CVT 41 8 Summary and Conclusions 43 III Appendices 44 A The Rotation Matrices 45 A . l Definition of/J^ m (a/? 7 ) and 4 , m ( / 3 ) 45 A. 2 Useful Relations for the £>^,T O(a/?7) and d3m,m{P) 45 B Addition of Angular Momenta 47 B. l Adding Two Angular Momenta 47 B.2 Definition of Clebsch-Gordan Coefficients 48 B.3 Relations Involving Clebsch-Gordan Coefficients 49 C Properties of Spherical Harmonics 50 D Restricting the form of Hi 52 D. l Imposing Isospin Rotation Invariance on Hi 52 D.2 Imposing Spatial Rotation Invariance on Hi 55 D.3 Imposing Space Inversion Invariance on Hi 58 D. 4 Imposing Time Reversal Invariance on Hi 60 E Restricting the form of Hi 63 E. l Imposing Isospin Rotation Invariance on Hi 63 E.2 Imposing Spatial Rotation Invariance on Hi 66 E.3 Imposing Space Inversion Invariance on Hi 69 E.4 Imposing Time Reversal Invariance on Hi 70 v F Imposing Charge Conjugation Invariance on Hi 74 F . l Imposing Charge Conjugation Invariance on Hi + Hi 74 F.2 Imposing Charge Conjugation Invariance on Hi 76 G Restricting the Form of Hi Under 111, Tl and CVT 79 H The Helicity Formalism 85 H . l Definition of Photon Helicity States 85 H . 2 Determining the Transformation Properties of the Photon Helicity Operators 86 H.2.1 Spatial Rotations 86 H.2.2 Space Inversion 86 H.2.3 Time Reversal 87 H.2.4 Charge Conjugation 87 I Restricting the form of Hi 89 I. 1 Imposing Spatial Rotation Invariance on Hi 89 1.2 Imposing Space Inversion Invariance on Hi 90 1.3 Imposing Time Reversal Invariance on Hi 91 J Restricting the form of Hi 94 J . l Imposing Spatial Rotation Invariance on Hi 94 3.2 Imposing Space Inversion Invariance on Hi 95 J.3 Imposing Time Reversal Invariance on Hi 96 K Imposing Charge Conjugation Invariance on Hi 99 K . l Imposing Charge Conjugation Invariance on Hi + Hi 99 K.2 Imposing Charge Conjugation Invariance on Hi : 100 Bibliography 102 vi List of Figures 4.1 A diagrammatic representation of the trilinear nucleon-antinucleon-pion interaction ac-counted for by Hi. Solid lines with upward-pointing arrows are nucleons and dashed lines are pions 15 4.2 A diagrammatic representation of the trilinear nucleon-antinucleon-pion interaction ac-counted for by Hj. Solid lines with downward-pointing arrows are antinucleons and dashed lines are pions 16 4.3 A diagrammatic representation of the trilinear nucleon-antinucleon-pion interaction ac-counted for by Hi. Solid lines with upward-pointing arrows are nucleons, those with downward-pointing arrows are antinucleons and dashed lines are pions 16 7.1 A diagrammatic representation of the trilinear electron-positron-photon interaction ac-counted for by Tii. Solid lines with upward-pointing arrows are electrons and wavy lines are photons 35 7.2 A diagrammatic representation of the trilinear electron-positron-photon interaction ac-counted for by Hi. Solid lines with downward-pointing arrows are positron and wavy lines are photons 35 7.3 A diagrammatic representation of the trilinear electron-positron-photon interaction ac-counted for by Tii. Solid lines with upward-pointing arrows are electrons, those with downward-pointing arrows are positrons and wavy lines are photons 36 vii Acknowledgments I would first like to thank my supervisor, Dr. Malcolm McMil lan , for always making himself available to answer my numerous questions about physics, as well as for providing me with the financial support which made my research possible. Secondly, I would like to thank my good friend, Mr . Michel Olivier, for being a constant source of inspiration to me over the last two years, both in physics and music. Finally, I would like to thank my mother and father, Peggy and J im Asman, for giving me the opportunity and the encouragement to pursue my education. vii i C h a p t e r 1 I n t r o d u c t i o n In the search for an accurate yet practical theory of the strong interactions of nucleons and pions at intermediate energies, Hearn [1] has developed a technique for finding potentials of direct fermion-fermion, boson-boson and fermion-boson interactions as well as boson production on two fermions. The technique entails defining a trilinear interaction Hamiltonian involving the integral JhF^FB, where and (the elementary fermion and boson creators, respectively) are taken as the fundamental dynamical variables of the theory. The function h, called the vertex function of the interaction, is then restricted in form by requiring the interaction to be invariant under spatial displacements and rotations, isospin rotations, space inversion and time reversal. Once h has been restricted in this way, it is shown that by applying a dressing transformation to the Hamiltonian and using the Cloudy Bag Model [2] vertex function, the usual One Pion Exchange Potential (OPEP) , slightly modified in strength, results. In this thesis, we take the first steps towards generalizing this approach. First of all, the vertex function considered by Hearn depends only on the momentum of the pion, and the theory does not treat antiparticles. Thus, in Part 1 of this thesis, we introduce a new interaction Hamiltonian involving the three integrals, J hF^FB, J hF^FB and J hF^F^B. The first integral describes interacting nucleons and pions, the second describes interacting antinucleons and pions, and the third describes interacting nucleons, antinucleons and pions. Also, each of the three vertex functions, h, h and h depends on the momentum of both the nucleon and the pion. Having defined our interaction Hamiltonian, the rest of Part 1 is concerned with restricting the form of the vertex functions by imposing the requirement that the interactions should be invariant under spatial translations and rotations, isospin rotations, space inversion, time reversal and charge conjugation. In Part 4.5, we introduce another interaction Hamiltonian describing interacting electrons, positrons and photons. The rest of Part 4.5 is concerned with restricting the form of the vertex functions contained within this Hamiltonian by requiring that the interaction be invariant under spatial translations and rotations, space inversion, time reversal and charge conjugation using the techniques developed in Part 1. Although such a theory does not replace quantum electrodynamics (QED), which is a Lagrangian based 1 Chapter 1. Introduction 2 local relativistic field theory, it does provide an alternative nonlocal semi-relativistic approach to the problem. It is also interesting to note that the diagrams of the elementary processes are similar for both theories. Finally, in Part 8, we include several appendices. The first three appendices provide an introduc-tion to some of the mathematical machinery used throughout the thesis. The remaining appendices provide a detailed treatment of the calculations performed in the course of restricting the forms of the Hamiltonians. Part I Nucleons, Antinucleons and Pions 3 C h a p t e r 2 I n t r o d u c t i o n Th i s part of the thesis in concerned w i t h defining a Hami l t on i an descr ib ing interact ing nucleons, ant i -nucleons and pions, and provides a general ization of the Ham i l t on i an considered by Hearn [1]. In Chapter 3, we define the fundamenta l dynamica l variables of our system, provide their commutat ion (or ant i commutat ion relations), and finally give their t ransformat ion properties. Then, in chapter 4, we define a general fo rm for a Hami l t on i an , H, describing strongly interact ing nucleons, antinucleons and pions. We then impose the requirement that i t be invariant under each of the fol lowing transformations: isospin rotations, spat ia l rotat ions, space inversion (parity), t ime reversal and charge conjugation. These requirements restrict the form of H. In the final two sections of this chapter, we consider two special cases: first we demonstrate how the fo rm of H is restr icted by requir ing it be invariant under t ime reversal and the combined act ion of space inversion and charge conjugation. Secondly, we show how the form of H is restr icted by requir ing i t be invar iant under the combined act ion of space inversion, t ime reversal and charge conjugation. 4 Chapter 3 Fundamental Dynamical Variables and Their Properties In this chapter, we define the fundamental dynamical variables of a system of interacting pions (spin 1, isospin 1), nucleons (spin 1/2, isospin 1/2), and antinucleons (spin 1/2, isospin 1/2), provide their commutation (or anticommutation relations), and finally give their transformation properties. 3.1 Fundamental Dynamical Variables Our system of interacting pions, nucleons and antinucleons are described by a Hilbert space which is a direct product of boson Fock space, fermion Fock space and antifermion Fock space. We choose the fundamental dynamical variables of this system to be the elementary particle creation and annihilation operators. 3.1.1 Nucleon Creators and Annihilators Fi^(x) (m = ±i,n = ± 5 ) , when operating on the vacuum state, |0) , creates a one-nucleon ket cor-responding to an elementary nucleon at position x, with z-axis spin projection m and z-axis isospin projection /x. Note, /x = i corresponds to a proton, and \i = — \ to a neutron. The adjoint operator, Fm M(x), when operating on this one-nucleon ket, returns the vacuum state. Equivalently, we can define the momentum space nucleon creators and annihilators, (p) and Fmfi(p). These create or destroy an elementary nucleon with momentum p, z-axis spin projection m, and z-axis isospin projection (i. The two creators are related by a Fourier transform: We define the operation of Fmil(x.) and Fm M(p) on the vacuum state as follows: Fmn (x) 10) = Fmil (p) 10) = 0. (3.2a) This property is usually described by saying that Fmil (x) and Fmil (p) annihilate the vacuum. Note, by 5 Chapter 3. Fundamental Dynamical Variables and Their Properties 6 taking the adjoint of the above expression, we also have ( 0 | F t (x) = <0|Ft ( p ) = 0 . (3.2b) 3.1.2 Antinucleon Creators and Annihilators The discussion for antinucleons is identical to that given above for nucleons, except that the creation and annihilation operators for antinucleons are denoted by (£) and - F m M ( £ ) , respectively, where £ represents either x or p. Here, [i = \ corresponds to an antineutron, and = — \ to an antiproton. 3.1.3 Pion Creators and Annihilators l?t(x) (/i = 0, ± 1 ) , when acting on the vacuum state, | 0 ) , creates a one-pion ket corresponding to an elementary pion at position x with z-axis isospin projection /x. The operators i?M(x), (p) and -BM(p) are defined analogously to the nucleon case. The pion position and momentum creators are also related by a Fourier transform: Note, we do not need to introduce a new symbol to describe antipions because the pion is its own antiparticle. (3.3) Chapter 3. Fundamental Dynamical Variables and Their Properties 7 3.2 Commutation Relations The fundamental dynamical variables satisfy the following commutation and anticommutation relations: {Fm^t),F* = <5(£ - 0 < W « V ( 3 - 4 a ) { F m M ( 0 , i W ( O } = { F ^ C F l ^ ' ) } = 0 (3.4b) {Fmv(£)AA£')} = S& ~ 0 < W < W (3.4c) {FmAO'Fm-= {FluMAA?)} = 0 (3"4d) {F,F}=0 (3.4e) [B^Bltf)] = J « - O W (3.4f) [B^), B». te')} = [Bite), Bl te')] = 0 (3.4g) [Btte),Fm;'te')] = [Bl^Fl^')] = 0 (3.4h) [Blte),Fm^'te')] = [Btte),Fl,'te')] = 0, (3.4i) where [ ] denotes a commutator, { } an anticommutator, £ represents either x or p, F = Fm^te) o r F^te) and F = F m V ( £ ' ) or i ^ v ^ ' ) 1 -3.3 Transformation Properties Here we discuss the effect of spatial displacements and rotations, isospin rotations, space inversion, time reversal and charge conjugation on the particle creation and annihilation operators. The results of this section, which are a consequence of imposing general Poincare invariance on our physical system [4], shall be used to determine the form which invariant particle interaction Hamiltonians describing nucleons, antinucleons and pions must have. Note, only the transformation equations for F^te), F^te), and B M (£ ) a r e given since the equations for - F m M ( £ ) , F m / 1 ( £ ) , and are easily obtained by Hermitian conjugation. xSee p. 174 of Weinberg [3]. Chapter 3. Fundamental Dynamical Variables and Their Properties 8 3.3.1 Spatial Translations To a spatial displacement of the system by an amount a, there corresponds a linear, unitary operator V(a) such that V(a)F^ (x)Z>t (a) = (x + a) (3.5a) P ( a ) F ^ ( x ) 2 ? t ( a ) = i ? ^ ( x + a) (3.5b) 2?(a )£ t (x )p t (a )=Bt (x + a). (3.5c) Thus, displaced creation operators create particles at position x + a rather than at x. If we use (3.1) and (3.3), we then have 2 ? ( a ) F t / 1 ( p ) C t ( a ) = i r t i / i ( p ) e - » - P M (3.6a) V^Fl^vHa) = F ^ ( p ) e - i a - p / f i (3.6b) D(a)£„(p)2>t( a) = Blx(p)eia^n. (3.6c) 3.3.2 Spatial Rotations To a spatial rotation of the system through Euler angles (ct,f3,j), there corresponds a linear, unitary operator TZ(aPj) such that s n{aM)Flli{x)ll\a(}1)= £ D^.m(aPi)Flll(y.--B) (3.7a) m' = — s s Tl(a^)Flll(x)lV(aP'r)= YI D^^aMWl.^-n) (3.7b) m' = — s ft(a/37Wx)7e+(a/?7) = 5 M ( x _ R ) , (3.7c) where we have followed the conventions for Euler angles and rotation matrices used in Rose [5]. For a discussion of the rotation matrices, see appendix (A). Note that x R = M x (3.8a) x _ R = M _ 1 x , (3.8b) Chapter 3. Fundamental Dynamical Variables and Their Properties 9 where x is a column vector and M is the matrix defined on p. 65 of Rose [5]. Using (3.1) and (3.3), we have s 7l(<xPl)F^(p)TlHal37) = ] T D ^ a B ^ F ^ p - n ) (3.9a) m> — — S S naPlWl^lZHapf) = J2 D^m(aPy)Fl,M(p-R) (3.9b) m' — — s n(afo)B» (P)7e+ (a/? 7) = B, ( p _ R ) . (3.9c) Thus, both momentum and spatial coordinates, as well as the direction of spin of the nucleons, are rotated. 3.3.3 I s o s p i n R o t a t i o n s To a rotation in isospin space through Euler angles (a, 8, 7), there corresponds a linear, unitary operator, 7?./(a/37) such that KdaMF^WJltiafr) = £ D ^ a ^ F ^ ) (3.10a) M' = - i n M M ^ i m ^ a M ) = £ D^iaP^Fl^i) (3.10b) V- — 2 + 1 7e/(a/37)7JM(^ )7e}(a/37) = £ 7 ^ ; ( a / ? 7 ) i V ( 0 - (3.10c) 3.3.4 S p a c e I n v e r s i o n To a space inversion transformation, there corresponds a linear, unitary operator V such that FFl^V^TrfF^i-Z) (3.11a) ^ ( ^ ^ A j - f l ( 3- l l b) p s ^ ( 0 ^ = nB»(-0, (3.11c) where 7T/ = 1 and 7T& = —1 are the space inversion phase factors for nucleons and pions, respectively, and £ represents either x or p . Notice that both the position and momentum vectors are reversed in Chapter 3. Fundamental Dynamical Variables and Their Properties 10 direction by this transformation. Also, the space inversion phase factors have the property -K) = 1 (3.12a) ir2b = 1. (3.12b) 3 . 3 . 5 Time Reversal To a time reversal transformation of the system there corresponds an antilinear, antiunitary operator T such that s TF^(x)T*=TF £ ^ m ( 0 , 7 r , 0 ) F ^ ( x ) (3.13a) m' = — s s TFI^WT* =TF £ D^m(0,n,0)Fl,^) (3.13b) m' = — s TBuWT* =TbB„{x) (3.13c) 77(x)T+ = . f ( x ) T T + = /*(x) , (3.13d) where / ( x ) is an arbitrary complex valued function and 77 = +1 and T& = — 1 are the time reversal phase factors for nucleons and pions, respectively. See p. 268 of Schweber [6] for a discussion of the determination of these phase factors. Notice that the time reversal phase factors satisfy T) = 1 (3.14a) rl = 1. (3.14b) Note that the time reversal transformation does not affect the position coordinates, but that it reverses the direction of both the momentum and spin vectors. Reversing the direction of spin is equivalent to rotating the observer through 180 degrees about an arbitrary axis. We have chosen this axis to be the y-axis. Thus, the spin rotation has Euler angles (0, IT, 0). Using (A.9), as well as (3.1) and (3.3), we then have T ^ M ( p ) r t = T / f - J - ^ f - p ) (3.15a) TFl^T* = r / ( - ) s - " 1 F t _ m M ( - P ) (3.15b) T B ^ T 1 = TbB^(-p) (3.15c) Tg{p)T* = g*(p)TT^ = g*(p), (3.15d) Chapter 3. Fundamental Dynamical Variables and Their Properties 11 where (^p) is an arbitrary complex valued function. 