Re: Wave Function of the Universe?

Arnold Neumaier a =E9crit :
> Seratend wrote:
>
> > Arnold Neumaier a =E9crit :
> >
> >>For example, if we measure the mass of a piece of metal, we
actually
> >>measure the expectation of the mass operator <M> (M is the sum of
> >>all the particle masses).
> >>
> > This is only true, in the absolute, if the piece of metal is made
of an
> > infinite number of particle masses (convergence in law). What we
really
> > measure is the value of M=3D sum_i Mi and not <M>.
>
> No. What we measure is an approximation of
> <integral_Omega dx a^*(x)Ma(x)> over the region Omega of interest.
> This is the expression that figures in statistical mechanics
> derivations of macroscopic elasticity theory.
>

We can apply the statistics to small independent areas. This does not
change the fact that at the end what we really measure on this local
are is M_area=3D sum_{i in the area} Mi and not <M>_{area} (what you call
the approximation when the area is not well defined). An instance of
the object has well defined (may be unkonwn to the observer) values of
Mi in all areas (independence of Mi random variables or commuting
observables). However, at the end one sums all over these areas (the
extensive property of M) to obtain the value we really measure.

In my opinion, this is simply the application of the law of large
numbers and the grand and micro canonical ensembles just use this
property with additional hypotheses.


Seratend.


> Statistical mechanics must use the grand canonical ensemble
> for calculations in local equilibrium such as elasticity theory
> or hydrodynamics, since the cell boundaries in the coarse graining
> are vague and hence the number of particles cannot be assumed
> constant in each cell. Hence all fields are expectations.
>
> Please resume discussion of this in the new thread
> ''Wave Function of the Universe?'' opened by Baugh,
> where I copied this answer.
>
>
> Arnold Neumaier

.

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