An Empty Chiral Rotation for the Adler-Bell-Jackiw Anomaly
aa r X i v : . [ phy s i c s . g e n - ph ] N ov An Empty Chiral Rotation for the Adler-Bell-Jackiw Anomaly
Israel Weimin Sun ∗ School of Physics, Nanjing University, Nanjing 210093, China (Dated: November 12, 2019)This is an article which intends to shake down the traditional belief that the celebrated Adler-Bell-Jackiw anomaly stems from the chiral rotation non-invariance of the fermionic measure. Thefermionic functional integration measure in quantum field theory should be defined so as to repro-duce the standard Feynman diagrammatic expansion. This implies that a plain definition of thefermionic measure automatically serves such a purpose. A dilemma then arises: how could oneidentify the ABJ anomaly as a nontrivial Jacobian factor for a chiral transformation ? The trueanswer is indeed surprising and unexpected, that is, the Jacobian factor is actually a random andindeterminate object, hence it carries no physical information. A true explanation for the ABJanomaly is suggested.
Introduction.
The quantum anomaly is a physical phe-nomenon deeply rooted in relativistic quantum field theo-ries, whose appearence is manifested as the breakdown ofsome classically valid symmetry after the field quantiza-tion procedure. The so-called Adler-Bell-Jackiw anomaly(or the chiral anomaly) [1] in the abelian quantum elec-trodynamics is just an example in this category. Inthe framework of the renowned path-integral quantiza-tion the raison d’etre of the ABJ anomaly is ascribedto the chiral rotational non-invariance of the fermionicfunctional integration measure. This seemingly splendidsuccess was established by Kazuo Fujikawa in his semi-nal work in 1979 [2]. A common wisdom of the physicscommunity is that Fujikawa’s theoretical framework is auniversal one since it has covered all the quantum anoma-lies known in field theory [3].But unfortunately there seems to be a huge stone in theway of our understanding of all these physical matters.It is based on a rather simple (but fatal) observation:in our present path-integral formalism of spinor QED(and any other field theoretical models of such types), bymeans of separating the total classical field action intoa free-field part plus an interacting part: S [ ¯ ψ, ψ, A µ ] = S [ ¯ ψ, ψ, A µ ] + S int [ ¯ ψ, ψ, A µ ], one could adopt a simplestdefinition of the Grassmann fermionic functional integra-tion measure (such as that in a free fermion theory), to-gether with a gauge-fixed path-integral measure of the U (1) gauge field, to establish a standard set of Feynmanrules which gives an order-by-order perturbative expan-sion of all the n -point Green functions for all types ofelementary interpolating field insertions (i.e., excludingthose of composite field insertions), which after some ap-propriate renormalization procedure would yield a com-plete set of transition amplitudes of all QED processes,i.e., all the S-matrix elements, within some physicallyreasonable accuracies. In short, a plain definition ofthe Grassmann fermionic functional integration measure,such as the one in a non-interacting situation, plus somestandard definition of the path-integral measure for the ∗ [email protected] gauge-field sector, would yield the standard machinery ofperturbative QED theory through our standard practicein theoretical physics, even though it strongly disagreeswith the equally ”standard” choice of defining the Grass-mann fermionic fields (and the relevant fermionic mea-sure) in one’s computation of the ABJ anomaly in theframework of path-integral method, whose basic spiritis to produce something representing the ABJ anomalyby ”rotating out” an A µ -dependent functional Jacobianfactor for a chiral rotation of the classical fermion field !In fact, such a disagreement is not just a purely formaldifference, since the standard mathematical practice tellsus an absolute truth: a plain definition of the classicalfermion fields ( ¯ ψ, ψ ), which has nothing to do with thegauge-field sector, could merely yield an A µ -independentJacobian factor for a finite chiral rotation so that onecould no longer expect to see an ”ABJ anomaly” term bymeans of such a calculation ! However, an entanglementwith the gauge-field sector is actually the ”absolute rai-son d’etre ” for Fujikawa’s evaluation of the ABJ anomalyusing the path-integral idea. So, such a dilemma seems tobe a severe damaging point against the standard path-integral explanation of the physical origin of the ABJanomaly which has been firmly established for 40 years !Needless to say, the ABJ anomaly is definitely there,and all these apparent difficulties need to have a trueexplanation. In this short note, I intend to provide afull answer to all these issues. I shall show that: (1) anatural definition of the ( ¯ ψ, ψ ) field in spinor QED (ac-tually in all other field theoretical models) should notbe entangled with the existence of the U (1) gauge fielddegrees of freedom, although Fujikawa’s original ”gauge-field-sector-entangled” definition could also be used todefine everything; (2) an evaluation of Fujikawa’s orig-inal ”functional Jacobian factor” within a larger familyof gauge-invariant regularizations reveals that this Ja-cobian is actually a totally random and indeterminateexpression, i.e, an object which is too random to rep-resent any physical reality; (3) as a result of all thesefacts, one should forget about the traditional Old Dogmathat the so-called ”chiral rotation non-invariance” of thefermionic measure is the raison d’etre for the celebratedABJ-anomaly in field theory. Finally, I would like topoint out that a natural explanation of the ABJ anomalyin the path-integral framework does exist, in which theso-called ”chiral rotation non-invariance” issues are sim-ply avoided, so that everything is saved from being dam-aged. An analysis of the real situation: how should one de-fine a fermionic functional measure?
