Zeta-Regularized Traces Versus the Wodzicki Residue as Tools in Quantum Field Theory and Infinite Dimensional Geometry

2004 ◽  
pp. 69-84 ◽  
Author(s):  
Sylvie Paycha
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Field ◽  
Wodzicki Residue ◽  
Infinite Dimensional

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 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.

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.

scholarly journals Infinite-dimensional Lie algebras in 4D conformal quantum field theory

2008 ◽  
Vol 41 (19) ◽  
pp. 194002 ◽  
Author(s):  
Bojko Bakalov ◽  
Nikolay M Nikolov ◽  
Karl-Henning Rehren ◽  
Ivan Todorov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Lie Algebras ◽  
Quantum Field ◽  
Infinite Dimensional

scholarly journals MODULAR THEORY AND GEOMETRY

2000 ◽  
Vol 12 (01) ◽  
pp. 139-158 ◽  
Author(s):  
B. SCHROER ◽  
H.-W. WIESBROCK
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Field ◽  
Hidden Symmetries ◽  
Modular Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional

In this communication we present some new results on modular theory in the context of quantum field theory. In doing this we develop some new proposals how to generalize concepts of finite dimensional geometrical actions to infinite dimensional "hidden" symmetries. The latter are of a purely modular origin and remain hidden in any quantization approach. The spirit of this work is more on a programmatic side, with many details remaining to be elaborated.

scholarly journals INFINITE QUANTUM GROUP SYMMETRY OF FIELDS IN MASSIVE 2D QUANTUM FIELD THEORY

1992 ◽  
Vol 07 (13) ◽  
pp. 2997-3022 ◽  
Author(s):  
ANDRÉ LECLAIR ◽  
F. A. SMIRNOV
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Group ◽  
Form Factors ◽  
Adjoint Action ◽  
Momentum Tensor ◽  
Group Symmetry ◽  
Quantum Field ◽  
Infinite Dimensional ◽  
Symmetry Generators

Starting from a given S-matrix of an integrable quantum field theory in 1 + 1 dimensions, and knowledge of its on-shell quantum group symmetries, we describe how to extend the symmetry to the space of fields. This is accomplished by introducing an adjoint action of the symmetry generators on fields, and specifying the form factors of descendents. The braiding relations of quantum field multiplets is shown to be given by the universal ℛ-matrix. We develop in some detail the case of infinite-dimensional Yangian symmetry. We show that the quantum double of the Yangian is a Hopf algebra deformation of a level zero Kac–Moody algebra that preserves its finite-dimensional Lie subalgebra. The fields form infinite-dimensional Verma module representations; in particular, the energy–momentum tensor and isotopic current are in the same multiplet.

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].

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