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arXiv:1803.06327 · AIP conference proceedings

A covariant approach to entropic dynamics

Selman Ipek, Mohammad Abedi, Ariel Caticha

Published Jan 1, 2017• 8 citations
arXivHTMLPDF

Abstract

Entropic Dynamics (ED) is a framework for constructing dynamical theories of inference using the tools of inductive reasoning. A central feature of the ED framework is the special focus placed on time. In [2] a global entropic time was used to derive a quantum theory of relativistic scalar fields. This theory, however, suffered from a lack of explicit or manifest Lorentz symmetry. In this paper we explore an alternative formulation in which the relativistic aspects of the theory are manifest.The approach we pursue here is inspired by the methods of Dirac, Kuchař, and Teitelboim in their development of covariant Hamiltonian approaches. The key ingredient here is the adoption of a local notion of entropic time, which allows compatibility with an arbitrary notion of simultaneity. However, in order to ensure that the evolution does not depend on the particular sequence of hypersurfaces, we must impose a set of constraints that guarantee a consistent evolution.

DOI
10.1063/1.4985371

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Entropic Dynamics (ED) is a framework for constructing dynamical theories of inference using the tools of inductive reasoning. A central feature of the ED framework is the special focus placed on time. In [2] a global entropic time was used to derive a quantum theory of relativistic scalar fields. This theory, however, suffered from a lack of explicit or manifest Lorentz symmetry. In this paper we explore an alternative formulation in which the relativistic aspects of the theory are manifest.The approach we pursue here is inspired by the methods of Dirac, Kuchař, and Teitelboim in their development of covariant Hamiltonian approaches. The key ingredient here is the adoption of a local notion of entropic time, which allows compatibility with an arbitrary notion of simultaneity. However, in order to ensure that the evolution does not depend on the particular sequence of hypersurfaces, we must impose a set of constraints that guarantee a consistent evolution.
In the Entropic Dynamics framework quantum theory is derived as an application of the method of maximum entropy. In previous work the entropic dynamics of relativistic quantum scalar fields was formulated in the Schroedinger functional representation in which the Lorentz symmetry is not manifest. Here the formalism is extended to curved spacetimes. We develop a manifestly covariant approach inspired by the Hamiltonian methods of Dirac, Kuchar, and Teitelboim. The key ingredient is the adoption of a local notion of entropic time in which instants are defined on curved three-dimensional surfaces and time evolution consists of the accumulation of changes induced by local deformations of these surfaces.
Entropic Dynamics (ED) is an inference-based framework that seeks to construct dynamical theories of physics without assuming the conventional formalism --- the Hamiltonians, Poisson brackets, Hilbert spaces, etc. --- typically associated with physics. In this work we develop an ED of scalar fields that is both quantum and manifestly covariant. The framework for accomplishing this is inspired by the covariant methods of Dirac, Teitelboim, and Kuchar. In addition to the ostensible result of a covariant quantum ED, we also show how the covariance requirement of path independence proposed by Teitelboim is sufficient for a derivation of Hamiltonian dynamics and also provides a proof of uniqueness for the quantum potential that leads to quantum theory.
Abstract The Entropic Dynamics reconstruction of quantum mechanics is extended to the quantum theory of scalar fields in curved space-time. The Entropic Dynamics framework, which derives quantum theory as an application of the method of maximum entropy, is combined with the covariant methods of Dirac, Hojman, Kuchař, and Teitelboim, which they used to develop a framework for classical covariant Hamiltonian theories. The goal is to formulate an information-based alternative to current approaches based on algebraic quantum field theory. One key ingredient is the adoption of a local notion of entropic time in which instants are defined on curved three-dimensional surfaces and time evolution consists of the accumulation of changes induced by local deformations of these surfaces. The resulting dynamics is a non-dissipative diffusion that is constrained by the requirements of foliation invariance and incorporates the necessary local quantum potentials. As applications of the formalism we derive the Ehrenfest for fields in curved-spacetime and briefly discuss the nature of divergences in quantum field theory.
The general framework of Entropic Dynamics (ED) is used to construct non-relativistic models of relational Quantum Mechanics from well-known inference principles-probability, entropy and information geometry. Although only partially relational-the absolute structures of simultaneity and Euclidean geometry are still retained-these models provide a useful testing ground for ideas that will prove useful in the context of more realistic relativistic theories. The fact that in ED the positions of particles have definite values, just as in classical mechanics, has allowed us to adapt to the quantum case some intuitions from Barbour and Bertotti's classical framework. Here, however, we propose a new measure of the mismatch between successive states that is adapted to the information metric and the symplectic structures of the quantum phase space. We make explicit that ED is temporally relational and we construct non-relativistic quantum models that are spatially relational with respect to rigid translations and rotations. The ED approach settles the longstanding question of what form the constraints of a classical theory should take after quantization: the quantum constraints that express relationality are to be imposed on expectation values. To highlight the potential impact of these developments, the non-relativistic quantum model is parametrized into a generally covariant form and we show that the ED approach evades the analogue of what in quantum gravity has been called the problem of time.
It has often been the case in history that the laws of physics have been used as the framework for understanding and implementing information processing. The tacit assumption is that the laws of physics are fundamental and that the notion of information is derived from these laws. Here we take the opposite view: the laws of physics are an application of the rules for processing information. The inferential framework of entropic dynamics (ED) has previously been developed by A. Caticha for the purposes of understanding and deriving quantum theory, in much the same way that E.T. Jaynes used the MaxEnt approach to derive the formalism of statistical mechanics. In this thesis we apply the ED framework to construct a quantum dynamics for scalar fields in space-time.
Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity later identified as the phase of the wave function. The central challenge is to specify how those constraints are themselves updated. In this paper we review and extend the ED framework in several directions. A new version of ED is introduced in which particles follow smooth differentiable Brownian trajectories (as opposed to non-differentiable Brownian paths). To construct ED we make use of the fact that the space of probabilities and phases has a natural symplectic structure (i.e., it is a phase space with Hamiltonian flows and Poisson brackets). Then, using an argument based on information geometry, a metric structure is introduced. It is shown that the ED that preserves the symplectic and metric structures—which is a Hamilton-Killing flow in phase space—is the linear Schrödinger equation. These developments allow us to discuss why wave functions are complex and the connections between the superposition principle, the single-valuedness of wave functions, and the quantization of electric charges. Finally, it is observed that Hilbert spaces are not necessary ingredients in this construction. They are a clever but merely optional trick that turns out to be convenient for practical calculations.