arXiv subjects

Rafael D. Sorkin

Publications and source records attributed to Rafael D. Sorkin.

At least 19 recordsLinked to original sources

arXiv2026

Measuring a Quantum Measure Exceeding Unity

The history based formalism known as Quantum Measure Theory (QMT) generalizes the concept of probability-measure so as to incorporate quantum interference. The resulting quantum measure $μ$ is defined for arbitrary events (sets of histories), not just for observables at a fixed moment of time. Thanks to interference effects, $μ$ can exceed unity, exhibiting its non-classical nature in a particularly striking manner. Here, in an optical experiment, we illustrate an ancilla based filtering scheme that gives operational meaning to the quantum measure. For a specific photonic event $E$, we report a measured value of $μ(E)=1.172^{+0.013}_{-0.019}$, which within errors agrees with the theoretical value of $5/4$, while exceeding the maximum value permissible for a classical probability (namely $1$) by $13.32$ upper or $8.89$ lower percentile widths. The directly observed quantity is an ordinary detector probability $p_D\le 1$ (or, with laser light, an equivalent power ratio); the value $μ(E)>1$ is inferred via the calibrated relation $μ(E)=2p_D$ for our filter. If an unconventional theoretical concept is to play a role in meeting the foundational challenges of quantum theory, it seems important to bring it into contact with experiment as much as possible. Our experiment does this for the quantum measure.

quant-ph↗

arXiv2026

From Green Function to Quantum Field

A pedagogical introduction to the theory of a gaussian scalar field which shows firstly, how the whole theory is encapsulated in the Wightman function $W(x,y)=\langleϕ(x)ϕ(y)\rangle$ regarded abstractly as a two-index tensor on the vector space of (spacetime) field configurations, and secondly how one can arrive at $W(x,y)$ starting from nothing but the retarded Green function $G(x,y)$. Conceiving the theory in this manner seems well suited to curved spacetimes and to causal sets. It makes it possible to provide a general spacetime region with a distinguished "vacuum" or "ground state", and to recognize some interesting formal relationships, including a general condition on $W(x,y)$ expressing zero-entropy or "purity".

gr-qc↗

arXiv2023

An intrinsic causality principle in histories-based quantum theory: a proposal

Relativistic causality (RC) is the principle that no cause can act outside its future lightcone, but any attempt to formulate this principle more precisely will depend on the foundational framework that one adopts for quantum theory. Adopting a histories-based (or "path integral") framework, we relate RC to a condition we term "Persistence of Zero" (PoZ), according to which an event $E$ of measure zero remains forbidden if one forms its conjunction with any other event associated to a spacetime region that is later than or spacelike to that of $E$. We also relate PoZ to the Bell inequalities by showing that, in combination with a second, more technical condition it leads to the quantal counterpart of Fine's patching theorem in much the same way as Bell's condition of Local Causality leads to Fine's original theorem. We then argue that RC per se has very little to say on the matter of which correlations can occur in nature and which cannot. From the point of view we arrive at, histories-based quantum theories are nonlocal in spacetime, and fully in compliance with relativistic causality.

quant-ph↗

arXiv2023

An inside view of the tensor product

Given a vector-space $~V~$ which is the tensor product of vector-spaces $A$ and $B$, we reconstruct $A$ and $B$ from the family of simple tensors $a{\otimes}b$ within $V$. In an application to quantum mechanics, one would be reconstructing the component subsystems of a composite system from its unentangled pure states. Our constructions can be viewed as instances of the category-theoretic concepts of functor and natural isomorphism, and we use this to bring out the intuition behind these concepts, and also to critique them. Also presented are some suggestions for further work, including a hoped-for application to entanglement entropy in quantum field theory.

math-ph↗

arXiv2021

Measurements and analysis of response function of cold atoms in optical molasses

We report our experimental measurements and theoretical analysis of the position response function of a cloud of cold atoms residing in the viscous medium of an optical molasses and confined by a magneto-optical trap (MOT). We measure the position response function by applying a transient homogeneous magnetic field as a perturbing force. We observe a transition from a damped oscillatory motion to an over-damped relaxation, stemming from a competition between the viscous drag provided by the optical molasses and the restoring force of the MOT. Our observations are in both qualitative and quantitative agreement with the predictions of a theoretical model based on the Langevin equation. As a consistency check, and as a prototype for future experiments, we also study the free diffusive spreading of the atomic cloud in our optical molasses with the confining magnetic field of the MOT turned off. We find that the measured value of the diffusion coefficient agrees with the value predicted by our Langevin model, using the damping coefficient. The damping coefficient was deduced from our measurements of the position response function at the same temperature.

