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The Schwinger model Link, Robert 1986

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T H E S C H W I N G E R M O D E L . by Robert Link B. Sc., University of Victoria, Victoria, 1984 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of M A S T E R OF S C I E N C E in The Faculty of Graduate Studies Department of Physics We accept this thesis as conforming to the required standard , The University of British Columbia July 1986 © Robert Link, 1986 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Physics The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall V ancouver, Canada V6T 1Y3 Date J u l y 29, 1986 11 Abstract The Schwinger model (quantum electrodynamics with massless fermions in two space-time dimensions) is investigated. The eigenenergies of the Dirac Hamil -tonian are found to depend on the gauge field in a gauge invariant manner. Some of the eigenstates of the full quantum Hamiltonian are found in the Fock space of the free fermionic Hamiltonian. The bosonization of the model is presented and used to obtain the axial anomaly equation, the exact two-point Wightman functions, and the current correlation function. Other two dimensional theories are discussed with emphasis on the aspects which are similar to the Schwinger model. iii Table of Contents Abstract ii Table of Contents iii Acknowledgments iv Section 1: Introduction 1 Section 2: Pure Electrodynamics on The Circle 3 Section 3: The M o d e l A n d It's Solution 5 Section 4: Bosonization 24 Section 5: Other Models 34 Section 6: Discussion 42 Bibliography 43 IV Acknowledgments I thank my research supervisor Dr. Gordon Semenoff for indicating an area requiring research, and providing guidance for that research. 1. Introduction l The Schwinger Model is quantum electrodynamics in two spacetime dimen-sions with vanishing fermion mass; the two independent Fermi fields, one left-handed and the other right-handed, are both of charge 1. It was introduced by Julian Schwinger in 1962 1 to provide an example of a vector gauge field with a nonzero mass particle. Working in the Lorentz gauge he found via functional methods the exact Green's functions; Brown 2 shortly afterwards obtained them in the Coulomb gauge. Schwinger derived the gauge field spectral function p(m2) = 6(m2 — e2/ir); i.e. the mass spectrum is localized at one point—describing a stable particle of mass e / 7 1 - 1 / 2 . The reason for this is that the theory is equivalent to a free massive scalar theory in two dimensions with the physical particle having mass e / T r 1 / 2 . Ex-hibiting the correspondence between the two theories is known as bosonization, and in fact many two dimensional fermion theories are equivalent to two dimensional boson theories. The bosonization of the Schwinger model is presented in section 4, the bosonization of other models is briefly discussed in section 5, and the physical meaning of bosonization is looked at in section 6. Bosonization does not provide a solution written in terms of the original vari-ables of the theory, and with the later goal in mind (section 3) we look at the Schwinger model in the canonical Hamiltonian formalism. We are partially success-ful in constructing the stationary states of the full Hamiltonian in the Fock space of the free Fermionic Hamiltonian. Following N.S. Manton's lead 3 we suppose space to be a circle of length 27r; the usual version of the model is easily obtained by letting the length be 2nR and taking the limit R >-*• 00. Pure electrodynamics on the circle is briefly discussed in section 2. A l l future references to Manton's work are meant to be reference [3]. Because the fermions are massless the Lagrangian is invariant under chiral symmetry, and in the classical theory the corresponding Noether current— the axial current—is therefore conserved. However the divergences which appear when quantizing a field often cause the classical conservation laws to be lost: the axial anomaly being a prime example. The axial current has a divergence proportional to the electric field strength; and can not, in contradiction to Jackiw's claim *, be induced purely via gauge transformations. 2. Pure Electrodynamics on The Circle 3 The Lagrangian is C=l-{dtAx-dzAt)\ (2.1) We choose the gauge At — 0 and are left with the residual freedom of time independent gauge transformations which are of the form A'x = AX- i{dxg)g-1 where g{x) = e '^ 1 ' . (2.2) Demanding that g be single-valued implies A(27r) = A(0) + 2nn where n is an integer. (2-3) When n = 0 the transformation is called a small gauge transformation; when n ^ 0 it is called a large gauge transformation. The zeroth Fourier component of Ax may be brought to the interval [0, 1] because 1 [ 2 i r A0 — — I Axdx 2ir JQ 1 f2w [2 4) ~ — / {Ax + dx\{x))dx K ' ; — A0 + n. A0 = 0 and A0 = 1 are gauge equivalent as they are related by the gauge transfor-mation g(x) — e,x. The electric field is the canonically conjugate momentum to Ax 8 £. Ex = — = dtAx (2.5) 6AX and there is no momentum conjugate to At. The Hamiltonian is given by •2TT ExdtAx - i 2.6 •2TT V ' Eldx. .If 2 Jo To canonically quantize this system we impose the commutator [Ax,Ex] = i. (2.7) We can represent Ax by the multiplication operator; Ex by the derivative operator Ex = -i-?