# Current Algebra and Anomalies

Sam B. Treiman
Roman Jackiw
Bruno Zumino
Edward Witten
Pages: 552
https://www.jstor.org/stable/j.ctt7ztmc8

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1. Front Matter
(pp. i-iv)
2. PREFACE
(pp. v-vi)
S. Treiman, R. Jackiw, B. Zumino and E. Witten
(pp. vii-xii)
4. CURRENT ALGEBRA AND PCAC
(pp. 1-80)
Sam B. Treiman

The basic ideas for the subject of current algebra were introduced by Gell-Mann [1] as long ago as 1961. But the development proceeded slowly for several years, until 1965, when Fubini and Furlan [2] suggested the appropriate techniques for practical applications and Adler [3] and Weisberger [4] derived their remarkable formula relatingβdecay parameters to pionnucleon scattering quantities. This inaugurated the golden age and the literature soon reflected what always happens when a good idea is perceived. In 1967 Renner [5] counted about 500 papers, and the number may well have doubled by now. Of course the number of...

5. FIELD THEORETIC INVESTIGATIONS IN CURRENT ALGEBRA
(pp. 81-210)
Roman Jackiw

The techniques of current algebra have been developed to circumvent two difficulties which hamper progress in particle physics. These are (1) a lack of knowledge of the precise laws which govern elementary processes, other than electromagnetism; (2) an inability of solving any of the realistic models which have been proposed to explain dynamics. It was in this context that Gell-Mann [1], in a brilliant induction from non-relativistic quantum mechanics, proposed his now famous charge algebra, which subsequently has been extended to the local algebra of charge and current densities. Just as the canonical, non-relativistic Heisenberg commutator between the momentum$p\equiv \delta L/\delta \dot{q}$...

6. TOPOLOGICAL INVESTIGATIONS OF QUANTIZED GAUGE THEORIES
(pp. 211-360)
Roman Jackiw

A relativistic quantum field theory provides a dynamical framework for explaining a vast variety of physical phenomena: not only are all properties of bound and scattering states contained in the theory, but also processes in which particle number changes are described. The ultimate model that physicists are seeking is the unified theory, which must account for everything around us — a variety which is practically unlimited. It comes as no surprise therefore that realistic field theories are not exactly solvable, and we must resort to approximate but accurate methods of calculation — as already we do in simple classical mechanical...

7. CHIRAL ANOMALIES AND DIFFERENTIAL GEOMETRY
(pp. 361-392)
Bruno Zumino

In these lectures I shall describe [0] a number of properties of chiral anomalies from a geometric point of view*. I follow mostly work done in collaboration with Raymond Stora [1]. Some of the results are contained in a recent paper written in collaboration with Wu Yong-Shi and Anthony Zee [2], to which I refer also for an extensive list of old and new references on chiral anomalies. It is possible that the methods and results described in these lectures are fully known in mathematics. On the other hand, several crucial formulas have not been given before (at any rate...

8. CONSISTENT AND COVARIANT ANOMALIES IN GAUGE AND GRAVITATIONAL THEORIES
(pp. 393-428)
William A. Bardeen and Bruno Zumino

The gauge principle is used as the fundamental basis for present theories of all known forces, from electromagnetism to gravitation. Anomalies [1–5] result when gauge invariance cannot be maintained in the quantum theory. A complete understanding of anomalies is essential for the full application of these theories to physical problems.

The anomaly is usually defined as the gauge variation of the connected vacuum functional in the presence of external gauge fields. When an anomaly occurs, this variation does not vanish and the vacuum functional is not gauge invariant. The gauge currents are no longer covariantly conserved but have the...

9. AN SU(2) ANOMALY
(pp. 429-436)
Edward Witten

It has been a long-standing puzzle to elucidate the properties of an SU(2) gauge theory with a single doublet of left-handed (Weyi) fermions. This theory defies simple phenomenological descriptions. There is no obvious attractive channel in which a fermion condensate could form, consistent with Fermi statistics and Lorentz invariance. But it is hard to believe that the fermions could remain massless in the presence of strong SU(2) gauge forces at long distances.

This puzzle persists (in the absence of other representations) whenever the number of elementary fermion doublets is odd. An even number of doublets, even if they have zero...

10. GLOBAL ASPECTS OF CURRENT ALGEBRA
(pp. 437-450)
Edward Witten

The purpose of this paper is to clarify an old but relatively obscure aspect of current algebra: the Wess-Zumino effective lagrangian [1] which summarizes the effects of anomalies in current algebra. As we will see, this effective lagrangian has unexpected analogies to some 2 + 1 dimensional models discussed recently by Deser et al. [2] and to a recently noted SU(2) anomaly [3]. There also are connections with work of Balachandran et al. [4].

For definiteness we will consider a theory with SU(3)L× SU(3)Rsymmetry spontaneously broken down to the diagonal SU(3). We will ignore explicit symmetry-breaking perturbations, such...

11. GRAVITATIONAL ANOMALIES
(pp. 451-514)
Luis Alvarez-Gaumé and Edward Witten

The fermion anomaly in (3 + l)-dimensional quantum field theory has a remarkable number of important applications. In the original version [1], one considers a massless fermion triangle diagram with one axial current and two vector currents. Requiring conservation of the vector currents, one finds, even for massless fermions, that the axial current is not conserved (fig. 1). This results in a breakdown of chiral symmetry in the presence of gauge fields that are coupled to the conserved vector currents. This breakdown is known to lead to an understanding ofπ0decay and to a resolution of the U(1) problem...

12. CURRENT ALGEBRA, BARYONS, AND QUARK CONFINEMENT
(pp. 515-528)
Edward Witten

The idea that in some sense the ordinary proton and neutron might be solitons in a non-linear sigma model has a long history. The first suggestion was made by Skyrme more than twenty years ago [1]. Finkelstein and Rubinstein showed that such objects could in principle be fermions [2], in a paper that probably represented the first use of what would now be calledθvacua in quantum field theory. A gauge invariant version was attempted by Faddeev [3]. Some relevant miracles are known to occur in two space-time dimensions [4]; there also exists a different mechanism by which solitons...

13. SKYRMIONS AND QCD
(pp. 529-537)
Edward Witten

Skyrme’s remarkable idea [1] that baryons are solitons in a meson theory has, in recent years, received partial confirmation as a result of developments in our understanding of QCD. In these notes, I will sketch the connection of Skyrmions with fundamental QCD theory and comment briefly on recent and possible future developments. Some of my comments overlap with those of Balachandran [2].

In the extremely low energy limit (energies much less than Λqcd), QCD is equivalent to a theory of Goldstone bosons (pions). This is, however, not the limit appropriate for Skyrmions or baryons, since the energy of a quark...