Contrary Inferences for Classical Histories within the Consistent Histories Formulation of Quantum Theory

TL;DR

This paper analyzes the time-of-arrival question within the consistent histories framework and shows that contrary inferences persist even in a quasi-classical, macroscopic regime. By constructing two distinct coarse-grainings for a quasi-classical Gaussian Klauder state in an infinite square well, the authors demonstrate that one CH partition assigns crossing of the origin with probability one while another assigns non-crossing with probability one, despite both sets being consistent. The results indicate that the CH consistency condition alone is insufficient to recover the expected quasi-classical limit and motivate additional set-selection criteria to constrain histories. The work highlights contextuality in CH at macroscopic scales and prompts reconsideration of how to select physically meaningful histories in closed quantum systems.

Abstract

In the histories formulation of quantum theory, sets of coarse-grained histories that are consistent obey the classical probability rules. It has been argued that these sets can describe the quasi-classical behaviour of closed quantum systems, e.g. Omnes (Rev. Mod. Phys. 64(2), 339, 1992) and Hartle (Les Houches1992). Most physical scenarios admit multiple different consistent sets and one can view each of these as a separate context. Using propositions from different consistent sets to make inferences leads to paradoxes such as contrary inferences, first noted by Kent (Phys. Rev. Lett. 78(15), 2874, 1997). In this contribution, we use the consistent histories to describe a quasi-classical and macroscopic system to show that paradoxes involving contextuality persist even in the quasi-classical limit. This is distinctively different from the contextuality of standard quantum theory, where the contextuality paradoxes do not persist in the quasi-classical limit. Specifically, we consider different consistent sets for the arrival time problem of a (quasi-classical) ball in an infinite square well. For this setting, we construct two different consistent sets. We find the probabilities that each consistent set assigns to the simple question of whether the ball ever crossed the middle of the interval. We show that one consistent set concludes with certainty that the ball crossed it while the other consistent set concludes with certainty that it did not. Our results point to the need for constraints on the histories sets, additional to the consistency condition, to recover the correct quasi-classical limit in this formalism and lead to the motto "all consistent sets are equal", but "some consistent sets are more equal than others".

Figures (7)

• Figure 1: Left Panel: Instant of the coarse-graining for the time-of-arrival problem for a free particle. The question is whether the particle will have for at least some moment $t$ positive position coordinate. A natural coarse-graining is to select the intervals $\Delta (t)= [0, +\infty )$ and $\bar{\Delta} (t)= (-\infty,0]$ as shown here. Right Panel: Various histories in the time-of-arrival problem in a specific partitioning -- red paths remain always in the region $\bar{\Delta}$, blue paths have being in $\Delta$ at some moment of time.
• Figure 2: Probability distribution of the coherent state in position and momentum space at $t=0$. The parameters were chosen as follows: $n_0 = 201$ and $\sigma_0 = 20$. The phase $\varphi_0$ is selected such that the wave function is localised at $x/L = -1/2$, as demonstrated in the figure.
• Figure 3: Left Panel: The time-of-arrival problem in the infinite square well; this is an instant of the partition for the first set of histories. The two subregions are $\Delta_1=[0,L]$ and $\bar{\Delta}_1 = [-L,0]$. The question is whether the particle, equipped with positive momentum will be even for only one moment in $\Delta_1$. The barrier at $x=0$ is fictitious and corresponds to infinitely many projections with the operator \ref{['projection_continuous']}. The projection takes place also at $x=-L$ but this also coincides with the real barrier of the square well. Right Panel: Various histories: the blue set belongs to $h_1$ while the red in $\bar{h}_1$
• Figure 4: The square of the overlap of the two wave packets as a function of time normalised with the classical time (left plot) and the "zoomed" plot near $\tau = 0.85 T_{cl}$ (right plot). The parameters the same as follows $n_{0}^1 = 100$, $n_0 = 2 n_{0}^1 + 1$, $\sigma_0^1 = 10$, $\sigma_0 = 2 \sigma_0^1$. The phases $\varphi_0^1$ and $\varphi_0$ are selected such that both wave functions are localised at $x/L = -1/2$. The superscript $1$ refers to the coherent state in the well spanning $[-L,0]$.
• Figure 5: Left Panel: Instant of the coarse-graining for the second histories set. In this case, the partition is $\Delta_2 (t)=[x_{cl} (t) -\lambda, x_{cl} (t) + \lambda]$ and changes in time, following the classical path of the particle. The two barriers move uniformly and continuously in the infinite square well due to the quasi-classical nature of the problem, see section \ref{['discussion']} and this one for detailed discussion on the continuity. Right Panel: Various histories: the blue set belongs to $h_2$ while the red in $\bar{h}_2$