So I'm currently reading Wald's QFT on curved spacetime. In section 2.3, he constructs the quantum theory of $n$ decoupled harmonic oscillators by choosing $\mathcal{H}$ as the set of positive frequency solutions and then choosing the Hilbert space to be $\mathcal{F}_s(\mathcal{H})$ - the symmetric Fock space of $\mathcal{H}$. He then defines the Heisenberg position operator to be $$ q^\prime_{iH}=\xi_i(t)a_i+\bar{\xi}_i(t)a^\dagger_i $$ where $\xi_i(t)$ is the solution with the $i$th harmonic oscillator excited and $a_i$ is the annihilation associated with said solution. Also, the momentum operator is defined as $p^\prime_{iH}=dq^\prime_{iH}/dt$. So the question is: how do I go about showing that, with these definitions, $$ [q^\prime_{iH},p^\prime_{jH}]=i\delta_{ij}? $$ And I ask this because, although Wald defined $[a_i,a^\dagger_j]$, I get terms like $[\xi_i(t)a_i,\bar{\xi}_j(t)a^\dagger_j]$ when I expand the equation above. And in all my ignorance, I don't know for certain what to do in this case.
$\begingroup$
$\endgroup$
3
-
1$\begingroup$ If I understand the notation correctly, $\xi_i(t)$ and $\bar \xi_j(t)$ are numbers, not operators. In that case $$[\xi_i(t)a_i,\bar{\xi}_j(t)a^\dagger_j]=\xi_i(t)\bar{\xi}_j(t)[a_i,a^\dagger_j]$$ and the only remaining commutator is known. Does this solves the issue? $\endgroup$– GoldCommented Feb 16, 2022 at 13:43
-
$\begingroup$ Well, that is something I'm a little bit confused. It's stated that $\xi_i(t) \in \mathcal{H}$, so it's a curve in phase space. If I consider it as a "number", as you suggested, this actually solves the issue. And thank you. $\endgroup$– Rafael ManciniCommented Feb 16, 2022 at 17:01
-
1$\begingroup$ Actually, now I see that it's supposed to be a number indeed. I got confused with the notation. $\endgroup$– Rafael ManciniCommented Feb 16, 2022 at 17:23
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
I AM confused If we considero p as operator in a curved soacetime It would have a relation [p,p]. Proporcional tô the Riemann curvature tensor which vanishes..in flat space ? I mean Heisenberg commutation relations would chance de tô tô Ricci identity no? Thanks
Garcia
