Noether's Theorem in Classical Field Theory

From Particles to Fields: A New Language

In classical mechanics, a system is described by a finite set of generalized coordinates $q_i(t)$ that are functions of time. In field theory, we promote these coordinates to fields, ${\varphi }_a(x^{\mu })$, which are functions of both space and time, $x^{\mu }=(t,\vec{x})$. The field itself becomes the dynamical object, with a degree of freedom at every point in spacetime.

This shift requires us to adapt our Lagrangian formalism. Instead of a Lagrangian $L$, we work with a Lagrangian density, $\mathcal{L}$, whose integral over space gives the total Lagrangian:

The action, $S$, is the integral of the Lagrangian density over all of spacetime:

The principle of least action, $\delta S=0$, then yields the equations of motion for the fields, known as the Euler-Lagrange equations:

Here, we're using Einstein summation convention, where repeated indices like $\mu$ are summed over (from 0 to 3). This compact, covariant notation is standard in field theory.

For a deeper review of this transition from classical mechanics to field theory, the following reading from David Tong's renowned QFT lecture notes is an excellent resource.

Noether's Theorem for Fields: Deriving the Conserved Current

With the field-theoretic language in place, we are ready to prove the main result of this lesson. As in the case of particle mechanics, we begin by defining a symmetry. A continuous field transformation ${\varphi }_a(x)\to {\varphi }_a(x)+\delta {\varphi }_a(x)$ is a symmetry if the action is invariant. This holds if the Lagrangian density changes by at most a total four-divergence:

for some set of functions $F^{\mu }$. If the Lagrangian density is strictly invariant, then $F^{\mu }=0$.

Now, let's consider the variation of the Lagrangian density under an arbitrary infinitesimal transformation $\delta {\varphi }_a$. Using the chain rule:

We can commute the variation and the derivative, so $\delta ({\partial }_{\mu }{\varphi }_a)={\partial }_{\mu }(\delta {\varphi }_a)$. Using a product rule "in reverse", we can write:

Grouping the terms gives:

Now, we impose the physics. If the fields obey the equations of motion, we are "on-shell". In this case, the term in the square brackets is exactly the Euler-Lagrange equation, which equals zero. Thus, on-shell, the variation of the Lagrangian density simplifies to:

We now have two expressions for $\delta \mathcal{L}$ for a symmetry transformation. Equating them gives:

Rearranging this equation, we find that the divergence of a specific quantity is zero:

This is the heart of Noether's theorem for fields. We have found a conserved four-vector, the Noether current:

which satisfies the conservation equation ${\partial }_{\mu }{j}^{\mu }=0$. For a one-parameter symmetry, we can write $\delta {\varphi }_a=\alpha X_a(\varphi )$, where $\alpha$ is the infinitesimal parameter. Factoring out this constant parameter leaves us with the definition of the current itself.

The following video provides a very clear, step-by-step walkthrough of this derivation.

The Physical Meaning: Local Conservation

The equation ${\partial }_{\mu }{j}^{\mu }=0$ is far more than a mathematical curiosity. It is a continuity equation. If we write out the components of the four-vector $j^{\mu }=(j^0,\vec{j})$, the equation becomes:

This is a statement of local conservation.

• $j^0$ is the density of a conserved quantity (e.g., charge density).
• $\vec{j}$ is the current density or flux of that quantity.

The equation states that the rate of change of the total amount of "stuff" in a volume is exactly balanced by the amount of "stuff" flowing across its boundary. A quantity cannot simply disappear from one point and reappear at another; it must flow continuously from one place to the next.

From this conserved current, we can recover a globally conserved charge, $Q$, which connects back to our previous lesson. We define the charge as the integral of the charge density over all of space:

Let's see if this charge is conserved in time:

Using the divergence theorem, we can convert the volume integral of the divergence into a surface integral of the flux over the boundary at infinity:

For any physically realistic field configuration, the fields (and thus the current $\vec{j}$) must vanish at spatial infinity. Therefore, the surface integral is zero, and we find that $\frac{dQ}{dt}=0$. The total charge $Q$ is constant in time.

The following video and text explain this crucial connection between the conserved current and the conserved charge.

Example: A U(1) Internal Symmetry

Let's apply this formalism to a concrete example that is central to your QFT studies: a complex scalar field. This is an "internal" symmetry, which transforms the fields into each other at the same spacetime point, unlike the spacetime symmetries we studied last time.

Consider the Lagrangian for a free, complex scalar field $\psi$:

This Lagrangian is invariant under a global U(1) phase transformation:

where $\alpha$ is a real, constant parameter. The Lagrangian is strictly invariant ($\delta \mathcal{L}=0$), so the $F^{\mu }$ term in our formula is zero. For an infinitesimal transformation, we have:

We treat $\psi$ and ${\psi }^{\ast }$ as independent fields. The Noether current is the sum of contributions from both:

Calculating the derivatives of the Lagrangian:

• $\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\psi )}={\partial }^{\mu }{\psi }^{\ast }$
• $\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\psi }^{\ast })}={\partial }^{\mu }\psi$

Plugging these in, we get:

Factoring out the constant $i\alpha$, we identify the conserved current:

Self-correction: The standard convention often uses a slightly different form. Let's re-derive carefully. $\mathcal{L}={\eta }^{\nu \mu }({\partial }_{\nu }{\psi }^{\ast })({\partial }_{\mu }\psi )-...$. So $\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\psi )}={\eta }^{\nu \mu }({\partial }_{\nu }{\psi }^{\ast })={\partial }^{\mu }{\psi }^{\ast }$. The derivation is correct. My result differs from some texts by a sign or overall factor, which is a matter of convention. Let's write it as $j^{\mu }\propto i({\psi }^{\ast }{\partial }^{\mu }\psi -({\partial }^{\mu }{\psi }^{\ast })\psi )$ to match common usage.

Let's adopt the common convention: $j^{\mu }=i(({\partial }^{\mu }{\psi }^{\ast })\psi -{\psi }^{\ast }({\partial }^{\mu }\psi ))$. This is the conserved Noether current associated with the global U(1) symmetry. The conserved charge $Q=\int d^{3}x{j}^0$ is interpreted in QFT as the net particle number (number of particles minus number of antiparticles). When this symmetry is promoted to a local (gauge) symmetry, this current becomes the source for the electromagnetic field, and $Q$ becomes the electric charge.

Conclusion

In this lesson, we have successfully generalized Noether's theorem to the framework of classical field theory. This was more than a simple mathematical extension; it revealed a deeper physical principle.

Key Takeaways:

• In field theory, continuous symmetries of the Lagrangian density lead to the existence of a conserved Noether current, $j^{\mu }$.
• This current satisfies a continuity equation, ${\partial }_{\mu }{j}^{\mu }=0$, which expresses the principle of local conservation. The change in a quantity's density in any region is precisely accounted for by its flow across the boundary of that region.
• The conserved current gives rise to a globally conserved charge, $Q=\int d^{3}x{j}^0$, connecting back to the conserved quantities of classical mechanics.
• The general formula for the Noether current is $j^{\mu }=\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\delta {\varphi }_a-F^{\mu }$.

We have bridged the gap between the conservation laws of particles and the continuity equations of fields. In our next lesson, we will immediately apply this powerful new tool. We will investigate the consequences of translational invariance in spacetime and derive one of the most important objects in all of physics: the energy-momentum tensor.