scholarly journals Entropic dynamics: From entropy and information geometry to Hamiltonians and quantum mechanics

2015 ◽  
Author(s):  
Ariel Caticha ◽  
Daniel Bartolomeo ◽  
Marcel Reginatto
Keyword(s):  
Quantum Mechanics ◽  
Information Geometry ◽  
Entropic Dynamics

Entropic Dynamics

2020 ◽  
pp. 185-212
Author(s):  
Ariel Caticha
Keyword(s):  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Information Geometry ◽  
Hamiltonian Mechanics ◽  
The Third ◽  
Laws Of Motion ◽  
Special Case ◽  
Entropic Dynamics

Entropic dynamics is a framework in which dynamical laws such as those that arise in physics are derived as an application of entropic methods of inference. No underlying laws of motion such as those of classical mechanics are postulated. Instead, the dynamics is driven by entropy subject to constraints that reflect the information relevant to the problem at hand. In this chapter I review the derivation of three forms of mechanics. The first is a standard diffusion, the second is a form of Hamiltonian mechanics, and the third form is an argument from information geometry used to derive the special case known as quantum mechanics.

scholarly journals The Entropic Dynamics Approach to Quantum Mechanics

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 943 ◽  
Author(s):  
Ariel Caticha
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Hilbert Spaces ◽  
Symplectic Structure ◽  
Superposition Principle ◽  
Information Geometry ◽  
Wave Functions ◽  
Hamiltonian Flows ◽  
Metric Structures ◽  
Entropic Dynamics

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.

scholarly journals Quantum Trajectories in Entropic Dynamics

Proceedings ◽  
2019 ◽  
Vol 33 (1) ◽  
pp. 25
Author(s):  
Nicholas Carrara
Keyword(s):  
Quantum Mechanics ◽  
Stochastic Equation ◽  
Bohmian Mechanics ◽  
Quantum Trajectories ◽  
Stochastic Mechanics ◽  
Scalar Particles ◽  
Double Slit ◽  
The Continuum ◽  
Entropic Dynamics

Entropic Dynamics is a framework for deriving the laws of physics from entropic inference. In an (ED) of particles, the central assumption is that particles have definite yet unknown positions. By appealing to certain symmetries, one can derive a quantum mechanics of scalar particles and particles with spin, in which the trajectories of the particles are given by a stochastic equation. This is much like Nelson’s stochastic mechanics which also assumes a fluctuating particle as the basis of the microstates. The uniqueness of ED as an entropic inference of particles allows one to continuously transition between fluctuating particles and the smooth trajectories assumed in Bohmian mechanics. In this work we explore the consequences of the ED framework by studying the trajectories of particles in the continuum between stochastic and Bohmian limits in the context of a few physical examples, which include the double slit and Stern-Gerlach experiments.

scholarly journals Quantum Mechanics as Hamilton–Killing Flows on a Statistical Manifold

2021 ◽  
Vol 3 (1) ◽  
pp. 12
Author(s):  
Ariel Caticha
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Information Geometry ◽  
Complex Numbers ◽  
Born Rule ◽  
Statistical Manifold ◽  
Finite Dimensional ◽  
Starting Point

The mathematical formalism of quantum mechanics is derived or “reconstructed” from more basic considerations of the probability theory and information geometry. The starting point is the recognition that probabilities are central to QM; the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold—a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini–Study metric, the linearity of the Schrödinger equation, the emergence of complex numbers, Hilbert spaces and the Born rule are derived rather than postulated.

scholarly journals An Entropic Dynamics Approach to Geometrodynamics

Proceedings ◽  
2019 ◽  
Vol 33 (1) ◽  
pp. 13
Author(s):  
Selman Ipek ◽  
Ariel Caticha
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
First Principles ◽  
Scalar Fields ◽  
Poisson Brackets ◽  
Classical Approach ◽  
Quantum Domain ◽  
Physical Information ◽  
Entropic Dynamics

In the Entropic Dynamics (ED) framework quantum theory is derived as an application of entropic methods of inference. The physics is introduced through appropriate choices of variables and of constraints that codify the relevant physical information. In previous work, a manifestly covariant ED of quantum scalar fields in a fixed background spacetime was developed. Manifest relativistic covariance was achieved by imposing constraints in the form of Poisson brackets and of intial conditions to be satisfied by a set of local Hamiltonian generators. Our approach succeeded in extending to the quantum domain the classical framework that originated with Dirac and was later developed by Teitelboim and Kuchar. In the present work the ED of quantum fields is extended further by allowing the geometry of spacetime to fully partake in the dynamics. The result is a first-principles ED model that in one limit reproduces quantum mechanics and in another limit reproduces classical general relativity. Our model shares some formal features with the so-called “semi-classical” approach to gravity.

scholarly journals Information geometry, dynamics and discrete quantum mechanics

2013 ◽  
Author(s):  
Marcel Reginatto ◽  
Michael J. W. Hall
Keyword(s):  
Quantum Mechanics ◽  
Information Geometry

scholarly journals Entropic Dynamics on Gibbs Statistical Manifolds

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 494
Author(s):  
Pedro Pessoa ◽  
Felipe Xavier Costa ◽  
Ariel Caticha
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Exponential Family ◽  
Information Geometry ◽  
Arrow Of Time ◽  
Quantum Field ◽  
Statistical Manifold ◽  
Statistical Manifolds ◽  
Gibbs Distributions ◽  
Entropic Dynamics

Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles. Here, we develop the entropic dynamics of a system, the state of which is described by a probability distribution. Thus, the dynamics unfolds on a statistical manifold that is automatically endowed by a metric structure provided by information geometry. The curvature of the manifold has a significant influence. We focus our dynamics on the statistical manifold of Gibbs distributions (also known as canonical distributions or the exponential family). The model includes an “entropic” notion of time that is tailored to the system under study; the system is its own clock. As one might expect that entropic time is intrinsically directional; there is a natural arrow of time that is led by entropic considerations. As illustrative examples, we discuss dynamics on a space of Gaussians and the discrete three-state system.

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