Anomalous chiral current commutators and divergence

Volume 211, number 4

PHYSICS LETTERS B

8 September 1988

A N O M A L O U S C H I R A L C U R R E N T C O M M U T A T O R S AND DIVERGENCE G.V. D U N N E The Blackett Laboratory, Imperial College, London SW7 2BZ, UK and A.C. KAKAS Institute for Theoretical Physics, University of Z~irich, Sch6nberggasse 9, CH-8001 Ziirich, Switzerland Received 16 May 1988

The anomalouschiral current commutators and divergenceare calculatedin the canonical formalism,within a proposedgeneral framework suitable for the study of anomalies. This frameworkincorporates a generalized form of normal ordering proposed by Faddeev and Shatashvili.

Since the discovery of chiral anomalies ~:l the direct computation of anomalous commutators has been problematical and usually the Bjorken-Johnson-Low method has been employed. Furthermore the calculation of the anomaly in the divergence of the classically conserved current was difficult (if not impossible) to do purely in a canonical framework. In this letter, we present a direct method for calculating the anomalous chiral current commutators and the current divergence within a canonical framework. This approach incorporates an idea of a generalized form of normal-ordering proposed recently by Faddeev and Shatashvili [ 2 ] in their study of the anomaly in the Gauss law commutator. (Unfortunately, the significance of this new idea for current commutators has been obscured by the particular complications in the example of the Gauss law commutator [ 3-5 ]. ) In order to amphasize the main ideas an abelian theory is studied here but similar results hold for non-abelian theories. These ideas and techniques have already been successfully applied to the study of the Virasoro algebra central charge [ 6], and to the problem of quantum canonical invariance [ 7 ]. The general scheme in which we shall work is the deformation theory approach to quantization [ 8,9 ]. In this approach one tries to construct a Lie algebra, whose bracket is denoted by ({, }}, on the space of classical observables with the following two properties: (i) For any pair of classical functions f, g on the phase space of a system {~, g}} = A ° ( f g) +hA~(f,, g) + h2A2(f,, g) +...,

( 1)

with A ° ~ g) = ~, g} the Poisson bracket of f and g;, {{, }} is said to be an h-deformation of the Poisson bracket

(,}.

(ii) The Lie algebra of ({,)} is isomorphic to the quantum one of the Dirac commutator ( - i / h ) [, ] under a generalized Weyl correspondence rule. For a finite dimensional bosonic quantum mechanical system such a generalized Weyl correspondence between a classical function f(q, p ) e C a (~2N, R ) and its corresponding opera t o r f i s defined by ~ For a review of anomalies see ref. [ 1]. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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J~=f drooldNfl~(a¢,fl)£2(a~, r)exp(ik~=~l (OLkqk+flk#k)), where jT(~, r ) is the Fourier transform o f f ( q , p) and the function £2 (o~, r ) reflects the particular chosen factor ordering [ 10]. Hence the bracket {{, }} is constructed to satisfy

{ ~ , , g } } ^ = - ( i / h ) f f , q] .

(2)

(in the rest of this paper we set h = 1 ). An important property of this construction is that both the classical Poisson algebra and the corresponding quantum algebra are defined on the same space of classical functions and hence such a framework is particularly well-suited to studying the relation between the classical and quantum theories. For example, ifa classical theory has a symmetry with current generators being the classical functions ja such that (/'a, jb} = f ~bjc then the corresponding quantum theory will have an anomaly if and only if the bracket {{ja, jb}} has non-zero h-corrections in its h-expansion of eq. ( 1 ). The non-zero corrections AI, A2... are identified with the anomaly of the commutator [fa, fb], as it is clear from eq. (2). The corrections d 1, A 2.... generally involve high (second, third,...) derivatives with respect to the basic canonical observables. In this letter we will be interested in the case of fermionic theories [ 11 ] with Grassmann basic observables ( (ak, a~), k = 1,..., N) such that {am, a*} = --iamn. The chosen factor ordering will be the normal ordering in which case the A ~(f, g) correction is - 2 . . . . ~ \SamSa~ 8a'roSa*

