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Zhi-qiang Shi



Joined: 15 Nov 2007
Posts: 19

PostPosted: Thu Nov 15, 2007 2:56 am    Post subject: I agree with you Reply with quote

Thurs 2007-08-09 12:55 PM

Dear Dr. J. C. Yoon,
Thank you very much for you helping me find the paper by E. Derman and W. J. Marciano.
After reading this paper, I agree with you that for SLAC E158 experiments the integrating cross section are not directly measured but implied. Therefore, this convinces us all the more that the integrating scattering cross section of weak interactions is not Lorentz invariant, but that of electromagnetic interactions is.
Sincerely yours,
Zhi-qiang Shi
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jcyoon



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PostPosted: Thu Nov 15, 2007 2:57 am    Post subject: Debate with Prof. Okun Reply with quote

Sun 2007-09-09 8:17 AM

Dear Professor Zhi-qiang Shi,

As my debate with Professor L.B. Okun seems over now, I have posted them on my website http://www.jcyoon.com/phpBB/viewforum.php?f=21.

I am not happy with my embarrassing mistake in the middle of discussion, but I think the discussion in general would be helpful for others.

Sincerely yours,
J.C. Yoon
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jcyoon



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PostPosted: Thu Nov 15, 2007 2:58 am    Post subject: Debate with Prof. Okun Reply with quote

Sun 2007-09-09 9:56 PM

Dear Professor Zhi-qiang Shi,

As Professor Okun resumed his discussion after a month of silence, I am afraid that I had to take down the discussion board to avoid any wrong impressions from incomplete record.

I apologize for my premature posting notice.

Sincerely yours,
J.C. Yoon
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jcyoon



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PostPosted: Thu Nov 15, 2007 2:59 am    Post subject: Debate with Prof. Okun 3 Reply with quote

Mon 2007-09-10 5:36 AM
Attachment: OkunDebate.pdf

Dear Professor Zhi-qiang Shi,

I am not sure if it helps, but here I attached pdf file of email debates with Prof. Okun, which is still in progress.

Sincerely yours,
J.C. Yoon
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Zhi-qiang Shi



Joined: 15 Nov 2007
Posts: 19

PostPosted: Thu Nov 15, 2007 3:01 am    Post subject: Re: Debate with Prof. Okun 3 Reply with quote

Fri 2007-09-21 1:05 PM

Dear Dr. J. C. Yoon,
I am very pleased to receive your three emails dated 9 and 10 September. I have also received the attached file of debates with Prof. Okun and have carefully read them. The discussion with Prof. Okun is helpful. I am very interested in the discussion. If you don’t mind, I would like to clarify my viewpoints.
1. We should be careful not to confuse terms helicity, polarization and polarization vector. In my paper (Calculation on the Lifetime of Polarized Muons in Flight), the helicity is defined by Eq. (10) and the polarization vector (four-polarization vector, polarization 4-vector or polarization 4-pseudovector) is defined by Eq. (5). The polarization (polarization degree) is the length of polarization vector and it is a scalar, not a vector. The helicity cannot be defined in the rest frame, where momentum of particle is zero. Strictly speaking, only in the rest frame can the polarization vector be most unambiguously defined. Hence the polarization state of fermions in flight can not be described by the polarization vector, and it must be described by the helicity which are closely related to directly observable quantity experimentally.
2. The integrating cross sections in SLD and SLAC E158 experiments and the lifetimes of fermions in flight are different for left- and right-handed helicity, i.e., they are the parity-violating cross-sections or they are not Lorentz scalars. However, we do not say that they violate the Lorentz invariance of weak interactions.
3. As mentioned in my paper (arxiv 0705.3711), the Lorentz invariance of weak interactions means that the Lagrangian of weak interactions is invariant under proper Lorentz transformation. The Lagrangian must be defined by using spin states or chirality states, but not helicity states. Therefore, the Lagrangian certainly is Lorentz invariant and the Lorentz invariance of weak interactions still holds true.
4. Why can’t the Lagrangian be defined by helicity states? Because the helicity states are only mathematical solutions of Dirac equation, not physical solutions.
Do you agree to my views? If you disagree with me, please let me know your detailed argument and I will be very grateful.
By the way, I can not enter your website (http://www.jcyoon.com/phpBB). Could you help me?
Sincerely yours,
Zhi-qiang Shi
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jcyoon



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PostPosted: Thu Nov 15, 2007 3:03 am    Post subject: RE: Debate with Prof. Okun 3 Reply with quote

Fri 2007-09-21 7:09 PM

Dear Professor Zhi-qiang Shi,

It is my pleasure to receive your kind comments.

