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Charged Quantum Fields Associated with Endomorphisms of CAR and CCR Algebras

Carsten Binnenhei

Published Jan 1, 1998• 0 citations
Publisher

Abstract

The appearance of the Cuntz algebras Od is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. In view of this apparent discrepancy, we develop a comprehensive theory of quasi-free endomorphisms of the CAR and CCR algebras which give rise to representations of the Cuntz algebras Od on Fock space. The number d is the statistics dimension of the endomorphism. It can be any power of 2 (including 1 and infinite) in the CAR case, but takes only the values 1 and infinite in the CCR case. We obtain necessary and sufficient conditions for implementability of quasi- free endomorphisms. By studying extensions of partial Fock states, we find that the semigroup of implementable quasi-free endomorphisms has a simple structure: It can be written as a product of a group of automorphisms which are close to the identity, and the semigroup of endomorphisms which leave the given Fock state invariant. We describe the construction of the implementers of a quasi-free endomorphism in terms of annihilation and creation operators in is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. We compare the structure of the semigroup of (gauge invariant) quasi-free endomorphisms with the generic superselection structure of quantum field theory.

DOI
10.17169/refubium-13590

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The appearance of the Cuntz algebras Od is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. In view of this apparent discrepancy, we develop a comprehensive theory of quasi-free endomorphisms of the CAR and CCR algebras which give rise to representations of the Cuntz algebras Od on Fock space. The number d is the statistics dimension of the endomorphism. It can be any power of 2 (including 1 and infinite) in the CAR case, but takes only the values 1 and infinite in the CCR case. We obtain necessary and sufficient conditions for implementability of quasi- free endomorphisms. By studying extensions of partial Fock states, we find that the semigroup of implementable quasi-free endomorphisms has a simple structure: It can be written as a product of a group of automorphisms which are close to the identity, and the semigroup of endomorphisms which leave the given Fock state invariant. We describe the construction of the implementers of a quasi-free endomorphism in terms of annihilation and creation operators in is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. We compare the structure of the semigroup of (gauge invariant) quasi-free endomorphisms with the generic superselection structure of quantum field theory.
The superselection sectors of three closely related models are studied and compared. I. The full Ising model with observable algebra [Formula: see text], the universal algebra of local even CAR algebras on the circle. [Formula: see text] has four sectors with Z(2) × Z(2) symmetry. II. The chiral Ising model with observable algebra [Formula: see text] (c = L or R), which is the universal algebra of local even Majorana algebras, has three sectors with quantum symmetry Gc ≅ M1⊕M1⊕M2. The Mack-Schomerus endomorphism creating the non-Abelian sector of [Formula: see text], respectively of [Formula: see text] is shown to coincide with the restriction [Formula: see text], respectively [Formula: see text] of a Z (2) charge creating automorphism ρ of [Formula: see text]. III. As an intermediate step the superselection sectors of the algebra [Formula: see text] — which can be interpreted as the observable algebra of the conformal Ising model — are found to have ordinary group symmetry described by the dihedral group D4. The relation between the sectors of [Formula: see text] and [Formula: see text] is explained in terms of a strange 'symmetry breaking': symmetry enhancement in the Neveu-Schwarz sector and symmetry breaking in the Ramond. Covariant charged fields are constructed in all three cases and the truncation in Gc is shown to arise from the failure of the Cuntz algebra relations for the chiral charged fields.
It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion structure . . . . . ....
Based on the treatment of the chiral Ising model by Mack and Schomerus, we present examples of localized endomorphisms ̺ loc 1 and ̺ loc 1/2. It is shown that they lead to the same superselection sectors as the global ones in the sense that unitary equivalence π0 ◦ ̺ loc 1 ∼ = π1 and π0 ◦ ̺ loc 1/2 ∼ = π1/2 holds. Araki’s formalism of the selfdual CAR algebra is used for the proof. We prove local normality and extend representations and localized endomorphisms to a global algebra of observables which is generated by local von Neumann algebras on the punctured circle. In this framework, we manifestly prove fusion rules and derive statistics operators. 1
