wick rotation

lim swee hoe

It seems to me that the whole fuss about switching to euclidean time to
remove wildly oscillatory behaviour in many integrals found in QFT to be
unnecessary. For instance if we calculate <0|0>j the result is a path
integral / <0|0>[j=0]. However we cheat in our mathematics by demanding that
<0|0>[j=0] is equal to 1. Mathematically <0|0>j=0 is actually some integral
with wildly oscilatory behaviour because we choose to take |0> at t'=+inf
and <0| at t=-inf. It is entirely reasonable to declare the amplitude to be
1 but there is no reason to say that the phase is zero. We could have simply
recognized the phase to be a meaningless quantity undetectable to
experiment. Now going back to <0|0>[j=0], we foolishly choose to integrate
to infinity when in fact all the action happens between t1 and t2, when
j~=0. Therefore, let us simply take the beginning time and the end time to
be large (again quite foolish) but restricted to the condition E0*(t' - t) =
2*pi*N, for some large N. Now all our integrals
end up with no phase contribution due to time evolution outside of [t1,t2].
If we did not artificially contrive t' and t as above, we end up with some
meaningless phase, so be it. Sure its not elegant but its saves a few pages
in your QFT textbook. Am i missing something?

Hendrik van Hees

lim swee hoe wrote:

> pages in your QFT textbook. Am i missing something?

Hm, it sounds to me as if that is what is usually done in textbooks,
isn't it?

Here you find my try to comprehend the very difficult topic around the
LSZ reduction formalism:

http://cyclotron.tamu.edu/hees/publ/lect.pdf
p. 111 ff

--
Hendrik van Hees Texas A&M University
Phone: +1 979/845-1411 Cyclotron Institute, MS-3366
Fax: +1 979/845-1899 College Station, TX 77843-3366
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