Rafael Sorkin

[Most references still to be added] OVERVIEW OF MY WORK More than anything else, my work in physics has been guided by a desire to overcome the disunity which characterizes our present conception of nature, and which shows itself most obviously in our failure to have reconciled our best theory of spacetime structure (general relativity) with the best dynamical framework we know of for describing the behavior of matter on small scales (quantum field theory). Ever since I encountered it in graduate school, the greater part of my efforts has been devoted directly or indirectly to this problem, which can also be viewed as the task of completing the twin revolutions initiated early in the last century in connection with the study of the very small (atoms), the very fast (light), and the very big (astronomy). In addressing this "quantum gravity" problem, I first explored the idea ("Regge Calculus") that spacetime can be represented as a simplicial complex, an essentially topological notion which today provides the basis for the causal dynamical triangulation and spin-foam approaches to quantum gravity. Regarded not just as an artificial finite-differencing of the Einstein equations, but as a hypothesis about the inner basis of spacetime, Regge Calculus fit in well with the expectation that the continuum is only an effective description of some more fundamental -- discrete -- structure. However, after some work, I came to feel that if topology was to be the ultimate source of spatio-temporal geometry, then a more flexible and natural structure than the simplicial complex was the finite topological space. By a kind of duality, such a space can also be regarded as a partially ordered set or "poset", and this links topology to the arguably more basic concept of order. But in the end the poset-type of topology seemed no more satisfactory as a substitute for spacetime than the simplicial complex had been. At best it could produce a metric-space with all distances positive, but not the kind of Lorentzian geometry which characterizes the spacetime of special and general relativity. Understood, however, not topologically but temporally or causally, an order-relation does contain the seeds of Lorentzian geometry, and this led me, finally, to the idea of the causal set as the deep structure of spacetime. At present, this is the road I am taking, and a separate web page describes causal set theory in more depth, including its recent great progress and current prospects. Although I no longer believe that topology enjoys a fundamental status, that does not mean that topological questions are unimportant for quantum gravity, and exploring the novel topological possibilities inherent in curved spacetime has always been an important strand in my work. Emerging from an underlying discreteness, potential topological effects will first arise on those length-scales -- possibly still close to Planckian -- where the spacetime manifold itself begins to make sense. In this regime spacetime topology possesses kinematic resources sufficient to reproduce all the degrees of freedom of the standard model. For example, extra spatial dimensions could give rise to gauge fields in the "Kaluza-Klein" manner, while topological geons (quasi-local excitations of the spatial topology) could serve as quarks, at least as far as kinematics is concerned. In particular, geons can exhibit fermionic statistics for purely topological reasons, without the need to introduce any fundamental fermions into the story. In this way, the causal set in its role as the source of an emergent spacetime topology could in principle be all there is. Beyond the potential to reproduce well established types of fields and particles, spacetime topology might also give rise to novel behaviours not so far observed or even theorized in other contexts, for example certain types of exotic statistics. But the study of spacetime topology, and especially of its temporal variation, also seems to me to have furnished important clues to the nature of quantum gravity in general. For example, my work with geons led me to believe that the spin-statistics correlation will break down unless quantum gravity provides for transitions from one topology to another. In the contrary case of frozen spatial topology, geons (including Kaluza-Klein monopoles) could not be pair-produced, and with the loss of that process one would lose the source from which, plausibly, the correlation between spin and statistics springs in general. Recovery of spin-statistics for geons thus points at the necessity for topology change, and in the context of the gravitational path integral, it turns out to place definite constraints on the amplitudes associated with the cobordisms involved in pair-creation. No less importantly, the need for topology-change reinforces the belief that a path-integral formulation of quantum mechanics is the only one that can meet the needs of quantum gravity, an encouraging conclusion from the point of view of causal sets, which seem to admit no other form of quantal dynamics. In this connection, I should also mention Skyrmions and anyons, two other instances of topological phenomena I have worked on. In the case of Skyrmions, the topology in question is that of the group SU2 or SU3, regarded as the "target manifold" of a meson field. The Skyrmion itself is a topological excitation of the