Algebras of local observables on a manifold

Dimock, J.

doi:10.1007/bf01269921

Cited by 205 publications

(309 citation statements)

References 14 publications

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“…By deriving suitable "consistency conditions" 24 on this anomaly, one can show [47] that an arbitrary renormalization prescription, consistent with all of the properties listed in thm. 2, can always be modified so as to remove this anomaly 25 .…”

Section: Yang-mills Fieldsmentioning

confidence: 99%

Quantum fields in curved spacetime

2015

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We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime. Our review emphasizes the conceptual issues arising in the formulation of the theory and presents known results in a mathematically precise way. Particular attention is paid to the distributional nature of quantum fields, to their local and covariant character, and to microlocal spectrum conditions satisfied by physically reasonable states. We review the Unruh and Hawking effects for free fields, as well as the behavior of free fields in deSitter spacetime and FLRW spacetimes with an exponential phase of expansion. We review how nonlinear observables of a free field, such as the stress-energy tensor, are defined, as well as timeordered-products. The "renormalization ambiguities" involved in the definition of time-ordered products are fully characterized. Interacting fields are then perturbatively constructed. Our main focus is on the theory of a scalar field, but a brief discussion of gauge fields is included. We conclude with a brief discussion of a possible approach towards a nonperturbative formulation of quantum field theory in curved spacetime and some remarks on the formulation of quantum gravity. Quantum field theory in curved spacetime (QFTCS) is the theory of quantum fields propagating in a background, classical, curved spacetime (M , g). On account of its classical treatment of the metric, QFTCS cannot be a fundamental theory of nature. However, QFTCS is expected to provide an accurate description of quantum phenomena in a regime where the effects of curved spacetime may be significant, but effects of quantum gravity itself may be neglected. In particular, it is expected that QFTCS should be applicable to the description of quantum phenomena occurring in the early universe and near (and inside of) black holesprovided that one does not attempt to describe phenomena occurring so near to singularities that curvatures reach Planckian scales and the quantum nature of the spacetime metric would have to be taken into account.It should be possible to derive QFTCS by taking a suitable limit of a more fundamental theory wherein the spacetime metric is treated in accord with the principles of quantum theory. However, this has not been done-except in formal and/or heuristic ways-simply because no present quantum theory of gravity has been developed to the point where such a well defined limit can be taken. Rather, the framework of QFTCS that we shall describe in this review has been obtained by suitably merging basic principles of classical general relativity with the basic principles of quantum field theory in Minkowski spacetime. As we shall explain further below, the basic principles of classical general relativity are relatively easy to identify and adhere to, but it is far less clear what to identify as the "basic principles" of quantum field theory in Minkowski spacetime. Indeed, many of the concepts normally viewed as fundamental to quantum field theory in Minkowski spacetime, such as Poinca...

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Section: Yang-mills Fieldsmentioning

confidence: 99%

Quantum fields in curved spacetime

2015

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“…21 For obviously local primitive causality entails entails (QNSP1) for any S. It has been veri…ed that local primitive causality holds for the quantized Klein-Gordon …eld and the Dirac …eld (see Dimock 1980Dimock , 1982. The upshot is that if (QNSP1) does adequately capture no superluminal propagation for quantum …elds, local primitive causality is su¢ cient to ensure no superluminal propagation even at very small scales despite the failure of various energy conditions at these scales.…”

Section: Causal Propagation For Quantum …Eldsmentioning

confidence: 99%

No superluminal propagation for classical relativistic and relativistic quantum fields

Earman

2014

Studies in History and Philosophy of Science Part B: Studies in

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A criterion is proposed to ensure that classical relativistic …elds do not propagate superluminally. If this criterion does indeed serve as a su¢ cient condition for no superluminal propagation it follows that various other criteria found in the physics literature cannot serve as necessary conditions since they can fail although the pro¤ered condition holds. The rejected criteria rely on energy conditions that are believed to hold for most classical …elds used in actual applications. But these energy conditions are known to fail at small scales for quantum …elds. It is argued that such a failure is not necessarily a cause for concern about superluminal propagation in the quantum regime since the pro¤ered criterion of no superluminal propagation for classical …elds has a natural analog for quantum …elds and, further, this quantum analog condition provably holds for some quantum …elds despite the violation of energy conditions. The apparatus developed here also o¤ers a di¤erent approach to treating the Reichenbach-Salmon cases of "pseudo-causal processes" and helps to clarify the issue of whether relativity theory is consistent with superluminal propagation.

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“…7 The C * -algebraic net O → A(O) then has the covariance property [9]α t (A(O)) = A(τ t (O)) . However, {α t } t∈R is not strongly continuous (because W(u) − W(v) = 2 for all u = v).…”

Section: The Dynamical Systemmentioning

confidence: 99%

“…[9]): first, let (S, σ) be the symplectic space of smooth real-valued Klein-Gordon solutions where the symplectic form is given by…”

Section: The Dynamical Systemmentioning

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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Algebras of local observables on a manifold

Cited by 205 publications

References 14 publications

Quantum fields in curved spacetime

Quantum fields in curved spacetime

No superluminal propagation for classical relativistic and relativistic quantum fields

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

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