- #1
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This has come up in a number of threads, so I made this thread to talk about it.
##\begin{eqnarray*}
\langle A(s)A(t)\rangle_\psi &=& \langle \psi| U^\dagger (s) A U(s) U^\dagger(t) A U(t)|\psi\rangle\\
&=& \langle \psi(s)| A U(s-t) A |\psi(t)\rangle\\
&=& \langle \psi(s)| \left( \sum_i |a_i\rangle \langle a_i| \right) A U(s-t) A \left( \sum_j |a_j\rangle \langle a_j| \right) \psi(t)\rangle\\
&=& \sum_{i,j} a_i a_j \langle \psi(s)| a_i\rangle \langle a_i| U(s-t) |a_j\rangle \langle a_j| \psi(t)\rangle\\
\end{eqnarray*}##
For each term in the sum, we have an initial probability amplitude ##\langle a_j| \psi(t)\rangle##, a propagator-like quantity ##\langle a_i| U(s-t) |a_j\rangle## and a final probability amplitude ##\langle \psi(s)| a_i\rangle##, and we sum over all possibilities weighted with the product of the corresponding eigenvalues. This doesn't look completely meaningless to me, although I'm scratching my head a bit about what it's actual meaning could be. ;-)
Any thoughts?
I think that an interpretation of this in the Schrödinger picture should be possible at least in the style of Feynman. This would go something like this.A. Neumaier said:To define a time correlation ##\langle A(s)A(t)\rangle## one needs a family of operators ##A(s)## that depend on time, hence the Heisenberg picture. One can convert the expression into one in the Schroedinger picture, but the resulting expression has no meaning without its interpretation in the Heisenberg picture!
##\begin{eqnarray*}
\langle A(s)A(t)\rangle_\psi &=& \langle \psi| U^\dagger (s) A U(s) U^\dagger(t) A U(t)|\psi\rangle\\
&=& \langle \psi(s)| A U(s-t) A |\psi(t)\rangle\\
&=& \langle \psi(s)| \left( \sum_i |a_i\rangle \langle a_i| \right) A U(s-t) A \left( \sum_j |a_j\rangle \langle a_j| \right) \psi(t)\rangle\\
&=& \sum_{i,j} a_i a_j \langle \psi(s)| a_i\rangle \langle a_i| U(s-t) |a_j\rangle \langle a_j| \psi(t)\rangle\\
\end{eqnarray*}##
For each term in the sum, we have an initial probability amplitude ##\langle a_j| \psi(t)\rangle##, a propagator-like quantity ##\langle a_i| U(s-t) |a_j\rangle## and a final probability amplitude ##\langle \psi(s)| a_i\rangle##, and we sum over all possibilities weighted with the product of the corresponding eigenvalues. This doesn't look completely meaningless to me, although I'm scratching my head a bit about what it's actual meaning could be. ;-)
Any thoughts?