Dynamical Locality and Covariance: What Makes a Physical Theory the Same in all Spacetimes?

Abstract

The question of what it means for a theory to describe the same physics on all spacetimes (SPASs) is discussed. As there may be many answers to this question, we isolate a necessary condition, the SPASs property, that should be satisfied by any reasonable notion of SPASs. This requires that if two theories conform to a common notion of SPASs, with one a subtheory of the other, and are isomorphic in some particular spacetime, then they should be isomorphic in all globally hyperbolic spacetimes (of given dimension). The SPASs property is formulated in a functorial setting broad enough to describe general physical theories describing processes in spacetime, subject to very minimal assumptions. By explicit constructions, the full class of locally covariant theories is shown not to satisfy the SPASs property, establishing that there is no notion of SPASs encompassing all such theories. It is also shown that all locally covariant theories obeying the time-slice property possess two local substructures, one kinematical (obtained directly from the functorial structure) and the other dynamical (obtained from a natural form of dynamics, termed relative Cauchy evolution). The covariance properties of relative Cauchy evolution and the kinematic and dynamical substructures are analyzed in detail. Calling local covariant theories dynamically local if their kinematical and dynamical local substructures coincide, it is shown that the class of dynamically local theories fulfills the SPASs property. As an application in quantum field theory, we give a model independent proof of the impossibility of making a covariant choice of preferred state in all spacetimes, for theories obeying dynamical locality together with typical assumptions. *

On the stress–energy tensor of quantum fields in curved spacetimes—comparison of different regularization schemes and symmetry of the Hadamard/Seeley–DeWitt coefficients

“…However, it is known that such an assignment of a state to a spacetime is impossible [HoWa01,BFV03,FeVe11]. Notwithstanding, this problem can be overcome if it is possible to restrict oneself to a limited class of spacetimes on which a unique state can be defined, this has been done in [Pin10].…”

Dynamical Locality of the Free Maxwell Field

“…A foundational problem for physics in curved spacetimes is to understand how a theory should be formulated such that its physical content is preserved across the various spacetimes on which it is defined; i.e., so that it represents the same physics in all spacetimes (SPASs) [FV12a]. This touches on what is actually meant by the physical content of a theory and it is not easy to make this mathematically precise.…”

“…On the other hand, all the theories discussed admit electromagnetic duality rotations as global symmetries. To conclude, we discuss three aspects in more detail, namely the status of dynamical locality, the categorical structure underlying some of our constructions, and the relation of our present work to the discussions of SPASs in [FV12a,FV12b].…”

Quantum estimation of parameters of classical spacetimes

“…(3.1) gives the infinitesimal movement of the observables in the Hilbert-space representation of A (M, g (0) ) due to an infinitesimal perturbation ds around s = 0. Further properties and interpretations of this functional derivative and the relative Cauchy evolution can be found in [5,23,24]. For the case of the Klein-Gordon field, with its corresponding Weyl algebra of observables, it can be shown that both elements of this algebra and polynomials of field operators constructed from it obey the following relation (Theorem 4.3 from [5]):…”

“…The approach has been used to develop a notion of "identical physics" on different spacetimes [23]. A key result is a method for calculating how quantum observables respond to local changes in the background spacetime.…”