Thursday, December 22, 2011

Relational Quantum Mechanics from Contextual Ontologies: Notes: leaned up a little

Suppose the x-spin of an electron, e, is going to be measured (in the sense of quantum mechanics) by a measuring apparatus, ma. The electron is represented by a vector 

in a Hilbert Space Hma, given by

This represents the maximal amount of physical information there is about the electron's x-spin in the measuring apparatus' frame of reference. Equivalently,

represents the maximal amount of physical information there is about the measuring apparatus' magnetic orientation along x in the electron's frame of reference. The electron necessarily instantiates a frame of reference because there is an object there, that exists, for which the rest of the universe must conform to the physical laws.

In either frame, if the measurement can happen at any time, then

I doubt the amplitudes (as opposed to the probabilities) have to be equal, since all we require is coordination upon observation. In this case there's a symmetry involved. Which group it is depends on the evolution (Schoedinger, Dirac, etc.) and the bases.

There is a transformation between the measuring apparatus' frame of reference and the electron's frame of reference that preserves the evolution equation and Plank's constant, exactly analogous to Lorentz transformations among classical but relatively moving frames of reference. (And it's non-trivial since the mass et. al. of the measuring apparatus is larger than the electron's mass.)

There is no such thing as "the" quantum state of the electron. Its "quantum state" depends on the (quantum) frame of reference.

That's how it can be that, for Wigner's friend, Schroedinger's cat is in the superposition

whilst the cat finds itself to be in one of the two classical states

(5) alive, or else, dead

at all times. (The cat's "times". Time is a parameter describing relatively quantum systems, so there is a different time-variable for each system and they are in a particular sense ontologically independent. See previous posts.)

In the cat's frame, Wigner's friend is in the state

Whilst Wigner's friend finds himself to actually be in one of the two states

(7) happy, or else, sad

at all (of his) times.

That's also how, after Wigner's friend opens the box, he can find the cat to be in one of the classical states

(8) alive, or else, dead

and the cat can find Wigner's friend to be in one of the classical states

(9) happy, or else, sad

whilst in Wigner's frame of reference the combined friend-and-cat system is still ("still" in Wigner's time) in the superposition

Note that a system S remains in a quantum state relative to another system O if they represent the other by a vector in a Hilbert space. This is so even if the vector happens to be an eigenvector of a relevant observable. A person observing the results of a Stern-Gerlach experiment does not represent himself as a vector in a Hilbert space (which he would then have to project onto to find the probability of his being in the state he thought he was in? No.)

I wrote up and tried to promulgate these ideas in 1996, independently of Carlo Rovelli.

[... quantum state is intransitive and what exists... the total amount of physical information... at some moment, in a system's own notion of time, does not form an equivalence class...]

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