Stochastic derivation of the chiral anomaly based on the generalized langevin equation

A generalized Langevin equation for fermion field is first derived within the framework of the stochastic quantization. Based on it, the chiral anomaly is derived directly from the stationary property of the pseudoscalar density. This approach is convenient to observe the quantum origin of anomalies. The conservation law of the vector current is also derived in a similar way.     

Dispersive derivation of the trace anomaly

We present a simple derivation of the one-loop trace anomaly in spinor QED through dispersion relations, avoiding completely any ultraviolet regularization. The anomaly can be expressed as a convergent sum rule for the imaginary part of a relevant formfactor. In the massless limit, the imaginary part produces a delta-function singularity at zero external momentum squared. Such a treatment reveals an “infrared face” of the trace anomaly, in striking similarity with the well-known case of the axial anomaly.     

Elementary derivation of the chiral anomaly

An elementary derivation of the chiral gauge anomaly in all even dimensions is given in terms of noncommutative traces of pseudo-differential operators.     

New perturbative derivation of the Fokker-planck equation for coloured multiplicative noise

Starting from a stochastic differential equation with coloured, multiplicative noise with a finite correlation time, a perturbative treatment of the phase-space density to second order leads in a straightforward way to the generalized Fokker-Plank equation.     

A proof of the axial anomaly

The local form of the axial anomaly with both left and right-handed gauge fields and a metric present is given and proved using the families index theoremResearch supported by an NSF Mathematical Sciences Postdoctoral Fellowship     

Renormalization of the axial anomaly operators

An expressionE(d ³ σ/dp ³)=Kp _T ^?n exp(?E/τ, wheren, K and τ are parameters to be fitted, has been used to represent the large-p _T hadron invariant single-particle inclusive cross sections. In most cases the fitting is within the quoted experimental uncertainties. An interpretation of the expression in terms of fireballs is given. Scaling is seen to be violated in large-p _T phenomena.     

A mathematical analysis of the axial anomaly

Letters in Mathematical Physics - As is well known to physicists, the axial anomaly of the massless free fermion in Euclidean signature is given by the index of the corresponding Dirac operator. We...     

Stochastic derivation of the Birula-Mycielski nonlinear wave equation

It is shown that the Schrödinger equation with a logarithmic nonlinearity proposed originally by Birula and Mycielski can be derived within the context of stochastic mechanics. Several important properties of this nonlinear model are easily analyzed by means of the stochastic interpretation, in particular the separability of the evolution equation and its classical limit. The expression for the energy is obtained on the basis of a stochastic variational principle developed by Yasue.     

Supersymmetric derivation of the Atiyah-Singer index and the chiral anomaly

We present an elementary derivation of the Atiyah-Singer formula for the index of the Dirac operator. This index is the space-time integral of the trace of the chiral anomaly. We calculate the full chiral anomaly using the supersymmetric path integral for a spinning particle moving through space-time.     

Ultraviolet and infrared aspects of the axial anomaly II

This is the second part of a brief review of some perturbative aspects of the Adler-Bell-Jackiw axial anomaly. To begin with, we discuss here the derivation of the anomaly based on the imaginary part of the VVA triangle graph and dispersion relations. Within such an approach the infrared properties of the VVA amplitude are substantial. Next we summarize the main results obtained by the present author and co-authors concerning some particular ultraviolet and infrared aspects of the axial anomaly. These results cover a wide variety of topics, ranging from dispersion relations to a systematic analysis of gauge invariant ultraviolet regularizations of the VVA triangle graph. In the latter context, one of the highlights is a reinterpretation of dimensional regularization of the VVA diagram as a continuous superposition of Pauli-Villars cut-offs.     

Ultraviolet and infrared aspects of the axial anomaly I

This is the first part of a brief review of some perturbative aspects of the Adler-Bell-Jackiw axial anomaly, described in terms of ultraviolet and infrared behaviour of the famous VVA triangle graph. Apart from a general overview of the diversified role played by the anomaly in quantum field theory and particle physics, we present here an elementary introduction to the subject of the anomaly, which hopefully should be comprehensible to an uninitiated reader with only a basic background in quantum field theory. In this article we stress the ultraviolet aspects of the anomaly and the topics covered here are the following: Vector and axial-vector Ward identities for the VVA triangle graph, the anomaly and several ways to derive it, namely the symmetric momentum cut-off and shifting the integration variables in linearly divergent integrals, the Adler-Rosenberg argument, the Pauli-Villars method and dimensional regularization.     

Quark-hadron duality,axial anomaly and mixing

Interplay between axial anomaly and quark-hadron duality in the presence of strong mixing is considered. The anomaly sum rule for meson transition form factors based on the dispersive representation of axial anomaly and quark-hadron duality in octet channel is analyzed. The comparison of this sum rule to the experimental data on the transition form factors of the η and η′ mesons shows that the interval of duality in this channel is rather small, contradicting the usual understanding of quark-hadron duality. The same values of interval of duality are supported by considering the two-point correlator in the local duality limit. This contradiction may be resolved by introducing of some nonperturbative non-one-pion exchange correction to the relevant spectral density. The form and value of this correction are discussed.     

The Schwinger model and its axial anomaly

The Schwinger model (quantum electrodynamics with massless fermions in one spatial dimension) is solved, supposing that space is a circle. This clarifies aspects of the usual version of the model, where space is a line, without changing the physics. The Hamiltonian formalism is used. On a circle, an abelian gauge field has one physical degree of freedom, and the gauge covariant Dirac operator, which couples the fermions to this degree of freedom, exhibits spectral flow. The relationship between the spectral flow and the axial anomaly is explained. Some variants of the Schwinger model are also discussed.     

The axial anomaly and anti-symmetric tensor fields

Spin-$\text{1}\text{2}$ fermions are coupled to external axial vector and rank-2 anti-symmetic tensor fields. The chiral U(1) Ward identity is shown to have anomalous structure given by $\text{F}{}_{\mathrm{\mu \nu }}\text{F}\text{?}{}^{\mathrm{\mu \nu }}$ only with F_μν the axial vector field strength tensor. No Additional axial anomalied are introduced due to the presence of the anti-symmetric tensor fields.     

The axial anomaly for spin 32 fields

The axial anomaly for a Majorana spin $\text{3}\text{2}$ loop with external gravitons in supergravity is ?_μJ^μ₅ = (192π²)^??^αβ?σR?σ^μνRμναβ. This is twice the value of the corresponding anomaly for a Dirac spin $\text{1}\text{2}$ loop.     

Scalar trace anomaly and anti-gravitational interaction in a perturbative approach to self-consistent cosmologies

We present a perturbative approach to the equations controlling the behavior of the recently proposed self-consistent, causal, singularity-free cosmologies. This approach sheds a new light on the threshold mass which governs both the (in)stability of empty Minkowski space and the existence of these cosmologies. An unexpected fact arises at the lower order of this perturbative scheme: the mass of the massive (scalar) field coupled non-minimally to gravitation is completely absorbed in a rescaling of the gravitational constant. The latter becomes negative, thereby causing an effective anti-gravitational interaction when the corresponding mass exceeds the minkowskian instability threshold. Moreover, the source of this effective antigravitational interaction is the usual scalar trace anomaly associated with the residual massless part of the matter field.