Appendix E. Slow measurements of small or sensitive systems

https://doi.org/10.48550/arXiv.2403.13981

The thesis on which most of this work is built is that what is observed at the macroscale is homogenized microstructure. The microstructure, in its most detailed and specific form, can be expressed as a pdf,

\begin{align*} \cfgdist:\configspace\to\realnonneg; \; x\mapsto \cfgdist(x), \end{align*}
where $x$ specifies the configuration of an unimaginably-large number of degrees of freedom, $\Ndof$. For almost all purposes, the degrees of freedom can be regarded as the ${\dimension=3}$ coordinates of ${\Nparticle=\Ndof/\dimension}$ particles. Therefore each configuration $x$ is a precise specification of all of the particles’ positions, and ${\configspace\subseteq\realone^{\Ndof}}$ is the set of all accessible configurations.

Some might argue that a classical microstate has the mathematical form of a point ${\Gammat\equiv (\pi_t, x_t)}$ in the particles’ phase space, ${\phasespace\cong\momspace\times\configspace}$, where ${\pi_t\in\momspace\subseteq\realone^{\Ndof}}$ is a specification of the $\Ndof$ momenta conjugate to the coordinates specified by $x_t$, and $\momspace$ is the set of accessible momenta. This is true, strictly speaking, but if classical particles had charges, masses, and separation distances comparable to those of electrons and nuclei, the average magnitudes of their accelerations would be enormous. Therefore they would move so fast that it would be impossible to observe the particles in a particular microstate; or even to observe, or measure properties of, short segments of their trajectory.

For example, in diamond, the average number density of electrons is approximately ${1\,\text{/ų}=e30\,\text{/m³}}$, and the shortest phonon period is approximately ${25\,\text{fs}=2.5e-14\,\text{s}}$ [Peckham, 1967]. If electrons were classical particles, the Coulomb repulsion between two electrons separated by a distance of ${1\,\text{Å}\equiv 10^{-10}\;\text{m}}$ would give them accelerations of ${\sim 250\,\text{Å/fs²}}$. Moreover, if $\Nelec$ particles are identical, exchange symmetry reduces the set of symmetry-inequivalent points in their phase space by a factor of ${\Nelec!}$. For example, if they were confined to a region ${\materialregion\subset\realone^3}$, the time that they would take to sample their configuration space would be less, by a factor of ${\Nelec!}$, than the time that each individual particle would take to explore ${\materialregion}$.

These considerations justify the assumption that, during each ‘moment of consciousness’ [Kent and Wittmann, 2021; Northoff and Lamme, 2020], and during most indirect observations (i.e., those not limited by our brains and senses because they are performed by artificial devices), the particles sample $\phasespace$ comprehensively. Consequently, observations of the set of particles, and measurements of their properties, are not observations and measurements of the particles in a microstate or in a small region of their phase space. They are observations and measurements of the particles’ statistical state.

SUBSECTION MISSING (mid rewrite)

E.1 Slow perturbative measurements of fast-moving classical systems

The outcome of a measurement of an observable $\Obs$ is determined by an interaction between a probe $\probe$ and the measurement subject, $\subject$. This section assumes that the duration $\tau$ of this interaction is much longer than the time scale on which the true instantaneous configuration $x_t$ of $\subject$ changes. It is so long, which is to say that $x_t$ changes so fast, that it is impossible to measure properties of short segments of the trajectory, ${x_t(t)}$; and the fraction of $\tau$ for which ${x_t}$ is in any substantial subset of the configuration space $\configS$ of $\subject$ is approximately equal to the fraction of time it would spend there in an infinite number of identical repetitions of the measurement.

In this context, ‘identical’ means that the information available to the observer (prior information, their readings, etc.) is the same in each repetition. For simplicity, let us assume that only one pdf is consistent with this information. In other words, no symmetries exist that make multiple symmetry-equivalent pdfs consistent with it.

Let us denote the configuration space of $\probe$ by $\configP$, and let us denote its true instantaneous configuration by $y_t$. Let us denote the composite system whose instantaneous microstate is ${(x_t,y_t)}$ by ${\combined\equiv\subject\cup\probe}$. As discussed above, for simplicity, let us assume that what is being measured does not depend on the momenta conjugate to $x_t$ and $y_t$. Therefore the measured value is the expecation value of some function,

\begin{align*} \Obs:\configS\times\configP\to\realone;\;(x,y)\mapsto \Obs(x,y). \end{align*}

Let us assume that the observer either has no information about $x_t$ prior to the measurement, or that they only possess a statistical description of $x_t$. Therefore, they do not know $x_t$ precisely immediately before $\probe$ and $\subject$ begin to interact, which implies that they cannot know ${(x_t,y_t)}$ precisely during the interaction. For the purposes of this section, it is unnecessary to assume that $y_t$ is not known precisely before or after the interaction, but of course nothing can be known to infinite precision.

