Quantum Foundations

In reference [1] R. D. Sorkin investigated a formulation of quantum mechanics as a generalized measure theory. Quantum mechanics computes probabilities from the absolute squares of complex amplitudes, and the resulting interference violates the (Kolmogorov) sum rule expressing the additivity of probabilities of mutually exclusive events.However, there is a higher order sum rule that quantum mechanics does obey, involving the probabilities of three mutually exclusive possibilities. We could imagine a yet more general theory by assuming that itviolates the next higher sum rule.An experiment is in progress in our laboratory which sets out to test the validity of this second sum rule by measuring the interference patterns produced by three slits and all the possible combinations of those slits being open or closed. We use either attenuated laser light or a heralded single photon source (using parametric down conversion) combined with single photon counting to confirm the single photon character of the measured light. We will show results that bound the possible violation of the second sum rule and will point out ways toobtain a tighter experimental bound.[1] R. D. Sorkin, Quantum Mechanics as Quantum Measure Theory,Mod. Phys. Lett. A 9, 3119 (1994).

It has sometimes - though usually informally - been suggested that the psychological arrow can be reduced to the thermodynamic arrow through information processing properties of the brain. In this talk we demonstrate that this particular suggestion cannot succeed, as, insofar as information processing (at least in the sense of a classical computer) has an arrow of time, it is not governed by the thermodynamic arrow.

In contrast to Heisenberg\'s position-momentum uncertainty relation, the status of the time-energy uncertainty relation has always remained dubious, For example, it is often said that \'time\' in quantum theory is not an observable and not represented by a self-adjoint operator. I will review the background of the problem and propose a view on the uncertainty relations in which the cases of position-momentum and time-energy can be treated in the same way.

In this talk I will discuss a feature of quantum state evolution in a relativistic spacetime, the feature that David Albert has recently dubbed \'non-narratability.\' This is: a complete state history given along one foliation does not always, by itself (that is, without specification of the dynamics of the system), determine the history along another foliation. The question arises: is this a deep distinction between quantum and classical state evolution, that deserves our fuller attention? I will discuss some results relevant to this question.

It has been a common viewpoint that the process of quantization ought to replace the singularities of classical general relativity by some chaotic-looking structure at the scale of the Planck length. In this talk I shall argue that whereas this is to be expected at black-hole singularities, Nature\'s true picture of what goes on at the Big Bang is very different, where clocks cannot exist and the conformal geometry is completely smooth.

The underlying motivation for rejecting Everett\'s many-worlds interpretation of quantum mechanics and instead exploring single-world interpretations is to make physical theory concordant with human experience. From this perspective, the wave function collapse and Bohm-de Broglie interpretations are anthropocentric in origin. But this does not lessen their importance. Indeed accounting for our human experience of the physical world is a key element of any physical theory. This is no less true for the theory of time where accounting for the anthropocentric notion of a unidirectional flow of time is a challenge. In this talk we examine a peculiar time asymmetry that may shed some light on this problem.The matter-antimatter arrow of time, which is associated with the weak force in neutral Kaon decay, has been an enigma for 40 years. While other arrows (cosmological, electromagnetic, thermodynamic and psychological) have been linked together, the matter-antimatter arrow stands alone. It is often regarded as having a negligible effect on time in our daily lives. The main reason for this view appears to be the relatively small violation of the Charge-Parity conjugation invariance (CP) involved. However the smallness of the violation is not necessarily an obstacle to the manifestation of macroscopic effects. For example, a small difference in a quantum-state fidelity for a single particle leads to a difference which grows exponentially with the number of particles. So provided sufficient numbers of particles are involved such a violation could yield significant effects.We examine the effect of the violation of CP invariance on the dynamics of a large system such as the universe. Provided the CPT theorem holds, the CP violation is equivalent to a violation of time reversal invariance (T). We impose the constraint that the violation should equivalent in both directions of time (past and future) with respect to the present. This implies that if H is the Hamiltonian for one direction of time, then THT the Hamiltonian for the opposite direction. We will see that any given quantum state a> that represents the present of our part of the universe is closer to its evolved state a+> in the future compared to its retro evolved state a-> in the past. In other words, our present state is more likely to be extended (slightly) into the future than the past. We will see that the end result is a never-ending extension of the present into the future. Moreover for a collection of a million neutral kaons, the fidelity between the present state and a slightly future-evolved state is a billion times larger than the fidelity between the present and an equivalent retro-evolved state. In this context, the seemingly insignificant kaons appear to be responsible for our anthropocentric view of moving through time.

