Representations of spacetime alternatives and their classical limits

Different quantum mechanical operators can correspond to the same classical quantity. Hermitian operators differing only by operator ordering of the canonical coordinates and momenta at one moment of time are the most familiar example. Classical spacetime alternatives that extend over time can also be represented by different quantum operators. For example, operators representing a particular value of the time average of a dynamical variable can be constructed in two ways: first, as a projection onto the value of the time-averaged Heisenberg picture operator for the dynamical variable and, second, as the class operator defined by a sum over those histories of the dynamical variable that have the specified time-averaged value. We show by both explicit example and general argument that the predictions of these different representations agree in the classical limit and that sets of histories represented by them decohere in that limit.

Figure 1

The real and imaginary parts of the function $E_{\Delta }(z,\ell )$ defined Eq. (4.7). These graphs are for $\delta ∕\lambda =15$. Already at this value the real part is a reasonable approximation to a top-hat function $e_{\Delta }(z∕\ell )$ and the imaginary part is small. As $\hslash$ approaches zero, the value of $\lambda$ becomes very small [cf. Eq. (4.6)], the approximation of the real part to a top-hat function in both Eq. (4.5) for $⟨x^{″}\mid {\widehat{P}}_{\Delta }\mid x^{\prime }⟩$ and Eq. (4.9) for $⟨x^{″}\mid {\widehat{C}}_{\Delta }\mid x^{\prime }⟩$ becomes better and better, and the imaginary part becomes increasingly negligible.

Figure 2

Graph of $\sqrt{d}\psi$ for the harmonic oscillator where $\psi$ equals $\psi (x^{″},0)$ (solid line), $\psi (x^{″},T)$ (small dashed line), $⟨x^{″}\mid {\widehat{C}}_{\Delta }\mid \psi ⟩$ (large dashed line), and $⟨x^{″}\mid {\widehat{P}}_{\Delta }\mid \psi ⟩$ (thick line). For (a) and (b), $\delta =d$. For (c) and (d), $\delta =10d$. In all graphs, $T=t_{\text{spread}}∕4$.

Figure 3

Graph of $\sqrt{d}\psi$ for the free particle where $\psi$ equals $\psi (x^{″},0)$ (solid line), $\psi (x^{″},T)$ (small dashed line), $⟨x^{″}\mid {\widehat{C}}_{\Delta }\mid \psi ⟩$ (large dashed line), and $⟨x^{″}\mid {\widehat{P}}_{\Delta }\mid \psi ⟩$ (thick line). For (a) and (b), $\delta =d$. For (c) and (d), $\delta =10d$. In all graphs, $T=t_{\text{spread}}∕4$ where $t_{\text{spread}}$ is given by Eq. (4.11).