Noether's Theorem: Symmetries and Conservation Laws

1. The Essence of Noether's Theorem

In 1915, while tackling puzzles in Einstein's newly formulated theory of general relativity, the mathematician Emmy Noether discovered a remarkable and universal connection. She proved that for every continuous symmetry of the action describing a physical system, there exists a corresponding conserved quantity.

This is a statement of incredible power. It tells us that the conservation of energy is a direct consequence of the fact that the laws of physics don't change over time (time-translation symmetry). The conservation of linear momentum comes from the laws being the same everywhere in space (spatial-translation symmetry).

The derivation hinges on a crucial insight we touched on last time: for the equations of motion to be invariant, the Lagrangian doesn't need to be strictly unchanged. It is sufficient for the Lagrangian to change by at most the total time derivative of some function, $G$. Let's see how this leads to a conservation law.

The following video provides a very clear, self-contained derivation of Noether's theorem in the context of classical mechanics.

The video's derivation leads to a general expression for the conserved quantity, or Noether charge, $Q$. For an infinitesimal transformation of the coordinates $\delta q_i$ that causes the Lagrangian to change by $\delta L=ϵ\frac{dG}{dt}$, the conserved charge is:

Here, $ϵ$ is the small parameter of the continuous transformation. Let's break this down:

• $\frac{\partial L}{\partial {\dot{q}}_i}$ is the generalized momentum $p_i$ conjugate to the coordinate $q_i$.
• $\frac{\delta q_i}{ϵ}$ represents how the coordinate $q_i$ changes under the symmetry transformation.
• $G$ is the function whose total time derivative gives the change in the Lagrangian. In many simple cases, the Lagrangian is strictly invariant, making $G=0$.

2. From Abstract Theorem to Physical Laws

The formula above is an engine for generating conservation laws. By plugging in the specific transformations for familiar symmetries, we can recover the conservation laws of classical mechanics. The same video provides excellent worked examples.

Let's summarize these foundational results in a table. This pattern—a spacetime symmetry leading to a mechanical conservation law—is fundamental.

This last case, time translation, is particularly insightful. The change in the Lagrangian under an infinitesimal time shift $t\to t+ϵ$ is $\delta L=ϵ\frac{dL}{dt}$. This means $G=L$, leading directly to the Hamiltonian as the conserved quantity associated with time-invariance.

3. Field Theory and Conserved Currents

As you are studying Quantum Field Theory, you know that the framework shifts from discrete coordinates $q_i(t)$ to fields $\varphi (x^{\mu })$. In this context, Noether's theorem is even more powerful. A continuous symmetry implies the existence of a conserved current, a four-vector $j^{\mu }(x)$ that satisfies the continuity equation:

This is a statement of local conservation. It says that the rate of change of the charge density $j^0$ within any given volume is precisely balanced by the flux of the current $\vec{j}$ across the boundary of that volume. Integrating $j^0$ over all space gives the total conserved charge $Q=\int d^{3}x{j}^0(x)$, which is constant in time.

The following video provides a clear derivation of the Noether current in the language of field theory, helpfully providing a "dictionary" to translate from the classical concepts we just discussed.

The derivation shows that if a transformation ${\varphi }_a\to {\varphi }_a+ϵ\Delta {\varphi }_a$ is a symmetry under which the Lagrangian density changes by $\delta \mathcal{L}=ϵ{\partial }_{\mu }{\mathcal{K}}^{\mu }$, then the conserved Noether current is:

This is the field-theoretic analogue of the classical formula. The most common case is when the Lagrangian density is strictly invariant, so ${\mathcal{K}}^{\mu }=0$.

4. Key Applications in Field Theory

Let's use this powerful result to derive two cornerstone results in field theory. We will use a set of QFT lecture notes to guide us.