3.3.6 Charge Conjugation To a charge conjugation transformation there corresponds a linear, unitary operator C such that CF^ip)^ = (3.16a) CFl^(pp = (3.16b) CBM(p)Ct = KbB^(p), (3.16c) where Kf = +1 and nb = +1 are the charge conjugation phase factors for nucleons and pions, respectively. Also, the charge conjugation phase factors satisfy 4 = 1 (3.17a) K\ = 1. (3.17b) 3.3.7 Charge Conjugation and Space Inversion Under the combined action of the operators C and V, the fundamental dynamical variables transform as follows: CVFl^VW = / C / T T / F ^ ^ - P ) (3.18a) CVFi^rW = KfVfFl^i-p) (3.18b) CVB^p^tf = KbirbB^(-p). (3.18c) 3.3.8 Charge Conjugation, Space Inversion and Time Reversal Under the combined action of the operators C,V, and T, the fundamental dynamical variables transform as follows: CVTFl^TtvW = rfnfKf(-)i-mFlm_^p) = ^ / R * - m F + _ r o _ » (3.19a) CVTTl^TivW = T / 7 r / « / ( - ) i - m F l m _ / 1 ( p ) = ¥ > / ( - ) i - ' n F i m _ » (3.19b) CVTB^T^VW = TbWB^{p) = tpbB-pte), (3.19c) Chapter 3. Fundamental Dynamical Variables and Their Properties 12 where tpj = T/nfKf and (pi, = TbitbKi,. Note that the phase factors iff and (pi, satisfy <p) = 1 (3.20a) tpl = 1. (3.20b) Finally, since TJ = — 1, 7Ti, = — 1 and KI, = +1, it follows that </?6 = +1. (3.21) Chapter 4 A Trilinear Nucleon-Antinucleon-Pion Interaction In this chapter, we define a general form for a Hamiltonian, H, describing strongly interacting nucleons, antinucleons and pions. We then impose the requirement that it be invariant under each of the following transformations: isospin rotations, spatial rotations, space inversion (parity), time reversal and charge conjugation. These requirements restrict the form of H. Although we impose the invariance of H under this set of transformations in a particular order (namely the order in which they are listed above), the final results do not depend on this choice 1. This follows from the fact that the operators associated with these transformations commute. 4.1 Defining the Total Hamiltonian, H We define the Hamiltonian, H, for our system of interacting nucleons, antinucleons and pions by H = H0 + XHi, (4.1a) where HQ is the free particle Hamiltonian, and Hi is the interaction Hamiltonian. Specifically, Ho = £ jd3p [eNo(p)Flfl(p)Fmtl(p) + e ^ ^ ^ F ^ (p) +e„0(p)Bj,(p)IV(p)] (4- l b) and HI=HI+HI+HI, (4.1c) 1 Note, however, that choosing a different order will , in general, make obtaining these results more difficult. In particular, charge conjugation invariance should be imposed last. 13 Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 14 where H l = ] T fd3p d3p' / i ™ 3 ( p , P ' ) ^ l M l (p)Fm^2(p')Bfi3(P - P') + adj (4.1d) 777-1 7712 1 / = £ / r f 3 P d V ^ ™ ( p , p O ^ L l M l ( p ) ^ m 2 M 2 ( p ' ) 5 M 3 ( P - p ' ) + ^ (4-le) mi 7712 £ / d ^ y ^ ™ ( P > P O ^ ( 4 . i f ) mi m 2 ^ M1M2M3 Also, £N 0(p) = • v / p 2 c 2 + m ^ o c 4 (4.2a) £jv 0(p) = ^P2c2+rnj^c* (4.2b) e»o(p) = yJp2c2+mloc* (4.2c) are the energies of the elementary nucleon, antinucleon and pion, respectively, and mjv 0: m j v 0 a n a - m , r a r e their rest masses. Note, for reasons to be explained in the next section, we shall assume that the neutron, proton, antineutron and antiproton all have the same rest mass and that the three pions are degenerate in mass as well. As can be seen by examining the energy expressions above, we treat the particles 'semi-relativistically' by including relativistic kinematics. We refer to h^\^3(p,p'), h^\^3(p,p') and hm\m23 (P> P') A S T N E vertex functions for the interaction. The symbol adj is used to denote the adjoint of the previous term in an expression2. Note, by including antiparticles in our discussion, as well as allowing the vertex functions depend on the momenta of both nucleons, H\ represents a generalization of the Hamiltonian considered by Hearn [1]. 4.2 Diagrams of the Interaction Hamiltonian, H\ Each of the three terms in Hi, and hence each vertex function, is associated with a different pair of elementary processes In the diagrams to follow, nucleons are represented by solid lines with upward-pointing arrows, antinucleons by solid lines with downward-pointing arrows, and pions by dashed lines. Note, the direction of the arrows is just a convenient way to distinguish nucleons and antinucleons in the diagrams and does not have any physical implication in this work. 2 B y including such adjoint terms in the definition, Hi is assured to be a Hermitian operator. Note, HQ is also Hermitian, and thus H is & Hermitian operator. Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 15 p,mi , / i i P ,m2,P2 / P - P',A*3 Hi = h£> (P,P') + \ \ P - P , A * 3 \ \ P,77li,/ii Figure 4.1: A diagrammatic representation of the trilinear nucleon-antinucleon-pion interaction ac-counted for by Hi. Solid lines with upward-pointing arrows are nucleons and dashed lines are pions. Figure 4.1 provides a diagrammatic representation of the processes described by the Hi term in the interaction Hamiltonian. The diagram on the left of the figure corresponds to the first term in (4.Id) and the diagram on the right to the adjoint term. Thus, Hi describes, interacting nucleons and pions. Figure 4.2 provides a diagrammatic representation of the process described by the Hi term in the interaction Hamiltonian. The diagram on the left of the figure corresponds to the first term in (4.1e) and the diagram on the right to the adjoint term. Thus, Hi describes interacting antinucleons and pions. Finally, Figure 4.3 provides a diagrammatic representation of the process described by the Hi term in the interaction Hamiltonian. The diagram on the left of the figure corresponds to the first term in (4.1f) and the diagram on the right to the adjoint term. Thus, Hi describes interacting nucleons, antinucleons and pions. 4.3 R e s t r i c t i n g t h e F o r m o f Hx U n d e r lli, 71, V, T a n d C At this point, the vertex functions h^\^{p,p'), fi&™(p,p'), and = < ™ 3 ( p , p ' ) are arbitrary, com-plex valued functions of p\, 113, m\, 7712, p and p ' . By requiring that H be separately invariant under isospin rotations, spatial rotations, space inversion, time reversal and charge conjugation, the form of the vertex functions will be greatly restricted. Note, it is easily shown that Ho is already invariant under all the transformations listed above. In fact, our reason for previously assuming the rest masses of nucleons and antinucleons (and pions) to be degenerate was to ensure this statement. Thus, we need only treat Hi in the following. Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 16 P,77li,//i Hi= h%X3 (P .P ' ) P ,m,2,p2 P , "22, M2 + \ \ P - P , A ^ 3 \ p,"ll , / l l t / Figure 4.2: A diagrammatic representation of the trilinear nucleon-antinucleon-pion interaction ac-counted for by Hi. Solid lines with downward-pointing arrows are antinucleons and dashed lines are pions. Figure 4.3: A diagrammatic representation of the trilinear nucleon-antinucleon-pion interaction ac-counted for by Hi. Solid lines with upward-pointing arrows are nucleons, those with downward-pointing arrows are antinucleons and dashed lines are pions. Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 17 More importantly, the form in which Hi has been expressed in (4.1c) allows us to impose invariance under all of the transformations except charge conjugation on each of Hi, Hi and Hi separately. This is possible because isospin rotations, spatial rotations, space inversion and time reversal do not mix the triplets of creation and annihilation operators associated Hi, Hi and Hi. 4.3.1 Restricting the Form of Hi Under Tli, K, V and T In this section, we restrict Hi by requiring that it be invariant under isospin rotations, spatial rotations, space inversion and time reversal. Although we simply quote the results here, a detailed treatment is given in Appendix D. Imposing Isospin Rotation Invariance on Hi By requiring that Hi be invariant under rotations in isospin space, it is meant that IZiiaB^HiTlliaBj) = Hj (4.3) should be satisfied. The details of this calculation are carried out in Appendix D . l . It is shown there that by requiring Hi to be invariant under isospin rotations, the vertex function hffl\^3(p,p') must be of the form ^ 2 M 3 ( P ' P ' ) = ( i l M 2 M 3 U / i l ) ^ ™ ( P , P ' ) , (4-4) where hmim2(p,p') is, as yet, an arbitrary complex valued function of m i , m.2, p and p'. Notice that all of the Hi, H2, A*3 dependence of h{^\^3 (p, p') resides in the Clebsch-Gordan coefficient ( A 1 fi2 Hz \ \ f-i) • Imposing Spatial Rotation Invariance on Hi At this point, Hi has been restricted to be of the form Hi= Y fd3pd3p' ( U / * 2 A * 3 U / i i ) ^ i m 2 ( p , p 0 ^ l M i ( P ) i r ™ 2 M ( p ' ) ^ , ( p - p ' ) + o 4 l - (4-5) mi 7712 M1M2M3 We now require that Hi be invariant under spatial rotations. That is, we require Tl{aM)HiV)(aM) = Hi (4.6) Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 18 be satisfied. As shown in Appendix D.2, imposing both isospin rotation and spatial rotation invariance on Hi restricts ft^^3 (p, p') to be of the form (P, P') = i ( h 1 M2 H3\l Mi) £ ( - ) - ( ™ + m ' ) ( i j m 2 7 | i mj) (Z Z'm m ' | j - 7 ) im I'm j • Yim(p)Yi<m>(p')hwj(p,p'), (4.7) where hu>j(p,p') is an arbitrary complex valued function of Z, I', j, p and p'. Here, j = 0,1. Note, all the m i m-i dependence is contained in the C-G coefficient, ( \ j iri2 ~f \ I mi). I m p o s i n g Space Invers ion Invariance o n Hi At this point, Hi has been restricted to be of the form Hi= Y, fd3pd3p' ( U w / i s U i « i ) / i m i m 2 ( p , p 0 ^ l M l ( p ) ^ m 2 M ( p ' ) S ; i , ( p - p ' ) + a*'. (4-8) M1M2M3 where /i m im 2 ( p , p ' ) is of the form given in (4.7). We now impose the requirement that Hi be invariant under space inversion. That is, we require VHiV* = Hi be satisfied. As is shown in Appendix D.3, imposing space inversion invariance on Hi implies (4.9) j — 1 is the only acceptable value for j, and I' = l±l. (4.10) Thus, imposing spatial rotation, isospin rotation, and space inversion invariance on Hi restricts the vertex function h^l\^3(p,p') to be of the form K \ ^ 3 (P ,P ' ) = i ( i 1M2 MaU Mi) £ R - ( m + m , ) ( i 1 ™2 7 | * mx) (Z I'mm'\ 1 - 7 ) (mm / '=;±i *Jm(p)*i'm' (P')hw(p,p'), (4.11) where hui(p,p') is, at this point, an arbitrary complex valued function of Z, I' = Z ± 1, p and p'. Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 19 Imposing Time Reversal Invariance on Hi At this point, Hi has been restricted to be of the form H,= £ fd3pd3p' ( i l M 2 / x 3 U M i ) / i m i m 2 ( p , p 0 ^ m i m ( p ) F ™ 2 ^ ( p O B M 3 ( p - p O + a ^ (4-12) where hmim2(p,p') is given by (4.11). We now impose the requirement that Hi be invariant under time reversal; that is, we require that T f f / T 1 = Hi be satisfied. As is show in Appendix D.4, this requirement implies that hu'(p,p') is a real valued function. (4.13) (4.14) Thus, imposing spatial rotation, isospin rotation, space inversion, and time reversal invariance on Hi restricts h ^ \ ^ 3 (p, p') to be of the form / « * (P. P') = i ( i 1 M s l i Mi) £ ( - ) - ( m + m ' } ( i 1 m2 7 | i mi) (I I'm m ' | 1 - 7 ) Imm' l'=l±l • Yim(p)Yi>m> (p')hU'(p,p'), (4.15) where hui(p,p') is an arbitrary real valued function of I, I' = I ± 1, p and p'. 4.3.2 Restricting the Form of Hi Under 1ZT, Tl, V and T By comparing the expressions defining Hi and Hi ((4.Id) and (4.1e), respectively), we see that they are identical in form, except that Hi involves anti-nucleon creators and annihilators. Since anti-nucleons transform the same way as nucleons under isospin rotations, spatial rotations, space inversion and time reversal, h^l1i^3(p,p') is restricted in exactly the same way as h^l1i^3(p,p'). That is, imposing spatial rotation, isospin rotation, space inversion, and time reversal invariance on Hi restricts h^\^3(p,p') to be of the form h£™> (P, P') = i ( i 1H2 A»3| i /*) E R - ( m + m , ) (* 1 m2 7 | J mi) (11'm m'| 1 - 7 ) Imm1 l'=l±l • Ylm(p)YVrn, {p')hw (p, p'). (4.16) where hw (p, p') is an arbitrary real valued function of I, V = I ± 1, p and p'. Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 20 4.3.3 Restricting the Form of Hi Under Ki, Tl, V and T In this section, we restrict Hi by requiring that it be invariant under isospin rotations, spatial rotations, space inversion and time reversal. Although we simply quote the results here, a detailed treatment is given in Appendix E . Imposing Isospin Rotation Invariance on Hi By requiring that Hi be invariant under rotations in isospin space, it is meant that TZi(ap1)Hilz\(aB1)=Hi (4.17) should be satisfied. The details of this calculation are carried out in Appendix E . l . It is shown there that by requiring Hi to be invariant under isospin rotations, the vertex function h^l\^3(p,p') must be of the form h m \ ^ 3 (P, P') = ( H Mi M211 Ms) h m i m 2 (p, p'), (4.18) where hmim2(p,p') is, as yet, an arbitrary complex valued function of m i , m2, p and p'. Notice that all of the pi, p,2, M3 dependence of h^l\^3(p, p') resides in the Clebsch-Gordan coefficient ( A A fix 11M3)-Imposing Spatial Rotation Invariance on Hi At this point, Hi has been restricted to be of the form Hi= £ fd3pd3p' ( H M i M 2 | l M 3 ) ^ ™ i m 2 ( p , p O ^ L l M l ( p ) ^ m 2 M 2 ( p ' ) ^ 3 ( P + P') + « ^ . (4.19) mim2 M1M2M3 We now require that Hi be invariant under spatial rotations. That is, we require •Jl{aj3j)Hj-R)(a/3j) =Hj (4.20) be satisfied. As shown in Appendix E.2, imposing both isospin rotation and spatial rotation invariance on Hi restricts h^l\^3(p,p') to be of the form h%™* (p, p') = i ( H Mi M2| 1M3) £ ( - ) - ( ™ + m ' ) ( H mx m 2 | j -o) (II'mm'| j o) Im I'm' j • Ylm(p)Yi<m> (p')h~u'j(p,p'), (4.21) where hwj{p,p') is an arbitrary complex valued function of /, I', j, p and p'. Here, j = 0,1. Note, all the m i m2 dependence is contained in the C - G coefficient ( | i m i m 2 | j — cr). Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 21 Imposing Space Inversion Invariance on Hj At this point, Hi has been restricted to be of the form H I = £ fd3pd3p' ( H M i M 2 | l ^ ) ^ m 1 ™ 2 ( p , p O ^ L l M l ( p ) ^ 2 M 2 ( p 0 ^ 3 ( P + P O + « ^ (4-22) 777.1 777.1 " mi m 2 /J1M2M3 where / imim 2 (P)P') is °f the form given in (4.21). We now impose the requirement that Hi be invariant under space inversion; that is, we require VHiV* =Hi be satisfied. As is shown in Appendix E.3, imposing space inversion invariance on Hi implies (4.23) j = 1 is the only acceptable value for j, and /' = / ± 1. (4.24) Thus, imposing spatial rotation, isospin rotation, and space inversion invariance on Hi restricts the vertex function h^\^3 (p, p') to be of the form * « s (P, P') = i ( H Mi U2\ 1Ms) £ ( - ) - ( m + ™ ' ) ( H m i m 2 | 1 - a ) (1 J 'm m ' | 1 a) /mm f = i ± l ' ^ W p ^ ' m ' (p')hll'(p,p'), (4.25) where /i«< (p,p') is, at this point, an arbitrary complex valued function of I, V = I ± 1, p and p'. Imposing Time Reversal Invariance on Hi At this point, Hi has been restricted to be of the form H,= £ fd3Pd3P' ( H M l M 2 | l / i 3 ) ^ m i m 2 ( p , p 0 ^ m i M I ( P ) f m 2 M 2 ( P 0 S M 3 ( P + P ' ) + « ^ (4-26) mim 2 J /Jl/J2M3 where /imim 2(p>p') is given by (4.25). We now impose the requirement that Hi be invariant under time reversal; that is, we require THjT* = Hj (4.27) Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 22 be satisfied. As is show in Appendix E.4, this requirement implies that hw(p,p') is a real valued function. (4.28) Thus, imposing spatial rotation, isospin rotation, space inversion, and time reversal invariance on Hj restricts h{fl\^3(p,p') to be of the form K\%?a (P» P') = *' ( H Mi M2I 1 Ms) £ (-)-("*+-') ( i i m i m 2 | i -a) (11' mm'\ la) Imm l'=l±l • Yim(p)YVm> (p')hw(p,p'), (4.29) where hw(p,p') is an arbitrary real valued function of I, I' = I ± 1, p and p'. 