Canonical Anticommutation Relations Research Articles

Factor representations of diffeomorphism groups

We give a new construction of seminite factor representations of the dieomorphism group of euclidean space. These representations are in canonical corre- spondence with the nite factor representations of the inductive limit unitary group. Hence, many of these representations are given in terms of quasi-free representations of the canonical commutation and anti-commutation relations. To establish this correspon- dence requires a generalization of complete positivity as developed in operator algebras. We also compare the asymptotic character formula for the unitary group with the ther- modynamic (N=V ) limit construction for dieomorphism group representations.

The Dirac–Ramond operator on loops in flat space

In this paper, a rigorous construction of the S 1-equivariant Dirac operator (i.e., Dirac–Ramond operator) on the space of (mean zero) loops in R d is given and its equivariant L 2-index computed. Essential use is made of infinite tensor product representations of the canonical anticommutation relations algebra.

A COMPLETE SET OF ISOMETRIC IMPLEMENTERS FOR A QUASI-FREE ENDOMORPHISM OF THE CAR ALGEBRA VIA UNITARY DILATIONS

In this paper we concern ourselves with non-surjective quasi-free endomorphisms of the CAR algebra. More precisely we give a construction of a complete set of isometric implementers for such an endomorphism in a fixed pure quasi-free state, the approach to which is more natural than has previously been proposed. The key idea in this construction is the use of the unitary implementers of an associated quasi-free automorphism in a set of carefully chosen quasi-free states. Our treatment is also novel in that we provide a sequence of implementers within the framework of the complex formalism of the CAR algebra.

Cuntz Algebra States Defined by Implementers of Endomorphisms of the CAR Algebra

AbstractWe investigate representations of the Cuntz algebra on antisymmetric Fock space Fa(𝒦1) defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states φP1. We pay special attention to the vector states on corresponding to these representations and the Fock vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra , we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states φP1. We proceed to consider the case for an arbitrary pair of implementers, and deduce that these Cuntz algebra representations are irreducible, as are their restrictions to .The endomorphisms of B(Fa(𝒦)) associated with these representations of are also considered.

Recursive Fermion System in Cuntz Algebra. I

Embeddings of the CAR (canonical anticommutation relations) algebra of fermions into the Cuntz algebra ?2 (or ?2 d more generally) are presented by using recursive constructions. As a typical example, an embedding of CAR onto the U(1)-invariant subalgebra of ?2 is constructed explicitly. Generalizing this construction to the case of ?2 p , an embedding of CAR onto the U(1)-invariant subalgebra of ?2 p is obtained. Restricting a permutation representation of the Cuntz algebra, we obtain the Fock representation of CAR. We apply the results to embed the algebra of parafermions of order p into ?2 p according to Green's ansatz.

On the Order Structure of Time Projection

The order structure of time projections associated with ran- dom times in a von Neumann algebra is investigated in the general setup as well as that of the CAR and CCR algebras. In the second case var- ious additional properties (such as e.g. the upper/lower continuity) of the lattice of time projections are also discussed.

UHF FLOWS AND THE FLIP AUTOMORPHISM

A UHF algebra is a C*-algebra A of the type [Formula: see text] for some sequence (ni) with ni≥2, where Mn is the algebra of n×n matrices, while a UHF flow α is a flow (or a one-parameter automorphism group) on the UHF algebra A obtained as [Formula: see text], where [Formula: see text] for some [Formula: see text]. This is the simplest kind of flows on the UHF algebra we could think of; yet there seem to have been no attempts to characterize the cocycle conjugacy class of UHF flows so that we might conclude, e.g., that the non-trivial quasi-free flows on the CAR algebra are beyond that class. We give here one attempt, which is still short of what we have desired, using the flip automorphism of A⊗A. Our characterization for a somewhat restricted class of flows (approximately inner and absorbing a universal UHF flow) says that the flow α is cocycle conjugate to a UHF flow if and only if the flip is approximated by the adjoint action of unitaries which are almost invariant under α⊗α. Another tantalizing problem is whether we can conclude that a flow is cocycle conjugate to a UHF flow if it is close to a UHF flow in a suitable sense. We give a solution to this, as a corollary, for the above-mentioned restricted class of flows. We will also discuss several kinds of flows to clarify the situation.

