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A. Neumaier
Apr21-11, 11:04 AM
Sci Advisor
P: 1,943
Quote Quote by strangerep View Post
Leaving aside the interpretation, and looking instead at how you formulate QM,
starting with an algebra of observables, then states and relying on a list of
axiomatic (Whittle-style) properties of expectations, I'm wondering whether one
can indeed account for all features of QM that way...

Consider the Cauchy (or Breit-Wigner) distribution that gives the probability
distribution of the lifetime of unstable particles. The usual expectation, variance,
etc, are undefined for that distribution but it's clearly an important part of quantum
physics. How then do you get a Breit-Wigner distribution if you've started the
theory from expectations?
Only _bounded_ quantities _must_ have an expectation. For unbounded quantities the expectation need not exist. Thus (as in C^*-algebras), one can always go to the exponentials e^{is A} of a self-adjoint but unbounded quantity. in probability theory, the function defined by the expectations f(s):=<e^{isa}> is called the characteristic function of A. it completely characterizes the distribution of functions of A, and is the right thing to study in case of cauchy-distributed A.