Refactoring physics
TL;DR: Before looking for new theories, we need to address the 'math smells' of the current ones
If you are a software engineer, you appreciate the need to refactor parts of your codebase. Refactoring is the art of taking a piece of software and cleaning up its internal structure while keeping the functionality intact. Over time, any codebase starts to morph under the addition of new functionality, ad-hoc fixes, and scope changes. Therefore, to move forward and make the project more sustainable, it may be useful to reorganize the code and rewrite some parts so that the overall project is more maintainable and easier to extend. I strongly believe that the mathematical foundations of physics needs a massive refactoring. And the more I work toward that goal, the more I realize how the problems people want to solve in the foundations of physics cannot be solved without it.
The telltale signs that your software project needs refactoring are the so called “code smells.” Typical examples are code duplications, very long and convoluted function implementations for things that, in principle, should not be complicated, unclear valid ranges for variables and data structures, and so on. None of these are technically bugs; they do not prevent the code from running… they just don’t feel right, make the code a lot harder to work on, and sometimes prevent the practical implementation of new functionality. We have similar problems in the mathematical structure of our physical theories. There are “math smells.”
In our work, we showed that classical mechanics, in the form of Hamilton’s equations, can be understood in 12 different equivalent ways. The basic physical assumption is that of deterministic and reversible evolution, and that can be implemented by requiring that the count of states is preserved through time, that the information entropy is preserved through time, that the thermodynamic entropy is preserved through time, or that the uncertainty (in some precise sense) is preserved through time. While the physical concept is the same, the mathematics one may use is different: it may be measure theory, information theory, differential geometry, or statistics. That is, roughly the same ideas are implemented in the math multiple times in different ways. This is a “math smell:” the math does not feel quite right. There should be a single unified mathematical structure underneath it all.
If one looks at all the core physical theories, there are some common structures. Conceptually, we always have a state and some laws of evolution. In classical mechanics (relativistic or not), quantum mechanics, and field theories (classical or quantum), we have the notion of conjugate quantities, the notion of transformations generated by a quantity, and a type of bracket (Poisson or commutator) that tells us how one quantity evolves under the transformation generated by a quantity. This is implemented slightly differently in all different theories. This is another “math smell:” there should be a common way to abstract this structure to share the math, and there should be a physical motivation that justifies that structure.
If one looks at quantum mechanics, it is clear that not all mathematical objects are physically well-defined. While we declare quantum states to be rays in a Hilbert space, not all rays can correspond to physical objects as this results in states with infinite expectation values. This is another “math smell:” it should be clear what is the valid range of mathematical objects in our physical theory.
As with code smells, math smells do not really prevent us from making calculations. They do not prevent us from getting the job done. But they make the work harder, they prevent us from a deeper understanding, and, likely, they make it practically impossible to find new well-posed theories.
In our physical mathematics approach, we have seen that, once the requirements of a physical theory are well-specified, they can be translated into mathematical definitions. We are seeing that simply requiring that we are able to describe ensembles, and statistical collections, constrains a theory so much that we are able to recover a lot of the currently used mathematical structure. While we develop new math, it should simply re-express the same concepts we already have in the current physical theories in a clearer and more general way, and that’s it. Only a better reformulation of the theories of today can point the way to the theories of tomorrow.