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Noether theorem + CP violation > Energy is not conserved? 
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#1
Mar2709, 03:32 PM

P: 2,456

Based on the Noether theorem http://en.wikipedia.org/wiki/Noether%27s_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 Tsymmetry is violated Hence, our spacetime is not isotropic: there is prefered direction in time Hence, stress energy tensor is not conserved. Any comments? 


#2
Mar2709, 03:39 PM

P: 2,828

There is nothing wrong with a timeasymmetric 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. 


#3
Mar2709, 03:41 PM

P: 2,456

Continious symmetries... yes, I did not think about it.
Thank you. But still, isnt Tsymmetry the most mysterious of all? It is like having a preferred direction in space... 


#4
Mar2709, 04:25 PM

P: 271

Noether theorem + CP violation > Energy is not conserved?
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. 


#5
Mar2809, 10:37 AM

P: 894

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. 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. 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. 


#6
Mar2809, 10:45 AM

P: 2,456

As we are talking about Tassymetry, I wanted to repeat my old question.
Do you think that Tassymentry is an intristic property of our spacetime, or it is like a weak 'echo' of the Big Bang  a result of the initial conditions? 


#7
Mar2809, 08:05 PM

P: 271




#8
Mar2909, 10:47 AM

P: 894

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 pseudovector field. With magnetic monopoles, this would not work the same way. Also, you'd need more than just the usual fourvector 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. 


#9
Mar3009, 11:13 AM

P: 271




#10
Mar3009, 05:35 PM

P: 894

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 astrophysicists. 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/Tsymmetry If what I said is making things worse instead of helping, I guess I'll just shut up and let an astrophysicist 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. 


#11
Mar3109, 05:01 AM

P: 271

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. 


#12
Mar3109, 02:27 PM

P: 102

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 timereversal symmetry would be something like [itex]\psi\rightarrow (i\sigma_y)\psi^*[/itex] where [itex]\psi[/itex] 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 timereversal symmetry as defined above). Another example is Anderson's theorem which states that nonmagnetic impurities can not suppress superconductivity (which can be thought of as a consequence of TR symmetry). In topological insulators the presence of timereversal 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 "timereversal 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. 


#13
Mar3109, 02:51 PM

P: 2,828

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.



#14
Mar3109, 07:18 PM

P: 894

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? 


#15
Mar3109, 07:50 PM

P: 2,828




#16
Mar3109, 08:36 PM

P: 894

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.



#17
Mar3109, 09:50 PM

P: 102




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