Canonical Anticommutation Relations Research Articles

Functional L1-Lp inequalities in the CAR algebra

In the present paper, we use the semigroup method to investigate various functional inequalities invoking L1 and Lp norms in the framework of canonical anti-commuting relations algebra (CAR algebra for short). As the main results, we obtain the Poincaré inequality for Talagrand type sum, Eldan-Gross inequality for projections, and the Talagrand influence inequality along with its strengthening form in the CAR algebra. All our results strengthen the noncommutative Poincaré inequality of Efraim and Lust-Piquard at several points. We conclude the paper with two applications of our inequalities. In the first application, we apply the noncommutative Eldan-Gross inequality to derive two KKL-type inequalities in the CAR algebra, which are closely related to the quantum KKL conjecture of Montanaro and Osborne. The second application is the CAR algebra counterpart of the superconcentration phenomenon derived from the noncommutative Talagrand influence inequality.

The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensor

Given a Cauchy surface in a curved spacetime and a suitably defined quantum state on the CCR algebra of the Klein-Gordon quantum field on that surface, we show, by expanding the squared spacetime geodesic distance and the 'U' and 'V' Hadamard coefficients (and suitable derivatives thereof) in sufficiently accurate covariant Taylor expansions on the surface that the renormalized expectation value of the quantum stress-energy tensor on the surface is determined by the geometry of the surface and the first 4 time derivatives of the metric off the surface, in addition to the Cauchy data for the field's two-point function. This result has been anticipated in and is motivated by a previous investigation by the authors on the initial value problem in semiclassical gravity, for which the geometric initial data corresponds, a priori, to the spatial metric on the surface and up to 3 time derivatives off the surface, but where it was argued that the fourth derivative can be obtained with aid of the field equations on the initial surface.

Twisted tensor products of field algebras

Let \mathcal{A} be a C*-algebra, {\boldsymbol h} a Hilbert space, and {\mathcal{C}}_{\boldsymbol h} the CAR algebra over {\boldsymbol h} . We construct a twisted tensor product of \mathcal{A} by {\mathcal{C}}_{\boldsymbol h} such that the two factors are not necessarily one in the relative commutant of the other. The resulting C*-algebra may be regarded as a generalized CAR algebra constructed over a suitable Hilbert \mathcal{A} -bimodule. As an application, we exhibit a class of fixed-time models where a free Dirac field (giving rise to the {\mathcal{C}}_{\boldsymbol h} factor) in general is not relatively local to a free scalar field (which yields the \mathcal{A} factor). In some of the models, gauge-invariant combinations of the two (not relatively local) fields form a local observable net.

Dissipative dynamics for infinite lattice systems

We study dissipative dynamics constructed by means of non-commutative Dirichlet forms for various lattice systems with multiparticle interactions associated to CCR algebras. We give a number of explicit examples of such models. Using an idea of quasi-invariance of a state, we show how one can construct unitary representations of various groups. Moreover in models with locally conserved quantities associated to an infinite lattice we show that there is no spectral gap and the corresponding dissipative dynamics decay to equilibrium polynomially in time.

Freedman’s Theorem for Unitarily Invariant States on the CCR Algebra

The set of states on CCR(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{CCR}({{\\mathcal {H}}})$$\\end{document}, the CCR algebra of a separable Hilbert space H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {H}}}$$\\end{document}, is here looked at as a natural object to obtain a non-commutative version of Freedman’s theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of CCR(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{CCR}({{\\mathcal {H}}})$$\\end{document} that are invariant under the action of all automorphisms induced in second quantization by unitaries of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {H}}}$$\\end{document}. We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least 1.

Equilibria when the temperature goes to zero

We present methods to construct flows with varying set of KMS∞ states on a given simple unital AF-algebra. It follows, for example, that for any pair D+ and D− of non-empty compact metric spaces there is a flow σ=(σt)t∈R on the CAR algebra whose set of KMS∞ states is homeomorphic to D+ while the set of KMS∞ states for the inverted flow (σ−t)t∈R is homeomorphic to D−. Remarkably the flows that realize all such pairs D± can be chosen to have isomorphic KMS bundles.

