# Noether theorem + CP violation -> Energy is not conserved?

#### Dmitry67

Based on the Noether theorem http://en.wikipedia.org/wiki/Noether's_theorem
there is a relationship between conseration laws and symmetries.

Conseration of energy and momentum are related to the isotropy of spacetime.

However, spacetime is NOT isotropic:
CP symmetry is violated: http://en.wikipedia.org/wiki/CP_symmetry

Based on the CPT theorem: http://en.wikipedia.org/wiki/CPT_theorem
T-symmetry is violated

Hence, our spacetime is not isotropic: there is prefered direction in time

Hence, stress energy tensor is not conserved.

Related Beyond the Standard Model News on Phys.org

#### humanino

Two. First, this is merely a weak analogy, which actually fails even before you made a first step if you tried to do it. I mean to say that you did not think very much before posting. Second, the reason it fails is that Noether theorem is for continuous symmetries, whereas the "T" in CPT is time inversion.

There is nothing wrong with a time-asymmetric process continuously described by laws which are uniform, that is to say the laws are the same yesterday and tomorrow, although we can tell in which direction time is flowing. Basically you are confusing uniform and isotropic.

#### Dmitry67

Continious symmetries... yes, I did not think about it.
Thank you.

But still, isnt T-symmetry the most mysterious of all? It is like having a preferred direction in space...

#### f-h

No, it is like having a preferred orientation in space. Which is exactly what P is. As charges know about orientation of space via the cross product in the Lorentz Law, already classical EM violates P, but not CP.

Thus CPT says (roughly, heuristically) that there is no preferred orientation in space time. The surprise is that the space orientation and the time orientation are linked though. Given a particular time direction you can measure the CP status of space via CP violating processes.

#### JustinLevy

Conservation of energy comes from time translation invariance, not "T" - time inversion. So no worries, the standard model doesn't violate energy conservation.

Does anyone know if there is some extension of Neother's theorem dealing with discrete symmetries?
We associate a "parity" quantum number with particles in interactions conserving parity, so it seems very plausible to me that there is something similar to Noether's theorem showing that all discrete symmetries lead to a conserved quantity.

No, it is like having a preferred orientation in space. Which is exactly what P is. As charges know about orientation of space via the cross product in the Lorentz Law, already classical EM violates P, but not CP.
I'm not following your logic here. QED interactions conserve parity, so I do not believe your statement is correct.

Our arbitrary choice of "handedness" for cross products does not affect the physics predicted by maxwell's equations. A lazy way of looking at this (maybe an incorrect oversimplification) is that there is the 'cross product' for the source, and the 'cross product' in the force law... so our arbitrary sign choice cancels.

Thus CPT says (roughly, heuristically) that there is no preferred orientation in space time. The surprise is that the space orientation and the time orientation are linked though. Given a particular time direction you can measure the CP status of space via CP violating processes.
Hmm... I never thought of it that way. Thanks for sharing.

To me, something seems fishy with that. I can't put my finger on it at the moment though. Can someone else comment on this? It would be good to hear any additional insights on this interpretation. I'm having trouble visualizing it.

#### Dmitry67

As we are talking about T-assymetry, I wanted to repeat my old question.

Do you think that T-assymentry is an intristic property of our spacetime, or it is like a weak 'echo' of the Big Bang - a result of the initial conditions?

#### f-h

I'm not following your logic here. QED interactions conserve parity, so I do not believe your statement is correct.
True, I shouldn't trustmy memory on stuff I don't work with. I wonder what it was I was remembering... Possibly magnetic monopoles? Would that mess with parity?

#### JustinLevy

True, I shouldn't trustmy memory on stuff I don't work with. I wonder what it was I was remembering... Possibly magnetic monopoles? Would that mess with parity?
Well, some people refer to a magnetic field as "breaking time reversal symmetry". The idea is if the source of the magnetic field is far enough away that you are not interested in interactions with the source, then you can treat it as an 'external' parameter which breaks the time reversal symmetry of the dynamics.

