Arxiv - Information Physics: The New Frontier provides examples of quantify partially-ordered sets and, in the process, derive physical laws.
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.
In this tutorial, I have shown how a variety of rules involving quantification arise as constraint equations to ensure that any quantification does not violate the underlying order. What is more striking is that this entire procedure is based on the quantification of order underlying our descriptions of physical reality—not necessarily physical reality itself. The consequence is that the physical laws we obtain are constraints on quantification imposed by our descriptions. This is where we arrive at Information Physics.
At the heart of this new methodology lies the valuation calculus which is applicable to any lattice. Associativity of the lattice join (or meet) gives rise to the sum rule. Associativity of the lattice product results in a product rule, which dictates how valuations are to be combined when taking lattice products. Associativity of changes of context result in a product rule for bi-valuations that dictates how valuations should be manipulated when changing context. The techniques based on projections are based on distinguishing a sub-lattice that can be used to employ valuations to quantify a poset in general.
Most exciting is the range of theories that have been successfully derived using this foundation: measure theory, probability theory, information theory, quantum mechanics, and special relativity. These results provide strong support for the claim that Information Physics, which relies on information about our descriptions of reality to derive physical laws, is a potentially useful general approach. With these positive examples as guideposts, we now aim to use these techniques to quantify new problems and derive new physical laws.
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