Ward identities and chiral anomalies in stochastic quantization

We derive the Ward identities (WI) for vector and axial currents in stochastic quantization at any given fictitious time t. This is achieved through a functional integral representation of the fermionic Langevin equations. The currents for this effective field theory differ in general from the naive ones; if stochastic regularization is used they are both conserved. We establish the connection between those WI and the field theory ones. The physical source of chiral anomalies is identified: these result from the quantum fluctuations in the fictitious time evolution of the system. In this context, both a traditional regularization method (Pauli-Villars) and stochastic regularization are considered.     

Stochastic regularization of QED

Stochastic quantization and stochastic regularization of QED are studied. The need to use gauge and chiral covariant Langevin equations is discussed. They allow the cancellation of all naive quadratic divergences of the photon vacuum polarization at the one-loop level. The leading logarithmically divergent contribution is then transverse, and the masslessness of the photon is easily checked. Furthermore, within stochastic regularization, a simple method for obtaining the chiral anomaly in a massive theory at large fictitious time is developed.     

Scaling behaviour of the fictitious time correlation function in stochastic quantization and its implications in Langevin simulation

The scaling behaviour of the fictitious time correlation length in stochastic quantization is investigated. Its implications in the numerical Langevin simulation are studied by the use of the two-dimensional 0(3) non-linear σ-model. It is discussed in connection with the measurement of the mass gap. Possible relations between the scaling behavior and higher order algorithms or the critical slowing down are also suggested.     

Conserved Noether currents in stochastic quantization

A general procedure for constructing Noether conserved currents in the stochastic quantization scheme corresponding to symmetries of the equilibrium theory is proposed. Two different regularizations — the Breit-Gupta-Zaks stochastic time regularization and a new supersymmetric regularization — are employed, and the origin of chiral anomalies is exhibited in this framework.     

Comments on chiral fermions in stochastic quantization

The subject of this Letter is twofold. First, a recently claimed failure of stochastic quantization scheme (SQS) to describe standard chiral anomalies at finite stochastic time is shown to result from an elementary mathematical error. The consistent approach of treating chiral fermions in SQS is briefly sketched and full agreement with previous investigations is established. Second, a serious new limitation on SQS is found, namely, its incapability to reproduce global chiral- and odd-dimensional parity-violating anomalies.     

Chiral anomalies in the stochastic quantization scheme

Stochastic quantization scheme with an invariant stochastic regularization is applied to models of chiral fermions in even-dimensional spacetime. In spite of the manifest preservation of chiral gauge symmetries by stochastic regularization of the stochastic averages, the anomalous chiral Ward identities are correctly reproduced in a covariant form after removing the regularization.This Letter covers part of the content of the preprint [11].Address since 15 December, 1985: CERN, Theory Division, CH-1211 Geneva 23, Switzerland.     

Renormalization and renormalization group behaviour of the O(N) non-linear sigma-model in stochastic quantization

The renormalization and renormalization group behaviour of the two-dimensional O(N) non-linear sigma-model has been discussed in the framework of stochastic quantization. For this purpose the path integral representation of the generating functional of the extended stochastic correlation function for constrained systems has been given. The renormalization is formulated by use of the stochastic effective action which can be defined from this generating functional. The renormalization group behavior of the correlation function with different fictitious time arguments has also been investigated. It reduces to the normal one if all external fictitious times are taken to be equal.     

Boundary Conditions for the Dynamical Equations of Quantum Mechanics

We discuss the problem whether the time evolution in quantum physics should be represented by the time-symmetric unitary-group evolution, i.e., whether time t extends over???∞?<?t?<?+∞ or it is more realistic to describe quantum systems by a mathematical theory, for which time t starts from a finite value t ₀: t ₀?≤?t?<?+∞, for which the mathematicians would choose t ₀?=?0,¹ but which could be any finite value. If the quantum system in the lab should be described by some kind of quantum theory, one should also admit the possibility that the solution of the dynamical equations needs to be found under boundary conditions that admit a semigroup evolution. It is remarkable that results in lab experiments indicate the existence of an ensemble of finite beginnings of time $ t_0^{(i) } $ for an ensemble of individual quanta.     

