Marcus: Penrose certainly believes that quantum theory needs a thorough reworking, but that is not what my question is about.
turbo-1: I was asking about something much more specific.
Thank you both for replying.
I have had a passing interest in Penrose's work for about eight months. The original reference for his ideas about gravity's role is
R. Penrose, "On Gravity's Role in Quantum State Reduction," General Relativity and Gravitation, 28(5), 1996, pp. 581-600.
This is reprinted as Chapter 13 of Physics Meets Philosophy at the Planck Scale.
Here's my take on his ideas about gravitationally induced state reduction.
Consider a single (extended) massive particle at rest, and two possible spatial positions of the particle, i.e., the particle is either "over here" (H) or "over there". Suppose the quantum states for "over here" and "over there" are stationary states \left| \Psi_{1} \right> and \left| \Psi_{2} \right>, respectively. If the the same gravitational potential energy is associated with both positions (no change in "height"), the stationary states have the same energy E.
So,
i \hbar \frac{\partial}{\partial t} \left| \Psi_{1} \right> = E \left| \Psi_{1} \right>
and
i \hbar \frac{\partial}{\partial t} \left| \Psi_{2} \right> = E \left| \Psi_{2} \right>.
Now, suppose that the quantum state of the mass is the superposition
\left| \Psi \right> = c_{1} \left| \Psi_{1} \right> + c_{2} \left| \Psi_{1} \right>
of "over here" and "over there". It would appear that \left| \Psi \right> is also a stationary state with energy E, but Penrose says "Not so fast!"
The states \left| \Psi_{1} \right> and \left| \Psi_{2} \right> refer to different spacetimes - the spacetime for the "over here" state is curved over here, because the mass is over here, while the spacetime for the "over there" state is curved over there. Consequently, \partial / \partial t for the "over here" spacetime and, \partial / \partial t for the "over there" spacetime are different as timelike vectors and as quantum operators. Thus, the superposition is ill-defined.
Superpose anyway, and try to find a quantative measure of just how different the two \partial / \partial t are. When taking the partial derivative with respect to t, spatial coordinates are held constant, so spatial variations cause changes in \partial / \partial t.
Thus, the spatial variation of some relevant quantity might serve as a measure of the ill-definedness of the superposition. What relevant quantity? For weak fields, maybe the difference between the Newtonian gravitation potentials for the two parts of the superposition, i.e., integrate over all space, the square (so plus and minus variations don't cancel) of the gradient (spatial variation) of the difference between the potentials.
Poisson's equation gives that this is proportional to the gravitational self-energy of the mass distribution that's left over when the two mass distributions are subtracted, which also seems like a good measure of just how different things are for the two spacetimes. This energy is then used to estimate, via the uncertainty principle, the time taken for state reduction.
It is here, I believe, that Penrose has had a new idea for further justification, and I was asking for details of this new "rigorous" idea.
Reduction to what state? Penrose claims that the states to which a "superposition" will reduce are solutions to the coupled system consisting of the Schrödinger equation including Newtonian gravitational potential energy together with Poisson's equation with the mass density represented by m \left| \psi \right|^2.
Penrose claims that this should be observable either by an experiment in space, or by a ground-based experiment being considered at the University of California. As Patrick Vanesch has said, if no decoherence is seen, everyone will interpret this as vindication for standard quantum mechanics. If decoherence is seen, then some people will claim that feedback with the environment is at work, not gravitationally induced state reduction.
http://arxiv.org/abs/quant-ph/0210001
Regards,
George