Here I’ll write the solution of my homeworks in QFT
Problems
1. Classical electromagnetism (with no sources) follows from the action
\[S = \int d^{4}x \Big(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\Big),\;\;\;\;\;\text{where}\;F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.\](a) Derive Maxwell’s equations as the Euler-Lagrange equations of this action, treating the components $A_{\mu\nu}$ as the dynamical variables. Write the equations in standard form by identifying $E^{i} = -F^{0i}$ and $\epsilon^{ijk}B^{k} = -F^{ij}$.
(b) Construct the energy-momentum tensor for this theory. Note that the usual procedure does not result in a symmetric tensor. To remedy that, we can add to $T^{\mu\nu}$ a term of the form $\partial_{\lambda}K^{\lambda\mu\nu}$, where $K^{\lambda\mu\nu}$ is antisymmetric in its first two indices. Such an object is automatically divergenceless, so
\[\hat{T}^{\mu\nu} = T^{\mu\nu} + \partial_{\lambda} K^{\lambda\mu\nu}\]is an equally good energy-momentum tensor with the same globally conserved energy and momentum. Show that this construction, with
\[K^{\lambda\mu\nu} = F^{\mu\lambda} A^{\nu},\]leads to an energy-momentum tensor T that is symmetric and yields the standard formulae for the electromagnetic energy and momentum densities:
\[\varepsilon = \frac{1}{2} (E^2+B^2);\;\;\;\; S = E\times B.\]2. The complex scalar field. Consider the field theory of a complex-valued scalar field obeying the Klein-Gordon equation. The action of this theory is
\[S = \int d^{4}x (\partial_{\mu}\phi^{\ast}\partial^{\mu}\phi-m^{2}\phi^{\ast}\phi) .\]It is easiest to analyze this theory by considering $\phi(x)$ and $\phi^{\ast}(x)$, rather than the real and imaginary parts of $\phi(x)$, as the basic dynamical variables.
(a) Find the conjugate momenta to $\phi(x)$ and $\phi^{\ast}(x)$ and the canonical commutation relations. Show that the Hamiltonian is
\[\mathcal{H} = \int d^{3}x (\pi^{\ast}\pi+\nabla\phi^{\ast}\cdot\nabla\phi+m^{2}\phi^{\ast}\phi).\]Compute the Heisenberg equation of motion for $\phi(x)$ and show that it is indeed the Klein-Gordon equation.
(b) Diagonalize $\mathcal{H}$ by introducing creation and annihilation operators. Show that the theory contains two sets of particles of mass $m$.
(c) Rewrite the conserved charge
\[Q = \int d^{3}x \frac{i}{2} (\phi^{\ast}\pi^{\ast}-\pi\phi)\]in terms of creation and annihilation operators, and evaluate the charge of the particles of each type.
(d) Consider the case of two complex Klein-Gordon fields with the same mass. Label the fields as $\phi_{a}(x)$, where $a=1,2$. Show that there are now four conserved charges, one given by the generalization of part (c), and the other three given by
\[Q^{i} = \int d^{3}x \frac{i}{2}(\phi^{\ast}_{a}(\sigma^{i})_{ab}\pi^{\ast}_{b}-\pi_{a}(\sigma^{i})_{ab}\phi_{b}),\]where $\sigma^{i}$ are the Pauli sigma matrices. Show that these three charges have the commutation relations of angular momentum $(SU(2))$. Generalize these results to the case of n identical complex scalar fields.
3. Evaluate the function
\[\bra{0} \phi(x)\phi(y) \ket{0} = D(x-y) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_{P}} e^{-ip(x-y)},\]for $(x-y)$ spacelike so that $(x-y)^2 = -r^2$, explicitly in terms of Bessel functions.