Semiclassical states for constrained systems

The notion of semiclassical states is first sharpened by clarifying two issues that appear to have been overlooked in the literature. Systems with linear and quadratic constraints are then considered and the group averaging procedure is applied to kinematical coherent states to obtain physical semiclassical states. In the specific examples considered, the technique turns out to be surprisingly efficient, suggesting that it may well be possible to use kinematical structures to analyze the semiclassical behavior of physical states of an interesting class of constrained systems.

Figure 1

The quotient $[(\Delta J_{1,2}{)}_{\mathrm{phy}}^2-(\Delta J_{1,2}{)}_{\mathrm{kin}}^2]/(\Delta J_{1,2}{)}_{\mathrm{kin}}^2$, plotted as a function of $x=|{\alpha }_2|$. The constant that fixes the energy difference is set to $k=10$. Note that the ratio approaches $1/2$, and is very close to $1/2$ even when $|{\alpha }_2|$ is not very large.

Figure 2

The quotient $⟨{\widehat{J}}_3{⟩}_{\mathrm{phy}}/(J_3{)}_{\mathrm{cl}}$, plotted as a function of $x=|{\alpha }_2|$. The constant that fixes the energy difference is set to $k=10$. Notice that the ratio is very close to 1 even for values of $|{\alpha }_2|$ that are not that large.

Figure 3

The quotient $[(\Delta {\widehat{J}}_3{)}_{\mathrm{kin}}^2-(\Delta {\widehat{J}}_3{)}_{\mathrm{phy}}^2]/(\Delta {\widehat{J}}_3{)}_{\mathrm{kin}}^2$, plotted as a function of $x=|{\alpha }_2|$. The constant that fixes the energy difference is set to $k=10$. Note that the physical fluctuations are smaller and very rapidly approach the kinematical ones.