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.
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.
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.
Entropic dynamics is a framework in which quantum theory is derived as an application of entropic methods of inference. Entropic dynamics on flat spaces has been extensively studied. The objective of this paper is to extend the entropic dynamics of N particles to curved spaces. The important new feature is that the displacement of a particle does not transform like a vector because fluctuations can be large enough to feel the effects of curvature. The final result is a modified Schrödinger equation in which the usual Laplacian is replaced by the Laplace-Beltrami operator.
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.
In the Entropic Dynamics (ED) approach the essence of quantum theory lies in its probabilistic nature while the Hilbert space structure plays a secondary and ultimately optional role. The dynamics of probability distributions is driven by the maximization of an entropy subject to constraints that carry the relevant physical information -- directionality, correlations, gauge interactions, etc. The challenge is to identify those constraints and to establish a criterion for how the constraints themselves are updated. In this paper the ED framework is extended to describe a spin-1/2 point particle. In ED spin is neither modelled as a rotating body, nor through the motion of a point particle; it is an epistemic property of the wave function. The constraint that reflects the peculiar rotational properties of spin is most effectively expressed in the language of geometric algebra. The updating of all constraints is carried out in a way that stresses the central importance of symmetry principles. First we identify the appropriate symplectic and metric structures in the phase space of probabilities, their conjugate momenta, and the spin variables. This construction yields a derivation of the Fubini-Study metric for a spin-1/2 particle which highlights its deep connection to information geometry. Then we construct an ED that preserves both the symplectic structure (a Hamiltonian flow) and the metric structure (a Killing flow). We show that generic Hamiltonian-Killing flows are linear in the wave function. Imposing further that the Hamiltonian be the generator of an entropic evolution in time leads to an entropic dynamics described by the Pauli equation. We conclude with a discussion of the new interpretation of the formalism which yields a physical picture that is significantly different from that provided by other interpretations.
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
Standard Quantum Mechanics, although successful in terms of calculating and predicting results, is inherently difficult to understand and can suffer from misinterpretation.Entropic Dynamics is an epistemic approach to quantum mechanics based on logical inference.It incorporates the probabilities that naturally arise in situations in which there is missing information.It is the author's opinion that an advantage of this approach is that it provides a clearer mental image with which to picture quantum mechanics.This may provide anWe then move on to applying Entropic Dynamics to several of the paradigmatic examples used to explain quantum mechanics.The first of these is wave packet expansion.The second is interference, which is the basis behind many of the important phenomena in quantum mechanics.The third is the double slit experiment, which provides some interesting insight into the subject of interference.In particular, we look at the way in which minima can occur without a mechanism for destructive interference, since probabilities only add.The idea of probability flow is very apparent at this point in the discussion.The next example is that of the harmonic oscillator.This leads to an interesting insight concerning rotation and angular momentum as it corresponds to the flow of probability.The last example explored is that of entanglement.The discussion begins with a review of EPR, but then comes to the interesting conclusion that many of the problems inherent in the traditional approach to entanglement do not exist in Entropic Dynamics.The last topic covered in this thesis consists of some remarks concerning the state of education research as it pertains to quantum mechanics and the ways in which Entropic Dynamics might address them.iii viii 9.3.