1. In the debates, what I meant by polarization was simply the orientation of spin relative to the fixed coordinate system, contrasting to helicity simplified as the spin orientation relative to the momentum. In my opinion the critical point remains the same with your more accurate definitions.

2. The difference in the integrating cross sections from SLD, SLAC E158 is related to the scattering or decay rates and lifetimes, which are supposed to be invariant in any reference frames. A scattering rate of one scattering event should be the same regardless of the reference frames.

Let us say we 6 out of 10 incoming left-handed helicity electrons decayed in the final states we want in one frame. In another frame, we may observe the same 10 physical events, but only 9, instead of 10, incoming electrons as left-handed helicity as one of particle’s momentum observed as flipped with still 6 final states we want. Then, we have a scattering rate for left-handed helcity electron as both 60% and 67%, which makes a scattering rate Lorentz violating.

3, 4. I agree the Lagrangian should be described in terms of spin states, but not in chiral states. In my opinion, the chirality of massive fermion is pure mathematical, since it does not have a corresponding physical measurement and its only measurement is by approximating the helicity of particle, while the helicity is physically observable as we can measure helicity from spin and momentum measurement in the experiments. The Lagrangian can be described by helicity, but its Lorentz-invariant physical observables should remain the same for left- and right-handed helicities. If the Lagrangians for left- and right-handed helicities have two different coupling constants and thus different scattering rates, then the Lagrangian is not proper.

Though I disagree with you at a couple of points, I am very grateful for asking my opinion.
Also, I am afraid I have not set my website unlocked yet and I think it is going to take some time.

Sincerely yours,
J.C. Yoon
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Zhi-qiang Shi



Joined: 15 Nov 2007
Posts: 19

PostPosted: Thu Nov 15, 2007 3:05 am    Post subject: Re: Debate with Prof. Okun 3 Reply with quote

Fri 2007-09-28 12:00 AM

Dear Dr. J. C. Yoon,

Thanks for your kind and elaborate reply.

1. In your definition, the polarization is the orientation of spin relative to the fixed coordinate system. However, this polarization is different from the helicity which is the orientation of spin relative to momentum. It is because under the Lorentz boost, a particle’s momentum would change its orientation but the fixed coordinate system would not when an observer's velocity exceeds the particle's one.

2. You said: “a scattering rate of one scattering event should be the same regardless of the reference frames”. For electromagnetic interactions or strong interactions, it is correct, but for weak interactions, it is not correct. SLD and SLAC E158 experiments have proved that the scattering rates of one scattering event are different for left- and right-handed helicity, i.e., it is related to reference frame. On the other hand, decay rate or lifetime is essentially Lorentz violating due to the time dilation effect predicted by the special theory of relativity.

3. I agree the chirality of massive fermion is pure mathematical, since it does not have a corresponding physical measurement. However, chirality states can also describe the Lagrangian. By using the relations
\bar{\psi}\gamma_\mu(1+\gamma_5)\psi=2\bar{\psi}_L\gamma_\mu\psi_L,
\bar{\psi}\gamma_\mu(1-\gamma_5)\psi=2\bar{\psi}_R\gamma_\mu\psi_R,
where \psi is spin state and \psi_L(R) is chirality state, we can make out that the expression for the Lagrangian can be described in two equivalent forms containing either spin states or chirality states.

4. If the Lagrangian can be described by helicity states, then its Lorentz-invariance will consequentially be violated. The Lagrangians of weak interaction for left- and right-handed chirality state have two different coupling constants. In the Lagrangian for the charged weak currents, for example, the coupling constant of right-handed chirality state is equal to zero, but that of left-handed chirality state is not equal to zero. Therefore, the Lagrangian should not remain the same for left- and right-handed helicity state.

Sincerely yours,
Zhi-qiang Shi
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jcyoon



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Posts: 213

PostPosted: Thu Nov 15, 2007 3:06 am    Post subject: RE: Debate with Prof. Okun 3 Reply with quote

Tue 2007-10-02 4:59 AM

Dear Professor Zhi-qiang Shi,

I am glad to receive your email, which shows that our perspectives are quite close.

