Quantum field theory 1, lecture 04

Noether’s theorems

There is a fundamental relation between symmetries and conservation laws, first formulated by Emmy Noether (1882 – 1935). We discuss this here in the context of classical field theory.

Global symmetries

Consider again the action for a classical field theory, which we write in the form \[S[\Phi] = \int_\Omega d^4 x \mathscr{L}(\Phi(x), \partial_\mu \Phi(x)).\] Here \(\Omega\) denotes some region in spacetime, for example bounded by an initial time \(t_\text{i}\) and final time \(t_\text{f}\). We now study continuous transformations of the fields parametrized by some real number \(\xi\) such that \(\xi=0\) is the identity transformation. It is enough to study infinitesimal transformations out of which we can build also finite transformations. We write them as \[\Phi(x) \to \Phi(x) + \delta_\xi \Phi(x).\] We assume that the action is invariant up to a total derivative or boundary term, \[S[\Phi] \to S[\Phi] + \delta S[\Phi] = S[\Phi] + \int_\Omega d^4 x \left\{ \partial_\mu \Lambda^\mu(x) \right\} = S[\Phi] + \int_{\partial \Omega} d\Sigma_\mu \Lambda^\mu(x).\] In such a situation we speak of a continuous global symmetry of the action.

Local conservation law

Noether’s first theorem states that for every continuous global symmetry of the action there exists a conserved current. So see this we consider the change in the action, written in the form \[\begin{split} \delta S[\Phi] = & \int_\Omega d^4 x \left\{ \frac{\partial \mathscr{L}}{\partial \Phi(x)} \delta_\xi \Phi(x) + \frac{\partial \mathscr{L}}{\partial (\partial_\mu \Phi(x))} \partial_\mu \delta _\xi \Phi(x) \right\} \\ = & \int_\Omega d^4 x \left\{ \left[ \frac{\partial \mathscr{L}}{\partial \Phi(x)} - \partial_\mu \frac{\partial \mathscr{L}}{\partial (\partial_\mu \Phi(x))}\right] \delta _\xi \Phi(x) + \partial_\mu \left[ \frac{\partial \mathscr{L}}{\partial (\partial_\mu \Phi(x))} \delta_\xi \Phi(x) \right]\right\} = \int_\Omega d^4 x \partial_\mu \left\{ \Lambda^\mu(x) \right\}. \end{split}\] We now use the equations of motion, \[\frac{\delta S}{\delta \Phi(x)} = \frac{\partial \mathscr{L}}{\partial \Phi(x)} - \partial_\mu \frac{\partial \mathscr{L}}{\partial (\partial_\mu \Phi(x))} = 0,\] and obtain the local conservation law \[\partial_\mu \left[ \frac{\partial \mathscr{L}}{\partial (\partial_\mu \Phi(x))} \delta_\xi \Phi(x) - \Lambda^\mu(x) \right] = 0.\]

Global U\((1)\) symmetry

As a first example we consider a global U\((1)\) transformation for the complex scalar field, \[\phi(x) \to e^{i \alpha} \phi(x), \quad\quad\quad \phi^*(x) \to e^{-i \alpha} \phi^*(x).\] In infinitesimal form this reads \[\delta_\alpha \phi(x) = i \alpha \phi(x), \quad\quad\quad \delta_\alpha \phi^*(x) = - i \alpha \phi^*(x).\] The Gross-Pitaevskii action \(\eqref{eq:actionGrossPitaevskii}\) is invariant under this transformation with \(\Lambda^\mu(x) = 0\). We find here \[\frac{\partial \mathscr{L}}{\partial (\partial_0 \phi)} = \frac{i \hbar}{2} \phi^*, \quad\quad\quad \frac{\partial \mathscr{L}}{\partial (\partial_0 \phi^*)} = - \frac{i \hbar}{2} \phi,\] and \[\frac{\partial \mathscr{L}}{\partial (\partial_j \phi)} = - \frac{\hbar^2}{m} \partial_j \phi^*, \quad\quad\quad \frac{\partial \mathscr{L}}{\partial (\partial_j \phi^*)} = - \frac{\hbar^2}{m} \partial_j \phi.\] Noethers theorem implies the conservation law \[\partial_0 \left[ - \alpha \hbar \phi^*(x) \phi(x) \right] + \partial_j \left[ i \alpha \frac{\hbar^2}{m} \phi^* \partial_j \phi - i \alpha\frac{\hbar^2}{m} \phi \partial_j \phi^* \right] = 0,\] or equivalently \(\partial_\mu N^\mu(x) = 0\) with the current composed out of the particle number density \[N^0(t, \mathbf{x}) = \phi^*(t, \mathbf{x}) \phi(t, \mathbf{x})\] and the particle number current \[\mathbf{N}(t, \mathbf{x}) = -\frac{i \hbar}{m} \left[ \phi^*(t, \mathbf{x}) \boldsymbol{\nabla} \phi(t, \mathbf{x}) - \phi(t, \mathbf{x}) \boldsymbol{\nabla} \phi^*(t, \mathbf{x}) \right].\] These expressions agree formally with the probability density and probability current in single-particle quantum mechanics, but have gained a different significance in the context of the classical Gross-Pitaevskii field theory.

Translations

Another interesting symmetry transformation is given by translations in spacetime, \(x^\mu\to x^\mu + a^\mu\). This should actually be seen as a set of four linearly independent transformations for the choices \(\mu=0,1,2,3\), which we discuss here together. Fields change according to \[\Phi(x) \to \Phi(x-a),\] such that for example a maximum at a position \(x_0^\mu\) before the transformation is moved to \(x_0^\mu + a^\mu\) after the transformation. When the transformation is infinitesimal we can write \[\Phi(x) \to \Phi(x) + \delta_a\Phi(x) = \Phi(x) - a^\mu \partial_\mu \Phi(x).\] The Lagrangian density transforms itself like a scalar field, \(\mathscr{L}(x) \to \mathscr{L}(x)- a^\mu \partial_\mu \mathscr{L}(x)\).

Energy-momentum conservation law

Accordingly we find a symmetry of the action with \(\Lambda^\mu(x) = - a^\mu \mathscr{L}(x)\), and the local conservation law is according to Noethers first theorem \[\partial_\mu\left[ \frac{\partial \mathscr{L}}{\partial \partial_\mu \Phi} a^\nu \partial_\nu \Phi - a^\mu \mathscr{L}\right] = 0.\] Because \(a^\mu\) is arbitrary we obtain the conservation law for the canonical energy-momentum tensor, \[\partial_\mu \mathscr{T}^{\mu\nu}(x) = 0,\] with \[\mathscr{T}^{\mu\nu}(x) = -\frac{\partial \mathscr{L}}{\partial \partial_\mu \Phi(x)} g^{\nu\rho}\partial_\rho \Phi(x) + g^{\mu\nu} \mathscr{L}(x).\]

Energy-momentum tensor for real relativistic scalar field

As an example we consider the real relativistic scalar field theory with action in eq. \(\eqref{eq:actionRealScalar}\). The energy-momentum tensor is given by \[\mathscr{T}^{\mu\nu}(x) = g^{\mu\rho} g^{\nu\sigma}\partial_\rho \phi(x) \partial_\sigma \phi(x) + g^{\mu\nu} \left[ - \frac{1}{2} g^{\rho\sigma} \partial_\rho \phi(x) \partial_\sigma \phi(x) - V(\phi(x)) \right].\] The zero-zero component is again the Hamiltonian density already derived by different means in equation \(\eqref{eq:01HamiltonianFieldTheory}\).