We elaborate on the existing idea that quantum mechanics is an emergent phenomenon, in the form of a coarse—grained description of some underlying deterministic theory. We apply the Ricci flow as a technical tool to implement dissipation, or information loss, in the passage from an underlying deterministic theory to its emergent quantum counterpart. A key ingedient in this construction is the fact that the space of physically inequivalent quantum states (either pure or mixed) has positive Ricci curvature. This leads us to an interesting thermodynamical analogy of emergent quantum mechanics.
We elaborate on the existing idea that quantum mechanics is an emergent phenomenon, in the form of a coarse—grained description of some underlying deterministic theory. We apply the Ricci flow as a technical tool to implement dissipation, or information loss, in the passage from an underlying deterministic theory to its emergent quantum counterpart. A key ingedient in this construction is the fact that the space of physically inequivalent quantum states (either pure or mixed) has positive Ricci curvature. This leads us to an interesting thermodynamical analogy of emergent quantum mechanics.
The emergent nature of quantum mechanics is shown to follow from a precise correspondence with the classical theory of irreversible thermodynamics. Specifically, the linear (or Gaussian) regime of the latter can be put in a 1-to-1 map with the semiclassical approximation to quantum mechanics. The very possibility of reinterpreting quantum mechanics as a thermodynamics proves that the former is an emergent phenomenon. That is, quantum mechanics is a coarse-grained description of some underlying degrees of freedom.
It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantization, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space ℂℙd-1 of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base ℂℙd-1. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.
Abstract It has been argued that, underlying any given quantum–mechanical model, there exists at least one deterministic system that reproduces, after prequantisation, the given quantum dynamics. For a quantum mechanics with a complex d–dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space CP d−1 of quantum states. Let R stand for the Ricci flow of the manifold SU(d − 1) down to one point, and let P denote the projection from the Hopf bundle onto its base CP d−1. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity. 1
The emergent nature of quantum mechanics is shown to follow from a precise correspondence with the classical theory of irreversible thermodynamics. Specifically, the linear (or Gaussian) regime of the latter can be put in a 1-to-1 map with the semiclassical approximation to quantum mechanics. The very possibility of reinterpreting quantum mechanics as a thermodynamics proves that the former is an emergent phenomenon. That is, quantum mechanics is a coarse-grained description of some underlying degrees of freedom.
Motivated by the conceptual problems concerning the quantisation of gravity, the Dutch theoretical physicist G. 't Hooft (1999 Nobel prize in physics) put forward the notion that quantum mechanics must be the emergent theory of some underlying, deterministic theory.This proposal usually goes by the name quantum mechanics as an emergent phenomenon.This research line, initiated by 't Hooft in the late 1990's, has been the subject of intense research over the last 15 years, by 't Hooft himself as well as by many other researchers.In this PhD thesis we present our own approach to quantum mechanics as an emergent phenomenon.According to the emergence paradigm for quantum mechanics, information-loss effects in the underlying deterministic theory lead to the arrangement of states of the latter into equivalence classes, that one identifies as quantum states of the emergent quantum mechanics.In brief, quantisation is dissipation, according to 't Hooft.Moreover it has been argued in the literature that, in the presence of weak gravitational fields, quantum effects must be indistinguishable from thermal effects.Since the latter are typically dissipative in nature, the presence of a weak gravitational field should provide a framework in which quantum effects can be explained as due to thermal, dissipative fluctuations.Furthermore, since gravitational effects can be locally gauged away (thanks to the equivalence principle), there should exist some kind of equivalence principle for quantum effects, i.e., some kind of relativity principle for the notion of quantumness as opposed to the notion of classicality.In this PhD thesis we elaborate on this idea.Once a reference frame is fixed, however, quantum effects cannot be gauged away, and the statement quantisation is dissipation lends itself to a thermodynamical treatment.In this thesis we also present one mechanism whereby quantum mechanics is seen to emerge, thus explicitly realising 't Hooft's proposal.This mechanism is based on a dictionary between semiclassical quantum mechanics, on the one hand, and the classical theory of irreversible thermodynamics in the linear regime, on the other.This thermodynamical formalism, developed by Nobel prize winners Onsager and Prigogine, can be easily mapped into that of semiclassical quantum mechanics. Chapter 3Emergent QM as a thermal ensemble This chapter is based on the article Emergent Quantum Mechanics as a Thermal Ensemble, by P.
This PhD thesis is submitted as a \emph{compendium} of the articles \cite{NS, holographic, topological}. The following has been adapted from their abstracts. Quantum mechanics has been argued to be a coarse--graining of some underlying deterministic theory. Here we support this view by establishing mappings between non-relativistic quantum mechanics and thermodynamic theories, since the latter are the paradigm of an emergent theory. First, we map certain solutions of the Schroedinger equation to solutions of the irrotational Navier--Stokes equation for viscous fluid flow. Although this is formally a generalization of Madelung's hydrodynamical interpretation, the presence of a viscous term leads to a novel interpretation. As a physical model for the fluid itself we propose the quantum probability fluid. It turns out that the (state--dependent) viscosity of this fluid is proportional to Planck's constant, while the volume density of entropy is proportional to Boltzmann's constant. Stationary states have zero viscosity and a vanishing time rate of entropy density. On the other hand, the nonzero viscosity of nonstationary states provides an information--loss mechanism whereby a deterministic theory (a classical fluid governed by the Navier--Stokes equation) gives rise to an emergent theory (a quantum particle governed by the Schroedinger equation). Then, we present a map of standard quantum mechanics onto classical thermodynamics of irreversible processes. In particular, the propagators of the quantum harmonic oscillator are mapped to the conditional probabilities that solve the Chapman-Kolmogorov equation for Markovian Gaussian processes. While no gravity is present in our construction, our map exhibits features that are reminiscent of the holographic principle of quantum gravity. Finally, the classical thermostatics of equilibrium processes is shown to possess a quantum mechanical dual theory with a finite dimensional Hilbert space of quantum states. Specifically, the kernel of a certain Hamiltonian operator becomes the Hilbert space of quasistatic quantum mechanics. The relation of thermostatics to topological field theory is also discussed in the context of the approach of emergence of quantum theory, where the concept of entropy plays a key role.