A. U(1) Symmetry and Conserved Charge

First, consider a complex scalar field $\varphi$ with the Lagrangian $\mathcal{L}=|{\partial }_{\mu }\varphi {|}^2-m^2|\varphi {|}^2$. As we saw in the last lesson, this has a continuous global U(1) symmetry: $\varphi \to e^{i\alpha }\varphi$. For an infinitesimal transformation, $\delta \varphi =i\alpha \varphi$ and $\delta {\varphi }^{\ast }=-i\alpha {\varphi }^{\ast }$. The Lagrangian is strictly invariant, so ${\mathcal{K}}^{\mu }=0$. We can now construct the current.

The resulting current, $j^{\mu }=i(\varphi {\partial }^{\mu }{\varphi }^{\ast }-{\varphi }^{\ast }{\partial }^{\mu }\varphi )$, is proportional to the 4-current you know from quantum mechanics. The associated conserved charge $Q=\int d^{3}x{j}^0$ can be interpreted as particle number or electric charge.

B. Spacetime Translations and the Energy-Momentum Tensor

Next, let's consider the symmetry of spacetime translation, $x^{\mu }\to x^{\mu }-{ϵ}^{\nu }$, where ${ϵ}^{\nu }$ is a set of four constant infinitesimal parameters. Under this transformation, a field transforms as $\varphi (x)\to \varphi (x+ϵ)\approx \varphi (x)+{ϵ}^{\nu }{\partial }_{\nu }\varphi (x)$, so $\Delta \varphi ={\partial }_{\nu }\varphi$. The Lagrangian density transforms as $\mathcal{L}(x)\to \mathcal{L}(x+ϵ)\approx \mathcal{L}(x)+{ϵ}^{\nu }{\partial }_{\nu }\mathcal{L}$. We can write this change as a total derivative: $\delta \mathcal{L}={ϵ}^{\nu }{\partial }_{\mu }({\delta }_{\nu }^{\mu }\mathcal{L})$, which means ${\mathcal{K}}^{\mu }={\delta }_{\nu }^{\mu }\mathcal{L}$. Plugging these into the master formula for the current gives a set of four conserved currents, one for each value of $\nu$, which we group into a single object: the energy-momentum tensor $T^{\mu \nu }$.

The derived energy-momentum tensor is:

The components of this tensor have direct physical meaning.

• $T^{00}$ is the energy density, which is the Hamiltonian density $\mathcal{H}$. Its integral is the total energy.
• $T^{0i}$ is the momentum density in the $i$-th direction. Its integral is the total momentum $P^i$.
• $T^{i0}$ is the energy flux in the $i$-th direction.
• $T^{ij}$ is the flux of $j$-momentum in the $i$-th direction, related to pressure and shear stress.

The conservation law ${\partial }_{\mu }{T}^{\mu \nu }=0$ expresses the local conservation of energy ($\nu =0$) and momentum ($\nu =1,2,3$) in a fully relativistic framework.

Conclusion

Today, we have seen how Noether's theorem provides a constructive and profound link between the symmetries of a system and its conservation laws. This is a cornerstone of the theoretical physicist's toolkit, allowing us to deduce fundamental physical principles directly from the mathematical structure of a theory's Lagrangian.

Key Takeaways:

• Noether's theorem states that every continuous symmetry of the action implies a corresponding conserved quantity.
• The invariance of the action is a less strict condition than the invariance of the Lagrangian; the Lagrangian can change by a total derivative.
• In classical mechanics, symmetries lead to conserved charges like energy and momentum, which you can derive using the formula $Q=\sum p_i(\delta q_i/ϵ)-G$.
• In field theory, symmetries lead to locally conserved currents $j^{\mu }$ satisfying ${\partial }_{\mu }{j}^{\mu }=0$. The spatial integral of the $j^0$ component gives the globally conserved charge.
• We applied this framework to derive the conserved U(1) charge for a complex scalar field and the conserved energy-momentum tensor $T^{\mu \nu }$ from spacetime translation invariance.

We have focused on the consequences of continuous symmetries. In our next lesson, we will broaden our scope to see how all symmetries, both continuous and discrete, can be used to derive powerful selection rules that determine which physical processes and transitions are allowed or forbidden.