Consistent quantization of a two-dimensional non-abelian chiral theory

A two-dimensional SU(N) gauge model coupled to Weyl fermions is studied following recent suggestions for the quantization of potentially anomalous chiral theories. The Weyl fermion determinant is evaluated and the fermionic current is shown to be conserved due to the gauge invariance of the resulting quantum theory. As in the abelian case, the vector meson acquires a mass and the model is consistent provided a regularization parameter is conveniently chosen.     

Gauge Anomaly in 'Arbitrary Even Dimensions from Stochastic Quantization

The stochastic quantization method is applied to evaluate gauge anomaly in arbitrary even dimensions. Stochastic regularization plays a distinct role in our method. The. anomaly structure of the anomalous model is studied.     

Effect of friction on the dynamics and kinematics of solitons in Frenkel-Kontorova systems

To describe the motion of a soliton in ideal Frenkel-Kontorova systems at energies above the amplitude of the Peierls-Navarro pinning potential (PNP), a dynamic Langevin equation with stochastic friction force was proposed: ÿ + α? ? βsin2πy = 0, where y is the dimensionless coordinate of the soliton in units of the substrate potential period, which is determined by the first moment of the soliton shape according to the standard procedure; differentiation is performed with respect to the dimensionless time τ = t(κ/m)^1/2, where κ is the force constant of the chain bond and m is the mass of a chain element. A procedure for determining the friction coefficient α is discussed. Numerical calculations of the dependences of the coordinate and velocity of the soliton on the time over wide ranges of initial conditions and parameters α and β. An analysis of the calculation results showed that the motion of the soliton slows down exponentially, being modulated as it does by the action of the PNP, and end up at one of the minima of the PNP.     

Approximate solution methods for linear stochastic difference equations: I. Moments

The cumulant expansion for linear stochastic differential equations is extended to the case of linear stochastic difference equations. We consider a vector difference equation, which contains a deterministic matrix A₀ and a random perturbation matrix A₁(t). The expansion proceeds in powers of ατ_c, where τ_c is the correlation time of the fluctuations in A₁(t) and α a measure for their strength. Compared to the differential case, additional cumulants occur in the expansion. Moreover one has to distinguish between a nonsingular and a singular A₀. We also discuss a limiting situation in which the stochastic difference equation can be replaced by a stochastic differential equation. The derivation is not restricted to the case where in the limit the stochastic parameters in the difference equation are replaced by white noise.     

Masses of the t and the b quark and their mass ratio m b /m t as a consequence of a dynamical breakdown of SU L (2) symmetry

A model (of the type of the Nambu-Jona-Lasinio model) where the t and the b quark simultaneously acquire dynamical masses m _t and m _b is constructed to describe a dynamical breakdown of chiral and weak SU _L(2) symmetry. That the ratio x=m _b/m _t is small may imply that, at high energies, the energy scales of isoscalar vector exchange and isoscalar scalar exchange between the quarks are markedly different $(M_V \sim \sqrt x M_S )$ . The spectrum of composite scalar states of the model and the mechanism that causes the transformation of Goldstone bosons of the system under consideration into components of vector bosons of local SU _L(2) symmetry are investigated.     

Finiteness of the chiral anomaly graph in higher dimensions

The chiral anomaly graph in 2n dimensions is shown to be completely finite, independent of any constraints which would be imposed from vector-current conservation or Bose-symmetry. There is an n-fold ambiguity present in the graph which guarantees that all current divergences are equivalent in all (self-consistent) perturbative regulating procedures. The chiral anomaly is shown to reside in the alternating sum of current divergences. The ambiguity structure of the chiral anomaly graph in the Pauli-Villars scheme is explicitly computed as a specific example of this general result.     