These assumptions mean that everything that can be known about $\subject$, $\probe$, and $\combined$ before, during, and after the measurement, can be expressed as the time-dependent probability density functions, ${\prs(x;t)}$, $\prp(y;t)$, and ${\prcm(x,y;t)}$ for ${x_t}$, ${y_t}$, and ${(x_t,y_t)}$, respectively. For simplicity, time ($t$) will sometimes be omitted as a parameter of these pdfs.

The probe is perturbative because it interacts with $\subject$. If it did not interact with $\subject$, nothing could be deduced about $\subject$ from ${\prp(y;t_0)}$ and ${\prp(y;t_0+\tau)}$, where ${t_0}$ is the earliest time at which the interaction between $\probe$ and $\subject$ becomes non-negligible, and ${t_0+\tau}$ is the earliest later time at which it becomes negligible again. The interaction implies that

\begin{align*} \prcm(x,y) &=\prsp(x|y)\prp(y) \\ &=\prps(y|x)\prs(x)\neq \prs(x)\prp(y), \end{align*}
where $\prsp$ is the conditional pdf for $x_t$ given $y_t$; and $\prps$ is the conditional pdf for $y_t$ given $x_t$.

The measurement is slow in the sense that the fraction of interval ${\intervalmeas\equiv(t_0,t_0+\tau)}$ for which $\prcm$ depends on the configuration ${(x_t(t_0),y_t(t_0))}$ of $\combined$ when ${\subject}$ and ${\probe}$ begin to interact is negligible. In other words, the transient is short-lived and has a negligible influence on the measurement outcome. For almost the entire duration of the measurement, $\combined$ is either in a steady state (the time dependence of $\prcm$ is periodic and its period is much less than $\tau$), a stationary state (the time dependence of $\prcm$ is negligible), or $\subject$ is responding adiabatically to $\probe$.

Let us treat cases in which the measurement is performed while $\combined$ is in a steady, but non-stationary, state by redefining the time-periodic pdf, $\prcm$, as the average of itself over one of its periods. For present purposes, this removes the distinction between steady states and stationary states, so steady states will not be discussed further.

Adiabatic response of $\subject$ to $\probe$ means that the influence of $\probe$ on $\prs$ changes so slowly that, on the timescale on which $x_t$ changes, it appears not to change at all. This does not imply that $y_t$ is not changing rapidly, but that, during the measurement interval ${\intervalmeas}$, the joint pdf ${\prcm(t)}$ changes so slowly that, instantaneously, it is almost the same as it would be if the probe’s influence was not changing.

For example, if $\probe$ was the moving tip of a scanning tunnelling microscope (STM), its constituent atoms and electrons would move rapidly. However we are assuming that the tip, as a whole, moves so slowly that, at each of its positions, $\prcm$ is the same as it would be if the tip was static and had been at that position for a long time. This assumption of adiabatic decoupling [Ott, 1979; Spohn and Teufel, 2001; Tangney, 2006] is like the Born-Oppenheimer approximation [Born and Oppenheimer, 1927] used to deduce the instantaneous statistical state of electrons from the instantaneous positions of the much slower-moving nuclei. However, for an STM tip it is likely to be a much better approximation, because the timescale of seconds on which an STM tip moves is about fifteen orders of magnitude larger than the timescale on which nuclei move.

On the other hand, if $\probe$ is an emitted wave pulse, its frequency would have to exceed the highest frequencies of visible electromagnetic waves for its adiabatic decoupling from classical electron-like particles to be less effective than the decoupling between nuclei and electrons that the Born-Oppenheimer approximation exploits.

It follows from all of these physical assumptions and considerations that the measurement is not a measurement of $\subject$ in a particular configuration, $x_t$. It is not even a measurement or observation of $\subject$ in one of the stationary statistical states it can be in when it is isolated. It is either a measurement or observation pertaining to a stationary state of $\combined$, or it is a property of the stationary state trajectory, ${\{\prcm(t):t\in\interval_\text{meas}\}}$, where each pdf ${\prcm(t)=\prcm(x,y;t)}$ along the stationary state trajectory is almost exactly the stationary state that $\combined$ would be in if the influence of $\probe$ on $\prcm$ was not changing.

