UNIVERSAL ALGEBRA OF SECTORS

2009 ◽  
Vol 19 (03) ◽  
pp. 347-371 ◽  
Author(s):  
KATSUNORI KAWAMURA
Keyword(s):  
Field Theory ◽  
Universal Algebra ◽  
Equivalence Classes ◽  
Cuntz Algebra ◽  
Quantum Field ◽  
Unitary Equivalence ◽  
C Algebra ◽  
Branching Laws ◽  
Common Unit ◽  
Algebraic Formulation

We show that a nontrivial example of universal algebra appears in quantum field theory. For a unital C *-algebra [Formula: see text], a sector is a unitary equivalence class of unital *-endomorphisms of [Formula: see text]. We show that the set [Formula: see text] of all sectors of [Formula: see text] is a universal algebra with an N-ary sum which is not reduced to any binary sum when [Formula: see text] includes the Cuntz algebra [Formula: see text] as a C *-subalgebra with common unit for N ≥ 3. Next we explain that the set [Formula: see text] of all unitary equivalence classes of unital *-representations of [Formula: see text] is a right module of [Formula: see text]. An essential algebraic formulation of branching laws of representations is given by using submodules of [Formula: see text]. As an application, we show that the action of [Formula: see text] on [Formula: see text] distinguishes elements of [Formula: see text].

ON THE INDEPENDENCE OF LOCAL ALGEBRAS IN QUANTUM FIELD THEORY

1990 ◽  
Vol 02 (02) ◽  
pp. 201-247 ◽  
Author(s):  
STEPHEN J. SUMMERS
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Relativistic Quantum ◽  
Mathematical Formulation ◽  
Quantitative Measure ◽  
Quantum Systems ◽  
Statistical Independence ◽  
Quantum Field ◽  
Relativistic Quantum Field Theory ◽  
C Algebra

A review is made of the multitude of different mathematical formalizations of the physical concept ‘two observables (or two systems) are independent’ that have been proposed in quantum theories, particularly relativistic quantum field theory. The most basic mathematical formulation of independence in any quantum theory is what one may call kinematical independence: the two observables, resp. the observables of the two quantum systems, which are represented by elements of a C*-algebra, resp. two subalgebras of a C*-algebra, are required to commute. This is related to a mathematical formulation of the notion of the coexistence (or compatibility) of two observables. Another basic notion of independence, generally called statistical independence in the literature, is, roughly speaking, two quantum systems are said to be statistically independent if each can be prepared in any state, how ever the other system has been prepared. There are numerous mathematical formulations of this notion and their interrelationships are explained. Statistical independence and kinematical independence are shown to be logically independent. Additional notions such as strict locality and their relation to statistical independence are discussed. The mathematics of a more quantitative measure of statistical independence, Bell’s inequalities, is reviewed and its relations with previously introduced notions are indicated. All of these notions are then viewed in application to relativistic quantum field theory.

scholarly journals THE USES OF CONNES AND KREIMER'S ALGEBRAIC FORMULATION OF RENORMALIZATION THEORY

2004 ◽  
Vol 19 (16) ◽  
pp. 2739-2754 ◽  
Author(s):  
HÉCTOR FIGUEROA ◽  
JOSÉ M. GRACIA-BONDÍA
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Algebraic Approach ◽  
Quantum Field ◽  
Feynman Amplitudes ◽  
Renormalization Theory ◽  
Algebraic Formulation

We show how, modulo the distinction between the antipode and the "twisted" or "renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes the proofs of equivalence of the (corrected) Dyson–Salam, Bogoliubov–Parasiuk–Hepp and Zimmermann procedures for renormalizing Feynman amplitudes. We discuss the outlook for a parallel simplification of computations in quantum field theory, stemming from the same algebraic approach.

scholarly journals AUTOMATA COMPUTATION OF BRANCHING LAWS FOR ENDOMORPHISMS OF CUNTZ ALGEBRAS

2007 ◽  
Vol 17 (07) ◽  
pp. 1389-1409 ◽  
Author(s):  
KATSUNORI KAWAMURA
Keyword(s):  
Semigroup Theory ◽  
Automata Theory ◽  
De Bruijn Graph ◽  
Cuntz Algebra ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Branching Laws ◽  
Branching Law ◽  
Mealy Machine

We study the representation theory of C*-algebras by using semigroup theory and automata theory. The Cuntz algebra [Formula: see text] is a finitely generated, infinite-dimensional, noncommutative C*-algebra. A certain class of cyclic representations of [Formula: see text] is characterized by words from the alphabet 1,…,N, which is called a cycle. A class of endomorphisms of [Formula: see text] is defined by polynomial functions in canonical generators and their conjugates. Such an endomorphism ρ of [Formula: see text] transforms a cycle π to π ◦ ρ which is a direct sum of cycles π1,…,πn unique up to unitary equivalence. The passage from π to π1,…,πn is called a branching law for ρ. In this article, we construct a Mealy machine from the endomorphism in order to compute its branching law. We show that the branching law is obtained as outputs from the machine for the input information of a given representation. Furthermore the actual computation of the branching law is executed by using a generalized de Bruijn graph associated with the Mealy machine.

scholarly journals Another formulation of the Wick’s theorem. Farewell, pairing?

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Igor V. Beloussov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Computer Algebra ◽  
Arbitrary Number ◽  
Statistical Physics ◽  
Computer Algebra Systems ◽  
Quantum Field ◽  
The Matrix ◽  
Wick’S Theorem ◽  
Algebraic Formulation

AbstractThe algebraic formulation of Wick’s theorem that allows one to present the vacuum or thermal averages of the chronological product of an arbitrary number of field operators as a determinant (permanent) of the matrix is proposed. Each element of the matrix is the average of the chronological product of only two operators. This formulation is extremely convenient for practical calculations in quantum field theory, statistical physics, and quantum chemistry by the standard packages of the well known computer algebra systems.

A Review of Conformal Field Theory in 2D

2014 ◽  
Vol 6 (2) ◽  
pp. 1079-1105
Author(s):  
Rahul Nigam
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Conformal Field Theory ◽  
Statistical Mechanics ◽  
Critical Behavior ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Quantum Field ◽  
Conformal Field ◽  
Insight Into

In this review we study the elementary structure of Conformal Field Theory in which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory. We briefly review CFT in dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary. We also describe the mapping of the theory from the cylinder to a complex plane which will help us gain an insight into the process of radial quantization and radial ordering. Finally we will develop the representation of the Virasoro algebra which is the well-known "Verma module".  

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