Jack Sarfatti science forum Guru
Joined: 29 Apr 2005
Posts: 487

Posted: Thu Jul 13, 2006 10:27 pm Post subject:
Signal Nonlocality in Curved Hilbert Space



Review of some basic points.
I. Feynman Lagrangian Histories
Uses world lines in configuration space for multiparticle entangled states.
The path amplitude ~ e^i(Classical Action)/(hbar)^?
Each symplectic phasespace 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,BA,B) = (A,B++)(++A,B) + (AB)(AB)
Because (++) = 0 (orthogonality)
Where completeness of the internal dichotomic qnumbers in pair Hilbert
space is
++)(++ + )( + +)(+ + +)(+ = 1
Note that
++)(++ + )( =/= 1
Even though
(++AB)^2 + (AB)^2 = 1
i.e.
(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
etc.
(A'BA'B) = (AB++)(++AB) + (AB)(AB) +
(AB++)(AB)(++U(A+)*U(A)) + cc
=/= (ABAB)
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 postquantum 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 pathdependent unitary
operator. That is, the unitarity may be anholonomic analogous to
parallel transport in a curved spacetime here we have a "curved Hilbert
space." Of course, one might argue that this is a new postquantum
theory. On the other hand, it may show an incompleteness in orthodox
quantum theory similar to the introduction of nonEuclidean geometries
in the 19th Century. 