You could do the same with an external force and claim that momentum is not conserved. It is more an issue of looking at an open instead of closed system, than it is an issue of actual symmetries.

I can't really comment about magnetic monopoles. Currently, we can consider electric fields as a vector field, and magnetic fields as a pseudo-vector field. With magnetic monopoles, this would not work the same way. Also, you'd need more than just the usual four-vector potential to describe the fields. I'm not sure the best or correct way to handle all this. Maybe a magnetic monopole would indeed break T symmetry.

As we are talking about T-assymetry, I wanted to repeat my old question.

Do you think that T-assymentry is an intristic property of our spacetime, or it is like a weak 'echo' of the Big Bang - a result of the initial conditions?
It is my understanding that the standard model lagrangian itself has CPT symmetry, but not CP or T symmetry. If it is broken at the level of the action, then no, I would not consider this a result of the initial conditions. (Unless one can formulate a testible model in which the standard model arises from said initial conditions.) At least that is my opinion. I'm sure others here can give a more detailed answer.

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#### f-h

Well, some people refer to a magnetic field as "breaking time reversal symmetry".
I guess you mean for static magnetic fields only? What about magnetic fields in an EM wave? Those can be consistently considered in a closed system without sources after all. as the QED vertex is P invariant that would mean parity would be spontaneously broken by EM radiation, right?

#### JustinLevy

I guess you mean for static magnetic fields only? What about magnetic fields in an EM wave? Those can be consistently considered in a closed system without sources after all. as the QED vertex is P invariant that would mean parity would be spontaneously broken by EM radiation, right?
I don't like the terminology that claims magnetic fields 'break time reversal symmetry', but if you do a literature search you'll see that people do use this phrasing. I have met more than one astro-physicist that have heard this so often that they forget in what context that claim even makes sense and they unfortunately start to believe that electrodynamics, when considering magnetic interactions, does not have time reversal symmetry. This of course is wrong. That is why I don't like such wording, because it can lead to misunderstanding ... which based on your questions seems to have happenned to you because of my poor wording as well (sorry).

Time reversal changes the sign of a magnetic field. Run everything backwards in time and it won't break any laws of electrodynamics.

ONLY in the very restricted case where you consider a magnetic field an 'external' parameter and time reverse everything 'local' do you get what appears to be a breaking of time reversal symmetry. I am not an astro-physicists. I don't understand why this means of thinking is useful, but the fact that it has survived so long must mean it is useful for something. I'd still prefer they worded it differently though.

EDIT:
This discusses some of those issues:
http://en.wikipedia.org/wiki/T-symmetry

If what I said is making things worse instead of helping, I guess I'll just shut up and let an astro-physicist explain in what sense they mean magnetic fields break time reversal symmetry, and how it is useful. Anyone?

EDIT:
A literature search also shows that some condensed matter people like to use that phrasing as well. I was not aware of that.
Can someone please explain the usefulness of such phrasing? It seems like it would cause more confusement than actually saying anything meaningful.

#### f-h

Yeah that makes more sense. So what you are saying is, if I treat the magnetic field as an external source which doesn't transform under P, then P is broken. Right? That is pretty nonsensical from a fundamental physics perspective but from a phenomenological perspective it does make sense to think that way. It just shows you what the origin of apparently time reversal invariant processes is.

And now that I've looked at old lecture notes that is actually how it was used there. As an example of an apparent violation which is resolved by a better understanding of the source of the field.

#### jensa

EDIT:
A literature search also shows that some condensed matter people like to use that phrasing as well. I was not aware of that.
Can someone please explain the usefulness of such phrasing? It seems like it would cause more confusement than actually saying anything meaningful.
I was hoping someone else would answer this but I guess the "Beyond the Standard Model" section is not really a hot-spot for condensed matter physics people. Anyway, I will give a shot at explaining how this phrasing is used among condensed matter physicists.

