Effective approach to the problem of time: general features and examples

Abstract

The effective approach to quantum dynamics allows a reformulation of the Dirac quantization procedure for constrained systems in terms of an infinite-dimensional constrained system of classical type. For semiclassical approximations, the quantum constrained system can be truncated to finite size and solved by the reduced phase space or gauge-fixing methods. In particular, the classical feasibility of local internal times is directly generalized to quantum systems, overcoming the main difficulties associated with the general problem of time in the semiclassical realm. The key features of local internal times and the procedure of patching global solutions using overlapping intervals of local internal times are described and illustrated by two quantum mechanical examples. Relational evolution in a given choice of internal time is most conveniently described and interpreted in a corresponding choice of gauge at the effective level and changing the internal clock is, therefore, essentially achieved by a gauge transformation. This article complements the conceptual discussion in arXiv:1009.5953.

Figures (9)

• Figure 1: A typical classical configuration space trajectory is a parabola with the peak value of $t$ dependent on ${p_t}_0$ and the separation of branches dependent on $p_0$. The orientation of evolution, indicated by the arrows, is consistent with $p_0<0$ and ${p_t}_0>0$. We refer to the left branch (solid) as "incoming" or "evolving forward in $t$", the right branch (dashed) as "outgoing" or "evolving backward in $t$".
• Figure 2: Schematic plots of the real part of $t$ (top) and the imaginary part of $t$ (bottom) against the flow parameter $s$.
• Figure 3: Top: evolution of moments $(\Delta q)^2$ (solid) and $\Delta(qp)$ (dashed) in $t$-gauge ($(\Delta p)^2 = {\rm const}$). Somewhere after $s=2.3$ the spread $\Delta q:=\sqrt{(\Delta q)^2}$ becomes comparable to the expectation values, as $\Delta q/\eta >.1$, and the semiclassical approximation breaks down in $t$-gauge. Bottom: corresponding effective trajectory (solid) and the related classical trajectory (dashed); the effective trajectory quickly diverges after $s=2.3$.
• Figure 4: Plot of the semiclassical trajectory evolved past the extremal point in $t$-gauge (solid part of the trajectory), by temporarily switching to the $q$-gauge (dashed part of the trajectory). Dotted vertical lines indicate the points where gauges were switched.
• Figure 5: Square amplitude of a coherent solution to the constraint (\ref{['quant-rov']}), with $M=50\hbar$, peaked about a circular configuration space trajectory.