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arXiv:1108.2629 · AIP conference proceedings

Momentum and uncertainty relations in the entropic approach to quantum theory

Shahid Nawaz, Ariel Caticha

Published Jan 1, 2012• 17 citations
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Abstract

In the Entropic Dynamics (ED) framework quantum theory is derived as an application of the method of maximum entropy. The particles have well-defined positions but since they follow non differentiable Brownian trajectories they cannot be assigned an instantaneous momentum. Nevertheless, four different notions of momentum can be usefully introduced. We derive relations among them and the corresponding uncertainty relations. The main conclusion is that in ED momentum is not to be interpreted as an attribute of the particles but as a statistical concept associated to the probability distributions.

DOI
10.1063/1.3703627

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In the Entropic Dynamics (ED) framework quantum theory is derived as an application of the method of maximum entropy. The particles have well-defined positions but since they follow non differentiable Brownian trajectories they cannot be assigned an instantaneous momentum. Nevertheless, four different notions of momentum can be usefully introduced. We derive relations among them and the corresponding uncertainty relations. The main conclusion is that in ED momentum is not to be interpreted as an attribute of the particles but as a statistical concept associated to the probability distributions.
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.
Non-relativistic quantum theory is derived from information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles: positions constitute a configuration space and the corresponding probability distributions constitute a statistical manifold. The dynamics follows from a principle of inference, the method of Maximum Entropy. The concept of time is introduced as a convenient way to keep track of change. A welcome feature is that the entropic dynamics notion of time incorporates a natural distinction between past and future. The statistical manifold is assumed to be a dynamical entity: its curved and evolving geometry determines the evolution of the particles which, in their turn, react back and determine the evolution of the geometry. Imposing that the dynamics conserve energy leads to the Schroedinger equation and to a natural explanation of its linearity, its unitarity, and of the role of complex numbers. The phase of the wave function is explained as a feature of purely statistical origin. There is a quantum analogue to the gravitational equivalence principle.
Non-relativistic quantum theory is derived from information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles so that the configuration space is a statistical manifold. The dynamics then follows from a principle of inference, the method of Maximum Entropy. The concept of time is introduced as a convenient way to keep track of change. The resulting formalism is close to Nelson’s stochastic mechanics. The statistical manifold is a dynamical entity: its (information) geometry determines the evolution of the probability distribution which, in its turn, reacts back and determines the evolution of the geometry. As in General Relativity there is a kind of equivalence principle in that “fictitious ” forces – in this case diffusive “osmotic ” forces – turn out to be “real”. This equivalence of quantum and statistical fluctuations – or of quantum and classical probabilities – leads to a natural explanation of the equality of inertial and “osmotic ” masses and allows explaining quantum theory as a sophisticated example of entropic inference. Mass and the phase of the wave function are explained as features of purely statistical origin.
We study quantum theory as an example of entropic inference. Our goal is to remove conceptual difficulties that arise in quantum mechanics. Since probability is a common feature of quantum theory and of any inference problem, we briefly introduce probability theory and the entropic methods to update probabilities when new information becomes available. Nelson's stochastic mechanics and Caticha's derivation of quantum theory are discussed in the subsequent chapters. Our first goal is to understand momentum and angular momentum within an entropic dynamics framework and to derive the corresponding uncertainty relations. In this framework momentum is an epistemic concept – it is not an attribute of the particle but of the probability distributions. We also show that the Heisenberg's uncertainty relation is an osmotic effect. Next we explore the entropic analog of angular momentum. Just like linear momentum, angular momentum is also expressed in purely informational terms. We then extend entropic dynamics to curved spaces. An important new feature is that the displacement of a particle does not transform like a vector. It involves second order terms that account for the effects of curvature . This leads to a modified Schrödinger equation for curved spaces that also take into account the curvature effects. We also derive Schrodinger equation for a charged particle interacting with external electromagnetic field on general Riemannian manifolds. Finally we develop the entropic dynamics of a particle of spin 1/2. The particle is modeled as a rigid point rotator interacting with an external EM field. The configuration space of such a rotator is R 3 × S3 (S 3 is the 3-sphere). The model describes the regular representation of SU(2) which includes all the irreducible representations (spin 0, 1/2, 1, 3/2,...) including spin 1/2.
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.
Quantum mechanics is derived as an application of the method of maximum entropy. No appeal is made to any underlying classical action principle whether deterministic or stochastic. Instead, the basic assumption is that in addition to the particles of interest x there exist extra variables y whose entropy S ( x ) depends on x . The Schrödinger equation follows from their coupled dynamics: the entropy S ( x ) drives the dynamics of the particles x while they in their turn determine the evolution of S ( x ). In this 'entropic dynamics' time is introduced as a device to keep track of change. A welcome feature of such an entropic time is that it naturally incorporates an arrow of time. Both the magnitude and the phase of the wavefunction are given statistical interpretations: the magnitude gives the distribution of x in agreement with the usual Born rule and the phase carries information about the entropy S ( x ) of the extra variables. Extending the model to include external electromagnetic fields yields further insight into the nature of the quantum phase.