Theorem: Joint spectral concentration precisely in the commuting sector

If $A_{ij}=0$ for every $i,j$, the self-adjoint fields $\Phi (v_j)$ strongly commute. Their scaled counterparts have a joint spectral law ${{\mu }_{ϵ}}^{\mathrm{joint}}$ in every state considered above.

For general residual states,

where $o=(o_1,\dots ,o_J)$ and the metric on ${\mathbb{R}}^J$ is Euclidean.

For the displaced vacuum,

Degenerate Gaussian covariance is allowed.

If some $A_{ij}\ne 0$, the corresponding operators do not strongly commute. In that case no common joint spectral law of these operators is asserted.

Proof. When $A_{ij}=0$, (eq:Weyl) shows that their unitary groups commute for all real parameters. Commuting spectral measures follow from the spectral theorem for commuting unitary groups. The squared Euclidean distance to $o$ is the sum of the squared coordinate distances. Applying (eq:mse) to every marginal proves (eq:joint-general).

Vacuum normal ordering gives

For $V={\sum }_j{t}_j{v}_j$, displacement therefore gives the joint characteristic function

This identifies (eq:joint-vacuum).

Conversely, strong commutativity would make the two unitary groups commute for every pair of parameters. Their Weyl interchange phase is $e^{-2\mathrm{i}st{A}_{ij}}$; it cannot be one for every $s,t$ when $A_{ij}\ne 0$.