Usually we are interested in the physics and properties of a particular subsystem of the universe, such as a piece of metal or other material. An important experimental probe for the properties of, lets say a metal, is the response to external fields. Since what we really are interested in is the metal, we don't want to write up the entire Hamiltonian of the experimental setup. So all the experimental pieces such as current/voltage generators, magnets etc, only enter the Hamiltonian of our system as external fields. Now, how the system responds to these external fields tells us something about how the electronic spectrum looks like, or how the electrons and phonons interact with each other, etc.

Since we are only interested in the degrees of freedom associated with the subsystem we are looking at, it makes sense to consider only symmetry transformations on these degrees of freedom. For example, the time-reversal symmetry would be something like $\psi\rightarrow (i\sigma_y)\psi^*$ where $\psi$ is the wavefunction of an electron. In the absence of magnetic fields this is a symmetry of the Hamiltonian (usually) and leads to e.g. Kramers degeneracy (degeneracy in the electronic spectrum due to time-reversal symmetry as defined above). Another example is Anderson's theorem which states that non-magnetic impurities can not suppress superconductivity (which can be thought of as a consequence of T-R symmetry). In topological insulators the presence of time-reversal symmetry ensures the stability of helical edge modes, etc.

So what I am trying to say is, that even if this usage of the term "time-reversal symmetry" is not consistent with fundamental time reversal symmetry (referring to action of the element of the Poincare group), it is certainly usefull to consider such a symmetry and to consider when it is broken (which it is when magnetic fields are applied).... At least in condensed matter physics.

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#### humanino

It is correct that you can use the breaking of parity as defined in one given referential by the (non-)symmetry of the source producing your magnetic field. But that does not solve the problem that the magnetic field is still an axial vector for a wave far far away from the sources. The trick is that QED is invariant under Poincare transformations, which itself is made of Lorentz (continuous) transformation plus additional discrete parity and time reversal (from which you can construct charge). But as soon as you talk about the magnetic field independently of the electric field, you have already broken Lorentz invariance. So I'm not sure what would be the general meaning of parity invariance in that case anyway, picking up a specific frame to define your "magnetic field" independently of the electric field has already broken the more elementary Lorentz invariance. What has broken parity is just this choice : you picked up in a specific frame the parity odd part. "Lorentz transformation" and "picking up the parity odd part of the field" do not commute.

#### JustinLevy

So what I am trying to say is, that even if this usage of the term "time-reversal symmetry" is not consistent with fundamental time reversal symmetry (referring to action of the element of the Poincare group), it is certainly usefull to consider such a symmetry and to consider when it is broken (which it is when magnetic fields are applied).... At least in condensed matter physics.
Well, neither is part of the Poincare group, but okay I see what you mean: they are referring to different symmetries. Gotcha. Thanks for taking the time to explain how considering such a symmetry can be quite useful.

Regarding Humanino's comments, it seems like since you are not applying the time reversal to the 'external' fields, it doesn't matter what frame we pick. If there is a magnetic field in one frame, there will be a magnetic field in all frames. And it doesn't matter that the electric and magnetic field behave differently to this operation ... since they don't apply the time reversal operation to any 'external' elements in the Hamiltonian anyway. Right?

#### humanino

If there is a magnetic field in one frame, there will be a magnetic field in all frames.
So what if you take a point charge at rest ? There will not be a magnetic field obviously, right ? Now change to a referential where the single charge is not at rest : is there a magnetic field ?

#### JustinLevy

Yes, my wording was poor, as that statement by itself is incorrect. In context though, the situation is one with an external magnetic field and no electric field. You can choose other inertial frames, which the external field will now be composed of an electric field as well, but the magnetic field cannot be zero.

#### jensa

Well, neither is part of the Poincare group, but okay I see what you mean: they are referring to different symmetries. Gotcha. Thanks for taking the time to explain how considering such a symmetry can be quite useful.
Ah, ok. I actually edited my post to add that comment about Poincare group to emphasize the different approach to symmetries...guess I messed up. But you got the point anyway so..glad I could help.

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