4.3.4 Imposing Charge Conjugation Invariance on Hi Having restricted Hi to be invariant under Hi, H, V and T , we now require that it be invariant under charge conjugation. That is, we demand CHitf = Hi (4.30) be satisfied. As is shown in Appendix F , this requirement imposes the following two conditions: (4.31) * « S ( P , P ' ) = - * « , S ( P . P ' ) and hw{p,p') = -hi>i(p',p). (4.32) Thus, we have our final result: imposing spatial rotation, isospin rotation, space inversion, time reversal and charge conjugation invariance on H\ = Hi + Hi + Hi restricts it to be of the form Hi= [ d"P d V hX3(P,P')FL,AP)Frn2,2(p')B^(P ~ P') m\rri2 + adj £ fd3P d V ^ ™ ( P . P ' ) ^ L 1 M 1 ( P ) ^ 2 M 2 ( P ' ) S M 3 ( P - P') rj,i mo J m\m2 + adj + £ /'d3pd3p' n ™ ( P . P 0 ^ L 1 M 1 ( p ) ^ a M ( P ^ , ( P + P 0 + o * ' , 771i 7T),o J m\m,2 (4.33) Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 23 where hm\^3(p,p') is given in (4.15) and h^\^3(p,p') is given in (4.29), with the additional require-ment that hu'(p,p') satisfy (4.32). 4.4 R e s t r i c t i n g t h e F o r m o f Hi U n d e r IZi, 71, T a n d CV In this section, we impose TZj, rotop, T and CV3 invariance on Hi assuming that Hi is already invariant under isospin rotations, IZi, spatial rotations, 71, and time reversal, T . As shown in previous sections, the requirement that Hx be invariant under 71 j, 71 and T implies that the vertex functions are of the form ^ 2 2 M 3 ( p ; p ' ) = ( n / i 2 M 3 U / i i ) ^ m i m 2 ( p ; p ' ) (4.34) fe?2"s(P.P') = ( H / * 2 / * 3 | J / i i ) ^ i m a ( p , p ' ) (4-35) >CAT(P.P') = U i / ' i / d l A * 3 ) £ m - i m 2 ( p , p ' ) , ( 4 - 3 6 ) where V r a J p , p O = ' E H " ( r a + m ' ) ( ^ i ^ 7 l ^ 1 ) ( n ' m m l i - T ) l i r a ( ^ (4.37) Im I'm'j ^ » 1 T O 2 ( P , P ' ) =i £ ( - ) - ( m + m , ) ( U m 2 7 l i m i ) ( H ' m m ' ^ (4-38) Im I'm'j and ^ m i m 2 ( p , p O = * E ( - ) " ( m + m ' ) ( H m i m 2 | j - a ) ( ; / ' m m ' | j V ) y ( m ( p ) ^ (4.39) Note, hu>j(p,p'), hu'j(p,p') and hu>j(p,p') are arbitrary real valued functions of I, I', j, p and p', and j = 0 , l . We now impose the condition that Hi be invariant under the action of CV. That is, we require CVHiV^tf = Hi (4.40) be satisfied. 3 That is, we shall require Hi be invariant under 7Zi,7l,T and the combined action of C and V. Note, this is a weaker condition than for Hi to be invariant under each of these five operators separately. Chapter 4. A Trilinear Nucleon-Antinucleon-Pion Interaction 24 Although we mearly quote the results here, the demonstration is very similar to that given in Ap-pendix G describing the restriction of Hi under CVT• Thus, (4.40) is satisfied if both hold, where hu>j(p,p'), hwj(p,p') and huij(p,p') are otherwise arbitrary real valued functions of I, /', j = 0,1,p and p'. 4.5 R e s t r i c t i n g the F o r m o f Hi U n d e r TZj, 1Z and CVT In this section, we impose TZi, TZ and CVT^ invariance on Hi assuming that Hi is already invariant under isospin rotations, TZj, and spatial rotations, TZ. That is, we require hwj&p1) = hWj(p,p') (4.41) and hwj(p,p') = -hi>tj(p',p) (4.42) CVTHiT^Vttf = Hi (4.43) be satisfied. As is shown in Appendix G , (4.43) is satisfied if both hwMp') = (-) , + I'^«V i(p,p') ti = 0,1) (4.44) and hwj(p,p') = -h*vlj(p',p) (j = 0,1) (4.45) hold. 4 That is, we shall require Hi be invariant under TZi, TZ and the combined action of C, V and T . Note, this is a much weaker condition than for Hi to be invariant under each of these five operators separately. Part II Electrons, Positrons and Phot 25 Chapter 5 Introduction This part of the thesis in concerned with defining a Hamiltonian describing interacting electrons, positrons and photons. In Chapter 6, we define the fundamental dynamical variables of our system, provide their commutation (or anticommutation relations), and finally give their transformation prop-erties. Then, in chapter 7, we define a general form for a Hamiltonian, H, describing interacting electrons, positrons and photons. We then impose the requirement that it be invariant under each of the fol-lowing transformations: spatial rotations, space inversion, time reversal and charge conjugation. These requirements restrict the form of Ti. 26 Chapter 6 Fundamental Dynamical Variables and Their Properties In this chapter, we define the fundamental dynamical variables of a system of interacting photons (spin 1), electrons (spin 1/2), and positrons (spin 1/2), provide their commutation (or anticommutation relations), and finally give their transformation properties. 6.1 Fundamental Dynamical Variables Our system of interacting photons, electrons and positrons are described by a Hilbert space which is a direct product of boson Fock space, fermion Fock space and antifermion Fock space. We choose the fundamental dynamical variables of this system to be the elementary particle creation and annihilation operators. 6.1.1 Electron Creators and Annihilators Fm(x) (m = ± i ) , when operating on the vacuum state, 10), creates a one-electron ket corresponding to an elementary electron at position x with z-axis spin projection m. The adjoint operator, Fm(x), when operating on this one-electron ket, returns the vacuum state. Equivalently, we can define the momentum space electron creators and annihilators, .Fm(p) and Fm(p). These create or destroy an elementary electron with momentum p and z-axis spin projection m. The two creators are related by a Fourier transform: F ™ ( p ) = ( d j v * / d 3 x e i p , x / B ^ w - ^ We define the operation of Fm(x) and Fm(p) on the vacuum state as follows: Fm(x)|0) =Fm(p)|0) = 0 . (6.2a) That is, Fm(x) and Fm(p) annihilate the vacuum. Note, by taking the adjoint of the above expression, we also have (0 |F t , ( x ) = < 0 | F i ( p ) = 0. (6.2b) 27 Chapter 6. Fundamental Dynamical Variables and Their Properties 28 6.1.2 Positron Creators and Annihilators The discussion for positrons is identical to that given above for electrons, except that the creation and annihilation operators for positrons are denoted by Fm(£) and Fm(£), respectively, where £ represents either x or p. 6.1.3 Photon Creators and Annihilators 35^ (x) (A = ± 1 ) , when acting on the vacuum state, |0) , creates a one-photon ket corresponding to an elementary photon at position x with helicity 1 A. The operators ®A(x)> ^ I ( P ) a n ( 1 ^A(P) a r e defined analogously to the electron case. The photon position and momentum creators are also related by a Fourier transform: Note, we do not need to introduce a new symbol to describe antiphotons because the photon is its own antiparticle. 6.2 Commutation Relations The fundamental dynamical variables satisfy the following commutation and anticommutation relations: (6.3) (6.4a) (6.4b) (6.4c) (6.4d) {F,F} = 0 (6.4e) (6.4f) [ B A ( « , S V ( € ' ) ] = [ s t ( 0 , s t , ( O ] = 0 [%{(t),Fm.(&] = [v{(t),FUt'j\=0 (6.4h) (6-4g) (6.4i) See Appendix (H) for a discussion of the helicity formalism. Chapter 6. Fundamental Dynamical Variables and Their Properties 29 where [ ] denotes a commutator, { } an anticommutator, £ represents either x or p, F = Fm(£) or F^te) a n d F = F m - ( £ ' ) or F m < ( £ ' ) -6.3 Transformation Properties Here we discuss the effect of spatial displacements and rotations, space inversion, time reversal and charge conjugation on the particle creation and annihilation operators. The results of this section, which are a consequence of imposing general Poincare invariance on our physical system [4], shall be used to determine the form which invariant particle interaction Hamiltonians describing electrons, positrons and photons must have. Note, only the transformation equations for FA(£) , F^te), and are given since the equations for Fm(£), Fm(£), and are easily obtained by Hermitian conjugation. 6.3.1 Spatial Translations To a spatial displacement of the system by an amount a, there corresponds a linear, unitary operator V (a) such that © ( a ) i ^ ( x ) 2 ? t ( a ) = + *) (6-5a) P (a )FL (x )P t (a )=F | „ (x + a) (6.5b) 2?(a)^(x)yD+(a)=S+(x + a). (6.5c) Thus, displaced creation operators create particles at position x + a rather than at x. If we use (6.1) and (6.3), we then have P(a)y7t l(p)pt ( a) = F t , ( p ) e - « - p / « (6.6a) X)(a)FL(p)2?t(a) = F ^ e - ^ (6.6b) 2>(a)2A(p)Z?t(a) = S A (p)e i " p / \ (6.6c) Chapter 6. Fundamental Dynamical Variables and Their Properties 30 6.3.2 Spat ia l Rotat ions To a spatial rotation of the system through Euler angles (a, ,$,7), there corresponds a linear, unitary operator H(aB^) such that s ft(a07)i*(x)ftt(a07) = £ D.m,m(aPl)Fl(x-n) (6.7a) m' = — s s KiafrpFlWKHafr) = £ £ > ^ m ( a / 3 7 ) ^ , ( x _ R ) (6.7b) m' = — s tt(a/37)BA(x)ftt(a07) = B A ( x _ R ) , (6.7c) where we have followed the conventions for Euler angles and rotation matrices used in Rose [5]. For a discussion of the rotation matrices, see appendix (A). Note that x R = M x (6.8a) x _ R = M _ 1 x , (6.8b) where x is a column vector and M is the matrix defined on p. 65 of Rose [5]. Using (6.1) and (6.3), we have s K(aPi)Flb)tf{aB>1)= £ ^ , m ( a ^ 7 ) F ^ ( p _ R ) (6.9a) m' — ~s s K{ah)F~l{v)'R>{aB1)= £ ^ m ( a / 5 7 ) ^ ( p - R ) (6.9b) m' = — s 7Y(a /? 7 )£ A (p)VJ (a/9 7) = £ A (p_ R) . (6.9c) Thus, both momentum and spatial coordinates, as well as the direction of spin of the nucleons, are rotated. However, the helicity of the photon is unaffected by spatial rotations. 6.3.3 Space Inversion To a space inversion transformation, there corresponds a linear, unitary operator V such that PFlMP* = *eFi(-£) (6.10a) VFli(T*)=ireFl(-S) (6.10b) VBxi&V* = ( - ) 1 T T P S a ( - £ ) , (6.10c) Chapter 6. Fundamental Dynamical Variables and Their Properties 31 where 7re = 1 and TTP = — 1 are the space inversion phase factors for electrons and photons, respectively, and £ represents either x or p. Notice that both the position and momentum vectors are reversed in direction by this transformation. Also, the space inversion phase factors have the property 7re2 = 1 (6.11a) •n* = 1. (6.11b) 6.3.4 Time Reversal To a time reversal transformation of the system there corresponds an antilinear, antiunitary operator T such that TFm(p)T^ =Te(-)^mFLm(-p) (6.12a) T F m ( P ) T + = r e ( - ) i - m F f _ m ( - P ) (6.12b) T S A ( p ) T t = r p ( - ) 1 S A ( - p ) (6.12c) Tg(p)T* = <?*(p)TTt = g*(p), (6.12d) where g(p) is an arbitrary complex valued function. r e and TP = +1 are the time reversal phase factors for electrons and photons, respectively. See p. 268 of Schweber [6] for a discussion of the determination of these phase factors. Notice that the time reversal phase factors satisfy rl = 1 (6.13a) r2p = 1. (6.13b) 6.3.5 Charge Conjugation To a charge conjugation transformation there corresponds a linear, unitary operator C such that CFt,(p)Ct = KeFl(p) (6.14a) CFl(pp = KeFl(p) (6.14b) CB A(p)C+ = K P S A ( P ) , (6.14C) Chapter 6. Fundamental Dynamical Variables and Their Properties 32 where Kf and KP = — 1 are the charge conjugation phase factors for electrons and photons, respectively. Also, the charge conjugation phase factors satisfy K} = 1 (6.15a) K\ = 1. (6.15b) 6.3.6 Cha rge C o n j u g a t i o n , Space Invers ion and T i m e R e v e r s a l Under the combined action of the operators C, V, and 7", the fundamental dynamical variables transform as follows: CVTFl{v)T^tf = r e 7 r e K e ( - ) * - m F L m ( P ) = (-)^mFlm(p) (6.16a) CVTFm(p)T*VW = r e 7 r e K e ( - ) 5 - ™ F l m ( p ) = ( - ) * - m F l m ( p ) (6.16b) CVT'Bx(p)T^C^ = TPKPKp3-X(P) = 3 _ A ( p ) . (6.16c) Chapter 7 A Trilinear Electron-Positron-Photon Interaction In this chapter, we define a general form for a Hamiltonian, Ti, describing interacting electrons, positrons and photons. We then impose the requirement that it be invariant under each of the following trans-formations: spatial rotations, space inversion (parity), time reversal and charge conjugation. These requirements restrict the form of Ti. Although we impose the invariance of Ti under this set of transfor-mations in a particular order (namely the order in which they are listed above), as in Part 1, the final results do not depend on this choice. This follows from the fact that the operators associated with these transformations commute. 7.1 Denning the Total Hamiltonian, Ti We define the Hamiltonian, Ti, for our system of interacting electrons, positrons and photons by where Tio is the free particle Hamiltonian, and Ti\ is the interaction Hamiltonian. Specifically, Ti — Tio + XTi\, (7.1a) Tio (7.1b) and Tii ^Tii+Tii + Ti!, (7.1c) where (7.1d) (7.1e) (7.1f) 3 3 Chapter 7. A Trilinear Electron-Positron-Photon Interaction 34 Also, £e0(p) £e 0 (p ) (7.2b) (7.2a) e 7(p) (7.2c) are the energies of the elementary electron, positron and photon, respectively, where meo and mj , are the rest masses of the electron and positron, respectively. Note, the electron and the positron have the same rest mass. As can be seen by examining the energy expressions above, we treat the particles 'semi-relativistically' by including relativistic kinematics. We refer to hmim2\(p, P')J ^mim 2 A(p ! P ' ) and hmim2x(Pi P') as the vertex functions for the interaction. 7.2 Diagrams of the Interaction Hamiltonian, Hi Each of the three terms in Hi, and hence each vertex function, is associated with a different pair of elementary processes In the diagrams to follow, electrons are represented by solid lines with upward-pointing arrows, positrons by solid lines with downward-pointing arrows, and photons by wavy lines. Note, the direction of the arrows is just a convenient way to distinguish electrons and positrons in the diagrams and does not have any physical implication in this work. Figure 7.1 provides a diagrammatic representation of the processes described by the Hj term in the interaction Hamiltonian. The diagram on the left of the figure corresponds to the first term in (7.Id) and the diagram on the right to the adjoint term. Thus, Hi describes interacting electrons and photons. Figure 7.2 provides a diagrammatic representation of the process described by the Hi term in the interaction Hamiltonian. The diagram on the left of the figure corresponds to the first term in (7.1e) and the diagram on the right to the adjoint term. Thus, Hi describes interacting positrons and photons. Finally, Figure 7.3 provides a diagrammatic representation of the process described by the Hi term in the interaction Hamiltonian. The diagram on the left of the figure corresponds to the first term in (7.1f) and the diagram on the right to the adjoint term. Thus, Hi describes interacting electrons, positrons and photons. Chapter 7. A Trilinear Electron-Positron-Photon Interaction 35 Figure 7.1: A diagrammatic representation of the trilinear electron-positron-photon interaction ac-counted for by Tii. Solid lines with upward-pointing arrows are electrons and wavy lines are photons. Figure 7.2: A diagrammatic representation of the trilinear electron-positron-photon interaction ac-counted for by Hi. Solid lines with downward-pointing arrows are positron and wavy lines are photons. Chapter 7. A Trilinear Electron-Positron-Photon Interaction 36 Ui = h, P',m2 '777,17712 Figure 7.3: A diagrammatic representation of the trilinear electron-positron-photon interaction ac-counted for by Hi- Solid lines with upward-pointing arrows are electrons, those with downward-pointing arrows are positrons and wavy lines are photons. 