On#n)

8a*Sa* 8a-~a~)

(3)

for any f, g arbitrary polynomials of (ak, a~). The main difficulty lies in the construction of a classical bracket {{, }} isomorphic to the one of the quantum commutator for the case of infinite-dimensional field theories. We will concentrate here on the four-dimensional spinor field theory whose lagrangian ~ = q,y'(i G -eA~, )u/has chiral symmetry under ¢ - , exp (ia¢75) ~, with A a an abelian stationary external electromagnetic field. The classical spinor field ¢ ( x ) -- ¢(t, x) on a Cauchy surface of constant t is expressed in terms of the basic fermionic (Grassmann) observables (b,~(k), b*(k) ) (d,,(k), d* (k)) as qJ(x)= f d3k ~

ot=l

[b,~(klUa(x,k)+d*(k)V,~(x,k)] ,

(4)

where U (V) are suitably normalized positive (negative) energy solutions of the interacting Dirac equation yu (i0, -eAa)v=O.

(5)

These solutions U, V can be generated perturbatively (see below eq. (9) ) from the plane waves of the free Dirac equation. Upon quantization, normal ordering with respect to (/~,/~*), (d, d*) is chosen to ensure the positivity of the hamiltonian. (For time-dependent Au the analysis and results follow formally in the same way but the usual difficulties of the interpretation of positive (negative) energy exist (see eq. [ 12 ] ). ) Our aim is to calculate the anomalous commutators of the quantized chiral currentj~' = iqJyuys~ as well as the quantum anomaly in 0~j~' in the canonical formalism by applying the above ideas to the infinite-dimensional system of ( ( b ( k ) , b* (k)), (d(k), d* (k)) ). (For clarity, we will drop the spin index c~and denote fd3kz~2= 1 by f [ d3k ]. ) Note that the calculation in the deformation framework will be purely in terms of classical quantities with no operators involved. (Throughout this letter operators are always denoted by a ..... . ) The generalization for infinite degrees of freedom of the {(, }} bracket can be done using a box normalization. It is easy to see from the form of the A ~ correction that the expansion ofeq. (4) when taken (as usual) with U, 446

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V solution of the free Dirac equation is not appropriate. We will show that the appropriate expansion is indeed that of eq. (4) with U, V solutions of the interacting Dirac equation (5) as has been suggested in refs. [ 2, 13 ]. Equivalently, this means that the normal-ordering is done with respect to the total energy and not the free energy. (A similar idea has also been studied in two-dimensional chiral theories in ref. [ 14 ]. ) As an example we present here the calculation for one commutator, [f°(x),j~(y)] where j ° ( x ) = ~)(x)y°~t(x), with other commutators computed in a similar way. For clarity of presentation, we will consider the smeared observables

j°(a, t ) : = f dax a(x)~(x)y°~t(x), j°(b, t ) : = i f d3y b(y)~(y)y°ySc/(y), where a, b are arbitrary smooth functions from N3 to R decaying to zero at infinity and x: = (t, x), y: = (t, y). (Note that for j(a, t) andj(b, t) to map to well-defined quantum operators we should in fact consider a pointsplit smearing (see e.g. refs. [2,3] ); however this does not affect the calculation which is done in terms of classical functions. ) Using a box normalization to generalize eq. (3) to the infinite-dimensional system of our theory and applying it to the observablesj°(a, t),j°(b, t) yields A'(j°,J°) = - 3 i

f d3xd3ya(x)b(Y)

f [d3k][dM

X [ lT"(x,k)y°U(x,p) O(y,p)y°7SV(y, k) - 17"(y,p)~,°),SU(y,k)U(x, k)~,°V(x,p) ],

(6)

with all other higher order corrections A 2, A3... identically zero. This can be simplified using the completeness of U, V f [d3p] [ U(x,p)@ [.](y, p) "k-V(x,p) @ l'~(y,p) ] =y°O3 ( x - y ) to

A l (j°,j ° ) = - 3 i f dax d3y[a(x)b(y)

-a(y)b(x)] f [d3k] 17"(x,k)7°~,SV(y, k)~3(x-y).

