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Rafael D. Sorkin

Publications and source records attributed to Rafael D. Sorkin.

At least 37 records · Page 2Linked to original sources
arXiv2014

Generalized Causal Set d'Alembertians

We introduce a family of generalized d'Alembertian operators in D-dimensional Minkowski spacetimes which are manifestly Lorentz-invariant, retarded, and non-local, the extent of the nonlocality being governed by a single parameter $ρ$. The prototypes of these operators arose in earlier work as averages of matrix operators meant to describe the propagation of a scalar field in a causal set. We generalize the original definitions to produce an infinite family of ''Generalized Causet Box (GCB) operators'' parametrized by certain coefficients $\{a,b_n\}$, and we derive the conditions on the latter needed for the usual d'Alembertian to be recovered in the infrared limit. The continuum average of a GCB operator is an integral operator, and it is these continuum operators that we mainly study. To that end, we compute their action on plane waves, or equivalently their Fourier transforms g(p) [p being the momentum-vector]. For timelike p, g(p) has an imaginary part whose sign depends on whether p is past or future-directed. For small p, g(p) is necessarily proportional to p.p, but for large p it becomes constant, raising the possibility of a genuinely Lorentzian perturbative regulator for quantum field theory. We also address the question of whether or not the evolution defined by the GCB operators is stable, finding evidence that the original 4D causal set d'Alembertian is unstable, while its 2D counterpart is stable.

hep-th↗
arXiv2013

Spacetime Entanglement Entropy in 1+1 Dimensions

arXiv:1205.2953 defines an entropy for a gaussian scalar field $ϕ$ in an arbitrary region of either a causal set or a continuous spacetime, given only the correlator $\langleϕ(x)ϕ(y)\rangle$ within the region. As a first application, we compute numerically the entanglement entropy in two cases where the asymptotic form is known or suspected from conformal field theory, finding excellent agreement when the required ultraviolet cutoff is implemented as a truncation on spacetime mode-sums. We also show how the symmetry of entanglement entropy reflects the fact that $RS$ and $SR$ share the same eigenvalues, $R$ and $S$ being arbitrary matrices.

hep-th↗
arXiv2013

Does a Quantum Particle Know its Own Energy?

If a wave function does not describe microscopic reality then what does? Reformulating quantum mechanics in path-integral terms leads to a notion of "precluded event" and thence to the proposal that quantal reality differs from classical reality in the same way as a set of worldlines differs from a single worldline. One can then ask, for example, which sets of electron trajectories correspond to a Hydrogen atom in its ground state and how they differ from those of an excited state. We address the analogous questions for simple model that replaces the electron by a particle hopping (in discrete time) on a circular lattice.

quant-ph↗
arXiv2012

A Ground State for the Causal Diamond in 2 Dimensions

We apply a recent proposal for a distinguished ground state of a quantum field in a globally hyperbolic spacetime to the free massless scalar field in a causal diamond in two-dimensional Minkowski space. We investigate the two limits in which the Wightman function is evaluated (i) for pairs of points that lie in the centre of the diamond (i.e. far from the boundaries), and (ii) for pairs of points that are close to the left or right corner. We find that in the centre, the Minkowski vacuum state is recovered, with a definite value of the infrared cutoff. Interestingly, the ground state is not the Rindler vacuum in the corner of the diamond, as might have been expected, but is instead the vacuum of a flat space in the presence of a static mirror on that corner. We confirm these results by numerically evaluating the Wightman function of a massless scalar field on a causal set corresponding to the continuum causal diamond.

hep-th↗
arXiv2012

A Distinguished Vacuum State for a Quantum Field in a Curved Spacetime: Formalism, Features, and Cosmology

We define a distinguished "ground state" or "vacuum" for a free scalar quantum field in a globally hyperbolic region of an arbitrarily curved spacetime. Our prescription is motivated by the recent construction of a quantum field theory on a background causal set using only knowledge of the retarded Green's function. We generalize that construction to continuum spacetimes and find that it yields a distinguished vacuum or ground state for a non-interacting, massive or massless scalar field. This state is defined for all compact regions and for many noncompact ones. In a static spacetime we find that our vacuum coincides with the usual ground state. We determine it also for a radiation-filled, spatially homogeneous and isotropic cosmos, and show that the super-horizon correlations are approximately the same as those of a thermal state. Finally, we illustrate the inherent non-locality of our prescription with the example of a spacetime which sandwiches a region with curvature in-between flat initial and final regions.

hep-th↗
arXiv2012

Expressing entropy globally in terms of (4D) field-correlations

We express the entropy of a scalar field phi directly in terms of its spacetime correlation function W(x,y) = , assuming that the higher correlators are of "Gaussian" form. The resulting formula associates an entropy S(R) to any spacetime region R; and when R is globally hyperbolic with Cauchy surface Sigma, S(R) can be interpreted as the entropy of the reduced density-matrix belonging to Sigma. One acquires in particular a new expression for the entropy of entanglement across an event-horizon. Thanks to its spacetime character, this expression makes sense in a causal set as well as in a continuum spacetime.