physics.atom-ph↗

arXiv2020

Lorentzian angles and trigonometry including lightlike vectors

We define a concept of Lorentzian angle that works even when one or both of the directions involved is null (lightlike). Such angles play a role in Regge-Calculus, in the boundary- and corner- terms for the gravitational action, and in the Lorentzian Gauss-Bonnet theorem (for which we provide a proof).

gr-qc↗

arXiv2019

Symmetry-breaking and zero-one laws

We offer further evidence that discreteness of the sort inherent in a causal set cannot, in and of itself, serve to break Poincar{é} invariance. In particular we prove that a Poisson sprinkling of Minkowski spacetime cannot endow spacetime with a distinguished spatial or temporal orientation, or with a distinguished lattice of spacetime points, or with a distinguished lattice of timelike directions (corresponding respectively to breakings of reflection-invariance, translation-invariance, and Lorentz invariance). Along the way we provide a proof from first principles of the zero-one law on which our new arguments are based.

gr-qc↗

arXiv2019

Manifold-Topology from K-Causal Order

To a significant extent, the metrical and topological properties of spacetime can be described purely order-theoretically. The $K^+$ relation has proven to be useful for this purpose, and one could wonder whether it could serve as the primary causal order from which everything else would follow. In that direction, we prove, by defining a suitable order-theoretic boundary of $K^+(p)$, that in a $K$-causal spacetime, the manifold-topology can be recovered from $K^+$. We also state a conjecture on how the chronological relation $I^+$ could be defined directly in terms of $K^+$.

gr-qc↗

arXiv2018

Cosmological Tests of Everpresent $Λ$

Everpresent $Λ$ is a cosmological scenario in which the observed cosmological "constant" $Λ$ fluctuates between positive and negative values with a vanishing mean, and with a magnitude comparable to the critical density at any epoch. In accord with a longstanding heuristic prediction of causal set theory, it postulates that $Λ$ is a stochastic function of cosmic time that will vary from one realization of the scenario to another. Herein, we consider two models of "dark energy" that exhibit these features. Via Monte Carlo Markov chains, we explore the space of cosmological parameters and the set of stochastic realizations of these models, finding that Everpresent $Λ$ can fit the current cosmological observations as well as the $Λ$CDM model does. Furthermore, it removes observational tensions with $Λ$CDM, for low redshift measurements of Hubble constant, and the Baryonic Acoustic Oscillations (BAO) in Lyman-$α$ forest at $z\sim 2-3$. However, we also find that Everpresent $Λ$ does not significantly help with the growth of ultramassive black holes at high redshift, and the Lithium problem in Big Bang Nuclesynthesis. Future measurements of "dark energy" at high redshifts will further test the viability of Everpresent $Λ$ as an alternative to the $Λ$CDM cosmology.

gr-qc↗

arXiv2018

Entanglement Entropy in Causal Set Theory

Entanglement entropy is now widely accepted as having deep connections with quantum gravity. It is therefore desirable to understand it in the context of causal sets, especially since they provide in a natural manner the UV cutoff needed to render entanglement entropy finite. Formulating a notion of entanglement entropy in a causal set is not straightforward because the type of canonical hypersurface-data on which its definition typically relies is not available. Instead, we appeal to the more global expression given in arXiv:1205.2953 which, for a gaussian scalar field, expresses the entropy of a spacetime region in terms of the field's correlation function within that region (its "Wightman function" $W(x,x')$). Carrying this formula over to the causal set, one obtains an entropy which is both finite and of a Lorentz invariant nature. We evaluate this global entropy-expression numerically for certain regions (primarily order-intervals or "causal diamonds") within causal sets of 1+1 dimensions. For the causal-set counterpart of the entanglement entropy, we obtain, in the first instance, a result that follows a (spacetime) volume law instead of the expected (spatial) area law. We find, however, that one obtains an area law if one truncates the commutator function ("Pauli-Jordan operator") and the Wightman function by projecting out the eigenmodes of the Pauli-Jordan operator whose eigenvalues are too close to zero according to a geometrical criterion which we describe more fully below. In connection with these results and the questions they raise, we also study the "entropy of coarse-graining" generated by thinning out the causal set, and we compare it with what one obtains by similarly thinning out a chain of harmonic oscillators, finding the same, "universal" behaviour in both cases.

