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

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 [M. Bojowald, P. A. Höhn, and A. Tsobanjan, Classical Quantum Gravity 28, 035006 (2011).].

Figure 1

A typical classical configuration space trajectory is a parabola with the peak value of $t$ dependent on $p_{t0}$ and the separation of branches dependent on $p_0$. The orientation of evolution, indicated by the arrows, is consistent with $p_0<0$ and $p_{t0}$. 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 the $t$ gauge ($(\Delta p{)}^2=\mathrm{const}$). 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 the $t$ 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 the $t$ 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 (60), with $M=50\hslash$, peaked about a circular configuration space trajectory.

Figure 6

Pictorial comparison of the classical relational Dirac observable $q_2(q_1)$ (full ellipse, blue curve) with the quantities $q_2(\Re [q_1])$ calculated in the effective theory using the $q_1$ gauge (violet dashed curve) and $⟨{\widehat{q}}_2⟩(q_1)$ in the Schrödinger regime (yellow solid curve). Where valid, the three curves agree perfectly. The Schrödinger regime breaks down earlier than the $q_1$ gauge of the effective framework. The initial data match in all three cases: we chose $q_{2_0}=0.7$ and $p_{2_0}=-0.7$ for the Schrödinger regime, which via Eq. (c1) yields $(\Delta q_2{)}^2(q_1=0)=(\Delta p_2{)}^2(q_1=0)=\frac{\hslash }{2}$ and $\Delta (q_2{p}_2)(q_1=0)=0$. We have set $M=10$ and, to amplify effects, $\hslash =0.03$. We take these values as initial data for the effective formalism as well, and, using Eq. (84), we determine the initial value for $p_{1_0}=-2.998$ (the minus sign is necessary here, since in Eq. (63) we quantized ${\tilde{C}}_{+}$ which evolves backwards in $q_1$). In the effective picture, due to the imaginary contribution to $q_1$ in the $q_1$ gauge, we have set the initial value of the clock to $q_1=-\frac{i\hslash }{2p_{1_0}}$, but employ $\Re [q_1]$ as relational clock (see also Fig. 8). The initial data for the classical curve has been chosen accordingly. As regards the axis labels: for the effective framework both $q_1$ and $q_2$ refer to the expectation values of the corresponding operators (for $q_1$ the real part), while for the internal time Schrödinger regime $q_2$ refers to the expectation value from Eq. (71) and $q_1$ is the real evolution parameter.

Figure 7

Comparison of the effective (black dotted curves) and internal time Schrödinger regime results (blue dashed curves) for the fashionables in $q_1$ time associated to moments: (a) $(\Delta q_2{)}^2(q_1)$, (b) $(\Delta p_2{)}^2(q_1)$ and (c) $\Delta (q_2{p}_2)(q_1)$. The curves agree perfectly to order $\hslash$. As explained in the main text, the Schrödinger regime breaks down earlier than the $q_1$ gauge of the effective framework. The breakdown of the latter is clearly demonstrated by the divergence of the effective moments near $|q_1|=3$. The initial data is identical to the one for Fig. 6.

Figure 8

Behavior of (a) the real and (b) the imaginary part of the local clock $q_1$ with respect to the gauge parameter $s$ of $C_H$ for the effective configuration with initial data as given in the caption of Fig. 6. Clearly, while $\Re [q_1]$ is monotonic along the flow of $C_H$ (as long as the $q_1$ gauge is valid) and, therefore, constitutes a useful local clock, $\Im [q_1]$ does not provide a suitable clock here. The divergence of both near $|s|=0.79$ signifies the breakdown of the $q_1$ gauge.

Figure 9

(a) Reconstruction of a semiclassical physical state via gauge switching in the effective framework. The jumps between the $q_1$ gauge (black dotted and dashed curves) and the $q_2$ gauge (blue solid curves) are a consequence of the $o(\hslash )$ jumps in the gauge transformations (86). The final evolution in $q_1$ Zeitgeist after the fourth clock change is given by the fat black dashed curve and coincides to $o(\hslash )$ with the initial evolution in the $q_1$ gauge prior to the first clock change. For convenience we have labeled the axes by $q_1$ and $q_2$. It should be noted that for the curves in the $q_i$ gauge, $q_i$ actually refers to $\Re [q_i]$. (b) Comparison of $\Delta (q_2{p}_2)(\Re [q_1])$ in the $q_1$ gauge before (dashed curve) and after (dotted curve) the complete revolution around the ellipse. The difference between the two curves is clearly of $o({\hslash }^2)$ or smaller. Initial data for both (a) and (b): $q_{1_0}=-\frac{i\hslash }{2}$, $p_{1_0}=q_{2_0}=p_{2_0}=1$, $(\Delta q_2{)}_0^2=(\Delta p_2{)}_0^2=\frac{\hslash }{2}$. Furthermore, $M=3$ and, to enhance effects, we have set $\hslash =0.01$. The initial value for $\Delta (q_2{p}_2)$ follows from Eq. (84).