Canonical Anticommutation Relations Research Articles

A FAMILY OF QUANTUM MARKOV SEMIGROUPS

For a given gauge invariant state <TEX>$\omega$</TEX> on the CAR algebra A isomorphic with the C<TEX>$\ast$</TEX> -algebra of <TEX>$2{\times}2$</TEX> complex matrices, we construct a family of quantum Markov semigroups on A which leave w invariant. By analyzing their generators, we decompose the algebra A into four eigenspaces of the semigroups and show some properties.

On *-representations of a deformation of a wick analogue of the car algebra

The classification of irreducible *-representations of deformations of canonical anti-commutation relations is given. A description of the corresponding enveloping C*-algebra is presented in terms of continuous matrix-functions satisfying certain boundary conditions.

A recursion relation for the Wick products of the CAR algebra

In this paper we obtain an explicit recursion relation for the Wick products of the CAR algebra in terms of Wick products of lesser order and the bracket of the Fermi fields. This formula extends to Fermi fields the recursion formula given by I.E. Segal for Bose fields (cf. §1.4 in [I.E. Segal, Nonlinear functions of weak processes. I, J. Funct. Anal. 4 (1969) 404–456] and §7.1–7.2 in [J.C. Baez, I.E. Segal, Z. Zhou, Introduction to Algebraic and Constructive Quantum Field Theory, Princeton Univ. Press, NJ, 1992]).

The one-cocycle property for shifts

The two-sided shift on the infinite tensor product of copies of the $n\times n$ matrix algebra has the so-called Rohlin property, which entails the one-cocycle property, useful in analyzing cocycle-conjugacy classes. In the case n = 2, the restriction of the shift to the gauge-invariant CAR algebra also has the one-cocycle property. We extend the latter result to an arbitrary $n\geq 2$ . As a corollary it follows that the flow $\alpha$ on the Cuntz algebra $\mathcal{O}_n=C^*(s_0,s_1,\dotsc,s_{n-1})$ defined by $\alpha_t(s_j)=e^{ip_jt}s_j$ has the Rohlin property (for flows) if and only if $p_0,\dotsc,p_{n-1}$ generate $\mathbb R$ as a closed sub-semigroup. Note that then such flows are all cocycle-conjugate to each other.

On the structure of the observable algebra of QCD on the lattice

The structure of the observable algebra ${\mathfrak O}_{\Lambda}$ of lattice QCD in the Hamiltonian approach is investigated. As was shown earlier, ${\mathfrak O}_{\Lambda}$ is isomorphic to the tensor product of a gluonic $C^{*}$-subalgebra, built from gauge fields and a hadronic subalgebra constructed from gauge invariant combinations of quark fields. The gluonic component is isomorphic to a standard CCR algebra over the group manifold SU(3). The structure of the hadronic part, as presented in terms of a number of generators and relations, is studied in detail. It is shown that its irreducible representations are classified by triality. Using this, it is proved that the hadronic algebra is isomorphic to the commutant of the triality operator in the enveloping algebra of the Lie super algebra ${\rm sl(1/n)}$ (factorized by a certain ideal).

Spectral Properties Of Weakly Inhomogeneous Bcs-Models In Different Representations

For a class of Bardeen-Cooper-Schrieffer (BCS)-models, with complex, weakly momentum dependent interaction coefficients, the representation dependent effective Hamiltonians and their spectra are reconsidered in order to obtain a consistent physical picture by means of operator algebraic methods. The starting point is the limiting dynamics, the existence of which had been proved in a previous work, in terms of a C*-dynamical system acting in a classically extended, electronic Canonical Anticommutation Relations (CAR)-algebra. The C*-algebraic KMS-theory, including the low temperature limit, specifies the order parameters. These appear as classical observables, which commute with all other observables, constituting elements of the center of the algebra. The algebraic spectral theory, in the sense of Arveson, is first applied to the dynamics in general pure energy state representations. The spectra of the finite temperature representations are analyzed, identifying the gap as the lowest of those energy values, which are stable under local perturbations. Further insights are obtained by decomposing the thermal dynamical systems into the pure energy state Heisenberg dynamics, after having first extended them to more comprehensive W*-dynamical systems. The decomposing orthogonal measure is transferred to the infinite product space of quasi-particle occupation numbers and its support is characterized in terms of 0-1-laws leading to an asymptotic ratio of quasi-particles and holes, which depends on the temperature. This ratio is connected with an algebraic invariant of the representation dependent observable algebra. Energy renormalization aspects and pair occupation probabilities are discussed. The latter reveal, beside other things, the difference between macroscopic term occupation and coherent macroscopic term occupation for a condensate.