vacuum to be identified with the nucleon or one of the other baryons, and fermionic statistics arises in close analogy to how it does for topological geons in quantum gravity. Even dynamically, the analogy with geons remains close, with the exception that (thanks to the absence of gauge-issues) one can actually compute the energy-spectrum and understand how the uncertainty principle can stabilize the Skyrmion against collapse. The resulting zero-parameter fit to the low lying baryon masses turned out to be surprisingly accurate. Thinking about geon statistics also led me to try to analyze the meaning of particle-statistics in general, as a kind of nonlocal interaction among particles which is expressed topologically. As a byproduct I noticed the possibility of so-called anyons, which actually had been pointed out a few years earlier by Leinaas and Myrheim. Along with topology and topology-change, black hole entropy has long interested me as another clue to the nature of quantum gravity. When I first learned that a black hole carries entropy and that (up to a coefficient of proportionality of order unity) this entropy equals the surface-area of the black hole in Planck units, the thought that suggested itself almost immediately was that this was telling us something important about the micro-structure of spacetime because the most natural reading of such an area law is that the horizon is somehow carrying about one bit of information per unit (Planck) area, and this in turn is most easily explained if the deep structure of spacetime is discrete so that the horizon is actually composed of large but still finite number of "molecules" of Planckian magnitude. In relation to causal sets, the lesson is that the fundamental discreteness scale must be near to 1e-32 cm, consistent with many other indications that sizes shorter than this lack physical meaning. (The subsequently popular idea of "holography" must have sprung from a train of thought similar to mine, but it has led in different directions and does not seem to have shed much light on the structure of the horizon, or on how the entropy can be "located there".) A correct counting of horizon "degrees of freedom" must of course be conducted quantum mechanically. To that end, I computed what is now called the entanglement entropy across the horizon, regarded as a surface separating the outside of the black hole from the inside. The calculation did indeed yield an entropy proportional to the area, or rather an entropy given by the horizon area in units of the cutoff that one needed to render finite an otherwise divergent answer. This result, together with others of a similar nature, furnishes to my way of thinking the single best argument in favor of spatio-temporal discreteness, and the most direct indication so far of where the discreteness-scale lies. If, for example, the kinematic counting of "horizon molecules" in causal set theory were to find a dynamical justification, one would be able to derive the coefficient of proportionality relating the true discreteness scale to the nominal Planck length of 0.81019e-32 cm. [mention fractal horizon, entropy as shapes] [mention Lambda prediction, or refer to accompanying web page on causets] [unimodular gravity?] The last strand of my work that I will sketch here concerns the path-integral and its interpretation. Or maybe it would be better to say that it concerns the possibility of re-formulating quantum mechanics entirely as a theory of quantal histories, without ever needing to call on state-vectors, measurements, or external agents as fundamental notions. But what purpose would such a re-formulation serve, and would it aid in the quest for quantum gravity? Part of the answer which I've already alluded to above is that quantum gravity seems to resist the Hamiltonian framework of "canonical quantization". This statement reflects first of all the so-called problem of time and the need to deal with diffeomorphism-invariant quantities, but when, as with causets, we add to the mix a discreteness which is not only spatial but temporal, the contradiction with the Hamiltonian as a generator of continuous time-evolution becomes severe. And when we allow for topology change (or do away with the continuum altogether) any thought of quantum gravity as a theory of operator-valued fields on a fixed manifold also fades out. A second part of the answer harks back directly to the question of the unity of physics. The lack of a theory of quantum gravity is an obvious symptom of the disunity in our present understanding, but it is not the only symptom one can point to. Another notorious instance is the "cut" which quantum mechanics supposedly forces on us, that severs the macroscopic world of classical physics from the microscopic world of quantal objects and processes. (Some authors have taken this cut so seriously as to deny reality to the micro-world altogether.) This issue of the quantal cut might appear to be remote from quantum gravity, but insofar as "quantum vs. gravity" reduces to "micro vs. macro", we are dealing with something rather similar in both situations; hence one might suspect that the two problems are in fact closely related. Even if such suspicions are mistaken, though, the practical need for a histories formulation remains, and further reasons to desire a more realistic re-formulation of quantum mechanics crop up when