Collapsing Superpositions

The standard description of quantum mechanics involves the state of a system being described by some vector in a Hilbert space, within which there are some state-vectors that correspond to simple-to-describe states of the sytem (e.g. a particle moving with a particular momentum, or passing through a particular position, at a given moment of time). Other states of the system need not be so simply described but can be expressed as results of scaling and summing vectors that do describe simple states. When we conduct an experiment that measures some property of the system, there are simple states that have definite values of the quantity we measure and the process of measurement forces the system into one of these simple states; in so doing, it effectively expresses the prior state of the system as a sum of scaled simple vectors, each of which has a definite value of the state; the probability that the experiment will find the system to have a particular value of the measured quantity is detemined by the scaling applied to the simple state, describing the system with that value of the measured quanity, in expressing the prior state as a sum – the bigger the scaling, the more probable the measured value. As normally described, that all sounds rather weird, bordering on the mystical. To make sense of what's really going on here, I believe it helps to talk about a simple system that can be described classically (by Maxwell's equations of electromagnetism): namely, the polarisation of light.

Sabine Hossenfelder explains the wave-function's collapse, decoherence and the flaw in it's not there unless you look notions (with a nicely ironic reference to a famous idiot along the way).

Polarised light

Light is described, in Maxwell's account of electromagnetism, as an oscillating variation in the electric and magnetic fields. At any given moment, at any given position in space that the light is shining through, the electric and magnetic fields vary in an on-going dance, that obeys Maxwell's equations describing how each of them varies. There are many ways for that dance to play out, but – as in the quantum systems mentioned above – there are some simple ways it can play out.

One of those simple ways is for the electric field to always point in a particular direction – which I'll call up, without caring whether gracity is involved; think of this as saying it's oriented in the direction that matches the observer's head's chin-through-nose-to-forehead direction – and its opposite (down, in the same terms); while the magnetic field always points in one pair of mutually-opposite directions sideways to this – which I'll call left and right – and the light propagates in a third direction perpendicular to both – forward. Let u, r and f be the names of unit vectors upwards, to the right and forwards respectively; any position in space can be written in terms of these, as v.u +s.r +e.f with v, s and e being numbers;. The simple case's values for the electric field E and magnetic field B don't depend on v or s, the scalings applied to u and r, but they do depend on e, the scaling applied to f; E is always some multiple of u and B some multiple of r, with the scalings being the Cos and Sin of some angle, that depends on e and time t as:

in which c is the speed of light (so c.t is, like e, a distance), w is an angle per unit distance (so w.c is an angle per unit time), a is some fixed angle and k is a constant scaling. Because the state depends on position and time only via c.t −e, the propatation in time is along trajectories with c.t −e constant; these move forward at the speed of light, as any increase T in t is accompanied by an increase c.T in e, to keep c.t −e constant.

That's (by the standards of these things) a simple solution:. We have free choice of the constants k, w and a, so there are simple solutions with any valuess for these three. We also have a free choice of the three directions u, r and f, provided only that they are mutually perpendicular. Let's describe this simple state as S(w, k, a, f, u, r). Maxwell's equations are linear in their electric and magnetic fields, so any sum of solutions is also a solution, as is the result of scaling any solution (i.e. varying k). For example, we can combine our original S(w, k, a, f, u, r) with S(w, k, b, f, r, −u), which rotates the original solution through a right angle about f, the forward direction, and changes the offset angle a to b. For the case b = a, we get

in which u +r and r −u are perpendicular to each other and to f, with length √2 each; so this combined solution is in fact S(w, k.√2, a, f, (u +r)/√2, (r −u)/√2), so already covered by our existing solution. However, b need not be equal to a; if b is a +turn/4, so Cos(x +b) = Cos(x +a +turn/4) = −Sin(x +a) and Sin(x +b) = Cos(x +a) for any angle x, we get

When light reflects off a surface

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