Commutator Theory on Hilbert Space

1987 ◽  
Vol 39 (5) ◽  
pp. 1235-1280 ◽  
Author(s):  
Derek W. Robinson
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Lie Group ◽  
Differential Operators ◽  
Quantum Field ◽  
Lie Group Theory ◽  
Commutator Theory ◽  
Constructive Quantum Field Theory ◽  
Partial Differential Operators

Commutator theory has its origins in constructive quantum field theory. It was initially developed by Glirnm and Jaffe [7] as a method to establish self-adjointness of quantum fields and model Hamiltonians. But it has subsequently proved useful for a variety of other problems in field theory [17] [15] [8] [3], quantum mechanics [5], and Lie group theory [6]. Despite all these detailed applications no attempt appears to have been made to systematically develop the theory although reviews have been given in [22] and [9]. The primary aim of the present paper is to partially correct this situation. The secondary aim is to apply the theory to the analysis of first and second order partial differential operators associated with a Lie group.

Dense Subalgebras of Left Hilbert Algebras

1982 ◽  
Vol 34 (6) ◽  
pp. 1245-1250 ◽  
Author(s):  
A. van Daele
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Cyclic Vector ◽  
Quantum Field ◽  
Von Neumann ◽  
Hilbert Algebras

Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]).

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 Universal bounds for large determinants from non-commutative Hölder inequalities in fermionic constructive quantum field theory

2017 ◽  
Vol 27 (10) ◽  
pp. 1963-1992 ◽  
Author(s):  
J.-B. Bru ◽  
W. de Siqueira Pedra
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Von Neumann Algebras ◽  
Perturbation Expansion ◽  
Perturbation Series ◽  
Interparticle Interaction ◽  
Quantum Field ◽  
Von Neumann ◽  
Constructive Quantum Field Theory ◽  
Strength Formula

Efficiently bounding large determinants is an essential step in non-relativistic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength [Formula: see text] of the interparticle interaction. We provide, for large determinants of fermionic covariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly [Formula: see text]. In particular, the convergence of perturbation series at [Formula: see text] of any fermionic quantum field theory is ensured if the matrix entries (with respect to some fixed orthonormal basis) of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use Hölder inequalities for general non-commutative [Formula: see text]-spaces derived by Araki and Masuda [Positive cones and [Formula: see text]-spaces for von Neumann algebras, Publ. RIMS[Formula: see text] Kyoto Univ. 18 (1982) 339–411].

scholarly journals QUANTUM FIELD THEORY ON CURVED BACKGROUNDS — A PRIMER

2013 ◽  
Vol 28 (17) ◽  
pp. 1330023 ◽  
Author(s):  
MARCO BENINI ◽  
CLAUDIO DAPPIAGGI ◽  
THOMAS-PAUL HACK
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Degrees Of Freedom ◽  
Algebraic Approach ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
System A ◽  
Step Procedure ◽  
Free Fields

Goal of this paper is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, it is assigned to a physical system a suitable algebra of observables, which is meant to encode all algebraic relations among observables, such as commutation relations. In the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.

scholarly journals The Hilbert space of quantum gravity is locally finite-dimensional

2017 ◽  
Vol 26 (12) ◽  
pp. 1743013 ◽  
Author(s):  
Ning Bao ◽  
Sean M. Carroll ◽  
Ashmeet Singh
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Quantum Gravity ◽  
Density Operator ◽  
Small Region ◽  
Quantum Field ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Model Independent

We argue in a model-independent way that the Hilbert space of quantum gravity is locally finite-dimensional. In other words, the density operator describing the state corresponding to a small region of space, when such a notion makes sense, is defined on a finite-dimensional factor of a larger Hilbert space. Because quantum gravity potentially describes superpositions of different geometries, it is crucial that we associate Hilbert-space factors with spatial regions only on individual decohered branches of the universal wave function. We discuss some implications of this claim, including the fact that quantum-field theory cannot be a fundamental description of nature.

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