Quantum Physics

The fact that quantum mechanics admits exact uncertainty relations is used to motivate an ‘exact uncertainty’ approach to obtaining the Schrödinger equation. In this approach it is assumed that an ensemble of classical particles is subject to momentum fluctuations, with the strength of the fluctuations determined by the classical probability density [1]. The approach may be applied to any classical system for which the Hamiltonian is quadratic with respect to the momentum, including all physical particles and fields [2]. The approach is based on a general formalism that describes physical ensembles via a probability density P on configuration space, together with a canonically conjugate quantity S [3]. Quantum and classical ensembles are particular cases of interest, but one can also ask more general questions within this formalism, such as (i) Can one consistently describe interactions between quantum and classical systems? and (ii) Can one obtain local nonlinear modifications of quantum mechanics? These questions will be briefly discussed, with respect to measurement interactions and spin-1/2 systems respectively. 1. M.J.W. Hall and M. Reginatto, “Schroedinger equation from an exact uncertainty principle”, J. Phys. A 35 (2002) 3289 (http://lanl.arxiv.org/abs/quant-ph/0102069). 2. M.J.W. Hall, “Exact uncertainty approach in quantum mechanics and quantum gravity”, Gen. Relativ. Gravit. 37 (2005) 1505 (http://lanl.arxiv.org/abs/gr-qc/0408098). 3. M.J.W. Hall and M. Reginatto, “Interacting classical and quantum systems”, Phys. Rev. A 72 (2005) 062109 (http://lanl.arxiv.org/abs/quant-ph/0509134).

Non-relativistic quantum theory is derived from information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles so that the configuration space is a statistical manifold with a natural information metric. The dynamics then follows from a principle of inference, the method of Maximum Entropy: entropic dynamics is an instance of law without law. The concept of time is introduced as a convenient device to keep track of the accumulation of changes. The resulting formalism is close to Nelson's stochastic mechanics. The statistical manifold is a dynamical entity: its (information) geometry determines the evolution of the probability distribution which, in its turn, reacts back and determines the evolution of the geometry. As in General Relativity there is a kind of equivalence principle in that “fictitious” forces – in this case diffusive “osmotic” forces – turn out to be “real”. This equivalence of quantum and statistical fluctuations – or of quantum and classical probabilities – leads to a natural explanation of the equality of inertial and “osmotic” masses and allows explaining quantum theory as a sophisticated example of entropic inference. Mass and the phase of the wave function are explained as features of purely statistical origin. Recommended Reading: arXiv:0907.4335 "From Entropic Dynamics to Quantum Theory" (2009)

I will consider physical theories which describe systems with limited information content. This limit is not due observer's ignorance about some “hidden” properties of the system - the view that would have to be confronted with Bell's theorem - but is of fundamental nature. I will show how the mathematical structure of these theories can be reconstructed from a set of reasonable axioms about probabilities for measurement outcomes. Among others these include the “locality” assumption according to which the global state of a composite system is completely determined by correlations between local measurements. I will demonstrate that quantum mechanics is the only theory from the set in which composite systems can be in entangled (non-separable) states. Within Hardy's approach this feature allows to single out quantum theory from other probabilistic theories without a need to assume the “simplicity” axiom. 1. Borivoje Dakic, Caslav Brukner (in preparation) 2. Caslav Brukner, Anton Zeilinger, Information Invariance and Quantum Probabilities, arXiv:0905.0653 3. Tomasz Paterek, Borivoje Dakic, Caslav Brukner, Theories of systems with limited information content, arXiv:0804.1423

Quantum theory is the most accurate scientific theory humanity has ever devised. But it is also the most mysterious. No one knows what the underlying picture of reality at quantum level is. This presentation will introduce you to some of the many interpretations of quantum theory that scientists have devised and discuss the infamous 'measurement problem'.

Violation of local realism can be probed by theory–independent tests, such as Bell’s inequality experiments. There, a common assumption is the existence of perfect, classical, reference frames, which allow for the specification of measurement settings with arbitrary precision. However, if the reference frames are ``bounded'', only limited precision can be attained. We expect then that the finiteness of the reference frames limits the observability of genuine quantum features. Using spin coherent states as reference frames, we determined their minimal size necessary to violate Bell’s inequalities in entangled systems ranging from qubits to macroscopic dimensions. In the latter, the reference frame’s size must be quadratically larger than that of the system. Lacking such large reference frames, precludes quantum phenomena from appearing in everyday experience.

Although most realistic approaches to quantum theory are based on classical particles, QFT reveals that classical fields are a much closer analog. And unlike quantum fields, classical fields can be extrapolated to curved spacetime without conceptual difficulty. These facts make it tempting to reconsider whether quantum theory might be reformulated on an underlying classical field structure. This seminar aims to demonstrate that by changing only how boundary conditions (BCs) are imposed on ordinary classical field equations, a psi-epistemic quantum theory naturally emerges. Uncertainty and basic quantization naturally result from imposing BCs on closed hypersurfaces (as in Lagrangian QFT); further quantization results from extending Hamilton's principle to restrict the BCs as well as the field equations. The partial dependence of field parameters on future BCs implies an effective contextuality, naturally avoiding the usual arguments against realistic quantum models. Successful applications to the relativistic scalar field will be presented, further motivating an ambitious research program of reformulating quantum theory in terms of ontic classical fields.

We give a mathematical framework to describe the evolution of quantum systems subject to finitely many interactions with classical apparatuus and with each other. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently, but may also interact. The evolution is coded in a mathematical structure in such a way that the properties of causality, covariance and entanglement are faithfully represented. The key to this scheme is to use a special family of spacelike slices -- we call them locative -- that are not so large as to permit acausal influences but large enough to capture nonlocal correlations. I will briefly describe how the dynamics can be described as a functor to a suitable category of Hilbert spaces and will also give some connections with logic.

We sketch some ideas about how higher-dimensional categories could be used to extend conventional quantum mechanics. The physical motivation comes from quantum field theory, for which higher-dimensional category theory is very relevant. We discuss how this new approach would affect familiar aspects of quantum theory, such as observables and the Copenhagen interpretation. Few solid answers will be given, but hopefully some discussion will be generated!

Tensor product is described in a family of categories that includes Set and Hilbert spaces. Such categories admit a "scalar" object which enables a definition of bi-arrows with two domains, generalizing functions of two variables. The tensor product is characterized by the expected universal property relating bi-arrows to arrows.