- + 9 (2.8) 6AX for some constant 9. This is the quantized field version of the Schrodinger picture. Since 2nA0 is an angular variable, the wavefunction must be periodic in A0. A stationary state has the form <p{A0) = e 2 n t n A " where n is an integer. (2.9) It is an eigenstate of the electric field operator with eigenvalue n + 6, and an eigen-state of the Hamiltonian with energy n{n + 9)2. These states are invariant under both large and small gauge transformations. In the following we will set 9 — 0 be-cause Man ton has shown 9 can be removed from the theory by a change of variables. 5 3. The Model and It's Solution The Lagrangian density of the Schwinger model is L = \{dtAx - dxAt)2 + + ieA^ (3.1) where 0 = ^ 7 ° 1° = <JX I'^-ia2 y» = y y = a 3 (3.2) the a's being the Paul i matrices. On the two-component spinors we impose the canonical anti-commutation relations (3.3) In the presence of matter Gauss' law reads dxEx = e ^ V ; (3.4) which when integrated implies the total electric charge must be zero. Since Ex = - » ' - — (3.5) 8AX ( ' we have the following constraint on physical states (where we write A(x) instead of Ax(x)). {idx JX[x) + ^^W)\*pi*»ic*i) = °- (3-6) Standard manipulations of the Lagrangian give us the Hamiltonian We want to solve the system as defined by (3.6) and (3.7). We will expand in the eigenmodes of the first quantized Dirac Hamiltonian h = a3[ij--eAx). (3.8) Solving the eigenvalue equation H^>„ = with ••-(SI) gives (3.9) V> (^x) — e x p ^ - t u . n x - ie j A(x') dx' and periodic boundary conditions imply un = n + eA0. (3.10) We define the operators aln to be the operator coeffcients in the Fourier expansion of the i'th component of the field operator (3.11) 7 Then the canonical anticommutators (3.3) imply the anticommutators it (3-12) and substituting eqn. (3.11) into the Hamiltonian (3.7) shows that it becomes H^Hl + H2 (3.13) where (3.14) We see that o p is the creation operator for a positive chirality particle of energy u>p which we call left-handed, and is the creation operator for a negative chirality particle of energy -u/p which we call right-handed. We could now redefine these operators to obtain the usual particle-hole description, in which case the minus sign in H2 would disappear as H2 would then be the sum of the energies of the particles and the antiparticles. This redefinition is however not necessary (we just remember an antiparticle is an empty negative energy state) and we follow Manton in not making it. We use the a operators to define the the Fock space of H2, and then use the basis of this space to construct physical states of the full Hilbert space. 8 The vacuum of Hi is defined by the following conditions: 4|0> 0 for UJn > 0 0 for un < 0 0 for u.„. < 0 0 for un > 0 Then applying these operators appropriately to |0) gives, for fixed Ax, a basis of states in which each energy level is specified as either empty or filled, and in which all but a finite number of negative energy levels are filled and only a finite number of positive energy levels are filled. Call this collection of states {|.F)}. We expand a solution to the full system in the basis of Solving the system entails finding Xf, and we start by showing \F) obeys the constraint (3.6). On our way to obtaining a derivative by A we show the following: if we define A (up to an additive c-number) by (3.15) F (3.16) where a is any of the elementary creation or annihilation operators, then 6\F) = *\F)- (3.17) 6A Proof: Since \F) — a ... o'jO) for some a operators SJFl 8A ISA , a... a |0> + o---«t^|0>. It is easily verified that eqn. (3.16) implies A,a ... a1 ; so ± ^ = ^ - . . . . . • ( . 1 1 0 ) - ± 1 0 ) ) and we need to show A\0) = {^|u)- Because A is only defined up to a constant we may set {0\A\0) = (0|^|0) (then demanding that (0\EX\0) = 0 gives {0\A\0) = 0). Now let \G) be an arbitrary basis state other than |0) : \G) = a[ .. .an|0) where a, means aj or a 2 . Then (G\A\0) = (0|o'B...oiyl|0) = (0\[atn...al,A\\0) (0| '6A |0) = ( 0 K - . . O ! —|0) and since \G) is arbitrary we have our claim (3.17). We proceed to find an explicit expression for A. Since the electric field operator and the fermion field operator commute we have [SA(y) = 0. Which by eqn. (3.16) implies and the orthogonality of the 0n's implies where With the use of the anticommutators (3.12) it is easily verified that to satisfy eqn. (3.18) A must be (up to an additive c-number function) A = E Mnn^m^n + f, (3-20) m,n,t where / is a c-number function. By substituting eigenmodes (3.9) into eqn. (3.19) Almn is easily evaluated; then relation (3.20) gives us (we set / — 0 to obtain (0|>l|0)=0) My) = ~E{(--y)E°C< + « E j^^ai); (3 .21) so that m,n (3.22) A n d by eqn. (3.17) \F) obeys the constraint (3.6). 11 Now let us pause and Fourier analyse the 2n periodic function A(x): oo A(x) = J2 A " e ~ l n X (3.23) 1 f2r An = — j A{x)einxdx. We demand our full solution (3.15) obey the constraint (3.6). Then since \F) obeys the constraint we have Which by the completeness of {(F)} implies d x ^ = 0. Now by the chain rule for differentiation 6XF dXF 6An dXFeinx 6A{x) dAn b~A{x) dAn 2TT ' d x. Therefore — 0 and we have XF — XF{A0). We Fourier analyse Gauss' law (3.6): / 0 8 ' e " , ' " * ( , ' d ^ + e ^ ( a ) ) ^ > = ° -Use of the chain rule in the first term and substitution of the expansion (3.11) in the second term yields the momentum space form of Gauss' law : d n- e £ 4 ' « U = 0. (3.24) Since \F) obeys the constraint (3.6), for n ^  0 this gives us the partial of \F) by An. Alternately, using the chain rule and eqn. (3.17) one can easily show that d\F) = / e-imxA{x)dx\F); (3.25) with A given by eqn. (3.21) this is in agreement with eqn. (3.24). As mentioned earlier, Gauss' law implies the total electric charge of a state must be zero. Using • t the fact that ^ aJm a3m is the charge, eqn. (3.25) for the case m — 0 implies that the partial derivative of \F) by A0 is zero. We are now ready to put our full Hamiltonian (3.13) into the momentum representation. By the chain rule and eqn. (3.24) W V- e i r e i v-. tt . . 