ENTROPY OF BOGOLIUBOV AUTOMORPHISMS OF CAR AND CCR ALGEBRAS WITH RESPECT TO QUASI-FREE STATES

We compute the dynamical entropy of Bogoliubov automorphisms of CAR and CCR algebras with respect to arbitrary gauge-invariant quasi-free states. This completes the research started by Størmer and Voiculescu, and continued in works of Narnhofer–Thirring and Park–Shin.

Residually finite dimensional C*-algebras and subquotients of the CAR algebra

It is proved that the cone of a separable nuclearly embeddable resid- ually finite-dimensional C*-algebra embeds in the CAR algebra (the UHF algebra of type 2 ∞ ). As a corollary we obtain a short new proof of Kirchberg's theorem asserting that a separable unital C*-algebra A is nuclearly embeddable if and only there is a semisplit extension 0 → J → E → A → 0 with E a unital C*-subalgebra of the CAR algebra and the ideal J an AF-algebra. The new proof does not rely on the lifting theorem of Effros and Haagerup.

COCYCLE PERTURBATION OF QUASIFREE ALGEBRAIC K-FLOW LEADS TO REQUIRED ASYMPTOTIC DYNAMICS OF ASSOCIATED COMPLETELY POSITIVE SEMIGROUP

We study the quasifree algebraic K-flow τ on the hyperfinite factor ℳ with the expanding subfactor [Formula: see text] generated by representations π of the C*-algebra of the canonical anticommutation relations (CAR) [Formula: see text] over separable Hilbert space ℋ. The type of ℳ and [Formula: see text] can be II1 or IIIλ, 0<λ<1, depending on π. The K-flow τ is obtained by the quasifree lifting of one-parameter group ST consisting of shifts in ℋ with the discrete parameter T=Z or the continuous one T=R. We prove that acting on τ by a quasifree inner Markovian cocycle, one can get the required asymptotic behavior of the perturbed group restriction on [Formula: see text].

Renormalization of the Regularized Relativistic Electron-Positron Field

We consider the relativistic electron-positron field interacting with itself via the Coulomb potential defined with the physically motivated, positive, density-density quartic interaction. The more usual normal-ordered Hamiltonian differs from the bare Hamiltonian by a quadratic term and, by choosing the normal ordering in a suitable, self-consistent manner, the quadratic term can be seen to be equivalent to a renormalization of the Dirac operator. Formally, this amounts to a Bogolubov-Valatin transformation, but in reality it is non-perturbative, for it leads to an inequivalent, fine-structure dependent representation of the canonical anticommutation relations. This non-perturbative redefinition of the electron/positron states can be interpreted as a mass, wave-function and charge renormalization, among other possibilities, but the main point is that a non-perturbative definition of normal ordering might be a useful starting point for developing a consistent quantum electrodynamics.

Weak Anticommutation Relations and Amalgams of Grassmann Algebras

The Clifford algebra constructed on a given linear space with a symmetric bilinear form is considered along with the family of its Grassmann subalgebras generated by all possible isotropic subspaces. The amalgam (i.e., the inductive limit) of this family is described. As an application, a modification of the canonical anticommutation relations (CAR) is examined. The modification is such that the conventional CAR are imposed only on pairs of space vectors that are orthogonal with respect to the form mentioned above, and each of the vectors is isotropic.

The Rohlin property for Z2-actions on UHF algebras

We define a Rohlin property for Z2-actions on UHF algebras and show a non-commutative Rohlin type theorem. Among those actions with the Rohlin property, we classify product type actions up to outer conjugacy. We consider two classes of UHF algebras. For UHF algebras in one class including the CAR algebra, there is one and only one outer conjugacy class of product type actions and for UHF algebras in the other class, contrary to the case of Z-actions, there are infinitely many outer conjugacy classes of product type actions.

