In a previous post, I calculated the Coulomb Law (gravity) in $d$-dimension. It is in fact also the free field propagator in $d$-dimension.
The equation of motion of a Klein-Gordon field is, \[ \left( \partial^2 - m^2\right) D_F(x-y) = \delta(x-y) \] where the Feynman propagator $D_F(x-y) = \left< 0 | \mathcal{T}\varphi(x) \varphi(y) | 0 \right>$ is also called Feynman propagator. Because of this equation, it is also called Green's function.
The equation of motion can be solved exactly in free space boundary condition. Fourier transform shows, \[
D_F(x-y) = - \int \small{\frac{\mathrm{d}^d k}{(2\pi)^d}} \frac{ e^{i k\cdot (x-y)} }{k^2+m^2 - i\epsilon}
\] Observation: $D_F(x-y)$ is a Lorentz invariant. Therefore, it must be only a function of the Lorentz invariant $(x-y)^2$. Do the contour integral of $k^0$ component first,
\[\
D_F(x-y) = i\int \small{\frac{\mathrm{d}^{d-1} k}{(2\pi)^{d-1}2k^0}} e^{i \mathbf{k}\cdot \mathbf{(x-y)} - i k^0 \cdot |x^0 - y^0|},
\]\[
= i\theta(x^0-y^0) \int \small{\frac{\mathrm{d}^{d-1} k}{(2\pi)^{d-1}2k^0}} e^{i k \cdot (x-y)} + i\theta(y^0-x^0) \int \small{\frac{\mathrm{d}^{d-1} k}{(2\pi)^{d-1}2k^0}} e^{i k \cdot (y-x)},
\] where $k^0 = \sqrt{\mathbf{k}^2 + m^2}$. Note that, $k \cdot x = -k^0 x^0 + \mathbf{k \cdot x}$.
Following the method in the previous post, \[
D_F(x-y) = i \frac{1}{2(2\pi)^\frac{d-1}{2} r^{d-2}} \int_0^\infty \mathrm{d}\xi \frac{ \xi^\frac{d-1}{2} }{\sqrt{\xi^2 + m^2 r^2}} J_{\frac{d-3}{2}}(\xi) e^{-i\sqrt{\xi^2+m^2r^2}\frac{|x^0-y^0|}{r}},
\] where $r = |\mathbf{x - y}|$, $J_\nu(x)$ is the Bessel function of the first kind.
It seems this integral does not have a closed form representation. Nevertheless, the closed-form expression can be obtained for several special cases,
\left. D_F(x-y) \right|_{x^0=y^0} = \frac{i}{(2\pi)^{\frac{d}{2}}} \frac{(m r)^{\frac{d}{2}-1} K_{\frac{d}{2}-1}(m r)}{r^{d-2}}
\]
Note that this is the same as Coulomb potential. Bessel functions of the second kind $K_\alpha(m r)$ are exponential decaying functions. At large $m r$, \[K_\alpha( m r) \sim \sqrt{\frac{\pi}{2 m r}} e^{- m r}.\]
Generally, power law is a sign of massless excitation (here from vacuum), exponential decay is a sign of excitation overcoming energy gap. In this instance, the energy gap is the particle mass (called mass gap).
\[
D_F(x-y) = \frac{i}{(2\pi)^{\frac{d}{2}}} \frac{m^{\frac{d}{2}-1} K_{\frac{d}{2}-1}\left(m \sqrt{(x-y)^2}\right)}{(x-y)^{d/2-1}}
\]
D_F(x-y) = \frac{i}{(2\pi)^\frac{d-1}{2}\Gamma\left(\frac{d-1}{2}\right)} \int_0^\infty \frac{\mathrm{d} k}{2\sqrt{k^2+m^2}} k^{d-2} e^{ - i \sqrt{k^2+m^2} |x^0 - y^0|},
\]
D_F(x-y) = \frac{i \Gamma\left( \frac{d}{2}-1 \right)}{4\pi^\frac{d}{2}}\frac{1}{ \left((x-y)^2\right)^{\frac{d}{2}-1}}
\]
where $(x-y)^2 = -(x^0-y^0)^2 + (\mathbf{x-y})^2$.