1 d 6Alx) ^ 2 n n ^ * k ~ n { 1 2*dA0 Then keeping in mind our expansion (3.15) for \if>) and using the fact that alk_n\F) is just another \F) state and obeys Gauss' law gives 62 ^ e 2 e i ( m + ' 1 ' 1 t • .t • 6A{x)2 ^-f. (2TT)* mn k.l.i.j e eimx A , d I d 2  + ^ 2 ^ ^ a ^ d A - o m + Integrating this expression from 0 to 2ir tells us the Hamiltonian may be written as ^ = ~ i ^ y + ^ W r l ( a " ' a ^ a ' t a " ) + 4 ^ ^ ^ ^ n ^ ^ - n - (3-26) Important operators are the vector current (we set e = l) 13 j» = ^ Yi> (3.27) and the axial current # = ^ y y V ; , (3.28) which have components (3.29) j1 = j* = 1>\1>i - 4>li>2-Let us define momentum space chiral charge densities rlir P M = / ^a{x)^xdx (3.30) which upon substituting the Fourier expansion for xp become P M = J24+Pl<, (3-31) Jfc so that J*°(p) = Pi{p) + P2{p) and jl{p) = Pi{p) - P2{p)- Then we may write the Hamiltonian (3.26) as * = -hj7~* + E t f » ( » i ' « i - *n*D + JZE ^J°(-n)j°(+n). (3.32) This agrees with Manton's Hamiltonian when the different gauges are acounted for; in fact one need only replace A0 by his Ax (which in his gauge is independent of x and lies in the interval [0, l]) to obtain his Hamiltonian. The first term represents the energy in the gauge field, the second term is the fermion kinetic energy, and the third term is the coulomb energy. From the form (3.10) for wn one sees as A0 increases from 0 to 1 the energies of the left-handed particles increase by one, while the energies of the right-handed particles decrease by one. This is a spectral flow of two which is due to the axial anomaly. The anomaly equation (divergence of the axial current) will be derived later. Jackiw * working with the same model (except on the infinite spatial line), in the same gauge, found the energy of the eigenmodes (3.9) to change when AX went from 0 to a constant 6AX . This resulted in a spectral flow induced merely by performing a gauge transformation. However his analysis invoked no boundary conditions. Our boundary conditions (which give us boundary conditions at infinity when the length of the circle is taken to infinity) yield the form (3.10) for the eigenenergies, which we will see to be responsible for their gauge invariance. For consider performing the gauge transformation AX >-* AX + dXA.(x)\ then by equations (2.4) and (3.10) un K-+ ujn+m where ra is the winding number of the transformation. We see that a small gauge transformation leaves the spectrum invariant, while a large one permutes it. Now suppose (like Jackiw did) that the background field is purely gauge (i.e. no electric field). Then intially Ax(X,T — 0) = dXA.p(x) where r is a parameter which may be thought of as time, and A P obeys condition (2.3). A spectral flow while maintaining a pure gauge field requires Ax(x, T — 1) = dXA.m(x) for m ^ p, with a continuous function A(x, r) joining A P and A M . A p ( i ) = A ( x , r = 0) A m ( x ) . = A(x, T = 1) these imply A(2TT,0) = A(0,0) + 2np A(2TT, 1) = A(0, 1) = 27rm. Now for any sufficiently small but finite dr the continuity of A along with the gauge condition (2.3) which applies to A for any r implies Increment r a finite number of times, repeating the argument each time, to find Thus rn — p contradicting our assumption of spectral flow. A mathematian would say the gauge groups labelled by the integers form distinct homotopy classes. A spectral flow requires an electric field , i.e. changes in Ax other than those due to gauge transformations. As mentioned in section 2 A0 may be brought to the interval [0,1], and if we make this gauge choice then only small gauge transformations are allowed. However consider making the large gauge transformation g(x) = elx. The eigenmodes (3.9) transform as ip'n[x) g~l(x)^xn(x) — i>ln+l[x) so that a}/ now creates ipln+x(x) and we write aln a l „ + 1 . So under g, the n'th level becomes the (n+l)'th level and it has energy wn+i. By the previous argument this must merely be a relabeling of the energy levels rather than a real change in particle energies. We will later demand observables to be invariant under g. We will find it desirable to have the second term of the Hamiltonian (3.32) written in terms of the chiral charge density operators. This is accomplished by the Sugawara-type formula A(2TT, dr) - A(0, dr) + 2irp. A(2TT,T) = A(0, T) + 2%p a - a n n whose proof is due to Manton and is as follows. Let \F) be a basis state; associated with it is an unexcited basis state, with the same numbers of left and right handed particles. Call this state | M , N) to indicate that the left-handed particles fill the levels < M and the right-handed particles fill the levels > TV. Now the chiral charges Qr, and QR are the eigenvalues of the charge operators p\ (0) and /92(0) both of which are infinite and therefore regulated in a gauge invariant manner (note the invariance under g): pj(0) = £ a j ' a j e ^ (3.34) ^ ( o ) = E < ^ - A w p -p The limit A \~* 0 will be taken eventually. Since exciting a particle does not change its charge, the charge of \F) is QR = 2s e n>N Evaluating the sums for small A yields Q i = x + ( m + a ° + h + \ w + a ° + \ y - A + ° ( a 2 ) (3.36) QR = I - [N + A0 - l-) + ±{N + A0 - l-f - A + 0 ( A 2 ) . The absolute regularized charge is defined by subtracting the divergent constant l / A and taking the limit A i — • 0: Q? = M + A 0 + \ (3.37) Q'i' = -N-A0 + l. (3.35) 17 Then demanding the total electric charge be zero implies N — M + 1 so that Q\eg = 2M + 2A0 + I. (3.38) Thus we see the dependence of the axial charge on the background field: as A0 increases from 0 to 1, the axial charge increases by two because one left-handed particle and one right-handed antiparticle are created. The kinetic energy of \M) = \M, M + 1), regularized in the same manner as the charges, is given by m<M m>M + l which can be evaluated by differentiating equations (3.36) by A, subtracting the divergent constant and taking A to zero. V£>(A0) = (M + A0 + 1/2) 2 - (<?"72)2 (3.39) - ^ ( ^ ( 0 ) ) 2 + ^ ( ^ ( 0 ) ) 2 To get the energy of \F) we must add the excitation energy. This is shown by Manton, by using the anticommutators (3.12) and doing a counting argument, to be ^ E " I ( P ) P I ( - P ) + \ YJPI{-P)PI{P)- (3-40) So we indeed have (3.33) with the appropriate regularizations implied. We have checked that both sides of the formula have the same commutator with a*. Our Hamiltonian now takes the form H — Hi + if 2 + H3, (3.41) 18 where Hx = -i d1 (3.42) 4*dA02 #2 = +^ £ P l ( p V l ( - p ) + *>2(-p)Pzfa) P # 3 - \{pl{p) + P2{P)){PI{1V) +Pl{-P)). p#0 r We will use the method of Mattis and Lieb 5 to put this Hamiltonian into a nicer form. From Manton we have the momentum space charge density commuta-tors: Pi{-p),p\{q) = P6; P9 P2[-p),P2[q) = - p * (3.43) p\{p),P2{q) = 0 . Manton demonstrated these for p, c/ ^ 0, however his method is easily extended to show they are true for p or q — 0 as well. Define a hermitian operator 5 = 1 ' E <^-p\{p)p2{-p) p P#0 (3.44) where <t>{p) is a real, even function of p to be determined. We will canonically transform H (3.45) H' = elSHe-lS using the well known operator equation eABe~A = B + [A,B\ + -\A,\A, B)\ + it • (3.46) Now from the commutators (3.43) we obtain the commutators 19 *S,Pi{p) =<f>{p)p2{p) iS,p2{p) = <t>{v)P\{p) (3.47) which used in the expansion (3.46) yield JS P\{PY -iS c1* p2(p)e %s = p2(p) cosh <p(p) + pi(p)s\nh <j>(p). Pi(p) cosh <f>(p) + p2{p) sinh <f>(p) (3.48) We are now ready to transform H. Well since [atk+pap\F)^ — 0 for arbitrary \F) states we have 3 ^ ~ ( a t + p a p ) = 0, which implies *S, -JL-* = 0. So by (3.46) Hi transforms to itself. Using re-lations (3.48) and defining <f>(0) = 0, H2 and # 3 are easily found to transform to H2 = ^J2[pi{p)pi[-p) + P2{p)P2{~p)) (cosh 2 <p + sinh 2 <f>) p + 2 Pi[p)P2[-p) cosh tftsinh <f> (3.49) 2 j ^ 3 = ^ E T ( ^ I ( P ) P I ( - P ) + PI{P)PI{-P) + 2pi{p)p2{-p)) (cosh 2 <f> + sinh 2 <j> + 2 cosh ^ sinh <f> j . We wish to eliminate the />i/»2 terms. By use of the identities 2 cosh <f> sinh <f> = sinh 2<£ 2 2 cosh </> -I- sinh <p — cosh 2<£ (3.50) (3.51) we find in order to do so we must choose <f> such that coth2<£ = - ( l + ^ y - ) - (3.52) This relation is always possible to satisfy, and in fact if we make this choice for ci the Hamiltonian assumes the remarkably simple form = + t f 7 (3.53) where - 1 & 1 2, . ' 1 2 , * _ ! v ( 3" 5 4) H1 = L , ^ ( e V * + P 2 ) 5 ( / » l ( p ) / > l ( - p ) +P2{P)P2{-P)). Let us define m = e/y/ir; then using the commutation relations (3.44) gives Hi = Y. ^ ^ ( A M P W - P ) + />2(-p)p2(p)) (3.55) p>0 p where we have dropped an infinite constant Vm2+P2- (3.56) p>0 The above ordering is desirable because states with no fermion ic excitations are an-nihilated by p\{ — p) and P2(p) f ° r p > 0. H p is identical to Hamiltonian (3.42) when no fermion ic excitations and no coulomb interaction are assumed. Hj therefore represents the excitation energy and the coulomb energy and is called the plasmon energy for reasons which will be clear in the next section. Since H0 and Hj com-mute, every eigenstate of H' may be assumed to be an eigenstate of H0 and Hj separately. 21 Recall that prior to the transformation by elS the full solution was given by \<P) = YI\F}XF(AO) with ^ = 0 dA0 The solution to our transformed system is \4>') = 52\F')XF{A0) • (3.57) F where \F') = eiS\F) (3.58) and j^- — 0 implies QA — 0. We proceed to find eigenstates of H0. Consider the collection of unexcited basis states which we will denote by (|M)}. From the form (3.55) of Hj it is clear that Hi\M) = 0, (3.59) so that any state constructed in the subspace spanned by {|M)} which is an eigen-state of H0 is also an eigenstate of H'. Let \xp) be such a state: \ip) = J2\M)XM(A°) (3-6°) M with X to be determined. Now from (3.54) and (3.39) H0\i>) = £ \M)[Z--— + ( M + A0 + -f)xM[A0). (3.61) Thus a stationary state \ip) is specified by the set of wavefunctions {XM(A0) : M C Z , 0 < A0 < 1} satisfying ( - 7 - I T T + (M + A0 + b2) XM(Ao) = £ J M K ) , (3.62) with boundary conditions *M(1) = ^M+i(0) (3.63) ^ I M ( 1 ) = ^ O X M + 1 ( 0 ) . The first boundary condition states that \M) when A 0 = 1 is the same state as \M + 1) when A0 — 0 (they have the same energy levels). The second says the two should join smoothly due to the angular nature of 27rA0. To solve the system (3.62), (3.63) define a wavefunction X(A) over the whole real line by XM{Ao) = X{M + A0) = X{A). (3.64) Then equations (3.62) reduce to the eigenvalue equation for the harmonic oscillator + [A+\)2)X{A) = E0X(A). (3.65) This oscillator has frequency m = e/yjn and it's wavefunctions are well known: Xn{A) = Hn{A)t-&X+W (3.66) where UN is the nth Hermite polynomial. The energy levels are EZ=(n+l-)m. (3.67) These states were constructed by Manton, however as he remarked they are not stationary states of the full original Hamiltonian. They are however stationary states of our transformed Hamiltonian and therefore