Foundations of space-time theories

Recently, my principal interests have concerned topics arising when one studies General Relativity in conjunction with the geometrized reformulation of Newtonian gravitation (sometimes called Newton-Cartan theory). There are several kinds of project here that I am interested in. One involves looking at how questions that arise in the context of one theory are (or are not) answered when one moves to the other theory. To take a concrete example, for a long time physicists and philosophers have been interested in the status of inertial motion in GR. Einstein and others (most recently Harvey Brown and some of his colleagues at Oxford) have suggested that inertial motion, described by the geodesic principle, has a special status in GR. This status is derived from the fact that the principle can be proven as a theorem, starting from the spacetime structure and some other central principles of the theory. There is no question that the geodesic principle in GR does have the ascribed status. But emphasizing this status as a point of contrast from other theories suggests a question: what is the status of the geodesic principle in geometrized Newtonian gravitation? It turns out that there is a precise sense in which here, too, the geodesic principle has the status of a theorem (see “The Motion of a Body in Newtonian Theories”). In fact, the relevant theorems are strikingly similar to one another, mutatis mutandis (see “On the Status of the Geodesic Principle in Newtonian and Relativistic Physics”).

Trying to better understand the role that the geodesic principle plays in general relativity and geometrized Newtonian gravitation has led me to a number of other questions concerning the status of other central principle of these theories, including the principles used as assumptions when one proves the geodesic principle as a theorem. These questions are of particular interest in the context what I have called the “puzzle ball view” of physical theories (or perhaps better, the “puzzle ball conjecture”), which holds that the foundations of a physical theory are best understood as an inter-dependent networks of mutually interdependent principles and assumptions—pieces that interlock as in a spherical puzzle (see “Inertial motion, explanation, and the foundations of classical spacetime theories”). Fore instance, I have recently been interested in claims to the effect that Einstein’s equation (and respectively, the geometrized Poisson equation) can be derived from other principles of GR (and respectively, GNG), and questions concerning the status of the conservation condition in geometrized Newtonian gravitation.

A closely related, but distinct, kind of project involves slightly more general questions that arise when one tries to understand the relationship between standard Newtonian gravitation, geometrized Newtonian gravitation, and GR. There are various well-known results in this neighborhood already: for instance, it is possible to prove a pair of geometrization/degeometrization theorems that systematically and precisely relate standard Newtonian gravitation with geometrized Newtonian gravitation. Meanwhile, there is a version of geometrized Newtonian gravitation that can be recovered in a precise way as the limit of a sequence of models of GR. It seems to me that careful study of these relationships can be quite fruitful. For instance, I have recently written a paper in which I explore a certain kind of cross-theoretical explanation that one can give by studying how certain characteristic expressions arise in the two-step limit from GR to standard Newtonian gravitation (see “On (Some) Explanations in Physics”). As an explicit example, I offer an explanation of why inertial and (passive) gravitational mass are equal in Newtonian gravitation. I argue that this kind of explanation has been overlooked by the standard philosophical accounts of explanation, but that it is important to the practice of physics because demands for the kind of explanation I describe shape inquiry into future theories. Cross-theoretic explanatory demands are useful for clarifying the extent to which we recognize certain shortcomings of our current theories, but nonetheless expect future theories to be sufficiently inter-translatable with current theories to answer questions that are expressed in the language of the current theories. There are also a cluster of technical questions that arise when one considers the details of the limiting process described above that I have begun to work on. I have also written a paper in which I argue that whether or not Newtonian gravitation and geometrized Newtonian gravitation are theoretically equivalent is a subtle issue that depends sensitively on how one understands the status of the gravitational potential in Newtonian gravitation (see “Are Newtonian gravitation and geometrized Newtonian gravitation theoretically equivalent?”).

