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Let me give an example at first: All of the calculations that are carried out in quantum chemistry rely on approximation methods to the Schrödinger equation. While these methods sometimes give quite good approximative results, they are computationally very hard.

I wonder whether one could find a theory, which agrees with the approximative results of the Schrödinger equation, but is computationally less hard.

This question may be asked in a more general context:

If you have a working, well known theory $T$ which predicts certain results $R$, can there be a theory $T^*$, which makes exactly the same predictions $R$ in a less computationally hard way?

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    $\begingroup$ I'm not sure if they would be strictly be considered distinct theories if they made exactly the same predictions. You can of course find theories that reduce to other theories in some given limit. Quantum mechanics > Classical mechanics. Statistical mechanics > Classical thermodynamics. Unless you count finding a less computationally expensive way of performing a particular calculation involved in a theory. $\endgroup$
    – Charlie
    Commented Aug 6, 2020 at 19:15
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    $\begingroup$ Take a QFT that describes nature, perform an extremely non-linear field redefinition. You get a new theory, which is equivalent to the old one, but computationally much harder. Now read this sentence backwards. $\endgroup$
    – Thomas
    Commented Aug 6, 2020 at 19:16
  • $\begingroup$ @Thomas Okay this technically answers the question, but artificially making harder theories harder is one thing, making them easier another. Do you think one can make the Schrödinger theory easier? Or may there on the contrary even be a way to prove that it is already the simplest form? $\endgroup$
    – user224659
    Commented Aug 6, 2020 at 19:29
  • $\begingroup$ Well, maybe QCD at strong coupling is dual to a weakly coupled string theory, which would obviously be computationally easier. $\endgroup$
    – Thomas
    Commented Aug 6, 2020 at 19:32
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    $\begingroup$ Or some non-relativistic many body theory (n-body Schrodinger equation) is dual to some suitably deformed weakly coupled string theory. $\endgroup$
    – Thomas
    Commented Aug 6, 2020 at 19:56

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I think that there could be such equations for each condition in particular. For this I can give you a very basic relatable example of energy conservation where energy conservation can be applied on all the systems and body but is less efficient in systems where the forces are not well identified and defined. To overcome this approximations the work energy theorum can be put in place where the forces are not needed to calculate the net energy of the body (i. e. work) which in comparison to conservation of energy can also give the close approximation answers as obtained from conservation of energy

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