The Klein-Gordon Propagator

1. Klein-Gordon Equation

The Klein-Gordon equation is a relativistic wave equation, related to the Schrödinger equation. It is second order in space and time and manifestly Lorentz covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles.

The free Klein-Gordon equation is

(1)

where . In the Heisenberg picture, the field operator can be written as (Each Fourier mode of the field is treated as an independent oscillator)

(2)

where , , , . acting on the vacuum creates a particle at . Then, the amplitude for a particle to propagate from to is , that is

(3)

Acting on will produce , thus is a solution of Klein-Gordon equation.

2. The Green’s Function

We may introduce a source to Klein-Gordon equation if the Klein-Gordon field coupled to an external field

(4)

And then, we need the Green’s function which satisfy

(5)

With the help of Fourier’s Transformation

(6)

(7)

Eq.4 can be transformed into

(8)

is just the inverse transformation of

(9)

or

(10)

We can use contour integral to find out the above integration over

3. The Contour Integration

Contour integration is an application of residue theorem. An integral over a closed path on the complex plane can be evaluated by its residues within this path

(11)

Let’s denote the integral over as

(12)

Depends on whether is larger or smaller than , there are two sets of contour as be chosen.

a. If , we chosen the contour with beneath the real axis. Since will contribute a factor on , which make the integral on this curve becomes 0. Besides the , we also have the freedom to choose and . But it turns out that it will make no difference. Take the contour (a) for example, the residue theorem is

(13)

The contour integral can be decomposed into

(14)

In the limit ,

(15)

In the limit ,

(16)

(17)

(18)

Take these results into Eq.13, we can get

(19)

We can get the same result if we take contour (b), although these is no singular points within this contour

(20)

The integral on and will be different since they are on the other half circles.

(21)

(22)

considering the above adjustment, is still . The contour (c) and (d) also give us the same result.

b. If , we chosen the contour with above the real axis to insure . The contour (a) will give us

(23)

(24)

(25)

thus

(26)

Contour (b),(c),(d) will give us the same result.

Take into Eq.10, we can get the Green’s function

(27)

4. Feynman Propagator

Green’s function plus any solutions of the homogeneous equation is still Green’s function. Add a to G(x-y) will give us the retarded Green’s function

(28)

We can get the Feynman propagator if we add to ,

(29)

Feynman propagator can also be written as

(30)

which has poles at and no poles on the real axis. This integral can be find out directly with the help of contour integral.
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Refferences

[1] https://en.wikipedia.org/wiki/Klein–Gordon_equation

[2] Peskin, Michael E., and Daniel V. Schroeder. “An Introduction to quantum field theory.” (1995).

[3] 梁昆淼. 数学物理方法(第四版). 高等教育出版社. 2010.