Category theory in the philosophy of science

Recently, Hans Halvorson and I have been collaborating on adapting tools from category theory for use in the philosophy of science, particularly for the purposes of understanding the relations between physical theories. This project is inspired by recent work in categorical logic (notably, Awodey and Forssell’s recent work on the categorical duality between syntax and semantics), and builds on Hans’ recent paper “What Scientific Theories Could Not Be” (2012) and my paper “Are Newtonian gravitation and geometrized Newtonian gravitation theoretically equivalent?”. Hans’ paper might be best understood as an argument that, at least for certain purposes, the technical tools associated with the so-called “semantic view” of theories are inadequate, and that category theory is a natural place to look for the more sophisticated tools one requires. Meanwhile, in my paper, I argue that a certain criterion of theoretical equivalence proposed by Clark Glymour (and, it seems, inspired by the semantic view) fails to capture an important sense in which two theories may be equivalent. I then show how one can capture the difference between Glymour’s criterion and an alternate criterion I propose very naturally by representing theories as the category of their models, with arrows corresponding to equivalences between models. This is at least one example in which category theory seems to help clarify an important question in the philosophy of science; presently, Hans and I are working to further develop these ideas by exploring how these tools might be applied more broadly.

Dressed Interacting Ground States (DIGS)

Although most of my recent work has been on foundational topics in space-time physics, I have continued to work on some of the features exhibited by a class of five-level atomic systems I invented in collaboration with Chris Search at Stevens Institute of Technology. These systems, which we dubbed Dressed Interacting Ground State (DIGS) systems, consist of an excited state along with two pairs of ground states, and . Transitions within the pairs are driven by rf/microwave control fields while transitions between the pairs are assumed to be forbidden; the excited state is strongly coupled to by a resonant control laser. We have principally been interested in the optical response of the atom as measured by a weak probe field near resonance with the transition. The system might be implemented in, for instance, cold atoms.

Our early work on this system concerned its the dispersive properties. My collaborators and I were able to show that under certain realistic parameter choices, one finds a wide frequency band of fully controllable dispersion and vanishing absorption. In fact, one can control the sign and magnitude of the dispersion by changing the relative populations of the ground states and , with maximum normal dispersion comparable to that exhibited by electromagnetically induced transparency and with maximum anomalous dispersion of the same order of magnitude (though of course with opposite sign). This latter result is two orders of magnitude or more than the largest anomalous dispersion yet observed, in a system with experimental difficulty comparable to EIT. The system also exhibits narrower windows in which the normal dispersion can be made several orders of magnitude larger than in standard EIT, again with vanishing absorption. For more on the dispersive properties of DIGS systems see “Lossless anomalous dispersion and an inversionless gain doublet via dressed interacting ground states” and “Quantum control of dispersion in electromagnetically induced transparency via interacting dressed ground states”. In collaboration with Markku Jääskeläinen, we also showed that one can instantiate the DIGS system in a double well Bose-Einstein condensate, with both lasers passing through a single well, and with tunneling between the two wells taking the place of the rf/microwave fields (see “Quantum control of electromagnetically induced transparency dispersion via atomic tunneling in a double-well Bose-Einstein condensate”). This system is of some potential foundational interest, since the optical response of the probe field depends on manifestly non-local coherences between spacelike separated atoms. The above work formed the basis of my physics dissertation, which also contains considerable background material on the theory of atom-light interactions.

More recently, Chris and I, in collaboration with Steven Sagona-Stophel, have been studying the refractive properties of the system. We have shown that the DIGS system exhibits regions of both coherently enhanced () and suppressed () index of refraction, with vanishing absorption. In fact, by varying the detuning of the rf field driving the transition, one can create a broad frequency window in which the absorption is everywhere zero and the index of refraction is sustained at a constant value that is substantially larger or smaller than 1, which means one can exhibit large refraction in either direction without pulse reshaping for pulses with appropriate bandwidth. We have also shown that one can produce frequency bands where both the index of refraction is less than 1, and the dispersion is anomalous, leading to simultaneously superluminal group and phase velocities. For more on the refractive properties of DIGS systems, see “Index of refraction engineering in five level DIGS atoms”.