Soliton quantization in lattice field theories

Quantization of solitons in terms of Euclidean region functional integrals is developed, and Osterwalder-Schrader reconstruction is extended to theories with topological solitons. The quantization method is applied to several lattice field theories with solitons, and the particle structure in the soliton sectors of such theories is analyzed. A construction of magnetic monopoles in the four-dimensional, compactU(1)-model, in the QED phase, is indicated as well.     

Relating field theories via stochastic quantization

This note aims to subsume several apparently unrelated models under a common framework. Several examples of well-known quantum field theories are listed which are connected via stochastic quantization. We highlight the fact that the quantization method used to obtain the quantum crystal is a discrete analog of stochastic quantization. This model is of interest for string theory, since the (classical) melting crystal corner is related to the topological A-model. We outline several ideas for interpreting the quantum crystal on the string theory side of the correspondence, exploring interpretations in the Wheeler–De Witt framework and in terms of a non-Lorentz invariant limit of topological M-theory.     

Canonical quantization of Galilean covariant field theories

The Galilean-invariant field theories are quantized by using the canonical method and the five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. This method is motivated by the fact that the extended Galilei group in 3 + 1 dimensions is a subgroup of the inhomogeneous Lorentz group in 4 + 1 dimensions. First, we consider complex scalar fields, where the Schrödinger field follows from a reduction of the Klein-Gordon equation in the extended space. The underlying discrete symmetries are discussed, and we calculate the scattering cross-sections for the Coulomb interaction and for the self-interacting term λΦ⁴. Then, we turn to the Dirac equation, which, upon dimensional reduction, leads to the Lévy-Leblond equations. Like its relativistic analogue, the model allows for the existence of antiparticles. Scattering amplitudes and cross-sections are calculated for the Coulomb interaction, the electron-electron and the electron-positron scattering. These examples show that the so-called ‘non-relativistic’ approximations, obtained in low-velocity limits, must be treated with great care to be Galilei-invariant. The non-relativistic Proca field is discussed briefly.     

A canonical structure for classical field theories

A general scheme of constructing a canonical structure (i.e. Poisson bracket, canonical fields) in classical field theories is proposed. The theory is manifestly independent of the particular choice of an initial space-like surface in space-time. The connection between dynamics and canonical structure is established. Applications to theories with a gauge and constraints are of special interest. Several physical examples are given.     

Measures in the geometric quantization of field theories

In this article, it is shown that for the standard symplectic form on the space of compactly supported sections of a symplectic fibre bundle, there is no locally-finite Borel measure which is preserved by the Hamiltonian flows of even a quite restricted set of functions on this space. As this means that some of the operators arising in geometric quantization associated to classical observables would not be Hermitean, the result suggests that one should consider quotients by gauge groups as classical phase spaces to avoid this problem.     

Covariant canonical quantization

We present a manifestly covariant quantization procedure based on the de Donder–Weyl Hamiltonian formulation of classical field theory. This procedure agrees with conventional canonical quantization only if the parameter space is d=1 dimensional time. In d>1 quantization requires a fundamental length scale, and any bosonic field generates a spinorial wave function, leading to the purely quantum-theoretical emergence of spinors as a byproduct. We provide a probabilistic interpretation of the wave functions for the fields, and we apply the formalism to a number of simple examples. These show that covariant canonical quantization produces both the Klein–Gordon and the Dirac equation, while also predicting the existence of discrete towers of identically charged fermions with different masses. Covariant canonical quantization can thus be understood as a “first” or pre-quantization within the framework of conventional QFT. PACS 04.62.+v; 11.10.Ef; 12.10.Kt     

Hopfion canonical quantization

We study the effect of the canonical quantization of the rotational mode of the charge Q=1$Q=1$ and Q=2$Q=2$ spinning Hopfions. The axially-symmetric solutions are constructed numerically, it is shown the quantum corrections to the mass of the configurations are relatively large.     

Equivalence of stochastic quantization to field theories from supersymmetry

Using the Ward-Takahashi identities from the hidden supersymmetry in Langevin equation we present a very simple proof of the equivalence of stochastic quantization to field theories.     