7.3 Restricting the Form of Hi Under Tl, V, T and C At this point, the vertex functions /im i77j.2A(p,p')7 ^mim2A(p>p')> a n d hmim2\(p, p') are arbitrary, com-plex valued functions of m i , m2, A, p and p ' . By requiring that H be separately invariant under spatial rotations, space inversion, time reversal and charge conjugation, the form of the vertex functions will be greatly restricted. Note, it is easily shown that Ho is already invariant under all the transformations listed above. Thus, we need only treat H\ in the following. More importantly, the form in which H\ has been expressed in (7.1c) allows us to impose invariance under all of the transformations except charge conjugation on each of Hi, Hi and Hi separately. This is possible because spatial rotations, space inversion and time reversal do not mix the triplets of creation and annihilation operators associated Hi, Hi and Hi-7.3.1 Restricting the Form of Hi Under Tl, V and T In this section, we restrict Hi by requiring that it be invariant under spatial rotations, space inversion and time reversal. Although we simply quote the results here, a detailed treatment is given in Appendix I. Chapter 7. A Trilinear Electron-Positron-Photon Interaction 37 Imposing Spatial Rotation Invariance on Hi We require that Hi be invariant under spatial rotations. That is, we require 1l(aM)HiVJ(aB7) = Hi (7.3) be satisfied. As shown in Appendix 1.1, spatial rotation invariance on Hi restricts hmim2\(p,p') to be of the form hmim2x(p, P ' ) = » E ( - ) - ( m + m , ) ( i j m2 T| i mx) (ll'mm'\ j -7) Im I'm'j • Yim(p)Yi-m> (p')hU'jx(p,p'), (7.4) where hwj\(p,p') is an arbitrary complex valued function of Z, I', j, A, p and p'. Here, j = 0,1. Note, all the m i m 2 dependence is contained in the C-G coefficient, ( 1 j m 2 7 | | m i ) . Imposing Space Inversion Invariance on Hi At this point, Hi has been restricted to be of the form H l = JI y ' d 3 p d V ^ i m 2 A ( p , p O ^ , 1 ( P ) ^ m » ( P , ) B A ( p - p ' ) + « & ' . (7-5) mi 7U2 X where hmim2^(p,p') is of the form given in (7.4). We now impose the requirement that Hi be invariant under space inversion. That is, we require VUiV* = Hi be satisfied. As is shown in Appendix 1.2, imposing space inversion invariance on Hi implies (7.6) hwjx{p,p') = {-)l+l'hWj-x(p,p'). (7.7) Thus, imposing spatial rotation and space inversion invariance on Hi restricts the vertex function hmim2\(p, p') to be of the form hmim2x(p,P') = » E ( - ) - ( m + m , ) ( i jm2 7 | J mi) (ll'mm'\ j -7) Ylm(p)Yj'm' (p')hw j\(p,p'), Im I'm'j (7.8) where /I/CJA(P,P') satisfies (7.7) but is otherwise an arbitrary complex valued function of I, I', j = 0,1, A, p and p'. Chapter 7. A Trilinear Electron-Positron-Photon Interaction 38 Imposing Time Reversal Invariance on Hi At this point, Hi has been restricted to be of the form Hl= £ y"d 3 pdV/« m i m 2 A(p,P^ l (p) i r m a (p ' )SA(p-p')+0*', (7-9) mi77l2A where hmim2x(p,p') is given by (7.8). We now impose the requirement that Hi be invariant under time reversal; that is, we require that THiT] =Hi be satisfied. As is show in Appendix 1.3, this requirement implies that hwj\(p,p') is a real valued function. (7.10) (7.11) Thus, by imposing spatial rotation, space inversion, and time reversal invariance on Hi, we have restricted hmim2\(p,p') to be of the form Ylm(p)Yl'm' (P')hwjx(p,p'), Im (7.12) where hu>j\(p,p') satisfies (7.7), but is otherwise an arbitrary real valued function of /, j = 0,1, A, p and p'. 7.3.2 Restricting the Form of Hi Under 71, V and T By comparing the expressions defining Hi and Hi ((7.Id) and (7.1e), respectively), we see that they are identical in form, except that Hi involves positron creators and annihilators. Since positrons transform the same way as electrons under spatial rotations, space inversion and time reversal, h m \ ^ 3 (p, p') is restricted in exactly the same way as hmxm2\(p,p'). That is, imposing spatial rotation, space inversion, and time reversal invariance on Hi restricts ^mim2A(p,p') to be of the form hmim2x(p,p') - i £ (-)-(m+m'> (j jm27| J m i ) (ll'mm'\j -7) Yim (p) Yv m , (p' jhw j\{p,p'), Im I'm'j (7.13) where hwjx{p,p') satisfies hu>jx(p,p') = (—hwj-x(p,p'), but is otherwise an arbitrary real valued function of /, / ', j = 0,1, A, p and p'. Chapter 7. A Trilinear Electron-Positron-Photon Interaction 39 7 .3 .3 R e s t r i c t i n g the F o r m of Tii U n d e r TZ, V and T In this section, we restrict Tii by requiring that it be invariant under spatial rotations, space inversion and time reversal. Although we simply quote the results here, a detailed treatment is given in Appendix J . Impos ing Spa t i a l R o t a t i o n Invariance on Tii We require that Tii be invariant under spatial rotations. That is, we require K(aBi)HJ'Ri(aB'y)=Hi (7.14) be satisfied. As shown in Appendix J . l , imposing spatial rotation invariance on Tii restricts the vertex function hmim2x(p,p') to be of the form hmimax(p, p ') = i E ( - ) - ( m + m , ) ( H m i rn2\j-l) {I I'm m'\j 7) hwjx{p,p')Ylm{p)Yi'm' (p'), Im V m' j (7.15) where h,wj\(p,p') is an arbitrary complex valued function of /, I', j, A, p and p'. Here, j = 0,1. Note, all the mi m2 dependence is contained in the Clebsch-Gordan coefficient ( i i m i m2\ j —7). I m p o s i n g Space Invers ion Invariance o n Tii At this point, Tij has been restricted to be of the form Ki= £ y ' d 3 p d V ^ i m 2 A ( p , p 0 ^ m 1 ( p ) ^ , a ( P , ) B A ( p + p') + o*' , (7-16) mimzX where hmim2\(p, p') is of the form given in (7.15). We now impose the requirement that Tii be invariant under space inversion. That is, we require VMiV^ = Tii (7.17) be satisfied. As is shown in Appendix J .2 , imposing space inversion invariance on Tii implies hwjx(p,p') = (-)l+l'hWj-x(p,p'). (7.18) Chapter 7. A Trilinear Electron-Positron-Photon Interaction 40 Thus, imposing spatial rotation and space inversion invariance on Tii restricts the vertex function hmim2x{p, P') t ° be of the form hmim2x(p, p') = * £ ( - ) - ( « ' ) ( i i m x m 2 | j -7) (/ /' mm'\j 7) ^m(p)^'m ' (p')^H ' jA(P,p'), Im I'm' j (7.19) where hwj\{jp,p') satisfies (7.18), but is otherwise an arbitrary complex valued function of I, j = 0,1, A, p and p'. Imposing Time Reversal Invariance on Tii At this point, Tii has been restricted to be of the form ni= £ j rf 3 p d V ftm.m^lp,?')^L, (P)4 ( P ' ) ^ ( P + p') + adj , (7.20) ; m\m2X  1 where hmim2\(p, p') is given by (7.19). We now impose the requirement that Tii be invariant under time reversal. That is, we require TTiiT* = Tii be satisfied. As is show in Appendix J.3, this requirement implies that hii'jx(p,p') = hu,jX(p,p'). (7.21) (7.22) Thus, imposing spatial rotation, space inversion and time reversal invariance on Tii restricts the vertex function hmim2\(p, p') to be of the form hmim2x(p,p') = i £ (-)-(m + m'> ( H " » i m 2 | j -l) (U'mm'ljj) • Ylm(p)Yl'm'(p')h[l>jx(p,p'), Im I'm'3 (7.23) where hu'jx(p,p') satisfies (7.18) but is otherwise an arbitrary real valued function of I, I', j = 0,1, p and p'. 7.3.4 Imposing Charge Conjugation Invariance on Tii Having restricted Tii to be invariant under TZ, V and T , we now require that it be invariant under charge conjugation. That is, we demand CHitf = Tii (7.24) Chapter 7. A Trilinear Electron-Positron-Photon Interaction 41 be satisf ied. A s is shown in A p p e n d i x K, th is requirement imposes the fo l lowing two condit ions: ^ m i m 2 A ( P > P ' ) = - ^ m 1 m 2 A ( P > P ' ) and hivjx{p,p') = W ( P ' , P ) (7.25) (7.26) Thus , we have our f inal result: impos ing spat ia l ro ta t ion , space inversion, t ime reversal and charge conjugat ion invar iance on H i — Hi + Hi +Hi restr icts i t to be of the form H l = £ Jd3Pd3P' / l m i m 2 A ( p , p ' ) ^ l l ( P ) ^ m 2 ( p ' ) ^ A ( p - p ' ) m i 7712A + adj - £ Jd3pd3P' / l m i m 2 A ( p , p ' ) ^ m i ( P ) ^ m 2 ( p ' ) ^ 3 ( p - p ' ) 7711 7712 A + adj + £ Jd3Pd3P' / l m i m 2 A ( p , P ' ) F L 1 ( p ) i l 2 ( p ' ) 3 a ( p + p ' ) + 0 ^ , 77 l i 7712 A (7.27) where hmim2\(p,p') is given in (7.12) and hmim2\(p,p') is given in ( J . 2 0 ) , w i t h the add i t iona l require-ment that hu'j\(p,p') satisfy (7.26). 7.4 Res t r ic t ing the F o r m of Hi U n d e r Tl a n d CVT In this sect ion, we impose CVT1 invariance on Hi assuming that Hi is a lready invar iant under spat ia l rotat ions, Tl. T h a t is, we require CVTHiT^V^C^ = H i (7.28) be satisfied. A l t h o u g h we s imp ly quote the result here, the demonstrat ion is completely analogous to that given in A p p e n d i x G , where CVT invariance is imposed upon Hi. Thus , (7.28) is satisf ied i f bo th hwjxM) = (-Y+l'+1hWj_x(p,p') (7.29) x T h a t is , we shal l require Hi be invar iant under the combined act ion of C, V and T. Note that this is a much weaker cond i t ion than for Hi to be invar iant under each of these three operators separately. Chapter 7. A Trilinear Electron-Positron-Photon Interaction and hwjx(p,p') = (-)%VA(P' ,P) 0' = 0,1) hold. Chapter 8 Summary and Conclusions In the first part of this thesis, we have defined a generalized version of the Hamiltonian considered by Hearn [1] describing interacting nucleons, antinucleons and pions. The allowed form of this new Hamiltonian is then restricted by requiring that it be invariant under spatial translations and rotations, space inversion, time reversal and charge conjugation. In the last two sections of chapter 4, we also treat two special cases: 1. How is Hi restricted if we require that it be invariant under isospin rotations, spatial rotations, time reversal and the combined action of space inversion and charge conjugation. 2. How is Hi restricted if we require that it be invariant under isospin rotations, spatial rotations and the combined action of space inversion, time reversal and charge conjugation. The Hamiltonians thereby obtained could facilitate including CP violating terms in a theory which seeks to describe the decay of 73-mesons. In the second part of this thesis, we define a new Hamiltonian analogous to that developed in Part 1, but which describes interacting electrons, positrons and photons. The allowed form of this Hamiltonian is then restricted by requiring that it should remain invariant under spatial translations and rotations, space inversion, time reversal and charge conjugation. This thesis represents the first few steps in generalizing the work of Hearn[l]. The next step will be to apply the dressing transformation to the Hamiltonians derived here and then, using suitably chosen vertex functions, calculate various processes (e.g. lifetime of positronium). Future research in this area might include examining the relationship between the electron-positron-photon Hamiltonian developed in Part 4.5, and the Hamiltonian of standard Q E D . Also, the techniques developed here could be extended to systems of interacting quarks, antiquarks and gluons. This could provide a complementary descriptions of quantum chromodynamics, although several new terms would need to be included in such a Hamiltonian. 43 Part III Appendices 44 Appendix A The Rotation Matrices In this appendix we first define the rotat ion matrices, £>;Lm(a/? 7). We then go on to list those relations invo lv ing the rotat ion matr ices which are used throughout this thesis. A . l Definition of Z?^,ro(a/37) and d^ J m(/3) Cons ider the basis of states w i th a given to ta l angular momen tum j and z-axis spin project ion m. A n element of this basis is denoted by \jm). The m a t r i x representation of the rotat ion operator, 7£(a/?7) = e~iaJ* e~l/3Jy e~nJ', in this basis is referred to as the rotat ion mat r i x and its elements are denoted D^^aBj). Specif ical ly, we have £>m<n>/?7) = (jm'\n(aB7)\jm) = {jm'\e-iaJ'e-ipj"e-i'lJ'\jm) ( A . l ) = e-im'aUm'\e-i0Jy\jm)e-im'r ^e-im'adi,m(B)e-im\ In the th i r d equal ity above, we have used the fact that | jm) is an eigenstate of Jz w i t h eigenvalue ra. Also, we have int roduced the def in it ion d^M^Um'le-^lJm). (A.2) A .2 Useful Relations for the Dm,m(aBj) and d3m,m{B) F rom p. 54 of Rose [5], we have DZmWl) = (-r'-mDim,,_m(aB~f). (A.3) T h e rule for combin ing two rotat ion matrices is given on p. 58 of Rose [5] to be Dm\mSa^)D^2m2(a^) = Y.^hm'lm'2\jm') (j tj2 m i m 2 | jm) D>,m(a/3 7), (A.4) 3 45 Appendix A. The Rotation Matrices 46 where m = m\ + m2, m' = m\ + m'2, (ji 32 ^ m 2 1 j ; m ) and ( j i j 2 m 2 | j m ' ) are Clebsch-Gordan coefficients, and the sum is over all values of j for which A(ji,j2,j) is satisfied. For the definition of A(ji,J2,j), see (B.13). See Appendix B for a discussion of Clebsch-Gordan coefficients. From p. 895 of Goldberger and Watson [7], we have dQ0(/3) = P/(cos/?) for I an integer, (A.5) where P; is a Legendre polynomial of the first kind. For the special case that I — 0, we then have doOoG0)=Po(cos/?) = l . (A.6) From (A.6) and (A . l ) it follows that D 0 ° 0 (a /? 7 ) = 1. (A.7) When Q = 7 = 0, then ^ ' m ( 0 , / M ) = d>, m 03) . (A.8) For the special case /? = 7r, we have from p. 885 of Goldberger and Watson [7] that dLmW - (-)j-mSm;-m. (A.9) From p. 54 of Rose [5], we have dlmV) = (-)m'-mdLm,_m(9). (A.10) Finally, from p. 884 of Goldberger and Watson [7], we have ^ ™ 0 ' ) = (-)J+,B'C.-m(*-«- ( A - l l ) Appendix B Addition of Angular Momenta In this appendix, we shall first discuss the general problem of adding two angular momenta. This discussion will lead us to define the so called Clebsch-Gordan (denoted C - G for short) coefficients. Finally, we list those relations involving C - G coefficients which are used in this thesis. B . l Adding Two Angular Momenta Consider a two component system in which each component has its own angular momentum degree of freedom. For definiteness, we refer to the two components as particle 1 and particle 2, and shall denote their angular momentum operators as J i and J2, respectively 1. Though it would be more mathematically precise to write the total angular momentum operator for the composite system as J = J i ® I +1 ® J2, such notation is cumbersome in practice. Hence, we simply write J = J i + J2 and keep in mind that J i and J2 operate on different spaces. The simultaneous2 eigenvectors of J i 2 and J\z are given by J i 2 | jimx) = 3\{j\ + l)h2\jimi) , (B. l ) Jiz\hmi) - rmhljimx) , m i = j\, jx - 1,... , (B.2) Similarly, the simultaneous eigenvectors of J 2 2 and J2z are given by J22|32m2) = j2(j2 + l ) f i 2 | J2m2) , (B.3) J2z\J2m2) = m2h\j2m2) , m2 = j2,j2 - 1,... ,-j2. (BA) According to Ballentine [9], it is possible to form a basis for the composite system by taking all binary products of vectors from both single particle spaces; that is, the following (2ji + 1) • (2j2 + 1) vectors 1 Although we refer to the components as separate particles in this discussion, it is quite possible for the components to relate to the orbital and spin degrees of freedom of the same particle. 2 A s shown on p. 42 of the book by Shankar [8], since J i 2 and J\z are two commuting Hermitian operators, there exists (at least) a basis of common eigenvectors in which both operators are diagonal. 