On M.M.Nieto's historical paper on quantum phase and quantum phase and time operators.
Nieto's discussion starts out a bit loosely at the beginning. For example
U = e^i(phase operator)
by itself is not really a time evolution operator.
For example, given
[P,X] = -ih
Treat P as the operator and a as the c-number
e^iPa/hbar generates the displacement a in the quantum wave Psi(x), i.e.
e^iPa/hbarPsi(x) = Psi(x + a)
Similarly in the momentum Fourier transform space
e^-iXkPsi(p) = Psi(p - hbark)
therefore, treating PHASE as the operator and n' as an integer
e^iPHASEn'Psi(n) = Psi(n + n')
Psi(n) is a basis in Fock space.
Nieto considers quantized phase eigenvalues in the conjugate problem which correspond to things like quantized fluxes for phase difference experiments like the Bohm-Aharonov electron interferometer showing the quantum reality of the connection field even in a region where the "curvature" vanishes! Of course the connection field is only defined locally mod a gauge transformation, or a general coordinate transformation if we are doing "quantum gravity".
Of course, the idea is to use these phase operators in some kind of Hamiltonian when you consider the time evolution - is it unitary or not?
For example in the Josephson and Bohm-Aharonov effects
The Hamiltonian H ~ C12, i.e. the cosine phase difference operator in Nieto's paper.
Is C12 "self-adjoint"?
If not, then there is signal nonlocality in e^Ht/hbar operating on some initial state.
So the math in Nieto's paper is concerned with finding the appropriate phase operators to put into some Hamiltonian.
Nieto is primarily interested in new low photon number experiments, but he does, at the end of his paper come across outside the light cone EPR nonlocal effects. Since I am primarily here interested in the opposite high N limit where is the connection? Also I am interested in off-mass-shell boson electron-positron pair vacuum condensates not the macro-quantum quantum optical states of real photons in laser radiation.
I am saying that the bottom -> up PW Anderson "More is different" emergence of Einstein's c-number general theory of relativity is in the collapse of vacuum phase space volume in the pre->post-inflationary phase transition that creates the Big Bang. Einstein's tetrad-based local c-number field equations then pop out of the ripples in the macro-quantum coherent "holographic universe" c-number phase from the Off-Diagonal-Long-Range Order (ODLRO) in the virtual boson condensate (center of mass coordinates of the virtual electron-positron pairs in a kind of BCS scenario).
So how do the micro-quantum phase operators that Nieto is interested come into this problem? Obviously they describe the exotic vacua "dark energy phases". The 1-sided non-unitarity of the time-evolution U operator generated by quantum phase in the naive attempt to get a self-adjoint phase operator, at the very beginning of Nieto's paper, is here a virtue not a vice. We do not expect unitarity at all here because the macro-quantum vacuum condensate is a reservoir for the outward and inward flow of virtual quanta. That is, the virtual electron and virtual positron bind together inside the vacuum condensate, but they are unbound outside the condensate in a kind of neutral ionized plasma, i.e. usual picture of vacuum polarization zero point fluctuations. Of course, lack of unitarity also implies signal-nonlocality - but you cannot make electromagnetic "clicks" with virtual quanta. So we are saved, as it were, by Bell's theorem, from signal-nonlocality using zero point fluctuations. However, because of the equivalence principle, these same ionized plasma micro-quantum fluctuations do gravitate - and even anti-gravitate as dark energy!
U = e^i(phase operator) = aN^-1/2
N = a*a
a is for the CM degrees of freedom of the virtual electron-positron pair as a boson oscillator
xcm = 2^-1/2[a + a*]
pcm = 2^-1/2[a - a*]
UNU* = N + 1
UU^-1 = 1
U^-1U =/= 1
Which is just what the Doctor ordered in our problem of "More is different" emergent creative complexity from spontaneous symmetry breakdown in the vacuum (or ground state).
This micro-quantum U is only for exotic vacuum "dark energy" part of our "two fluid" model.
Einstein's cosmological constant is small because most of the large-scale zero point fluctuations are swept under the rug of vacuum coherence.
On Dec 12, 2004, at 11:24 AM, Jack Sarfatti wrote:
On Dec 12, 2004, at 1:51 AM, firstname.lastname@example.org wrote:
This pretty much sums up Jack's position: any person who proposes an interpretation of GR that is not fully consistent with the Einstein equivalence hypothesis -- whatever that may have been -- is "ipso facto off his rocker".
Indeed, Paul your writing "the Einstein equivalence hypothesis -- whatever that may have been" says it all. It is very revealing. You have not connected the dots properly. You are missing what Michael Polyani calls "tacit knowing" a kind of subconscious heuristic probably connected with "signal nonlocality" - what Bierman measures as "presponse".
GODD (I.J. Good's version of P.K. Dick's VALIS) is subtle but not malicious and moves in strange ways such as my discovery of Martin Nieto's 1993 paper from Los Alamos describing my role with Lenny Susskind and Johnny Glogower (who I brought to Cornell) in the discovery of the C & S operators for micro-quantum phase.
Click NOW on the very important http://arxiv.org/pdf/hep-th/9304036
That's an order soldier! ;-)
I am now able to look on the first really important problem I worked 40 years ago with a new perspective! The issue is what the nonunitarity of the time evolution operator for quantum phase is really telling us - especially in the macro-quantum case of spontaneous broken ground state symmetry with the Goldstone phase and the Higgs field that controls the dark energy of the universe! "Signal nonlocality" is the clue. More on that anon.
BTW Nieto's history is incomplete. I was also working on the quantum phase operator problem BEFORE returning to Cornell in the Autumn of 1963 when, with Phil Morrison's help, I brought Johnny Glogower with me and rightaway "The Three Stooges" Lenny Susskind, Johnny and me became a team. We were all equally dysfunctional megalomaniacs who complimented each other with the whole greater than the sum of its parts. I had been in George Parrent's Jr Tech/Ops group on spy satellites, either NSA or CIA, in a building shared with Mitre on Route 2, Burlington, Mass in early 1963. George was a student of Emil Wolf's and we also spent a lot of time in the Boston University physics department. Roy Glauber was developing quantum coherence theory at Harvard and George assigned me to learn all that stuff to develop the quantum version of Wolf's classical partial coherence theory. Also lasers were still in their infancy right then. George was more a hands on guy and I was his resident boy genius theorist. They did not want me to leave Tech/Ops and go back to Cornell. I was guaranteed a fast PhD at BU if I stayed and high pay in defense work. But then I never would have met Lenny Susskind. Indeed, Carruthers probably posed the problem because of me. I think Nieto put the cart before the horse.
Violation of cause and effect backwards in time requires nonunitary time evolution.