物理代写|PHYS11065 Quantum field theory

Yukawa potential.
(a) Calculate the equations of motion for a massive vector $A_\mu$ from the Lagrangian
$$
\mathcal{L}=-\frac{1}{4} F_{\mu \nu}^2+\frac{1}{2} m^2 A_\mu^2-A_\mu J_\mu,
$$
where $F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. Assuming $\partial_\mu J_\mu=0$, use the equations to find a constraint on $A_\mu$.
(b) For $J_\mu$ the current of a point charge, show that the equation of motion for $A_0$ reduces to
$$
A_0(r)=\frac{e}{4 \pi^2 i r} \int_{-\infty}^{\infty} \frac{k d k}{k^2+m^2} e^{i k r} .
$$
(c) Evaluate this integral with contour integration to get an explicit form for $A_0(r)$.
(d) Show that as $m \rightarrow 0$ you reproduce the Coulomb potential.
(e) In 1935 Yukawa speculated that this potential might explain what holds protons together in the nucleus. What qualitative features does this Yukawa potential have, compared to a Coulomb potential, that make it a good candidate for the force between protons? What value for $m$ might be appropriate (in $\mathrm{MeV}$ )?

Photon polarizations (this problem follows the approach in [Feynman et al., 1996]).
(a) Starting with $\mathcal{L}=-\frac{1}{4} F_{\mu \nu}^2+J_\mu A_\mu$, substitute in $A_\mu$ ‘s equations of motion. This is called integrating out $A_\mu$. In momentum space, you should get something like $J_\mu \frac{1}{k^2} J_\mu$.
(b) Choose $k_\mu=(\omega, \kappa, 0,0)$. Use current conservation ( $\left.\partial_\mu J_\mu=0\right)$ to formally solve for $J_1$ in terms of $J_0, \omega$ and $\kappa$ in this coordinate system.
(c) Rewrite the interaction $J_\mu \frac{1}{k^2} J_\mu$ in terms of $J_0, J_2, J_3, \omega$ and $\kappa$.
(d) In what way is a term without time derivatives instantaneous (non-causal)? How many causally propagating degrees of freedom are there?

Graviton polarizations. We will treat the graviton as a symmetric 2-index tensor field. It couples to a current $T_{\mu \nu}$ also symmetric in its two indices, which satisfies the conservation law $\partial_\mu T_{\mu \nu}=0$.
(a) Assume the Lagrangian is $\mathcal{L}=-\frac{1}{2} h_{\mu \nu} \square h_{\mu \nu}+\frac{1}{M_{\mathrm{P} 1}} h_{\mu \nu} T_{\mu \nu}$. Solve $h_{\mu \nu}$ ‘s equations of motion, and substitute back to find an interaction like $T_{\mu \nu} \frac{1}{k^2} T_{\mu \nu}$.
(b) Write out the 10 terms in the interaction $T_{\mu \nu} \frac{1}{k^2} T_{\mu \nu}$ explicitly in terms of $T_{00}, T_{01}$, etc.
(c) Use current conservation to solve for $T_{\mu 1}$ in terms of $T_{\mu 0}, \omega$ and $\kappa$. Substitute in to simplify the interaction. How many causally propagating degrees of freedom are there?
(d) Add to the interaction another term of the form $c T_{\mu \mu} \frac{1}{k^2} T_{\nu \nu}$. What value of $c$ can reduce the number of propagating modes? How many are there now?

Calculate the transition matrix element $T_{i j}$ for the process $e^{+} e^{-} \rightarrow \gamma \rightarrow \mu^{+} \mu^{-}$.
(a) Write down the $\frac{1}{E_i-E_0}$ terms for the two possible intermediate states, from the two possible time slicings.
(b) Show that they add up to $\frac{2 E_\gamma}{k^2}$, where $k_\mu$ is now the 4-momentum of the virtual off-shell photon.

Show that the differential cross section for $2 \rightarrow 2$ scattering with $p_i^\mu+p_A^\mu \rightarrow$ $p_f^\mu+p_B^\mu$ in the rest frame of particle $A$ can be written as
$$
\frac{d \sigma}{d \Omega}=\frac{1}{64 \pi^2 m_A}\left[E_B+E_f\left(1-\frac{\left|\vec{p}_i\right|}{\left|\vec{p}_f\right|} \cos \theta\right)\right]^{-1} \frac{\left|\vec{p}_f\right|}{\left|\vec{p}_i\right|}|\mathcal{M}|^2,
$$
where $\theta$ is the angle between $\vec{p}_i$ and $\vec{p}_f, E_B=\sqrt{\left(\vec{p}_f-\vec{p}_i\right)^2+m_B^2}$ and $E_f=$ $\sqrt{\vec{p}_f^2+m_f^2}$