Some branches of mathematics like functional analysis do not on first encounters seem to have any possible applications.Can someone please give me some examples of applications of functional analysis in phsics?I believe there can hardly be any absolutely unapplied branch of mathematics:it is just I am curious to know how this particular field of study in mathematics has been instumental in physics or possibly other sciences.Thank you.
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7Essentially all of quantum mechanics? Or quantum field theory in particular? – hmakholm left over Monica Jan 31 '14 at 18:05
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1Isn't the whole of Lagrangian mechanics founded on the idea "hey, newton's laws can be reformulated in terms of functional analysis"? – Jack M Jan 31 '14 at 18:47
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2@JackM: this is completely wrong, unfortunately. – Alexander Grothendieck Jan 31 '14 at 18:54
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Historically functional analysis emerged from applications; it is therefore obvious that it has "applications". In quantum mechanics the obserables are operators on a hilbert space, and one wants in particular to realize the "canonical commutation relation" $[q,p]=1$ for self-adjoint operators $q$ and $p$ (position and momentum) on a hilbert space; it is easily seen that this implies that at least one of them must be unbounded and hence cannot be defined on the whole space (Hellinger-Töplitz). From this it follows that one must take care of all technicalities which come along with unbounded operators. Also the spectrum of the operator is related to the measurement of the observable which it represents.
Alexander Grothendieck
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