Path integral quantization of field theories with second-class constraints

Faddeev's Hamiltonian path integral method for singular Lagrangians is generalized to the case when second-class constraints appear in the theory. The general formalism is then applied to several problems: quantization of the massive Yang-Mills field theory, light-cone quantization of the self-interacting scalar field theory, and quantization of a local field theory of magnetic monopoles.     

Topological field theories,Nicolai maps and BRST quantization

We establish a connection between topological field theories, Nicolai maps, BRST quantization and Langevin equations. In particular we show that there is a one-to-one correspondence between global unbroken supersymmetric theories which admit a Nicolai map and theories which arise as the BRST quantization of the square of the Langevin equation, setting the random field to zero. As such they are topological in nature. As an example we consider the topological quantum field theory of Witten in the Labastida-Pernici form and show that it is the first example of a theory admitting a complete Nicolai map in four dimensions. We also consider the topological sigma models of Witten and show that they too arise from the BRST quantization of the square of the Langevin equation.     

Consistent canonical quantization of a model field theory with Schwinger term

Canonical quantization of a finite two-dimensional model field theory is consistently carried out, in spite of the occurrence of a non-canonical Schwinger term. Heisenberg field operators are defined perturbatively by means of the dimensionally regularized Feynman chronological product, while their equal-time commutators may be related to free-field equal-time commutators through the use of the non-covariant Dyson chronological product. A concrete agreement (including the axial anomaly) is thus found between the quantum Lagrangian and Hamiltonian formulations.     

Remarks on unified field structures,spin structures and canonical quantization

Unified field structures are defined and reviewed. Under certain conditions these are shown to be dynamical systems. And quantizable dynamical systems are shown to be unified field structures with invariant Riemannian metric. Spin structure is reviewed and manifoldsM ^8k+4 with spin structure are shown to be symplectic.This research was supported in part by NSF GP-13375.     

Propositional systems in local field theories

We investigate prepositional systems for local field theories, which reflect intrinsically the uncertainties of measurements made on the physical system, and satisfy the isotony and local commutativity postulates of Haag and Kastler. The space-time covariance can be implemented in a natural way in these propositional systems. New techniques are introduced to obtain these propositional systems: the lattice-valued logics. The decomposition of the complete orthomodular lattice-valued logics shows that these logics are more general than the usual two-valued ones and that in these logics there is enough structure to characterize the classical and quantum, nonrelativistic and relativistic local field theories in a natural way. The Hubert modules give the natural inner product spaces (modules) for the realization of the lattice-valued logics.The author is also with the Publishing House of the Hungarian Academy of Sciences, H-1363, Budapest, P.O.B. 24.     

Stochastic quantization of gauge theories

The equivalence between a scalar quantum field theory in D dimensions and its classical counterpart in D + 2 dimensions which is coupled to an external random source with Gaussian correlations was observed by previous authors. This stochastic quantization is extended to gauge theories. The proof exploits the supersymmetry formalism suggested by Parisi and Sourlas.     

Baby Skyrmions stabilized by canonical quantization

We analyse the effect of the canonical quantization of the rotational mode of the O(3)$O(3)$σ  -model which includes the Skyrme term. Numerical evidence is presented that the quantum correction to the mass of the rotationally-invariant charge n=1,2$n=1,2$ configurations may stabilize the solution even in the limit of vanishing potential. The corresponding range of values of the parameters is discussed.     

On local field products in special Wightman theories

We shall try to define local field products under assumptions imposed only on the four-point-function. This idea is based on the work of Schlieder and Seiler [1]. In our framework we shall prove that the two-point-function carries the strongest singularity whenever two arguments in a Wightman function coincide. This will be generalized to the case when more arguments coincide. We shall define “regulated”n-point-functions and study their properties in detail. This will lead us to the definition of arbitrarily high powers of the field-operators as operator-valued distributions overD(?⁴) in the center coordinate with a dense domain of definition.