1. I agree with your statement 1 and I am fully aware of the difference between helicity and my definition of polarization. Though I have tried to make this distinction clear, there might have been some points I was not clear.

2. The Lorentz invariance of some physical observables such as scattering rate is not a matter of right or wrong, but rather a matter of consistence as Lorentz invariance is accepted as a priori for physical observations. Without Lorentz invariance, the corresponding experimental measurements become inconsistent and physically meaningless. The parity-violating asymmetry, for example, differs for different reference frames as theoretically the value of A_{pv} could vary from -1 to +1, while the very reason that the parity-violating asymmetry supports the Standard Model is that the value of A_{pv} in SLAC E158 measurements is predicted by the coupling constants of the Standard Model sin^{2} \theta ~0.023. If the scattering rate is not Lorentz invariant, then the value of A_{pv} and thus sin^{2} \theta varies for each reference frames so that we cannot confirm the prediction of the Standard Model for the value of sin^{2} \theta as 0.0023. Therefore, to consider the scattering rate as Lorentz variant while claiming that the validity of the Standard Model as the constants of the Standard Model to be a fixed value is inconsistent. In other words, if we conclude the scattering rate Lorentz variant, then we should not claim the Standard Model is supported by such experimental measurements.

According to the conventional particle physics, the decay rate and lifetime measured in the SLAC E158 is independent of the time dilation effect from the special relativity, since the asymmetry effectively cancels out relativistic factors in the nominator and denominator and it is the very reason the asymmetry is often preferred as experimental measurements. Also, conventionally, the decay rate and lifetime(more accurately, mean proper lifetime) is defined in the proper frame where the object is at rest so that it is diretly proportional to the matrix element without relativistic factor. Please refer to http://pdg.lbl.gov/2007/reviews/kinemarpp.pdf

3. The Lagrangian described by chiral state is only valid in terms of the chiral structure ( 1 +- \gamma^{5}) acting on the fermion field \psi in spin state, as you have stated, but the description is ambiguous and requires an approximation of helicity to be valid as accounted in my preprint. Let us consider an electron in spin state \psi. Since the helicity of electron could be left- or right-handed depending on the reference frames, whether (1 + \gamma^{5}) or (1 - \gamma^{5}) will act on the electron is unknown. Therefore, regardless of the helicity measured in the lab frame, we should have the probability of both (1 +- \gamma^{5}) acting on the electron and thus whether the electron is left- or right-handed chilarity is also unknown. To set up a experimental tests to compare between two chiralities, it is inevitable to separate the chiral structure ( 1 +- \gamma^{5}) and the separation is from the approximation based on the helicity. If we have a left-handed helicity in high-energy measurements, we can have (1 - \gamma^{5}) only approximately ignoring (1 + \gamma^{5}). Without the helicity, the chirality structure is indistinguishable and thus remains detached from the experimental measurements.

4. For practical measurements, the description of Lagrangian in terms of helicity is inevitable, as mentioned above. If whether the interaction structure has (1 + \gamma^{5}) or (1 - \gamma^{5}) is not determined by approximating from helicity, then the chiralities of electron and thus their coupling constants are also not known and the experimental comparison between ( 1 +- \gamma^{5}) or two chiralities is not valid.

Sincerely yours,
J.C. Yoon
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Zhi-qiang Shi



Joined: 15 Nov 2007
Posts: 19

PostPosted: Thu Nov 15, 2007 3:07 am    Post subject: about Lagrangian Reply with quote

