Against Collapse: A Mathematical Argument

[Peter Morgan]

One sees "Collapse" language all the time, and yet it seems there's a very simple argument that shows that it's not necessary. Suppose we measure $\hat A$ twice, at an earlier and at a later time, then the joint probability density for the measurement results being $u$ and $v$ respectively, in a state $\rho(\hat A)=\mathsf{Tr}[\hat A\hat\rho]$ is $\rho\left(\delta(\hat A-u)\delta(\hat A-v)\right)$. If we take the Fourier transform of that and perform a single change of variable, we obtain
\begin{eqnarray*}
\quad\rho\!\left(\delta(\hat A-u)\delta(\hat A-v)\right)
&=&\int\hspace{-0.7em}\int\!\!\rho\!\left(\mathrm{e}^{\mathrm{j}\lambda\hat A}\mathrm{e}^{\mathrm{j}\mu\hat A}\right)
\mathrm{e}^{-\mathrm{j}\lambda u-\mathrm{j}\mu v}\frac{\mathrm{d}\lambda}{2\pi}\frac{\mathrm{d}\mu}{2\pi}\cr
&=&\int\hspace{-0.7em}\int\!\!\rho\!\left(\mathrm{e}^{\mathrm{j}\alpha\hat A}\right)
\mathrm{e}^{-\mathrm{j}\alpha u-\mathrm{j}\mu(v-u)}\frac{\mathrm{d}\alpha}{2\pi}\frac{\mathrm{d}\mu}{2\pi}
\quad\mbox{(substituting $\lambda=\alpha-\mu$)}\cr
&=&\rho\!\left(\delta(\hat A-u)\right)\delta(v-u)
\end{eqnarray*}
[we can think of the integrands above as generating functions for moments $\rho\left(\hat A^n\right)$.] This asserts that there's a 100% correlation between the results one obtains for the two measurements, because $v$ must be exactly equal to $u$, and yet there's been no "collapse" of the state $\rho(\cdot)$, we've just used the one state at different times. If $\hat A$ commutes with the Hamiltonian, the same measurement procedure will give the same result, whereas if $[\hat A,\hat H]\not=0$, a different measurement procedure will have to be used to give the same measurement result as was obtained at the earlier time, but if we make the same measurement at different times we will, according to this calculation, obtain precisely the same result. Conversely, if we don't obtain the same, perfectly correlated results, we haven't made the same measurement (the proof of how well we understand what measurements we've made is in the measurement results).

This is a very simple calculation, but it seems to say that just using a multi-time formalism correctly (which is very natural for quantum fields such as quantized EM/quantum optics, but trickier, though it can be done, if we use a one-time Hilbert space unitarily evolved to different times) is enough, we don't ever have to collapse the wavefunction. Note that although the wavefunction does not determine what the measurement result for either the first or the second measurement will be, the wave function determines the conditional probability density for both the first and the second measurement, given the measurement result of either one, which does determine the measurement result of the other. Indeed, adopting this conditional probability approach is perhaps less likely to lead our intuition astray, insofar as one might not want to say that the second measurement "collapsed" the result for the first measurement.

Of course it has been argued before that no-collapse interpretations are possible, but this is an entirely mathematical argument that seems to me to have relatively little metaphysical/ontological baggage. I haven't seen the argument reduced to the three line derivation above (short is how I like 'em), but is it out there in a closely comparable form?

 

[A. Neumaier]

$\hat A$ twice, at an earlier and at a later time, then the joint probability density for the measurement results being $u$ and $v$ respectively, in a state $\rho(\hat A)=\mathsf{Tr}[\hat A\hat\rho]$ is $\rho\left(\delta(\hat A-u)\delta(\hat A-v)\right)$

No, this is the joint density for simultaneously getting $u$ and $v$. For consecutive measurement you need first to apply a filter operation $\rho\to P\rho P^*$ corresponding to the first measurement, then the unitary dynamics for the time between the measurements, and then another filter operation for the second measurement. This is the generally accepted procedure.

Note also that rephrasing collapse in terms of conditional probability is essentially von Neumann's 1932 argument showing that the Heisenberg cut can be places anywhere. The collapse happens at the Heisenberg cut when you match a theoretical quantum description of part of the universe with a classical description of the remainder in terms of preparation and measurement. As long as you stay on the level of conditional probabilities you are on the level of theory. To make your argument work in a longer sequence of activities you need to consider more and more complex conditional probabilities. But somewhere you need to start with having prepared something and end with having observed something, and collapse there is unavoidable and objective.