(7)

Hence the singular part (as x-,y) of f [d3k] 12(x, k)y°y 5V(y, k) will determine A ~U °, jo). Its behaviour as x--,y can be studied by solving eq. (6) perturbatively in the coupling constant e. We then have

f [d3k]lT"(x,k)y°,SV(y,k)= f d3k ~, lT"~°)(x,k),°ysV~°)(y,k) ot= l

+ e f d3k ~

ot=l

[ P~')(x,

k)~,°7'V~°)(y, k)+ I?~°)(x, k)),°~,sV~')(y, k)] + O ( e 2 ) ,

(8)

where

V.(x, k) = V~°) (x, k) + eV~ ~) (x, k) + e: V~.2) (x, k) + .... with the zero-order solution V (o) (x, k) = exp ( - i [k It) exp (ik.x) V. (k) (i.e. the negative energy solution to the free Dirac equation) and

V(.~)(x, k) = f d4ZSF(X--Z)d(z) V~°)(z, k), V~2)(x, k) = f d4z d4WgF(x-z)4(Z)SF(Z--W)4(w)V(O)(w, k) ,

(9)

etc..., where Sv(x-z) is the free Feynman propagator. (Note that by defining A~(j °, jo) as lim A ~(j°(a, t~ ),j°(b, t) ) this can also be equivalently expressed as "~'+

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A ~(j°,J °)= tl~t +lim( - 3i f d3xd3y[a(x)b(y)-a(y)b(x) ]" I [d3k ]sp(7075SA(x' tx ;y' t) )~53(x-y) ) ' where SA (X, tt ;y, t) is the Feynman propagator of the full interacting Dirac equation ( 5 ). A perturbation expansion Of SA yields the same expansion forA t (jo, jo ) as the one above following from eq. (8.) The zero-order term, which is the result for A l if the expansion (4) is done with respect to the free Dirac equation, is identically zero. The singular part of (8) (as xoy) is (see e.g. ref. [3] ) of first order in e (for the abelian case) and equals

i

(X--y)i eiikOjAk"

4n 2 e - ~ - - ~

Then letting r/: = y - x

A tO'°,j °) = ~-3e f d3x d3q[a(x)b(x+q)-a(x+rl)b(x) ] rli eijkOjAkCj3(Ti) -3e --

47~2

f d3xd3q[a(x)Otb(x)_b(x)Ota(x)]~ff_(7 •l•i

eiJkOsAk~53(q)=~- e l d3xa(x)[Olb(x)]eokOjAk,

(10)

where we have adopted the definition of -~31ifor fd3q 6 3(q) ~hr/i/I q 12. (Here we must point out that for this fourdimensional theory the limit as q--,0 of qd/i/Iq[ 2 depends on the direction along which q approaches zero. We have chosen the radial limit as the physically appropriate one (c.f. ref. [ 3 ] ). ) Hence we have shown that the generalized bracket

{{j°(a, t),j°(b,

t)}} = f d3x d3y a(x)b(y)({j°(x),j°(y)}}

has a correction to the Poisson bracket equal to

2:2 f d3x d3y a(x)b(Y) (eako~Ak + ¢j3(x-Y) ) for any arbitrary functions a, b and thus the corresponding commutator

[f°(x),j~(y)

] has an anomaly equal to

ie eijk O_.O_AkO_~.ir53(x_y), 2n 2 OxJ Oy which agrees with the standard result [ 1 ]. To study the anomaly in 0u/"~ we note that in the above framework of isomorphically describing the quantum commutator algebra using a deformation of the Poisson bracket, the quantum time evolution of an operator f (i.e. 0w~= - i [/1,]] ) is correspondingly defined by

Oof={{H,f}} in terms of the classical observable f a n d hamiltonian H. Hence the divergence off~'(x) can be studied correspo ndingly using the classical currents j ~'(x) as 8M'g = 0iJ~ + 0 oj ° = 0~j~ + {{H, jo }} = 0d'~ + {H, jo } + A' (H, jo ) + A 2(H, jo ) +... =0+A

t(H,j°) +A2(H,j °) + ....

