List of published papers on non-unitary time evolution.
Jack Sarfatti
We have seen from Asher Peres that nonlinear (implies nonunitary)
extensions of quantum mechanics will not only permit quantum connection
communication but will also let us beat the second law of thermodynamics
giving a new energy source. Peres also points out that the second law at
the quantum level may not be all that secure in the formal nitty gritty.
So, at the very least, this line of inquiry will subject the second law to
more stringent tests than heretofore.
I am indebted to Creon Levit of NASA Ames Super Computer Facility for
finding these papers for me.
In response to e-mail I will list some published papers that talk about
nonunitarity in several contexts not all quantum (e.g. Prigogine). I will
not discuss details of the papers here. Would appreciate others posting
references on more recent "nonunitary" papers.
The order is (not quite) random:
*Oh! WOW! Here's a paper:
On a non-unitary evolution of quantum systems, by W. Daniel, Nicholas
Copernicus University, Poland, Hev. Physica Acta, 55, 330 (1982) "the
deterministic evolution of a quantum system does not need to preserve the
orthogonality of states. an example of non-linear Schrodinger-like
equation governing such an evolution is indicated." If you remember it was
the preservation of the orthogonality of the transmitter photon spin states
that prevented my attempt at quantum connection communication. If we can
squash the initial orthogonality then we can get a controllable (at a
distance) polarization for the twin receiver photon.
Daniel starts from Piron's quantum logic. He uses a semigroup not a group
for time evolution. On p.336 "However there are physically interesting
situations where the orthogonality of states is not preserved during the
evolution. This is the case when the evolution is described by an equation
with a 'perturbed' generator
dx/dt = -iHx - kBx
... it has an unpleasant feature that it does not preserve the norm of a
vector, which is the usual demand for the evolution equation in quantum
theory......the nnormalized solution. X(t) .. .satisfies
dX/dt = -iHX + k(<B> - B)X
<B> = <x|B|x>/<x|x>
this equation with B = H was proposed in (N Gisin, Journ Phys, 14A, 2259
(1981)). It was hown there that the evolution governed by this equation is
dissipative .. provided the initial state is not an eigenstante of H ,,,,
... Gisins equation has been successfully applied to the damped harmonic
oscillator, spin 1/2 sustem and to the description of quantum measurement
...it ...may be derived using the general formalism of master equations.."
Daniel concludes:" one obtains .. a generalization of a unitary evolution
....the orthogonality of the states is not preserved, although the
superposition of states is..."
Non-unitary Scattering and Capture, E.B. Davies, Comm Math Phys. 71,277
(1980) semigroups on Hilbert space for complex optical potential
it's phenomenology and non-unitarity comes from coupling to environment so
in principle everything is unitary - but???
Canonical Description of the Solitary Quantum Decay, Garbaczewski 7
Vitiello, Nuovo Cimento, 44A , 108 1 March 1978
nonunitary transformation for irreversible behavior of soliton
Time, Structure, and Fluctuations, I. Prigogine, Science, 201 No 4358,
p.777 1 Sept. 1978 nonunitarity as irreversibility in classical stat mech
but he insinuates it's important for QM also, three levels of time 1)
dynamical (QM&classical) 2) irreversible Lyapounov 3) history via
bifurcations. I am not clear on whether this is bogus or not despite the
Nobel prize - anyone understand what Prigogine is talking about? Does the
Emperor have clothes? I'm sure he does. I will make an effort to study
Prigogine's ideas soon. I was his guest for two weeks in 1973 when I was
at ICTP, Trieste (invited by Abdus Salam). I found another preprint of
Prigogine in which he indicates that the non-unitary transformations are
needed when the dynamics is "weakly unstable". So I think the general idea
is something like this:
Standard unitary time evolution of closed quantum systems between
measurements is fine when the classical limit is stable so that canonical
transformations and all that jazz you learned in Goldstein's book work -
but when the classical dynamics is chaotic then the corresponding quantum
dynamics requires non-unitarity in a fundamental way - that is more than a
truncation procedure. Thus - quantum connection communication and beating
second law of thermodynamics (i.e. Prigogine + Peres) involves classical
chaos with fractal strange attractros in phase space. The quantum analog
to classical chaos means going beyond standard QM to a nonlinear theory
like Weinberg's perhaps.