Relative Entropy of Fermion Excitation States on the CAR Algebra

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

Particle-Hole Transformation in the Continuum and Determinantal Point Processes

Let X be an underlying space with a reference measure sigma . Let K be an integral operator in L^2(X,sigma ) with integral kernel K(x, y). A point process mu on X is called determinantal with the correlation operator K if the correlation functions of mu are given by k^{(n)}(x_1,dots ,x_n)={text {det}}[K(x_i,x_j)]_{i,j=1,dots ,n}. It is known that each determinantal point process with a self-adjoint correlation operator K is the joint spectral measure of the particle density rho (x)=mathcal A^+(x)mathcal A^-(x) (xin X), where the operator-valued distributions mathcal A^+(x), mathcal A^-(x) come from a gauge-invariant quasi-free representation of the canonical anticommutation relations (CAR). If the space X is discrete and divided into two disjoint parts, X_1 and X_2, by exchanging particles and holes on the X_2 part of the space, one obtains from a determinantal point process with a self-adjoint correlation operator K the determinantal point process with the J-self-adjoint correlation operator widehat{K}=KP_1+(1-K)P_2. Here P_i is the orthogonal projection of L^2(X,sigma ) onto L^2(X_i,sigma ). In the case where the space X is continuous, the exchange of particles and holes makes no sense. Instead, we apply a Bogoliubov transformation to a gauge-invariant quasi-free representation of the CAR. This transformation acts identically on the X_1 part of the space and exchanges the creation operators mathcal A^+(x) and the annihilation operators mathcal A^-(x) for xin X_2. This leads to a quasi-free representation of the CAR, which is not anymore gauge-invariant. We prove that the joint spectral measure of the corresponding particle density is the determinantal point process with the correlation operator widehat{K}.

Interacting Stochastic Schrödinger Equation

Being the annihilation and creation operators on the space h of square integrable Bernoulli functionals, quantum Bernoulli noises (QBN) satisfy the canonical anti-commutation relation (CAR) in equal time. Let K be the Hilbert space of an open quantum system interacting with QBN (the environment). Then K⊗h just describes the coupled quantum system. In this paper, we introduce and investigate an interacting stochastic Schrödinger equation (SSE) in the framework K⊗h, which might play a role in describing the evolution of the open quantum system interacting with QBN (the environment). We first prove some technical propositions about operators in K⊗h. In particular, we obtain the spectral decomposition of the tensor operator IK⊗N, where IK means the identity operator on K and N is the number operator in h, and give a representation of IK⊗N in terms of operators IK⊗∂k*∂k, k≥0, where ∂k and ∂k* are the annihilation and creation operators on h, respectively. Based on these technical propositions as well as Mora and Rebolledo’s results on a general SSE, we show that under some mild conditions, our interacting SSE has a unique solution admitting some regularity properties. Some other results are also proven.

Ergodicity of exclusion semigroups constructed from quantum Bernoulli noises

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. This paper aimed to discuss the classical reduction and ergodicity of quantum exclusion semigroups constructed by QBN. We first study the classical reduction of the quantum semigroups to an Abelian algebra of diagonal elements and the space of off-diagonal elements. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the semigroups of classical Markov chains. Finally, we prove that the asymptotic behavior of the quantum semigroups is equivalent to one of its associated Markov chains, and that the semigroups restricted to the off diagonal space of operators have a zero limit.

