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 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.

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.

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