47 Appendix B. Addition of Angular Momenta 48 form a basis for the composite system: \j1j2mun2) = | i i m i ) <g> \j2m2) mi = j i , j i - 1 , . . . , m 2 = j2,j2 - 1 , . . . ,-j2. (B.5) This basis is referred to as the product basis. It is sometimes preferable3, however, to work in the simultaneous eigenbasis of the mutually com-muting operators J i 2 , J2 2, J 2 and Jz (where Jz = J\z + J2z): J i 2 | J i J 2 j w ) = + l)h2\jiJ2Jm) (B.6) J22\jiJ2Jm) - j2{J2 + l)ti2\jiJ2Jm) (B.7) J2\ji32jm) = j(j + l)h2\jij2jm) - j2\ <j< 31 + 32) (B.8) Jz\hhjm) = mh\jij2jm) m = j,j - 1 , . . . ,-j. (B.9) This basis is referred to as the total-j basis. B.2 Definition of Clebsch-Gordan Coefficients Since both the product basis and the total-j basis are complete, it is possible to express the total-j kets as \j1j2jm) - £ \j1j2m1m2) (jiJ2mim2\jiJ2Jm). (B.10) m i m - 2 The coefficients of this expansion, (jiJ2iTiim2\jiJ2J'm,), are called the Clebsch-Gordan coefficients (or C-G coefficients for short), and they shall be denoted (ji 32 mi m2\j m) = (jiJ2mim2\jiJ2Jm). ( B - H ) The C-G coefficients have the following properties: (jihmim2\jm)=0 unless A(ji,j2,j), (B.12) where by A(ji, j2, j) it is meant that ji, 32, and jz satisfy v \jl-J2\<j<jl+J2- (B.13) 3See p. 417 of the book by Shankar [8] for examples in which it is preferable to use either the | jiJ2Jm) basis or the | j i J 2 ^ i ^ 2 ) basis. Appendix B. Addition of Angular Momenta 49 When equation (B.13) is satisfied, we say that ji, j2, and j3 form a triangle. (ji 32 m\ m2| j m) = 0 unless m = m\ + m2. (ji 32 mi m2\j m) € IR (conventional). (3i 32 ji j - ji is positive (conventional). B.3 Relations Involving Clebsch-Gordan Coefficients The following symmetry relations were taken from pages 38-39 of Rose [5]. (J1J2WI1 m2\j3m3) = _yi+n n (j1 j2 _ m j -m2\j3 _m3) _yi+ji-n (j2jx m 2 mi\j3 m3) mi\l'i^~{3\hrni-mz\j2-m2) y2+m2 , / 2 J 3 + 1 0 . , T (h 32 - m 3 m21 j i -mi ) 2j i + 1 •)* " m i ^ | ^ ( J 3 j i m 3 - m i I j 2 m 2) ,J2 + m 2 h _ h ± l ( . , {32 h ~m2 m3\j\ mi). V2^' I + 1 From (B.17a), (B.17b), and (B.17d), it follows that Ui h m i m 2 | j3 m 3 ) = (-)**>+V»-*+™*^l±i ( j 2 j 3 _ m 2 m 3 | j, mi) From (B.17f), (B.17b), and (B.17e), if follows that (-r 3"m i(|l"»2 - m 3 | i - m i ) = ( - ) i y | ( H " » i m 2 | l m 3 ) . Also, from p. 42 of Rose [5], we have (ji Omi 0| j3 m 3 ) = 5jli38mim3. Using (B.17c) in (B.20) and relabeling indices yields ( _ ) J 2 + m 2 (3i 32 m i m 2 | 00) = -j===6jlJ2Smi-m2. (B.14) (B.15) (B.16) (B.17a) (B.17b) (B.17c) (B.17d) (B.17e) (B.17f) (B.18) (B.19) (B.20) (B.21) Appendix C Properties of Spherical Harmonics In this appendix are gathered those properties of spherical harmonics which are used throughout this thesis. The spherical harmonics are to be denoted by Yim{p) = Y,m{9,ip), ( C l ) where 6 and ip are the polar and azimuthal angles, respectively, of the unit vector p. From p. 61 in Rose [5], we have Ym(p) = (-ry(,-m(P). (C.2) To obtain the next result, we shall use the relation between spherical harmonics and rotation matrices, as given on p. 60 of Rose [5]: A-rr D U v M ^ d y ^ Y ^ i p ) . (C.3) Using (C.3), ( A . l ) , and ( A . l l ) , gives I A-TT = \f^e~imlv+*~"H-y+mdu* - 9) = { w T l H 2 m + ' e - i m ( * + ^ o ( - - 9) = H ' V / 2 T T I I ) - o ( 7 r + n ~ d>0 ) = (-),YZn(*-0,* + <p) = ( - ) ' ^ m ( - p ) , (C.4) 50 Appendix C. Properties of Spherical Harmonics 51 which, upon complex conjugation and rearrangement yields Yim(-P) = (-)lYim(p). (C.5) From p. 60 in Rose [5], we see that under rotations, the spherical harmonics transform according to Ylm(pR) = £ D i . , m ( a / 3 7 ) t f m < (p). (C.6) m' Finally, from p. 75 of Rose [5], we see that the spherical harmonics satisfy the following orthogonality relationship: / Ylm{p)Yi*m' ( p ) = r ^ r d e Yi™(9> ^ y ^ ' & =Smm'6"' • ( c j ) J Jo Jo A p p e n d i x D R e s t r i c t i n g the fo rm o f Hj In this appendix, we impose the requirement that Hi, as defined in (4.Id), be invariant under the following set of transformations: isospin rotations, spatial rotations, space inversion (parity) and time reversal. These requirements restrict the form of Hi. D . l Impos ing Isospin R o t a t i o n Invariance o n Hi As given in (4.Id), Hi is initially of the form H,= £ f d3pd3p' h£™>(P,P')Fmiltl {p)Fm(p')BM(P - p') + adj. (D.l) 77117712 ^ By requiring that Hi be invariant under rotations in isospin space, it is meant that IZiiaB^HiKliaB-y) = Hi (D.2) should be satisfied. The condition given by (D.2) then leads to a restriction on the allowed form of the vertex function, h}^\^3(p,p'). To this end we now calculate 1li(aB'y)HiTl\(aB'y). Using (3.10a) and (3.10c), we have TliiaP^HiVJjia^) = £ f d3p d3p' hm\^{p,V,)D\^{aM)D^2{aB1) • 7 7 7 1 7 7 1 2 ^ Ml M'lM2M3 ' D^s ( * 0 7 ) i ! l M l (p)Fm2»'2 ( p ' ) ^ i (P - p') + adj = £ /'d3pd3p' ^ ™ ( P , P , ) ( - ) " I " M + / I I " ' ' S " (D-3) 77717772 ^ M1M2M3 l*'il*'2l*3 " ^ t 1 ( ^ 7 ) ^ i i _ w ( « ) 8 7 ) ^ l M i - M ( ^ 7 ) • • ( P ) * ^ {p')B,a (p - p') + adj, 52 Appendix D. Restricting the form of Hj 53 w h e r e w e h a v e u s e d ( A . 3 ) i n t h e s e c o n d e q u a l i t y . U s i n g ( A . 4 ) t w i c e a n d o m i t t i n g t h e a n g l e s i n t h e D - m a t r i c e s a n d r o t a t i o n o p e r a t o r s t o save s p a c e , w e h a v e j?JU D V/* D V / . . = DL E ( * 1 - w. J - M ) ( i i -A V s i J V ) DL^ i = £ ( I 1 -U2 -uz | j V ( i 1 V -/x313 V) • (°-4) • ( i j Mi V j' tr)(ljn[ VI j'o 1 )D*, a . T h u s , 77117712 M 1 M 2 M 3 (D .5 ) • (i 1M2 /isl J » ( - ) i + 1 _ j (11M 2 Msl V) (i J Mi -Ml j ' tr) • • (| j Mi V | j' a ' ) r>i'(>^ 1M, ( p )F r o 2 / 1 . (p')B^ (P - p') + ^j. F o r ( D . 2 ) t o b e s a t i s f i e d , i t is c l e a r t h a t HjHi7l\ m u s t h a v e n o d e p e n d e n c e o n a, 8 o r 7. T o e n s u r e t h i s , w e r e q u i r e t h a t j' = a' = o = 0 i n ( D . 5 ) . U s i n g ( A . 7 ) , w e t h e n h a v e -2j+l . KIHIK]= £ / d 3 p d V ^ ™ ( P ' P ' ) ( - ) M ' " M " W 77117712 M1M2M3 (D .6 ) • ( i l M 2 M 3 | J M ) ( i l M 2 M 3 l J M ' ) ( U M i V 0 0 ) (J j ^ VI 0 0 ) • • < l M i (P)^n 2 M a (p ' )^(p - P') + adj. Appendix D. Restricting the form of Hj 54 Us ing (B.21) twice then gives KM*] = £ fd3Pd3P' /CA 2f (p>p')R/J'-M~2j+1 • mi mo i 7712 M1M2M3 M'IM 2M 3 i • ( 3 lM2M3 | j» ( 2 1 p'2 fJ.'3\ j p') + 1 ^ l j ^ M l M y2 j- + VtM' • " ^ ^ ( P ^ M ^ P ' ^ M s t P - P') + (D.7) H l - 2 , , i = £ / d 3 * d V £ > « s ( P . P ' ) -m i m 2 7 e 7 ^ } = E fd3Pd3p' E ^ ' ^ ( P . P ' ) -m i m2 ,,' i,' m\ TTl2 (D.8) • ( 3 I M 2 M 3 I 2 Mi) U 1/4 M s l 5 Mi) • • FllM> (p)Fm2^(P')B^3(p - p') + adj. Interchanging the d u m m y indices /*» and //,• (i = 1, 2,3) yields M1M2M3 • - F m , m (p)^m2M2 ( P ' ) 5 M 3 (P - P') + adj. Compar ing (D.8) and (D . l ) , we see that (D.2) is satisfied if /CAT (P. P') = E ^ ™ 3 (P. P') ( ~ ) X 2 " 2 M l ( 4 1 ^ /41 I Mi ) ( i 1 M2 M3 U Ml) = ( i l/i2/i3 | i / n ) E (P, P') ( ~ } 1 2 2 ^ ( U M 2 M 3 U Mr) • M1M2M3 Hence, the result of impos ing isospin invariance on Hi is that the vertex funct ion, hm\'^2U3(p,p'), must be of the form (D.9) / C A T W ) = U 1 M 2 M 3 I AMlMmim2 (p ,p ' ) , (D.10) where hmim2(p,p') is, as yet, an unrestr icted complex valued funct ion of m i , m2, p and p'. Appendix D. Restricting the form of Hj 55 D .2 I m p o s i n g Spa t i a l R o t a t i o n Invar iance o n Hj At this point, Hi has been restricted to be of the form m i 7712 We now impose the requirement that Hi be invariant under spatial rotations. That is, we require that K(aBy)HIlV(aB'y) = Hi (D.12) be satisfied. Applying the spatial rotation operator to Hi and using (3.9a) and (3.9c) yields 7Z(aBy)HIT^(aBj)= Y Jd3p d3p' ( \ 1 /x2 fi3\* m) hmim2(p,p') • (D.13) m i 7712m j TII^ • • D ^ m i ( ^ 7 ) ^ . m a ( ^ 7 ) -• ^ m i M 1 ( P - R ) i 7 , - 2 M 2 ( P - R ) ^ 3 ( P - R - P ' - R ) + adf Making the change of variables p —> P R , p ' —> p'R (since an integral over all momenta is unaffected by such a change) and omitting the angular arguments of the rotation operators and D-matrices gives KHiR}= Y fd3pd3p' (il^2to\i^i)hmima{PB.,PH)DiimD^ma-/ 1 J m i m2Tnlm2 (D.14) • ( P)^ 2M2 (P')BM (P - P') + adj. Comparing ( D . l l ) and (D.14), we see that (D.12) is satisfied if hm\m'2{P,P') = Y / l m i m 2 ( P R , P R ) £ ) ^ i m i ^ r m 2 , (D.15) 77117712 or making the replacements m; m\ (i — 1,2), ^m,m a (p ,p ' )= £ hm\m'2{Y>nMDlimlDl*2m,2. (D.16) 777 , 7712 Let us now expand hmim2(p,p') as a double infinite series of spherical harmonics: V m s ( P , P ' ) = i Y hrnim2a?0'(P>P')Y<x0(P)Y<*'l3>(p'), (D-17) apa'0' Appendix D. Restricting the form of Hi 56 where p = | p | , p = p/p, and the leading factor of i = v—T has been added for future convenience. Using (D.17) on both sides of (D.16) and canceling the common factor of i yields £ hmim2 "?0,(p,P')Ya0(p)Ya,t3,(p') = £ ^ m > 2 ^3<(Pfl,Pfl)^a /3(pR)^<H^ a0a'0' m[m2 a0a'0' (D.18) Because rotating a vector doesn't change its magnitude, it follows that PR = p. Also, according to (C.6), we know how the spherical harmonics transform under rotations. Thus, we can write £ hmim2%(p,p')Yag(p)Yal0,(p')= £ hm[ml2 aJ0,(p,p')Y^(p)Ya,0,(p')-a0a'0' m'xm.'2 a0a'0'00' • D t o D ^ D l r n > D £ m . . (D.19) Operating on both sides of this equation with J dClp dflp' Ylm(p)Yi?m, (p') and using the orthogonality of the spherical harmonics (see (C.7)), we have a0a'0' m\m'2 7 7 1 j 7 7 1 2 ^ m i ™ 2 ! ^ ( P , P ' ) = £ ^ ^ f e p O ( - ) m 2 - m ^ m / 3 ^ / 3 ^ i i m i ^ l 2 _ m M (D.22) 771 7712 /3/3' Appendix D. Restricting the form of Hi 57 where we have used (A.3) in the last equality. By repeated application of (A.4), this becomes h m i m ^ f a p ' ) = £ hm-iTn-2 ; ^ , ( p , p ' ) ( - ) m 2 ~ m ' 2 - D m / 3 - D m ' / 3 ' ( H ml ~ m 2 I J V) ( 2 I m l -™2\ j l) D 3 77' = E ^ m > J ^ ( P . P 0 ( - r 2 _ m i ^ m ^ ( H m i - m 2 | J 7 0 ( H ^ l - m 2 | j 7 ) -m ' j m 2 /3/3'^ " •(0 ' i 8 7 ' l i ' O ( i j ' m 7 | j V ) I>£ , = E ^ m > J ^ ( p , p O ( - r 2 " m i ( H ^ i - " » 2 l J 7 ' ) ( i i " » i - m 2 | J 7 ) -m ' j r n 2 • (/ J /? 7' I j" (lj m 7 | j ' ff) (/' / a' | j" 8') (V j'm' a\ j" S) D&. (D.23) For (D.12) to be satisfied, it is clear that TZ(aB^)HilV (aBj) must have no dependence on a, 8 or 7. To ensure this, we require that j" = S = S' = 0 in (D.23). Using (A.7) and (B.21), we then have h m i m 2 ^ ( P , P ' ) = E ! v 3 ' ( P , P ' ) ( - ) m 2 " m 2 - ™ 2 I H ) ( H ™ i - m 2 | j 7 ) • / / 771 j 7712 • (lJPi\j'(r')(lJmj\j' <r)(l'j'8' CT'|00) (/'j"m'«r|00) D°0 = E V ™ J^ ' (P ' ^ ( - r a " m 2 (H " » i - " » 2 | j 7 , ) ( i i " » i - m 2 | j 7 ) -771 j 7 7 1 2 /3/3'w' • (lJPi\j' <*') (/J j'cr) -j==8iljlS0'-v>-^==Si'r6m>-<r = E ^ ' I M 2 ^ ( P ' P 0 ( - r 2 " m ' 2 ( H m i - m 2 | J 7 0 ( H m 1 - m 2 | J 7 ) -/3j (D.24) / \2i"-m'+CT' (IjBfll'o^iljmjll'-m') y ! 21'+ 1 = EH™2"™' (H™i - m 2 | j 7 ) (/ jm^l'-m') • j ( _\2l' +<r'-m'2 • E [LMPT 2L, + 1 U W l - m 2 | j 7 ' ) (Z j / ? 7 ' l *V ) • mim 2/3 Appendix D. Restricting the form of Hj 58 Using (B.18) and (B.17c), this becomes h m i m 2 ^(p ,p ' ) = £ H m 2 - m ' ( - ) f _\2l'+a' -m'2 (ll'mm'lj-j) h^XMp') 2V + l(iim'1-m'\j1')(lj01'\l'a') ^_^+2j+l+2l' +a'-m'2 = E ( - ) " ( m ' + m ) ( ^ ' m 2 7 l i m 1 ) ( / / ' m m ' | i - 7 ) £ ^ 2 ( 2 / ' + 1) V ™ 2 j - 3 - , (P, p') ( H m[ -m 2 1 j 7 ' ) {ljBi\l'o'), (D.25) or, denoting the bracketed term by hwj(p,p'), we can write h m i m 2 fa, (p,p') = E ( - ) " ( m ' + m ) ( U ' "»2 7l i n»i) " i m ' | j - 7 ) /Hr,-(p,p')- (D,26) j Now, according to (B.12), the Clebsch-Gordan coefficient (\ jm27|^mi) vanishes unless A ( i , j , \ ) is satisfied. This condition is met only for j = 0,1. Hence, we need only sum over j = 0,1 since all other terms vanish. By using (D.26) in (D.17), we obtain the result that separately imposing both spatial rotation and isospin rotation invariance on Hi restricts h^l\'^2U3(p,p') to be of the form KTrZZ3 (P, P') = i ( i 1M2 A*3| i Mi) £ (-)-<"*+"*'> ( m 2 7l i " » ! ) ( / / ' m m ' | j -7) Im I'm' j ' Yim(p)Y[>m> (p')hwj(p,p'), (D.27) where hu'j{jp,p') is an arbitrary complex valued function of I, /', j, p and p'. Here, j = 0,1. D.3 Imposing Space Inversion Invariance on Hi At this point, Hi has been restricted to be of the form H,= £ fd3pd3p' ( H / * 2 / i 3 | i / i i ) / » m i m a ( p , p 0 ^ l M l ( p ) i r m a « ( p 0 B w ( P - P 0 + ^ ' , (D.28) 777.-1 771,0 ^ 7 7 1 1 7 7 1 2 Appendix D. Restricting the form of Hi 59 where hmim2 (p,p ') is of the form given in (D.27). We now impose the requirement that Hi be invariant under space inversion; that is, we require VHiV* = Hi (D.29) be satisfied. Applying the space inversion operator to Hi and using (3.11a) and (3.11c) yields mi m 2 ' ^/-FmiMl ( - p ) 7 T / ^ m 2 M 2 ( " P ' ) ^ - 8 ^ (~P + p') + adj (D.30) = £ / d3p d3p' ( J I M 2 M 3 I i Mi)7rfc^™i™2(P>P') • • ^ i M l ( - P ) F - 2 M 2 ( - p ' ) 5 / i 3 - p + P' + adj, where, in the second equality, we have used (3.12a). Now, because an integral over all momenta is unaffected by the transformation p —> — p, we have VHjP* = ] T j d3pd3p' U l M 2 M 3 l 2 M i ) 7 r ^ m i m 2 ( - p , - p ' ) -77X17712 ^ M l M 2^3 (D.31) • Fmin ( P ) F - 2 M 2 ( P ' ) S M , (P - P') + adj. By comparing (D.28) and (D.31), we see that for (D.29) to be satisfied, we require ^mim 2 (P>P') = 7 r f c / » m i m 2 ( - P » - p ' ) - (D.32) From (D.17) we have that hmim2(-p,-p') = i £ h m i m 2 { ^ ' ( p , p ' ) 5 W - p ) 5 W ( - p ' ) Iml'm' = i £ ftmim2!w(P,P')(-)W'^m(p)^m'(p')-(D.33) Iml'm' Thus, (D.32) is satisfied if irb(-) l+i' = 1. (D.34) Using the fact that TT& = —1 (see section (3.3.4)), this condition becomes (-) '+' ' = - 1 , (D.35) Appendix D. Restricting the form of Hj 60 which implies that / + / ' must be an odd integer. (D.36) Now, consider the expression for hmim2(p,p') obtained from (D.27) and (D.10): h m i m 2 (P, p')=«E ( - ) - ( m + m < ) (J j m 2 7 U ™i) (I I' m m ' | j -7) Ylm(p)Y,,ml (p')hw ,(p,p'), (D.37) where j = 0,1. We see that for j = 0, the requirement that A(l,l',0) be satisfied (so that the C-G coefficient (IVrnm'\ j-j) does not vanish) implies that I' = I. This is not consistent with the requirement that I + I' be odd. Hence, we must reject the j = 0 term. For j = 1, the requirement that A(l,l', 1) be satisfied implies that I and I' must differ by no more than 1 in absolute value. That is, I' = I or / ' = / ± 1 must hold. The only one of these conditions consistent with (D.36) is V = l ± l . (D.38) Thus, using (D.38) in (D.27), we obtain the result that separately imposing spatial rotation, isospin rotation and space inversion invariance on Hi restricts h^^3(p,p') to be of the form /CAT (P, P') = *" ( i 1 M2 Mali Pi) £ R-<m+m') (11 m 2 7 | * m i ) (IV m m'\ 1 -7) Imm' •Ylm(p)YVm,(p')hlv(p,p'). (D.39) where hw(p,p') = hwi(p,p') is, at this point, an arbitrary complex valued function of I, I' = I ± 1, p and p'. D.4 Imposing Time Reversal Invariance on Hi At this point, Hi has been restricted to be of the form H'= lL fd3Pd3p' ( U M 2 M 3 U M l ) / i m 1 m 2 ( p , p 0 ^ m 1 M 1 ( P ) F - 2 M 2 ( p 0 5 M 3 ( p - p O + adj, (D.4.0) m\m2 J where hmim2(p, p') is given by (D.39). We now impose the requirement that Hi be invariant under time reversal; that is, we require that THiT* = Hi (DAI) Appendix D. Restricting the form of Hj 61 be satisfied. Applying the time reversal operator to Hj and using (3.15a), (3.15c) and (3.15d) yields THiT^ = £ fd3pd3p' (iln2to\ifn)hmim2{p,p')-777 i 777 o J m\m.2 M1M2M3 • r / ( - ) ^ m i F l m i M l ( - p ) r / ( - ) 5 - ^ F _ m 2 M 2 ( - p ' K B ; J 3 ( - p + p') + adj (D.42) £ fd3pd3p' ( i l M 2 / U 3 | 3 M i ) r 6 ( - ) 1 + m i + m ^ l m i _ m 2 ( - p , - p ' ) -H i 777 o « mi ni2 (P')- bMS(-P + P') + a d j , where we have made the replacements m, - m i (i = 1,2), p -> —p, p ' -> —p' (since an integral over all momenta is unaffected by such a change of variables), and have used (3.14a) in the second equality. Comparing this to (D.40), we see that for (D.41) to be satisfied, we require hmimAP,p')=n(-)x+mi+m2hlmi_m2(-p,-p'). (D.43) From (D.32), we have h*_mi_ma(-p,-p')=nbhlmi_m3(p,p'), (D.44) which, upon substitution into (D.43), yields hmimAP,P')=n^b(-)1+mi+m2htmi_m2(p,p'). (D.45) As discussed in subsections 3.3.4 and 3.3.5, Trb = rb = —1, and so hmim2(P,P') = ( - ) 1 + M I + M 2 ^ M I - M 2 ( P , P ' ) (D.46) must hold if (D.41) is to be satisfied. Using (D.39) on the right hand of this expression yields ( - ) 1 + m i + r o ^ - T O l - r o a ( P , P ' ) = (-)1+m>+mH-i) £ ( - ) - ( « ' ) ( 1 1 —772.2 - 7 l i - m r ) • Imm' l'=l±l • (Z Z'm m ' 11 7 ) ( p ) y ; m , (p')hH, (p, p') = j ( _ ) m 1 + m 2 ( - ) ™ 2 - m i ( _ ) 2 + l - 2 ( I l m 2 7 | i m i ) . I ' S i i i (D.47) • (ll'mm'\l^)(-)m+m'Yi-m{p)Yi^m'(p')hul{p,p') = i(-)2m2+1 £ ( - ) m + m ' ( 2 1 ^ 2 71 2 m i ) -Imm l'=l±l • (I V m m ' | 1 7 ) Y,-m(p)Y,.-m. (p')h*u, (p,p'), Appendix D. Restricting the form of Hj 62 where we have used (B.17a) and (C.2) in the second equality. Making the index change m -» —m, m! —> —m' gives ( - ) 1 + m i + m 2 ^ m i - m 2 ( P , P ' ) = i E (-)-(ro+m,)(* 1^271 i^x)-Imm' l'=l±l •(U'-m-m'\l 7) Yim(p)Yt.m. (p')hw (p,p') = i E (-)"(m+m,) (i 1^2 7 1 ^ ! ) • Imm' l'=l±l • +1(ll'mm'\l -i)Ylm{-p)Yi'm'{-p')h*w{P,P') = i Y (il"»2 7 l i m i ) -Imm l'=l±l (D.48) • (1Z 'mm' | 1 -7) * W p ) * i ' m ' ( p ' ) h , V ( p , p ' ) , where we have used (B.l7a) in the second equality, and that Z + Z' + l is even integral in the third equality. Also, in the first equality, we have used the fact that m-2 is half odd integral, and thus ( _ ) 2 " l 2 + 1 = i . Comparing the last equality with (D.39), we see that (D.41) is satisfied if hw(P,p') - hu,(p,p'), (D.49) and so hw(p,p') is a real valued function. (D.50) Thus, we obtain the result that separately imposing spatial rotation, isospin rotation, space inversion and time reversal invariance on Hi restricts h!