 

[Peter Morgan]

No, this is the joint density for simultaneously getting $u$ and $v$. For consecutive measurement you need first to apply a filter operation $\rho\to P\rho P^*$ corresponding to the first measurement, then the unitary dynamics for the time between the measurements, and then another filter operation for the second measurement. This is the generally accepted procedure.

Note also that rephrasing collapse in terms of conditional probability is essentially von Neumann's argument showing that the Heisenberg cut can be places anywhere. The collapse happens at the Heisenberg cut when you match a theoretical quantum description of part of the universe with a classical description of the remainder in terms of preparation and measurement. As long as you stay on the level of conditional probabilities you are on the level of theory. But to make your argument work in a longer sequence of activities you need to consider more and more complex conditional probabilities. But somewhere you need to start with having prepared something and end with having observed something, and collapse there is unavoidable and objective.

Thanks, Arnold, very useful as always. I started thinking in these terms in a free field theoretic setting, inevitably for me, where $\hat\phi_f$ is equivalent to $\hat\phi_g$ if $(f-g,f-g)=0$ (using the pre-inner product $(f,g)=\langle 0|\hat\phi_f^\dagger\hat\phi_g|0\rangle$), which can be true when $f$ and $g$ are time-like separated. Then $\rho\left(\delta(\hat\phi_f-u)\delta(\hat\phi_g-v)\right)=\rho\left(\delta(\hat\phi_f-u)\delta(\hat\phi_f-v)\right)$ is consecutive in the sense that $g$ has support that is later or earlier in time than the support of $f$, but is simultaneous in the sense of a block world model (to which I personally don't attach an ontological significance, though some people do). In a one-Hilbert-space-at-each-time model, the usual nonrelativistic approach, $\hat A$ now is equivalent to the unitarily evolved operator $\hat U^\dagger\hat A\hat U$ at a later time, so that in $\rho\left(\delta(\hat A-u)\delta(\hat A-v)\right)$, $\hat A$ can refer to both $\hat A$ now and $\hat U^\dagger\hat A\hat U$ at a later time, which again is as much as to say at the same time "yes, simultaneous", but also that this is how one can represent consecutive measurements.
Your second paragraph seems to me much more interesting, although a classical physicist would seem to be equally bound by the argument, insofar as the expression for $\rho\left(\delta(\hat A-u)\delta(\hat A-v)\right)$ is equally true for a classical physicist (defining such as an observer who only has access to a commutative algebra of observables $\hat A$, for whom all states can be assumed to be mixed states because they cannot distinguish mixed states from pure states). Both a classical and a quantum physicist can measure what there is (what the universe presents to us without our intervention) or they can work with prepared states that have been calibrated and tuned by using standardized measurement devices. Yes there is a separation into state $\rho:\hat A\mapsto \rho(\hat A)$ and measurements $\hat A$, but the state doesn't necessarily have to be "prepared", with "collapse" invoked as providing that separation.
When I look at it in the light of your comment, my title, 'Against "Collapse" ', more clearly alludes to Bell's 'Against "Measurement" ' than I thought when I wrote the OP, though of course that was in my mind. I'm particularly against saying that collapse —the application of a projection operator as a dynamical or nondynamical process— happens at the moment of appearance of a measurement event such as an Avalanche PhotoDiode current spike. I think that's become a common way of talking about measurement and there's no need for it, but, as you say in your second paragraph, anyone sophisticated will not think in such terms in any case, since they will follow von Neumann in just applying the Heisenberg cut in whatever way they think is most convenient. Which strongly suggests that I can stop worrying about it. Thank you!

 

[A. Neumaier]

but also that this is how one can represent consecutive measurements.

No. Try your recipe on a harmonic oscillator with measurements of momentum $p$ followed by a measurement of position $q$, or vice versa, and you'll see that you get nonsense.

 

[Peter Morgan]

No. Try your recipe on a harmonic oscillator with measurements of momentum $p$ followed by a measurement of position $q$, or vice versa, and you'll see that you get nonsense.

I'm not sure I can see how to fix this, but I would take a measurement of $\hat p$ at later and earlier times, translated by the Hamiltonian to the same time as a measurement of $\hat q$, $$\exp(i(\hat p^2+\hat q^2)t/2)\hat p\exp(-i(\hat p^2+\hat q^2)t/2)=\hat p\cos{t}+\hat q\sin{t},$$ modulo constants, to be sometimes, indeed for the SHO periodically, the same as the measurement of $\hat q$ at a given time.