# Why is Perturbation Theory named “a theory”?

From Wikipedia:

In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. Usually a deductive system is understood from context. An element ϕ ∈ T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms ( T ⊢ ϕ ) is called a theorem of the theory. Every axiom is also a theorem. A first-order theory is a set of first-order sentences.

On the other hand,

A scientific theory is an explanation of an aspect of the natural world that can be repeatedly tested, in accordance with the scientific method [...]. Are testable and make falsifiable predictions. They describe the causes of a particular natural phenomenon and are used to explain and predict aspects of the physical universe or specific areas of inquiry

Perturbation Analysis or Perturbation Methods sound reasonable, but why Theory? How is it a theory?

Just a physicist asking mathematicians. Looking forward to interesting insights.

Thank you

• There are other fields than "mathematical logic" and "science in general". Theory could mean something different there. – Arthur Jan 10 '18 at 23:19
• – littleO Jan 10 '18 at 23:22
• Mathematical fields are often called "theory" : group theory, field theory,... even when they're not, say, first order theories : ergodic theory for instance, information theory. The word "theory" can have many meanings, even formally defined one (a Lawvere theory is very different from a first order theory, and they're both different from a homology theory) – Maxime Ramzi Jan 10 '18 at 23:50
• Assuming the word theory in "perturbation theory " means something precise is misguided. Theory means in that context, and in many others, just "bunch of results, methods, heuristics and what not that's e use to understand something". – Mariano Suárez-Álvarez Jan 11 '18 at 0:02
• (misguided because we rarely use words in technical meanings) – Mariano Suárez-Álvarez Jan 11 '18 at 0:03

## 3 Answers

Perturbation theory is a mathematical theory. It is a set of mathematical methods for solving problems, that has been used long before quantum mechanics, general relativity or quantum field theory. It has applications in purely mathematical problems as well as the physical sciences. Therefore, I'd side with the mathematical definition for the term "theory".

Perturbation Theory is specific to physics as a theory because it's useful in delivering tractable, approximate solutions for complex physical models. While it is fully justified using mathematics, I suppose you can call it a theory because it does a good job of describing natural phenomena, especially in quantum mechanics/field theory and general relativity. It gives results in excellent agreement with experiment for multiparticle and multiforce interactions, along with studying the effects of splitting degenerate states.

• That is the first thought that comes to mind for a physicist, but I wanted to know the point of view of a pure mathematician, who might not even care about the behavior of nature. – Á. F. López de Quadros Jan 11 '18 at 0:18
• "fully justified using mathematics" - In QFT we don't know whether the perturbation series converges and there are reasons to think not. – Keith McClary Jan 13 '18 at 18:33

Another meaning of "theory", both in mathematics and in English in general, is something like "branch of study". In math, we have "theory of groups", "matrix theory", "number theory", "theory of equations", "probability theory", and so on, including "perturbation theory". In literature, there is just plain "theory". My dictionary gives as one meaning: "That department of an art or technical subject which consists in the knowledge or statement of the facts on which it depends, or of its principles or methods, as distinguished from the practice of it" (illustrating this with an example from Grove's Musical dictionary, about music theory). Others: "mental view", "a conception or mental scheme of something to be done, or of the method of doing it; a systematic statement of the rules or principles to be followed" (with an illustration from a fencing manual: "Theorie without Practice will serve but for little"), etc.