^1^3(p,p') to be of the form ^ f ( P . P ' ) = i(i l^/isl i/ii) E ( - T ( m + m , ) (i 1">2 7| i m O (ll'mm'\ 1 -7) Imm' l'=l±l Yim{p)YVml(p')hw(p,p'), (D.51) where hw (p,p') is an arbitrary real valued function of /, I' = I ± 1, p and p'. A p p e n d i x E R e s t r i c t i n g the f o r m o f Hi In this appendix, we impose the requirement that Hi, as defined in (4.1f), be invariant under the following set of transformations: isospin rotations, spatial rotations, space inversion (parity) and time reversal. These requirements restrict the form oi Hi. E . l I m p o s i n g Isospin R o t a t i o n Invar iance o n Hi As given in (4.1f), Hi is initially of the form Hi= £ [d"P d V ( P > P ' ) ^ L 1M 1 ( p) i ? T O 2M 2 ( P')^3 ( P + P') + adj. (E. l ) mi m 2 By requiring that Hi be invariant under rotations in isospin space, it is meant that III (aBj)HiTZ\ (api) = Hi (E.2) should be satisfied. The condition given by (E.2) then leads to a restriction on the allowed form of the vertex function, hm\^3(p,v')- Thus, using (3.10a),(3.10b) and (3.10c) and omitting the angles in the 63 Appendix E. Restricting the form of H/ 64 rotation matrices and D-matrices to save space, we have m\m2 J M1M2M3 ( p ^ m 2 ^ ( p ' ) ^ ( p + P ' ) + ^ i = £ / d ^ d V ^ r C P ' P O C - ^ - ^ ^ M ^ M t ^ ^ - M a -77117712 ^ M1M2M3 ( p ) ^ L 2( p'^3 ( p + p O + ^ = £ / d 3 p d V h £ ^ ( P , P 0 H ^ ^ ( E ' 3 ) 777 1 7772 ^ /J1/72M3 / j iM 2M3 • ^ ' ' X i M i ( P ) ^ M i ( P ' ) ^ (P + P') + = £ / £ ™ M P , P 0 ^ 77117712 M1M2M3 /Ji/J2M3 ( l i f t Ml / A) ( i j Mi M'l j' A') I^X^, ( p ) ^ , ( p ' ) B M i (P + P') + adj, where we have used (A.3) in the second equality and used (A.4) twice in the third and fourth equalities. For (E.2) to be satisfied, it is clear that HiHjll\ must have no dependence on the rotation angles a, /? Appendix E. Restricting the form of Hj 65 or 7. To ensure this, we require that j' = A' = A = 0 in (E.3). Using (A.7) and (B.21), we then have KIHIK\= E fd3pdV^»np.pO(-)Mi"MMU^-wljiu)(H^-i«3lJ>')-mi 111 " mi 7712 M1M2M3 M'IM 2 M 3 = E / d 3 p d V ^ ™ ( P , P O ( - ^ 777.1 777. <i "X m i m 2 M1M2M3 f_')l-*«i-Mi _ t , p r mi/j'i (p)^ a M i (p ' )^(p + p') + odj E / d 3 * d V H ^ " " ' 1 ( i i / 4 1 - / 4 ) -mi i7i2 M 1 M 2 M 3 (E.4) E ^ ^ ( P . P ' ) ^ ^ — u i/*2-/x3i ^ - M I ) ( P) Fm2M2 ( P')^3 ( P + P ' ) + a ^ ' = E / d 3 P d V ( - ) * \ / | ( H / i i / 4 | l / i 3 ) -mi 7712 MiM2^3 •( E / t m 1 A 2 2 M 3 (P ; P ' ) ( ) 1 ^ 1 " M 3 ( ^ l M 2 - M 3 U - M i ) ) • \M1M2P3 / ( P ) < , 2 ( p ' ) S M 3 ( p + p')+adj, where we have used (B.19) in the last equality. Thus, interchanging the dummy indices \n <-> \i\ and absorbing the numerical constants into the parenthetical factor, we have niHI1l\= E fd3pd3p' ( H M 1 M 2 I I M 3 ) mi 7712 • ( H 4 " ^ ! E ^ ^ ( P . P ' ) ^ " ^ 1 " " 3 ( U M 2 - M 3 U - M 1 ) ) • \ M1M2M3 / (p)^m2 . 2 (P ' )5M 3 (P + P') + adj = E [d3Pd3p' ( H / i l / ^ | l/i3 ) h m 1 m 2 ( p , P ^ L l M i ( p ) ^ L 3 M a ( P 0 5 M , ( P + P0+^' . mi T7i2 *^  (E.5) Appendix E. Restricting the form of Hj 66 where h m i m 2 ( p , p ' ) is an arbitrary complex valued function of m i , 1712, p and p ' defined by the second equality above. If (E.2) is to be satisfied, then comparing (E . l ) and (E.5), it follows that (p,p') = ( H ^ i ^ | l A i 3 ) h m i r o a ( p , p ' ) (E-6) must hold. Hence, the result of imposing isospin invariance on Hi is that the vertex function, / i ^ ^ 3 ( p , p ' ) , must be of the form hm\^3(P,P') = ( 3 2 Mi M 2 | l M 3 ) / i m i m 2 ( p , p ' ) , (E.7) where hmirri2(p, p') is, as yet, an unrestricted complex valued function of m i , m2, p and p ' . E.2 Impos ing S p a t i a l R o t a t i o n Invariance o n Hi At this point, Hi has been restricted to the form Hi= £ j dzpd3p'\li^to\lto)hmim2{p,p')^ (E.8) miTTl2 ^ M1M2M3 We now impose the requirement that Hi be invariant under spatial rotations. That is, we require that K(aBy)HIK*(aB~() = HI (E.9) be satisfied. Applying the spatial rotation operator to Hi and using (3.9a), (3.9b) and (3.9c) yields H(aB7)HInHaB7)= £ J d3p d V (\\^  p2\ 1 /i3) ^ i m a ( p , p ' ) ^ , l i m i i<*fo) • mim 2 m' 1 m 2 • Dm,m2 W-rWl-n ( P - R ) < W ( P ' - R ) ^ ( P - R + P - R ) + adj (E.10) / d3Pd3P' (2 2 M i M 2 | l M 3 ) ^ m I M 2 ( P R , P R ) ^ i m i ( a ^ 7 ) -777-177127712 m2 ' ^ ™>2 («^LiMl ( P ) ^ M 2 ( P ' ) 5 M 3 ( P + P ' ) + adj, where, in the second equality, we have made the change of variables p —> P R , p ' —> p'R (since an integral over all momenta is unaffected by such a change). Omitting the angular arguments of the .D-matrices to save space, we see that (E.9) is satisfied if Amjmi (P,p')= £ ^im2(PR,PR)£>i i m i^| 2 m 2, ( E . l l ) Appendix E. Restricting the form of Hi 67 i i or making the replacements <-> m • (i = 1,2), hmim2(p,p')= E ^ " ^ P R ' P R ) ^ . ™ ; ^ ™ - ( E - 1 2 ) m',m'2 Let us now expand hmim2(p,p') as a double infinite series of spherical harmonics: hmim2(P,P')=i Y ftm.maiw(P.P')5Wp)W(p'). (E-13) Zm/'m' where p = |p|, p = p /p , and the leading factor of i = \ / ~ T has been added for future convenience. Following the same procedure as given on page 56 of section D.2, substituting (E.13) into both sides of (E.12) yields h m i m 2 {^(p,p') = Y ^ > ' 2 \?0>(P,P')K0Dil0,Dlm,Dlm,. (E.14) 00' Next, we repeatedly use (A.4) to combine the D-matrices to give - W 2 ^ ( P , P ' ) - Y ftmim2{^(p,pO(»'mm'|ja)(Zi'/3/3'|ja')^ • 771 j 7712 • ( H m i m 2 | j ' 7) ( H m i m 2 | j ' 7') D^, = Y hm>m'2\?0,(p,p')(ll'rnm'\ja)(ll'8 8'\jcT')-(E.15) fW'tf'j" • ( J J mi m21 j ' 7 ) ( H m i m21 / 7') ( j j V 7 | j" 5) ( j j ' a' 7' | j " D\s,. For (E.9) to be satisfied, it is clear that H(aB~f)HilV (a.8~f) must have no dependence on the rotation angles, a, 8 or 7. To ensure this, we require that j" — 6 = 5' = 0 in (E.15). Using (A.7) and (B.21), we Appendix E. Restricting the form of Hj 68 then have J W » a | ^ ( p , p ' ) = £ hm,m,J0,(p,p')(ll'mm'\jo)(ll'p/3'\jo') • ( \ \ m i m 2 | j'7) ( H ™ i TO2li'7') ^ 2 - , + 1 % ' ^ - 7 ^ T T t ^ " ' ^ ' " 7 ' E ^ i m 2 r^(p ,p ' ) (hhm1m2\j -a) (ll'mm'\ja) • 00'i .L^^-(ll'BB'\ja')(Um'1rnil\j-<T') 2j + 1 = E ( - ) -(m+m') /• ! , ( i A m i m 2 | j -cr) (ll'mm'ljo) E (P. P')(S,\ {W P P'\3 a') ii irn^m'^j a') 2j + l ( E . 1 6 ) If we denote the bracketed term by hu'j(p,p'), we then have h m i m 2 \Z' (P,P') = £ ( - r ( m W ) ( I i m i m 2 | j -c r ) ( l l ' m m ' l j a) hWj{p,p'). (E.17) Now, according to (B.12), the Clebsch-Gordan coefficient ( A A M I m 2 | j —,7) vanishes unless A ( A , I , j) is satisfied. This condition is met only for j = 0,1. Hence, we need only sum over j = 0,1 since all other terms vanish. Using (E.15) in (E.13), we obtain the result that separately imposing both spatial rotation and isospin rotation invariance on Hi restricts h^l\'^2U3(p,p') to be of the form ^AT(p.p') = i( A A Mi M211 Ms) E ( - ) " ( m + m , ) ( H m i ™,2\j-a) (ll'mm'l 3 • Im I'm'j • Ylm(p)Yl'm'(p')hwj(p,p'), (E.18) where hwj(p,p') is an arbitrary complex valued function of I, I', j, p and p'. Here, j = 0,1. Appendix E. Restricting the form of Hj 69 E.3 I m p o s i n g Space Invers ion Invariance o n Hj At this point, Hj has been restricted to be of the form Hi= Y fd3pd3p' \lp3)hmima(p,p')fLlW(p)4w(P')Bw(P + P') + '«(l- (E.19) mi m.i J m i m 2 where hmim2(p,p') is of the form given in (E.18). We now impose the requirement that Hj be invariant under space inversion. That is, we require VHiV^ = Hi (E.20) be satisfied. Applying the space inversion operator to Hi and using (3.11a), (3.11b) and (3.11c) yields VHiP* = Y fd3Pd3P' ( H/*i/*2 | l/i3 ) f t m i m 2 ( p , p ' ) -m i m 2 J • T/^L,W ( - P W ^ m 2 M 2 ( " P ' K ^ M ( - P - P') + ADJ (E.21) = Y / ^P ^P' ( H f t M2| l/U3)7T6frmim2(-P, ~p ' ) " • * t l M l (P)^™ 2M 2 ( P ' ) ^ 3 (P + P') + adj, where, in the second equality, we have used (3.12a) and the fact that an integral over all momenta is unaffected by the transformation p ->• —p. Comparing (E.19) and (E.21) reveals that for (E.20) to be satisfied, we require hmim2 (P, P') = Kbhmim2 ( - p , - p ' ) - (E.22) From (E.13) we have that hmim2{-P, ~P') = i Y ^ " » i m 2 llZi'(P>P')Ylm{-p)Yl'm'(-p') Iml'm' = i Y hmim2^,(P,p')(-y+l'Ylm(p)Yllm,(p^ (E.23) Iml'm' Thus, (E.22) is satisfied if 7 r 6 ( - ) , + l ' = 1. (E.24) Using the fact that 7rt = —1 (see section (3.3.4)), this condition becomes (_)'+«' = _ i , (E.25) Appendix E. Restricting the form of Hj 70 which implies that I + I' must be an odd integer. (E.26) Now, consider our expression for hmim2(p, p') obtained from (E.18) and (E.7): hmim2(P,P')=i E R " ' " ' ^ 2 - m i m 2 | j - C T ) ( Z Z ' m m ' | j ^ (E.27) Im I'm j where j = 0,1. B y the exact same reasoning as in D.3, we see that that we must reject the j = 0 term in (E.18). For j = 1, the requirement that A(/ ,Z' ,1) be satisfied implies that I and /' must differ by no more than 1 in absolute value. That is, I' = I or I' = I ± 1 must hold. The only one of these conditions consistent with (E.26) is l' = l±l. (E.28) Thus, using (E.28) in (E.18), we obtain the result that separately imposing spatial rotation, isospin rotation and space inversion invariance on Hj restricts /CITTT (P> P') to be of the form /CAT (P. P') = » ( H Mi M2I 1 U3) £ ( - ) - ( m + m ' ) ( H m i m211 -cr) (I / ' mm'| 1 a) Imm' l'=l±l Yim{p)Yi'm'(p')hw{p,p'), (E.29) where hui(p,p') = ha>i{p,p') is, at this point, an arbitrary complex valued function of I, I' = I ± 1, p andp' . E.4 I m p o s i n g T i m e R e v e r s a l Invariance o n Hi At this point, Hi has been restricted to be of the form Hi= £ Idzpdzp> {\\uiu2\luz)hm,m2(p,p')Fm^ (E.30) T77 1 777.0 7TilT l2M1M2M3 where hmim2(p,p') is given by (E.29). We now impose the requirement that Hi be invariant under time reversal. That is, we require TH1T1 = Hi (E.31) Appendix E. Restricting the form of Hj 71 be satisfied. Applying the time reversal operator to Hi and using (3.15a), (3.15b), (3.15c) and (3.15d) yields THiT*= £ fd3pd3P' ( H M i M 2 | l M 3 ) / C i m 2 ( P , p ' ) -mi ni2 J • T/(-)5- r a i FL m i / I 1 ( - p ) r /(-)5-^F l m 2 M 2 ( - p ' ) r 6 JB M 3 ( - p - p') + adj r _ (E.32) = £ d3pd3p' ( H M i M 2 | l M 3 ) r 6 ( - ) 1 + ^ + m 2 / i I m i _ m 2 ( - p , - p ' ) -777-17712 ^  Ml J*2M3 • ^ n 1 M l ( P ) ^ 2 M 2 ( P ' ) S M 3 (P + P') + adj, where we have made the replacements mj —• - m ; (i = 1, 2), p —> —p, p' —> —p' (since an integral over all momenta is unaffected by such a change of variables), and have used (3.14a) in the second equality. Comparing this to (E.30), we see that for (E.31) to be satisfied, we require ^m i m 2(p,p') = r 6 ( - ) 1 + m ' + " l ^I m i _ m 2 ( -p , -p ' ) . (E.33) From (E.22), we have ft_TOl_TOa(-p,-p') =7r 6 f t l m i _ m 2 (p,p') , (E.34) which, upon substitution into (E.33), yields fim,m2 (P,p ' ) = r 6 7 r 6 ( - ) 1 + - ' + ' " ^ l m i _ m 2 ( p , p ' ) . (E.35) As discussed in subsections 3.3.4 and 3.3.5, TTI, = rb = —1, and so hmim2(p,p') = ( - ) 1 + m i + m 2 / i l m i _ m 2 ( p ) p ' ) (E.36) Appendix E. Restricting the form of Hi 72 must hold if (D.41) is to be satisfied. Using (E.29) on the right hand of this expression yields R 1 + m i + m 2 ^ l m i - m 2 ( p , p ' ) = ( - ) 1 + m ' + m * ( - i ) £ (_)-(™+™') ( 1 1 _ m i 777.2 | 1 ff) • Imm l'=l±l •(ll'mm'\l-a) Ylm(p)Yl7m,(p')h;l,(p,p') = i^_ymi+m2 ^ ( _ ) - ( " l i + ™ 2 ) ( _ ) 5 + 5 - 1 ( i i m i m2\ 1 -a) • S i (E-37) • ( » ' m m ' | 1 -ff) (-)m + m 'yi_m(p)y ( '_m-(p')h* u,(p,p') = * £ ( - ) - (™+™) ( j i m i m 2 | 1 -ff) • /mm /'=/±l . (Z i ' _ m - m ' l 1 -ff) Ylm(p)Y,,ml(p')hu, (p,p'), where we have used (B.17a) and (C.2) in the second equality and made the index change m —> —m, m! —> —m' in the third equality. Again using (B.17a), this becomes ( - ) 1 + m i + m a f c * - m a - r o a ( P , P ' ) = * £ ( - ) - ( m + m , ) (I | m 1 m 2 | 1-ff)-i' ti ( / / 'mm' | lcr)Ylm(p)Yi'm'(p')hw(p,p') Imm' l'=l±l (E.38) = * £ ( - ) - ( m + m ' ) ( H m i m 2 | l - f f ) -/ m m ' f ' = / ± l • (i v m m ' | i ff) y , m ( p ) y r m . (p')ftii. (P.PO. where we have used the fact that / + V — 1 is even integral in the second equality. Comparing the last equality with (E.29) and using (E.7), we see that (E.31) is satisfied if hw(p,p') = h;,(p,p'), (E-39) and so hw(p,p') is a real valued function. (E.40) Thus, we obtain the result that separately imposing spatial rotation, isospin rotation, space inversion Appendix E. Restricting the form of Hj 73 and time reversal invariance on Hj restricts hj^^3(p,p') to be of the form ft W (P, P') = * ( H Mi M2I 1 Ms) £ (-)-(™+™') (1 1 m i m211 -a) ( i i ' m m'| 1 a) imm' l'=l±l (E.41) Yim(p)Yi'm'(p')hw{p,p'), where hu>(p,p') is an arbitrary real valued function of 1,1' = I ± 1, p and p'. Appendix F Imposing Charge Conjugation Invariance on H\ In this appendix, we impose charge conjugation invariance on the interaction Hamiltonian, H\ = Hi + HI+HI. A t this point, we assume that Hi, Hi and Hi have already been restricted to be invariant under TZi,TZ,V and T . See the results of section 4.3.1 for details. As will become apparent in the discussion to follow, demanding that H\ be invariant under charge conjugation imposes a relation between the vertex functions of Hi and Hi and places a constraint on the vertex function oi Hi. In light of this, we shall impose charge conjugation invariance first upon Hi + Hi, and then on Hi. Note, the same results are obtained if we impose charge conjugation invariance on all three terms of Hi simultaneously, it is just much more cumbersome to present. F . l Imposing Charge Conjugation Invariance on Hi + Hi We now impose the requirement that Hi + Hi be invariant under charge conjugation. That is, we require that be satisfied. Applying the charge conjugation operator to Hi + Hi as defined in (4.Id) and (4.1e), and using (3.16a), (3.16b) and (3.16c) gives C(Hi + Hi)tf = (Hi + Hi) (F.l) C(HI + HI)C*= £ d3pd3p' { "•77117712 WlP 7771 - / J 1 (P)KfF., 7 7 l 2 - / i 2 (p ' )K 6 /3_ M 3 (p - p ' ) T T l l T712 + ^ m ^ 3 ( P . P 0 « / ^ l - M l ( P ) « / ^ 2 - M 2 ( P ' ) « 6 S - M 3 ( P - P')} + adj (F.2) £ d3pd3p' { / i - ^ 7 - ^ ( p , p ' ) ^ i M i ( p ) F m 2 M 2 ( p')^3 ( p - p ' ) 77li 7712 M1M2M3 + « 7 2 " P 3 ( P , P ' ) ^ i / J i ( p ) i ; l ™ 2 M 2 ( p ' ) 5 M3 ( P - P')} + adj, 74 Appendix F. Imposing Charge Conjugation Invariance on Hi 75 where, in the second equality, we have made the change of dummy index m —• — (i = 1,2,3), and used (3.17a) plus the fact that Kb = +1 (see page 11). Now, as shown in (4.4), /CATte'P') = U l M 2 M 3 U M l ) ' W 2 ( P , P ' ) , and so using (B.17a), we have that ^ { ^ " " ( P . P ' ) = ( i 1 - A * — A*s| I - M i ) A ™ ( P , P ' ) = + ( i 1/U2/U3I 5 M l ) / l m 1 m 2 ( p , p ' ) = - / l « 3 ( P ' P ' ) -Similarly, te;"2_M8(p.p') = - ^ « s ( P . P ' ) -Using these two results in (F.2), we obtain c(Hi+%)ci= Y d3pd3p' { - ^ r ( p > p ^ L 1 p 1 ( p ) ^ 2 ( p ' ) ^ 3 ( p - p ' ) (F.3) (F.4) (F.5) m\rri2 M1M2M3 (F.6) -hZ^HP,P')FL^(P)F^(p')B,3(p - P ' )} + adj, and so, comparing this with (4.Id) and (4.Id), we see that for (F . l ) to be satisfied, ^ 2 i 8 ( P , P ' ) = - ^ a M s ( P . P ' ) (F.7) must hold. Thus, imposing spatial rotation, isospin rotation, space inversion, time reversal and charge conjuga-tion invariance on Hj + Hj restricts it to be of the form HJ + HJ = Y [ D3PD3P' m i m 2 ^ h&W (P, P ' ) ^ 1 M 1 iP)Fm2,2 (P')B,3 (p - P') + adj - Y [d3pd3p> mi m 2 •* / J 1 M 2 / J 3 h&W (P. P'WIM (p)Fm2ti2 (P')B,3 (p - P ' ) + adj, (F.8) where h^^3(p,p') is of the form given in (D.51). Appendix F. Imposing Charge Conjugation Invariance on Hi 76 F.2 Impos ing Cha rge C o n j u g a t i o n Invariance on Hj We now impose the requirement that Hj be invariant under charge conjugation. That is, we require CHitf = Hj (F.9) be satisfied. Applying the charge conjugation operator to Hi as defined in (4.1f) and using (3.16a), (3.16b) and (3.16c) gives (F.10) = £ / d 3 p d V fc-^2_M(P>P^ minis where, in the second equality, we have made the change of dummy index Ui —> — Ui (i = 1,2,3), and used (3.17a) plus the fact that K& = +1 (see page 11). Making the change of dummy indices ui <-> H2, m\ o m 2 , and interchanging the variables of integration, p o p ' , (F.10) becomes CHjtf = £ fd3pd3p' / ^ ^ ( ^ p ) ^ 77117712 M1M2M3 = £ / d 3 p d y / > - « 7 " - ^ ^ m. 1 m.n J mi77l2 where in the second equality, we have used the fact that Fmi (p) and Fm2fi2(pf) anticommute (see (3.4e)). Comparing ( F . l l ) with (4.1f), we see that (F.9) is satisfied if « r ( p , p ' ) = -K^T~"3(P'^)- (P-12) Appendix F. Imposing Charge Conjugation Invariance on Hi 77 Using (4.29), we see that - t e 7 1 " M 3 ( p ' ! P ) = - i ( H - M 2 - M i | l - / i 3 ) £ (-r{m+m'Hiim2m1\l-a) Imm l'=l±l •(ll'mm'\la) Ylm{p')YVm, (p)hu, (p',p) = - i ( - ) H 4 - i ( H A l 2 A 1 1 | i M 3 ) £ ( - ) - ( ™ + " 0 ( H m 2 m i 1 1 - C T ) -i'=r±i (F.i3) •(ll'mm'\l(T) Ylm{p')YVm, (p)hw (p',p) / m m ( ' = (±1 • ( i i m i m 2 | 1 - c r ) ( / / ' m m ' | l c r ) r < m ( p ' ) ^ ' m ' ( p ) / i H ' ( p ' . P ) , where we have used (B.17a) in the second equality and (B.17b) twice in the second equality. Making the change of dummy indices I <4 I', m f> m ' in this expression gives -^m 2 M m7 1 " M 3 (p' .P) = - U H M l M 2 | l M 3 ) £ R ( H TUl m 2 | 1 -a) • / m m ' / ' = ( ± 1 • ( J ' / m ' ml 1 cr) YVm.