Classical Field Equations Research Articles

Trace anomaly through the Schwinger-Dyson equation

In analogy to what happens with the chiral anomaly, we show that the trace (conformal) anomaly can also be understood as a discrepancy between classical and quantum field equations. We work out explicitly a simple example using a regularization similar to that employed by Tsutsui. We also show how to generalize this kind of regularization in order to establish a connection with other ones, like the generalized zeta -function method.

An example of semiclassical instanton-like scattering: (1 + 1)-dimensional sigma model

A solution to the classical field equations in the massless (1 + 1)-dimensional O(3) sigma model is found, which describes a multiparticle instanton-like transition at high energy. In the limit of a small number of initial particles, the number of final particles is shown to be also small, and the probability of the transition is suppressed by exp(−2 S 0), where S 0 is the instanton action. This solution, however, does not correspond to the maximum transition probability among all states with given number of incoming particles and energy. Unless the limit g 2 n initial → 0 is exponentially sensitive to the structure of the initial state, our results imply that well above the sphaleron energy, the instanton-induced cross section becomes again suppressed by the instanton exponent, and the number of final particles is again small.

Homogeneous quantum electrodynamic turbulence

The electromagnetic field equations and Dirac equations for oppositely charged wave functions are numerically time-integrated using a spatial Fourier method. The numerical approach used, a spectral transform technique, is based on a continuum representation of physical space. The coupled classical field equations contain a dimensionless parameter which sets the strength of the non-linear interaction (as the parameter increases, interaction volume decreases). For a parameter value of unity, highly nonlinear behavior in the time-evolution of an individual wave function, analogous to ideal fluid turbulence, is observed. In the truncated Fourier representation which is numerically implemented here, the quantum turbulence is homogeneous but anisotropic and manifests itself in the nonlinear evolution of equilibrium modal spatial spectra for the probability density of each particle and also for the electromagnetic energy density. The results show that nonlinearly interacting fermionic wave functions quickly approach a multi-mode, dynamic equilibrium state, and that this state can be determined by numerical means.

Quantum field theories on manifolds with curved boundaries: Scalar fields

A framework allowing for perturbative calculations to be carried out for quantum field theories with arbitrary smoothly curved boundaries is described. It is based on an expansion of the Green function for second-order differential operators valid in the neighbourhood of the boundary and which is obtained from a corresponding expansion of the associated heat kernel derived earlier for arbitrary mixed Dirichlet and Neumann boundary conditions. The first few leading terms in the expansion are sufficient to calculate all additional divergences present in a perturbative loop expansion as a consequence of the presence of the boundary. The method is applied to a general renormalisable scalar field theory in four dimensions using dimensional regularisation to two loops and expanding about arbitrary background fields. Detailed results are also specialised to an O( n) symmetric model with a single coupling constant. Extra boundary terms are introduced into the action which give rise to either Dirichlet orgeneralized Neumann boundary conditions for the quantum fields. For plane boundaries the resulting renormalisation group functions are in accord with earlier results but here the additional terms depending on the extrinsic curvature of the boundary are found. Various consistency relations are also checked and the implications of conformal invariance at the critical point where the β-function vanishes are also derived. For a general scalar field theory, where the fieldsø attain specified values ϕ in the boundary, the local Schrödinger equation for the wave functional defined by the functional integral under deformations of the boundary is also verified to two loops. The perturbative expansion for the wave functional is defined by expansion around the solution of the classical field equations satisfying the required boundary values and the counterterms necessary to derive a finite hamiltonian operator, which includes a functional Laplace operator on the fields ϕ, are found to the order considered. Consistency of the local Schrödinger equation with the renormalisation group to all orders in perturbation theory is also discussed.

Vacuum polarization and the black hole singularity

In order to investigate the effects of vacuum polarization on mass inflation singularities, the authors study a simple toy model of a charged black hole with cross flowing radial null dust which is homogeneous in the black hole interior. In the region r2<<e2 they find an approximate analytic solution to the classical field equations. The renormalized stress-energy tensor, for conformally coupled fields, is evaluated on this background and they find the vacuum polarization backreaction corrections to the mass function m(r). Asymptotic analysis of the semiclassical mass function shows that the mass inflation singularity is much stronger in the presence of vacuum polarization than in the classical case.

Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing

Hamiltonian classical systems are the farthest from variational systems. Formally they cannot decay because the equation of motion is reversible. Thus the long time behavior of their solutions is naturally described by statistical methods. For instance, a large assembly of interacting particles in a box follow the Gibbs-Boltzmann statistics, as far as average properties are concerned. As noticed already by Jeans, this statistical approach does not hold for the long time behavior of classical nonlinear field equations, because of the infinite heat capacity of a classical field. Thus nonlinear and nonintegrable classical fields cannot relax for long times toward an ill defined thermal equilibrium. The specific example considered here is the long time dynamics of solutions of the nonlinear Schrödinger equation, in the defocusing case. I show, under some assumptions, that for long times a cascade of energy toward smaller and smaller scales introduces a kind of dissipation in a system that is formally reversible, and I give the scaling laws for this. Besides this relaxation by weak interaction between phonons, there is also a relaxation of localized vortical structures emitting phonons.