The unparalleled empirical success of quantum theory strongly suggests that it accurately captures fundamental aspects of the workings of the physical world. The clear articulation of these aspects is of inestimable value --- not only for the deeper understanding of quantum theory in itself, but for its further development, particularly for the development of a theory of quantum gravity. However, such articulation has traditionally been hampered by the fact that the quantum formalism consists of postulates expressed in an abstract mathematical language to whose elementary objects (complex vectors and operators) our physical intuition cannot directly relate. Recently, there has been growing interest in elucidating these aspects by expressing, in a less abstract mathematical language, what we think quantum theory might be telling us about how nature works, and trying to derive, or reconstruct, quantum theory from these postulates. In this talk, I describe a very simple reconstruction of the finite- dimensional quantum formalism. The derivation takes places with a classical probabilistic framework equipped with the information (or Fisher-Rao) metric, and rests upon a small number of elementary ideas (such as complementarity and global gauge invariance). The complex structure of quantum formalism arises very naturally. The derivation provides a number of non-trivial insights into the quantum formalism, such as the extensive nature of the role of information geometry in determining the quantum formalism, and the importance (or lack thereof) of assumptions concerning separated systems.

A reference frame can be treated as a physical quantum object internal to the theory. Quantum reference frames whose size, and therefore accuracy, are bounded in some way necessarily limit one\'s ability to prepare states and to perform quantum operations and measurements on a system. The nature of these limitations is similar in many ways to that of decoherence. We investigate how a quantum reference frame of bounded size can be \'dequantized\', i.e., treated as external to the quantum formalism, in such a way as to induce an effective decoherence on any system described relative to it. In particular, we show that this decoherence has an interpretation as a lack of classical information about an ideal (infinite size) reference frame.

It is widely believed that the dynamical mechanism of decoherence plays a key role in understanding the emergence of classicality from quantum systems, via the environment-induced superselection of a preferred set of subsystem states, the density matrices for which are approximately diagonal in the pointer basis. In this talk, I prove that the vast majority of subsystems do *not* exhibit this behavior, regardless of the Hamiltonian. This shows that the emergence of classicality is highly state-dependent (as suggested by related work of Hartle and others).

In deBroglie-Bohm theory the quantum state plays the role of a guiding agent. In this seminar we will explore whether this is a universal feature shared by all hidden variable theories, or merely a peculiarity of the deBroglie-Bohm theory. We present the bare bones of a theory in which the quantum state represents a probability distribution and does not act as a guiding agent. The theory is also psi-epistemic according to Spekken\'s and Harrigan\'s definition. For simplicity we develop the model for a 1D discrete lattice but the generalization to higher dimensions is straightforward. The ontic state consists of a definite particle position and in addition possible non-local links between spatially separated lattice points. These non-local links comes in two types: directed links and non-directed links. Quantum superposition manifests itself through these links. Interestingly, this ontology seems to be the simplest possible and immediately suggested by the structure of quantum theory itself. For N lattice points there are N*3^(N(N-1)) ontic states growing exponentially with the Hilbert space dimension N as expected. We further require that the evolution of the probability distribution on the ontic state space is dictated by a master equation with non-negative transition rates. It is then easy to show that one can reproduce the Schroedinger equation if an only if there are positive solutions to a gigantic system of linear equations. This is a highly non-trivial problem and whether there exists such positive solutions or not is not clear at the moment. We end by speculating how one might incorporate gravity into this theory by requiring permutation invariance of the dynamical evolution law.

One approach to the problem of time in canonical quantum gravity is to use correlations between a carefully chosen physical system and all other physical systems to provide a simulacrum of time. Time emerges as an ordering of correlated measurement results. In many ways this is an echo of an idea introduced by Poincare to give a geometric description of dynamical systems. Pullin and Gambini have addressed some objections to this approach using a consistent discretization, but in so doing introduce an intrinsic decoherence mechanism into physical theories. In this talk I will show how this approach leads to a simple form of intrinsic decoherence that has possible experimental consequences, among which are modifications to the dispersion relations of the electromagnetic field. This form of decoherence enables the emergence of semiclassical behavior of large systems.

The presumed irreversibility of quantum measurements (whatever they are) leads, in conventional approaches to quantum theory, to an asymmetry between state preparation and post-selection. Is it possible that a trajectory can be predicted from the former, yet not inferred from the latter? Especially in light of the exciting applications of non-unitary operations (i.e., postselection) in quantum information, it becomes timely to reconsider how much one can say about a post-selected subensemble. I will review the weak-measurement formalism of Aharonov, Vaidman et al., and discuss some applications and extensions. These will include a proposed experiment to study the duration of the tunneling process (a question controversial since the 1930s) and a recently completed experiment aiming to \'resolve\' Hardy\'s retrodiction paradox.