See the scan attached below. Brown, in his QFT book, argues a certain way to do
ID: 1384077 • Letter: S
Question
See the scan attached below. Brown, in his QFT book, argues a certain way to do an integral. I understand that 1.8.13 or equivalently 1.8.14 can be performed once analytic continuation is done. I also understand that under this transformation the LHS of 1.8.12 would not change when viewed as an integral over C2. But, I do not understand why the value of the integral should not alter when viewed as an integral over R2 as Brown claims and then uses to finish the evaluation of the integral.
A similar kind of argument is used while doing complex scalar field theory where we treat ? and ?? as independent fields. For a while, I have understood it as first complexifying them such that the the two fields combined now live in C2 and then solving the variational problem in the complexified space. And then in the end we identify the two fields as conjugate of each other, that is, we project them over to C again. The fact that extremas in the variation in complexified space maps bijectively to extrema in the variation in the original problem looks kind of a mathematical accident to me. But, as often happens in mathematics, there is a deep reason behind such coincidences. So, I wonder if somebody can shed the light on some deep reason for the fact that we can use this kind of trick of complexification and projection. It isn't obviously clear to me why this trick can be used unless this above argument is produced which seems like some magic is happening behind the curtains.
Explanation / Answer
The only two "somewhat nontrivial" facts that the trick relies upon is: a) the independence of de facto closed contour integrals of holomorphic functions on the paths in the complex space (I say "de facto closed" because infinite curves combined with functions that quickly, e.g. in a Gaussian way, decrease at infinity allows us to consider the curve closed at infinity, too; this method also applies to two-dimensional integrals as long as they are correctly rewritten or rethought as contour integrals of contour integrals); b) the fact that the contour integrals over the real axes are the same thing as the integrals of a real function of a real variable. To see the latter, just realize that ?dzf(z) may literally be interpreted in the Riemannian way, as the sum of many infinitesimal products ?z?f(z), and this definition may be used whether ?z is complex or real. When it's real, it's just a special case.
In all such real-vs-complex considerations, the right way to think about it is that you allow all variables to be complex for as long a time as possible, and you only impose the reality conditions such as x=x? at the very end. In a wide variety of contexts and variables, it's right to consider the complex variables to be more fundamental than the real ones. Only with this attitude, one can "internalize" the analytic continuation that is so natural in so many contexts in modern physics. It seems to me that your general concerns boil down to your efforts not to make this transition to the "thinking in complex numbers" and to keep on demonizing and tabooing the methods based on complexification, so it's a self-inflicted wound. Your comment slightly sounds as "I won't admit that complex numbers play a key role, anyway" which wouldn't be a good starting point to understand modern physics.
Concerning the variational problem, the complexification of the variables may at most produce additional solutions, i.e. additional stationary points of the action