Local Lorentz covariance in finite-dimensional local quantum physics

Abstract

We show that local Lorentz covariance arises canonically as the group of transformations between local thermal states in the framework of Local Quantum Physics, given the following three postulates: (i) Local observable algebras are finite-dimensional. (ii) Minimal local observable algebras are isomorphic to M2(C), the observable algebra of a single qubit. (iii) The vacuum restricted to any minimal local observable algebra is a non-maximally mixed thermal state. The derivation reveals a new and surprising relation between spacetime structure and local quantum states. In particular, we show how local restrictions of the vacuum can determine the connection between different local inertial reference frames.

Spacetime granularity from finite-dimensionality of local observable algebras

“…Of course, when gravity becomes relevant we should not expect to have global Lorentz symmetry in general, but (at most) local Lorentz covariance in agreement with the equivalence principle. In [12] we showed how local Lorenz covariance may appear in the locally finite-dimensional context as transformations between local thermal Hamiltonians: If the local algebras associated to minimal spatial regions are isomorphic to the observable algebra of a qubit (i.e., a 2-by-2 matrix algebra), then local thermal states on any two of these minimal local algebras can be transformed to each other via a unique SL(2, C) transformation of the thermal Hamiltonian. In this way we can recover a Lorentz connection on the minimal local spacetime regions.…”