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I wonder how you would describe a correct use of the term "ansatz" in English physics literature? In German, "ein Ansatz" simply means "an approach". Could we say, in the context of theoretical physics, that an ansatz stands to a solution like a hypothesis stands to theory? (E.g. we make an ansatz that a solution of differential equation is given by an exponential function, and then we find out that this indeed is the case.)

user373714
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    Short answer is yes. Ansatz basically means guess. – Matt Hanson Sep 13 '23 at 11:20
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    from this definition, my answer is no. https://www.merriam-webster.com/dictionary/ansatz "an initial estimate of the solution to a mathematical or technical problem that is used to guide work to a more precise answer" . A hypothesis leading to a theory is more like an axiom in mathematics, necessary for the calculations, IMO – anna v Sep 13 '23 at 11:25
  • I was perhaps a bit crude, but that is, ultimately, an educated guess. I think the analogy presented here works quite nicely. – Matt Hanson Sep 13 '23 at 13:46

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I would say no, but it depends on what you mean by a "theory".

An ansatz is a guess or trial answer to an equation. The value of ansätze is that it is often much easier to check if they are actual solutions than it is to come up with the solution in the first place.

Sometimes you see a sharp distinction made between hypotheses and scientific theories, where the latter have received and withstood some testing. If you use this definition of theory, there are indeed some similarities between the ansatz-solution and hypothesis-theory pairs, although a solution is much more final than a theory, which can be updated or overturned upon discovering new evidence.

However, in physics (and especially theoretical physics), the word "theory" is usually used in different ways, closer in meaning to a mathematical theory. For example, quantum field theory is viewed as a framework, rather than a specific theory. You don't experimentally test QFT as much as you test specific models within it. And often people study a specific QFT Lagrangian and call that a theory, whether or not it's particularly realistic, see for example $\phi^4$ theory.

Anyon
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By my interpretations "Ansatz" is more like "one of the ways to skin the cat" or "approach". Often there are different ways to approach the problem and each of those are an "Ansatz".

A simple example of this is "choice of coordinate system". You can solve a certain 3D problem in rectangular, spherical, cylindrical or other coordinate systems. These are all different "Ansätze". They are all valid approaches but some are more practical then others for your specific problem.

Is Ansatz compared to Solution like Hypothesis compared to Theory?

No. A hypothesis is either true or false. An Ansatz is a potential way to solve a problem. They are generally all "true" in a sense that they are based on correct physicals laws and/or mathematical rules. The difference is practical: what is the amount and type of work required to solve the problem. There are can indeed be "bad" Ansätze. It's not "factually wrong": it's just that the work required and the math around it becomes too unwieldy to solve.

It that sense the quality of an Ansatz lies in the eyes of the beholder. If you are really good with a hammer, you will prefer Ansätze that turn everything into nails. If you are better with a screw driver you want everything to look like a screw.

Hilmar
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    An ansatz can be wrong. If, for example, the solution depends on radius but you don't have radius in your ansatz, it won't work. – John Doty Sep 13 '23 at 13:42
  • Thank you. You are right - «Ansatz» means an approach (my bad - it’s been a long time). Have fixed this in the original post.) – user373714 Sep 13 '23 at 14:24
  • @John Doty Exactly. That’s my issue here. Ansatz can be wrong and lead to nowhere, like a hypothesis can be wrong. We surmise that the solution is given by an exponential function but actually it should be something else. – user373714 Sep 13 '23 at 14:30
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    @user373714 Theory has the same difficulty. We surmise that the chosen theory applies to the physical situation we're attempting to model. Sometimes it doesn't. – John Doty Sep 13 '23 at 16:08