hep-th↗
arXiv2011

Ten Theses on Black Hole Entropy

I present a viewpoint on black hole thermodynamics according to which the entropy: derives from horizon "degrees of freedom"; is finite because the deep structure of spacetime is discrete; is "objective" thanks to the distinguished coarse graining provided by the horizon; and obeys the second law of thermodynamics precisely because the effective dynamics of the exterior region is not unitary.

hep-th↗
arXiv2011

Two-Site Quantum Random Walk

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure $μ_n$ on the space of $n$-paths, and the $μ_n$ in turn induce a quantum measure $μ$ on the cylinder sets within the space $Ω$ of untruncated paths. Although $μ$ cannot be extended to a continuous quantum measure on the full $σ$-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of $Ω$ in a systematic way. We begin an investigation of this problem by showing that $μ$ can be extended to a quantum measure on a "quadratic algebra" of subsets of $Ω$ that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the $n$-path space.

quant-ph↗
arXiv2011

Toward a "fundamental theorem of quantal measure theory"

We address the extension problem for quantal measures of path-integral type, concentrating on two cases: sequential growth of causal sets, and a particle moving on the finite lattice Z_n. In both cases the dynamics can be coded into a vector-valued measure mu on Omega, the space of all histories. Initially mu is defined only on special subsets of Omega called cylinder-events, and one would like to extend it to a larger family of subsets (events) in analogy to the way this is done in the classical theory of stochastic processes. Since quantally mu is generally not of bounded variation, a new method is required. We propose a method that defines the measure of an event by means of a sequence of simpler events which in a suitable sense converges to the event whose measure one is seeking to define. To this end, we introduce canonical sequences approximating certain events, and we propose a measure-based criterion for the convergence of such sequences. Applying the method, we encounter a simple event whose measure is zero classically but non-zero quantally.

hep-th↗
arXiv2010

Hilbert Spaces from Path Integrals

It is shown that a Hilbert space can be constructed for a quantum system starting from a framework in which histories are fundamental. The Decoherence Functional provides the inner product on this "History Hilbert space". It is also shown that the History Hilbert space is the standard Hilbert space in the case of non-relativistic quantum mechanics.

quant-ph↗
arXiv2010

Logic is to the quantum as geometry is to gravity

I will propose that the reality to which the quantum formalism implicitly refers is a kind of generalized history, the word history having here the same meaning as in the phrase sum-over-histories. This proposal confers a certain independence on the concept of event, and it modifies the rules of inference concerning events in order to resolve a contradiction between the idea of reality as a single history and the principle that events of zero measure cannot happen (the Kochen-Specker paradox being a classic expression of this contradiction). The so-called measurement problem is then solved if macroscopic events satisfy classical rules of inference, and this can in principle be decided by a calculation. The resulting conception of reality involves neither multiple worlds nor external observers. It is therefore suitable for quantum gravity in general and causal sets in particular.

quant-ph↗
arXiv2009

Light, Links and Causal Sets

After sketching a context in which to seek observable signals of spatio-temporal discreteness, I briefly review the status of the causal set program for quantum gravity, concluding with a simple model for the field produced by a moving charge in a background causal set.

gr-qc↗
arXiv2007

Is the cosmological "constant" a nonlocal quantum residue of discreteness of the causal set type?

The evidence for an accelerating Hubble expansion appears to have confirmed the heuristic prediction, from causal set theory, of a fluctuating and ``ever-present'' cosmological term in the Einstein equations. A more concrete phenomenological model incorporating this prediction has been devised and tested, but it remains incomplete. I will review these developments and also mention a possible consequence for the dimensionality of spacetime.

gr-qc↗
arXiv2007

An exercise in "anhomomorphic logic"

A classical logic exhibits a threefold inner structure comprising an algebra of propositions `A', a space of ``truth values'' `V', and a distinguished family of mappings `phi' from propositions to truth values. Classically A is a Boolean algebra, V=Z_2, and the admissible maps phi:A-->Z_2 are {\it homomorphisms}. If one admits a larger set of maps, one obtains an anhomomorphic logic that seems better suited to quantal reality (and the needs of quantum gravity). I explain these ideas and illustrate them with three simple examples.

quant-ph↗
arXiv2007

Does Locality Fail at Intermediate Length-Scales

If quantum gravity implies a fundamental spatiotemporal discreteness, and if its ``laws of motion'' are compatible with the Lorentz transformations, then physics cannot remain local. One might expect this nonlocality to be confined to the fundamental discreteness scale, but I will present evidence that it survives at much lower energies, yielding for example a nonlocal equation of motion for a scalar field propagating on an underlying causal set.

gr-qc↗
arXiv2007

The Disjointed Thermodynamics of Rotating Black Holes With a NUT Twist

We study the solutions of the Euclidean-signature Einstein equations (gravitational instantons) whose line-element takes the Kerr-bolt form, characterized by three real parameters: a size, a NUT charge, and a spin rate. The exclusion of singularities eliminates most combinations of these parameters, leaving only separated solution-manifolds between which continuous transitions are impossible (The angular velocity divided by the temperature is forced to be a rational multiple of $2π$). This ``quantization'' prevents the free variations presupposed by an equation like $T dS=dM + ΩdJ$, and thereby renders the first law true, false, meaningless, or tautological, depending on how one approaches it.

hep-th↗