Quantizing the damped harmonic oscillator

We consider the Fermi quantization of the classical damped harmonic oscillator (dho). In past work on the subject, authors double the phase space of the dho in order to close the system at each moment in time. For an infinite-dimensional phase space, this method requires one to construct a representation of the CAR algebra for each time. We show that unitary dilation of the contraction semigroup governing the dynamics of the system is a logical extension of the doubling procedure, and it allows one to avoid the mathematical difficulties encountered with the previous method.

On quasifree representations of infinite dimensional symplectic group

We consider an infinite dimensional generalization of metaplectic representations (Weil representations) for the (double covering of) symplectic group. Given quasifree states of an infinite dimensional CCR algebra, projective unitary representations of the infinite dimensional symplectic group are constructed via unitary implementors of Bogoliubov automorphisms. Complete classification of these representations up to quasi-equivalence is obtained.

Method for Solving Quantum Field Theory in the Heisenberg Picture

This paper is a review of the method for solving quantum field theory in the Heisenberg picture, developed by Abe and Nakanishi since 1991. Starting from field equations and canonical (anti)commutation relations, one sets up a (q-number) Cauchy problem for the totality of d-dimensional (anti)commutators between the fundamental fields, where d is the number of spacetime dimensions. Solving this Cauchy problem, one obtains the operator solution of the theory. Then one calculates all multiple commutators. A representation of the operator solution is obtained by constructing the set of all Wightman functions for the fundamental fields; the truncated Wightman functions are constructed so as to be consistent with all vacuum expectation values of the multiple commutators mentioned above and with the energy-positivity condition. By applying the method described above, exact solutions to various 2-dimensional gauge-theory and quantum-gravity models are found explicitly. The validity of these solutions is confirmed by comparing them with the conventional perturbation-theoretical results. However, a new anomalous feature, called the field-equation anomaly, is often found to appear, and its perturbation-theoretical counterpart, unnoticed previously, is discussed. The conventional notion of an anomaly with respect to symmetry is reconsidered on the basis of the field-equation anomaly, and the derivation of the critical dimension in the BRS-formulated bosonic string theory is criticized. The method outlined above is applied to more realistic theories by expanding everything in powers of the relevant parameter, but this expansion is not equivalent to the conventional perturbative expansion. The new expansion is BRS-invariant at each order, in contrast to that in the conventional perturbation theory. Higher-order calculations are generally extremely laborious to perform explicitly.

Construction of a family of quantum Ornstein–Uhlenbeck semigroups

For a given quasi-free state on the CCR algebra over one dimensional Hilbert space, a family of Markovian semigroups which leave the quasi-free state invariant is constructed by means of noncommutative elliptic operators and Dirichlet forms on von Neumann algebras. The generators (Dirichlet operators) of the semigroups are analyzed and the spectrums together with eigenspaces are found. When restricted to a maximal Abelian subalgebra, the semigroups are reduced to a unique Markovian semigroup of classical Ornstein–Uhlenbeck process.

Entropy growth of shift-invariant states on a quantum spin chain

We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length N are typically mixed and have therefore a nonzero entropy SN which is, moreover, monotonically increasing in N. We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterized by a measurable subset of the unit interval. As the entropy density is known to vanish, SN is sublinear in N. For states corresponding to unions of finitely many intervals, SN is shown to grow slower than log2 N. Numerical calculations suggest a log N behavior. For the case with infinitely many intervals, we present a class of states for which the entropy SN increases as Nα where α can take any value in (0,1).

DIRICHLET FORMS AND SYMMETRIC MARKOVIAN SEMIGROUPS ON ℤ2-GRADED VON NEUMANN ALGEBRAS

We extend the construction of Dirichlet forms and symmetric Markovian semigroups on standard forms of von Neumann algebras given in [1] to the case of ℤ2-graded von Neumann algebras. As an application of the extension, we construct symmetric Markovian semigroups on CAR algebras with respect to gauge invariant quasi-free states and also investigate detailed properties such as ergodicity of the semigroups.