Summing tree graphs at threshold.

The solution of the classical field equation generates the sum of all tree graphs. We show that the classical equation reduces to an easily solved ordinary differential equation for certain multiparticle threshold amplitudes and compute these amplitudes.

Black hole evaporation in 1+1 dimensions

The formation and quantum mechanical evaporation of black holes in two spacetime dimensions can be studied using effective classical field equations, recently introduced by Callan et al. We find that gravitational collapse always leads to a curvature singularity, according to these equations, and that the region where the quantum corrections introduced by Callan et al. could be expected to dominate is on the unphysical side of the singularity. The model can be successfully applied to study the back-reaction of Hawking radiation on the geometry of large mass black holes, but the description breaks down before the evaporation is complete.

Baryon-number violation in the one-instanton sector: A classical procedure of calculation

By choosing a particular form for zero-mode constraints we show that initial-state-initial-state corrections can be eliminated. With these constraints, initial-state interactions with a single, soft, final-state particle reduce to the solution of a classical field equation with a source. Interactions involving more than one final-state particle are given by applying the KRT procedure to the solution of the classical equation mentioned above.

Change of signature in classical relativity

The authors point out that the classical Einstein field equations, suitably interpreted, allow a change of signature of spacetime. Specific examples of such changes are constructed in the case of Robertson-Walker geometries. They obtain classical solutions that have properties similar to those obtained in quantum cosmologies obeying the Hartle-Hawking 'no boundary' condition: these singularity-free universes have no beginning, but they do have an origin of time. They can be regarded either as classical analogues of the quantum cosmology results, or as classical solutions where a quantum cosmology era is avoided.

High-energy anomalous scattering: Is it semiclassical?

We discuss the possibility of a semiclassical evaluation of baryon-number-nonconserving scattering amplitudes at nonperturbative'' energies of order {ital M}{sub {ital W}}/{alpha}{sub {ital W}}. Semiclassical expansions around standard instanton configurations are known {ital not} to be valid at these energies: multiloop radiative corrections are not suppressed relative to tree-graph contributions. Despite this pathology, we present a conjecture showing how anomalous scattering at such nonperturbative energies may nevertheless remain semiclassically calculable, and discuss partial results supporting this conjecture. To determine the correct weak-coupling behavior of high-energy anomalous scattering amplitudes, we argue that one must solve a modified set of classical field equations, or equivalently sum suitably modified tree graphs.

A solvable four-dimensional model with gauge symmetry

Recently, interest has been focused on quantum field theories in which a Lagrange multiplier field occurs in the classical Lagrangian. This has the effect of restricting the sum over classical paths in the path integral to solutions of the classical field equation. Examples of such theories are the dynamical theory of two-dimensional gravity proposed by Jackiw and Teitelboim, the Chern–Simons formulation of gravity in 2 + 1 dimensions by Witten, and the path-integral formulation of classical systems by Gozzi. We examine, in this paper, a gauge theory in which a vector field [Formula: see text] acts as a Lagrange multiplier in the classical Lagrangian, ensuring that a vector field [Formula: see text] satisfies the Yang–Mills equations of motion. Quantization can be carried out either using BRST quantization or by using the Faddeev–Popov procedure. Either by explicitly integrating over the field [Formula: see text] and its associated ghost fields, or by directly examining the Feynman perturbation theory, it can be established that all diagrams beyond one-loop order vanish, allowing one to compute the one-particle irreducible generating functional exactly. Background-field quantization is introduced to simplify the renormalization program. The β function is computed in closed form. In an appendix we show how our interaction can be derived from Yang–Mills theory based on a group G or by considering the Yang–Mills theory for a group IG. This can be extended to deal with four interacting gauge fields. A second appendix deals with a scalar model that superficially resembles our vector model.

Asymptotic time behaviour of nonlinear classical field equations

Hamiltonian systems with many degrees of freedom, like large assemblies of interacting particles in a box, are described by Gibbs-Boltzmann statistics, as far as their average properties are concerned. This does not hold for the long-time behaviour of classical nonlinear field equations, as has been already noticed by Jeans, because of the infinite heat capacity of this field. Thus, nonlinear (and nonintegrable) classical fields cannot relax for long times towards an ill-defined thermal equilibrium. The author considers an example of this relaxation problem: the long-time evolution of solutions of the nonlinear Schrodinger equation, in the defocusing case. Under some assumptions that for long times there is a cascade towards smaller and smaller scales, he introduces a kind of dissipation in a system that is formally reversible, and he gives the scaling laws for this.

Yang–Mills theory and nonirreducible representations of the Lorentz group

We start from an O(4) = SU(2) × SU(2) Yang–Mills theory and argue that the O(4) indices can, in fact, be space-time indices. The resulting theory is that of a tensor, Cμαβ = −Cμβα, which is in a reducible representation of the Lorentz group.fj This is a special case of the extended Yang–Mills formalism of Gabrielli. Some special solutions of the classical field equations are found.