Sun 2007-10-07 5:07 PM

Dear Dr. J. C. Yoon,
It is my pleasure to receive your email.
1. Yes, the lifetime of particles is defined in the proper frame where the particle is at rest. In the laboratory frame, however, the lifetime of particles in flight is Lorentz violated. Eq. (38.19) in http://pdg.lbl.gov/2007/reviews/kinemarpp.pdf is an expression in its rest frame. In an arbitrary frame, M in the denominator of this expression should be E. Please see Eq. (10.130) in Relativistic Quantum Mechanics by Bjorken and Drell. The lifetime asymmetry shows that the lifetime of particles in flight is not only Lorentz violated, but also parity violated.
2. We limit our discussion to within weak interactions, therefore, there are the chiral structure ( 1 +- \gamma^{5}) in the Lagrangian density. Theoretically, the Lagrangian density of weak interactions is defined by using spin states or chirality states. As you have stated, it is pure mathematical, experimentally, the description of the Lagrangian in terms of helicity states is inevitable. (note: the definition is different from the description) Therefore, we need to establish the relations between chirality states and helicity states. These relations are just Eq. (18) and (19) in my paper (calculation on the lifetime of polarized muons in flight). They are rigorously mathematical relations, not approximative relations.
3. The left-right integrated cross section asymmetry in SLD and E158 experiment and the lifetime asymmetry indicate that the Lagrangian density of weak interactions is parity violated. However, it is still a question whether it indicate that the Lagrangian is Lorentz violated. In fact, this question includes a paradox. If the Lagrangian density of weak interactions is defined by using helicity states, then the parity violation will require that the Lagrangian is helicity dependent because a fermion's helicity can changes sign under space inversion, meanwhile, Lorentz invariance will require that the Lagrangian is helicity independent. Obviously, these two viewpoints contradict each other. It presents a rather puzzling question of whether the Lagrangian is helicity dependent, or rather there would be a paradox logically.
4. To decide unequivocally the helicity dependence puzzle, we have to understand further the helicity states. In your debate with Professor Max, Max said: “For massive free fermions, plane wave solutions are not eigenstates of helicity.” do you agree with his this argument? By the way, at which university is he working?
5. The Lagrangian must be Lorentz invariant. Therefore, we conclude that the Lagrangian is helicity independent and the Lagrangian can not be defined by using helicity states.
Sincerely yours,
Zhi-qiang Shi
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jcyoon



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PostPosted: Thu Nov 15, 2007 3:08 am    Post subject: RE: about Lagrangian Reply with quote

Tue 2007-10-09 9:49 PM

Dear Professor Zhi-qiang Shi,

Thanks for your kind and elaborate reply.

1. I must have not been clear about the experimental measurements of asymmetry in SLAC E158 and SLD. The lifetime in an arbitrary frame does have E , i.e., E_{P} in Eq. (10.130) in Relativistic Quantum Mechanics by Bjorken and Drell. However, it is Lorentz variant, not Lorentz violating. In my definition, the term 'Lorentz violation' is legitimate when the quantity found not Lorentz-invariant is supposed to be Lorentz invariant, while the lifetime in a boosted frame is not supposed to be Lorentz invariant, but the matrix element is.

Also, the lifetime asymmetry in SLD shows the difference between the Lorentz-invariant part of lifetimes, since such Lorentz variant factor E is effectively the same (its variation has been considered in experimental analysis) and cancels out so that only the Lorentz-invariant part remains: A_{PV} = [{…L}/E - {…R}/E]/[{…L}/E + {…R}/E] = [{…L} - {…R}]/[{…L } + {…R}]. Since the factor E will be canceled out, the theoretical calculation for differential cross sections, for example, Eq. (1.52) and the like in “Measurement of the Polarized Forward-Backward Asymmetry of B Quarks Using Momentum-Weighted Track Charge at SLD” by Thomas Robert Junk (http://www.slac.stanford.edu/pubs/slacreports/slac-r-476.html), omit the energy dependent factor and thus the asymmetry in Eq. (1.58) contains only the matrix element part, which is equivalent of comparing the lifetimes in the rest frame. For the asymmetry of SLAC E158, we can see that the energy dependent factor is also omitted as shown in E. Derman and W. Marciano, Ann. Phys. 121, 147 (1979).

Since the energy of electron beam could vary in experimental measurements, the accuracy of cancellation has been well-studied and estimated and, for example, statistical weighting for energy was taken into account for each run’s asymmetry, in “A Precision Measurement of Parity Violation in Moller Scattering”, by David R. Relyea, 110p, (http://www.slac.stanford.edu/pubs/slacreports/slac-r-717.html).

Therefore, what the SLAC E158 and SLD measurement has shown is the asymmetry of Lorentz-invariant matrix element, which also implies the difference of lifetimes in the rest frame.

2. I agree that Eq. (18) and Eq. (19) in your paper are mathematically rigorous, but the approximation of weak interactions I mentioned occurs in considering only one the chiral structure V-A, for example, ignoring V+A, which to act on Eq. (18) and (19).

3. To investigate parity, the Lagrangians for left- and right-handed helicity should be compared. And if the Lagrangians are the same regardless of helicity, then parity is conserved and Lorentz invariance holds. It is not a matter of how the Lagrangian is defined, but whether the Lagrangian is the same or not in terms of helicity.