{p')Yim(p)hvi{jp\p) = - i ( H M l M 2 | l M 3 ) £ ( - ) - ( m + m , ) ( H m i m 2 | l - C r ) . i {=i±i (F.14) . ( - ) ' + " - 1 ( r Z m ' m | l c r ) y , - m - ( p ' ) y / m ( p ) ^ / ( p ' , p ) = - « ( i 4 M i / i 2 | l A i 3 ) £ ( - ) - ( m + m , ) ( H m i m 2 | l - c r ) . /77 lTTl ' C=(±I • ( / ' Z m ' m | lcr)y i m (p)yj' m '{p ' )hi ' i (p ' ,p) , where we have used (4.24) in the last equality. Comparing the last equality with (4.29), we see that (F.12) is satisfied if hu>(p,p') = -hl,l(pl,p). (F.15) Appendix F. Imposing Charge Conjugation Invariance on Hx 78 Thus, imposing spatial rotation, isospin rotation, space inversion, time reversal and charge conjuga-tion invariance on Hj restricts it to be of the form Hi= Y, [d3p d3p' ^™ 3 (P 'P ')^L i m (p)^ m 2M 2 (p ')s M 3 (p+p ' )+^j , m i 7712 (F.16) where h^11^^3 (P> P') 1 S °^ the form given in (E.41), but with the extra requirement that hu< (p,p') satisfy (F.15). Appendix G Restricting the Form of Hx Under IZi, 1Z and CVT In this appendix, we impose the requirement that Hi, as defined in (4.1c), be invariant under the action of the operators IZi, 1Z and CVT. The restrictions placed on Hi by requiring it be invariant under Hi and H have already been discussed in section (4.3), so we simply quote the result that imposing both isospin rotation and spatial rotation invariance on Hi implies that the vertex functions are of the form KTrZ?3 (P, P') = ( 4 1 M2 M3 | } Ml) hmim2(P, P ' ) (G. l ) ^ S f (P» P') = (* 1 A*2 Ms| i Mi) Ttmim2 (p, p') (G.2) £ & £ a M ( P > P ' ) = ( H M i M 2 | l M 3 ) ^ n i m 2 ( p , p ' ) ; (G-3) where /i m im 2 (P,P' ) = i £ ( - ) - ( m + ™ ' > ( i j m 2 7 | i mx) (ZZ'mm'| j -7) y , m ( p ) 5 W ( p ' ) ^ i ( P , p ' ) , (G.4) Im I ra 3 ^ m i m 2 ( p , p O E ( - ) " ( m + m ' ) ( 5 i m 2 7 | i m 1 ) ( / Z ' m m ' | j - 7 ) ^ r o ( ^ (G.5) Im I'm'j and / i m i m 2 ( p , p O = * E ( - ) _ ( m + m ' ) ( H m i m 2 | i - a ) ( Z Z ' m m ' | j V ) y ( m ( p ) ^ ^ (G.6) Im I'm'j Note, hwj(p,p'), hu'j(p,p') and ha>j(p,p') are arbitrary complex valued functions of Z, Z', j, p and p', and j = 0,1. We now impose the requirement that Hi is invariant under CVT. That is, we require CVTHXT^V^CX = Hi (G.7) be satisfied. 79 Appendix G. Restricting the Form of Hi Under TZj, 1Z and CVT 80 Applying CVT to Hx and using (3.19a), (3.19b) and (3.19c) yields CVTHiT^V^C^ = CVTH^V^tf + CVTE^V^tf + CVTHiT]V^C] = £ ld3Pd3p' U l / i 2 M 3 U M l ) / 4 1 m 2 ( P > P ' ) < M - ) i M1M2M3 • F t _ m i _ M i ( p ) < p / ( - ) i - m 2 F _ m 2 _ M 2 ( p ' ) ^ B - - J 3 ( P - P') + adj + Y fd3pd3p' (ilU2^\itii)t*mim2(p,p')<pf(-)i-mi • ro. i n i n J M1M2M3 • J F ! m i _ / J 1 ( p ) ^ / ( - ) 5 - ^ F _ m 2 _ / J 2 ( p ' ) ^ J B _ ^ ( p - p') + adj + Y fd3Pd3P' ( H M i M 2 | l / x 3 ) ^ m i m 2 ( p , p ' V / ( - ) i " m i • m \ni.-2 Mi/t2M3 • F l m i _ M 1 (p ) ( P / ( - ) i-^F t _ m 2 _ M 2 (p')'P6 / 3-M3 (P + P') + adj (G.8) £ / " d ' p d V ( i l - ^ - / i 3 | i - A i i ) ( - ) 1 + m i + m ^ - m i - m a ( P , P ' ) -m\m% M1M2M3 • Flltll (p)Fm2tl2 ( p ' ) B M S (p - p') + adj + fd3Pd3p' ( U - M 2 - M 3 U - M i ) ( - ) 1 + r o i + m 2 ^ - m i - m 2 ( P , P ' ) m i m i » i 7712M1M2M3 • ^ m i ( J l ( P ) i ^ 2 / J 2 ( p ' ) ^ 3 ( P - P') + adj + Y, fd3pd3p' ( H - M i - M 2 | l -Ms) ( - ) 1 + m i + m ^ i m i - m 2 ( p , p ' ) -m i m2 M1M2M3 • (P )FL 2 M 2 (P')^3 (P + P') + adj, where, in the second equality, we have used (3.20a), (3.21), and made the change of dummy indices Appendix G. Restricting the Form of Hi Under TZ], TZ and CVT 81 mi -> —mi (i = 1,2), /x, —> - t i j (j = 1,2,3). Using (B.17a) and (3.4e) in this expression gives C 7 > T # I T W = £ / d 3 P d V H i + 1 - * ( n M 2 M 3 U M i ) ( - ) 1 + m i + m 2 ^ m i - m 2 ( P , p ' ) TT> i TTJ.o mim-2 • * t l M l ( P ) ^ ™ 2 M 2 (P')-BM 3 (P - P') + adj + £ f d 3 P d V ( - ) ^ + 1 ^ ( 5 l M 2 ^ | 5 M i ) ( - ) 1 + m i + m 2 ' ' - m i - m 2 ( P 1 p ' ) m i ro.o v m i 7712 M1M2M3 • ^mi / i i ( P ) ^ 2 M 2 ( P ' ) S M 3 (P - P') + a ^ + £ / d 3 p d V ( - ) i + i - M H w / * 2 | l / i 3 ) ( - ) 1 + m i + r o ^ I m i - r o 2 ( p , p ' ) m i m o m i 7712 M1M2M3 ( - ) 1 ^ L 2 M 2 ( p ' ) i ? l ( x 1 ( p ) ^ 3 ( P + P ' ) + a d j = £ fd3pd3P' Q i / w i i M I ) ( - r i + m 2 ^ - m i - m 2 ( p , p ' ) -m 1 m o « m\m,2 M1M2M3 • ^ L l M l ( p ) i r ™ 2 M 2 ( p ' ) ^ 3 ( P - P') + adj + £ fd3pd3p' ( 4 l M 2 M 3 | 2 M i ) ( - ) m i + m 2 / i I m i - m 2 ( P , P ' ) •m i m o m\m2 M 1 M 2 M 3 • ^ , l M i ( P ) F " » a w ( P ' ) ^ / . s ( P - P') + adj + £ fd3pd3p' ( H M i M 2 | l M 3 ) ( - r i + m ^ I m i _ m 2 ( P , p ' ) -m i m 2 M1M2/J3 ( p ' ) / 4 l f l l ( p)^3 ( P + P') + adj. (G.9) If we now make the change of dummy indices mi <-> m 2 , Mi M2 phis the change of variables of Appendix G. Restricting the Form of Hi Under TZj, H and CVT 82 integration p p' in the last term, this becomes CVTHI-PV*&= Y, fd3pd3p' ( n M 2 M 3 U M i ) ( - ) m i + m ^ - r a i - m 2 ( P ; P ' ) ro.i rn i J 7711 7712 m 2 M2 (p')^3(p-p')+adj + Y fd3Pd3p' (ii/x2/i3u/ii)(-ri+m='/liTni_ma(p>p'). m i 7722 ^ M 1 M 2 M 3 (G.10) • F m 1 M l (P)^m a M 2 ( P ' ) # „ 3 (P - P') + adj + Y fd3Pd3p' ( H M 2 M l | l M 3 ) ( - ) m i + m 2 P m 2 _ m i ( p ' , p ) . m i 771 o J i m 2 • ^ Lm (P)^™ 2M 2 (P')^M3 (P + P') + adj. mim Vf/- 1 m 2 / i 2 /  > Thus, using (B.17b) in the last term yields CVTHiTWtf= Y fd3pd3p'(ilp2U3\iui)(-)mMh*_mi_m2(p,p') m i m o ^ i T712 M l M 2 ^ 3 (P)^m a w ( p ' ) ^ a (P - P') + adj + „ S a /d3pdV ( i 1 ^ ^ l i ^ ) ( - ) m i + m ' S l T O l _ m 8 ( p , p ' ) . (G.ll) • Fmiftl (p)Fm2fl2 (p')BM3 (p - p') + adj + JL Id3pd3p' (-^^^^(-r^-^^cp' .p). mi 7712 M1M2 /J3 • ^ m l W ( P ) ^ a W ( P ' ) ^ . ( P + P') + adj. Examining this equation, we see that (G.7) is satisfied provided / * ™ ( P , P ' ) = (-ri+m^iroi_m2(p,p') (G.12) and ftmima(p,p') = (-)mi+m2^-m2-mi(p',P) (G.13) hold. Appendix G. Restricting the Form of Hi Under TZi, TZ and CVT 83 Starting from the right hand side of (G.12) and using (G.4), we have ( _ r + ™ . / T i r a i _ m a ( p > P ' ) = ( -r>+-H) £ (-r(m+m,) ( i i - r i * - ™ ' ) • Zm • ( H ' m m' | j 7) >?m (p)Yi*m< ( p ' f e (P' P1) _ j j _ j H - m i + m 2 ^ ( _ ) - m i + m 2 ) ( - ) i + i - i ( 1 j m 2 7| i mi) I 'm ' j (G.14) - (ll'mm'l Jl) (-)m+m'yi-m(p)Yr-ml(p')hulj(p,P'), where we have used (B.17a) and (C.2) in the second equality. Also, because the only values of m and m! for which the Clebsch-Gordan coefficient, (ll'mm'l J7)> does not vanish are those for which m + m' = —m 2 + m i , we are able to replace ( - ) ~ ( m + m ) with ( - ) - m ^ + m ^ in the second equality. Thus making the change of dummy indices m —» —m and m' —> —m', we have ( - r ' + ^ ! m i _ m a ( p , p ' ) = i ( - ) 2 m 2 + 1 £ ( - ) - ( « ' ) ( - ) ^ ( i j m 2 7 | 5 m i ) • Im I'm' j • (W -m-m'lJ^)Yim(p)YVml(p')h*Wj(p,p') = i £ (-)-(m+m'H-y (l j m 2 7 | i mi) (-)'+''-(G.15) . (W _ m _ m ' I 7 ) y I m ( p ) y , , m , (p')^*,.( p,p») = i £ ( - ) - ( m W ) (J j m 2 7 | j m i ) (I / ' - m - m ' | j 7) 1771 tl 1 • i m j •Ylm(p)Yi'm'{p'){-)l+l'hnij(p,p'). Comparing the last equality with (G.4), we see that for (G.12) to be satisfied, hwj(p,P') = (-)l+l'h*wAp,p') (j = 0,1) (G.16) must hold. Note, we can equate the coefficients of the spherical harmonics in this way since they are orthonormal functions. Appendix G. Restricting the Form of Hi Under TZi, TZ and CVT 84 Next, starting from the right hand side of (G.13) and using (G.6), we have ( - ) m M h - m 2 - m i (P', P) = R m i + m 2 (-») E (-)-(•»+-»') ( i i - m 2 - m x | j a) Im I'm'j (ll'mm'lj - a ) y ^ ( p ' ) ^ ^ ( P K 0 - ( p ' . p ) 1 x 1 . ( _ i ) ( _ ) " l i+ m2 ^ ( _ ) - ( " » i + m 2 ) ( - _ ^ 5 + 3 - J ( i l m 2 m i | j - c r ) • • (ll'mm\j-(j){-)m+m'Yi-m (f>')Yv _ m . ( p f o r > (p', p) (_i) £ (_ ) - ( ">+« . ' ) (_ )2 ( i - j ) ( i i m i m 2 | j-o) • Im I'm'j •(11' -m -m'\j - c r ) YVm>(ft)Yim(p')h*Wi(p',p), where we have used (C.2) in the second equality and (B.17a) in the second and third equalities. Making the change of dummy indices m o m ' and I H I ' , this becomes ( - ) m i + m a « . m 2 - m i (P', P) = i-i) E ( - r ( m W ) ( H mi m 2 | j - c r ) ( / ' / -m! - m | j -cr) • n V i T O ( p ) y i ' m ' ( P ' ) ^ y (P'.P) Im (G.18) = ( - 0 E (-)"(ni+m') (i m a l j - a ) (-)'+''-•>•(-)'+''->• • • (J J'mm'| j a) Ylm(p)YVm, (p')hrij(p',p), where we have used (B.17a) and (B.17b) in the second equality. Since (-) 2 ( '+' ~i) = 1 and the spherical harmonics are orthonormal, by comparing this last equality with (G.6), we see that for (G.13) to be satisfied hii'j(p,p') = -h;,lj(p',p) 0' = u,l) (G.19) must hold. A p p e n d i x H T h e H e l i c i t y F o r m a l i s m In this appendix, we introduce the helicity formalism and then determine the transformation properties of the operators which create photons of a given momentum and helicity. H . l D e f i n i t i o n o f P h o t o n H e l i c i t y States Until this point, we have been describing the momentum states of particles with spin using the z-axis spin projection basis. For example, to describe photons (spin 1) with a given momentum and z-axis spin projection, we can introduce the creation (annihilation) operator B^p) which, when operating on the vacuum state 10), creates a one-photon ket corresponding to a photon of momentum p and z-axis spin projection m. The adjoint operator B\(p), when operating on this one-photon ket, returns the vacuum. Although such operators generate a perfectly valid basis of states, for a massless particle like the photon, it is not the most elegant. Instead we consider the so called helicity basis. The helicity A of a particle (in this discussion, a photon) is defined as the component of the particle's spin in the direction of it's momentum. Here we introduce the operator 23A(p) (A = ±1) which, when operating on the vacuum state 10), creates a one-photon ket corresponding to an elementary photon with momentum p and helicity A. The adjoint operator, H\(p), when operating on this one-photon ket, returns the vacuum state. The basis of states associated with these operators is much more appropriate for describing massless particles. The reason for this will become apparent when we determine the transformation properties of the helicity operators in the next section. However, we shall first express the helicity operators for photons in terms of the z-axis spin projection operators. Following the discussion which starts on p. 119 of Taylor [10], we define (H.l) m or using ( A . l ) (H.2) m 85 Appendix H. The Helicity Formalism 86 H.2 Determining the Transformation Properties of the Photon Helicity Operators H.2.1 Spatial Rotations Based on Wigner's [11] article, we know that for a massless particle, " . . . there are only two states of polarization, and even the existence of the second one can be inferred only on the basis of reflection symmetry.". Specifically, " . . . the statement that 'velocity and spin are parallel' is invariant under rotations.". Thus, we require that ft(a/37)£A(p)ftt(a/?7) = SA(p_R). (H.3) H.2.2 Space Inversion Under space inversion, the operators -BA(p) transform (as in the pion case) according to VBl(p)pi = npBU-p). (H.4) Using this result, (H.l) and (H.2) , we have VVl(p)V* = V (^DUv,8,0)Bm(pJj 7>+ . =Ye-im<PdmX(9)VBUp)Vi m = £ e - ^ O 0 ) v 4 ( - p ) m = TTPY e-im^-^{-)1+mdlm_^ - 9)Bm(~p) (H.5) = np ]T eim"(-r+1e-im(*+"Ui_x(n - 0)Bl(-p) 771 = *P E ( - ) 2 " l + l e " " i m ( V ' + ' r ) ^ - A ( 7 r - S)BU-P) rn = M-) 1 £ e-imiv+n)dm-x(* - 9)Bl(-p) m = T T ^ - J ^ L ^ - P ) , where we have used (A.11) in the fourth equality, and the fact that (—)2m = 1 in the sixth equality. Thus, since np = —1, we have 7'«t i(p)7>t=S t_A(-p). (H.6) Appendix H. The Helicity Formalism 87 H . 2 . 3 T i m e R e v e r s a l Under time reversal, the operators B\(P) transform (as in the pion case) according to TBm(p)T* = r P (-) 1 - m 73l m (-p) . (H.7) Using this result and (H.2) and recalling that T is and antihermitian operator, we have T'BUP)^ = r (£e-™*o«i)i4(p)j rt = Yeimvd^Tp(-r~mB-m(-p) m = Y(-r-m^miv,+n~w)(-)1+md^-X(7r - 6)(-y-mBlm(-p) m = ^ E ( - ) " " l e i m ( ^ ) d - - A ( 7 r - 6)B-m{-p) (H.8) = rp E ( - ) m e - i m ( v + 7 r ) d l m - A ^ - O)Bl(-p) m = rP E ( - r e - i m ( v + 7 r ) ( - r m + A ^ - e)BU-p) m m = T J ) ( - ) * 2 t ( - p ) , where we have used (A.11) in the third equality, (A.10) in the sixth equality, and made the change of dummy index m —> — m in the fifth equality. Thus, since r p = 1, and A = ±1, we have TVlMT* = (-) 1S1(-P). (H.9) H . 2 . 4 C h a r g e C o n j u g a t i o n Under charge conjugation, the operators -B^(p) transform (as in the pion case) according to CBj,(p)Ct = KpBl{p). (H.10) Appendix H. The Helicity Formalism Using this result and (H. l ) , we have = £ e - i m ^ m A ( 0 ) C B m ( p ) C t m = KpJ2e~im'pdmX(e)Bl(p) m = KpS ]» (p ) . Since KP = —1, we thus have A p p e n d i x I R e s t r i c t i n g the fo rm of Hi 1.1 Impos ing Spa t i a l R o t a t i o n Invariance on Hi In this appendix, we impose the requirement that Hi be invariant under spatial rotations. That is, we require that K(a,9i)'Hi'R} {aBi) = Hi (1.13) be satisfied. Applying the spatial rotation operator to Hi and using (6.9a), and (6.9c) yields TKaPWtfWi) = £ Jd3pd3p' /lmim2A(p,p')/ji;mi(a/37)^ 4*m2(a/37) • mi m2 A (1.14) • F t , , ( p _ R ) F m 2 ( p ' _ R ) S A ( p _ R - p'_R) + adj. Making the change of variables p —>> P R , p ' —> p ' R (since an integral over all momenta is unaffected by such a change) and omitting the angular arguments of the rotation operators and D-matrices gives HHj-R)= £ fd3pd3p' I £ hmim2X(Pn,p'R)Di,mD^l m\m'2\ W i n ! (1.15) • F l ^ F ^ p ' ^ p - p ^ + adj. If this expression is compared with (7.Id), we see that (1.13) is satisfied if holds. Now, comparing this condition to that given in (D.16) of section (D.2), we see that the two are identical (except for the additional spin index A in hmim2\(p,p') and the isospin indices ui, U2, and p.3 in hmllm>2U3(p,p'), neither of which are involved here). Thus, we can immediately conclude the result of that section: (1.13) is satisfied if hmim2\(p,p') is of the form hmim2x(p,p')=i ^ ( - ) - ( m W ) ( ^ i m 2 7 | (l/'mm'lj - ^ / i ^ A ^ p K ^ m ' t p ' ) , Im I'm'j (1.17) 89 Appendix I. Restricting the form of Hi 90 where h[i>j\(p,p') is an arbitrary complex valued function of I, I', j, A, p and p'. Note, because of the Clebsch-Gordan coefficient (± j m.2 7| \ m i ) , j is only summed over the values j = 0,1. 