GNS Representation Research Articles

The Hilbert Space Structure Condition for Quantum Field Theories with Indefinite Metric and Transformations with Linear Functionals

The sufficient Hilbert space structure condition (abbr. HSSC) (Ĥ) is introduced such that, if a linear functional T defined on some tensor algebra satisfies (Ĥ), then the transformed functional p(T)|s obtained by s-product power series satisfies the usual HSSC. Consequently, the GNS representation with respect to p(T)|s yields a Krein space as the space of state vectors. This generalizes the sufficient HSSC for truncated Wightman functionals due to Albeverio, Gottschalk and Wu.

On the Spectra of the Kink for Ferromagnetic XXZ Models

We consider the kink (infinite volume nontranslationally invariant ground states) of the ferromagnetic XXZ models on the multi-dimensional lattices. We obtained the following results: (i) The pure states satisfying the local zero energy condition are necessarily product states. (ii) The Hamiltonian in the GNS representation of the kink has no gap between the ground-state eigenvalue and the rest of the spectra.

On the GNS representation of generalized free fields with indefinite metric

The GNS representation leading to generalized free fields on state spaces with indefinite metric is considered. Along these lines, the domain D of field operators, a fundamental decomposition D = D + [ + ̇ ] D − into positive and negative definite subspaces, and a fundamental symmetry J such that the completed hull K = D ∥ · ∥J is a Krein space are explicitely described. These results are finally applied to a rigorous treatment of the charged free fields of mass m > 0.

On certain states on free *-algebras and their GNS representations

States on free *-algebras expressed in terms of ordered partitions are studied. To this class belong twisted limit states that are derived from a convolution type q- analog of the quantum central limit theorem (qclt) for twisted free *-bialgebras 𝒞q, studied previously in the case of quantum groups. A special form of the GNS representation in terms of generalized creation and annihilation operators, A(v) and A(v*), respectively, living in the infinite tensor product of Hilbert spaces is given leading to a generalization of the Fock space. It is also shown that another family of states of similar combinatorics can be obtained from the convolution series.

Quantum spin chains with quantum group symmetry

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasi-local) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The non-locality arises from the ordering of factors in the quantum groupC *-algebra, and can be made one-sided, thus allowing semi-local actions on a half chain. Under such actions, localized quantum group invariant elements remain localized. Hence the notion of interactions invariant under the quantum group and also under translations, recently studied by many authors, makes sense even though there is no global action of the quantum group. We consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state. Under weak locality assumptions, its GNS representation carries no unitary representation of the quantum group.

An explicit description of the fundamental unitary for SU(2) q

We give a concrete description of an isometryv from l2(ℕ×ℤ×ℤ×ℤ) to l2(ℕ×ℤ×ℕ×ℤ) whose existence has recently been discovered by Woronowicz [11]. The isometryv gives the comultiplication δ on the C*-algebraA of the quantum group SU(2) q through the formula δ(x)=v(x⊗1)v*(x∈A), where 1 is the identity operator on l2(ℤ×ℤ). The matrix entries ofv are described in terms of littleq-Jacobi polynomials. Usingv, we give a concrete description of a unitary operatorV onH η ⊗H η such that (πη⊗πη)δ(x)=V(πη(x⊗1)V*, whereHη=l2(ℕ×ℤ×ℕ) and πη:A→L(Hη) is the GNS representation associated with the Haar state η onA. The operatorV satisfies the pentagonal identity of Baaj and Skandalis [1].

The limiting dynamics of an imperfect Bose gas

For an imperfect Bose gas consisting of massive particles and interacting through a long-range pair potential in a mean-field approximation, both in equilibrium and far from equilibrium, the dynamics are derived in the thermodynamic limit. This is done in the framework of the operator algebraic quantum theory. The limiting dynamics is formulated in the Schrödinger picture as a group of affine transformations of certain folia of physical states. Similar to the mean-field models for spin lattices, the algebra of quasilocal observables is not invariant under the Heisenberg dynamics in the GNS representation of general time invariant states. It is known that above critical density the infinite system exhibits condensation, and the condensate is characterized by the classical parameters density and phase. For these parameters a classical nonlinear equation of motion is derived.

