Posted by Pasti on May 30, 2003 at 01:03
(64.10.124.136)
Re: Noether theorem vs. Runge_Lenz (cont'd) (Uncle Al)
"A fully discretized Noether theorem is quite a pickle indeed! This monstrous gap has not prevented mainstream physics from including parity and charge conjugation as symmetry-coupled conserved physical properties. CPT conservation is *the* fundamental underpinning of all quantum field theories without exception. We've both got to be missing something."
We are not missing anything.CPT is a discrete symmetry,and cannot be described via Noether theoremBut the fact that you cannot associate a Noether current with this symmetry, and correspondingly a Noether charge does not mean that lagrangeans cannot have these symmetries.They do have them.
Noether theorem was developed for continuous symmetries, and in the present frame works only for that(whether gauge symmetries or diffeos).
In my previous posting I was being sarcastic, because you cannot represent a discrete parity as a Lie group element in the neighborhood of the identity element(the only ref that comes to mind is Peskin and Schroeder,Intro to QFT, and you can work your way up the refs in there).Therefore you cannot approach these symmetries via a Noether formalism.And in order to understand this you don't need any academic matematician.Lie group theory suffices.
"If a qualified Eotvos balance gives reproducible net output, the Equivalence Principle is empirically counterdemonstrated and falsified. There are no footnotes, there is no waffling, there is no escape. This is an irrefutable fact. The conclusion remains unchanged whether test mass composition or geometry is contrasted."
Al, try to think logically and unbiased.First of all, as I said before, the equivalence principle is a local principle, i.e. a principle concerned with what happens at ONE POINT (MATHEMATICAL POINT)IN THE SPACETIME.Which means that it is concerned with POINT MASSES (AGAIN,MATHEMATICAL POINT),in which case,the internal structure of these point masses becomes irrelevant.That's how you get(in a simplistic way) to the Weak Equivalence Principle (WEP) which says that the acceleration in free fall is independent of the type and structure of the masses.However, the WEP has one other drawback, that makes it "weak":you need a uniform gravitational field for it to be true (no tidal effects), which in real life does not exist (you can find/approximate in certain regions the gravitational field with a quasiuniform one, but strictly speaking, there is no truly uniform gravitational field).
That is where the SEP comes into place, telling you that you can always find locally(AGAIN, MATHEMATICAL POINT) a Lorentz frame in which the laws of physics take their special relativistic form.
Let me make the difference clearer.For the WEP, the trajectory of an extended (real life mass) must be a geodesic.In other words each constituent of the mass -atom,molecule,etc.- moves along a geodesic, and all these individual geodesics form a parallel congruence(all these geodesics are parallel) such that the extended mass moves along the "average" of these geodesics.
But since in real life there is no uniform gravitational field,testing the equivalence principle reduces to locally account for all non-local effects (structure of the mass, etc), calculate its trajectory, and then experimentally see how the measured trajectory compares with the calculated one.
So testing the EP accounts for the structure of the mass,and experimentally the is valid with an absolute error for the Eotvos mass parameter of ~10^(-12) (Braginsky&Panov).
After all the above, think about it:these limits have been determined without taking into account the parity symmetry of the test masses,which means, at least for a physicists, that parity is a smaller order effect.This on one hand.
On the other hand, assuming that your experiment works indeed with the results you expect, you are far from disproving the EP.You will only show that one has to better account for parity in performing the experiment,but your experiment will not be able to infirm all the previous testing results.
That is why I said that your ideea has some merit, but not as much as you think.It is worth trying to do it, to test the equivalence principle against parity,but in view of what has been already done,chances are 99% at least that you will only confirm the EP within the experimental errors.
"The worst the parity Eotvos experiment can do is fail. That puts it squarely on the curve of the best researcher's best efforts. Because it is radically different it might succeed. Somebody should look."
I fully agree with you on this, although following the reasoning above,many will consider it is not worth the effort.You will have to fight against that ideea quite hard, if you want to actually do the experiment.
Your second post:
"In Hamiltonian formalism: Any symmetry S (discrete or not) i.e. [S,H] = 0, gives a conserved quantity: [H,S] = 0. If S=parity then a parity invariant (symmetric) Hamiltonian implies that parity eigenstates remain such eigenstates while evolving in time."
Assuming that you went to quantum mechanics in the Scyhroedinger picture, your statement above is not exactly true.If S commutes with H,then S is an operator that is constant in time.This does not mean that the eigenfunctions of the system are constant in time, but only that the eigenvalues (not eigenstates) of S are eventually constant in time.However, the time dependence of the eigenstates is determined by the Schroedinger equation
df/dt=Hf f-eigenvector(eigenstate)
and the eigenstates of H,S (they admit a complete set of common eigenstates since they commute)depend on time(if H is not zero).
"Lagrangians and Hamiltonians are proven equivalent in a tensor analysis by an academic Canadian researcher's PhD thesis. I've got to go track it down."
That is intersting, although I not true as you state it(let me know if you find the ref you mentioned). Maybe the Lagrangean formalism and the Hamiltonian formalism are equivalent, but the hamiltonian and the lagrangean are not equivalent.They are related through a Legendre transform,and for a lagrangean that admits time translations as symmetry, the canonical energy (hamiltonian) is the corresponding conserved Noether charge.
I suspect you wanted to say something else.
Re: Noether theorem vs. Runge_Lenz (cont'd) Uncle Al 30/5 12:19
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