Abstract The closing chapter details the application of noncommutative Orlicz spaces to Quantum Physics. We specifically show that the (Köthe dual) pair of Orlicz spaces Llog(L+1) and Lcosh−1 may be better suited as home for regular states and observables respectively than the more standard pair of spaces L∞ and L∞. Specifically the geometry of the space L1∩Llog(L+1) is shown to be well suited to identifying those states which admit a well-defined notion of entropy. In fact the geometry of this space may be used to posit a clear definition of entropy valid for the most general von Neumann algebras. Parallel to this the space Lcosh−1 proves to be home for possibly unbounded observables which nevertheless are very regular in the sense of for example having all their moments finite. It is further pointed out that in the case of B(H) this approach yields exactly the Dirac style elementary Quantum Mechanical formalism. The technology thus developed is also used to gain a clearer understanding of relative entropy of states. The fact that the space Lcosh−1 admits unbounded observables allows for an application to the necessarily unbounded field operators of Quantum Field Theory. We specifically show that under very natural criteria, a large number of quantum fields which satisfy a so-called generalised H-bound condition automatically also satisfy an Lcosh−1 regularity condition namely that one may canonically embed the field operators into Lcosh−1 - the natural superspace for the regular observables.
Abstract The closing chapter details the application of noncommutative Orlicz spaces to Quantum Physics. We specifically show that the (Köthe dual) pair of Orlicz spaces Llog(L+1) and Lcosh−1 may be better suited as home for regular states and observables respectively than the more standard pair of spaces L∞ and L∞. Specifically the geometry of the space L1∩Llog(L+1) is shown to be well suited to identifying those states which admit a well-defined notion of entropy. In fact the geometry of this space may be used to posit a clear definition of entropy valid for the most general von Neumann algebras. Parallel to this the space Lcosh−1 proves to be home for possibly unbounded observables which nevertheless are very regular in the sense of for example having all their moments finite. It is further pointed out that in the case of B(H) this approach yields exactly the Dirac style elementary Quantum Mechanical formalism. The technology thus developed is also used to gain a clearer understanding of relative entropy of states. The fact that the space Lcosh−1 admits unbounded observables allows for an application to the necessarily unbounded field operators of Quantum Field Theory. We specifically show that under very natural criteria, a large number of quantum fields which satisfy a so-called generalised H-bound condition automatically also satisfy an Lcosh−1 regularity condition namely that one may canonically embed the field operators into Lcosh−1 - the natural superspace for the regular observables.
We present a new rigorous approach based on Orlicz spaces for the description of the statistics of large regular statistical systems, both classical and quantum. The pair of Orlicz spaces we explicitly use are, respectively, built on the exponential function (for the description of regular observables) and on an entropic type function (for the corresponding states). They form a dual pair (both for classical and quantum systems). This pair has the advantage of being general enough to encompass regular observables, and specific enough for the latter Orlicz space to select states with a well-defined entropy function. Moreover for small quantum systems, this pair is shown to agree with the classical pairing of bounded linear operators on a Hilbert space, and the trace-class operators.
Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [W. A. Majewski, L.E. Labuschagne, Ann. H. Poincare. 15, 1197-1221, (2014)] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces ⟨Lcosh−1,Llog(L+1)⟩. The present paper therefore in some sense "completes" the picture by showing that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition, canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation technique, specifically suited to the above context, for extending the action of such maps to the appropriate intermediate spaces of the pair ⟨L∞,L1⟩. Moreover, it is shown that quantum dynamics in the form of Markov semigroups described by some Dirichlet forms naturally extends to the context proposed in [W. A. Majewski, L.E. Labuschagne, Ann. H. Poincare. 15, 1197-1221, (2014)].
We present a new rigorous approach based on Orlicz spaces for the description of the statistics of large regular statistical systems, both classical and quantum. The pair of Orlicz spaces we explicitly use are, respectively, built on the exponential function (for the description of regular observables) and on an entropic type function (for the corresponding states). They form a dual pair (both for classical and quantum systems). This pair has the advantage of being general enough to encompass regular observables, and specific enough for the latter Orlicz space to select states with a well-defined entropy function. Moreover for small quantum systems, this pair is shown to agree with the classical pairing of bounded linear operators on a Hilbert space, and the trace-class operators.
Quantum Field Theory (QFT) represents a vast generalization of Quantum Mechanics (QM), as it deals with systems that have an infinite number of degrees of freedom. The Stone-von Neumann theorem, which establishes the equivalence of irreducible representations of the canonical commutation relations (CCR) in QM, does not extend to QFT. Consequently, QFT admits multiple inequivalent irreducible representations, leading to a much richer algebraic structure. This essay aims to explore the physics of QFT from the operator algebra perspective, particularly focusing on entanglement entropy. We discuss the role of von Neumann algebras of different types in QFT, describe the local operator algebra approach to QFT, and explain how entanglement entropy can be defined in terms of the algebra of observables. Additionally, we explore the benefits of this approach in concrete applications, specifically in quantum field theory on curved spacetime.
This paper presents a generalization of quantum mechanics from conventional Hilbert space formalism to Banach space one. We construct quantum theory starting with any complex Banach space beyond a complex Hilbert space, through using a basic fact that a complex Banach space always admits a semi-inner product. Precisely, in a complex Banach space X with a given semi-inner product, a pure state is defined by Lumer \cite{Lumer1961} to be a bounded linear functional on the space of bounded operators determined by a normalized element of X under the semi-inner product, and then the state space S(X) of the system is the weakly closed convex set spanned by all pure states. Based on Lumer's notion of the state, we associate a quantum system with a complex Banach space X equipped with a fixed semi-inner product, and then define a physical event at a quantum state ω∈S(X) to be a projection P (bounded operator such that P2=P) in X satisfying the positivity condition 0≤ω(P)≤1, and a physical quantity at a quantum state ω to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at ω. The Born formula for measurement of a physical quantity is the natural pairing of operators with linear functionals satisfying the probability conservation law. A time evolution of the system is governed by a one-parameter group of invertible spectral operators determined by a scalar type operator with the real spectrum, which satisfies the Schrödinger equation. Our formulation is just a generalization of the Dirac-von Neumann formalism of quantum mechanics to the Banach space setting. We include some examples for illustration.
Introduction The aim of information geometry is to introduce a suitable geometrical structure on families of probability distributions or quantum states. For parametrised statistical models, such structure is based on two fundamental notions: the Fisher information and the exponential family with its dual mixed parametrisation, see for example (Amari 1985, Amari and Nagaoka 2000). For the non-parametric situation, the solution was given by Pistone and Sempi (Pistone and Sempi 1995, Pistone and Rogantin 1999), who introduced a Banach manifold structure on the set P of probability distributions, equivalent to a given one. For each µ ∈ ρ, the authors considered the non-parametric exponential family at µ. As it turned out, this provides a C ∞ -atlas on ρ, with the exponential Orlicz spaces Lφ(µ) as the underlying Banach spaces, here φ is the Young function of the form φ(x) = cosh(x) — 1. The present contribution deals with the case of quantum states: we want to introduce a similar manifold structure on the set of faithful normal states of a von Neumann algebra M. Since there is no suitable definition of a non-commutative Orlicz space with respect to a state φ, it is not clear how to choose the Banach space for the manifold. Of course, there is a natural Banach space structure, inherited from the predualM. But, as it was already pointed out in (Streater 2004), this structure is not suitable to define the geometry of states: for example, any neighbourhood of a state φ contains states such that the relative entropy with respect to φ is infinite.