In my opinion, parity violation reported in SLAC E158 and SLD is equivalent of Lorentz violation. Parity is symmetry between left- and right-handed helicity electrons. Since a left-handed helicity can be observed as right-handed helicity, the symmetry between these particles, parity, also means Lorentz invariance. If we conclude parity is violated for electron, then we should also assert that Lorentz invariance is violated. Note that parity violation and thus Lorentz violation does not confirm Lorentz violation, but leads us to seek other possible Lorentz-invariant explanation.

4. I disagree that for massive free fermion plane wave solutions are not eigenstates of helicity. See Peskin and Schroeder, “An Intro. To Quantum Field Theory”, page 47. And I have no information about him.

5. I disagree that the Lagrangian can not be defined or expressed in terms of helicity to be Lorentz invariant. To be Lorentz invariant, the Lagrangian expressed in terms of helicity is required to be the same regardless of helicity.

Sincerely yours,
J.C. Yoon
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Zhi-qiang Shi



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Posts: 19

PostPosted: Thu Nov 15, 2007 3:10 am    Post subject: about helicity states Reply with quote

Thurs 2007-10-11 11:25 AM

Dear Dr. J. C. Yoon,

Thank you very much for your email. I like our discussion because both of us can learn something.

1. I agree that the factor E has be canceled out in SLAC E158 and SLD experiment. The Lorentz invariance of the Lagrangian is different from that of the matrix element. Our discussion should center on the problem of the Lorentz invariance of the Lagrangian.

2. You said: if parity is violated, then we should also assert that Lorentz invariance is violated. Does professor Okun agree with this viewpoint? The Lorentz invariance of interaction Lagrangian is the essence of law, and so it is one of basic principle of physics. To negate the Lorentz invariance of interaction Lagrangian will not be accepted by mainstream physicists.

3. I think that helicity states are not the plane wave solutions of Dirac equation. The other important reason is as follows:

In Quantum Mechanics, the variables of wave functions are separable and then a wave function can be written as \Psi(x,y,z,\sigma)=\psi(x,y,z)\chi(\sigma), where wave function \psi(x,y,z) is only related to the position variables and spin wave function \chi(\sigma) dependents only on a single variable \sigma. Spin is the intrinsic property of particles and then the eigenvalue of spin wave function is Lorentz invariant. A particle which does not possess determinate spin value is can not be accepted in physically. On the other hand, Spin variable \sigma has three spatial components, \sigma_x, \sigma_y and \sigma_z. The \sigma_z commutes with Hamilton operator H and so it is conventionally designated as the variable of spin wave function. In this way, a wave function can be rewritten as \Psi(x,y,z,\sigma)=\psi(x,y,z)\chi(\sigma_3). Therefore, the solution of Schrodinger equation is simultaneously an eigenfunction of H and of \sigma_z.

In Relativistic Quantum Mechanics, Hamilton operator H still commute with \sigma_z in its rest frame (where the momentum p=0). In an arbitrary frame, however, they do not commute anymore and so \sigma_z can not serve as spin variable. We need to construct new spin operator and its eigenfunctions. Based on discussion above, obviously, spin wave function should satisfy two conditions as follows: (1) its eigenvalue is Lorentz invariant; (2) in its rest frame it will reduce to the eigenfunction of \Sigma_z, where \Sigma is spin operator in four-dimensional space. In order to satisfy the two conditions, we must refer to the Pauli-Lubanski covariant spin vector \omega, four-polarization vector e and spin states. (See Eqs. (4) and (5) in my paper, calculation on the lifetime of polarized muons in flight). The plane wave solution of Dirac equation is simultaneously an eigenfunction of H, of momentum p and of scalar operator \omega\cdot e. Spin states can also reduce to the eigenfunction of \Sigma_z in its rest frame.

The helicity state is meaningless in its rest frame and its eigenvalue is not Lorentz invariant. The helicity states do not satisfy the two conditions, and so they are not the plane wave solutions of Dirac equation.

Sincerely yours,
Zhi-qiang Shi
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jcyoon



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PostPosted: Thu Nov 15, 2007 3:12 am    Post subject: RE: about helicity states Reply with quote

Sat 2007-10-20 7:20 AM
Attachment: OkunDebate_Update.pdf

Dear Professor Zhi-qiang Shi,

I am glad that we are making progress in this constructive discussion, which is a great opportunity for me to learn and speculate on the issue.

