Another Proof of Born’s Rule on Arbitrary Cauchy Surfaces

In 2017, Lienert and Tumulka proved Born’s rule on arbitrary Cauchy surfaces in Minkowski space-time assuming Born’s rule and a corresponding collapse rule on horizontal surfaces relative to a fixed Lorentz frame, as well as a given unitary time evolution between any two Cauchy surfaces, satisfying that there is no interaction faster than light and no propagation faster than light. Here, we prove Born’s rule on arbitrary Cauchy surfaces from a different, but equally reasonable, set of assumptions. The conclusion is that if detectors are placed along any Cauchy surface $$\Sigma $$ Σ , then the observed particle configuration on $$\Sigma $$ Σ is a random variable with distribution density $$|\Psi _\Sigma |^2$$ | Ψ Σ | 2 , suitably understood. The main different assumption is that the Born and collapse rules hold on any spacelike hyperplane, i.e., at any time coordinate in any Lorentz frame. Heuristically, this follows if the dynamics of the detectors is Lorentz invariant.

Time-independence of gravitational Rényi entropies and unitarity in quantum gravity

The Hubeny-Rangamani-Takayanagi surface γHRT computing the entropy S(D) of a domain of dependence D on an asymptotically AdS boundary is known to be causally inaccessible from D. We generalize this gravitational result to higher replica numbers n > 1 by considering the replica-invariant surfaces (aka ‘splitting surfaces’) γ of real-time replica-wormhole saddle-points computing Rényi entropies Sn(D) and showing that there is a sense in which D must again be causally inaccessible from γ when the saddle preserves both replica and conjugation symmetry. This property turns out to imply the Sn(D) to be independent of any choice of any Cauchy surface ΣD for D, and also that the Sn(D) are independent of the choice of boundary sources within D. This is a key hallmark of unitary evolution in any dual field theory. Furthermore, from the bulk point of view it adds to the evidence that time evolution of asymptotic observables in quantum gravity is implemented by a unitary operator in each baby universe superselection sector. Though we focus here on pure Einstein-Hilbert gravity and its Kaluza-Klein reductions, we expect the argument to extend to any two-derivative theory who satisfies the null convergence condition. We consider both classical saddles and the effect of back-reaction from quantum corrections.

Information loss paradox revisited: Farewell firewall?

Unitary evolution makes pure state on one Cauchy surface evolve to pure state on another Cauchy surface. Outgoing Hawking radiation is the only subsystem on the late Cauchy surface. The requirement that Hawking radiation should be pure amounts to requiring purity of the subsystem when the total system is pure. We will see that this requirement will lead to firewall even in flat spacetime, and thus is invalid. Information is either stored in the entanglement between field modes inside black hole and the outgoing modes or stored in correlation between geometry and Hawking radiation when singularity is resolved by quantum gravity effects. We will give a simple argument that even in semi-classical regime, information is (at least partly) stored in correlation between geometry and Hawking radiation.

Classical phase space and Hadamard states in the BRST formalism for gauge field theories on curved spacetime

We investigate linearized gauge theories on globally hyperbolic spacetimes in the BRST formalism. A consistent definition of the classical phase space and of its Cauchy surface analogue is proposed. We prove that it is isomorphic to the phase space in the ‘subsidiary condition’ approach of Hack and Schenkel in the case of Maxwell, Yang–Mills, and Rarita–Schwinger fields. Defining Hadamard states in the BRST formalism in a standard way, their existence in the Maxwell and Yang–Mills case is concluded from known results in the subsidiary condition (or Gupta–Bleuler) formalism. Within our framework, we also formulate criteria for non-degeneracy of the phase space in terms of BRST cohomology and discuss special cases. These include an example in the Yang–Mills case, where degeneracy is not related to a non-trivial topology of the Cauchy surface.

On the initial value problem for the wave equation in Friedmann–Robertson–Walker space–times

The propagator W ( t 0 , t 1 )( g , h ) for the wave equation in a given space–time takes initial data ( g ( x ), h ( x )) on a Cauchy surface {( t , x ) : t = t 0 } and evaluates the solution ( u ( t 1 , x ),∂ t u ( t 1 , x )) at other times t 1 . The Friedmann–Robertson–Walker space–times are defined for t 0 , t 1 >0, whereas for t 0 →0, there is a metric singularity. There is a spherical means representation for the general solution of the wave equation with the Friedmann–Robertson–Walker background metric in the three spatial dimensional cases of curvature K =0 and K =−1 given by S. Klainerman and P. Sarnak. We derive from the expression of their representation three results about the wave propagator for the Cauchy problem in these space–times. First, we give an elementary proof of the sharp rate of time decay of solutions with compactly supported data. Second, we observe that the sharp Huygens principle is not satisfied by solutions, unlike in the case of three-dimensional Minkowski space–time (the usual Huygens principle of finite propagation speed is satisfied, of course). Third, we show that for 0< t 0 < t the limit, lim t 0 → 0 + W ( t 0 , t ) ( g , h ) = W ( 0 , t ) ( g ) exists, it is independent of h ( x ), and for all reasonable initial data g ( x ), it gives rise to a well-defined solution for all t >0 emanating from the space–time singularity at t =0. Under reflection t →− t , the Friedmann–Robertson–Walker metric gives a space–time metric for t <0 with a singular future at t =0, and the same solution formulae hold. We thus have constructed solutions u ( t , x ) of the wave equation in Friedmann–Robertson–Walker space–times which exist for all − ∞ < t < 0 and 0 < t < + ∞ , where in conformally regularized coordinates, these solutions are continuous through the singularity t =0 of space–time, taking on specified data u (0,⋅)= g (⋅) at the singular time.

HADAMARD STATES, ADIABATIC VACUA AND THE CONSTRUCTION OF PHYSICAL STATES FOR SCALAR QUANTUM FIELDS ON CURVED SPACETIME

Quasifree states of a linear Klein-Gordon quantum field on globally hyperbolic spacetime manifolds are considered. After a short mathematical review techniques from the theory of pseudodifferential operators and wavefront sets on manifolds are used to develop a criterion for a state to be an Hadamard state. It is proven that ground- and KMS-states on certain static spacetimes and adiabatic vacuum states on Robertson-Walker spaces are Hadamard states. A counterexample is given which shows that the idea of instantaneous positive energy states w.r.t. a Cauchy surface does in general not yield physical states. Finally, the problem of constructing Hadamard states on arbitrary curved spacetimes is solved in principle.

On the instability of a class of generalized strong curvature singularities

Cosmic censorship hypothesis [1] is a major unsolved problem in classical general relativity. According to this hypothesis singularities occurring in generic space-times should not be naked. This means that there should not be singularities to the future of a regular initial surface that are visible to observers at infinity A mathematically precise statement of the hypothesis is that a space-time should be future asymptotically predictable from a partial Cauchy surface ([2] p. 310).