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 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 COMPLEX MODULI OF PHYSICAL QUANTA

2005 ◽  
Vol 20 (12) ◽  
pp. 869-874
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Symplectic Structure ◽  
Classical Mechanics ◽  
Differentiable Structure ◽  
Complex Moduli ◽  
Classical Phase Space

Classical mechanics can be formulated using a symplectic structure on classical phase space, while quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This paper is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space.

Observables, interference phenomenon and Born’s rule in the probability representation of quantum mechanics

2020 ◽  
Vol 18 (01) ◽  
pp. 1941021 ◽  
Author(s):  
Margarita A. Man’ko ◽  
Vladimir I. Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Probability Distributions ◽  
Superposition Principle ◽  
Quantum States ◽  
Wave Functions ◽  
Harmonic Oscillators ◽  
Probability Representation ◽  
Interference Phenomenon ◽  
Born’S Rule ◽  
System States

The description of quantum states by probability distributions of classical-like random variables associated with observables is presented. An invertible map of the wave functions and density matrices onto the probability distributions is constructed. The relation of the probability distributions to quasidistributions like the Wigner function is discussed. The interference phenomenon and superposition principle of pure quantum states are given in the form of nonlinear addition of the probabilities identified with the quantum states. The probability given by Born’s rule is expressed as a function of the probabilities describing the system states. The suggested probability representation of quantum mechanics is presented using examples of harmonic oscillators and qubits.

scholarly journals On the Geometry of Relativistic Anyon

1997 ◽  
Vol 12 (24) ◽  
pp. 1783-1789 ◽  
Author(s):  
A. Nersessian
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Irreducible Representations ◽  
Hamiltonian Reduction ◽  
Poincare Group ◽  
Covariant Model ◽  
Lobachevsky Plane ◽  
Spin Generator

A twistor model is proposed for the free relativistic anyon. The Hamiltonian reduction of this model by the action of the spin generator leads to the minimal covariant model; whereas that by the action of spin and mass generators leads to the anyon model with free phase space which is a cotangent bundle of the Lobachevsky plane with twisted symplectic structure. Quantum mechanics of that model is described by irreducible representations of the (2+1)-dimensional Poincaré group.

scholarly journals Quantum Physics in Phase Space: an Analysis of Simple Pendulum

2018 ◽  
Vol 1 (2) ◽  
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Schrödinger Equation ◽  
Wigner Function ◽  
Schrodinger Equation ◽  
Quantum Physics ◽  
Symplectic Structure ◽  
Unitary Representations ◽  
Galilei Group ◽  
Simple Pendulum

In this work we present a brief review about quantum mechanics in phase space. The approach discussed is based in the notion of symplectic structure and star-operators. In this sense, unitary representations for the Galilei group are construct, and the Schrodinger equation in phase space is derived. The connection between phase space amplitudes and Wigner function is presented. As a new result we solved the Schrodinger equation in phase space for simple pendulum. PACS Numbers: 11.10.Nx, 11.30.Cp, 05.20.Dd

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 Formulation Of Quantum Mechanics On Poincaré Disks

2021 ◽  
Author(s):  
Arturo Tozzi
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Gauge Theories ◽  
Temporal Evolution ◽  
Wave Functions ◽  
Mathematical Framework ◽  
Genus Zero ◽  
Proper Understanding ◽  
Oscillatory Dynamics ◽  
Topological Theorem

The unexploited unification of general relativity and quantum mechanics (QM) prevents the proper understanding of the micro- and macroscopic world. Here we put forward a mathematical approach that introduces the problem in terms of negative curvature manifolds. We suggest that the oscillatory dynamics described by wave functions might take place on hyperbolic continuous manifolds, standing for the counterpart of QM’s Hilbert spaces. We describe how the tenets of QM, such as the observable A, the autostates ψa and the Schrodinger equation for the temporal evolution of states, might work very well on a Poincaré disk equipped with rotational groups. This curvature-based approach to QM, combined with the noncommutativity formulated in the language of gyrovectors, leads to a mathematical framework that might be useful in the investigation of relativity/QM relationships. Furthermore, we introduce a topological theorem, termed the punctured balloon theorem (PBT), which states that an orientable genus-1 surface cannot encompass disjoint points. PBT suggests that hyperbolic QM manifolds must be of genus ≥ 1 before measuring and genus zero after measuring. We discuss the implications of PBT in gauge theories and in the physics of the black holes.

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