I am currently reading the Feynman and Hibbs about Quantum mechanics and path integrals and I found something pretty confusing ( for me ) at page 72. At this page, they are replacing an integration on all paths by an integration on all the coefficient of their Fourier series. I know all continuous function ( and so are the paths we are talking about ) can be represented by a Fourier serie. But I also know that some non-continuous functions like a square-wave signal can also be approximated almost everywhere ( only at the discontinuity is it not correct anymore but we work in $L^2$ so we don't mind of discontinuities on a negligible set ) by a fourier serie.

Furthermore, searching some information on this topic, I found in the chapter "The uses of instantons" ( chapter 7 of the book Aspects of symmetry of Sydney Coleman I think ) I found that they made the same decomposition using the spectrum of the operator $ (-\frac{d^2}{dx^2} + V"(x))$ $$ (-\frac{d^2}{dx^2} + V"(x))\psi_n = \lambda_n \psi_n $$

where $V"$ is the second derivative of the potential. They were pretending it was equivalent to the integral along all paths... But what are the condition ( in addition to the need to be hermitian ) to the operator in order that his spectrum generate all the continuous functions.

Furthermore, this includes more terms than it should since non-continuous functions can be represented by Fourier series for instance or by orthogonal polynomials. In addition, the spectrum of the operator $x$ contain distribution so that we create lot more "paths" ( not only the continuous one ).

I am not sure I was clear enough but all I want to understand is how to be sure the integrals over the fourier coefficient ( or decomposition in another basis )is equivalent to the integral over all paths since we consider additionnal paths and how to be sure the spectrum of an hermitian operator generate all the paths...



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