Define the partition function, \[
Z[J] = Z[0] \int \mathcal{D}\varphi \exp\{i S[\varphi] + \int \mathrm{d}^d x J(x) \varphi(x)\}.
\] The two-point time-ordering correlation function, \[
\left< 0 | \mathcal{T}\varphi(x) \varphi(y) | 0 \right> = \left. Z^{-1}[0] \frac{\delta}{\delta J(x)}\frac{\delta}{\delta J(y)} Z[J] \right|_{J \to 0}.
\] The equation of motion (the first Dyson-Schwinger equation) reads, \[
\left< 0 | \right. \mathcal{T}\varphi(x) \frac{\delta S}{\delta \varphi(y) } \left. | 0 \right> = i \delta(x-y). \] Now, Dyson-Schwinger Equations (DSEs) are normally a series of tower. Fortunately for free field theory, the equation of motion is self-contained. In free field theory, \[
S_0 = \frac{1}{2}\int \mathrm{d}^d x \varphi(x) \left( \partial^2 - m^2 \right) \varphi(x).
\] The partition function of the free field theory can be expressed in terms of Feynman propagator, \[
Z[J] = Z[0] \exp\{-\frac{i}{2}\int \mathrm{d}^d x \mathrm{d}^d y J(x) D_F(x-y) J(y) \}.
\] To prove this, just substitute $\chi(x) = \varphi(x) + \int \mathrm{d}^d y J(y) D_F(x-y) $. That's why propagator (classical Green's function) is important in free field theory (subsequently perturbation theory).
The equation of motion of a Klein-Gordon field is, \[ \left( \partial^2 - m^2\right) D_F(x-y) = \delta(x-y) \] where the Feynman propagator $D_F(x-y) = \left< 0 | \mathcal{T}\varphi(x) \varphi(y) | 0 \right>$ is also called Feynman propagator. Because of this equation, it is also called Green's function.
The equation of motion can be solved exactly in free space boundary condition. Fourier transform shows, \[
D_F(x-y) = - \int \small{\frac{\mathrm{d}^d k}{(2\pi)^d}} \frac{ e^{i k\cdot (x-y)} }{k^2+m^2 - i\epsilon}
\] Observation: $D_F(x-y)$ is a Lorentz invariant. Therefore, it must be only a function of the Lorentz invariant $(x-y)^2$. Do the contour integral of $k^0$ component first,
\[\
D_F(x-y) = i\int \small{\frac{\mathrm{d}^{d-1} k}{(2\pi)^{d-1}2k^0}} e^{i \mathbf{k}\cdot \mathbf{(x-y)} - i k^0 \cdot |x^0 - y^0|},
\]\[
= i\theta(x^0-y^0) \int \small{\frac{\mathrm{d}^{d-1} k}{(2\pi)^{d-1}2k^0}} e^{i k \cdot (x-y)} + i\theta(y^0-x^0) \int \small{\frac{\mathrm{d}^{d-1} k}{(2\pi)^{d-1}2k^0}} e^{i k \cdot (y-x)},
\] where $k^0 = \sqrt{\mathbf{k}^2 + m^2}$. Note that, $k \cdot x = -k^0 x^0 + \mathbf{k \cdot x}$.
Following the method in the previous post, \[
D_F(x-y) = i \frac{1}{2(2\pi)^\frac{d-1}{2} r^{d-2}} \int_0^\infty \mathrm{d}\xi \frac{ \xi^\frac{d-1}{2} }{\sqrt{\xi^2 + m^2 r^2}} J_{\frac{d-3}{2}}(\xi) e^{-i\sqrt{\xi^2+m^2r^2}\frac{|x^0-y^0|}{r}},
\] where $r = |\mathbf{x - y}|$, $J_\nu(x)$ is the Bessel function of the first kind.
