Foundations of Physics

Background Philosophy (top)

I am interested in the foundations of physics.
The goals are admittedly ambitious and bold.
Let's face it, life is short!

My belief is that most foundational research either assumes too much or is too focused on specific sub-fields of physics. For example, I do not believe that one can effectively study the foundations of quantum mechanics and ignore probability theory, gravity, electromagnetism, and other related phenomena. The universe is a package deal, and to understand it requires understanding the package as a whole. Certainly progress is made in relatively small steps, but if one is serious about solving the puzzle, one has to keep in mind the whole picture while one is trying to place a particular piece.

I bring to this work my experience in machine learning, which amounts to effective and efficient problem-solving. The more that is assumed in a theory, the more likely it is to be wrong. And perhaps more importantly, what is assumed cannot be understood. For example, studying the foundations of quantum mechanics by assuming all of the mathematics of a Hilbert space, basically assumes half the problem, and in doing so prevents one from achieving deep insight.

I take the advice given by Galileo to heart:
"Measure that which is measurable, and make measurable that which is not so."

In my research to date, I have found that apt consistent quantification of any set of entities is often constrained by symmetries and order, and that the resulting constraint equations tend to reflect what we conceive of as physical laws. That is, underlying order results in orderly laws. I, often in collaboration with others, have applied these ideas to probability theory, information theory, quantum mechanics, space-time physics, and relativistic quantum mechanics. The progress my colleagues and I have made can be followed below in a series of papers.

How far this approach can take us is anyone's guess, but one must admit that it important to know just how much of physics is derivable as being contingent on underlying symmetry and order.

Quantification (top)

The topic of apt consistent quantification has a long history with many players and examples and I cannot begin to do it justice here. The main different in our approach, is that we treat this as a central philosophy toward understanding foundations, and not simply a toolbox of disconnected examples throughout history.

Janos Aczel at the Universty of Waterloo and other researchers in the field of Functional Equations have clearly been aware of the critical importance symmetries in the derivation of laws. Perhaps one of the first texts to treat quantification as a foundational principle is the book by Pfanzagl:

Pfanzagl J. "Theory of Measurement", John Wiley & Sons, 1968.

Our relevant papers range from early:

Knuth K.H. 2003. Deriving laws from ordering relations. In: G.J. Erickson, Y. Zhai (eds.), Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Jackson Hole WY 2003, AIP Conference Proceedings 707, American Institute of Physics, Melville NY, pp. 204-235. arXiv:physics/0403031v1 [] (pdf 206K)

to more recent:

Knuth K.H. 2009. Measuring on lattices. P. Goggans, C.-Y. Chan (eds.) Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Oxford, MS, USA, 2009, AIP Conference Proceedings 1193, American Institute of Physics, Melville NY, 132-144. (pdf 227K)

Inference and Probability Theory (top)

In many ways, physics is about making optimal inferences about the world around us. To understand this aspect of physics, which is critical to statistical mechanics and quantum mechanics, one must properly understand the foundations of inference. The inspiration for this research approach came from Richard T. Cox's derivation of probability theory from the foundation of Boolean logic.

Cox R.T. 1946 “Probability, Frequency, and Reasonable Expectation”, American Journal of Physics, 14, 1-13.

Since then, these ideas have evolved and matured as we have employed the more general and powerful formalism of order theory to expose the relevant concepts and expand the applicability of the results.

Knuth K.H. 2005. Lattice duality: The origin of probability and entropy. Neurocomputing. 67C: 245-274. DOI: 10.1016/j.neucom.2004.11.039 (pdf 477K)

Knuth K.H., Skilling J. 2012. Foundations of Inference. 1(1), 38-73; doi:10.3390/axioms1010038 (Free Full-Text at Axioms)

Quantum Mechanics (top)

Philip Goyal and John Skilling and I have demonstrated that the concepts involved in the derivation of probability theory via quantification can be used to derive the Feynman path integral formulation of quantum mechanics. This was inspired in part by the efforts of Tikochinski, Tikochinski and Gull, and Caticha's experimental setups.

Goyal P., Knuth K.H., Skilling J. 2010. Origin of complex quantum amplitudes and Feynman's rules, Phys. Rev. A 81, 022109. arXiv:0907.0909v3 [quant-ph]

The following year, Philip Goyal and myself showed how quantum mechanics and probability theory are related. Not only is quantum mechanics consistent with probability theory (and the underlying logic), but it is dependent on it:

Goyal P., Knuth K.H. 2011. Quantum theory and probability theory: their relationship and origin in symmetry, Symmetry 3(2):171-206.