E.2 Why measured values are eigenvalues

Let us consider the case in which what is measured is not determined by a stationary state trajectory of $\combined$, but by a single stationary state of $\combined$. Let ${\{\prcm_\gamma\}}$ be the set of all stationary states of $\combined$ that might be measured, where index $\gamma$ distinguishes between different stationary states. Then the $\gamma^\text{th}$ possible measured value of observable $\Obs$ is
\begin{align*} \obs_\gamma &\equiv \int_{\configS}\dd{x}\int_{\configP}\dd{y} \prcm_\gamma(x,y) \Obs(x,y) \\ & = \int_{\configS}\dd{x}\prs_\gamma(x)\overbrace{\left(\int_{\configP}\dd{y} \prps_\gamma(y|x) \Obs(x,y)\right)}^{\displaystyle \tObs_\gamma(x)} \\ & = \int_{\configS}\dd{x} \prs_\gamma(x) \tObs_\gamma(x). \tag{113} \end{align*}
Eq. (113) implies that, under the physical assumptions stated above, ${\obs_\gamma}$ is not a property of $\subject$, but a property of the $\gamma^\text{th}$ stationary state of the act of measurement of $\Obs$.

However, the purpose of most measurements is to reveal information about $\subject$, not $\combined$. Therefore most observables of interest are functions

\begin{align*} \Obs:\configS\to\realone;\; x\mapsto \Obs(x), \end{align*}
rather than functions
\begin{align*} \Obs:\configS\times\configP\to\realone;\; (x,y)\mapsto \Obs(x,y). \end{align*}
The instantaneous value of such an observable is ${\Obs(x_t)}$, rather than ${\Obs(x_t,y_t)}$, and what is measured is its average, $\obs_\gamma$, in one of the stationary states of ${\combined}$.

When ${\Obs=\Obs(x)}$, Eq. (113) simplifies, because ${\int_{\configP}\dd{y}\prps_\gamma(y|x)=1}$ implies that ${\tObs_\gamma(x)=\Obs(x), \;\forall \gamma}$. Therefore,

\begin{align*} \obs_\gamma = \int_{\configS}\dd{x} \prs_\gamma(x)\Obs(x). \tag{114} \end{align*}
Eq. (114) means that the result of the slow perturbative measurement of property $\Obs$ of the fast-moving system $\subject$ is an average, over a very large number of configurations of $\subject$, while $\combined$ is in one of the stationary states of the act of measurement of $\Obs$.

Note that, by definition of a stationary state, $\prcm_\gamma$ is independent of time. Therefore, ${\prs_\gamma(x)\equiv \int_{\configP} \prcm_\gamma(x,y)\dd{y}}$ is also independent of time, and is a stationary state of subsystem $\subject$ of $\combined$ during the act of measurement of $\Obs$. As discussed in Appendix C, we can specify this stationary state pdf with a function ${\psicm_\gamma\in\lebesgue(\configspace\times\ydomain)}$ such that ${\big|\psicm_\gamma\big|^2=\prcm_\gamma}$.

Let us specify $\prs$, which is an arbitrary pdf for $x_t$, with the function ${\psis\in\lebesgue(\configspace)}$, where ${\big|\psis\big|^2=\prs}$. Let us represent ${\psis}$ as an element ${\ket{\psis}}$ of an abstract Hilbert space ${\hilbert}$, and let ${\hO:\hilbert\to\hilbert}$ denote the Hermitian operator for which