Let H1{\mathcal H}_1H1​ be a finite dimensional complex Hilbert space. Let ψ↦Z(ψ)ψ\mapsto Z(ψ)ψ↦Z(ψ) be a canonical anti-commutation relations (CAR) field over H1{\mathcal H}_1H1​ acting irreducibly on a Hilbert space K{\mathord{\mathscr K}}K. The ∗*∗-algebra AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ generated by the Z(ψ)Z(ψ)Z(ψ), ψ∈H1ψ\in {\mathcal H}_1ψ∈H1​, is simply all operators on K{\mathscr K}K. However, the CAR field endows AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ with additional structure, and we are concerned with quantum operations acting in harmony with this structure. In particular, there is a "gauge" automorphism group generated by "second quantizing'' ψ↦eitψψ\mapsto e^{it}ψψ↦eitψ. The fixed point algebra of the gauge group, GH1{\mathscr G}_{{\mathcal H}_1}GH1​​, is a sub-algebra of AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ studied by Araki and Wyss. It contains the density matrices of an important class of states, the {\em gauge invariant Gaussian states}, SGIG{\mathfrak S}_{GIG}SGIG​. Our focus is on semigroups {etL}t≥0\{e^{t{\mathscr L}}\}_{t\geq 0}{etL}t≥0​ of quantum operations on AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ that map SGIG{\mathfrak S}_{GIG}SGIG​ into itself. Each etLe^{t{\mathscr L}}etL is one-to-one, and our first main result is a structure theorem for such quantum operations on GH1{\mathscr G}_{{\mathcal H}_1}GH1​​ that map SGIG{\mathfrak S}_{GIG}SGIG​ into itself. We apply this to study semigroups of quantum operations on GH1{\mathscr G}_{{\mathcal H}_1}GH1​​ that map SGIG{\mathfrak S}_{GIG}SGIG​ into itself. Our second main result is a structure theorem showing that they are parameterized by pairs (G,A)(G,A)(G,A) where GGG is a contraction semigroup generator on H1{\mathcal H}_1H1​, and 0≤A≤−G−G∗0 \leq A \leq -G -G^*0≤A≤−G−G∗. We then show that each of these semigroups has a natural extension to the full CAR algebra AH1{\mathscr A}_{{\mathcal H}_1}AH1​​. Further results are obtained under further assumptions on the pair (G,A)(G,A)(G,A).
Abstract In the previous sections we treated the irreducible representation space associated with the vacuum state. In this chapter we treat other representation spaces necessary to describe particle systems with the vacuum as a background. Such representation spaces are characterized as transportable localized excitations on the vacuum background. If we assume Haag’s duality of the vacuum representation, such a localized excitation is described by a localized endomorphism of the observables and its equivalence classes correspond to different representations called sectors(eigensubspaces of the superselection rules in each of which unrestricted superposition principle holds). From the discussion about composition and conjugation of localized endomorphisms, we define the statistics (of a sector) which represents the behaviour of more than two equivalent localized excitations under permutation, and we derive the connection of spin and statistics. We also introduce the lattice of local field algebras as operators intertwining different sectors, and the gauge group acting on them. Then the local observable algebras are the gauge invariant part of the local field algebras and sectors are inequivalent representations of the gauge group. The local field algebras can be chosen to satisfy the standard commutation or anticommutation relations between space-like separated regions and we can derive the connection of spin and statistics of sectors. In this way we obtain a deep understanding of the reason why Fermi fields in spatially separated regions anticommute.
A general theoretical framework is developed for the treatment of a class of quantum field theories that are explicitly exactly solvable, but require the use of C*-algebraic techniques because time-dependent scattering theory cannot be constructed in any one natural representation of the observable algebra. The purpose is to exhibit mechanisms by which inequivalent representations of the observable algebra can arise in quantum field theory, in a setting free of other complications commonly associated with the specification of dynamics. One of two major results is the development of necessary and sufficient conditions for the concurrent unitary implementation of two automorphism groups in a class of quasifree representations of the algebra of the canonical commutation relations (CCR). The automorphism groups considered are induced by one-parameter groups of symplectic transformations on the classical phase space over which the Weyl algebra of the CCR is built; each symplectic group is conjugate by a fixed symplectic transformation to a one-parameter unitary group. The second result, an analog to the Birman–Belopol’skii theorem in two-Hilbert-space scattering theory, gives sufficient conditions for the existence of Mo/ller wave morphisms in theories with time-development automorphism groups of the above type. In a paper which follows, this framework is used to analyze a particular model system for which wave operators fail to exist in any natural representation of the observable algebra, but for which wave morphisms and an associated S matrix are easily constructed.