Undergrad Thermal Interpretation and Consistent Histories

[Morbert]

TL;DR

The thermal interpretation frames linear functionals of a system's density operator, and any complex-valued functions constructed from these, as beables. This approach is quite intuitive for conventional q-observables on microscopic systems. This thread is for discussing the thermal interpretation of quantities constructed from history operators.

A q-expectation value of observable $A$ is $\langle A\rangle = \mathrm{Tr}\rho A$ and has a statistical interpretation. The thermal interpretation instead reframes the q-expectation value as the uncertain value of a Hermitian quantity, where "uncertain" has a fundamental character, akin to the uncertainty of the center of a country with ill-defined borders. The uncertain value is a beable.

This interpretation is quite intuitive for typical observables, but perhaps less so when histories are introduced. Consider a set of consistent histories $\{ C_\alpha \}$ defined on the state space of the microscopic system. We can write down a quantity $C= \sum_\alpha \lambda(\alpha) C_\alpha$ and hence the uncertain value $\langle C \rangle = \mathrm{Tr} \rho C$. We might want to invoke an infinite ensemble of histories of the system and interpret the uncertain value as an expectation value over this ensemble, but this is not possible under the thermal interpretation if I understand it correctly.

One possibility is to observe that if the histories obey medium decoherence, $\mathrm{Tr} C^\dagger_\alpha \rho C_\beta = 0$, then there must exist some observable $R = \sum_\alpha r_\alpha\Pi_{r_\alpha}$ where $C^\dagger_\alpha \rho C_\alpha = \Pi_{r_\alpha}\rho\Pi_{r_\alpha}$ and so we might be tempted to associate $\langle C\rangle$ with $\langle R \rangle$. For most realistic cases, this would require an extension of the state space to the measurement apparatus. We might also deconstruct the histories into conventional microscopic quantities and consider these to be the beables. E.g. If the histories are constructed from the spin-x and spin-y of a particle, we would consider the uncertain values of these spins to be the beables, and histories as the time-evolution of these values. However, this introduces a hierarchy of beables, where some uncertain values are fundamental and others are not.

 

[A. Neumaier]

Consider a set of consistent histories {Cα} defined on the state space of the microscopic system. [...] We might want to invoke an infinite ensemble of histories

In the thermal interpretation, one models statistical samples instead of ensembles, which are their idealized abstractions. See Section 3 of my quantum tomography paper, where complete details can be found.

To see how to model a statistical sample of measurements of an - if multiple measurements are done vector valued - quantity $A$ on $N$ identically prepared systems we note that the $N$ measurements differ in their spacetime position, hence actually correspond to $N$ different quantities $A_k$ ($k=1,…,N$) acting on the Hilbert space of quantum field theory, where $A_k$ is the quantity $A$ in the $k$th measurement. Thus the sample is actually a measurement of $N$ different quantities $A_k$ ($k=1,…,N$). That they are identically prepared means that the expectations $⟨f(A_k)⟩$ equal $⟨f(A)⟩$ for all sufficiently smooth functions $f$.

The statistical sample expectation of $f(A)$ in an ideal measurement is

${\bf E} f(A):=\displaystyle\frac{1}{N}\sum_{k=1}^N\langle f(A_k)\rangle= \Big\langle\frac{1}{N}\sum_{k=1}^N f(A_k)\Big\rangle.$

Under the standard independence assumptions, this satisfies a weak law of large numbers, so that the uncertainty is $O(N^{−1/2})$ for large $N$.
This is discussed in much more detail in Section 3 of my quantum tomography paper.

We might want to invoke an infinite ensemble of histories of the system and interpret the uncertain value as an expectation value over this ensemble, but this is not possible under the thermal interpretation if I understand it correctly.

A history is simply the case where the vector consists of observables supported in slightly different regions of spacetime (where the individual measurements of the history were made), but otherwise everything remains unaltered. In particular, one may consider statistical samples of histories, and gets analogous results.

 

[A. Neumaier]

@Morbert: By the way, I noticed the existence of this thread only by chance, today; hence the late answer. If you start a thread and want someone particular to answer, you need to add a ping!