On Deriving Space–Time from Quantum Observables and States

Abstract

We prove that, under suitable assumptions, operationally motivated quantum data completely determine a space-time in which the quantum systems can be interpreted as evolving. At the same time, the dynamics of the quantum system is also determined. To minimize technical complications, this is done in the example of three-dimensional Minkowski space.

Spacetime-Free Approach to Quantum Theory and Effective Spacetime Structure

“…5 As an example, the locally covariant QFT framework [20,34], which is arguably the most general formulation of ordinary QFT today, follows this logic in defining QFT models. Following ideas in earlier works [1,4,10,11,29,51,78], we suggest trying to invert the arrows leading from the spacetime to the equilibrium state in order to recover spacetime geometry from the equilibrium state. 6 The purpose of this reversal of logic is that a state of the system can be determined and represented algebraically without referring to any spacetime structure.…”

“…The second arrow from the dynamics to the equilibrium state is already known (in some cases) to be inverted by the Tomita-Takesaki modular theory, as reviewed in Section 2.4. The first arrow leading from the dynamics to the spacetime is more enigmatic, although some ideas and hints on how to achieve the inversion have appeared in the literature (see, e.g., [4,10,11,24,51,78]). In Section 3 we will develop certain physically motivated ideas and methods for studying the effective spacetime structure induced by the dynamics.…”

“…Accordingly, the existence of the modular involution is strongly related to Lorentz invariance through the CPT theorem and the symmetry between matter and antimatter (i.e., retarded and advanced solutions to the equations of motion), since the CPT transformation in QFT maps particles to antiparticles, and vice versa [38]. Indeed, taking advantage of the relation between the modular involutions and CPT transformations, it has been shown in [22,78,91] that the modular involutions induced by the QFT vacuum restricted to Rindler wedges can be used to generate a representation of the proper Poincaré group, thus recovering spacetime structure from the purely algebraic data of a vacuum state and a suitable family of subalgebras of observables. However, it is unclear whether this method for deriving spacetime from algebraic data can be extended to less symmetric situations.…”

“…The earliest works we have found addressing the issue are [4,51], where the inverse problem of recovering spacetime topology and causal structure from the net of local observable algebras is considered. On the other hand, [22,78,91] show that it is possible to recover the symmetry group of spacetime from the modular structure of subalgebras in some highly symmetric cases. These results already show that quite a lot of information about spacetime structure is encoded into quantum field theory.…”

Spacetime granularity from finite-dimensionality of local observable algebras

“…In order to guarantee background-independence without explicit diffeomorphism-invariance, it should be possible to formulate the theory in a way that does not directly refer to spacetime geometry, but only to the algebraic and statistical relations between quantum operators, while the effective spacetime geometry is extracted a posteriori. (See, e.g., [16,17,18,19,20,21,22,23] for a small subset of works in this direction.) With this goal in mind, in [24] we formulated a spacetime-free framework for quantum theory.…”