1.2 Imposing Space Inversion Invariance on Hi At this point, Hi has been restricted to be of the form Hi= Jd'pd'p' ^ m i m 2 A ( p , p O ^ ™ 1 ( p ) - P , m 2 ( p ' ) ^ ( p - p ' ) + ^ (1-18) m 1 m 2 A where hmim2\(p,p') is of the form given in (1.17). We now impose the requirement that Hi be invariant under space inversion; that is, we require VHiVx =Hi (1.19) be satisfied. Applying the space inversion operator to Hi and using (6.10a) and (6.10c) yields VUiV*= £ JdVdVVm2A(p,p')'r^iI(-P)'r^m2(-p>P(-)1S-A(-p + p ' ) + 4 m i m 2 A (1.20) = £ Jd3pd3p' / 7 m i m 2 _ A ( - p , - p O F m i ( p ) F m 2 ( p O S A ( p - p O + « d ? ' , 7 7 l l 7 7 l 2 A where in the second equality we have used (6.11a), made the change of dummy index A -> —A, and used the fact that an integral over all momenta is unaffected by the transformation p —• —p. In the second equality, we have also used np(—)* = 1. Comparing this expression with (7.Id), we see that for Hi to be invariant under V we require /lmim 2A(P,p') = hmim2-\(-p, -p ' ) (1.21) hold. Starting from the right hand side of (1.21), and using (1.17), we have hmima-x(-p,-p') = * E ( - ) - ( r o + m ' » U j m 2 7 | imi ) (ll'mm'\ j-j) hWj-x(P,p')Ylm(-p)Yllml(-p) Im I'm'j = * £ ( - ) " ( m + m ' U 3 J m 2 7 U m i ) ( / / ' m m ' l j - 7 ) / l « ' j - A ( P ! P ' ) ' 1 • ( - ) ' + ' V ( m ( p)rr m - ( p ' ) . (1.22) Im I'm'j Appendix I. Restricting the form of Tii 91 Hence, for (1.19) to be satisfied, we require hWjX{p,p')^{-)l+l'hWj-x{p,p'). (1.23) Thus, imposing spatial rotation and space inversion invariance on Tii restricts the vertex function hmim2\{p,p') to be of the form hmim2x(p, P') = i Y (-)-{m+m,) (ijm2l\im1)(ll'm m ' | j -7) • • Yim(p)Yym>(p')hWjx(p,p'), Im I'm'j (1.24) where hu>jx(p,p') satisfies (1.23) but is otherwise an arbitrary complex valued function of I, I', j = 0,1, p and p'. 1.3 I m p o s i n g T i m e R e v e r s a l Invariance o n Tii At this point, Tii has been restricted to be of the form Tii= Y jd3pd3p' C , m a A ( p , p ' ) ^ 1 ( p ) ^ n 3 ( p ' ) B A ( p - p ' ) + « & - > (1.25) m\m2\ • where hmim2x(p,p') is of the form given in (1.24). We now impose the requirement that Tii be invariant under time reversal; that is, we require that TTiiT1 = Tii (1.26) be satisfied. Applying the time reversal operator to Tii and using (6.12a), (6.12c) and (6.12d) yields TTiiT^= Y Jd3Pd3p' h*mim2X(p,p'). mim2\ • r e ( - ) i - m ' F ! T O l ( - p ) r e ( - ) 5 - ^ F _ m 2 ( - p ' ) r p ( - ) 1 S A ( - p + p') + adj (1.27) = Y jd3pd3p' ( - r i + r o 2 f t _ m i _ m 2 A ( - P , ^ m\m2\ where we have made the replacements mj —> —mi (i = 1,2), p —> —p, p ' -> —p' (since an integral over all momenta is unaffected by such a change of variables), and have used (6.13a) in the second equality. Comparing this to (1.25), we see that for (1.26) to be satisfied, we require hmim2x(p,p') = ( - ) m i + m 2 / > l m i - m 2 A ( - P , - p ' ) - (1.28) Appendix I. Restricting the form of Hi 92 Starting from the right hand side of this expression and using (1-24), we have ( _ ) m 1 + m 2 / l l m i _ m 2 A ( _ P ; _ p / ) = ( _ ) m 1 + m 2 ( _ . ) £ ( i i _ m 2 _ 7 | I _ m i ) . / m m ' • ( i f m m ' | j 7 ) y ^ ( - p ) y ; m , (-P')/I«V,-A(P,P') = i ( _ ) l + m i + m 2 ^ ( - ) - ( m + m , ) ( - ) 5 + J - 2 ( l j m 2 7 | l m i ) . / m m ' t'i • (ii'mm'i ji) (-)'+' ' ( - ) ™ + ™ ' y ( _ m ( p ) y r _ r a , ( P ' ) • •/>,*'j A (P» P') = £ (_)<+''+i ( i i m 2 7 U mi) ( i V -m - m ' | j 7 ) • Imm' i'i •Ylrn{mVrn>{i>')h*WjX(p,p') = i(-)1+m>+m* Y, ( - ) l + l ' + i ( U r n 2 l \ i m i ) • Imm i'i . (_)'+!'-> ( H ' m m ' l j -7) y , r a (p) W(p') /»iV J-A(P.P ' ) = i ( _ ) i + m i + m a £ ( i j m 2 7 | i m 1 ) ( / / ' m m ' | j - 7 ) -Imm' i'i •Ylm(p)Yi'm'(p')hu,jX(P,p'), (1.29) where we have used (B.17a) in the first and forth equalities, (C.2) and (C.5) in the second equality, made the change of dummy indices m —> — m, m ' —> — m' in the third equality, and used the fact that ( _ ) 2 ( W ) — i ( s i n C e both I and /' are integers) in the fifth equality. Now, since the two Clebsch-Gordan coefficients in the fifth equality impose the relation m 2 — m i = — (m + m') (otherwise one or both of Appendix I. Restricting the form of "Hi 93 them vanish), it follows that ( - ) m 2 - m i ( - ) - K m ' ) = 1 and so (-r*+m*hlmi_maX(-p, -p') = £ ( - ) - ( « ' ) ( i j m 2 7 | i m O Imm' i'i • (H' mm'\j -7)Ylm(p)Yllml(P')h;iljX{p,p') = i E ( - r ( m + m , ) ( i J m2 7 | | m x ) (11'm m' | j -7) Imm' i'i (1.30) • y i m ( p ) y r m ' ( p ' ) ^ A ( P > P ' ) , where, in the second equality, we have used the fact that ( — ) 1 + 2 m 2 = l (since m 2 is half odd integral). Comparing this result with (1.24), we see that for (1.28) to be satisfied, we require that hwj\(p,p') = h*WiX{p,p'). (1.31) Thus, hu'jX(p,p') is a real valued function of I, I', j = 0,1, A, p and p'. Hence, imposing spatial rotation and space inversion and time reversal invariance on Tii restricts the vertex function hmim2X(p, p') to be of the form hmim2x(p,P') = t E R - ( m + m , ) ( \ J m2 7| 4 ™i) (U'mm'\ j -7) • Yim{p)YVmi (p')hu,jX(p,p'), Im I'm'j (1.32) where hu'jX(p,p') satisfies (1.23) but is otherwise an arbitrary real valued function of I, V, j = 0,1, p and p'. A p p e n d i x J R e s t r i c t i n g the f o r m of Hi J . l I m p o s i n g S p a t i a l R o t a t i o n Invar iance o n "Hi In this appendix, we impose the requirement that Hi be invariant under spatial rotations. That is, we require that n(aBi)HiV) (aBi) =Hi (J. l) be satisfied. Applying the spatial rotation operator to Hi and using (6.9a), (6.9b), and (6.9c) yields TliaB^HillHaPy) = £ Jd3p d3p' h^^p^D^fap^Dl^apj) • 7711 777-2 A • Fl, ( P - R ) ^ , ( P'_R)SA (P-R + P-R) + adj. Making the change of variables p —> PR , p ' —>• p R (since an integral over all momenta is unaffected by such a change) and omitting the angular arguments of the rotation operators and Z?-matrices gives TlHirt= £ fd3pd3p' Y^im2x(Pn,p'n)Di,mDi,m2-m i m 2 A m i m 2 p ^ . F ^ ( p ) F ^ ( p ' ) S A ( p + p')+adj. If this expression is compared with (7.1f), we see that (J . l) is satisfied if _ _ K _ i i h m i m 2 \ { P , p ' ) = 2^  ^ I m i x t P R . P l l ) ^ ^ ; ^ ™ ! , ( J - 4 ) holds. Now, comparing this condition to that given in (E.12) of section (E.2), we see that the two are identical (except for the additional spin index A in hmim2x(p,p') and the isospin indices ui, /i2, and u3 in hm\^3(p,p'), neither of which are involved here). Thus, we can immediately conclude the result of that section: (J . l ) is satisfied if hmim2x(p,p') is of the form hmim2x{p, P') = i £ ( - ) - ( M + M ' ) ( I I m i ma | j - 7 ) ( J J 'm m ' | j 7) hWjx(p,p')Ylm(p)Yllm.(p'), Im i'm'j (3.5) 94 Appendix J. Restricting the form of Tii 95 where hwjx(p,p') is an arbitrary complex valued function of I, V, j, A, p and p'. Note, because of the Clebsch-Gordan coefficient ( 1 1 m\ m 2 | j —j),j is only summed over the values j = 0,1. J.2 I m p o s i n g Space Invers ion Invariance on Tii At this point, Tii has been restricted to be of the form Hi= Y y*d 3p d V ^ i m 2 A ( p , p O^L 1 ( p ) ^ a ( p O S A ( p + pO + «&", (J-6) mim2A where hmim2x(p, p') is of the form given in (J.5). We now impose the requirement that Tii be invariant under space inversion; that is, we require F H / p t = Ui (J.7) be satisfied. Applying the space inversion operator to Tii and using (6.10a), (6.10b) and (6.10c) yields 7 W > + = £ j ^pd^p'h^rn^p'^eF^i-p)^ (J.S) = Y / d 3 P d V ^ m i m 2 - A ( - p , - p 0 ^ m i ( p ) ^ m 2 ( p ' ) ^ A ( p + p') + a ^ mim 2 A where in the second equality we have used (6.11a), made the change of dummy index A -¥ - A , and used the fact that an integral over all momenta is unaffected by the transformation p —> — p. In the second equality, we have also used 7TP(—)x = 1. Comparing this expression with (7.1f), we see that for Tii to be invariant under V we require hmim2x(p,p') = hmim2-\(-p, -p ' ) (J.9) hold. Starting from the right hand side of (J.9), and using (J.5), we have hmim2-x(-p,-p') = i Y (-yim+m'} i m i m * \ i V Im I'm'j = i E ( - r ( m + m , ) (i im im 2 | j-l) {IV'mm'\j j)hwj-x(p,p') • (-)w'Ylm(p)Yllml(p''). {3 AO) Im I'm'j Appendix J. Restricting the form of Hi 96 Hence, for (J.7) to be satisfied, we require hwj\(p,p') = hWj-x(p,p'). (J.ll) Thus, imposing spatial rotation and space inversion invariance on Hi restricts the vertex function hm1m2\(Pi P') to be of the form hmim2x(p,p') = i £ ( - ) - ( m + m ' » ( ! ^ i m 2 | j -7) ( i / ' m m ' l j 7 ) • Ylm (p)yj'm> {p')hlV jX(P, P'), tm I'm'j (J.12) where hwj\(p,p') satisfies ( J .H) but is otherwise an arbitrary complex valued function of /, / ' , j = 0,1, p and p'. J.3 Imposing Time Reversal Invariance on Hi At this point, Hi has been restricted to be of the form Hi= | d 3 p d V ^ m i m 2 A ( p , p O ^ L 1 ( p ) ^ m 2 ( p O ® A ( P + P O + a ^ (J-13) mim2A where hmim2\(p, p') is of the form given in (J.12). We now impose the requirement that Hi be invariant under time reversal; that is, we require that THiT* = Hi (J-14) be satisfied. Applying the time reversal operator to Hi and using (6.12a), (6.12b), (6.12c) and (6.12d) yields T H , T t = £ f d3P d*P> h m i m 2 X ( P , P > ) . mi 7712 A • Te(-)*-m>Fimi (-p)re(-)*-ro2^m2(-p')rp(-)1SA(-p - p') + adj (J.15) = J£jd*P d3P' (-)m]+m^-mi-m2A(-p, -p')^L1(p)^ 2(P')SA(p + p') + adj, mim2\ 1 where we have made the replacements rrij —> —mi (i — 1,2), p —> —p, p' —> —p' (since an integral over all momenta is unaffected by such a change of variables), and have used (6.13a) in the second equality. Comparing this to (J.13), we see that for (J.14) to be satisfied, we require fcmim2A(p,P') = ( - ) m i + m 2 / > i m i - m 2 A ( - p , - p ' ) - (J.16) Appendix J. Restricting the form of "Hi 97 (J.17) Starting from the right hand side of this expression and using (J-12), we have ( - ) m i + r a ^ l m i - m 2 A ( - P > - p ' ) = (-)m i + m a(-0 £ ( - ) - ( m + m , ) ( H -mi -m 2 | i 7) • Imj I'm' •(U'mm'lj -7) ^(-P)^,(-p')h; i l j X(p,p') = j ' _ ) i + « i + m 2 ^ ( . ) - ( ™ W ) ( _ ) H r i ( i i m i m 2 | j - 7 ) • Imj I'm' • (ll'mm'l j - 7 ) ( - ) ' + ' ' ( - ) m + r a ' y ; _ m (p )F r _ r a - (p ' ) • •hwjxiPiP1) = i(-)m^+m> £ ( - ) ' + ' ' - > ( H m i m 2 | j - 7 ) • Imj I'm' •(ll'-m - m ' | j - 7 ) y<m(p)11<M<(P')hhjX(P,p') = i(-)m*+m> £ (_ ) '+ ' ' - • . ( H m i " » 2 | j - 7 ) • I'm' . (-) '+' '-* ( / Z ' m m ' l j>)r,m(p)yj.T O.(p')>i,VJ-A(P.P') = i ( _ ) m 1 + m 2 J2(^m1m2\j-j)(ll'mm'\j7)-Imj I'm' • Ylm(p)Yi,m>(p')hu,jX(p,p'), where we have used (B.17a) in the first and forth equalities, (C.2) and (C.5) in the second equality, made the change of dummy indices m -> -m, m' -> —m' in the third equality, and used the fact that ( _ ) 2 ( W - j ) _ i ( s i n c e j ; // a n d j are integers) in the fifth equality. Now, since the two Clebsch-Gordan coefficients in the fifth equality impose the relation mi + m 2 = —(m + m') (otherwise one or both of them vanish), it follows that ( - ) ™ i + ™ 2 = (_)-(m+m') _ x a n d s 0 ( - ) m i + m 2 ^ - m i - m 2 A ( - P ; - P ' ) = i Y(-rim+m,) ( H m i m2\j - 7 ) • TJ' (J.18) • (/ I' m m'| j 7 ) y < m (p)yi 'm' (P ' ) ^ K ' J A ( P . P ' ) -Comparing this result with (J.12), we see that for (J.16) to be satisfied, we require that hu>jx(p,p') = hu,jX(p,p'). (J.19) Appendix J. Restricting the form of Hi 98 Thus, hwj\(p,P') is a real valued function of I, I', j = 0,1, A, p and p'. Thus, imposing spatial rotation and space inversion invariance on Hi restricts the vertex function /imim2A(p,p') to be of the form hmim2x(p,P') = * £ (-)-< r o + m '> ( i 2-m1m2\j -7) (II'mm'\ j7) • I'm'j Ylm(p)Yl'm' (P')hll'j\(P,P'), (J.20) where hu>j\(p,p') satisfies ( J - l l ) but is otherwise an arbitrary real valued function of I, V, j = 0,1, p and p'. Appendix K Imposing Charge Conjugation Invariance on Hi In this appendix, we impose charge conjugation invariance on the interaction Hamiltonian, Hi = Hi 4-Hi + Hi. A t this point, we assume that Hi, Hi and Hi have already been restricted to be invariant under TZ, V and T. See the results of section 7.3.1 for details. As will become apparent in the discussion to follow, demanding that Hi be invariant under charge conjugation imposes a relation between the vertex functions of Hi and Hi and places a constraint on the vertex function of Hi- In light of this, we shall impose charge conjugation invariance first upon Hi + Hi, and then on Hi. Note, the same results are obtained if we impose charge conjugation invariance on all three terms of Hi simultaneously, it is just much more cumbersome to present. K . l Imposing Charge Conjugation Invariance on Hi +Hi We now impose the requirement that HI+HI be invariant under charge conjugation. That is, we require that be satisfied. Applying the charge conjugation operator to Hi + Hi as defined in (7.1d) and (7.1e), and using (6.14a), (6.14b) and (6.14c) gives C(Hi + H!)& = (HI+HI) (K.l) (K.2) +( - ) 1 / im 1 m 2 A(p , p')Fmi (p)Fm2 (p')BA(p - p')) + adj, 99 Appendix K. Imposing Charge Conjugation Invariance on Hi 100 where in the second equality, we have used (3.17a) plus the fact that KP = —1 (see page 11). Comparing this result with (7.Id) and (7.1e), we see that ( K . l ) is satisfied if hm,m2x{P,p') — -hmim2x(P,p')- (K.3) Thus, imposing spatial rotation, space inversion, time reversal and charge conjugation invariance on Hi + "Hi restricts it to be of the form + H ' = E Jd3pd3p' {hmimaX(p,p')Fmi(p)Fma(p')'Bx(p-p') / T i l 7712 A - /lm1m2A(p,p')^L1(p)^m2(p')^A(p-p')} + adj, (K.4) where hmim2\(p,p') is of the form given in (1.32). K.2 Imposing Charge Conjugation Invariance on Hi We now impose the requirement that Hi be invariant under charge conjugation. That is, we require CHitf =Hi (K.5) be satisfied. Applying the charge conjugation operator to Hi as defined in (7.1f) and using (6.14a), (6.14b) and (6.14c) gives CHjtf = E jd3P d V ^ 1 m 2 A ( p , p ' ) « e i ? m i ( p ) « e ^ L 2 ( p ' ) K p B A ( p + p ' ) + adj 77117712 A = E Jd3pd3p' K P ^ m i m 2 A ( p , p O ( - ) 1 f L 2 ( p ' ) ^ m i ( p ) % ( P + P') + adj (K.6) mi 777-2 A = E fd3P d V ^ m 2 m 1 A ( p ' , p ) f L 1 ( p ) i l 2 ( p ' ) B A ( p + P ' ) + adj, 7711 7772 A where in the second equality, we have made the change of dummy indices m i f>m2 and interchanged the variables of integration, p «-> p'. We have also used the fact that Fmi(p) and -F m 2(p') anticommute (see (6.4e)) in the second equality. In the third equality we have substituted KP = —1 (see page 32). Thus, (K.5)is satisfied if /IT7II77I2A(P)P') - hm2Tnix(p',p) (K.7) holds. Appendix K. Imposing Charge Conjugation Invariance on Tii 101 Using (7.23), we see that hm2mi A(P\ P) = i £ ( - ) - ( ' " + " ' ) ( I I m2 mi\j -7) (I V m m'\ j 7) • Imj I'm* • Yim(p')Yi>m> (p)hwj\(p',p) = i ^ ( - ) - ( m + m ' ) ( - ) 5 + 5 - j ( H m i m2lj-7)('''m'm|i7)-;'m' (K.8) •YVml(p')Yim{p)hvlj\{p' ,p), where we have used (B.17b) and made the change of dummy index I /' in the second equality. Thus, using (B.17b) a second time, we have hm2mix(p',P) = i £ ( - ) - ( ™ + ™ ' ) ( - ) 1 ^ ( H m i m 2 | j -7) (-)'+''->' ( H ' m m ' | j7) • Imj I'm' • Yim(p)Yiim< {p')hVij\{p' ,p) = i Y,(-r{m+m'] ( i i m i m 2 | j -7) {W m m ' l j 7 ) • /mj • r i m ( p ) r i . m ' ( p ' ) ( - ) , + , ' + 1 ^ i ' i i A ( p , , p ) So, for (K.5) to be satisfied, we require (K.9) hw3x(p,p') = (-)'+' +lhvliX(p\p) (K.10) holds. Thus, imposing spatial rotation, space inversion, time reversal and charge conjugation invariance on Hi restricts it to be of the form Til= £ J d 3 P d 3 P ' /lmim2A(p,p')^L1(p)^m2(p')^A(p + p') + adj, mi 7712 A (K.ll) where hmim2\(p,p') is of the form given in (J.20), but with the extra requirement that hu'j\(p,p') satisfy (K.10). Bibliography [1] D . J . Hearn. The dressing transformation and its application to a fermion-boson trilinear interaction. Master's thesis, The University of British Columbia, 1981. [2] D . J . Hearn, M . McMil lan , and A . Raskin. Dressing the cloudy bag model: Second-order nucleon-nucleon potential. Physical Review C, 28(6), December 1983. [3] S. Weinberg. The Quantum Theory of Fields. Cambridge University Press, 1995. [4] P. A . Kalyniak. Consequences of space-time invariances in quantum mechanics and direct interac-tion theories. Master's thesis, The University of British Columbia, 1978. [5] M . E . Rose. Elementary Theory of Angular Momentum. John Wiley & Sons, Inc., 1957. [6] S. S. Schweber. An Introduction to Relativistic Quantum Field Theory. Harper and Row, 1961. [7] M . L . Goldberger and K . M . Watson. Collision Theory. John Wiley & Sons, Inc., 1964. [8] R. Shankar. Principles of Quantum Mechanics. Plenum Press, 1980. [9] L . E . Ballentine. Quantum Mechanics. Prentice Hall , 1990. [10] J . R. Taylor. Scattering Theory, chapter 7, pages 119-127. John Wiley & Sons, Inc., 1972. [11] Eugene P. Wigner. Relativistic invariance and quantum phenomena. Reviews of Modern Physics, 29(3), July 1957. 102 

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