e~xS\ip) are stationary states of the original full Hamiltonian, with energies given by (3.67). e~lS is the Bogoliubov transformation Manton mentions must exist. As previously mentioned the eigenstates of H' may be assumed to be eigen-states of H0 and Hj separately. Now suppose we have an eigenstate constructed outside the span of {|M)}, i.e. we have an excitation. Then the eigenenergy of Hj will be seen to be at least \Jm2 + 1 and since H0 is positive (it is the electric field squared plus the axial charge squared), we will have an eigenenergy greater than m / 2 . So the lowest energy state, i.e. the ground state of H' is the n — 0 state of solution (3.66). Explicitly it is Suppose we let the coupling constant go to zero; then we may choose A0 — 1/2 in which case \GS) >—> | — 1). This corresponds to the vacuum of the free theory because the energy levels of energy < - 1 / 2 are filled for both left and right components. When the coupling constant is nonzero there is a finite probability of finding any particular number of particles, this is due to the polarizability of the vacuum of which we will speak more of later. In the next section we will see the states constructed here correspond to the zero momentum scalar particles of the bosonized theory. (3.68) M 4. B o s o n i z a t i o n In this section the equivalence of the Schwinger model to a free massive scalar theory will be demonstrated. Following Mattis and Lieb 5 we define for p > 0 p-1/2P2(-P) = Bt(-p) p-1/2Pi(-p) = A(p) p-1/2p2(p) = B(-P). (4.1) Then Al{p) and Bt{—p) have the properties of Boson raising operators, while A{p) and B( — p) have the properties of Boson lowering operators; for the commutators (3.43) imply the commutators A, B = A ( p ) , A ' ( 9 ) = B{-p),Bt{-g) - 8. These definitions allow us to write the Hamiltonian (3.55) as Hj = £ ( m 2 + p 2 ) 1 / 2 ( ^ (p )A(p) + B ' ( - p ) B ( - p ) ) . p>0 (4.2) (4.3) Declare the B field to be the continuation of the A field to negative p; together they define a single Boson field defined for all p ^ 0. The Hamiltonian (4.3) may then be written as p/0 u{p)At{p)A{p) (4.4) 25 where u,(p) = (m 2 + p 2 ) 1 / 2 . (4.5) To complete the bosonization we must consider H0. Well using the expression (3.39) for the axial charge, H0 as given by (3.54) can be written d2 47rdA 0 2 [A+1/2Y (4.6) Let us define (4.7) then and up to an additive constant m/2 y/TT (4.8) Furthermore A(0),A(p) = A(0),Al(p) = 0 and ,4(0),^(0) = 1. So we have the completely bosonized form of the Hamiltonian: H'= ^uj(p)At(p)A(p) (4.9) with A(p) ,A ' ( j ) ] =6 p , A(p),A(a)] = [ A ' ( P ) , ^ ( 9 ) (4.10) - 0. 26 If we replace the constants we had previously dropped, namely the one given by (3.56), and m/2, then we may write H'^l-J2^(p){At(p)Mp) + Mp)At(p)). (4.11) Now let us define $(p) = (2 W (p) ) - 1 / 2 (A(p) + A £ ( - P ) ) (4.12) Note that n'(p) — IT(-p) and $e(p) = $(-p). Definitions (4.12) are easily inverted: A(p) = 2- 1 / 2 (u/ 1/ 2$(p) + zu;-1/2n(p)) A*{p) = 2" 1 / 2 (u; 1/ 2$(-p) - tu;- 1/ 2n(-p)), Substituting these into the Hamiltonian (4.11) yields (4.13) H'= ^ £ ( n t ( p ) n ( p ) 4 - ( p 2 + m 2)$ t(p)$(p)), (4.14) and the commutators (4.10) become *(?),*(*) = n(p),n( ?) n(-p),*(g) - o — —id pq-(4.15) (4.14) and (4.15) are indeed the theory of a free scalar field of mass ra, on a circle of length 2n, in the momentum representation. For Fourier transforming $(p) and 27 n(p) 1 re* $(p) = - = / eipx${x)dx V 2 7 T JQ 1 f2* n{p) = -f= elpxXi{x)dx V 2 7 T 7o (4.16.) yields the configuration space Hamiltonian H ' = - [ ( n 2 ( z ) + ( V $ ( x ) ) 2 + m 2 $ 2 ( x ) ) dx (4.17) 2 Jo with the canonical commutators $ ( x ) , n ( y ) = i ' 6 ( x - y ) * (x ) , * (y)l = \n{x),n{y) (4.18) - 0. We have seen that H0 upon bosonization becomes the zero momentum field oscillator; so the previously constructed states (those with no fermionic excitations) correspond to multiparticle states of the free scalar field where all particles have zero momentum. Notice that the zero momentum field oscillator in the bosonized form of H' has frequency m, the same frequency of the harmonic oscillator (3.65), thus the energy levels in both descriptions agree. States with fermionic excitations correspond to non-zero momentum free scalar particles and these are the plasmons of Mattis and Lieb 5 . However due to the Sugawara formula (3.33) we have none of their quasiparticles. As was shown by Manton, the original Hamiltonian can be put in the form (4.17) without doing the canonical transformation (3.45). Define for p ^ 0 HP) = ^ M P ) + ^(P)) n ( P ) = ~ ( P l ( P ) - P2(p)). (4.19) Then it easily verified that the Hamiltonian as given by (3.42) becomes H = H» + \T, {^(P^iP) + (P2 + " » 2 ) * t ( p ) * ( p ) ) (4-20) with H0 given by eqn.