## Date of Award

8-1-2021

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## College/School/Department

Department of Physics

## Content Description

1 online resource (xvii, 441 pages) : color illustrations.

## Dissertation/Thesis Chair

Ariel Caticha

## Committee Members

Justin Curry, Daniel Robbins, Oleg Lunin

## Keywords

Entropy, Information Geometry, Pauli Equation, Probability, Spin, Inference, Maximum entropy method, Quantum theory, Pauli exclusion principle, Rotational motion

## Subject Categories

Numerical Analysis and Scientific Computing | Other Physics | Physics

## Abstract

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 and constraints and imposing that the physically relevant flows in this space are those which preserve a particular metric and symplectic form. These flows lead to a linear Pauli equation for one and two particles with spin.

## Recommended Citation

Carrara, Nicholas Matthew, "The foundations of inference and its application to fundamental physics" (2021). *Legacy Theses & Dissertations (2009 - 2024)*. 2649.

https://scholarsarchive.library.albany.edu/legacy-etd/2649