Recently, Stefan on Backreaction put together a beautiful and informative post on some experiments involving Schroedinger’s equation for neutrons in a situation where the gravitational field could be modeled as in Newton’s equations, that is, as mgz where z is height, and g is the acceleration of gravity at earth’s surface.
The experiment consists of a plane which neutrons cannot penetrate. A beam of very low temperature neutrons is sent just above the plane. The force of gravity causes the neutrons to bounce off the plane. Schroedinger’s equation turns this classical problem into a wave function. One finds various solutions with various energy. As he notes, this is not quantum gravity, but in the post, he put a link to an article that I think gives a hint on how to write gravity as a quantum effect:
Spontaneous emission of graviton by a quantum bouncer
G. Pignol, K.V. Ptotasov, V.V. Nesvizhevsky
Spontaneous emission of graviton rates for the quantum bouncer states are evaluated.
The above article calculates the probability that a neutron that is known to be in one of the energy eigenstates of Schroedinger’s equation for the bouncing neutron problem, makes a spontaneous transition to another solution, via the emission of a graviton. This calculation reads directly on how I’ve been thinking about deriving gravity as a quanum effect the past few weeks.
Though I’ve been thinking about this sort of thing for the last week or so, and I downloaded and read the above short article when reading Stefan’s post, I somehow didn’t notice that the calculation was similar to what I have been playing with recently. Instead, I found the article again when I googled arXiv for spontaneous+emission, and put two and two together when Google showed that I’d read the article in the past 24 hours.
There are two things to note here. The first is that if we want to write gravity as a particle interaction on a flat space in the usual method of QFT, the mgh term is just the overall effect of very large numbers of virtual gravitons. Any one of these gravitons has to change the energy of the neutron by negligible amounts, otherwise they would already have been detected. The overall effect of these gravitons is to act as the mgh gravitational potential term in Schroedinger’s equation, just as the overall effect of virtual photons between an electron orbiting a proton can be modeled in Schroedinger’s equation as the 1/r electric potential.
The second is that the paper discusses not these billions and billions of virtual gravitons, but instead the possibility of a graviton spontaneously emitting from the neutron system due to a jump in the energy eigenvalue of Schroedinger’s equation.
In physics, when one sees “spontaneous emission”, it is natural for one’s brain to consider the subject of “stimulated emission”. It was Einstein who determined the relationship between the rates of spontaneous and stimulated emission. One can also make difficult stimulated emission calculations for gravitons. The results are cross sections that are a multiple of the square of the Planck length, i.e. very small.
The standard model puts the elementary fermions as left and right handed terms that are coupled together by a Higgs field. If one wishes to couple virtual gravitons to standard model particles in such great numbers that their overall effect is to appear as a classical potential, then one must pick a coupling that occurs a great number of times per second. The Higgs coupling is a natural for this.
So my idea is to think of the virtual gravitons that are responsible for the mgh potential term as being due not to the exchange of virtual gravitons, but instead due to the stimulated emission of a sort of Higgs boson. This sort of thing has to have zero mass, so that it can travel great distances. And the primary interaction needs to be a stimulated emission, rather than an absoption cross section, because when one brings masses together, the result is that the gravitational field they produce is not canceled.