Two similar questions related to analytic continuation of a complex variable and its conjugate

Question

1. 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 $C^2$. But, I do not understand why the value of the integral should not alter when viewed as an integral over $R^2$ as Brown claims and then uses to finish the evaluation of the integral.
2. A similar kind of argument is used while doing complex scalar field theory where we treat $\phi$ and $\phi^*$ as independent fields. For a while, I have understood it as first complexifying them such that the the two fields combined now live in $C^2$ 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.

Answer 1

1. 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 $\int dz\,f(z)$ may literally be interpreted in the Riemannian way, as the sum of many infinitesimal products $\Delta z\cdot f(z)$, and this definition may be used whether $\Delta z$ is complex or real. When it's real, it's just a special case.

2. 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 – because we're dealing with a "larger space" that contains the original "physical space" as a subspace. The solutions of the "real problem" are among the solutions to the "complex problem" as long as the action is analytic (both in the real sense and the complex sense) near these solutions. At the end, the toy model for this statement is simple and based on this assertion: if $S'(x)=0$ for a real $x$ in the real sense, then also $S'(x)=0$ for the same real value of complex $x$ in the complex sense.

On the contrary, even if additional solutions exist in the complex plane, they have physical consequences, so it's good to know about them. (It's analogous to a key result here, the fundamental theorem of algebra, that says that in complex numbers, the $n$-th order polynomials simply allow one to find all $n$ roots; those that are not real – and complex roots often exist even for real polynomials – shouldn't really be discriminated against.) For a simpler example of this point, note that the scattering amplitudes $A(k)$ as functions of the momentum $k$ often have singularities for complex values of $k$ and they know about the energies of the bound states. Why it's so – and why many other statements dependent on analytic continuation are right – may look like "miraculous coincidences" or "magic behind the scenes" at the beginning but these insights are clearly so important that one should better learn and exercise sufficiently so that this "magic" will stop to look mysterious and will become a mundane part of the physicist's thinking. This part depends on some "somewhat advanced" mathematics and an intuitively, emotionally based person interested in physics has probably never expected that he would have to learn such mathematical things to understand the behavior of physical systems. Nevertheless, it's true.

Because the mathematical facts about the analytic continuation are so useful and universally applicable, one should better get rid of any negative emotions and hostility towards them as quickly as possible. I have personally considered complex numbers to be more fundamental than the real numbers since my 8th birthday or so – but even if it's not your case, it's never too late. One must ultimately make this transition to understand modern physics and especially to be able to do calculations. As one moves towards more complex theories, especially quantum field theory and string theory, the importance of various analytical continuations increases.

Complex numbers are "truly fundamental" in some physical contexts and at least "very useful tricks" in others. For example, the probability amplitudes (and matrix elements etc.) everywhere in quantum mechanics have to be complex for many reasons. The Wick rotation is a way to switch to the Euclidean spacetime where the integrals relevant for the calculation of Green's functions and probability amplitudes are more well-behaved. The correct calculations in quantum field theory involve various $i\epsilon$ prescriptions in the propagators and the right choice of contours in the complex plane – even if one is trying to stay in the infinitesimal vicinity of the real axes. Metastable states have energies $E-i\Gamma/2$; the imaginary part is related to the decay rate so instabilities and wide resonances "genuinely" move some singularities that existed on the real axis into the complex plane (by a finite, nonzero amount). Thermal calculations in quantum mechanics may be converted to the evolution by an imaginary time – so they admit a path integral calculation with a periodic, Euclidean time.

Also, representations of groups – in physics as well as mathematics – should be thought of as complex representations. It's the default type. When they are real, they should be viewed as complex representations with an extra structure that may be imposed, a "structure map", something that allows one to impose reality projections. A different kind of structure maps creates quaternionic or pseudoreal representations – so quaternions are, much like real numbers, less fundamental than complex numbers (despite their being "larger"). One can't do modern physics without complex representations of groups: most of the representations appearing in quantum physics are complex.

On the other hand, in classical electromagnetism, the complex amplitudes for various electromagnetic waves and oscillating circuits are just a "useful tool" that doesn't have to be considered fundamental or real. But they're still useful. Already classical physicists had to learn to think in terms of $\exp(\pm ix)$ rather than $\cos x$ and $\sin x$ because the exponentials are simpler: $\exp(A+B)=\exp(A)\exp(B)$ is simpler than the formulae for $\cos(A+B)$ etc. And I could continue for a while. The overall message is that the insights of modern physics have shown the we have thought about these matters in a wrong way when we were thinking that the real numbers were fundamental. Complex numbers aren't just a "trick" to find various things; the analytic continuations of many things into the complex realm is a natural setup to fully describe many physical situations and the restriction to real values is often just an ad hoc procedure that hides most of the relevant structures hiding behind the laws of physics. And some objects in physics can't be required to be real at all.