As usual I willconsider a spinor QED theory consisting of one speciesof Dirac fermion and an abelian gauge field with whichit has a nontrivial interaction. In all applications of thepath-integral method, one needs to input a primary defi-nition of the fermionic measure into the whole formalism.In fact, the usual perturbative expansion of all n -pointfunctions of QED is solely based on the Wick’s theorem,namely, the expression for the many-point moments ofa free Gaussian measure. For the Dirac fermion sector,the relevant Grassmann integration with sources has theform Z D ¯ ψ D ψe − ¯ ψK ψ + ¯ ψη +¯ ηψ . (1)Practically, this Gaussian integration would generate ev-erything one needs in a nontrivial perturbative calcula-tion. No matter how one chooses to define such a sym-bolic expression, it should posses two basic properties: (1)translation-invariance of the form R D ¯ ψ D ψF [ ¯ ψ, ψ ] = R D ¯ ψ D ψF [ ¯ ψ + ¯ ψ , ψ + ψ ]; (2) under a linear transforma-tion of ( ¯ ψ, ψ ), the functional Jacobian equals the inverseof the relevant determinant factor.A practical definition of such a Grassmann integrationconsists in introducing a countable set of Grassmann gen-erators and expanding the ( ¯ ψ, ψ ) field as follows ψ ( x ) = X n ξ n ( x ) a n ¯ ψ ( x ) = X n ¯ b n ξ † n ( x ) (2)where { ξ n ( x ) } is some set of basis functions. One thentacitly assumes D ¯ ψ D ψ = Q n d ¯ b n da n . This definitionguarantees the above mentioned two properties.All we need to know is just the fundamental Gaussianintegration identity: Z D ¯ ψ D ψe − ¯ ψK ψ + ¯ ψη +¯ ηψ = Z D ¯ ψ D ψe − ¯ ψK ψ × e ¯ ηK − η , (3)which is readily established by a simple translation ofvariables. Here a most essential fact appears: (3) is a uni-versal formula, i.e., it holds irrespective of the concretechoice of the basis function in (2). Then, an expansion ofboth sides of (3) in terms of the anticommuting sourcesyields the usual Wick’s theorem of the following format R D ¯ ψ D ψψ ( x ) ¯ ψ ( x ) · · · ψ ( x N − ) ¯ ψ ( x N ) e − S [ ¯ ψ,ψ ] R D ¯ ψ D ψe − S [ ¯ ψ,ψ ] = X pairs Y S F ( x i − x j ) . (4) This expression is valid for all choices of the basis func-tion { ξ n ( x ) } .Now let me turn to Fujikawa’s calculation of the ABJanomaly. When a classical background gauge field A µ ex-ists, Fujikawa insists that one should expand the classicalfermion fields in a gauge-field-sector-entangled manner ψ ( x ) = X n ϕ n ( A ; x ) a n ¯ ψ ( x ) = X n ¯ b n ϕ † n ( A ; x ) , (5)where his basis functions ϕ n ( A ; x ) are just the completeset of eigenfunctions of the Euclidean Dirac operator: i D ( A ) ϕ n ( A ; x ) = λ n ( A ) ϕ n ( A ; x ). The merit of the ex-pansion (5) is merely a formal mathematical beauty: thegauge-invariant fermionic action has an explicitly diago-nalized form so that a direct application of the Berezinintegration rule yields Z D ¯ ψ D ψe − R d x ¯ ψ ( D ( A )+ m ) ψ = Z Y n d ¯ b n da n e − P n ( − iλ n ( A )+ m )¯ b n a n = det( D ( A ) + m ) . (6)However, I intend to point out a simple mathematicalfact: any definition of the form (2) with an arbitrarily chosen basis function { ξ n ( x ) } is equally capable of pro-ducing the same functional determinant in (6)! The ar-gument is simple. One could formally separate out the j · A interaction term from the gauge-invariant fermionicaction and obtain N Z D ¯ ψ D ψe − R d x ¯ ψ ( D ( A )+ m ) ψ = N Z D ¯ ψ D ψe − R d x ¯ ψ ( ∂ + m ) ψ X n ( − n n ! (cid:2) Z ( ie ¯ ψγ µ ψA µ ) (cid:3) n (7)A formal functional integration using the Wick’s theorem(4) just produces a formal Feynman diagram expansionwhich represents exactly the functional determinant in(6). Therefore, it is clear that every choice of the basisfunction in (2) can do the job, while Fujikawa’s definition(5) is just a specific one among all these possible choices.In the same manner, one could also readily show thatthe whole machinery of perturbative QED