Super-exponential distinguishability of correlated quantum states

In the problem of asymptotic binary i.i.d. state discrimination, the optimal asymptotics of the type I and the type II error probabilities is in general an exponential decrease to zero as a function of the number of samples; the set of achievable exponent pairs is characterized by the quantum Hoeffding bound theorem. A super-exponential decrease for both types of error probabilities is only possible in the trivial case when the two states are orthogonal and hence can be perfectly distinguished using only a single copy of the system. In this paper, we show that a qualitatively different behavior can occur when there is correlation between the samples. Namely, we use gauge-invariant and translation-invariant quasi-free states on the algebra of the canonical anti-commutation relations to exhibit pairs of states on an infinite spin chain with the properties that (a) all finite-size restrictions of the states have invertible density operators and (b) the type I and the type II error probabilities both decrease to zero at least with the speed e^{-nclog n} with some positive constant c, i.e., with a super-exponential speed in the sample size n. Particular examples of such states include the ground states of the XX model corresponding to different transverse magnetic fields. In fact, we prove our result in the setting of binary composite hypothesis testing, and hence, it can be applied to prove super-exponential distinguishability of the hypotheses that the transverse magnetic field is above a certain threshold vs. that it is below a strictly lower value.

De Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products

Local actions of [Formula: see text], the group of finite permutations on [Formula: see text], on quasi-local algebras are defined and proved to be [Formula: see text]-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of [Formula: see text]-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon [Formula: see text] in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here.

CCR and CAR Algebras are Connected Via a Path of Cuntz–Toeplitz Algebras

For q in {mathbb {R}}, |q| < 1 we consider the universal enveloping C^*-algebra of a *-algebra of q-canonical commutation relations (q-CCR), which is generated by a_1, ldots , a_n subject to the relations ai∗aj=δij1+qajai∗.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} a_i^* a_j = \\delta _{ij} 1 + q a_j a_i^* . \\end{aligned}$$\\end{document}It has a distinguished representation pi _F called the Fock representation, which is believed to be faithful. In this article we denote the image of the universal enveloping C^*-algebra of q-CCR in the Fock representation by beth _{n,q}. The question whether C^*-isomorphism beth _{n,q} simeq beth _{n,0} holds has been considered in the literature and proved for |q| < 0.44. In this article we show that beth _{n,q} simeq beth _{n,0} for |q| < 1.

Failure of the Ryll-Nardzewski theorem on the CAR algebra

Spreadability of a sequence of random variables is a distributional symmetry that is implemented by suitable actions of JZ, the unital semigroup of strictly increasing maps on Z with cofinite range. We show that JZ is left amenable but not right amenable, although it does admit a right Følner sequence. This enables us to prove that on the CAR algebra CAR(Z) there exist spreadable states that fail to be exchangeable. Moreover, we also show that on CAR(Z) there exist stationary states that fail to be spreadable.

AF-algebras with lattice-ordered K0: Logic and computation

Following Bratteli's original definition, an AF-algebra A is the norm closure of the union of an ascending sequence of finite-dimensional C⁎-algebras, all with the same unit. Elliott proved that the unital dimension (Grothendieck) group K0(A) uniquely determines A up to isomorphism. AF-algebras with lattice-ordered K0, (AFℓ-algebras) have a preeminent role in the AF-algebraic literature. The Elliott classifier E(A) of any AFℓ-algebra A—i.e., the unit interval of K0(A), or equivalently, the countable set of Murray-von Neumann equivalence classes of projections of A—has the structure of a countable MV-algebra. Every countable MV-algebra arises as E(A) for some AFℓ-algebra A. Since E(A) is the Lindenbaum algebra of some countable set of formulas in Łukasiewicz logic Ł∞, every Ł∞-formula ϕ naturally codes an equivalence class ϕA of projections in A. The deductive-algorithmic machinery of Ł∞ can be thus applied to decide, among many others, the following basic problems, with ⪯ the Murray-von Neumann order:Is ϕA⪯ψA?IsϕA={0}? Is ϕA central ? For any AFℓ-algebra A we construct mutual polytime reductions of the first two problems. The third problem is polytime reducible to both problems. If E(A) is finitely generated and A either has no quotient isomorphic to C or is simple, then the first two problems are polytime reducible to the third one. The complexity of these problems is polytime for many AFℓ-algebras in the literature, including the Behncke-Leptin algebras Am,n, the Farey AFℓ-algebra M1 and all its principal quotients, the CAR algebra and Glimm's universal UHF algebra, and every Effros-Shen algebra Fθ for θ∈[0,1]∖Q a real algebraic integer, or for θ=1/e. For any fixed n=1,2,… we consider the class Kn of AFℓ-algebras presented by generators X1,…,Xn and relations θ1=0,…,θm=0, for an arbitrary finite set Θ={θ1,…,θm} of Ł∞-formulas in the variables X1,…,Xn. We prove that all decision problems studied in this paper are polytime decidable for Kn—uniformly over the presentation Θ.

