Noether’s Theorem

Emmy Nöter theorem is a theorem proved by Emmy Nöter in 1918. It was first defined in the works of scientists of Göttingen school D. Hilbert, F. Klein and Emmy Nöter herself.

This article was sent to us by Muhannad Qasim Muhammad, Bachelor of Physics (Salah al-Din Province, Samarra District, Iraq).

Symmetry…Of What?

In a nutshell , Noether’s Theorem tells us that for each symmetry of a system there exists a conserved quantity.

For example , if the system is symmetric with respect to time translations , energy is conserved ; or, if the system is symmetric with respect to spatial translations , momentum is conserved , etc …..

We’ll prove these later on . But , what is a symmetry? Imagine that we lock a physicist inside a box and demand that he perform an experiment:

Then , we rotate the box as a whole , If the physicist repeats the same experiment with the exact same initial configurations and cannot find any difference in the results , then we say the laws governing the experiment are rotationally symmetric.

Similarly, if no differences are present under spatial or time translations then the laws describing the experiment are spatially or temporally invariant.

“Temporal translation is when the physicist repeats the experiment after waiting for some time“

The physicist describes the experiment using an equation of motion . To compare the results after the translation , he uses the exact initial configuration :

If the equation of motion (EOM) is the same and valid before and after , then the transformation is a symmetry of the EOM. This way , there’d be no way to find differences in the laws of physics . we shall use this as the symmetry condition.

Canonical Transformations

In particular, good transformations q→Q(q,p) and p→P(q,p) that keep Hamilton’s equations

valid are said to be canonical transformations. However, when it comes to Noether’s Theorem, we care about symmetries, whose condition is :

Where because only then would Hamilton’s equations actually be invariant before and after the transformation.

As an example , we have a Hamiltonian . After the constant shift x→X=x+s , it  reads :

Which is the same as H(X),and therefore the shift is indeed a symmetry. While often glossed over, it is important for us to understand the difference between active and passive transformations and that between symmetry and redundancy.

Symmetry vs. Redundancy

There are two  interpretations for x→X=x+s :

1-The system is at rest, but the coordinate system moved by the passive transformation X=x+s :

2-The system moved, but the coordinate system stays the same. The coordinate in the newX system is given by the active transformation X=x+s:

An invariance under passive transformations is a redundancy, while that under active transforma-tions is a symmetry. The difference is that a sym-metry is

 a real feature of a system, while a redun-dancy arises from our mathematical description of the system, such as using curvilinear coordinates.

Noether’s Theorem

Thus, we’ll only consider active transformations. Under the infinitesimal canonical transformation:

Generated by the generator G, the symmetry condition HQ,P=HQ,P can be rewritten as :

We also know that the time-evolution of G is :

And that the Poisson bracket is antisymmetric ,so that G,H=-H,G=0 . This direcly implies :

G={G,H}=0

“we therefore learn that if the generator generates a symmetry 0={H,G}, it automatically describes a conserved quantity . This is Noether’s Theorem.

Conservation of Momentum

The Hamiltonian for a single free object reads :

H=p²/2m

Suppose that the generator is momentum itself :

G(q,p) = p

We can check to see that momentum generates a symmetry as .The generator acts on phase space coordinates via the Poisson bracket:

Since the generator is momentum , these become :

This means that the location coordinate q is shifted by a constant amount , and therefore p generates spatial translations . Recall that 0={H,G} implies that , so we have we learn here that momentum is conserved whenever the system is invariant under spatial translations.

Conservation of Energy

For this example, suppose that the generator of the transformation is the Hamiltonian itself:

G(q,p)=H

We can check that {H,H} is indeed 0 because the Poisson bracket of anything with itself is 0 . This means that H generates a symmetry , but what kind of symmetry ? The transformation generated reads:

This implies (substituting Hamilton’s equations ) :

This means that the change in coordinates generated by H is the same as the change we get if we wait for  units of time, as q and p are rates of changes of the coordinates . in other words , the Hamiltonian generates temporal translations . And , since {H,H} is 0 , =0 : The Hamiltonian (usually total energy) is conserved whenever the system possesses time translation symmetry generated by the Hamiltonian itself . How amazing!!

The Extended Theorem

The symmetry condition that we discovered on page 4 is a bit too harsh. We can relax it by remembering that the gauge transformation:

keeps Hamilton’s equation completely invariant as it produces constant shifts to action functionals:

With further calculations ( which I shall omit ) the symmetry condition boils down to we immediately discover a con-served quantity because clearly :

What this shows us is that when a transformation doesn’t fulfill the strict symmetry condition but instead the extended one, the conserved quantity is the generator G(q,p) plus the function F.

The Converse Theorem

Previously, we derived a conserved quantity for each symmetry. Let’s see if the opposite is true. Given a conserved quantity, we have:

Written in terms of Hamilton’s general equation:

Moreover , G,H=0 is the symmetry condition. From above, we see that this condition is always fulfilled by the conserved quantity. Therefore, each conserved quantity indeed generates a symmetry.

No only does Noether’s Theorem uncover an intimate link between conservation laws and the symmetries of nature, it also provides a systematic procedure to search for conserved quantities, a connection that physicists have exploited ever since. To quote Emmy Noether herself:

“My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously.”