Hi all, I have a question about observables and the absolute value
I am considering an expectation value of a density operator of some system, and it appears as though taking the absolute value prematurely gives the wrong results
Why is this? I always thought the absolute value may be considered distributive
It also seems as if taking it prematurely makes the system non-integrable
@1010011010 I'm not sure what you mean by "distributive", but $\langle \lvert A\rvert\rangle \neq \lvert \langle A\rangle \rvert$.
In other words, you need to be clear whether you want to take the absolute value of the expectation value or the expectation value of the absolute value; those are not the same thing.
Certainly, though suppose my density operator is given by two factors, certainly one would expect the absolute value of the 2 state matrix element to be the absolute value of the individual factors, no?
In technical terms $\lvert <\hat\rho> \rvert = \lvert A\rvert \cdot \lvert B\rvert$, where $A$ and $B$ are those two factors
It appears as though in my system these two factors "communicate" for the lack of a better wording, or so the math suggests
Oh, sorry, ok, so basically the quantum operator is calculated between two states, which I'm calling a matrix element of the state space, so the matrix element could be $<\Psi|\hat\rho|\Phi>$ for any two states psi and phi
$\lvert f(x) g(x)\rvert$ = $\lvert f(x) \rvert \lvert g(x)\rvert$ is rather trivially true - you'll have to describe more precisely what you're doing here and how your result differs from that.
I calculated my density operator of the 1d bose gas by the conventional method and by using that identity I too thought was trivial (using the absolute value in that way) and by keeping all the phases
If I keep the phases and use a geometrical argument, parts of the expression become Dirac deltas after the thermodynamic transition
This imposes constraints that don't seem to be there when I use that rule
I have made a plan and will start with the derivation of the two state matrix element to see if the absolute value is already applied, that would explain a lot