January 2021

Since the appearance of Einstein’s paper {em”On the Electrodynamics of Moving Bodies”} and the birth of special relativity, it is understood that the theory was basically coded within Maxwell’s equations. The celebrated mass-energy equivalence relation, $E=mc^2$, is derived by Einstein using thought experiments involving the kinematics of the emission of light (electromagnetic energy) and the relativity principle. Text book derivations often follow paths similar to Einstein’s, or the analysis of the kinematics of particle collisions interpreted from the perspective of different inertial frames. All the same, in such derivations the direct dynamical link with hypothetical fundamental fields describing matter (e.g. Maxwell theory or other) is overshadowed by the use of powerful symmetry arguments, kinematics, and the relativity principle. Here we show that the formula can be derived directly form the dynamical equations of a massless matter model confined in a bo(which can be thought of as a toy model of a composite particle). The only assumptions in the derivation are that the field equations hold and the energy-momentum tensor admits a universal interpretation in arbitrary coordinate systems. The mass-energy equivalence relation follows from the inertia or (taking the equivalence principle for granted) weight of confined field radiation. The present derivation offers an interesting pedagogical perspective on the formula providing a simple toy model on the origin of mass and a natural bridge to the foundations of general relativity.

In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime QRF, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of describing the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the evolution parameter coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition. Concretely, we consider N relativistic quantum particles in a weak gravitational field and introduce a timeless formulation in which the global state of the N particles appears “frozen”, but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fithe QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, keeping the proper time in the local frame of the particle. This fully relational construction shows how the remaining particles evolve from the perspective of the QRF and includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we observe a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations in the QRF.

A cornerstone of quantum mechanics is the characterisation of symmetries provided by Wigner’s theorem. Wigner’s theorem establishes that every symmetry of the quantum state space must be either a unitary transformation, or an antiunitary transformation. Here we extend Wigner’s theorem from quantum states to quantum evolutions, including both the deterministic evolution associated to the dynamics of closed systems, and the stochastic evolutions associated to the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time symmetric formulation of quantum theory must either restrict the set of the allowed evolutions, or modify the operational interpretation of quantum states and processes. Here we propose a time symmetric formulation of quantum theory where the allowed quantum evolutions are restricted to a suitable set, which includes both unitary evolution and projective measurements, but excludes the deterministic preparation of pure states. The standard operational formulation of quantum theory can be retrieved from this time symmetric version by introducing an operation of conditioning on the outcomes of past experiments.