Representations of CCR Algebras in Krein Spaces of Entire Functions

Representations of CCR algebras in spaces of entire functions are classified on the basis of isomorphisms between the Heisenberg CCR algebra A_H and star algebras of holomorphic operators. To each representations of such algebras, satisfying a regularity and a reality condition, one can associate isomorphisms and inner products so that they become Krein star representations of A_H, with the gauge transformations implemented by a continuous U(1) group of Krein isometries. Conversely, any holomorphic Krein representation of A_H, having the gauge transformations implemented as before and no null subrepresentation, is shown to be contained in a direct sum of the above representations. The analysis is extended to infinite dimensional CCR algebras, under a spectral condition for the implementers of the gauge transformations.

Path integral of spin models

A path-integral representation of 1D Ising and XY spin models is investigated. Short-time propagator algorithms and a discrete time formalism are used in combination with Grassmann variables non-orthogonal coherent states to get a many-body analytic propagator. Fermion operators satisfying the canonical anticommutation relations are constructed from the rising and lowering spin operators via the Jordan–Wigner transformation. Computation of the partition function and thermodynamic properties follows from an appropriate tracing over Grassmann variables in the imaginary time domain.

THE STRUCTURE OF SPIN SYSTEMS

A spin system is a sequence of self-adjoint unitary operators U1, U2, … acting on a Hilbert space H which either commute or anticommute, UiUj = ±UjUi for all i, j; it is called irreducible when {U1, U2, …} is an irreducible set of operators. There is a unique infinite matrix (cij) with 0, 1 entries satisfying [Formula: see text] Every matrix (cij) with 0, 1 entries satisfying cij = cji and cii = 0 arises from a nontrivial irreducible spin system, and there are uncountably many such matrices. In cases where the commutation matrix (cij) is of "infinite rank" (these are the ones for which infinite dimensional irreducible representations exist), we show that the C*-algebra generated by an irreducible spin system is the CAR algebra, an infinite tensor product of copies of [Formula: see text], and we classify the irreducible spin systems associated with a given matrix (cij) up to approximate unitary equivalence. That follows from a structural result. The C*-algebra generated by the universal spin system u1, u2, … of (cij) decomposes into a tensor product [Formula: see text], where X is a Cantor set (possibly finite) and [Formula: see text] is either the CAR algebra or a finite tensor product of copies of [Formula: see text]. We describe the nature of this decomposition in terms of the "symplectic" properties of the ℤ2-valued form [Formula: see text]x, y ranging over the free infninite dimensional vector space over the Galois field ℤ2.

On the product system of a completely positive semigroup

Given a W ∗ -continuous semigroup φ of unital, normal, completely positive maps of B( H), we introduce its continuous tensor product system E φ . If α is a minimal dilation E 0-semigroup of φ with Arveson product system F, then E φ is canonically isomorphic to F. We apply this construction to a class of semigroups of B(L 2( R)) arising from a modified Weyl–Moyal quantization of convolution semigroups of Borel probability measures on R 2 . This class includes the heat flow on the CCR algebra studied recently by Arveson. We prove that the minimal dilations of all such semigroups are completely spatial, and additionally, we prove that the minimal dilation of the heat flow is cocyle conjugate to the CAR/CCR flow of index two.

Dirichlet forms and symmetric Markovian semigroups on CCR algebras with respect to quasi-free states

Employing the construction method of Dirichlet forms on standard forms of von Neumann algebras developed in Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2000, Vol. 3, No. 1, pp. 1–14 (Ref. 1), we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let A(h0) be the CCR algebra over a complex separable pre-Hilbert space h0 and let ω be a quasi-free state on A(h0). For any normalized admissible function f and complete orthonormal system (CONS) {gn}⊂h0, we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of (A(h0),ω). It turns out that the form is independent of admissible function f and CONS {gn} chosen. By analyzing the spectrum of the generator (Dirichlet operator) of the semigroup, we show that the semigroup is ergodic and tends to the equilibrium exponentially fast.

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.