Standard perturbation theory - what exactly is meant?

omg!

"standard perturbation theory" - what exactly is meant?

hi,

standard perturbation theory can be viewed as a special case of the
semiclassical method, where one quantises fluctuations around trivial
classical solutions, whereas in soliton or instanton physics, one does the
same thing around non-trivial, non-perturbative classical solutions.
the quantization around trivial classical solutions can be done via the minkowskian path integral, while instanton solutions arise in the euclidean formalism (using a semiclassical approximation, in the sense of stationary phase approximation and steepest descent).

however, i'm not sure what the author (Rajamaran) means with standard perturbation theory and the assertion that instanton solutions are non-perturbative classical solutions.

Thanks a lot.

(i hope my question makes sense, i'm quite new at qm)

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DarMM

Gold Member
Instantons are lowest action solutions of the Classical Field Theory that live in a different topological sector from the typical lowest action solution, the Classical "vacuum" of $A_{\mu} = 0$. Different Topological sector means that the Fiber Bundle of which they are a section is not homeomorphic to the Fiber Bundle of the vacuum solution (despite them both having the same base space). It is not really possible to discuss Instantons without the language of Fiber Bundles, so I cannot simplify it beyond that without explaining the theory of Fiber Bundles.

They are "nonperturbative" because typically these Classical solutions contain information about the corresponding Quantum Field Theory that is not accessible from Perturbation theory about the Classical vacuum.

"Standard perturbation theory - what exactly is meant?"

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