The purpose of this post is to solve some problems touching on various aspects of classical field theory. Most are just taken from this problem sheet, with solutions here, but there are also a few extra problems meant to clarify aspects of classical field theory.
Problem #\(1\): A string of length \(L\), mass per unit length \(\mu\), under uniform tension \(T\) is fixed at each end. The Lagrangian \(\mathcal L\) governing the time evolution of the transverse displacement \(\psi(x,t)\) is:
\[\mathcal L=\int_{0}^L\left(\frac{\mu}{2}\dot{\psi}^2-\frac{T}{2}\psi’^2\right)dx\]
where \(x\in[0,L]\) identifies position along the string from one endpoint. By expressing the transverse displacement as a Fourier sine series:
\[\psi(x,t)=\sqrt{\frac{2}{L}}\sum_{n=1}^{\infty}q_n(t)\sin\frac{n\pi x}{L}\]
Show that the Lagrangian \(\mathcal L\) becomes:
\[\mathcal L=\sum_{n=1}^{\infty}\left(\frac{\mu}{2}\dot{q}^2_n-\frac{T}{2}\left(\frac{n\pi}{L}\right)^2q_n^2\right)\]
Derive the equations of motion. Hence, show that the string is equivalent to an infinite set of decoupled harmonic oscillators with frequencies:
\[\omega_n=\sqrt{\frac{T}{\mu}}\frac{n\pi}{L}\]
Solution #\(1\):
Problem #\(2\): Show that the solution space of the Klein-Gordon equation is closed under Lorentz transformations.
Solution #\(2\):
Problem #\(3\): The motion of a complex scalar field \(\psi(X)\) is governed by the Lagrangian density:
\[\mathcal L=\partial_{\mu}\psi^*\partial^{\mu}\psi-m^2\psi^*\psi-\frac{\lambda}{2}(\psi^*\psi)^2\]
Write down the Euler-Lagrange field equations for this system. Verify that the Lagrangian density \(\mathcal L\) is invariant under the infinitesimal transformation:
\[\delta\psi=i\alpha\psi\]
\[\delta\psi^*=-i\alpha\psi^*\]
Derive the Noether current \(j^{\mu}\) associated with this transformation and verify explicitly that it is conserved using the field equations satisfied by \(\psi\).
Solution #\(3\):
Problem #\(4\): Verify that the Lagrangian density:
\[\mathcal L=\frac{1}{2}\partial_{\mu}\phi_{\alpha}\partial^{\mu}\phi_{\alpha}-\frac{1}{2}m^2\phi_{\alpha}\phi_{\alpha}\]
for a triplet of real fields \(\phi_{\alpha}(X)\), \(\alpha=1,2,3\), is invariant under the infinitesimal \(SO(3)\) rotation by \(\theta\):
\[\phi_{\alpha}\mapsto\phi_{\alpha}+\theta\varepsilon_{\alpha\beta\gamma}n_{\beta}\phi_{\gamma}\]
where \(n_{\beta}\) is a unit vector. Compute the Noether current \(j^{\mu}\) associated to this transformation. Deduce that the three quantities:
\[Q_{\alpha}=\int d^3\textbf x\varepsilon_{\alpha\beta\gamma}\dot{\phi}_{\beta}\phi_{\gamma}\]
are all conserved.
Solution #\(4\):
Problem #\(5\): By requiring that Lorentz transformations \(\Lambda^{\mu}_{\space\space\nu}\) should preserve the Minkowski norm of \(4\)-vectors \(\eta_{\mu\nu}X’^{\mu}X’^{\nu}=\eta_{\mu\nu}X^{\mu}X^{\nu}\), show that this implies:
\[\eta_{\mu\nu}=\eta_{\sigma\tau}\Lambda^{\sigma}_{\space\space\mu}\Lambda^{\tau}_{\space\space\nu}\]
Show that an infinitesimal transformation of the form \(\Lambda^{\mu}_{\space\space\nu}=\delta^{\mu}_{\space\space\nu}+\omega^{\mu}_{\space\space\nu}\) is specifically a Lorentz transformation iff \(\omega_{\mu\nu}=-\omega_{\nu\mu}\) is antisymmetric.
Write down the matrix for \(\omega^{\mu}_{\space\space\nu}\) corresponding to an infinitesimal rotation by angle \(\theta\) around the \(x^3\)-axis. Do the same for an infinitesimal Lorentz boost along the \(x^1\)-axis by velocity \(v\).
Solution #\(5\):
Problem #\(6\): Consider a general infinitesimal Lorentz transformation \(X^{\mu}\mapsto X’^{\mu}=X^{\mu}+\omega^{\mu}_{\space\space\nu}X^{\nu}\) acting at the level of the \(4\)-vector \(X\). How does this perturbation manifest at the level of a scalar field \(\phi=\phi(X)\)? What about at the level of the Lagrangian density \(\mathcal L=\mathcal L(\phi)\)? What about at the level of the action \(S=S(\mathcal L)\)? In particular, show that the perturbation \(\delta\mathcal L\) at the level of \(\mathcal L\) is a total spacetime derivative, and hence describe the Noetherian implication of this.
Solution #\(6\):
Problem #\(7\): Maxwell’s Lagrangian for the electromagnetic field is:
\[\mathcal L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\]
where \(F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\) and \(A_{\mu}\) is the \(4\)-vector potential. Show that \(\mathcal L\) is invariant under gauge transformations \(A_{\mu}\mapsto A_{\mu}+\partial_{\mu}\Gamma\) where \(\Gamma=\Gamma(X)\) is a scalar field with arbitrary (differentiable) dependence on \(X\). Use Noether’s theorem, and the spacetime translational invariance of the action \(S\) to construct the energy-momentum tensor \(T^{\mu\nu}\) for the electromagnetic field. Show that the resulting object is neither symmetric nor gauge invariant. Consider a new tensor given by:
\[\Theta^{\mu\nu}=T^{\mu\nu}-F^{\rho\mu}\partial_{\rho}A^{\nu}\]
Show that this object also defines \(4\) conserved currents. Moreover, show that it is symmetric, gauge invariant and traceless.
Solution #\(7\):
Problem #\(8\): In Problem #\(3\), a classical field theory involving two complex scalar fields \(\psi(X),\psi^*(X)\) with Lagrangian density:
\[\mathcal L(\psi,\psi^*,\partial_{\mu}\psi,\partial_{\mu}\psi^*)=\partial_{\mu}\psi^*\partial^{\mu}\psi-V(\psi^*\psi)\]
was analyzed (there the potential \(V\) was taken to be analytic in \(\psi^*\psi\) and so Taylor expanded to quadratic order). In particular, the Noether current \(j^{\mu}\) was obtained explicitly for the continuous global, internal \(U(1)\) \(\mathcal L\)-symmetry \(\psi\mapsto e^{i\alpha}\psi\) for a constant \(\alpha\in\textbf R\). By promoting \(\alpha=\alpha(X)\) but still acting infinitesimally across \(X\) in spacetime, recompute the Noether current \(j^{\mu}\), and notice in particular that the Noether current \(j^{\mu}\) doesn’t care about the potential terms \(V(\psi^*\psi)\), only the kinetic terms.
Solution #\(8\):
Problem #9: