这是一份nottingham诺丁汉大学PHYS2003作业代写的成功案例

$$
\partial_{t}^{2} G\left(t, t^{\prime}\right)=\delta\left(t, t^{\prime}\right)
$$
If we try to solve this using the Fourier method, we end up with an integral of the form
$$
G\left(t, t^{\prime}\right)=\int \frac{\mathrm{d} \omega}{2 \pi} \frac{\mathrm{e}^{-\mathrm{i} \omega\left(t-t^{\prime}\right)}}{-(\omega \pm \mathrm{i} \epsilon)^{2}} .
$$
This integral has a second-order pole and cannot be used to solve an equation involving $\partial_{t}^{2}$. For example, the equation for the position of a Newtonian body
$$
\partial_{t}^{2} x(t)=F / m,
$$
cannot be solved in this way since it is not homogeneous in the source $\mathrm{F} / \mathrm{m}$. The solution is easily obtained by integration
$$
x(t)=\frac{1}{2} \frac{F}{m} t^{2}+v t+x_{0} .
$$
Since there are terms in this solution which are not proportional to $\mathrm{F} / \mathrm{m}$, it is clear that the Green function method cannot provide this full answer. However, the equation can still be solved by the Green function method in two stages.

PHYS2003 COURSE NOTES ：

Consider the relativistic scalar field as an example. The complex scalar field satisfies the equation
$$
\left(-\square+m^{2}+V\right) \phi(x)=J(x) .
$$
Now let $\varphi_{n}$ be a complete set of eigenfunctions of the operator in this equation, such that a general wavefunction $\phi(x)$ may be expanded in terms of a complete set of these with coefficients $c_{n}$,
$$
\phi(x)=\sum_{n} c_{n} \varphi_{n}(x)
$$
such that
$$
\left.\int \mathrm{d} \sigma_{x}\left(\varphi_{n}, \varphi_{m}\right)\right|{t=t^{\prime}}=\delta{n m}
$$
The wavefunction $\phi(x)$ and the eigenfunctions $\varphi_{n}(x)$ are assumed to be oneparticle wavefunctions. The discrete indices $n, m$ denote any bound state quantum numbers which the wavefunction might have. The eigenfunctions satisfy
$$
\left(-\square+m^{2}+V\right) \varphi_{n}(x)=0 .
$$
The eigenfunctions can also be expressed in terms of their positive and negative frequency parts,
$$
\varphi_{n}(x)=\varphi_{n}^{(+)}(x)+\varphi_{n}^{(-)}(x)
$$