Is there a mathmatical theory to explain all physical phenomena?

Question

Does anybody know of any physicists or mathematicians who have put forward a theory to everything, for which this mathematical theory would explain all physical phenomena in the universe in one equation?

Answer 1

In one context, not yet. In another context, no, it does not make sense.

The not-yet context is the so-called theory of everything.

https://en.wikipedia.org/wiki/Theory_of_everything

This is a goal of a theory that combines all physical forces, namely, electromagnetic, strong and weak nuclear, and gravity.

String theory is a candidate, but so far, it has not got any strong experimental evidence. And it does have some significant theoretical challenges, specifically, non-locality in that it has finite sized fundamental objects. This is a feature that produces a lot of push back.

https://www.sciencedirect.com/science/article/pii/0550321389904616

There are some other candidates, but they have their own problems. Gravity is a tough thing to add to the other forces. So far, nobody has managed to produce a theory and match it up to achievable experimental tests.

The doesn't make sense context is a question of scale. The theory of everything just discussed won't help with a lot of things.

For example, if you are studying the behavior of crystals, you want to know about things that determine the size, composition, configurations, etc., of crystals. This tends to be at the size scale, and energy scale, of atoms and molecules. Say a few eV to a few KeV kind of scale, and one-atom sorts of distances out to a few 1000 atoms distance. I may be getting the low end wrong, because there may be important features of crystals at lower energy. Maybe phonons or some such thing.

The important thing in this context is, worrying about quarks when you are doing crystal structure is just a distraction.

On the other hand, if you are doing nuclear physics, you are often interested in what is happening in a single nucleus. In this case, you are again interested in energies of nuclear transitions. And you are typically interested in distances typically of at most a few times the diameter of a nucleus. You may be interested in quarks. And your energy range is possibly a few 100 MeV down to very small energy transitions.

In each of these example, gravity is a very small effect, at least in the usual experiment. It is possible to construct some very special experiments to see gravity effects on nuclear interactions, but it's not the usual thing. So worrying about gravity in trying to work out the energy levels of a nucleus is going to be seriously distracting and use up your effort without adding much.

Lots of other possible levels. For example, if you are trying to get a probe to Jupiter, you probably want to think about gravity, maybe inter-planetary gas, maybe a little bit about solar wind, maybe about some electromagnetic fields. But probably you are not going to be worrying about quarks.

So we are going to be keeping different explanations at different scales of distance, energy, time, and application. So we don't want a "theory of everything" in that context.

Answer 2

Firstly, physical theories use mathematical structures to model reality - a mathematical theory is about mathematical abstraction. Max Tegmark's lofty claim of the Mathematical Universe, in which anything that exists mathematically must exist physically, is an extreme case of what you ask, where by definition mathematical structure would always precede physical conception. https://arxiv.org/abs/0704.0646

So, you seem to be asking if an abstract mathematical theory has ever turned out to be a good Theory of Everything (TOE)? The answer is: No, not yet. One could say that string theory is a mathematical model that we are trying to bend to fit our current picture of reality, but has not acheived it yet. String theory is a concoction of numerous physicists and mathematicians, some trying to find the TOE and some just following their curiosities. To call string theory a scientific theory is an insult to any real scientific theory, like General Relativity (GR) or Evolution.

Secondly, it seems everyone and their uncle has their own (TOE) these days... Of course Einstein tried using GR (which is a good starting point due to its background independence) to find a unified theory of all of physics, and it is now a so called "Holy Grail" of modern physics research (hence, why everyone and their uncle nowadays has their own TOE).

I appreciate puppetsock's answer, since they explain how difficult it is to package ALL of physics - across all scales of energy - into one theoretical framework. It's interesting to note, that Sir Arthur Eddington spent the last years of his life creating a TOE, where his major focus was on deriving theoretically the values of the fundamental constants of nature, but due to his rather unorthodox (and sometimes flawed) understandings of quantum mechanics the community was largely uninterested in his attempt. https://arxiv.org/ftp/arxiv/papers/1510/1510.04046.pdf

A more modern attempt is the Wheeler-DeWitt equation https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equation which tries to unify quantum mechanics and GR. This approach is kind of what you ask for: one equation that determines the evolution of everything in the universe. Of course, this has not yet achieved the TOE status, and it has some serious philosophical issues