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Reviews in Mathematical Physics

ENTROPY DENSITY FOR RELATIVISTIC QUANTUM FIELD THEORY

Heide Narnhofer

Published Jan 1, 1994• 15 citations
Publisher

Abstract

We show how the nuclearity condition of Buchholz and Wichmann allows to define in the ground state a local entropy with the desired properties despite the fact that local algebras are type III. Generalization to temperature states is also possible so that thermodynamic functions also exist in the context of relativistic quantum field theory.

DOI
10.1142/s0129055x94000390

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We show how the nuclearity condition of Buchholz and Wichmann allows to define in the ground state a local entropy with the desired properties despite the fact that local algebras are type III. Generalization to temperature states is also possible so that thermodynamic functions also exist in the context of relativistic quantum field theory.
This essay was completed as partial fulfillment of the requirements forthe MASt in Applied Mathematics degree (Part III of the Mathemat-ical Tripos) carried out at the DAMPT, Cambridge. After reviewingthe essentials of C*-algebras and von Neumann algebras, we present theMurray–von Neumann classification of factors into types I, II and III.We then introduce the algebraic approach to Quantum Field Theory(QFT) and discuss existing arguments for the association of type IIIalgebras to local regions of spacetime. We explain why, as a conse-quence, the entanglement entropy of a region is not well defined, andhow relative entropy offers an alternative, reviewing the fundamentalsof entanglement entropy in QFT. Finally, we see how including grav-ity turns the algebra of observables outside a black hole into one oftype II∞, where a well-defined notion of entropy can be recovered. This work should be regarded as introductory notes to operator al-gebras with a particular emphasis on their applications in theoreticalphysics. They are intended to synthesize the mathematical content fora physics-oriented audience, using worked-out examples throughout toset a foundation and gradually build up to some direct examples of theusage of the framework. They are aimed at advanced undergraduatestudents or early postgraduate students, and only some basic knowl-edge on quantum mechanics, QFT, group theory and linear algebra areassumed
Within the framework of relativistic quantum field theory, a novel method is established which allows to distinguish non-equilibrium states admitting locally a thermodynamic interpretation. The basic idea is to compare these states with global equilibrium states (KMS states) by means of local thermal observables. With the help of such observables, the states can be ordered into classes of increasing local thermal stability. Moreover, it is possible to identify states exhibiting certain specific thermal properties of interest, such as a definite local temperature or entropy density. The method is illustrated in a simple model describing the spatio-temporal evolution of a “big heat bang”. 1
Simple example of local temperature stateAiming at a general formulation of non-equilibrium states in the framework of QFT, we start from a simple model-example of local temperature state: Let φ(x)\varphi(x)φ(x) be a massless free scalar field (in four dimensions) characterized by(Given a two-point function ω(2)(φ(x)φ(y))\omega^{(2)}(\varphi(x)\varphi(y))ω(2)(φ(x)φ(y)) , we can define a quasi-free state ω\omegaω of φ\varphiφ through the Wick formula:When a two-point function ω(2)\omega^{(2)}ω(2) is chosen consistently with Eqs.(1,2)(1, 2)(1,2) asthe corresponding quasi-free state ω=ωβ\omega=\omega_{\beta}ω=ωβ​ describes a global thermal equi- librium satisfying the KMS condition for any pair of polynomial fields A=A=A= φ(x1)⋯φ(xr)\varphi(x_{1})\cdots\varphi(x_{r})φ(x1​)⋯φ(xr​) and B=φ(y1)⋯φ(ys)B=\varphi(y_{1})\cdots\varphi(y_{s})B=φ(y1​)⋯φ(ys​) , ωβ(Aα(iβ,O)(B))=ωβ(BA)\omega_{\beta}(A\alpha_{(i\beta,\mathrm{O})}(B))=\omega_{\beta}(BA)ωβ​(Aα(iβ,O)​(B))=ωβ​(BA) , (5) *Talk based on a joint work of D.Buchholz, H.Roos and I.O\mathrm{I}.\mathrm{O}I.O . in progress
We review the heuristic arguments suggesting that any thermal quantum field theory, which can be interpreted as a quantum statistical mechanics of (interacting) relativistic particles, obeys certain restrictions on its number of local degrees of freedom. As in the vacuum representation, these restrictions can be expressed by a `nuclearity condition'. If a model satisfies this nuclearity condition, then the net of von Neumann algebras representing the local observables in the thermal representation has the split property.