Thread: MWI bugging me
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Hurkyl is offline
Dec4-07, 04:33 AM
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Vanesh, in your view... do you ever attempt to explain where the complex amplitude formalism comes from, or is it just taken to be a fact, and you are trying to interpret it?
For the record...

If, given a quantum state and a measurement operator, you have some means of extracting the "expected value" of the operator... then the Gelfand-Naimark-Segal construction says that you can take the 'square root' of the state, giving you a bra and a ket that represent your quantum state. Such objects live in Hilbert spaces.

This applies to any state -- including statistical mixtures. In the case of a statistical mixtrue, the Hilbert spaces1 produced by the GNS construction has a special form; it is reducible. You can split the Hilbert space into irreducible state spaces (e.g. you can split "particle with unit charge" into "particle with charge +1" and "particle with charge -1"). The state corresponding to a statistical mixture can always be decomposed into its individual parts.

The same cannot be said for pure states; the GNS construction provides you with an irreducible Hilbert space. Thus we see that pure states cannot be reinterpreted as statistical mixtures. (At least, not in any direct way)

1: more precisely, it's a unitary representation of the measurement algebra.