Enclosed is the update version of the debates with Prof. Okun, which seems to be over by now.

I agree that mainstream physicists would be reluctant to acknowledge Lorentz violation. But, I think most physicists in 1950s also found parity violation hard to accept.

And I agree with your elaborate account on helicity that gives me a different but helpful point of view, though it would be more comfortable for me to state that the helicity state of plane wave solutions of the Dirac equations is physically meaningless.

Sincerely yours,
J.C. Yoon
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Zhi-qiang Shi



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PostPosted: Thu Nov 15, 2007 3:13 am    Post subject: our common points Reply with quote

Tue 2007-10-23 11:35 PM

Dear Dr. J. C. Yoon,
I am very pleased to receive your email and the update version of the debates with Prof. Okun,. I think that we have made an agreement about Lorentz invariance. They are summarized as follows:

1. In weak interactions, the integrating scattering cross section or the decay rate is different for left- and right-handed helicity, or it is parity violated due to parity unconservation. The decay rate or lifetime of fermions in flight is neither Lorentz scalar nor parity invariant. In other words, it is neither a four-dimensional scalar, nor a scalar under the three-dimensional space inversion. In one word, the cross section or decay rate is related to the reference frames.

2. The helicity states are not the plane wave solutions of Dirac equation. Strictly speaking, the helicity states are only mathematical solutions of Dirac equation, not physical solutions. Therefore, the Lagrangian can not be defined by using helicity states and so it is helicity independent.

3. The Lorentz invariance of weak interactions means that the Lagrangian density of weak interactions is invariant under proper Lorentz transformation. The Lagrangian must be defined by using spin states or chirality states, but not helicity states. Therefore, the Lagrangian certainly is Lorentz invariant and the Lorentz invariance of weak interactions still holds true, though the Lagrangian density of weak interactions is parity violated because there are the chiral structure ( 1 +- \gamma^5) in the Lagrangian density. Therefore, the parity violated cannot violate the Lorentz invariance.

4. Theoretically, the Lagrangian density of weak interactions is defined by using spin states. It is pure mathematical. Experimentally, however, the description of the Lagrangian in terms of helicity states is inevitable. Therefore, we need to establish the relations between chirality states and helicity states.

Do you agree with the above statements? I wish it would be helpful for your debates with Professor L.B. Okun.
Sincerely yours,
Zhi-qiang Shi
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jcyoon



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PostPosted: Thu Nov 15, 2007 3:14 am    Post subject: RE: our common points Reply with quote

Wed 2007-10-24 5:22 AM

Dear Professor Zhi-qiang Shi,

Thanks for you kind reply to confirm my perspective.
In fact, I am afraid that there are a couple of points I was not clear about.

As for your statement 1, I would say it is Lorentz violation, as long as we call it parity violation. This may concern your statement 3.

> 3. The Lorentz invariance of weak interactions means that the Lagrangian
> density of weak interactions is invariant under proper Lorentz
> transformation.

The Lagrangian density of weak interactions with the chiral structure and exact Dirac massive fermion fields only appears to remain the same under proper Lorentz transformation of fermion fields, as it fails in the normalization factor. The failure of normalization can be checked by explicit calculation for the rest frame and boosted frame as one yields normalization factor of m and the other some factors such as \sqrt{p \cdot \sigma}. And this is also related to the fact that the axial vector current is conserved only when m=0. We may ignore the failure of normalization for convenience, but in weak theory it is the very reason we can ignore one chirality of massive electrons for the claim of parity violation; parity violation is valid when we ignore one of chirality structure leaving only v-a interactions.

For this subtle issue in theoretical point, in order to determine Lorentz invariance it would be best and inevitable to include the practical calculations and measurements, which consistently show the failure of Lorentz invariance of Lagrangian in more explicit way.

Sincerely yours,
J.C. Yoon
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Zhi-qiang Shi



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PostPosted: Thu Nov 15, 2007 3:16 am    Post subject: normalization factor Reply with quote

Tue 2007-10-30 9:22 FM

Dear Dr. J. C. Yoon,
Thank you for your email.
First, the cross section or decay rate is different from the Lagrangian, and so that the cross section or decay rate is related to the reference frames is not equal to the Lorentz violation of the Lagrangian.
Second, I do not understand “The failure of normalization” you said. Would you please tell me its meaning by using formula?
Sincerely yours,
Zhi-qiang Shi
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