scholarly journals Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?

1997 ◽  
Vol 09 (08) ◽  
pp. 993-1052 ◽  
Author(s):  
T. Schmitt
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Field Model ◽  
Classical Field ◽  
Mathematical Work ◽  
Quantum Field ◽  
Infinite Dimensional ◽  
Fermion Fields ◽  
Classical Field Model ◽  
Conceptual Difficulties

We discuss the conceptual difficulties connected with the anticommutativity of classical fermion fields, and we argue that the "space" of all classical configurations of a model with such fields should be described as an infinite-dimensional supermanifold M. We discuss the two main approaches to supermanifolds, and we examine the reasons why many physicists tend to prefer the Rogers approach although the Berezin–Kostant–Leites approach is the more fundamental one. We develop the infinite-dimensional variant of the latter, and we show that the superfunctionals considered in [44] are nothing but superfunctions on M. We propose a programme for future mathematical work, which applies to any classical field model with fermion fields. A part of this programme will be implemented in the successor paper [45].

scholarly journals Infinite dimensional Langevin equation and Fokker-Planck equation

1981 ◽  
Vol 81 ◽  
pp. 177-223 ◽  
Author(s):  
Yoshio Miyahara
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Partial Differential Equations ◽  
Langevin Equation ◽  
Planck Equation ◽  
Theory Theory ◽  
Quantum Field ◽  
Filtering Theory ◽  
Infinite Dimensional

Stochastic processes on a Hilbert space have been discussed in connection with quantum field theory, theory of partial differential equations involving random terms, filtering theory in electrical engineering and so forth, and the theory of those processes has greatly developed recently by many authors (A. B. Balakrishnan [1, 2], Yu. L. Daletskii [7], D. A. Dawson [8, 9], Z. Haba [12], R. Marcus [18], M. Yor [26]).

scholarly journals The relation between quantum and classical field theory with a classical source

2020 ◽  
Vol 476 (2243) ◽  
pp. 20200656
Author(s):  
S. A. Fulling ◽  
A. G. S. Landulfo ◽  
G. E. A. Matsas
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Classical Field Theory ◽  
Classical Field ◽  
Quantum Field ◽  
Schwarzschild Solution ◽  
Classical Source ◽  
Time Quantum ◽  
Perturbative Quantum ◽  
Classical Picture

Classical field theory is about fields and how they behave in space–time. Quantum field theory, in practice, usually seems to be about particles and how they scatter. Nevertheless, classical fields must emerge from quantum field theory in appropriate limits, and Michael Duff showed how this happens for the Schwarzschild solution in perturbative quantum gravity. In a series of papers, we and others have shown how classical radiation from an accelerated charge emerges from quantum field theory when the Unruh thermal effect is taken into account. Here, we sharpen those conclusions by showing that, even at finite times, the quantum picture is meaningful and is in close agreement with the classical picture.

scholarly journals Area-Dependent Quantum Field Theory

2020 ◽  
Author(s):  
Ingo Runkel ◽  
Lóránt Szegedy
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Field ◽  
Two Dimensional ◽  
Positive Real ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Topological Theories

AbstractArea-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real number—interpreted as area—which behaves additively under glueing. As opposed to topological theories, in area-dependent theories the state spaces can be infinite-dimensional. We introduce the notion of regularised Frobenius algebras in Hilbert spaces and show that area-dependent theories are in one-to-one correspondence to commutative regularised Frobenius algebras. We also provide a state sum construction for area-dependent theories. Our main example is two-dimensional Yang–Mills theory with compact gauge group, which we treat in detail.

Quantum field theory

1955 ◽  
Vol 231 (1186) ◽  
pp. 321-335 ◽  
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Variational Principle ◽  
Schrodinger Equation ◽  
Classical Field Theory ◽  
Functional Expression ◽  
Scattering Matrix ◽  
Classical Field ◽  
Quantum Field ◽  
Relativistic Analogue

A functional expression resembling the scattering matrix is introduced into classical field theory, and with this foundation a postulate of quantization is introduced analogous to the definitions of Feynmann. From this are derived some alternative and more familiar forms of field theory. A variational principle is introduced which provides a relativistic analogue of the familiar non-relativistie variational principle for the Schrödinger equation.

scholarly journals An Approach to Classical Quantum Field Theory Based on the Geometry of Locally Conformally Flat Space-Time

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
John Mashford
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vector Bundles ◽  
Principal Bundle ◽  
Adjoint Action ◽  
Flat Space ◽  
Quantum Field ◽  
Euclidean Topology ◽  
Fermion Fields ◽  
Classical Quantum

This paper gives an introduction to certain classical physical theories described in the context of locally Minkowskian causal structures (LMCSs). For simplicity of exposition we consider LMCSs which have locally Euclidean topology (i.e., are manifolds) and hence are Möbius structures. We describe natural principal bundle structures associated with Möbius structures. Fermion fields are associated with sections of vector bundles associated with the principal bundles while interaction fields (bosons) are associated with endomorphisms of the space of fermion fields. Classical quantum field theory (the Dirac equation and Maxwell’s equations) is obtained by considering representations of the structure group K⊂SU(2,2) of a principal bundle associated with a given Möbius structure where K, while being a subset of SU(2,2), is also isomorphic to SL2,C×U(1). The analysis requires the use of an intertwining operator between the action of K on R4 and the adjoint action of K on su⁡(2,2) and it is shown that the Feynman slash operator, in the chiral representation for the Dirac gamma matrices, has this intertwining property.

ASYMPTOTICS OF INFINITE-DIMENSIONAL INTEGRALS WITH RESPECT TO SMOOTH MEASURES I

1999 ◽  
Vol 02 (04) ◽  
pp. 529-556 ◽  
Author(s):  
S. ALBEVERIO ◽  
V. STEBLOVSKAYA
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Probability Measures ◽  
Quantum Field ◽  
Laplace Method ◽  
Large Parameter ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Smooth Measures ◽  
Minimum Points

This is the first part of a work on Laplace method for the asymptotics of integrals with respect to smooth measures and a large parameter developed in infinite dimensions. Here the case of finitely many (nondegenerate) minimum points is studied in details. Applications to large parameters behavior of expectations with respect to probability measures occurring in the study of systems of statistical mechanics and quantum field theory are mentioned.

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