It seems this integral does not have a closed form representation. Nevertheless, the closed-form expression can be obtained for several special cases,
1. $x^0 - y^0 = 0$
\[\left. D_F(x-y) \right|_{x^0=y^0} = \frac{i}{(2\pi)^{\frac{d}{2}}} \frac{(m r)^{\frac{d}{2}-1} K_{\frac{d}{2}-1}(m r)}{r^{d-2}}
\]
Note that this is the same as Coulomb potential. Bessel functions of the second kind $K_\alpha(m r)$ are exponential decaying functions. At large $m r$, \[K_\alpha( m r) \sim \sqrt{\frac{\pi}{2 m r}} e^{- m r}.\]
Generally, power law is a sign of massless excitation (here from vacuum), exponential decay is a sign of excitation overcoming energy gap. In this instance, the energy gap is the particle mass (called mass gap).
(Corollary) $x^0 = y^0, m = 0$
\[ \left. D_F(x-y) \right|_{x^0=y^0} = \frac{i \Gamma\left(\frac{d}{2}-1\right)}{4\pi^{\frac{d}{2}}} \frac{1}{r^{d-2}}, \quad (m=0) \](Corollary) $(x-y)^2 > 0$, space-like
As we said above, $D_F(x-y)$ is only a function of $(x-y)^2$. If $(x-y)^2 = (\mathbf{x}-\mathbf{y})^2 - (x^0-y^0)^2 > 0 $, we can Lorentz-transform to a frame in which $x'^0 = y'^0$, $ r'^2 = \left| \mathbf{x'} - \mathbf{y'}\right|^2 = (x-y)^2$. Following 1,\[
D_F(x-y) = \frac{i}{(2\pi)^{\frac{d}{2}}} \frac{m^{\frac{d}{2}-1} K_{\frac{d}{2}-1}\left(m \sqrt{(x-y)^2}\right)}{(x-y)^{d/2-1}}
\]
2. $\mathbf{x} - \mathbf{y} = 0$
\[D_F(x-y) = \frac{i}{(2\pi)^\frac{d-1}{2}\Gamma\left(\frac{d-1}{2}\right)} \int_0^\infty \frac{\mathrm{d} k}{2\sqrt{k^2+m^2}} k^{d-2} e^{ - i \sqrt{k^2+m^2} |x^0 - y^0|},
\]
3. $ m = 0 $
\[D_F(x-y) = \frac{i \Gamma\left( \frac{d}{2}-1 \right)}{4\pi^\frac{d}{2}}\frac{1}{ \left((x-y)^2\right)^{\frac{d}{2}-1}}
\]
where $(x-y)^2 = -(x^0-y^0)^2 + (\mathbf{x-y})^2$.
 |
| Figure 1. The image of the free scalar field propagator in $d=4$D. $x$-axes is $\mathbf{x-y}$, $y$-axes is $x^0-y^0$. |
Appendix: The Feynman Propagator
The following derivation for scalar field (Klein-Gordon field) is standard-textbook.Define the partition function, \[
Z[J] = Z[0] \int \mathcal{D}\varphi \exp\{i S[\varphi] + \int \mathrm{d}^d x J(x) \varphi(x)\}.
\] The two-point time-ordering correlation function, \[
\left< 0 | \mathcal{T}\varphi(x) \varphi(y) | 0 \right> = \left. Z^{-1}[0] \frac{\delta}{\delta J(x)}\frac{\delta}{\delta J(y)} Z[J] \right|_{J \to 0}.
\] The equation of motion (the first Dyson-Schwinger equation) reads, \[
\left< 0 | \right. \mathcal{T}\varphi(x) \frac{\delta S}{\delta \varphi(y) } \left. | 0 \right> = i \delta(x-y). \] Now, Dyson-Schwinger Equations (DSEs) are normally a series of tower. Fortunately for free field theory, the equation of motion is self-contained. In free field theory, \[
S_0 = \frac{1}{2}\int \mathrm{d}^d x \varphi(x) \left( \partial^2 - m^2 \right) \varphi(x).
\] The partition function of the free field theory can be expressed in terms of Feynman propagator, \[
Z[J] = Z[0] \exp\{-\frac{i}{2}\int \mathrm{d}^d x \mathrm{d}^d y J(x) D_F(x-y) J(y) \}.
\] To prove this, just substitute $\chi(x) = \varphi(x) + \int \mathrm{d}^d y J(y) D_F(x-y) $. That's why propagator (classical Green's function) is important in free field theory (subsequently perturbation theory).
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