Noether's Theorem: Symmetry and Conservation Laws

1. From Symmetry to Conservation: The Quantum Statement

Emmy Noether's original 1915 theorem was formulated in the language of Lagrangian mechanics. It states that for every continuous symmetry of the action, there exists a corresponding conserved quantity. While the Lagrangian formulation is powerful, especially in field theory, the Hamiltonian formulation—which is more natural to quantum mechanics—provides a more direct path to the same conclusion.

Let's begin with our definition of a continuous symmetry from the last lesson. A one-parameter continuous symmetry is represented by a family of unitary operators $U(\alpha )$, where $\alpha$ is a continuous parameter (e.g., a translation distance or a rotation angle). The symmetry condition is that the Hamiltonian $H$ is invariant under these transformations:

for all values of $\alpha$.

Now, consider an infinitesimal transformation, where $\alpha$ is a very small value, let's call it $\delta \alpha$. As we've seen, any such unitary operator can be expressed in terms of a Hermitian operator $G$, known as the generator of the transformation:

If the commutation relation $[H,U(\alpha )]=0$ holds for all $\alpha$, it must hold for this infinitesimal transformation as well:

Since $H$ commutes with the identity operator and $\delta \alpha$ is a non-zero scalar, this simplifies to:

This is a pivotal result: a continuous symmetry of the Hamiltonian implies that the Hamiltonian commutes with the generator of that symmetry.

The following reading provides a concise statement of this derivation and frames it as the quantum mechanical version of Noether's Theorem.

2. The Heisenberg Picture and the Dynamics of Observables

The notes you just read made a jump to dG/dt. To make this step completely rigorous and to understand what dG/dt truly means for an operator, we must introduce the concept of time evolution for observables. The most natural framework for this is the Heisenberg picture of quantum mechanics.

You are familiar with the Schrödinger picture, where state vectors $|\psi (t)⟩$ evolve in time according to the Schrödinger equation, while operators like position $x$ and momentum $p$ are typically fixed. The Heisenberg picture offers an equivalent but alternative perspective:

• State vectors are fixed in time: $|{\psi }_H⟩=|\psi (t=0)⟩$.
• Operators evolve in time: An operator $A_S$ in the Schrödinger picture becomes a time-dependent operator $A_H(t)$ in the Heisenberg picture.

The relationship between the two is defined such that the physical predictions (i.e., expectation values) are identical in both pictures:

Given that $|\psi (t)⟩=U(t)|\psi (0)⟩$, where $U(t)=e^{-iHt/\hslash }$ is the time-evolution operator, we can deduce the definition of the Heisenberg operator:

To see how these operators evolve, we can differentiate $A_H(t)$ with respect to time. This leads to the Heisenberg equation of motion.

The following video lecture derives the Heisenberg picture from first principles and then arrives at the crucial Heisenberg equation of motion.

For a Schrödinger operator $A_S$ that does not have explicit time dependence (like the generators we are considering), the Heisenberg equation of motion is:

3. Deriving the Conservation Law

We now have all the pieces to complete our derivation.

1. A continuous symmetry implies its generator $G$ commutes with the Hamiltonian $H$: $[G,H]=0$.
2. The time evolution of the corresponding Heisenberg operator $G_H(t)$ is governed by the Heisenberg equation of motion.

If the Hamiltonian itself is time-independent, then $H_H(t)=U^{†}HU=H$. The commutator in the Heisenberg picture is then:

Since $H$ commutes with $U(t)$ and $U(t{)}^{†}$, we can write:

Since our symmetry condition gives us $[G,H]=0$, we find that:

Substituting this into the Heisenberg equation of motion:

This is the formal statement of a conservation law in quantum mechanics. It says that the operator $G_H$ is constant in time. Consequently, its expectation value for any (time-independent Heisenberg) state is also constant:

The following short clip from the same MIT lecture summarizes this specific point beautifully.

4. Foundational Examples

This theorem provides the theoretical underpinning for the most fundamental conservation laws in physics.

• Conservation of Momentum: If a system's Hamiltonian is invariant under spatial translations ($x\to x+a$), it must not depend on the position operator $x$. The generator of spatial translations is the momentum operator, $p$. Therefore, translational invariance implies $[H,p]=0$, and momentum is conserved.
• Conservation of Angular Momentum: If a system's Hamiltonian is invariant under rotations, it is spherically symmetric. The generators of rotations are the components of the angular momentum operator, $J_x,J_y,J_z$. Therefore, rotational invariance implies $[H,J_i]=0$, and angular momentum is conserved.
• Conservation of Energy: If a system's Hamiltonian has no explicit time dependence, it is invariant under time translations ($t\to t+\tau$). The generator of time evolution is the Hamiltonian $H$ itself. The condition $[H,H]=0$ is trivially true, which implies that energy is conserved for any closed system.

The connection between momentum and the generator of translations is definitional, as explained in this short video.

5. An Algebraic Perspective on Noether's Theorem

Given your background in mathematics and interest in formal structures, it's worth taking a moment to appreciate the deep algebraic structure underlying Noether's theorem. The proof you've seen relies on the Heisenberg equation of motion, which in turn depends on the commutator.

Consider two observables, $A$ and $B$, which are generators of continuous symmetries.

• "A generates a symmetry of B" means that B is unchanged by the transformation generated by A. This is true if and only if $[A,B]=0$.
• "B generates a symmetry of A" means that A is unchanged by the transformation generated by B. This is true if and only if $[B,A]=0$.

Because the commutator is antisymmetric, $[A,B]=-[B,A]$, the condition $[A,B]=0$ is perfectly equivalent to $[B,A]=0$. Therefore, the statement "A is conserved under the symmetry generated by B" is equivalent to "B is conserved under the symmetry generated by A". This beautiful reciprocity is the essence of Noether's theorem in the Hamiltonian/quantum framework.

For a deeper exploration of this algebraic viewpoint, you may find the following paper insightful. It's an optional read, but it elegantly reframes the theorem in a more abstract light that connects classical and quantum mechanics.

Conclusion

In this lesson, we have moved from the static consequences of symmetry to its dynamic implications. We have rigorously demonstrated how the existence of a continuous symmetry in a quantum system logically necessitates the existence of a corresponding conserved quantity.

Key Takeaways:

• A continuous symmetry of a Hamiltonian $H$, represented by unitary operators $U(\alpha )$, implies that $H$ commutes with the symmetry's Hermitian generator $G$: $[H,G]=0$.
• In the Heisenberg picture, operators evolve in time according to the Heisenberg equation of motion: $i\hslash \frac{d{A}_H}{dt}=[A_H,H_H]$.
• Combining these two facts shows that if $[G,H]=0$, then the Heisenberg operator $G_H$ is constant in time ($\frac{d{G}_H}{dt}=0$), meaning $G$ is a conserved quantity.
• This provides the quantum mechanical basis for fundamental laws like the conservation of momentum, angular momentum, and energy, which arise from symmetries of space and time.
• At its core, the theorem's reciprocity in the Hamiltonian formalism stems from the antisymmetry of the commutator.

You now understand how symmetries act as powerful constraints not only on what states can exist, but also on how those states can evolve. In our next lesson, we will see another dynamical consequence of symmetry: the existence of selection rules, which dictate which transitions between quantum states are allowed or forbidden.