theory, i.e.,the relativistic covariant Feynman diagrammatic expan-sion of all its n -point Green functions (or the Schwingerfunctions in a Euclidean spacetime) could be establishedon the bases of the definition (2) and the appropriatedefinition of the functional integration measure of thegauge-field sector. The most important point is that allchoices of the basis function { ξ n ( x ) } in (2) are equallycapable of achieving this purpose.Then, since a natural definition of the fermionic mea-sure suffices and the relevant functional Jacobian fora chiral transformation is necessarily A µ -independent,how could one ”explain” the very existence of the ABJanomaly within the framework of the path-integral for-malism ? If one accepts the basic fact that the choice ofthe basis function in (2) is arbitrary, one needs to make auniversal judgment: whether Fujikawa’s original evalua-tion of the functional Jacobian factor is absolutely correctso that his background-gauge-field-dependent definition(5) should be a mandatory choice, or one needs to in-voke some other mechanism to explain the existence ofthe ABJ anomaly ?Needless to say, such a judgment should not be self-contradictory or misleading in any sense. In the followingI will provide a careful and critical analysis of Fujikawa’soriginal evaluation of the Jacobian factor and show thathis regularized ”Jacobian factor” is a totally ambiguousobject and thus should not carry any physical informa-tion with it. I hope my argument is clear enough to washaway all the misunderstanding of this issue for so manyyears. The evaluation of the Jacobian factor: how to make adivergent series to be a convergent one ?
Here let mefirst recall some basic facts. If one considers a chiraltransformation of ( ¯ ψ, ψ ) ψ ′ ( x ) = e iα ( x ) γ ψ ( x )¯ ψ ′ ( x ) = ¯ ψ ( x ) e iα ( x ) γ , (8)the relevant Jacobian factor `a la Fujikawa would be J = e − i R d xα ( x ) P n ϕ † n ( A ; x ) γ ϕ n ( A ; x ) . (9)When one uses a natural definition (2), the correspond-ing Jacobian factor is obtained by a simple substitution ϕ n ( A ; x ) → ξ n ( x ) in (9).As it stands, the formal summation S [ A ; x ] = X n ϕ † n ( A ; x ) γ ϕ n ( A ; x ) (10)is an ill-defined process. In order to extract somethingmeaningful from it, one needs to regularize it. In a for-mal sense, the summation (10) is a gauge-invariant pro-cess, since ϕ n ( A U ) = U ϕ n ( A ), and a good regularizationshould respect the gauge-invariance. Fujikawa choosesthe following regularization S reg [ A ] = lim M →∞ X n ϕ † n ( A ; x ) γ ϕ n ( A ; x ) e − λ n ( A ) /M , (11)where the large λ n contributions in the summation aresuppressed by the Gaussian cut-off factor. This regular-ization is gauge-invariant because of λ n ( A U ) = λ n ( A ).The regularized sum is calculated as S reg [ A ] = e π ǫ µνρσ F µν F ρσ , (12)which produces the expected ABJ anomaly.This is the standard story. However, there is no rea-son to believe that the regularization (11) should be the unique one. In fact, all one needs to do is to speed upthe rate of convergence for the series (10) by damping itslarge eigenvalue contributions. If one works in the frame-work of gauge-invariant regularizations, one could try toregularize it in a different manner S reg | ( A → ˜ A ) [ A ] = lim M →∞ X n ϕ † n ( A ; x ) γ ϕ n ( A ; x ) e − λ n ( ˜ A ) /M , (13)where λ n ( ˜ A ) is the relevant eigenvalues of some differentDirac operator i D ( ˜ A ). Here one should assume that thetwo set of eigenvalues { λ n ( A ) } and { λ n ( ˜ A ) } are in one-to-one correspondence with each other and at the sametime both λ n ( A ) and λ n ( ˜ A ) grow unboundedly when n gets large. In order to guarantee this property, one needsto assume the pair of gauge field configurations ( A, ˜ A ) tobe sufficiently close to each other. Needless to say, such adeformation A → ˜ A is not a pure gauge transformation,i.e., it