Proper condensates and long range order

Within the framework of the algebra of canonical commutation relations in position space, a long range order between particles in bounded regions is established in states with a sufficiently large particle number. It occurs whenever homogeneous proper (infinite) condensates form locally in the states in the limit of infinite densities. The condensates are described by eigenstates of the momentum operator, covering also those cases where they are streaming with a constant velocity. The arguments given are model independent and lead to a new criterion for the occurrence of condensates. It makes use of a novel approach to the identification of condensates, based on a characterization of regular and singular wave functions.

Locally equivalent quasifree states and index theory

We consider quasifree ground states of Araki’s self-dual canonical anti-commutation relation algebra from the viewpoint of index theory and symmetry protected topological (SPT) phases. We first review how Clifford module indices characterise a topological obstruction to connect pairs of symmetric gapped ground states. This construction is then generalised to give invariants in with A a -algebra of allowed deformations. When A = C*(X), the Roe algebra of a coarse space X, and we restrict to gapped ground states that are locally equivalent with respect X, a K-homology class is also constructed. The coarse assembly map relates these two classes and clarifies the relevance of K-homology to free-fermionic SPT phases.

Finite dimensional systems of free fermions and diffusion processes on spin groups

In this article, we are concerned with “finite dimensional fermions,” by which we mean vectors in a finite dimensional complex space embedded in the exterior algebra over itself. These fermions are spinless but possess the characterizing anticommutativity property. We associate invariant complex vector fields on the Lie group Spin(2n + 1) to the fermionic creation and annihilation operators. These vector fields are elements of the complexification of the regular representation of the Lie algebra so(2n+1). As such, they do not satisfy the canonical anticommutation relations; however, once they have been projected onto an appropriate subspace of L2(Spin(2n + 1)), these relations are satisfied. We define a free time evolution of this system of fermions in terms of a symmetric positive-definite quadratic form in the creation–annihilation operators. The realization of fermionic creation and annihilation operators brought by the (invariant) vector fields allows us to interpret this time evolution in terms of a positive self-adjoint operator that is the sum of a second order operator, which generates a stochastic diffusion process, and a first order complex operator, which strongly commutes with the second order operator. A probabilistic interpretation is given in terms of a Feynman–Kac-like formula with respect to the diffusion process associated with the second order operator.

The Dirac sea for the non-separable Hilbert spaces

We give a rigorous construction of the Dirac sea for the fermionic quantization in non-separable Hilbert spaces. These canonical anticommutation relations and representations depend on the axiom of choice, and hence, they are not unique; nevertheless, they are unitarily equivalent to the classic Fock representation.

Relative Entropy of Coherent States on General CCR Algebras

For a subalgebra of a generic CCR algebra, we consider the relative entropy between a general (not necessarily pure) quasifree state and a coherent excitationthereof. We give a unified formula for this entropy in terms of single-particle modular data. Further, we investigate changes of the relative entropy along subalgebras arising from an increasing family of symplectic subspaces; here convexity of the entropy (as usually considered for the Quantum Null Energy Condition) is replaced with lower estimates for the second derivative, composed of “bulk terms” and “boundary terms”. Our main assumption is that the subspaces are in differential modular position, a regularity condition that generalizes the usual notion of half-sided modular inclusions. We illustrate our results in relevant examples, including thermal states for the conformal U(1)-current.