Entropy, Topological Theories and Emergent Quantum Mechanics

Abstract

Abstract: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 the emergence of quantum theory, where the concept of entropy plays a key role.Keywords: emergence of quantum theory; topological field theory MotivationThe approach of the emergence of quantum mechanics has provided interesting clues into the deeper structure of the theory. The statement that standard quantum mechanics is an emergent phenomenon [1-4] has found further support in a series of papers, some of which have been reviewed in Reference [5]. Although this is a huge topic to summarize here, let us briefly mention some key points of this approach. The underlying notion is that it provides a coarse-grained version of some deeper theory, out of which quantum mechanics emerges as a kind of effective description. This effective description-in using variables that arise as averages over large collections of individual entities carrying the truly fundamental degrees of freedom-ignores the underlying fine structure. These fundamental degrees of freedom have been identified in References [3,4] as those of cellular automata.This state of affairs is reminiscent of the relation between thermodynamics (as an emergent phenomenon) and statistical mechanics (the corresponding underlying theory). Based on this analogy, we have in previous publications (see [5] and references therein) established a bijective map that one can define between quantum mechanics, on the one hand, and the classical thermodynamics of irreversible processes, on the other [6,7]. It must be stressed that the classical thermodynamics of irreversible processes [6,7] is conceptually quite different from the usual thermostatics of equilibrium as presented in the standard textbooks [8]. Specifically, in the theory of irreversible processes, the continual production of entropy provides a rationale for the dissipation-or information loss-that has been argued to lie at the heart of quantum mechanics [3,4]. The relevance of thermodynamical concepts to quantum theory and gravity has been emphasized recently in references [9][10][11][12][13].It might thus appear that the usual quasistatic thermodynamics [8] (i.e., the thermostatics of equilibrium processes) possesses no quantum mechanical dual theory at all. In this letter, we point out that such a conclusion is not true: the thermostatics of equilibrium processes does have a quantum mechanical dual; namely, a quasistatic quantum mechanics. By quasistatic, we mean that the kinetic term in the mechanical Lagrangian can be neglected compared to the potential term.Neglecting the kinetic term in the Lagrangian function forces one to look elsewhere for the dissipative mechanism that is characteristic of quantum theory [...

On the van der Waals Gas, Contact Geometry and the Toda Chain

“…when a = 0 and b = 0, systems (10) and (11) correctly reduce to the corresponding system of PDEs for the ideal gas, obtained in Ref. [5].…”

“…This limit has been called the topological limit in Ref. [11]; roughly speaking, it amounts to cancelling the kinetic term while keeping only the potential term in the Hamiltonian. This fact allows us to sharpen our previous correspondence, which we can now state more precisely as follows: the classical thermodynamics of the (ideal or van der Waals) gas has a dual theory which, to first order of approximation, coincides with the topological limit of a thermal ensemble of waves in the Toda chain.…”