Thursday, July 13, 2006

Signal Nonlocality in Curved Hilbert Space

Review of some basic points.

I. Feynman Lagrangian Histories
Uses world lines in configuration space for multi-particle entangled states.
The path amplitude ~ e^i(Classical Action)/(hbar)^?
Each symplectic phase-space area element gets a factor of hbar.

II. Hamiltonian theory
Use "Unitary" Operators of generic form e^i(Hamiltonian)(Time)/(hbar)^?

Consider the generic pair entangled quantum state

|A,B) = |++)(++|A,B) + |--)(--|A,B)

(A,B|A,B) = (A,B|++)(++|A,B) + (AB|--)(--|AB)

Because (++|--) = 0 (orthogonality)

Where completeness of the internal dichotomic q-numbers in pair Hilbert space is

|++)(++| + |--)(--| + |+-)(+-| + |-+)(-+| = 1

Note that

|++)(++| + |--)(--| =/= 1

Even though

|(++|AB)|^2 + |(--|AB)|^2 = 1


(AB|++)(++|AB) + (AB|--)(--|AB) = 1

Consider only Alice's (A) evolution starting from

|A,B) = |++)(++|A,B) + |--)(--|A,B)

|A,B) -> |A'B) = U(A+)|++)(++|AB) + U(A-)(--|AB)

Note, for now do not assume that

U(A+) = U(A-)

U(A+)*U(A+) = 1

U(A+)*U(A-) =/= 1


(A'B|A'B) = (AB|++)(++|AB) + (AB|--)(--|AB) + (AB|++)(--|AB)(++|U(A+)*U(A-)|--) + cc

=/= (AB|AB)

in the general case.

This allows signal nonlocality because the effective transformation is not unitary.

One needs an additional postulate that for all possible total experimental arrangements.

U(A+) = U(A-)

This is a possible loophole in orthodox QM for signal nonlocality to creep back in without going to a post-quantum covering theory. For example, with long coherence times and retardation plates in alternate paths for the same quantum one may have different travel times for interfering alternatives each with a different path-dependent unitary operator. That is, the unitarity may be anholonomic analogous to parallel transport in a curved space-time here we have a "curved Hilbert space." Of course, one might argue that this is a new post-quantum theory. On the other hand, it may show an incompleteness in orthodox quantum theory similar to the introduction of non-Euclidean geometries in the 19th Century.

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