(p.473) B Useful complex analysis
(p.473) B Useful complex analysis
For every complex problem there is an answer that is clear, simple, and wrong.
H. L. Mencken (1880–1956)
Throughout the book we have tried to keep the amount of complex analysis to a minimum. This appendix provides a simple guide to some of the complex analysis commonly employed in quantum field theory. The guide is illustrated by examples drawn from the subject, including the most important function of a complex variable in quantum field theory: the propagator.
B.1 What is an analytic function?
If a function is analytic in a region close to a point z, then it has a derivative at every point in that region. We define the derivative of a complex number as
Importantly, for the function to be analytic, the derivative shouldn’t depend on the way the interval in the complex plane is selected.
The function is analytic; is not. To see the first write(B.2)
just as for a normal derivative. Note that we didn’t have a choice of , this procedure works whatever we choose.
On the other hand for we have(B.3)
If (with y real) then you will get a different derivative to the case of , with x real. See Boas for more details.
(p.474) B.2 What is a pole?
A pole is a type of singularity1 that behaves like the singularity of at . Let be analytic between two circles and . In the region between them we can write as a so-called Laurent series expanded about a point :
The part with b coefficients is known as the principal part of the series. Don’t confuse this with the principal value of an integral, which is different, and discussed below.
A result and some definitions:
• If all b’s are zero then is analytic at .
• If all b’s after are zero then we say that we have a pole of order n at . If we have a simple pole.
• The coefficient is called the residue of at .
An important example is the function
which has residue and all other and zero. This function has a simple pole at .
Let’s examine the pole structure of two of our propagators. The non-relativistic, retarded, free electron propagator is given by(B.6)
This has a first-order pole at . This is shown in Fig. B.1(a).
The Feynman propagator for the free scalar field is usually written(B.7)
this is helpfully rewritten (see Chapter 17) as a function of the complex variable :(B.8)
The first (particle) part has a simple pole at . The second (antiparticle) part has a simple pole at . This is shown in Fig. B.1(b).
B.3 How to find a residue
You can find a residue at the pole by writing a Laurent series. There are more direct methods too. For example, when we have a simple pole, we can write
(p.475) Example B.3
The residue of at the simple pole is given by(B.10)
B.4 Three rules of contour integrals
A contour C is a closed path in the complex plane with a finite number of corners which doesn’t cross itself. Integrals around such contours have a number of useful properties.
We can get some practice with a contour integral by calculating(B.11)
where the contour is shown in Fig. B.2. We split the contour into two parts and start with the straight line along the real axis. Take and this part becomes(B.12)
Now for the semicircle, described by , where . We have giving(B.13)
Adding the contributions we have(B.14)
There are three useful theorems for evaluating integrals taken around contours. The first is Cauchy’s theorem:
If is analytic on and inside C then
This is good news, since it says that if the region in a contour contains no poles then the integral gives zero. It also explains why the previous example gives zero: is analytic on and inside the contour.
The second theorem is known as Cauchy’s integral formula:
(p.476) If is analytic on and inside a simple closed curve C, and the point a is inside C, then the value of is given by
The third is the residue theorem:
If has singularities at points , then, for a closed curve enclosing these points we have
where the integral around C is performed in the anticlockwise direction. (You merely change the sign of the answer if you perform the integral in the clockwise direction.)
Often we want to do difficult integrals over real variables. These may be turned into easier integrals if we form a contour in the complex plane which includes the original domain of integration and use the rules given above. The art is in choosing the best contour to do the integral.
We can use Cauchy’s theorem along with the residue theorem to justify some of the more seemingly cavalier tricks employed in the discussion of propagators in Chapters 16 and 17. Let’s find the inverse Fourier transform of the retarded propagator , given by(B.18)
for which the integration path is along the real axis. To use our contour integral rules we must complete the contour by joining up this path with a further section of path which will either be in the upper half of the complex E plane or in the lower half.
Suppose we take it in the lower half-plane. Then, as we take the limits along the real axis to the semicircular path gets larger and larger. This will make a large, negative imaginary contribution to E. Let’s call it . The exponential will then involve a contribution . If is positive then this contribution gets smaller, eventually vanishing as the contour becomes infinitely large.2 We conclude that, for the case , the integral above is equivalent to the contour shown in Fig. B.3(a).
