On a Quantized Scalar Field in Some Bianchi-Type I Universe. II: DeWitt's Two Vacuum States Connected Causally

H. Nariai
1977 Progress of theoretical physics  
This is a sequel to the previous paper on the relation between Parker-Fulling's approach and DeWitt's one to a quantized scalar field in a special Bianchi-type I universe. An attempt is made to dissolve such a complexity in our previous stipulation of DeWitt's two vacuum states connected causally that the resulting elementary solution includes information at the two limiting epochs appearing in it. By virtue of the refined stipulation which is based on a suitable extension of the space-time
more » ... fold, it is shown that the revised elementary solution becomes identical with ours relying on the former approach, whose coincidence limit is a local quantity (as it should be so). Several problems related to the above procedure are also touched upon. O'(x, x') between two points x = (t, x) and x' = (t', x') and the associated hi-scalar propagators, except for Hadamard's elementary solution, 3 J were already known. 4 J) On the basis of the former approach which leads to a time-independent set of annihilation and creation operators and the associated vacuum state, an elementary solution G(l) (x, x') was defined by the vacuum expectation value of field operators at x and x', and a time-dependent vacuum state at any time was also introduced by a suitable Bogoliubov transformation. It was expected that some limit of the latter vacuum states at two separate epochs leads to De Witt's vacuums I in, vac) and I out, vac) connected causally, in terms of which his elementary solution Gn (1) (x, x') can be defined. However, it was shown that, contrary to ours, the Gn OJ (x, x') thus defined has such a regular part that it inevitably depends upon infor· mation at the limiting two epochs t = t; C>O) and t = t 1 (> t;). This is undesirable,
doi:10.1143/ptp.58.842 fatcat:3zzgi7xotbc4toasawxcgnjvwq