# 极端环境下的物理学|PHYS11066 Physics of Extreme Environments代写 Edinburgh代写

The propagator $G\left(r-r^{\prime}\right)$ appeared in the calculation when we did the Gaussian integral and had to invert the discrete, lattice Laplacian on a twodimensional square array,
$$\Delta^{2} G(r)=\delta_{r, 0}$$
where $\Delta^{2}=\Delta_{x}^{2}+\Delta_{y}^{2}$ and $\Delta_{x}^{2}$ is the simplest discrete form of the second derivative,
$$\Delta_{x}^{2} G(r)=G(r+x)+G(r-x)-2 G(r)$$
We solve for the propagator $G$ by passing to momentum space, as in the corresponding calculation in continuum field theory,
$$G(r)=\int_{-\pi}^{\pi} \frac{\mathrm{d} k_{x}}{2 \pi} \int_{-\pi}^{\pi} \frac{\mathrm{d} k_{y}}{2 \pi} \frac{\mathrm{e}^{\mathrm{i} \vec{k} \cdot \vec{r}}}{4-2 \cos k_{x}-2 \cos k_{y}}$$

We will need some of the properties of $G(r)$. First, it is easy to check that, at large $|r| \gg 1, G(r)$ is well approximated by the continuum massless propagator,
$$G(r) \approx-\frac{1}{2 \pi} \ln (r / a)-\frac{1}{4}, \quad|r| \gg 1$$
The behavior of $G(r)$ for small $r$ can also be determined from the integral above,
$$G(0) \approx \frac{1}{2 \pi} \ln (L / a)$$

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