Anyons and the Bose-Fermi duality in the finite-temperature thirring model

Solutions to the Thirring model are constructed in the framework of algebraic QFT. It is shown that for all positive temperatures there are fermionic solutions only if the coupling constant is $\lambda=\sqrt{2(2n+1)\pi}, n\in {\bf N}$. These fermions are inequivalent and only for $n=1$ they are canonical fields. In the general case solutions are anyons. Different anyons (which are uncountably many) live in orthogonal spaces and obey dynamical equations (of the type of Heisenberg's Urgleichung) characterized by the corresponding values of the statistic parameter. Thus statistic parameter turns out to be related to the coupling constant $\lambda$ and the whole Hilbert space becomes non-separable with a different Urgleichung satisfied in each of its sectors. This feature certainly cannot be seen by any power expansion in $\lambda$. Moreover, since the latter is tied to the statistic parameter, it is clear that such an expansion is doomed to failure and will never reveal the true structure of the theory. The correlation functions in the temperature state for the canonical dressed fermions are shown by us to coincide with the ones for bare fields, that is in agreement with the uniqueness of the $\tau$-KMS state over the CAR algebra ($\tau$ being the shift automorphism). Also the $\alpha$-anyon two-point function is evaluated and for scalar field it reproduces the result that is known from the literature.

NEW EXAMPLES OF CONTINUOUS SPATIAL SEMIGROUPS OF *-ENDOMORPHISMS OF ${\mathfrak B} ({\mathfrak h})$

This paper concerns the structure of continuous spatial semigroups of *-endomorphisms of [Formula: see text]. These are semigroups of *-endomorphisms of [Formula: see text] for which there exist a strongly continuous one parameter semigroup of intertwining isometries. New examples of such semigroups are constructed. Unlike all previous examples these semigroups are not cocycle conjugate to semigroups constructed using the CAR or CCR algebras.

On representations of finite type.

Representations of the (infinite) canonical anticommutation relations and the associated operator algebra, the fermion algebra, are studied. A "coupling constant" (in (0,1]) is defined for primary states of "finite type" of that algebra. Primary, faithful states of finite type with arbitrary coupling are constructed and classified. Their physical significance for quantum thermodynamical systems at high temperatures is discussed. The scope of this study is broadened to include a large class of operator algebras sharing some of the structural properties of the fermion algebra.

Quantum Anosov flows: A new family of examples

A quantum version is presented for the Anosov system defined by the time evolution implemented by the geodesic coflow on the cotangent bundle of any compact quotient manifold obtained from the Poincaré half-plane. While the canonical Weyl algebra does not close under time evolution, the symplectic structure of these classical systems can be exploited to produce objects akin to the CCR algebras encountered in quantum field theory. This construction allows one to lift both the geodesic and the horocyclic flows to a Weyl algebra describing the quantum dynamics corresponding to the systems under consideration. The Anosov relations as proposed in Ref. Reference 1 are found to be valid for these models. A quantum version of the classical ergodicity of these systems is discussed in the last section.

CFT fusion rules, DHR gauge groups and CAR algebras

It is demonstrated that several series of conformal field theories, while satisfying braid group statistics, can still be described in the conventional setting of the DHR theory, i.e. their superselection structure can be understood in terms of a compact DHR gauge group. Besides theories with only simple sectors, these include (the untwisted part of) c = 1 orbifold theories and level-two WZW theories. We also analyse the relation between these models and theories of complex free fermions.

Semidirect product of CCR and CAR algebras and asymptotic states in quantum electrodynamics

A C*-algebra containing the CCR and CAR algebras as its subalgebras and naturally described as the semidirect product of these algebras is discussed. A particular example of this structure is considered as a model for the algebra of asymptotic fields in quantum electrodynamics in which Gauss’ law is respected. The appearance in this algebra of a phase variable related to electromagnetic potential leads to the universal charge quantization. Translationally covariant representations of this algebra with energy-momentum spectrum in the future lightcone are investigated. It is shown that vacuum representations are necessarily nonregular with respect to total electromagnetic field. However, a class of translationally covariant, irreducible representations is constructed explicitly, which remain as close as possible to the vacuum, but are regular at the same time. The spectrum of energy-momentum fills the whole future lightcone, but there are no vectors with energy-momentum lying on a mass hyperboloid or in the origin.

Unitary implementations of one-parameter squeezing groups

For the case of infinitely many photon (Boson) modes we investigate the unitary implementability of a class of symplectic one-parameter groups (more exactly, of the associated groups of Bogoliubov automorphisms on the CCR algebra) in the Fock representation and in representations of the CCR algebra, which are symplectically related and inequivalent to Fock. Furthermore, the existence of the associated (squeezing) quadratic Hamiltonians is discussed. Finally, applications in the theory of quantization in QED and in the Luttinger model are pointed out.