Is it possible to write down step-by-step instructions for inventing a new theory?
I've been wondering if there exists some 'recipe' or proceedure for inventing a new theory. Presumably some theorists must have different approaches to others but are there some general pointers that would always apply? Realistically, I should point out, I'm not anticipating stumbling upon anything more significant than an interesting little toy model but it would certainly be good practice.
In terms of QFT, I'm familiar with Lagrangian-mechanics, extremization-of-action and the Euler-Lagrange equations. I've also got a vague appreciation of what renormalization and gauge invariance imply but I don't really know how one would go about testing either condition.
If I had to guess I'd say, within the construct of QFT at least, that it would proceed like:
1/ add a new term to the Lagrangian,
2/ check if it breaks the gauge or Lorentz invariance,
3/ fiddle to see what new properties have manifested themselves.
Is this roughly how one would proceed? Futhermore, if I'm embarrissingly honest, I'm not really sure exactly how to do any of these things. If anyone could offer some elucidation I'll gladly split the Nobel Prize money with you!
Furthermore, if one wanted their theory to be a little more revolutionary (ie quantum gravity of course), what could we say about possible approaches to finding such a theory? Are there some similarities between the processes-of-invention that occurred for GR, QFT, String-theory, etc, that we can draw on to serve as a guide for inventing new theories?
For instance, GR required differential geometry and tensor calculus. Whilst these areas of mathematics existed pre-Einstein, it took a man of his extra-ordinary genius to use them to invent a completely new perspective of gravity that facilitated a model that does indeed describe nature. Here it seems his initial gut-feeling was that, since inertial motion was preferred in SR, there should exist a theory which prefers no state of motion, from the outset. It seems his 'hunch', when applied to gravity led him straight to what later became the equivalence principle. Oh, what I would do for a 'hunch' like that!
The invention of string theory (as a potential TOE), originally, appears to have been an accident. Veneziano, followed closely by Nambu, Nielsen and Susskind noticed that nuclear interactions of point-particles viewed instead as 1D stings could be modelled by the Euler-Beta function. Then followed a number of couple 'revolutions' and fantastic number of incredible breakthoughs leading to what is now, collectively, refered to as M-theory (thanks to Witten.) In this case it appears the starting point was Veneziano noticing that certain properties of the strong force resembled something he had used in another area of physics.
Please forgive my brutally selective account of these histories but I do so in an effort to explain my query. I've only (very briefly) mentioned the invention of GR and string-theory here as I know less-still about the stories of QM, QFT, EM, LQG,... I imagine almost every physicist has once dreamed that they might be the next Newton/Einstein/Dirac/Feynman etc.. I know there are some very learned people that frequent this site and I'd really love to hear you thoughts on this matter.