(4.6). Then defining #(0) = $(0) and fl(0) = 11(0) puts H in the form (4.14) with $ and n replaced by $ and ft. A l l is consistent, for we have checked that if the canonical transformation (3.45) is applied to H as given by (4.20), and ($,A) are written in terms of ( $ , E ) , then the Hamiltonian (4.14) results. Furthermore $ and fl have all the hermiticity and commutation relations of $ and n . Since it is preferable to work in the original space, we use $ and fl and drop the overlines. Then all previous formulas in this section (with H replacing H') are valid, except definitions (4.19) replace definitions (4.1). From definitions (4.7) and (4.12) of the. zero momentum field oscillators it can be shown that $(p = 0) is the zero momentum Fourier transform of the electric field, while H{p — 0) is the regularized axial charge. As a result the definitions (4.19) for $ and n extend sensibly to p = 0. We are ready to present the derivation, due to Manton, of the anomaly equa-tion. Hamilton's equations as derived from the Hamiltonian (4.17) are $ ( i ) = n(x) (4-21) n(x) = - ( m 2 - 6 - 2 ) $ ( x ) . In momentum space these equations are $( P ) = n(p) li(p) = - ( m 2 + p 2)$(p) (4-22) and definitions (4.19) allow these to be expressed in terms of the currents. The first equation then reads 2jJ°{p)-iP31{p) = 0, (4.23) which is the Fourier transform of d^j11 — 0. Gauss' law (3.4) in momentum space is -ipE(p) = e{px{p) + P2{p)) (4.24) where E(p) is the Fourier transform of Ex. With definition (4.19) this becomes E{p) = \/2$(p). Then the second equation of (4.22) along with definitions (4.19) yield Jt£ ~ ipiliv) =rn2E{p), (4.25) which is the Fourier transform of the anomaly equation dtf* = -Ex. (4.26) 7T The above described Bosonization does not give us a solution in terms of the original variables in the sense of section 3. However as we shall see it does make the finding of the Green's functions easy. To facilitate upcoming discussion of this and other two dimensional theories let us state that the Fourier transform of the mo-mentum space charge density commutators (3.43) imply the current commutators \jo(x),3o(y)\ = o \3o(x)Mv)\ = -*'{*-v) (4-27) 7T [ii(x),ji(y)] = 0. A n d the Fourier transform of the definitions of the Bose field (4.19) imply the correspondence (4.28) where is the antisymmetrical tensor with € 0 1 = 1. G r e e n ' s F u n c t i o n s The Green's functions for the Schwinger model have been found by numerous authors 1 2 6 7 in both the Coulomb and the Lorentz gauge. We will just indicate how they are found by using bosonization, doing the calculation of Casher et.al. in our gauge A% — 0. The field equations derived from the Lagrangian (3.1) are the Dirac- Maxwell equations which in component form are (4.29) i 'd_V 2 = -Axip2 3° = dxdtAx (4.30) j1 = -dtdtAx where d+ — dt + dx and d- — dt — dx. If we let x be a solution to the free Dirac equation d+x1 = d _ x 2 = 0, the solution to the full Dirac equation is j>1 = e x p ( - i d ; 1 A I ) X 1 (4.31) V 2 = exp[idZlAs)X2. Let us define j+ — j° + j1 and j _ = j° — j1. Then Maxwell's equations (4.30) read dtd+Ax = j'_ (4.32) dtd-Ax = 31 Recall = 0 (4.33) dtf* = m2E. Multiplying the first of these by do, the second by d\, and subtracting; along with Maxwell's equations implies A2*0 ^2-0 2 -0 °o3 - °\3 = ~™ 3 • Similarily multiplying the first by d\ and the second by do implies d2jl-d2jl=m2j\ So we have in component form the Klien-Gordon equation + m2)j = 0, which in the plus-minus notation has the forms d+d-j- = -m2j_ d+d-3+ = (4.35) Using these and the current conservation equation d+j+ 4- d-j- — 0 in equations (4.32) results in d+lAx = m~2d~lj+ (4.36) dZlAx = -m.-2d;lj_. 32 So that from (4.31) we have the field ip in terms of the currents and the free field x rp1 = e x p ( - - ^ y I i + ) x 1 (4.37) , / i r \ -•2 = - P ( - ^ / _ J - ) X 2 -Then using the fact that the current operators conserve particles; we may write, for example, the vacuum expectation of ip\ip\ {4>[{xi,h)ipi{x2,t2)) = (xti{xuti)xi{x2,ti)) (eXP(-^/_ljVMeXP(^/_li+(l8)))- ( ) Using the definitions (4.28) of the scalar field $ allows us to write i+ = 4 = ( 3 x * - 3 , * ) . (4.39) We then expand $ in terms of the operators defined in (4.13), and if we go to the infinite spatial line this expansion is where u>(k) is as in (4.5) and the time dependence of the A's has been moved into the exponential factors. Then using this expansion in equation (4.38) allows one to calculate fairly easily that M f s . O V ' i ( 0 , 0 ) ) = <xi(-t, 0X1 ( 0 , 0 ) ( 4 . 4 1 ) where K(x,t) = c j (eW-W - l ) ^ I ' ^ d k . (4.42) c is a positive real constant. For t — 0 the integral converges and tends to zero as i « 0 and thus the propagator is like the free field propagator 7 . However for t ^ 0 the integral diverges for all values of x and the propagator vanishes. The reason being is that if we have a single fermion in the system its electric field does not fall off at large distances and the probability to create a fermion-antifermion pair from the vacuum is proportional to the volume of space. So at any time after putting a fermion into the system there is vanishing probability to find only one fermion. Finally we note that the current correlation function is immediately obtainable via the correspondence (4.28). Putting in a factor e2 to compensate our definition of j, and using T to indicate time ordering, the correlation function is T>"(x-y) = -te2(Tj»(x)f(y)) = -i-t^datv'd9 (r*(x)*(y)> (4.43) 7T = m V ^ d " - d"G>")A F (m 2 , x - y), where A p is the Feynman propagator for the scalar field. Therefore the intermediate spectrum is exhausted by the boson of mass m , 2 and this explains why Schwinger obtained the spectral function mentioned in the introduction. 5. Other Two Dimensional Models In this section we briefly review work done on other two dimensional models. Free Massless Fermi Field One may set the coupling constant of the Schwinger model to zero and imme-diately deduce that the free massless Fermi theory is equivalent to the free massless scalar theory. In fact if one makes definitions (4.28) then it follows that the La-grangians of the two theories are the same— t ^ y ^ = V ^ ^ - (5-i) Li There is another approach to the bosonization of this theory due to Coleman, Gross, and Jackiw. Use point-splitting to define the current y ( x ) — lim(V'(x + e ) W ( x — e) — vacuum expectation); (5-2) i—*o then the canonical Fermi anticommutators imply the current commutators (4.27) (which are the same commutators Sugawara gave in his field theory of currents 8 ) . Coleman et.al . 