By Abhay Ashtekar
Chosen subject matters on quantum gravity are taken care of emphasizing extra geometrical me- tools and conceptual matters than useful research, perturbative expansions and computation of numbers, The lirst half bargains with Asymptotic Quantization sheding gentle at the foundation of the Bondi-Meizner-Sachs team within the gravitational radiation idea. the second one half is dedicated to canonical quantization, which supplies the de- distinctive quantum dynamics and enhances the basically kinematic;)] asympotic descrip- description preset"!Led within the lirsf half.
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Additional info for Asymptotic quantization: based on 1984 Naples lectures
We are now ready to calculate the transition amplitude. Let us recall that in the Heisenberg picture, for tf > U, we have U (tf,Xf]ti,Xi) = H(Xf,tf\Xi,ti)H. Let us divide the time interval between the initial and the final time into N equal segments of infinitesimal length e. i8) In other words, for simplicity, we discretize the time interval and in the end, we are interested in taking the continuum limit e —> 0 and N —> oo such that Eq. 18) holds true. ,(iV-l). 19) Introducing complete sets of coordinate basis states for every intermediate time point (see Eq.
In other words, (bx 0 0 - ^ 0 b2 0 ••• BD o o \\ h ••• i i = UBU]. 40) we obtain T / d C eJC BDC &neiriTBl1 / / dCi • • • dCiv-iiE^bnC e N-l -n(r" N 71=1 = (z7r)^f i (det J B)~5 . 41) Here we have used the familiar fact that the Jacobian for a change of variable by a unitary matrix is unity. Using this result in Eq. 42) 40 Field Theory: A Path Integral Approach It is clear from this analysis that the transition amplitude can be denned only if the matrix B does not have any vanishing eigenvalue.
In this integration, the end points are held fixed and only the intermediate coordinates are integrated over the entire space. Any spatial configuration of the intermediate points, of course, gives rise to a trajectory between the initial and the final points. Thus, integrating over all such configurations (that is precisely what the integrations over the intermediate points are supposed to do) is equivalent to summing over all the paths connecting the initial and the final points. Therefore, Feynman's path integral simply says that the transition amplitude between an initial and a final state is the sum over all paths, connecting the two points, of the weight factor eft ^J.