log entry ID: .a.e..... 2024-02-11 23:03:25 EST

This log entry also requires knowledge of basic quantum mechanics. Sorry, not sorry.

Measurement: What and When

There is a concept in English called a bucket list. This is undoubtedly a concept in Finnish too, given the significance of the bucket in Finnish culture, but in English, it means something different. It is a list of all of the things you want to do before you “kick the bucket”, which is another English idiom meaning “die”. My bucket list has exactly one item: I must solve the quantum measurement problem.

This is why I want to complete a Master’s in Theoretical Physics at Helsingin yliopisto. I have already applied, and am awaiting their response. Their response is not set to arrive until April 15, so this is how I am spending the intervening days; writing log entries like this.

My original plan was to move to Finland as a first priority and work on the measurement problem as, more or less, a means of getting to Finland. I have since changed my mind. If I can’t do theoretical physics in Finland, I’ll just look for somewhere else where I can. I won’t be happy about it, but those are my priorities.

Which quantum measurement problem?

My life’s mission is not aided by the fact that the measurement problem itself is not well-defined in the first place. Thus I must limit the scope of my mission. I am not going to try to answer philosophical questions like what does measurement “mean”, or anything like that. Here’s what I want to know.

First, some terms, so that I can refer to specific parts of the problem.

A measurement is defined by a point in time, t_m, and a countable list of projectors Pi[n], with one projector per possible outcome. The state, psi(t), is one way before the measurement, and another way after the measurement:

lim(t_m, NEGATIVE, psi) = psi_before   # 0.
psi(t_m) = psi_after   # 1.

where lim(t_m, NEGATIVE, psi) is the limit as t approaches t_m from the negative side of psi(t). The transition from psi_before to psi_after happens randomly with the following probabilities:

P[n] = hc(psi_before) * Pi[n] * psi_before   # 2.

where hc is the Hermitian conjugate. Once a particular outcome, n, is decided, the state transitions according to the formula:

psi_after = normalise(Pi[n] * psi_before)   # 3.

where normalise(v) = v/norm(v).

psi_after then forms the initial condition for the continuation of the Schrödinger equation after the measurement. (See equation 1.)

A measurement corresponds to an experimental observation. n is the result of this observation. This is the only information that physicists can recover from the prior quantum state, psi_before. It is connected to the value of psi_before, but only randomly. It is impossible to fully reconstruct psi_before from just this measurement.

This is what all physicists agree on.

Well, there is a generalisation of this called weak measurement, but it’s the same basic idea.

Here’s the problem:

what is Pi, and what is t_m?

In other words,

What measurements happen and when do they happen?

Ultimately, right now, we don’t know. Quantum mechanics is obviously incomplete. This is unacceptable!

What we do know.

There are makeshift half-answers to this problem that physicists use to get their work done. I mean, of course there are. How else would there have been agreement on what a measurement is, in the first place? There’s a half-answer for spectroscopy; a different half-answer for statistical mechanics; a different half-answer for each and every quantum computer that has been built so far; etc.; but there is no one rule that applies in all circumstances!

This is what I am after.

This is my life’s mission.