I should note that some of the older order-theoretic concepts were published by Knuth in 2003:

Knuth K.H. 2003. Deriving laws from ordering relations. In: G.J. Erickson, Y. Zhai (eds.), Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Jackson Hole WY 2003, AIP Conference Proceedings 707, American Institute of Physics, Melville NY, pp. 204-235. arXiv:physics/0403031 [] (pdf 206K)

Those ideas were left out of our two first publications above in favor of the more familiar algebraic relations.

Space-time Physics (top)

Everything that is detected or measured is the direct result of something influencing something else. By considering both the act of influencing and the response to such influence as a pair of events, we can describe a universe of interactions as a partially-ordered set of events. We take the partially-ordered set of events as a fundamental picture of influence and aim to determine what interesting physics can be recovered. This is accomplished by identifying a means by which events in a partially-ordered set can be aptly and consistently quantified. Since, in general, a partially-ordered set lacks symmetries to constraint any quantification, we distinguish a chain of events, which represents an observer, and quantify some subset of events with respect to the observer chain. Consistent quantification with respect to pairs of observer chains exhibiting a constant relationship with one another results in a metric analogous to the Minkowski metric and that transformation of the quantification with respect to one pair of chains to quantification with respect to another pair of chains results in the Bondi k-calculus, which represents a Lorentz transformation under a simple change of variables. We further demonstrate that chain projection induces geometric structure in the partially-ordered set, which itself is inherently both non-geometric and non-dimensional. Collectively, these results suggest that the concept of space-time geometry may emerge as a unique way for an embedded observer to aptly and consistently quantify a partially-ordered set of events.

Knuth K.H., Bahreyni N. 2012. The physics of events: A potential foundation for emergent space-time, arXiv:1209.0881 [math-ph].

Fermion Physics, the Feynman Checkerboard,
and the Dirac Equations (top)

We consider describing a particle by focusing on the fact that it influences others. Such a model results in a partially ordered set where a particle is modeled by a chain of influences. As described above, these interactions give rise to an emergent spacetime where the particle influences can be viewed as the particle taking paths through spacetime. We illustrate how this framework of influence-generated events gives rise to some of the well- known properties of the Fermions, such as the uncertainty relation and Zitterbewegung. We can take this further by making inferences about events, which is performed by employing the process calculus, which coincides with the Feynman path integral formulation of quantum mechanics. This results in the Feynman checkerboard model of the Dirac equation in a 1+1 dimensional space describing a Fermion at rest.

Knuth K.H. 2012. Inferences about interactions: Fermions and the Dirac equation. U. von Toussaint (ed.) Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Garching, Germany, July 2012, AIP Conference Proceedings, American Institute of Physics, Melville NY., arXiv:1212.2332 [quant-ph].

Essays (top)

Knuth K.H. 2013. Information-based physics and the influence network. 2013 FQXi Essay Entry (
Download Essay

Papers (top)

Knuth K.H. 2003. Deriving laws from ordering relations. In: G.J. Erickson, Y. Zhai (eds.), Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Jackson Hole WY 2003, AIP Conference Proceedings 707, American Institute of Physics, Melville NY, pp. 204-235.

Knuth K.H., Skilling J. 2012. Foundations of Inference. Axioms 1(1), 38-73.

Goyal P., Knuth K.H., Skilling J. 2010. Origin of complex quantum amplitudes and Feynman's rules. Phys. Rev. A 81, 022109.

Goyal P., Knuth K.H. 2011. Quantum theory and probability theory: their relationship and origin in symmetry. Symmetry 3(2):171-206.'''

Knuth K.H., Bahreyni N. 2012. The Physics of Events: A Potential Foundation for Emergent Space-Time. arXiv:1209.0881 [math-ph].

Knuth K.H. 2012. Inferences about Interactions: Fermions and the Dirac Equation. MaxEnt 2012.

Knuth K.H. 2013. Information-based physics: an observer-centric foundation. Contemporary Physics, (Invited Submission).
arXiv:1310.1667 [quant-ph]

Talks (top)

Knuth K.H. 2014. FQXI 2014: Foundations of Probability and Information.
Opening Panelist Discussion on the Perspectives of Information at the FQXi 2014 Conference on Physics and Information, Vieques Island, Puerto Rico, USA on 6 Jan 2014.

Knuth K.H. 2013. Information-Based Physics: An Intelligent Embedded Agent's Guide to the Universe.
Presented to the Santa Fe Institute , Santa Fe NM on 26 Mar 2013.
Presented to Complexity Sciences Center at UC Davis, Davis CA on 9 Apr 2013.
Presented to Stanford Physics, Stanford University, Stanford CA on 12 Apr 2013.