one tries to generalize to the quantal case the condition of "Bell Causality" that figures so heavily in connection with classical models of "sequential growth" for causal sets. How then does the path-integral offer an alternative to the textbook formalism of state-vectors, Hamiltonians, and external observers? A first answer is that from the path integral one can derive a functional mu_quantum -- the quantal measure -- which directly furnishes the probability of any desired "instrument-event" E. (This measure is closely related to the so called decoherence functional.) In saying this, I am presupposing that the Born rule (or rule of thumb!) is correct, and then just taking note of the fact that the Bornian probabilities for any specified set of "pointer readings" are furnished directly by mu_quantum, without any appeal to Schroedinger evolution of the wave-function or its "collapse" during the measurement. In this way mu_quantum is analogous to the classical measure mu_classical that furnishes the probability of a set of histories -- an "event" -- in the case of a purely classical stochastic process like diffusion or Brownian motion. If one construes the path-integral in this way, namely as a generalized measure on a space of "histories", then one sees not only how quantal processes differ from classical stochastic processes, but also how closely the two resemble each other, the primary difference being simply that mu_classical and mu_quantum satisfy different sum-rules. The former obeys a "2-slit" sum-rule that expresses the absence of interference between alternative histories, the latter obeys a weaker, "3-slit" sum-rule that expresses the absence of "higher order" interference beyond pairwise. (This 3-slit sum-rule, which reflects the quadratic nature of Bornian probabilities, has now been tested directly in a literal 3-slit experiment with individual photons.) The formal framework I have just sketched rests mathematically on a space of histories and a notion of integration thereon that allows one to compute the quantal measure mu(A) for any (sufficiently regular) subset of the history-space. Satisfactory as this framework is in some ways, the interpretation of mu(A) as a probability still does not take us beyond the realm of "pointer events" that occur in laboratory instruments or other macroscopic objects. But the laboratory, or even the observatory, is not the place where we expect to encounter quantum gravity. There are no laboratories deep inside black holes or in the very early universe. Not instrument-events, but events like the "big bang" are the natural domain of quantum gravity, and one would like to be able to reason about these events in direct physical terms. When one ventures into such realms, however, the physical significance of the quantal measure becomes uncertain: the Born rule loses its force and, because of interference, one can no longer interpret mu(A) as a probability in any ordinary sense. The question then is whether mu admits of a broader interpretation that remains serviceable outside the city limits of Copenhagen. (If it does not, then perhaps an entirely different type of dynamical framework will be called for.) My current belief is that the quantal-measure (meaning in effect the path-integral) does have a direct microscopic significance, not as a probability per se, but as an arbiter of whether a given event can occur at all. In other words, we concede that an event A of vanishing measure mu(A) will never occur. Drawing out the implications of this "preclusion principle", one soon finds that it conflicts with the classical conception of reality as a single history (a single point in the "history space" over which the path-integral integrates). Of course, such a contradiction with classical conceptions of reality is just what one would have expected, but an escape route also opens up. It turns out that -- at least so far -- the preclusion principle does not seem to conflict with a modified conception of reality that replaces the single classical history with a set of one or more histories. Such a set is a special case of what is called an "anhomomorphic coevent", and the new point of view is that reality is best described by such a coevent. (In the resulting re-formulation of quantum theory, it is most natural to reason about events using rules of inference that differ from those of classical logic. One could even regard these modified rules as the essence of the new formulation, but one should not confuse them with what has previously been called "quantum logic".) We thus acquire an interpretation of the quantal measure in conjunction with a particular answer to the question, What is quantum mechanics telling us about the nature of reality? As a byproduct of this development, one solves the "measurement problem", or more properly, one obtains a solution of that problem if instrument-events can be proved to obey a certain separability condition. It remains to be seen whether this condition can be established in sufficient generality. It also remains to be seen whether the new formulation will be be able to accomplish the task for which it was ultimately intended, namely to provide a more precise framework for thinking about quantum gravity, and thereby to help clear away some of the obstacles standing between us and that theory. [mention QBC (Quantal Bell Causality) here or above?]