should change the field strength F µν . I will nowshow that this method could serve as a gauge-invariantregularization when structured appropriately.The recipe is very simple. One first notes that theeigenvalue { λ n ( A ) } only depends on the ”gauge orbit”(i.e., the gauge equivalence class) to which a particulargauge field configuration A belongs, so that one couldeffectively write λ n ( A ) = λ n ([ A ]) where [ A ] denotes thecorresponding ”gauge equivalence class”. With this athand, one then arbitrarily picks a map F from the gaugeorbit space to itself whose effect is to establish a ”physicalcorrespondence” between the various gauge orbits. Soone writes effectively F : [ A ] [ ˜ A ] = [ A ] F . Here onehas to make the additional assumption that under the F -action each gauge orbit [ A ] is only slightly changedso that the two eigenvalue sets { λ n ( A ) } and { λ n ( ˜ A ) } are appropriately matched. Therefore, one sees clearly achain of correspondence: A [ A ] [ A ] F = [ ˜ A ], thuseach one of the eigenvalues λ n ( ˜ A ) = λ n ([ ˜ A ]) = λ n ([ A ] F )could be effectively regarded as a ”gauge-invariant body”made up of the background gauge field A .With all such preparations, I can generalize the orig-inal Fujikawa’s regularization (11) to a whole family ofnew regularizations which I call F -action Gaussian cut-off regularization, or F -regularization for short. Whenthe gauge-orbit-space map F is degenerate to the iden-tity map, this regularization would coincide with Fu-jikawa’s original one, hence it is an essential enlargementof the original framework. Such an F -regularization, asI mentioned previously, preserves explicit gauge invari-ance, and all of my further analysis will be based on it.Now let me describe all the necessary steps. First ofall, since A and ˜ A are sufficiently close to each other,one naturally assumes the two sets of basis functions,namely, { ϕ n ( A ; x ) } and { ϕ n ( ˜ A ; x ) } are exactly in one-to-one correspondence. They are connected by a formalunitary transformation which reads ϕ n ( A ; x ) = X m K nm ϕ m ( ˜ A ; x ) . (14)Then, using the relation (14) I establish a chain of deriva-tions X n ϕ † n ( A ; x ) γ ϕ n ( A ; x ) e − λ n ( ˜ A ) /M = X n (cid:0) X m ϕ † m ( ˜ A ; x ) K ∗ nm (cid:1) γ (cid:0) X ¯ m K n ¯ m ϕ ¯ m ( ˜ A ; x ) (cid:1) e − λ n ( ˜ A ) /M = X m ¯ m ϕ † m ( ˜ A ; x ) γ ϕ ¯ m ( ˜ A ; x ) (cid:0) X n K ∗ nm K n ¯ m e − λ n ( ˜ A ) /M (cid:1) = X m ¯ m ϕ † m ( ˜ A ; x ) γ ϕ ¯ m ( ˜ A ; x ) (cid:0) δ m ¯ m e − λ m ( ˜ A ) /M + R m ¯ m ( M ) (cid:1) = X m ϕ † m ( ˜ A ; x ) γ ϕ m ( ˜ A ; x ) e − λ m ( ˜ A ) /M + Remainder . (15)Finally, I shall argue that Remainder ( M ) M →∞ −− −→ S reg | ( A → ˜ A ) [ A ] = S reg [ A ] | A → ˜ A .To see why this is so, let me rewrite R m ¯ m as R m ¯ m ( M ) = X n (cid:0) K ∗ nm K n ¯ m − δ nm δ n ¯ m (cid:1) e − λ n ( ˜ A ) /M . (16)I will show that lim M →∞ R m ¯ m ( M ) = 0. The argumentis like this. Denote a n = K ∗ nm K n ¯ m − δ nm δ n ¯ m . Sincethe K matrix is a unitary one, one has automatically P n a n = 0. Now, if the sequence { a n } is absolutelysummable, i.e., P n | a n | < ∞ , then I could invoke theso-called dominated convergence theorem to interchangethe order of the summation and the M → ∞ limit:lim M →∞ X n a n e − λ n ( ˜ A ) /M = X n a n = 0 , (17)which is the desired result. This absolute summabilitycan be readily verified. For m = ¯ m , since P n | K nm | =1, the sequence { K ∗ nm K nm } is absolutely summable,hence this is also the case for { K ∗ nm K nm − δ nm δ nm } . For m = ¯ m , the absolute summability of { K ∗ nm K n ¯ m } can beestablished in the following way. First note that bothof the two sequences {| K ∗ nm |} and {| K n ¯ m |} are squaresummable, i.e., they all belong to l . Then, an elemen-tary application of the Cauchy-Schwarz inequality on the l space gives X n | K ∗ nm K n ¯ m | ≤ (cid:0) X n | K ∗ nm | (cid:1) / (cid:0) X n | K n ¯ m | (cid:1) / = 1 , (18)which establishes the desired absolute summability. Withthe result lim M →∞ R m ¯ m ( M ) = 0 at hand, one couldsafely conclude that the ”Remainder” term in (15) shouldvanish in the M → ∞ limit.Now, since to a large extent the physical correspon-dence F : [ A ] [ ˜ A ] is arbitrary, one may conclude thatthe regularized ”Jacobian factor” is a totally random andindeterminate expression, i.e, an object which is too ran-dom to represent any physical reality. This implies acrucial fact, that is, Fujikawa’s original evaluation of thefunctional Jacobian is a fake process, hence one is unable to extract anything sensible from this process which couldstand for a genuine ABJ anomaly term. Of course, no-body is willing to say it is a real damaging point againstthe existing knowledge of quantum anomalies, since theABJ anomaly is a true physical existence and shouldnever be washed away in the realm of physics. Nev-ertheless, an essential modification to our conventionalwisdom of QFT seems to be unavoidable, that is, oneshould abandon the old belief that the ABJ anomaly iscaused by the so-called ”chiral rotation non-invariance”of the fermionic functional integration measure, since therelevant ”Jacobian factor” is totally random, even thoughit has a perfectly finite expression ! This opinion shouldnot be rejected by the long-period practice of high en-ergy physics experiments. Any endeavor in theoreticalhigh energy physics should have its foundation in gen-uine mathematics, instead of just living on purely tech-nical advances.With all the previous facts being established, a lastquestion necessarily arises: how could one identify atrue mechanism in the path-integral formalism which cor-rectly explains the physical existence of the celebratedABJ anomaly? Such a mechanism does exist, whichis clearly described in the monograph [3]. One onlyneeds to introduce a compensating Pauli-Villars regula-tor field with a large mass to regularize everything. Inthis method, the ABJ anomaly term emerges naturallyas the result of the compensating effect of the two typesof loop diagrams: the one corresponding to the origi-nal fermion field together with an opposite one corre-sponding to the regulator field. Now, since the regulatorfield should be a bosonic one, the total ”Jacobian fac-tor” equals one automatically, thus in this scheme oneneeds not to resort to a nontrivial Jacobian to explainthe physical existence of the ABJ anomaly. Therefore,the ABJ anomaly is definitely there, but the conven-tional path-integral interpretation for its origin needs tobe changed. In connection with this, I would also liketo mention a recent e-print [4] in which it is shown thatfor the so-called transverse anomalies Fujikawa’s path-integral method would yield a result different from thatof the one-loop perturbative calculation. This also showsthe weakness of Fujikawa’s approach. Concluding remarks.
In this note I show that thedefinition of fermionic functional integration measure inquantum field theory should not be entangled with thegauge-field sector. It is also rather unexpected to see thatFujikawa’s path-integral evaluation of the ABJ anomalyis actually a fake process since his ”Jacobian factor” istoo random to represent any physical reality. I believethis trend is a healthy one even though it has shaken theold basis of one’s understanding of quantum anomaliesestablished in the past 40 years. The more general situ-ations seem to be essentially the same. I plan to discussthem in the future works.I thank Fan Wang for useful conversations. Thiswork is supported in part by the Natural ScienceFunds of Jiangsu Province of China under Grant No.BK20151376. [1] For a splendid account of the historical matters of quan-tum anomalies in field theory, see the two contributed ar-ticles of Stephen L. Adler together with Roman Jackiwin