In this talk, I propose an approach to understanding the foundations of physics by considering the optimal inferences an intelligent agent can make about the universe in which he or she is embedded. Information acts to constrain an agent’s beliefs. However, at a fundamental level, any information is obtained from interactions where something influences something else. Given this, the laws of physics must be constrained by both the nature of such influences and the rules by which we can make inferences based on information about these influences. I will review the recent progress we have made in this direction. This includes: a brief summary of how one can derive the Feynman path integral formulation of quantum mechanics from a consistent quantification of measurement sequences with pairs of numbers (Goyal, Skilling, Knuth 2010; Goyal, Knuth 2011), a demonstration that consistent apt quantification of a partially-ordered set of events (connected by interactions) by an embedded agent results in space-time geometry and Lorentz transformations (Knuth, Bahreyni 2012), and an explanation of how, given the two previous results, inferences (Knuth, Skilling 2012) about a direct particle-particle interaction model results in the Dirac equation (in 1+1 dimensions) and the properties of Fermions (Knuth, 2012). In summary, critical aspects of quantum mechanics, relativity, and particle properties appear to be derivable by considering an embedded agent who consistently quantifies observations and makes consistent inferences about them.

Knuth K.H. 2013. The foundations of probability Theory and quantum theory.
Presented at NASA Ames Research Center, 11 Apr 2013.
Presented at Google on 10 Apr 2013.

Probability theory is a calculus that enables one to compute degrees of implication among logical statements. Quantum theory is a calculus that enables one to compute the probabilities of the possible outcomes of a measurement performed on a physical system. Since the development of quantum theory (and probability theory), there have been many questions regarding the relationship between the two theories; some going as far as to question whether quantum theory is even compatible with probability theory. In this talk, I demonstrate precisely the relationship between probability theory and quantum theory by deriving both theories from first principles. This is accomplished by observing how consistent quantification of logical statements (Knuth, Skilling 2012) and quantum measurement sequences (Goyal, Skilling, Knuth 2010) are constrained by the relevant symmetries in each of the two domains (Goyal, Knuth 2011). It will be shown that the derivation of quantum theory is not only consistent with, but also relies on probability theory. In addition, these results highlight some important differences between inference in the classical and quantum domains.

Knuth K.H. 2010. Information Physics: The Next Frontier, MaxEnt 2007, Chamonix, France, July 2007.

At this point in time, two major areas of physics, statistical mechanics and quantum mechanics, rest on the foundations of probability and entropy. The last century saw several significant fundamental advances in our understanding of the process of inference, which make it clear that these are inferential theories. That is, rather than being a description of the behavior of the universe, these theories describe how observers can make optimal predictions about the universe. In such a picture, information plays a critical role. What is more is that little clues, such as the fact that black holes have entropy, continue to suggest that information is fundamental to physics in general.
In the last decade, our fundamental understanding of probability theory has led to a Bayesian revolution. In addition, we have come to recognize that the foundations go far deeper and that Cox’s approach of generalizing a Boolean algebra to a probability calculus is the first specific example of the more fundamental idea of assigning valuations to partially-ordered sets. By considering this as a natural way to introduce quantification to the more fundamental notion of ordering, one obtains an entirely new way of deriving physical laws. I will introduce this new way of thinking by demonstrating how one can quantify partially-ordered sets and, in the process, derive physical laws.
The implication is that physical law does not reflect the order in the universe, instead it is derived from the order imposed by our description of the universe. Information physics, which is based on understanding the ways in which we both quantify and process information about the world around us, is a fundamentally new approach to science.

Knuth K.H. 2010. The role of order in natural law, Workshop on the Laws of Nature: Their Nature and Knowability, Perimeter Institute, Waterloo, Canada, May 2010.

In the last four and a half centuries, we have found that we are able to identify laws of nature that are generally applicable, and because of this we have inferred that there is an underlying order to the structure and dynamics of the universe. In many cases we have been able to identify this order as being related to symmetries, which have enabled us to derive various laws, such as conservation laws. But in most cases, the role that order plays in determining natural law remains obscured. In this talk I will rely on order theory to demonstrate how symmetries among our descriptions of various states of a physical system result in constraint equations, generally called sum and product rules, which are ubiquitous in natural laws. The fact that much of the order that determines the structure of natural laws arises from relationships inherent in our particular description of a physical system implies that the laws of nature are more closely related to what we choose to say about the universe and how we say it rather than being fundamental governing principles.