# Noether's theorem in field theory

## Energy and momentum in scalar field theory

Let \(\mathcal{F}\) denote the space of fields \(\phi\colon\mathbf{R}^n\to\mathbf{R}\). The phase space for the classical theory of scalar fields is the cotangent bundle \(T^*\mathcal{F}\). Since \(\mathcal{F}\) is just a vector space whose elements are field configurations \(\phi\), its cotangent fibre is also a vector space of functions \(\pi\) (technically, this should probably be a space of distributions, as it's the dual of \(\mathcal{F}\); we will ignore this issue). If \(A\) is a vector on \(T^*\mathcal{F}\) then it has a component \(A_\phi\) in the \(\mathcal{F}\)-directions and a component \(A_\pi\) in the cotangent directions. We can think of \(A_\phi\) and \(A_\pi\) as functions. The symplectic structure \(\Omega\) on \(T^*\mathcal{F}\) is given by \[\Omega(A,B)=\int d^nx(A_\phi(x)B_\pi(x)-A_\pi(x)B_\phi(x)).\] Given a function \(H\colon T^*\mathcal{F}\to\mathbf{R}\), we can write down a corresponding Hamiltonian vector field \(A\); if \(\delta H(B)\) denotes the directional (functional) derivative of \(H\) in the \(B\)-direction then \(A\) is defined by Hamilton's equations \[\Omega(A,B)=-\delta H(B),\] in other words \[\delta H(B_\phi)=\int d^nx A_\pi B_\phi,\quad\delta H(B_\pi)=-\int d^nx A_\phi B_\pi.\]

*momentum*associated to the \(\mathbf{R}\)-action which changes functions (fields) by translating their argument in the \(e_i\)-direction.

## Internal symmetries

Consider a free complex scalar field \(\psi=\frac{1}{\sqrt{2}}(\phi_1+i\phi_2)\). The Hamiltonian which describes the time evolution of this system is \[H=\int d^nx\left(|\pi|^2+\nabla\psi^*\cdot\nabla\psi+m^2|\psi|^2\right).\] Writing this out in terms of \(\phi_1\) and \(\phi_2\) we get the Hamiltonan for two free Klein-Gordon fields with mass \(m\): \[H=\sum_{k=1}^2\frac{1}{2}\int d^nx\left(\pi_k^2+|\nabla\phi_k|^2+m^2\phi_k^2\right).\] Here, \(\pi_1\) and \(\pi_2\) are the conjugate momenta to \(\phi_1\) and \(\phi_2\). One slightly confusing aspect of this is that \(\pi=\frac{1}{\sqrt{2}}(\pi_1-i\pi_2)\).

Consider the circle action on the space of fields where \(e^{i\alpha}\in U(1)\) acts on \(\psi\) to produce \(e^{i\alpha}\psi\); equivalently, \(e^{i\alpha}\) acts on \((\phi_1,\phi_2)\) to produce \((\phi_1\cos\alpha-\phi_2\sin\alpha, \phi_1\sin\alpha+\phi_2\cos\alpha)\). This action induces an action by symplectomorphisms on \(T^*\mathcal{F}\): \[\left(\begin{array}{c}\phi_1 \\ \phi_2 \\ \pi_1 \\\pi_2\end{array}\right) \stackrel{e^{i\alpha}}{\to} \left(\begin{array}{c}\phi_1\cos\alpha-\phi_2\sin\alpha \\ \pi_1\sin\alpha+\pi_2\cos\alpha \\ \pi_1\cos\alpha-\pi_2\sin\alpha \\ \pi_1\sin\alpha+\pi_2\cos\alpha\end{array}\right).\] The infinitesimal action (i.e. the vector field on \(T^*\mathcal{F}\) generating this action) is \[V=(-\phi_2,\phi_1,-\pi_2,\pi_1).\]

## Noether currents

Up till now, everything we have said is just like in the case of
finite-dimensional Hamiltonian systems: when we have a symplectic
phase space, Hamiltonian functions give rise to Hamiltonian symmetries
of the phase space and vice versa. Now we come to a phenomenon which
only really makes sense in the field theory setting: the idea of
*Noether currents*.

We start with a Hamiltonian \(H=\int d^nx\mathcal{H}\) which generates
the time-evolution of our system. Suppose we have another function
\(G=\int d^nx\mathcal{G}\) which generates a Hamiltonian flow that
commutes with \(H\), in other words \(\{H,G\}=0\). An example would be
the free complex scalar field \(\psi\) with its free Hamiltonian and
the circle action rotating the phase of \(\psi\). The claim is that,
not only does the time derivative of \(G\) vanish along the flow of
\(H\), but the time derivative of \(\mathcal{G}\) is equal to the
divergence of a vector field \(J\) on space. This is a stronger
assertion: if you integrate \(\dot{\mathcal{G}}=\nabla\cdot J\) over
space then you get \(\dot{G}=0\) because the integral of a divergence
is zero. Moreover, you can figure out what the vector field \(J\) is
using the time-honoured trick of integrating by parts. The 4-vector
\((\mathcal{G},J)\) is then called the *Noether current*
associated to the symmetry. It is not uniquely determined, since we
can always add a gradient vector field to \(J\) without changing the
property that \(\nabla\cdot J=\dot{\mathcal{G}}\). Nonetheless,
there's usually a sensible-looking choice.

The key point which allows us to find these currents is the well-known fact in symplectic geometry that if you have Hamiltonian functions \(A\) and \(B\) (with Hamiltonian vector fields \(V\) and \(W\)), you let \(\phi_t\) be the Hamiltonian flow associated to \(A\), and you consider the function \(B(\phi_t(x))\) then \(\dot{B}=\{A,B\}=\Omega(V,W)\). Armed with this fact, we will study the simplest example.

This trick only makes sense in field theory, where our Hamiltonian functions have the form \(\int d^nx\mathcal{G}\): otherwise we cannot insert a cheeky factor of \(\alpha\). It also relies heavily on the equations of motion for \(G_\alpha\), i.e. \(\dot{G}_\alpha=\{H,G_\alpha\}\).