New Directions in the Foundations of Physics Conference in Washington DC 2013 (part 4)
“Quantum information and quantum gravity” by Seth Lloyd
A thought provoking talk at the conference was that of Seth Lloyd. He showed how to derive Einstein’s general relativity equation from quantum limits for measuring space-time geometry and an additional black hole assumption.
One way to think of measuring the geometry of space-time is to think of a comprehensive
Measuring time amounts to measuring the number of clock ticks and this requires
energy. Everybody is familiar with the position-momentum uncertainty principle,
but the energy-time uncertainty principle is not so clear cut. This is because
in quantum mechanics time is a parameter, and not an operator, and care has to
be exercised in interpreting the energy-time uncertainty principle.
Margolus and Levitin had obtained a bound of quantum evolution time in terms of the initial mean energy of the system E: E delta t >= hbar pi/2. From this the total possible number of clock ticks in a bounded region of space time (of radius r and time span t) cannot exceed 2Et/pi hbar. In principle, quantum mechanics does not limit the accuracy for measuring time, and all you need is to do is add enough energy. But in general relativity, adding energy in a bounded region will eventually lead to the creation of a black hole. So here is a general relativity assumption: we want the radius of the bonded region to be larger than the Schwarzschild radius Rs = 2GM/c^2
From this (in terms of the Plank time Tp and Plank distance Lp) one obtains the maximum number of clock ticks achievable in a bounded region of space time before creating a black hole: r t / pi Lp Tp
Now r*t is an area and naive field theory would suggest r^3 t. Also naïve string theory would suggest at first sight r^2 t.
From those kinds of area considerations, Seth was able to deduce general relativity equations inspired in part by Ted Jacobson’s ideas (in fact Seth collaborated with Ted on this result). Now you may ask (as I certainly did) if you start with Schwarzschild’s radius and you derive Einstein’s equations, are you not vulnerable to charges of circularity? Perhaps, but the result is still interesting.
(I have one more story to tell from the conference. Please stay tuned for part 5-the last one.)