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Zeitschrift für Naturforschung A

Decomposition of Positive Sesquilinear Forms and the Central Decomposition of Gauge-Invariant Quasi-Free States on the Weyl-Algebra

Reinhard Honegger

Published Jan 1, 1990• 14 citations
Publisher

Abstract

Abstract A decomposition theory for positive sesquilinear forms densely defined in Hilbert spaces is developed. On decomposing such a form into its closable and singular part and using Bochner's theorem it is possible to derive the central decomposition of the associated gauge-invariant quasifree state on the boson C*-Weyl algebra. The appearance of a classical field part of the boson system is studied in detail in the GNS-representation and shown to correspond to the so-called singular subspace of a natural enlargement of the one-boson testfunction space. In the example of Bose-Einstein condensation a non-trivial central decomposition (or equivalently a non-trivial classical field part) is directly related to the occurrence of the condensation phenomenon.

DOI
10.1515/zna-1990-0105

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Abstract A decomposition theory for positive sesquilinear forms densely defined in Hilbert spaces is developed. On decomposing such a form into its closable and singular part and using Bochner's theorem it is possible to derive the central decomposition of the associated gauge-invariant quasifree state on the boson C*-Weyl algebra. The appearance of a classical field part of the boson system is studied in detail in the GNS-representation and shown to correspond to the so-called singular subspace of a natural enlargement of the one-boson testfunction space. In the example of Bose-Einstein condensation a non-trivial central decomposition (or equivalently a non-trivial classical field part) is directly related to the occurrence of the condensation phenomenon.
Specifying their (normally ordered) characteristic functions we determine all states of the boson C *-Weyl algebra which satisfy Glauber's coherence condition and are not realizable as density operators in Fock space. The pure ones are shown to be just the eigenstates of the annihilation operators in their GNS-representations (in contrast to the Fock case) and are characterized in many equivalent manners. The central decomposition of an arbitrary coherent state has the macroscopic phase variable as parameter and is supported by the pure coherent states, which is in fact the only way for a maximal decomposition. The set of all coherent states with the same absolute factorizing function is proven to be a Bauer simplex. The appearence of a classical coherent field part is studied in detail in the GNS-representations and shown to correspond to an enlargement of the set of one boson states by just one additional mode.
We present a comprehensive analysis of the ideal structure of the resolvent algebra for interacting bosonic quantum field theories. The resolvent algebra, introduced by Buchholz and Grundling as a C∗C^*C∗-algebraic alternative to the Weyl algebra, possesses a rich ideal structure that faithfully reflects the quantum-mechanical properties of bosonic many-body systems. Our main results establish a complete characterization of the closed two-sided ideals for the van Hove model and the generalized spin-boson model, proving that the singular subspace associated with infrared divergences corresponds to a unique maximal proper ideal. We show that the quotient by this ideal cleanly separates regular and singular contributions, and that Bose-Einstein condensation (BEC) can be characterized entirely in terms of the emergence of a nontrivial center in the von Neumann algebra obtained from the GNS representation of equilibrium states. Furthermore, we prove a no-go theorem establishing that quasiparticle excitations in the spin-boson model do not undergo Bose-Einstein condensation at finite temperature, resolving a long-standing question in the rigorous formulation of quantum statistical mechanics. The proofs draw on spectral analysis of the van Hove and spin-boson Hamiltonians, Tomita-Takesaki modular theory, the Araki-Woods direct integral decomposition, and functional integra representations of thermal equilibrium states.
We analyze the possibility of Bose-Einstein condensation (BEC) at finite temperature in the spin-boson model within the frameworks of functional integral representations and the resolvent algebra. Because a sesquilinear form arising from the zero mode appears, analogous to the case of the free Bose gas, a BEC-type component is also formally present in the spin-boson model. However, according to arxiv:1207.4621, quasiparticles do not undergo BEC, so we have to exclude this possibility. A no-go theorem for BEC is obtained, but we find that the spin-boson model is not a model that follows the criterion in arxiv:1207.4621.
We present a systematic description of the structure of Bose-Einstein condensation (BEC) in the free Bose gas from the viewpoint of the correspondence between the operator-algebraic formulation based on the resolvent algebra and the functional integral representation. By clarifying the representation-theoretic structure of finite-temperature BEC states and rigorously analyzing the correspondence between their direct integral decomposition and the ergodic decomposition of the associated probability measures, we provide a framework in which general features of phase transitions-such as the emergence of order parameters, the decomposition of states, and clustering properties-are explicitly described using BEC in the free Bose gas as a concrete example. Furthermore, we construct in detail the correspondence between the decomposition of measures in the functional integral approach and that of operator-algebraic representations, thereby establishing the equivalence between the probabilistic and algebraic aspects, and providing a guiding principle for isolating the essential structures by disentangling the additional mathematical complications arising from the treatment of infrared singularities in interacting systems. These results lay a foundation for the rigorous analysis of phase transitions in non-relativistic constructive quantum field theory and quantum statistical mechanics, and serve as a starting point for extensions to interacting models.
We consider entanglement entropy between regions of space in lattice gauge theory. The Hilbert space corresponding to a region of space includes ``edge states'' that transform nontrivially under gauge transformations. By decomposing the edge states in irreducible representations of the gauge group, the entropy of an arbitrary state is expressed as the sum of three positive terms: a term associated with the classical Shannon entropy of the distribution of boundary representations, a term that appears only for non-Abelian gauge theories and depends on the dimension of the boundary representations, and a term representing nonlocal correlations. The first two terms are the entropy of the edge states, and depend only on observables measurable at the boundary. These results are applied to several examples of lattice gauge theory states, including the ground state in the strong coupling expansion of Kogut and Susskind. In all these examples we find that the entropy of the edge states is the dominant contribution to the entanglement entropy.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.