Let’s do that integral. The contour contains the pole at so we use the residue theorem to say(B.19)
where the minus sign follows from our attempt to take the integral in the clockwise direction. The residue at the pole is and the answer is(B.20)
which we stress applied for .
(p.477) What if we tried to complete the contour in the upper half-plane? Then we would have obtained a large, positive, imaginary contribution to the exponential resulting in a contribution which blows up for . Such a badly behaved integral is certainly not suitable for evaluating the Fourier transform. However, for the semicircular contour has a vanishing contribution at infinity and we again have the equivalence of the Fourier transform and the contour shown in Fig. B.3(b). Notice that contains no poles, so Cauchy’s theorem says that the integral is zero.
We conclude that for and noting that is an infinitesimal quantity, we may replace both (i) zero for and (ii) for with and conclude(B.21)
just as we had in Chapter 16 without the need for adding damping factors by hand!
The meaning of the factors now becomes clear. These infinitesimals position the poles of the propagators in such a way as to ensure the correct causality relationships. Returning to the scalar field propagator of Fig. B.1 we see that closing the contour in the lower half-plane picks up the positive energy pole, leading to a factor , while closing the contour in the upper half-plane picks up the negative energy pole and leads to the factor . This motivates the definition of the Feynman propagator in Chapter 17.
B.5 What is a branch cut?
The function is multivalued. We are always at liberty to add whole numbers of to the argument in the exponential, i.e. where n is an integer. The fact that this function is multivalued means that care must be taken when taking roots and logarithms.
For the case of the logarithm we have, taking :(B.22)
Clearly, for fixed r, takes different values for and , even though both choices correspond to the same z.
We therefore agree that we should only consider angles in some interval in of size , known as a branch of the function. In order to make this clear in the complex plane, we can define lines that we will agree not to cross with any of our operations. These are known as branch cuts. The points from which these emerge are known as branch points.
For the branch point is the origin and the branch cut may be taken along the positive real axis as shown in Fig. B.4(a). It may also be taken along the negative real axis, or indeed any convenient line. Crossing the branch cut makes the function jump by .
which we consider in the complex plane. The square root in this equation must be restricted to a single branch. The square root vanishes for . which are therefore the branch points. For convenience, we take the branch cuts to extend along the imaginary axis as shown in Fig. B.4(b). Notice that when we do the integral in Chapter 8 we can’t cross the cuts with our contour, so we must direct the contour around the cuts.
Another occasion where we must consider a function with a branch cut is the full propagator discussed in Chapter 31, given by(B.24)
which has a pole structure shown in Fig. B.5. The second term in the expression tells us to expect multiparticle contributions. These lead to a line of poles with infinitesimal separation between them. Such a line of poles is another way of describing a branch cut and so we draw a cut extending along the real axis from the branch point given by the two-particle production threshold .
B.6 The principal value of an integral
The Cauchy principle value is a method of giving improper integrals a value. This is not complex analysis, but is often used in integrals involving the propagator. Suppose we want to evaluate the integral but we have the problem that the integrand diverges at (where
and so both and will also blow up.3 In that case we may take the Cauchy principal value of the integral, denoted by and defined by
This gives the integral an unambiguous value, as demonstrated in the example below.
(p.479) is not well defined as the integrand diverges at . You can see that the two integrals and both diverge, giving and respectively.
To get around this we integrate from up to and then from up to , thereby cutting out the troublesome part of the problem. We obtain(B.27)
We find that , independent of . We may therefore take the limit and obtain an unambiguous result. We conclude that(B.28)
The principal value arises in quantum field theory when we want to do integrals of the form where is a complex-valued function and a and b are real, obeying . In this case we use the following theorem which says that
Often we take and obtain the identity,4
(1) By singularity, we mean a point where a mathematical object is not defined.
(2) This disappearance of semicircular contours in the limit of infinite radius is a result of Jordan’s lemma. This says that the integral along the infinite upper semicircle is zero, provided (i) , and (ii) is a well-behaved function satisfying
(3) Specifically we require
for . (That is, one sign in front of the ∞ is plus and one minus.)
(4) Sometimes called the Dirac relation in the physics literature.