## Measurement Problem

December 11th, 2007A commonplace computational practice in quantum mechanics generates the most profound conceptual challenge to the theory. The challenge is called *the measurement problem*. Here are some quotes summarizing the problem.

“The quantum measurement parodox.. stated succinctly… In quantum mechanics all possibilities… are left open whereas in … experience a definite outcome always (occurs).”

A. J. Leggett in Foundations of Physics. 18, 939 (1988)

“How is the measuring instrument proded into making up its mind which value it has observed?”

Bryce S. Dewitt, Physics Today 23, 30 (1970)

“Some explanation must be provided for the fact that the Hilbert—space vector… collapses onto a certain eigenvector during a measurement process…”

J. Bub, Nuovo Cimento v. 57, Nr.2, 503 (1968)

The probability amplitudes evolve deterministically until a measurement is made: the measurement stops the evolution. What is the essential element that changes the evolution of the system from being in a state

|S> = (superposition sum of many states |n>),

into being in a state, say, |n=3>, one from among the many in the superposition?

Marvin Chester, never published

In quantum mechanics amplitudes for events progress quite deterministically – until a measurement is made. Then the amplitudes for all but the measured event are simply discarded. And the universe begins anew starting with the conditions found in the measurement. So at each measurement the old universe disappears and a new universe begins!

I offer below an animation to portray the idea. Press the particle release button to inject a vertically polarized particle into the magnetic field gradient region. Each press of the release button yields a new particle.

Made visible, here by the red ball-and-pole icons, is something intrinsically invisible and not measurable; the computational element called the particle amplitude. The particle amplitude is split into two by the field gradient through which the particle passes. From this amplitude split arises the particle’s potential to materialize in one or the other detector. But even though it has a 50% chance of appearing in each detector, only one detector registers. That detection tells us that the particle is known, with 100% certainty, to be in the registering detector. So the universe must be reset – from 50/50 amplitude split before detection to certainty for one of the amplitudes at detection.

Thus, on measurement, the particle’s spatial probability distribution is revised and from these new initial conditions a new universe begins its deterministic evolution. In the figure each press of the release button repeats the experiment with a new particle. The counters accumulate the detection events thus revealing the statistics. On repeating the experiment many times one finds particles are registered as often in one detector as the other. This is how the pre-detection 50/50 amplitude split reveals itself experimentally.

The figure shows an idealized laboratory measurement. But measurement events are taking place interminably everywhere. When a photon of sunlight falling on a leaf gets absorbed in photosynthesis, that is a measurement event. The leaf is a photodetector. Every chemical reaction is a measurement event; the reactants disappear and the products appear. Isn’t every inelastic scattering a measurement event?

An important feature of the measurement problem is thus: What constitutes a measurement? Is any elemental process that procedes irreversibly a measurement? These are topics to be explored in further posts.

I always thought that the “measurement problem” was simply a critique of the “projection postulate.” The act of measurement, whatever it is, must somehow be describable as a process taking place according to the rules of QM, but since these dynamics are unitary and hence reversible, how do you get out of them the irreversible effect of projection? All attempts to derive the projection/collapse postulate seem to founder. My problem with it is not so serious since I do not believe in the universal applicability of the Schrödinger equation – time must also be quantized so p.d. equations like that of Schrödinger can at best be only semi-classical approximations – nor the concomitant universality of irreversibility: though I realize this begs the question.

- Stephen @ 11 January 2008What a pleasure to read your comment. It is at the core of the issue. I would summarize what I understand you to say thusly:

1. The ‘projection postulate’: measurement suspends the reversible dynamics of state evolution forcing the state to become a mere projection of its original self – an intrinsically irreversible event.

2. How can we understand the world if quantum mechanics does not have “universal applicability”?

I have some questions to offer for consideration that might, at least, let us acquire a physical intuition on the matter.

- marvin chester @ 11 January 2008I concur with 1. As for 2 I have to admit upon reflection that I do not believe that the dynamical portion of QM yet has “universal applicability.”

- Stephen @ 12 January 2008If one assumes that the temporal evolution of a quantum results in the action of a one-parameter group via unitary operators on the Hilbert space of “states” of the quantum, then abstract theorems of von Neumann et. al. will force the usual form of Schrödinger dynamics. But what an idealization this is, even without relativity! Firstly, how can time be a continuum in the quantum deep? Secondly, could the Hilbert space itself not change from instant to instant, making moot the entire contraption?

The operationalist view of the Hilbert space is that its vectors represent descriptors of experimental acts performable upon the quantum and are not strictly speaking attributes of the quantum itself. Quanta are generally not objects in the sense that they have objective states of being. For instance, a superposition does not describe the state of an ordinary object but rather a particular experimental arrangement. Nothing is observed or measured (indeed nothing is observable or measurable) until some such superposition is resolved or collapsed. So we don’t see any dials or meters registering anything until this happens. And this happens after the Schrödinger ball is over, assuming it even started. This collapse or contraction of the Hilbert space is just a change in the experimenter’s repertoire of available experiments concomitant with that particular “run” of the experiment: it is the weeding out of possible outcomes rather than the promiscuous multiplexing of worlds. Another experimenter, or the same one on a different occasion, might find a different eigenspace is selected.

This view of collapse appears to divorce it from dynamics. It is a more primitive sort of kinematical phenomenon possibly underlying actual dynamics, of which Schrödinger’s is just a semiclassical approximation. I think this is not far from some views of Penrose.

If time weren’t continuous would that solve the measurement problem?

What would it mean physically for the Hilbert-space to “change from instant to instant”?

You raise such fabuously interesting issues.

- marvin chester @ 13 January 2008