High-energy multileg electroweak processes and unitarity.

Because of the failure of the dilute-instanton-gas approximation (DIGA) at high energies, recent calculations of ($B+L$)-violating processes in the standard model show blatant violations of unitarity, and suggest that these processes may be relatively unsuppressed at multiplicities $N\ensuremath{\sim}{\ensuremath{\alpha}}_{W}^{\ensuremath{-}1}$. We show how to cure the DIGA failure and restore the high-energy behavior necessary for consistency with unitarity in two ways: one in Minkowski space and the other in Euclidean space. In Euclidean space this is done by solving the classical field equations in the presence of space-time-dependent sources; we work out an explicit example. The same techniques allow us to investigate a similar failure of high-energy behavior in perturbation theory [($B+L$)-conserving processes] as studied with the DIGA in the manner of Lipatov. An independent Minkowski-space analysis, also dealing with classical solutions in the presence of sources, confirms these results and shows that even with the right high-energy behavior, factors growing rapidly with $N$ when $N\ensuremath{\ge}{\ensuremath{\alpha}}_{W}^{\ensuremath{-}1}$ still violate unitarity in the ($B+L$)-conserving sector. Within the framework of a simple model which automatically restores unitarity, we investigate whether dispersion integrals relating high- and low-energy ($B+L$)-violating processes can restrict the size of the high-energy $B+L$ violation, and find that they cannot.

Von Hove model of quantum field theory-from the point of view of infinite dimensional stationary Markov process theory

There are two ways to quantize the classical von Hove model. One way is to quantize the classical von Hove field equation, and the other is to quantize the classical energy functional. In this paper, their relation to each other is pointed out. As an application the second-order approximate quantization of (φ4)2-kink field is studied.

Recoil effects in a quantum theory of the Skyrmion

The purpose of this work is to illustrate how the Kerman-Klein method of quantization, introduced originally for the restoration of broken symmetries in the non-relativistic many-body problem and later utilized by Goldstone and Jackiw (1975) for a one-dimensional soliton, can be applied to the study of Skyrmion physics. In this method, one replaces the piecemeal quantization of collective coordinates by a formal canonical quantization of a full classical Hamiltonian, collectivity arising from the structure of the associated Hilbert space. In this first account, the restoration of translational invariance is studied using a quantized form of the original Skyrme Hamiltonian. This involves the study of matrix elements of the Heisenberg equations of motion by means of the completeness relation in a restricted Hilbert space consisting of all possible momentum eigenstates of the Skyrmion. In the limit in which the ratio of the Compton wavelength of the Skyrmion to its size is negligible (corresponding to the large Nc limit of QCD), the resulting classical field equation properly describes a boosted hedgehog. Higher order terms in this ratio describe quantum recoil corrections to the Skyrmion mass. In contrast to (1+1)-dimensional field theories studied previously, a gradient expansion for the calculation of these corrections diverges. In a more careful evaluation, there is a finite but large contribution from the Skyrme term. The dependence of the result on the ordering ambiguities that occur in the quantization of the field kinetic energy is also described.

Topological Half-Flat Gravity

The topological half-flat gravity, which has been proposed as a topological model of four-dimensionaI Einstein gravity, is explained. The classical field equations restrict manifold into a half-flat space that is the space with selfdual Riemann curvature. Ashtekar's new variables are used, which are more suitable for studying the half-flat space than using the metric or the vier-bein. The canonical formulation is discussed and it is shown that the Hamiltonian is BRST exact as expected in topological field theories

QUANTUM COSMOLOGY IN R1×S1×S2 TOPOLOGICAL SPACE

We have discussed classical and quantum cosmological phenomena for anisotropic Kantowski–Sachs model. The classical Lorentzian field equations are reduced to an autonomous system and are analyzed near critical point. The minisuperspace path-integral reduces to a single ordinary integration over the lapse, and we study this integral by lapse method. We find that the integral has an infinite number of saddle points, each one has identical contribution to the integral. Also matter-gravity wave function is considered for this metric.

Two-loop instabilities of gauge vacua and topological symmetry breaking on R n × S 1

The general problem of symmetry breaking by nonintegrable phase factors, or Wilson loops, on a non-simply connected spacetime of the form R n × S 1 is considered. The background field method is used to compute the effective potential as a function of a constant gauge field which solves the classical field equations. We show, in a general setting, how Wilson loops may be used to reduce this problem to a vacuum energy calculation where the fields satisfy non-trivial boundary conditions. Analysis of the one-loop effective potential is inadequate in many cases to determine stability of solutions to the effective field equations. We perform an analysis of some models where the two-loop effective potential must be used. One particular example considered is N = 4 supersymmetric Yang-Mills theory. We show that the local SU(2) symmetry in this theory breaks to U(1), but that the supersymmetry is left intact.