Causal independence and the energy-level density of states in local quantum field theory

Buchholz, Detlev; Wichmann, Eyvind H.

doi:10.1007/bf01454978

Cited by 181 publications

(307 citation statements)

References 30 publications

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“…9 Despite its rather technical definition, the condition of nuclearity is wellmotivated from a physical viewpoint as the discussion in [3] makes plain: the nuclearity index can be interpreted as a local partition function, and the form of the nuclearity bound (15) is suggested by the requirement that the associated pressure should remain finite in the thermodynamic limit and scale polynomially with temperature (as is the case, for example, in the StefanBoltzmann law).…”

Section: Connections With Nuclearitymentioning

confidence: 99%

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Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

Fewster

2007

Rigorous Quantum Field Theory

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Summary. We give a brief review of quantum energy inequalities (QEIs) and then discuss two lines of work which suggest that QEIs are closely related to various natural properties of quantum field theory which may all be regarded as stability conditions. The first is based on joint work with Verch, and draws connections between microscopic stability (microlocal spectrum condition), mesoscopic stability (QEIs) and macroscopic stability (passivity). The second direction considers QEIs for a countable number of massive scalar fields, and links the existence and scaling properties of QEIs to the spectrum of masses. The upshot is that the existence of a suitable QEI with polynomial scaling is a sufficient condition for the model to satisfy the Buchholz-Wichmann nuclearity criterion. We briefly discuss on-going work with Ojima and Porrmann which seeks to gain a deeper understanding of this relationship.

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Section: Connections With Nuclearitymentioning

confidence: 99%

“…Namely, the sets N β,r are nuclear if [3] and only if [5] j exp(−βm j ) < ∞ for all sufficiently small β; furthermore, the nuclearity index may be estimated from above by…”

Section: Connections With Nuclearitymentioning

confidence: 99%

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

Fewster

2007

Rigorous Quantum Field Theory

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“…Recall that, in a n-dimensional spacetimes, n ≥ 3, we have d(ρ) ∈ N∪∞. The above formula holds also for finitely many charges, and we regard (3) as a physical description of (9). It can be read as a quantum index theorem (or, more appropriately, "QFT index theorem" as it concerns infinitely many degrees of freedom) where the quantum Fredholm index log d(ρ) − log d(σ) is expressed in terms of dF and the geometric quantity κ.…”

Section: On Black Hole Entropymentioning

confidence: 99%

“…log-ellipticity is essentially the nuclearity condition of Buchholz and Wichmann [9] (and we fix the dimension).…”

Section: Spectral Invariants Associated With Lmentioning

confidence: 99%

Noncommutative Spectral Invariants and Black Hole Entropy

Kawahigashi

Longo

2005

Commun. Math. Phys.

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We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the "heat kernel" semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system with infinitely many degrees of freedom, we regard these coefficients as noncommutative geometric invariants. Under a natural modularity assumption, the leading term of the entropy (noncommutative area) is proportional to the central charge c, the first order correction (noncommutative Euler characteristic) is proportional to log µ A , where µ A is the global index of A, and the second spectral invariant is again proportional to c.We give a further general method to define a mean entropy by considering conformal symmetries that preserve a discretization of S 1 and we get the same value proportional to c.We then make the corresponding analysis with the proper Hamiltonian associated to an interval. We find here, in complete generality, a proper mean entropy proportional to log µ A with a first order correction defined by means of the relative entropy associated with canonical states. * Supported in part by JSPS. † Supported in part by GNAMPA and MIUR. 1By considering a class of black holes with an associated conformal quantum field theory on the horizon, and relying on arguments in the literature, we indicate a possible way to link the noncommutative area with the BekensteinHawking classical area description of entropy.

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“…Thus, Sec. 4 Namely, it was shown in [12] (see also [15]) that free quantum field states fulfilling the microlocal spectrum condition satify QWEIs. In the present work, we show that QWEIs imply the existence of passive states.…”

Section: Introductionmentioning

confidence: 99%

Stability of Quantum Systems at Three Scales: Passivity, Quantum Weak Energy Inequalities and the Microlocal Spectrum Condition

Fewster

Verch

2003

Communications in Mathematical Physics

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Abstract. Quantum weak energy inequalities have recently been extensively discussed as a condition on the dynamical stability of quantum field states, particularly on curved spacetimes. We formulate the notion of a quantum weak energy inequality for general dynamical systems on static background spacetimes and establish a connection between quantum weak energy inequalities and thermodynamics. Namely, for such a dynamical system, we show that the existence of a class of states satisfying a quantum weak inequality implies that passive states (e.g., mixtures of ground-and thermal equilibrium states) exist for the time-evolution of the system and, therefore, that the second law of thermodynamics holds. As a model system, we consider the free scalar quantum field on a static spacetime. Although the Weyl algebra does not satisfy our general assumptions, our abstract results do apply to a related algebra which we construct, following a general method which we carefully describe, in Hilbert-space representations induced by quasifree Hadamard states. We discuss the problem of reconstructing states on the Weyl algebra from states on the new algebra and give conditions under which this may be accomplished.Previous results for linear quantum fields show that, on one hand, quantum weak energy inequalities follow from the Hadamard condition (or microlocal spectrum condition) imposed on the states, and on the other hand, that the existence of passive states implies that there is a class of states fulfilling the microlocal spectrum condition. Thus, the results of this paper indicate that these three conditions of dynamical stability are essentially equivalent. This observation is significant because the three conditions become effective at different length scales: The microlocal spectrum condition constrains the short-distance behaviour of quantum states (microscopic stability), quantum weak energy inequalities impose conditions at finite distance (mesoscopic stability), and the existence of passive states is a statement on the global thermodynamic stability of the system (macroscopic stability).

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Causal independence and the energy-level density of states in local quantum field theory

Cited by 181 publications

References 30 publications

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

Noncommutative Spectral Invariants and Black Hole Entropy

Stability of Quantum Systems at Three Scales: Passivity, Quantum Weak Energy Inequalities and the Microlocal Spectrum Condition

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