How Does Trace Relations in Fock Space Reflect on Quantum Expectations?

[naima]

Consider the state $\rho_n$ obtained by applying n times the same creation operator on the vacuum and an observable A.
How is $Tr(\rho_n A)$ related to $Tr(\rho_1 A)$ ?
And if $\rho$ is a coherent state?

 

[A. Neumaier]

How is $Tr(\rho_n A)$ related to $Tr(\rho_1 A)$ ?

It depends on the form of $A$. If it is a 1-particle operator, it is $n$ times the single-particle trace. Otherwise there is no simple relationship.

 

[naima]

I think that this is not valid when the one particle operator is a projector.

 

[A. Neumaier]

I think that this is not valid when the one particle operator is a projector.

Let $B$ be an operator on single-particle space and $O_1(B)$ be the corresponding 1-particle operator. Then, according to the rules of statistical mechanics, $O_1(B)\psi$ where $\psi(x_1,\ldots,x_n)=\psi(x_1)\otimes \cdots \otimes\psi(x_n)$ is obtained by applying $B$ to only one 1-particle wave function in turn and summing the results. This is independent of the form of $B$ and implies my claim for separable states, and by linear combination for general states.

Note that your $A$ is my $O_1(B)$. It cannot be a projector. Only $B$ can be one.

 

[naima]

And what about if the state $\rho$ is a coherent state? Have you to invoke an infinite sum on a serie?
How can you describe things like:
$<\alpha | n |\alpha> = |\alpha |^2$

 

[A. Neumaier]

And what about if the state $\rho$ is a coherent state? Have you to invoke an infinite sum on a series?
How can you describe things like:
$<\alpha | N |\alpha> = |\alpha |^2$

If one defines coherent states via an infinite sum of number states, one can calculate expectations via the evaluation of a corresponding infinite double sum. But number states are very clumsy to work with...

If one defines coherent states as normalized eigenstates of the annihilator operator, $a |z\rangle = z|z\rangle$, and expresses the operators whose mean is taken in terms of a normally ordered expression in creation and annihilation operators one can work out the expectations in a far more natural, finite way:
$\langle z| | N |z\rangle = \langle z| | a^*a |z\rangle = |a |z\rangle|^2 = |z |z\rangle |^2 = |z|^2 ||z\rangle |^2 = |z|^2.$

 

[naima]

In your old Arxiv paper, you wrote that
"An ensemble is a mapping − that assigns to each quantity f ∈ E its
expectation <f> ∈ C such that... E1, E2, E3, E4,"
Now that you take into account creation and annihilation operators, did you add properties after E4 so that the expectation value of a single system appears as a peculiarity of a general case?

 

[A. Neumaier]

In your old Arxiv paper, you wrote that
"An ensemble is a mapping − that assigns to each quantity f ∈ E its
expectation <f> ∈ C such that... E1, E2, E3, E4,"
Now that you take into account creation and annihilation operators, did you add properties after E4 so that the expectation value of a single system appears as a peculiarity of a general case?

The creation and annihilation operators are part of the algebra E of quantities; you should think of a harmonic oscillator where number states are states with a definite number of excitations, not with a definite number of particles. A general Fock space is just the Hilbert space for a system of arbitrarily many oscillators - see my post in another thread and its subsequent discussion.

Nothing changes in the general properties of an ensemble as defined in that paper. In my thermal interpretation the notion of ensemble is to be understood not as an actual repetition by repeated preparation. It should be understood instead in the original sense used by Gibbs - who coined the notion of an ensemble as a collection of imagined copies of which only one is actually realized -, giving an intuitive excuse to be able to use the statistical formalism for a single system. What is conventionally called expectation (also in my old paper) becomes in the thermal interpretation simply the uncertain value.