Right Prigogine says "irreversible processes correspond to an extension of
dynamics and not to some defined approximation as was often assumed in the
past." (e.g. some of the papers I list below) I think Prigogine may be on
to something really good! There is no classical Liouiville theorem for
chaos...hence quantum version will not conserve probability in Hilbert
space either?
Representations of the Poincare Group by Higher Order Field Equations.., H.
Stumpf 0378-4371/82 North-Holland (I only have xerox without journal ref)
uses nonunitary reps of Poincare with indefinite metric in Hilbert space. I
think he was student of Heisenberg. (I was at Heisenberg's Institute in
summer of 66. Wheeler taught course on geometrodynamics NATO Institute.)
He suggests that quarks violate causality inside the nucleon but since they
are trapped it does not matter. What about early universe before quarks
get confined?
On the Reduction of a Contraction Semigroup to a Completely Non Unitary
Semigroup, N Levan & L Rigby, J Math Analysis & Applications 67, 1 (1979)
deals with control theory - says non-unitarity is "state feedback"...
"a unitary operator is never strongly stable.... it is always desirable to
stabilize an unstable system by means of appropriate feedback... the
semigroup of the feedback system is a completely non-unitary contraction
semigroup on the Hilbert space..."
Space-like fields and non-unitary representations of the Poincare group, by
Marcel Guenin and Jacques Simon (Univ. Geneva) about 1978 I only have xerox
with no journal title - but see Phys. Letters 62B, 81 (1976) tachyons need
nonunitary representations - that's good. They also say quarks are
tachyons. Somehow quark confinement is connected to tachyons! "phase
symmetry"..."complex mass representations of the Poincare group" ...OK
nonunitary representations of the Poincare group have been studied in G
Rideau, Rep Math Phys 4, 47 (1973), L.S. Shulmann, Ann Phys. 59, 201 (1970)
the tachyon propagators do not exponentially decay outside the light cone
...see M Flato & M Guenin , Helv Phys Acta 50, 117 (1977)
Matrix elements for infinitesimal operators of the groups U(p+q) and U(p,q)
in a U(p)xU(q) basis, Klimyk & Gruber, J Math Phys 20, 1995 (Oct
1979)..."representations of compact Lie groups in a noncannonical basis ..
by means of ... the principle nonunitary series representations of a ,,,,
noncompact semisimple Lie group...G is... real noncompact Lie group...every
irreducible finite dimensional representation of G is.. a subrepresentation
in a .... principle nonunitary series representation of G ... obtained from
the principle unitary series ... by means of analytic continuation.. of the
parameters..." So this may connect with wavelets! They get everything
unitary at end though.
Zwanzig_Feshabach projection... Landau-Zener approx.... Turner & Dahler, J.
Phys. B Atom Molec 13,161 (1980) it's a chemistry paper but they have a
"non-unitary time-ordered memory operator" for collisions.
Time-dependent theory of non-Hemitian Schrodinger equation: application to
multiphootn-induced ionization decay of atoms, Faisal & Moloney, J Phys. B
Atom Molec Phys. 14, 3603 (1981) reduced problem is non-unitary -- total
probability still conserved
Non-linear approximation to the Schrodinger Time Evolution in a Truncated
Subspace, Schultheis, Annal of Physics 141, 179 (1982) "the solutions are
non-unitary and allow for a flux of probability between the subspace and
the excluded space..."
*I suppose the idea to consider is that irretrevable leakage of probability
to a very random environment will give an effective non-unitary time
evolution in the open truncatedd system far off equilbrium that permits
using EPR nonlocality as a communication channel within separated parts of
the open system.