The general form of the microscopic coherent boson states

Recently the classical coherent states on the photonic C ∗-Weyl algebra have been classified. Non-classical coherence occurs only for states which are normal with respect to the Fock representation of the CCR. Here, the Fock-normal (=microscopic) coherent states are characterized completely. It suffices to consider Fock coherence only on the one-mode Weyl algebra W ( C ). The smoothness and coherence properties of a regular state on W ( C ) are expressed by the diagonal elements of the associated density matrix. With a Kolmogorov decomposition the off-diagonal matrix elements are replaced by a unique sequence of normalized vectors in a Hilbert space, which leads to the GNS representation and to a construction procedure for the whole set of all (microscopic) coherent states of arbitrary order. The variety of non-classical fully coherent states on W ( C ) is shown to be much larger than the one for classical fully coherent states. Moreover, it is proved that there exist (non-classical) elements of the extreme boundary of the weak ∗-compact, convex set of fully coherent states, which are not pure states.

Subalgebras of infinite C*-algebras with finite Watatani indices I. Cuntz algebras

Using fusion rules of sectors as a working hypothesis, we construct endomorphisms of the Cuntz algebra $$\mathcal{O}_n $$ whose images have finite Watatani indices. Quasi-free KMS states on $$\mathcal{O}_n $$ appear in a natural way associated with the endomorphisms, and we determine the Murray-von Neumann-Connes types of their GNS representations.

ON THE WEAK COUPLING LIMIT FOR A FERMI GAS IN A RANDOM POTENTIAL

General conditions are derived for the essential self-adjointness of –∆ + V, where V is a translation-invariant random potential, and for the existence of the perturbation expansion. A sequence of graphs is exhibited violating Dell'Antonio's bound for skeleton graphs. For a translation-invariant and clustering Gaussian random potential V, and a translation-invariant and clustering initial state S of the Fermi gas, uncorrelated with the random potential, the weak coupling limit (Van Hove limit) yields increase of entropy, propagation of chaos, convergence of the state for sufficiently small values of the parameter τ to a gauge-invariant and quasi-free asymptotic state, and the semigroup describing the evolution of the two-point function. The asymptotic system is Bernoulli. Results are obtained not only for the average over the random potential but also with probability one. If the random potential V′ is absolutely continuous with respect to V, and if the state S′ is given by a density matrix in the GNS representation for S, then the weak coupling limit is the same as for V and S.

An Explicit Realization of a GNS Representation in a Krein-Space

On the tensor-algebra over the basic space \boldsymbol C^2 , a P -functional is constructed. Using methods which are due to J.P. Antoine and S. Ôta, a Krein-space theory (i.e., a *-algebra of operators which are defined on a common, dense, and invariant domain in a Krein-space \mathcal K ) is obtained via the GNS construction. It is shown that \mathcal K does not contain any π -invariant dual pairs. This gives an answer to a problem first posed by J.P. Antoine and S. Ôta. The theory so obtained describes the complex superposition of two harmonic oscillators. With this in mind, the annihilation and creation operators, the operator of total electric charge, and the gauge group are explicitly given.

Partial *-Algebras of Closable Operators. II. States and Representations of Partial *-Algebras

This second paper on partial Op*-algebras is devoted to the theory of representations. A new definition of invariant positive sesquilinear forms on partial *-algebras is proposed, which enables to perform the familiar GNS construction. In order to get a better control of the corresponding representations, we introduce and study a restricted class of partial Op*-algebras, called partial GW*-algebras, which turn up naturally in a number of problems. As an example, we extend Powers' results about the standardness of GNS representations of abelian partial *-algebras.

Unbounded GNS representations of a *-algebra in a Krein space

Given a *-algebra $$\mathfrak{A}$$ and a linear, nonpositive definite, functional on it, the GNS construction yields a representation of $$\mathfrak{A}$$ in an indefinite metric space. We show that, if the functional satisfies a suitable condition, expressed in terms of a conditional expectation on $$\mathfrak{A}$$ , then the representation space is a Krein space. This provides an abstract setup for the description of a massless quantum field theory.

On the microscopic derivation of the finite-temperature Josephson relation in operator form

As a microscopic description of the Josephson junction, two BCS models, are studied in the strict pair formulation with quite an arbitrary weak coupling potential. The modular formalism, the separate gauge transformations, and the limiting dynamics are analyzed for the interacting system in terms of the GNS representation of the uncoupled limiting Gibbs state. By means of the Connes theory the condensed Cooper pair and the quasiparticle spectrum is shown to be stable against weak perturbations. The modular formalism is used to construct a local approximation to the renormalized particle number operator and, by this, its time dependence, in spite of this observable not being affiliated with the von Neumann algebra of the temperature representation. The time derivation from this unbounded operator-valued function coincides with the limit of the local currents and splits under a natural assumption into a sum of the Josephson and the quasiparticle current operator extending the two-fluid picture also to the coupled model.

Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas

We consider the local perturbation $$V = \varepsilon \sum\limits_{x,y \in \mathbb{Z}^v } {V(x,y)\chi _\Omega (x)\chi _\Omega (y)a * (x)a * (y)a(y)a(x)} $$ of the ideal Fermi-gas on the lattice ℤv, where Ω is a finite subset of ℤv and χΩ is its indicator. The invertibility of Möller morphisms for small ɛ is proven. It follows that in the cyclic GNS representation with respect to KMS states the dynamics of ideal and locally perturbed Fermi-gas are unitary equivalent.

Remarks on positive semigroups

Let ℳ be a von Neumann algebra with a cyclic and separating vector Ω and let ω(·) denote the corresponding vector state, i.e., ω(A)=(Ω, AΩ) A ∈ ℳ. We have proved that a positive semigroup τ on ℳ can induce the dynamical semigroup in the GNS representation associated with ω if the state ω is a τ-invariant one. Some applications are given.

Multiplicative and additive cluster expansions for the evolution of quantum spin systems in the ground state

It is shown that for the v-dimensional quantum Ising model in the high temperature region e − tH in the GNS representation admits a “multiplicative” N-particle cluster expansion and H admits an “additive” N-particle cluster expansion.

Conditional Probability and the Axiomatic Structure of Quantum Mechanics

The paper, summarizing the results of recent papers of the author, is devoted to a detailed analysis of postulates of nonrelativistic quantum theory appearing within two classical axiomatic frameworks the “quantum logic” and the “algebraic approach” respectively, which are presently the two main alternatives for quantum axiomatics. The first part of the paper concerns the structure of two important sets, the set of questions (propositional logic) and the set of pure states. It is shown how the quantum logic axiomatic scheme can be modified and improved in order to overcome the well known troubles connected with the lack of satisfactory physical justification for some postulates of this approach. In particular, an axiom system is developed, related closely to the quantum logic axiomatic scheme, in which the questions of the complete lattice structure, the atomisticity, and the validity of the covering law in the propositional logic do not appear so problematic. The particular attention is directed to the covering law, which is here obtained as a consequence of physically clear properties of the experimental procedures (“filters”) associated with the quantum-mechanical propositions. In Section 1.5 we formulate an axiom system for quantum mechanics based exclusively on the concepts of pure state, transition probability, and pure filter. In the second part of the paper the axiomatic scheme presented in Part 1 is developed with the aim to analyse the algebraic structure of some subspaces of the set of simple observables. The quantum logic axiomatic framework is here shown closely connected with the Jordan-Banach algebraic scheme. In the third part of the paper, similarly as it was done in Parts 1 and 2, the correspondence between the propositions and the experimental (measuring) procedures verifying these propositions is put as the basic assumption of the axiomatization of quantum mechanics. As a consequence of this assumption, there is established the structure of a real Jordan-Banach algebra in the set Ob of the bounded observables associated with a physical system, so that we can apply the GNS representation theorem proved recently for such algebras by Alfsen et al. (1978) to obtain the Hilbert space (or, to be more precise, the C*-algebraic) representation for Ob.

Extension of Tomita-Takesaki theory to the unbounded algebra of the canonical commutation relations

We consider the unbounded CCR algebra in infinitely many degrees of freedom equipped with a suitable faithful state. We prove that this state satisfies the KMS condition with respect to a certain time evolution and the associated unbounded GNS representation πβ has the structure encountered in Tomita-Takesaki theory; what is more, the commutant πβU′ is a standard von Neumann algebra, invariant under the time evolution.

Dynamics and phase transitions for a continuous system of quantum particles in a box

A particular type of continuous quantum system with infinitely many particles is analyzed, and the existence of dynamics is proven in the GNS representations of certain states. The dynamics is not a group of automorphisms on the original algebra, so equilibrium states are defined in terms of the KMS condition in the representations of the states. The basic theorems about KMS states do not apply here. Nevertheless, for a special class of interactions it is proven that the central decomposition of an equilibrium state is concentrated on a Borel set of equilibrium factor states and that such factor states are precisely the extremal equilibrium states. Furthermore, the equilibrium factor states are in one-to-one correspondence with sets of functions satisfying a certain system of trace equations. This explicit correspondence is then used to show that there are no phase transitions for high temperature, and an example of a phase transition is constructed for low temperature. The phase transition also provides an example of continuous symmetry breaking.