1. Re: Simple description of general relativity. Physics tries not to waste time :) (Veit Elser)
Message: 1 Date: Thu, 19 May 2016 16:46:07 +0000 From: Veit Elser <ve10@cornell.edu> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Simple description of general relativity. Physics tries not to waste time :) Message-ID: <A2861FB7-5047-4610-B002-149D653C8CEC@cornell.edu> Content-Type: text/plain; charset="utf-8"
On May 19, 2016, at 10:27 AM, Warren D Smith <warren.wds@gmail.com> wrote:
You've heard of the "principle of least time" in optics... light follows a path which minimizes the time it takes to get there... in quantum field theory there is a more general action principle, which is that the integral, over all paths followed by all particles, of exp(i*T*m*c^2/hbar) dT is stationarized, where T is the proper time consumed by that particle (of rest mass m) in following its trajectory. (Actually things a bit trickier for spinor particles, and there are certain constant factors inserted whenever two particles merge or bifurcate, but I will ignore that here.)
Taking seriously -- in quantum mechanics -- the sum over all paths, one has to include particle world-lines parts of which are not time-like and over which T is pure imaginary. With the right choice of sign, ?tachyonic? propagation is thereby suppressed in a direct way.
If the Hilbert action formulation is the right way to think about classical GR, i.e. the Einstein equations are just a consequence, then it seems reasonable that action-stationarity in that setting is also the hbar->0 asymptotics of a sum-over-whatever (e.g. metrics) principle. Nobody has yet made sense of that sum. Are there real-exponentially suppressed configurations?
-Veit
---The idea you just mentioned is the Wheeler-DeWitt equation and had already been explored by Bryce DeWitt (starting in the 1950s or 1960s I think?). To put it in a nutshell, the space of all 3+1 metrics is (if I recall right) called "superspace." It is an infinite dimensional space which seems pretty damn hard to comprehend or work with. Anyhow, the Wheeler-DeWitt equation is basically Schrodinger's equation set in superspace (which they figured out how to do somehow). Then you proclaim that the universe is a quantum system whose state is a wave function on superspace. In the classical limit hbar-->0, a single trajectory in superspace gets all the probability density, just like in quantum mechanics where (allegedly) a particle in the classical limit moves along a well defined trajectory. That trajectory is the one stationarizing the action, which explains classical mechanics action principles. This "fact" is "seen" because the Feynman path-integral formulation of QM Schrodinger equation really puts in all trajectories with a complex phase arising from exp(i*action/hbar). Only the extrema-of-the-action trajectories get to interfere constructively, the rest cancel out. I am using the scare quotes to indicate that the rigor in this whole line of handwaving jive, aka theoretical physics, has pretty much never managed to come... but it is a great story. So in the Wheeler-DeWitt version of the story, the Einstein GR equations giving a unique evolution of our 3+1 metric are an approximate fiction arising in classical limit, since they arise from Hilbert action principle... and the real story is the Wheeler-DeWitt equations of quantum gravity, and the real arena of physics is not 3+1 space, it is superspace. [By the way, do not confuse this superspace with supersymmetry, which also has sometimes got something they call "superspace", totally different.] OK, sounds like a great story. As far as I understand (which is not much) they did succeed in a few cases in solving the WDW equation to get some exact solutions, but for the most part this whole theory of quantum gravity (if it is one) has been unusable. Further, I do not know if they were able to put in source terms (in fact I doubt they were able to do it in any successful way). Later so called "loop theory" [totally unrelated to abstract-algebra "loops" aka nonassociative groups] was cooked up by Abhay Ashtekar and followers as a way to try to quantize GR. And it presumably in some sense must be equivalent to the Wheeler-DeWitt equation, albeit expressed very differently... but now you are getting totally above my head. I have reason to suspect those guys do not know what they are doing. But they know more than me at least about some stuff. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)