9 again use point-splitting to define products of currents juji/{x) = lim(j^(x + t)ju{x — t) — vacuum expectation), (5-3) £-+0 and go on to show that the usual energy-momentum tensor O i^/ — - : (iplnd^ + tpl^d^ - dytfrdpip - d^dytp) : (where the vertical double dots indicate normal ordering) is equal to the Sugawara expression for the energy-momentum tensor: 7T Tpv = -{JuJv + jvjp - g^jXj\)- (5.4) The equations of motion for the currents are Heisenberg's equations [ y „ ( s ) , P M ] - idpjv[x) (5.5) where F/x — - J ToM(x) dx. Sugawara showed these to be < V = o (5.6) Spiv - dvjii = °-On the other hand a free massless scalar theory with Lagrangian (5.1) and cur-rents given by (4.28), has an energy-momentum tensor given by (5.4). The canonical commutators for $ imply the current commutators (4.27), and the equations of mo-tion (5.6) are trivially satisfied. We conclude the two theories are equivalent. The equivalence can be extended beyond the currents since the Fermi fields themselves are determined by the currents. 1 0 The method used here is similar to the methods used in the Schwinger model. Here point-splitting was used to regularize the Fermi currents, whereas in the Schwinger model the charge density operators were regularized as in equation (3.34). Here the Sugawara formula was — 0jt„, there the Sugawara formula was equa-tion (3.33). To write the Schwinger model completely in terms of currents, definition (4.7) was necessary to write the gauge field as a current. The Thirring Model The massless Thirring model is the theory of a Dirac field in two space-time dimensions with dynamics determined by the Lagrangian density £ = i ^ d ^ - ^gf3ll (5.7) where j** is the current, and g is the coupling constant. Like the Schwinger model it is exactly solvable and has received numerous investigations. We summarize the solution given by Dell'Antonio, Frishman, and Zwanziger 1 1 because they show the model to be determined by the current and its properties. The vector and axial currents are as before (3.27),(3.28), and the equation of motion of the spinor field derived from the Lagrangian is il^d^x) = 07*7,1 (z)0 (a:). (5.8) Conservation of the vector and axial currents follows. The current commutators are the same as in equations (4.27) with T T - 1 replaced by c, a constant which fixes the normalization of j. Introduce the variables u = t+x, v — t — x and the currents7+ = jo+ji, 3- — 3o~3\- Then current conservation implies j+ = j+(u), j_ = j-(v), and the current commutators imply [y +(u),jV(u')] = 2 i c * ' ( t t - « ' ) [j.(v),j.(v')] = 2ic6'(v-v') (5.9) [j+(u),j_(v)} = 0. A set of current-spin or commutators were obtained by Johnson 1 2 [y+(«) ,V' (« '»v ' ) ] = - {a + a^)tp(u',v')6{u - u') (5-10) []-{v),rl>{u',v')\ = - (a - ai<i)ip{u',v')6(v - v'), where a, a are constants to be determined. As in the Schwinger model, the current j satisfies the Klien-Gordon equation and can therefore be decomposed into positive and negative frequency parts in an invariant manner. Thus one can define normal ordering for current-spinor products: : j„(z)0(z) := ji+)(x)^(z) + V(x)ijf}(x). (5.11) Normal ordering equation (5.8) and choosing the Dirac matrices as in (3.2) yields the equations of motion i d^2{u,v) = --g : j+ {u)ii>2(u, v) : (5.12) i dvi>i (u, v) =• --g : j- {v)ipi [u, v) : . The equations of motion dvj+ — 0, duj~ — 0 along with the commutators (5.9) define a quantum dynamical system for j+ and j _ . The energy-momentum tensor can be built out of currents alone: r „ „ = ^ - : 2 ; ' ^ - glil/jaja : . (5.13) However this is the same as the energy-momentum tensor (5.4) for the free field, as are the current commutators and current field equations. The program for solving the model is therefore as follows.^The products ^i{x)^\{y) will be found as operators which are functions of the currents only, and these can be used to construct the Green's functions which will be vacuum expectations of operators depending on the currents only. These are easily obtained because the currents are free fields. The Poincare generators for time translations (H) and space translations (F) are given by H + P = i J : / * ( « ) : in H - P - ^ f :.&<•)-dv. (5.14) The space-time translations for the charged field ip follow from d^{u,v) = -[H + P,iP{u,v) duip(u,v) = l- H - P,ip{u,v) (5.15) which are found from straightforward use of normal ordering and the current-spin or commutators (5.10) to be d^[u,v) = : ]+(u)1>(u,v) : Ic a i I \ t(a ~ aq) • , w , x dvip(u,v) = : j-{v)ip[u,v) : Ic (5.16) These are consistent with equations (5.12) for a — a — gc. The equations for dvip2 and duip\ are extra and provide additional information which takes the place of point-splitting the equations of motion. Demanding that ip transform like a spinor gives another relation between the constants so that a and a are fixed in terms of g and c. From the equations of motion (5.16) the products iptipl can be found. They are 0x(u,t;)V>i(u , v ) U [i{u - u') + e](1 + ^ T>[t(t; - v') + e]^T : exp /g2 TT \ 1/2 fu ig fv ( 7 + 7 ) 7 / ' - W * (5.17) and similarity for i\>i with u «-» v and jf+ <-• j _ . / 0 is a positive real number whose value determines the normalization of tp. Notice we have a c-number function singular on the lightcone multiplying a regular operator depending only on the currents. A s already mentioned, the Green's functions can be constructed