hep-th↗

arXiv2017

A quantum diffusion law

We analyse diffusion at low temperature by bringing the fluctuation-dissipation theorem (FDT) to bear on a physically natural, viscous response-function R(t). The resulting diffusion-law exhibits several distinct regimes of time and temperature, each with its own characteristic rate of spreading. As with earlier analyses, we find logarithmic spreading in the quantum regime, indicating that this behavior is robust. A consistent R(t) must satisfy the key physical requirements of Wightman positivity and passivity, and we prove that ours does so. We also prove in general that these two conditions are equivalent when the FDT holds. Given current technology, our diffusion law can be tested in a laboratory with ultra cold atoms.

cond-mat.stat-mech↗

arXiv2017

Is the spacetime metric Euclidean rather than Lorentzian?

My answer to the question in the title is "No". In support of this point of view, we analyze some examples of saddle-point methods, especially as applied to quantum "tunneling" in nonrelativistic particle mechanics and in cosmology. Along the way we explore some of the interrelationships among different ways of thinking about path-integrals and saddle-point approximations to them.

gr-qc↗

arXiv2017

An equation for a time-dependent profit rate

Taking as a hypothesis a form of the labour theory of value, and $without$ $assuming$ $equilibrium$, we derive an equation that yields the profit-rate $π$ as a function of time. For a mature economy, $π(t)$ reduces to the product of two factors: ($i$) a certain $retarded$ $average$ of the sum of the growth-rates of productivity and of the size of the labour-force measured by hours worked, and ($ii$) the ratio of the current rate of surplus value to its own retarded average. We also suggest an empirical test of the equation.

econ.GN↗

arXiv2017

Scalar Field Theory on a Causal Set in Histories Form

We recast into histories-based form a quantum field theory defined earlier in operator language for a free scalar field on a background causal set. The resulting decoherence-functional resembles that of the continuum theory. The counterpart of the d'Alembertian operator is nonlocal and is a generalized inverse of the discrete retarded Green function. We comment on the significance of this and we also suggest how to include interactions.

gr-qc↗

arXiv2016

Gravitational action with null boundaries

We present a complete discussion of the boundary term in the action functional of general relativity when the boundary includes null segments in addition to the more usual timelike and spacelike segments. We confirm that ambiguities appear in the contribution from a null segment, because it depends on an arbitrary choice of parametrization for the generators. We also show that similar ambiguities appear in the contribution from a codimension-two surface at which a null segment is joined to another (spacelike, timelike, or null) segment. The parametrization ambiguity can be tamed by insisting that the null generators be affinely parametrized; this forces each null contribution to the boundary action to vanish, but leaves intact the fredom to rescale the affine parameter by a constant factor on each generator. Once a choice of parametrization is made, the ambiguity in the joint contributions can be eliminated by formulating well-motivated rules that ensure the additivity of the gravitational action. Enforcing these rules, we calculate the time rate of change of the action when it is evaluated for a so-called "Wheeler-deWitt patch" of a black hole in asymptotically-anti de Sitter space. We recover a number of results cited in the literature, obtained with a less complete analysis.

hep-th↗

arXiv2015

When is an area law not an area law?

Entanglement entropy is typically proportional to area, but sometimes it acquires an additional logarithmic pre-factor. We offer some intuitive explanations for these facts.

hep-th↗

arXiv2015

To What Type of Logic Does the "Tetralemma" Belong?

Although the so called tetralemma might seem to be incompatible with any recognized scheme of logical inference, its four alternatives arise naturally within the anhomomorphic logics which have been proposed in order to accommodate certain features of microscopic (i.e. quantum) physics. This suggests that the possibility of similar, "non-classical" logics might have been recognized in India at the time when Buddhism arose.

math.LO↗

arXiv2015

Onset of the Asymptotic Regime for Finite Orders

We describe a Markov-Chain-Monte-Carlo algorithm which can be used to generate naturally labeled n-element posets at random with a probability distribution of one's choice. Implementing this algorithm for the uniform distribution, we explore the approach to the asymptotic regime in which almost every poset takes on the three-layer structure described by Kleitman and Rothschild (KR). By tracking the n-dependence of several order-invariants, among them the height of the poset, we observe an oscillatory behavior which is very unlike a monotonic approach to the KR regime. Only around n=40 or so does this "finite size dance" appear to give way to a gradual crossover to asymptopia which lasts until n=85, the largest n we have simulated.

math.CO↗