Stochastic quantization of supersymmetric field theory

The Parisi-Wu stochastic quantization method is applied to supersymmetric field theory. The Langevin equation, which reproduces the Green functions of euclidean field theory, is written in terms of superfields. Supersymmetric U(1) theory under gauge fixing and the large N reduction in chiral SU(N) theory are discussed. Regularization based on the stochastic method is studied also.     

A Particle-Picture Approach to Anomalies in Chiral Gauge Theories

The system of a chiral fermion field coupled to a background gauge field is considered. By taking what we call the particle picture and carefully defining the S-matrix in the Heisenberg picture, we investigate anomalous phenomena in this system. It is shown by explicit calculations that the gauge-field configuration with nonvanishing topological-charge causes anomalous production of particles that is directly responsible for the chiral U(1) anomaly. Unlike the chiral U(1) anomaly, the gauge anomaly, that is, gauge non-invariance of the S-matrix is a problem that arises in the phase of the S-matrix. It is shown that this phase is related to the freedom existing in the quantization method, and that a suitably chosen phase which of course is consistent with the equation of motion can remove the gauge anomaly. Finally, a modified form of path-integral quantization for this system is proposed.     

Transport of a Passive Tracer by an Irregular Velocity Field

The trajectories of a passive tracer in a turbulent flow satisfy the ordinary differential equation x′(t)=V(t,x(t)), where V(t,x) is a stationary random field, the so-called Eulerian velocity. It is a nontrivial question to define the dynamics of the tracer in the case when the realizations of the Eulerian field are only spatially Hölder regular because the ordinary differential equation in question lacks then uniqueness. The most obvious approach is to regularize the dynamics, either by smoothing the velocity field (the so-called ε-regularization), or by adding a small molecular diffusivity (the so-called κ-regularization) and then pass to the appropriate limit with the respective regularization parameter. The first procedure corresponds to the Prandtl number Pr=∞, while the second to Pr=0. In the present paper we consider a two parameter family of Gaussian, Markovian time correlated fields V(t,x), with the power-law spectrum. Using the infinite dimensional stochastic calculus we show the existence and uniqueness of the law of the trajectory process corresponding to a given field V(t,x) for a certain regime of parameters characterizing the spectrum of the field. Additionally, this law is the limit, in the sense of the weak convergence of measures, of the laws obtained as a result of any of the described regularizations. The so-called Kolmogorov point, that corresponds to the parameters characterizing the relaxation time and energy spectrum of a turbulent, three dimensional flow, belongs to the boundary of the parameter regime considered in the paper.     

Dynamical correlations for vicious random walk with a wall

A one-dimensional system of nonintersecting Brownian particles is constructed as the diffusion scaling limit of Fisher's vicious random walk model. N Brownian particles start from the origin at time t=0 and undergo mutually avoiding motion until a finite time t=T. Dynamical correlation functions among the walkers are exactly evaluated in the case with a wall at the origin. Taking an asymptotic limit N→∞, we observe discontinuous transitions in the dynamical correlations. It is further shown that the vicious walk model with a wall is equivalent to a parametric random matrix model describing the crossover between the Bogoliubov–deGennes universality classes.     

Managing <Emphasis Type="Italic">γ</Emphasis><Subscript>5</Subscript> in Dimensional Regularization II: the Trace with more <Emphasis Type="Italic">γ</Emphasis><Subscript>5</Subscript>’s

In the present paper we evaluate the anomaly for the abelian axial current in a non abelian chiral gauge theory, by using dimensional regularization. This amount to formulate a procedure for managing traces with more than one γ ₅. The suggested procedure obeys Lorentz covariance and cyclicity, at variance with previous approaches (e.g. the celebrated ’t Hooft and Veltman’s where Lorentz is violated). The result of the present paper is a further step forward in the program initiated by a previous work on the traces involving a single γ ₅. The final goal is an unconstrained definition of γ ₅ in dimensional regularization. Here, in the evaluation of the anomaly, we profit of the axial current conservation equation, when radiative corrections are neglected. This kind of tool is not always exploited in field theories with γ ₅, e.g. in the use of dimensional regularization of infrared and collinear divergences.