Quantum features in statistical observations of "timeless" classical systems

We pursue the view that quantum theory may be an emergent structure related to large space-time scales. In particular, we consider classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the discrete physical time. This is motivated by studies of ``timeless'' reparametrization invariant models, where discrete physical time has recently been constructed based on coarse-graining local observables. Describing such deterministic classical systems with the help of path-integrals, primordial states can naturally be introduced which follow unitary quantum mechanical evolution in suitable limits.Comment: 7 pages. Invited talk at Int. Workshop Trends and Perspectives on Extensive and Non-Extensive Statistical Mechanics, Angra dos Reis (Brazil), Nov. 2003. To appear in Physica

Quantum Mechanics and Discrete Time from "Timeless" Classical Dynamics

We study classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the physical time, which is assumed to be discrete. - This is motivated by the ``timeless'' reparametrization invariant model of a relativistic particle with two compactified extradimensions. In this example, discrete physical time is constructed based on quasi-local observables. - Generally, employing the path-integral formulation of classical mechanics developed by Gozzi et al., we show that these deterministic classical systems can be naturally described as unitary quantum mechanical models. The emergent quantum Hamiltonian is derived from the underlying classical one. It is closely related to the Liouville operator. We demonstrate in several examples the necessity of regularization, in order to arrive at quantum models with bounded spectrum and stable groundstate.Comment: 24 pages, 1 figure. Lecture given at DICE 2002. To be published in: Decoherence and Entropy in Complex Systems, Lecture Notes in Physics (Springer-Verlag, Berlin 2003). - Comprises quant-ph/0306096 and gr-qc/0301109, additional reference

Is there a relativistic nonlinear generalization of quantum mechanics?

Yes, there is. - A new kind of gauge theory is introduced, where the minimal coupling and corresponding covariant derivatives are defined in the space of functions pertaining to the functional Schroedinger picture of a given field theory. While, for simplicity, we study the example of an U(1) symmetry, this kind of gauge theory can accommodate other symmetries as well. We consider the resulting relativistic nonlinear extension of quantum mechanics and show that it incorporates gravity in the (0+1)-dimensional limit, where it leads to the Schroedinger-Newton equations. Gravity is encoded here into a universal nonlinear extension of quantum theory. The probabilistic interpretation, i.e. Born's rule, holds provided the underlying model has only dimensionless parameters.Comment: 10 pages; talk at DICE 2006 (Piombino, September 11-15, 2006); to appear in Journal of Physics: Conference Series (2007

Does quantum mechanics tell an atomistic spacetime?

The canonical answer to the question posed is "Yes." -- tacitly assuming that quantum theory and the concept of spacetime are to be unified by `quantizing' a theory of gravitation. Yet, instead, one may ponder: Could quantum mechanics arise as a coarse-grained reflection of the atomistic nature of spacetime? -- We speculate that this may indeed be the case. We recall the similarity between evolution of classical and quantum mechanical ensembles, according to Liouville and von Neumann equation, respectively. The classical and quantum mechanical equations are indistinguishable for objects which are free or subject to spatially constant but possibly time dependent, or harmonic forces, if represented appropriately. This result suggests a way to incorporate anharmonic interactions, including fluctuations which are tentatively related to the underlying discreteness of spacetime. Being linear and local at the quantum mechanical level, the model offers a decoherence and natural localization mechanism. However, the relation to primordial deterministic degrees of freedom is nonlocal.Comment: Based on invited talks at Fourth International Workshop DICE2008, held at Castello Pasquini / Castiglioncello, Italy, 22-26 September 2008 and at DISCRETE'08 - Symposium on Prospects in the Physics of Discrete Symmetries, held at IFIC, Valencia, Spain, 11-16 December 2008 - to appear in respective volumes of Journal of Physics: Conference Serie

Deterministic models of quantum fields

Deterministic dynamical models are discussed which can be described in quantum mechanical terms. -- In particular, a local quantum field theory is presented which is a supersymmetric classical model. The Hilbert space approach of Koopman and von Neumann is used to study the classical evolution of an ensemble of such systems. Its Liouville operator is decomposed into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. Thus, choosing suitable variables, the classical Liouville equation becomes a functional Schroedinger equation of a genuine quantum field theory. -- We briefly mention an U(1) gauge theory with ``varying alpha'' or dilaton coupling where a corresponding quantized theory emerges in the phase space approach. It is energy-parity symmetric and, therefore, a prototype of a model in which the cosmological constant is protected by a symmetry.Comment: 6 pages; synopsis of hep-th/0510267, hep-th/0503069, hep-th/0411176 . Talk at Constrained Dynamics and Quantum Gravity - QG05, Cala Gonone (Sardinia, Italy), September 12-16, 2005. To appear in the proceeding

The Attractor and the Quantum States

The dissipative dynamics anticipated in the proof of 't Hooft's existence theorem -- "For any quantum system there exists at least one deterministic model that reproduces all its dynamics after prequantization" -- is constructed here explicitly. We propose a generalization of Liouville's classical phase space equation, incorporating dissipation and diffusion, and demonstrate that it describes the emergence of quantum states and their dynamics in the Schroedinger picture. Asymptotically, there is a stable ground state and two decoupled sets of degrees of freedom, which transform into each other under the energy-parity symmetry of Kaplan and Sundrum. They recover the familiar Hilbert space and its dual. Expectations of observables are shown to agree with the Born rule, which is not imposed a priori. This attractor mechanism is applicable in the presence of interactions, to few-body or field theories in particular.Comment: 14 pages; based on invited talk at 4th Workshop ad memoriam of Carlo Novero "Advances in Foundations of Quantum Mechanics and Quantum Information with Atoms and Photons", Torino, May 2008; submitted to Int J Qu Inf

On supercorrelated systems and phase space entrainment

It is demonstrated that power-laws which are modified by logarithmic corrections arise in supercorrelated systems. Their characteristic feature is the energy attributed to a state (or value of a general cost function) which depends nonlinearly on the phase space distribution of the constituents. A onedimensional dissipative deterministic model is introduced which is attracted to a supercorrelated state (phase space entrainment). Extensions of this particular model may have applications in the study of transport and equilibration phenomena, particularly for supply and information networks, or for chemical and biological nonequilibrium systems, while the qualitative arguments presented here are believed to be of more general interest.Comment: 12 pages - some technical material removed from second version, several explanatory comments and references added instea

Quantum fields, cosmological constant and symmetry doubling

Energy-parity has been introduced by Kaplan and Sundrum as a protective symmetry that suppresses matter contributions to the cosmological constant [KS05]. It is shown here that this symmetry, schematically Energy --> - Energy, arises in the Hilbert space representation of the classical phase space dynamics of matter. Consistently with energy-parity and gauge symmetry, we generalize the Liouville operator and allow a varying gauge coupling, as in "varying alpha" or dilaton models. In this model, classical matter fields can dynamically turn into quantum fields (Schroedinger picture), accompanied by a gauge symmetry change -- presently, U(1) --> U(1) x U(1). The transition between classical ensemble theory and quantum field theory is governed by the varying coupling, in terms of a one-parameter deformation of either limit. These corrections introduce diffusion and dissipation, leading to decoherence.Comment: Replaced by published version, no change in contents - Int. J. Theor. Phys. (2007