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.
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.
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.
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 Schr\"odinger 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 wave function 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.
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.
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 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.
This thesis concerns the foundations of inference – probability theory,entropic inference, information geometry, etc. – and its application to the Entropic Dynamics (ED) approach to Quantum Mechanics (QM) [21, 22, 41, 53, 56–61, 150–153, 165, 195, 196, 268]. The first half of this thesis, chapters 2-6, concern the development of the inference framework. We begin in chapter 2 by discussing de- ductive inference, which involves formal logic and it’s role in access- ing the truth of propositions. We eventually discover that deductive inference is incomplete, in that it can’t address situations in which we have incomplete information. This necessitates a theory of inductive inference (probability theory), which is developed in chapter 3. Prob- ability theory is derived as a framework for manipulating degrees of belief of propositions, in a way which is consistent with its deduc- tive counterpart [47]. In chapter 4 we review the construction of en- tropic inference as a means for updating our beliefs in the presence of new information. The entropy functional is designed through the pro- cess of eliminative induction by imposing a principle of minimal updating (PMU) and various constraints [47, 238, 242, 266, 267]. Chapter 5 con- siders the design of another entropic functional, the total correlation and all its variants, for the purposes of ranking join distributions with respect to their correlations [46]. Finally, in chapter 6, we discuss the application of a special case of the correlation functionals from chap- ter 5, the mutual information, to problems in experimental physics and machine learning [42–45]. The second half of this thesis, chapters 7-9, concerns the ED ap- proach to QM. In particular, chapters 8 and 9 involve the inclusion of particles with spin 1/2 into the framework [41, 61]. These devel- opments are the main contribution of this thesis to the body of work in the ED approach. The problem is defined as an application of in- ference to the dynamics of quantum particles which have definite yet unknown positions and follow continuous trajectories. Through the method of maximum entropy developed in chapter 4, we can deter- mine the transition probability that these particles will move from one location to another. Geometric algebra (GA) [90, 140] is chosen as the preferred representation for the algebra of spin, which is then introduced through constraints in the maximum entropy method. A quantum mechanics is subsequently developed by constructing an epistemic phase space of probabilities