\begin{align*} \expval{\Obs}\equiv \int_{\configS}\dd{x}\prs(x)\Obs(x) = \frac{\expvaltwo{\hO}{\psis}}{\braket{\psis}{\psis}}. \end{align*}
The set of eigenstates of a Hermitian operator is a complete basis of the space on which it acts and it is always possible to choose this set such that they are mutually orthogonal. Therefore we can express ${\ket{\psis}}$ as
\begin{align*} \ket{\psis}=\sum_{\alpha} a_\alpha \ket{\uppsis_\alpha}, \end{align*}
where each ${\ket{\uppsis_\alpha}}$ is one of the mutually-orthogonal eigenstates of ${\hO}$; and ${a_\alpha\in\complex}$. Let us denote the eigenvalue of $\hO$ corresponding to eigenstate ${\ket{\uppsis_\alpha}}$ by $\obs_\alpha$. That is,
\begin{align*} \hO\ket{\uppsis_\alpha}&=\obs_\alpha\ket{\uppsis_\alpha} \Rightarrow \obs_\alpha = \frac{\expvaltwo{\hO}{\uppsis_\alpha}}{\braket{\uppsis_\alpha}{\uppsis_\alpha}} \end{align*}
Therefore, when $\subject$ is in state ${\ket{\psis}}$, the expectation value of $\Obs$ can be expressed as
\begin{align*} \expval{\Obs}&= \frac{\expvaltwo{\hO}{\psis}}{\braket{\psis}{\psis}}=\frac{\sum_\alpha \abs{a_\alpha}^2 \obs_\alpha}{\sum_\beta \abs{a_\beta}^2}. \end{align*}
Now let us consider an arbitrary change of state,
\begin{align*} \ket{\psis}\mapsto \ket{\psis+\zeta\delta\psis} \equiv \ket{\psis}+\zeta\ket{\delta\psis}, \end{align*}
where ${\zeta\in\realone}$ and
\begin{align*} \ket{\delta\psis}=\sum_\alpha \delta a_\alpha\ket{\uppsis_\alpha}\in\hilbert. \end{align*}
The corresponding change in ${\expval{\Obs}}$ as a function of $\zeta$ is
\begin{align*} \delta\expval{\Obs}(\zeta) &= \frac{\expvaltwo{\hO}{\psis+\zeta\delta\psis}}{\braket{\psis+\zeta\delta\psis}{\psis+\zeta\delta\psis}} - \frac{\expvaltwo{\hO}{\psis}}{\braket{\psis}{\psis}} \\ &= \frac{\sum_\alpha \abs{a_\alpha+\zeta\delta a_\alpha}^2 \obs_\alpha}{\sum_\beta \abs{a_\beta+\zeta\delta a_\beta}^2} - \frac{\sum_\alpha \abs{a_\alpha}^2 \obs_\alpha}{\sum_\beta \abs{a_\beta}^2}. \end{align*}
After a little algebra, we find that the lowest-order contribution to ${\delta\expval{\Obs}(\zeta)}$ is linear in $\zeta$ and proportional to
\begin{align*} \sum_{\alpha\beta} \obs_\alpha &\bigg[ \abs{a_\beta}^2\left(a_\alpha\delta a_\alpha^*+a_\alpha^*\delta a_\alpha\right) -\abs{a_\alpha}^2\left(a_\beta\delta a_\beta^*+a_\beta^*\delta a_\beta\right) \bigg] \\ &= 2\sum_{\alpha\beta} \Re\{a_\alpha^*\delta a_\alpha\} \abs{a_\beta}^2 \left(\obs_\alpha-\obs_\beta\right) \\ &= 2\sum_{\alpha} \Re\{a_\alpha^*\delta a_\alpha\} \left(\obs_\alpha\sum_\beta\abs{a_\beta}^2-\sum_\beta \abs{a_\beta}^2 \obs_\beta\right) \\ &= 2\left(\sum_\beta\abs{a_\beta}^2\right)\sum_{\alpha} \Re\{a_\alpha^*\delta a_\alpha\} \left(\obs_\alpha-\expval{\Obs}\right) \tag{115} \end{align*}
The set of values of ${\expval{\Obs}}$ in states of $\subject$ that are stationary during the act of measurement of $\Obs$ is the set of possible measured values of $\Obs$. These values depend only on the time-independent function ${\Obs(x)}$ and on whichever stationary state $\subject$ is in during the measurement. The $\gamma^\text{th}$ such state can be represented by $\prs_\gamma$ or by an element ${\ket{\uppsis_\gamma}}$ of $\hilbert$.

If $\zeta$ is sufficiently small, the condition that our arbitrary state ${\ket{\psis}}$ must satisfy to be one of these stationary states is that ${\zeta\ket{\delta \psis}}$ does not change ${\expval{\Obs}}$ to linear order in $\zeta$. Therefore ${\mathrm{d}\expval{\Obs}/\mathrm{d}\zeta=0}$, which implies that the set of possible measured values of ${\expval{\Obs}}$ is the set of its values for which

\begin{align*} \sum_{\alpha} \Re\{a_\alpha^*\delta a_\alpha\} \left(\obs_\alpha-\expval{\Obs}\right) =0 \end{align*}
for any choice of ${\ket{\delta\psis}\in\hilbert}$, and therefore for any set ${\{\delta a_\alpha\}}$. This implies that, for each $\alpha$, either ${\expval{\Obs}=\obs_\alpha}$ or ${a_\alpha=0}$.

This concludes the demonstration that the set of possible measured values of any observable $\Obs$ is the set of eigenvalues of a Hermitian operator ${\hO}$.

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