from these. This derivation shows that the Thirring model is determined by the dynamics of the currents and the commutation relations between the currents and the spinor fields. The Quantum Sine-Gordon Equation The Sine-Gordon equation is the name for the theory of a massless scalar field in two space-time dimensions with dynamics determined by the Lagrangian density £ = V < ^ < £ + ^ c o s / 3 < £ - jr (5.18) S. Coleman 1 3 has found the following: if fi2 exceeds 8w the energy per unit volume is unbounded below and the theory has no ground state. However if /32 is less than 8TT the theory is equivalent to the charge-zero sector of the massive Thirring model, and in the case 02 — in the theory is equivalent to the charge-zero sector of a free massive Dirac field theory. Further explanation is necessary. Within the massless Thirring model one may define a renormalized scalar density a — Zipxp with Z a cutoff dependent constant. The massive Thirring model is defined by adding a term proportional to a to the Lagrangian density (5.7): £ = i ^ y d^ - ^gj^j/i - rn'a (5-19) Li where m' is a real parameter. The model is not exactly solvable so a perturbation about m' — 0 is necessary. Every term in the perturbation series for the Green's functions is well denned except for infared divergences. These divergences disappear if instead of (5.19) one considers the theory defined by Z = ixp^d^ - ^-gj11^ ~ m'af{x) (5.20) where / is some function of space-time with compact support. A perscription for solving the theory would be to sum up the the perturbation series in m ' and then go to the limit / = 1. The above perturbation series for the massive Thirring model is term by term identical to a perturbation series in a for the Sine-Gordon equation if the following identifications are made: 1 + - (5.21) 7T 1* (5.22) - m ' a . (5.23) Note that if ft 2 — 4TT then g = 0 and we have the free massive Dirac field; accordingly equation (5.22) is identical to equation (4.28) of the free Dirac field theory. S. Mandelstam 1 4 later reestablished these results of Coleman without the use of perturbation theory. He constructed the creation and annihilation operators of quantum Sine-Gordon solitons and showed these operators satisfy the anticommu-tation relations and field equations of the massive Thirring Model. S U ( N ) T h i r r i n g M o d e l s Finally we mention the bosonization of the SU(N) Thirring Models developed by Banks, Horn, and Neuberger 1 5 . The Lagrangian density for these models is C = i W d ^ - ^J'Ju - £ (5-24) 2TT a J2 47T cos /3(f> = where the U(N) currents are defined by (with point splitting understood) i (5-25) a,b The N x N matrices A' 1), I = l,...,N form the regular representation of SU(N). These currents can be written in terms of Bose fields by formulas analogous to (4.28). Using these Bose expressions for the currents and using point-splitting as in definition (5.3) to define products of currents, the interaction Lagrangian in (5.24) can be written in boson language. For the case of SU (2) the Lagrangian (5.24) turns out to be equivalent to the Lagrangian of one free massless scalar field plus one Sine-Gordon field, with a fixed relation between the Sine-Gordon couplings a and /?. 6. D i s c u s s i o n The original purpose of this work was to construct the stationary states of the Schwinger model in terms of the fermionic variables. The canonical transformation (3.45) turned out to give the bosonization of the theory, which is actually more economically expressed by definitions (4.28). However it did decouple the excitation and Coulomb energy from the energy of the electric field and the Fermi-Dirac sea. This led to solution (3.66), which Manton constructed by supposing there were no fermionic excitations and no Coulomb interaction. The fact that the Schwinger model can be written as a boson theory is due to the polarizability of the vacuum which is responsible for the complete screening of the electric charge. Casher, Kogut and Susskind in their attempts to explain the absence of free quarks have developed the following physical picture of an e + e~ annihilation 7 . The virtual photon decays into an e+e~ pair which move apart at almost the speed of light. A n electric field develops between them and begins producing e + e~ pairs out of the vacuum. As the original pair separate, a line of polarized pairs forms between them. The polarization charge eventually catches up with the outgoing pair and neutralizes them. The net result is the production of bound states which we see as bosons. Bibliography 1. Schwinger, J., Phys. Rev. 128, 2425 (1962). 2. Brown, L . , Nouvo Cimento 29, 617 (1963). 3. Manton, N . , Annals of Physics 159, 220 (1985). 4. Jackiw, R., Effects of Dirac's Negative Energy Sea on Quantum Numbers, (M. I. T . preprint, 1985). 5. Mattis, D . and Leib, E . , J . M a t h . Phys. 6, 304 (1965). 6. Lowenstein, J. and Swieca, J., Annals of Physics 68, 172 (1971). 7. Casher, A . , Kogut, J., and Susskind, L . , Phys. Rev. D 1 0 , 732 (1974). 8. Sugawara, H . , Phys. Rev. 170, 1659 (1968). 9. Coleman, S., Gross, D . , and Jackiw, R., Phys. Rev. 180, 1359 (1969). 10. Affleck, I., Nuclear Physics B265, 409 (1986). 11. Dell'Antonio, G . , Frishman, Y . , and Zwanziger, D . , Phys. Rev. D6, 988 (1972). 12. Johnson, K . , Nouvo Cimento 20, 773 (1961). 13. Coleman, S., Phys. Rev. D l l , 2088 (1975). 14. Mandelstam, S., Phys. Rev. D l l , 3062 (1975). 15. Banks, T . , Horn, D . , and Neuberger, H . , Nuclear Physics B108, 119 (1976). 


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