( 11)

It can easily be seen from the form of eq. (3) that only the interacting part of the hamiltonian H, i.e. H~ (t) = efd3x #(x)7UV(x)Au(x), contributes to At(H, jo). (Also A 2, A3... are all identically zero since jo and H are quadratic in the basic canonical fields. ) The calculation ofA t (Hi(t), jo (b, t) ) is similar to A t (jo, jo) above, leading to (c.f. eq. (10) ) 448

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e 2

27t2 f d3x b(x) (OjAu)euJ~OkA~. Hence A l(Hi(t),j°(x) ) is

( eZ /2n2)tuJ~( OjAu) (OkA~) =_( eZ/16n2)tu~'~PF~pFu~ , recovering from ( 11 ) the familiar result for the a n o m a l y in the current divergence [ 1 ]. We have presented a general canonical f r a m e w o r k within which it is possible to study a n d calculate anomalies. The a p p e a r a n c e o f an a n o m a l y is seen concretely to be a simple consequence o f the fact that the classical Poisson bracket is (in general) not i s o m o r p h i c to the q u a n t u m c o m m u t a t o r bracket algebra. The a n o m a l o u s terms are naturally identified with the d e f o r m a t i o n s o f the classical algebra. The c o m p u t a t i o n s are done entirely in terms o f classical functions with no operators appearing explicitly. O u r results also show that the generalized n o r m a l ordering p r o p o s e d by F a d d e e v a n d Shatashvili [2] is a p p r o p r i a t e for the study o f current anomalies b u t note that in the o p e r a t o r f r a m e w o r k this idea is not sufficient, since even if the o p e r a t o r currents are regulated using this generalized form o f n o r m a l ordering their cofiamutators r e m a i n unaffected. F u r t h e r work is u n d e r progress to a p p l y the ideas presented here to field theories in an external gravitational field, e.g. to the ghost n u m b e r a n o m a l y o f strings [ 15 ] a n d conformal trace a n o m a l y in q u a n t u m field theory on a curved spacetime [ 12 ]. We w o u l d like to t h a n k I. Bakas a n d B. K a y for m a n y useful discussions. G.V. D u n n e acknowledges the support o f a C o m m o n w e a l t h Scholarship from the British Council. A.C. K a k a s wishes to thank the Royal Society a n d the Swiss N a t i o n a l Science F o u n d a t i o n for their financial support.

References [ 1] R. Jackiw, in: Current algebra and anomalies, eds. S.B. Treiman, R. Jackiw, B. Zumino and E. Witten (World Scientific, Singapore, 1985). [ 2 ] L.D. Faddeev and S.L. Shatashvili, Phys. Lett. B 167 ( 1986 ) 225. [3] S.G. Jo, Phys. Rev. D 35 (1987) 3179. [4] I.G. Halliday, E. Rabinovici, A. Schwimmer and M. Chanowitz, Nucl. Phys. B 268 (1986) 413. [ 5 ] R. Percacci and R. Rajaraman, Phys. Lett. B 201 ( 1988 ) 256. [6] I. Bakas and A.C. Kakas, Class. Quant. Grav. 4 (1987) 67. [7] G.V. Dunne, J. Phys. A 21 (1988), to be published. [8 ] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Ann. Phys. (NY) 111 (1978 ) 61, 111. [9] I. Bakas and A.C. Kakas, J. Phys. A 20 (1987) 3713. [ 10] G. Agarwal and E. Wolf, Phys. Rev. D 2 (1970) 2187. [ 11 ] F.A. Berezin and M.S. Marinov, Ann. Phys. (NY) 104 (1977) 336. [ 12] N.D. Birrel and P.C.W. Davies, Quantum fields in curved space (Cambridge U.P., Cambridge, 1982). [ 13] L.D. Faddeev, Phys. Lett. B 145 (1984) 81. [ 14] K. Isler, C. Schmid and C.A. Trugenberger, Nucl. Phys. B 301 (1988) 327. [ 15 ] K. Fujikawa, Phys. Rev. D 25 ( 1982 ) 2584.

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