Jack Sarfatti
Debate with Zielinski on the physics of over-complete non-orthogonal eigenfunctions http://bit.ly/q3YVeX
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Zohar Ko PZ is obviously right.
Jack Sarfatti How much did Zielinski pay you to say that? ;-) Polemics are inappropriate here. Before I delete your remark, defend it so I can see if you know what you are talking about or merely trying to pi$$ me off.
Jack Sarfatti There is nothing "obvious" about Z's generally dark remarks.
Jack Sarfatti If you think Z is correct then you do not understand quantum measurement theory of Bohr to von Neumann. A switch in the base eigenfunctions is not a passive formal affair, but is a real change in Bohr's "total experimental arrangement" e.g. changing relative orientations of Stern-Gerlach magnets, introducing quarter-wave plates etc.
Zohar Ko It's the first axiom of QM that everybody learns in school: physics is independent of the basis, orthogonal or not.
Jack Sarfatti You are a good example of the saying a little bit of knowledge is a dangerous thing.
Jack Sarfatti The invariance of the trace of the density matrix with operators is only for unitary transformations that preserve inner products with similarity transformations on the operators.
Jack Sarfatti In particular, unitary transformations on an orthogonal basis will give a new orthogonal basis. However, unitary transformations on a non-orthogonal basis will give a new nonorthogonality basis - actually they are in same unitarily equivalence class.
Jack Sarfatti However, a non-unitary transformation is needed to connect an orthogonal basis with a non-orthogonal basis. Therefore, the traces of density matrices with operators in a non-orthogonal basis are different from those with those operators in an orthogonal basis. We have two different non-overlapping unitarily equivalent equivalence classes. In particular there is entanglement signaling in entangled systems with a physically demanded non-orthogonal basis, and none in those with physically demanded orthogonal basis.
Jack Sarfatti Let U be a unitary transformation. Let N be a non-unitary transformation.
UU* = U*U = 1
NN* =/= 1
Let {|Oi)} = orthogonal basis
(Oi|Oj) = 0 , i =/= j
Let {|Zi)} be a non-orthogonal basis
|Z) = N|O)
For any operator A
A' = UAU^-1
For the appropriate density matrix R
(A) = Tr{RA}
R = sum over eigenvalues P(oj)|Oj)(Oj|
P(oj) = probability to find oj in a statistical mixture
R' = URU^-1
(A') = Tr{R'A'} = Tr{RA} = (A)
i.e. invariance under unitary transformations of all physical expectation values.
On the other hand
S = NRN^-1 is a completely new physical object that is not unitarily equivalent to R - it means the emergence of qualitatively new physics.
In particular for the same operator A
Tr{SA} =/= Tr{RA}
However, under a unitary U
S' = USU^-1 = UNRN^-1U^-1
TrS'A' = TrSA
In particular if A describes a nonlocal signal
Tr{RA} = 0
Tr{SA} =/= 0