What is the Noether's theorem and how it break the law of conservation?

Question

I was in http://worldbuilding.stackexchange.com when I read a question about how to explain magic breaking physical laws, and in one answer they talk about the Noether's theorem and how to break the law of conservation. I couldn't understand nothing (I am not sure if I can't understand the law or it's a translation barrier) so I decided to search a little more and I found a lot of information in Physics.SE but their are too complicate and I can't understand anything. Could someone explain me what is the Noether's theorem in a simple way? I mean a basic explanation, no hard-science.

Answer 1

Noether's theorem relates conservation laws to symmetries of a system.

What is a conservation law?

A conservation law is a statement that a measurable property of a system does not change with time. For example, if we denote the total energy of a system to be $H$, then energy conservation is the statement that,

$$\frac{dH}{dt} = 0$$

which if you're not familiar with calculus is essentially saying the change in energy over some period of time is zero, as expected if it were conserved.

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What is a symmetry?

Mathematically, we often describe systems in physics using a Lagrangian, and one can make precise the meaning of a symmetry.

In layman's terms, it essentially means that the physics of a system remain the same if we make a certain change. For example, time translation invariance means that if we shift time $t\to t+c$, then the physical laws remain the same.

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Noether's Theorem

Noether's theorem relates to every conservation law, a physical symmetry of a system. In fact, in general, it shows us how to compute the conserved quantity given a symmetry. For example, in the case of time translation, one can show the conserved quantity is $H$, the total energy.

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Symmetry Breaking

As it now relates to your world-building concern, if you want to break a conservation law, this can only be possible if the system no longer possesses the symmetry, or a more subtle case.

We have explicit symmetry breaking which means the system flat out does not possess the symmetry. The other case is spontaneous symmetry breaking which is when the equations that describe a system have the symmetry, but a state of the system does not.

In the more complicated case of spontaneous breaking, Goldstone's theorem shows there are rather deep implications; the Higgs boson in the Standard Model relates to this, for example.

Answer 2

Could someone explain me what is the Noether's theorem in a simple way? I mean a basic explanation, no hard-science.

(1) Start with something called the Lagrangian of a system and the rules for deriving the equations of motion from the Lagrangian.

(2) A symmetry of the Lagrangian is a transformation (of coordinates etc.) that leaves the Lagrangian unchanged and thus, the equations of motion unchanged.

(3) Noether showed that there is a conservation law associated with a (differentiable) symmetry of the Lagrangian.

Here are some examples.

If the Lagrangian is unchanged by a translation in space (loosely, moving the origin of the coordinate system) we say that the Lagrangian has a spatial translation symmetry and the associated conservation law is conservation of (linear) momentum.

If the Lagrangian is unchanged by a rotation, there is a rotational symmetry and the associated conservation law is conservation of angular momentum.

If the Lagrangian is unchanged by a translation in time (change the zero on the clock), there is a temporal translation symmetry and the associated conservation law is conservation of energy.

So, for example, to 'break' conservation of (linear) momentum would require that physical law somehow depend on location so that the Lagrangian is changed by a spatial translation.

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On a more advanced note, the Standard Model (the best model we have of the elementary particles and their interactions) is based in part on the idea that the fundamental interactions are most elegantly introduced by promoting a global symmetry of a Lagrangian to a local symmetry which requires the introduction of new terms in the Lagrangian called gauge fields.

For example, the electromagnetic field is a gauge field that is required to give the Lagrangian a local U(1) symmetry

This is well beyond the scope of your question but I put this here just to emphasize how important symmetries of the Lagrangian and Noether's theorem are to fundamental physics.

Answer 3

The short version is this: every conserved quantity in physics (like energy and momentum) corresponds to some way in which the physical laws that govern reality are invariant (don't change) when you transform the situation somehow. For example, if I hit a puck on an air hockey table why does it move in a straight line at constant speed (ignoring friction)? The answer is because the table is flat - it's invariant under translations. If the table were curved then the puck wouldn't move in a straight line, even without gravity.

Answer 4

See Wikipedia's article on "Noether's Theorem"

Noether's (first)1 theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.

For example, I would think, if one has a ring of charge and rotates that ring in its plane, the charge is still conserved:

From: https://en.wikipedia.org/wiki/Charge_conservation#Connection_to_gauge_invariance

Charge conservation can also be understood as a consequence of symmetry through Noether's theorem, a central result in theoretical physics that asserts that each conservation law is associated with a symmetry of the underlying physics...

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