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Lorentz Violation of the Standard Model
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Warren Siegel



Joined: 28 Sep 2006
Posts: 14

PostPosted: Sat Oct 21, 2006 2:15 pm    Post subject: Re: Lorentz Violation of the Standard Model Reply with quote

Wed 2006-05-17 8:03 PM

[quote="On May 16, 2006, at 7:51 PM, J.C. Yoon"]We agree that the unification of the Standard Model is an
approximation with the implication of Lorentz violation and the SLAC
E158 measurement is not Lorentz invariant.[/quote]

No, the Standard Model is completely Lorentz covariant.
I don't know what you mean by a measurement being noninvariant.

[quote="J.C. Yoon"]And the issue here is whether the unification is inconsistent due to
its approximation.[/quote]

There is no issue.
You do some calculations based on certain approximations, and you check they are accurate.

[quote="J.C. Yoon"]Please note that the inconsistency I would like to point out is not in
the approximation itself but in the implication of this approximation,
i.e. the rigorous unification of electroweak interaction, the way I
found most of people have accepted the Standard Model; We cannot claim
the unification in the rigorous manner when its scientific method
turns on an approximation implying a potential contradiction against
the fundamental assumption.[/quote]

The ultrarelativistic approximation is NOT part of the Model.
It is an approximation that is sometimes applied to the model to make calculations simpler.


Last edited by Warren Siegel on Wed Apr 25, 2007 8:38 pm; edited 1 time in total
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jcyoon



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Posts: 213

PostPosted: Sat Oct 21, 2006 2:16 pm    Post subject: RE: Lorentz Violation of the Standard Model Reply with quote

Thu 2006-05-18 12:32 AM

Dear Professor Warren Siegel,

Thanks for clarifying your perspective.
Please let me be more specific with the details.

[quote="On Wednesday, May 17, 2006 4:03 AM, Warren Siegel"] No, the Standard Model is completely Lorentz covariant.

The ultrarelativistic approximation is NOT part of the Model.
It is an approximation that is sometimes applied to the model to make
calculations simpler.[/quote]


The Standard Model Lagrangian after the Higgs mechanism
can be given by either

L_{SM1} = i \overline{E}_{L} \gamma^{\mu} \partial_{\mu} E_{L}
+ i \overline{E}_{R} \gamma^{\mu} \partial_{\mu} E_{R}
- m ( \overline{E}_{L} E_{R} + \overline{E}_{R} E_{L}
+ ...

or

L_{SM2} = i \overline{E} \gamma^{\mu} (1 - \gamma^{5} )
\partial_{\mu} E
+ i \overline{E} \gamma^{\mu} (1 + \gamma^{5} )
\partial_{\mu} E
- m ( \overline{E}_{L} E_{R} + \overline{E}_{R} E_{L}
+ ...

But, both L_SM1 and L_SM2 fail to derive the massive Dirac
equations while the Lagrangian for the massive Dirac
equations is

L = i \overline{E} \gamma^{\mu} \partial_{\mu} E + m E E

Without the ultrarelativistic approximation, L_{SM1} is not L
since there is no two independent particles to be described
by E_L and E_R separately.

It seems like that you would prefer to understand the
Standard Model in virtue of L_{SM2} with E without the
ultrarelativistic approximation. However, L_{SM2} is not
L and the mass term vanishes when described by E.

The only consistent way to describe the Standard
Model is to use L_SM1 with the ultrarelativistic approximation.
Therefore, the ultrarelativistic approximation is inevitable
for the Standard Model and thus it is not completely but
approximately Lorentz covariant.

[quote="Warren Siegel"] I don't know what you mean by a measurement being noninvariant.[/quote]

Basically, the asymmetry of SLAC E158 is
A_PV ~ (|M_L|^2 - |M_R|^2) / (|M_L|^2 + |M_R|^2)
where M_L is the matrix element of incoming electron with
left-handed helicity. For example, in one frame we have 2
incoming electrons with left-handed helicity and no right-handed
one and thus A_PV = 1. But, in the other frame boosted ahead
of both incoming electrons, we have two right-handed ones and
A_PV = -1.

Thanks,
J.C. Yoon
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Warren Siegel



Joined: 28 Sep 2006
Posts: 14

PostPosted: Sat Oct 21, 2006 2:18 pm    Post subject: Re: Lorentz Violation of the Standard Model Reply with quote

Thu 2006-05-18 12:40 AM

[quote="On May 17, 2006, at 11:31 AM, J.C. Yoon"] The Standard Model Lagrangian after the Higgs mechanism can be given
by either

L_{SM1} = i \overline{E}_{L} \gamma^{\mu} \partial_{\mu} E_{L}
+ i \overline{E}_{R} \gamma^{\mu} \partial_{\mu} E_{R}
- m ( \overline{E}_{L} E_{R} + \overline{E}_{R} E_{L}
+ ...

or

L_{SM2} = i \overline{E} \gamma^{\mu} (1 - \gamma^{5} )
\partial_{\mu} E
+ i \overline{E} \gamma^{\mu} (1 + \gamma^{5} )
\partial_{\mu} E
- m ( \overline{E}_{L} E_{R} + \overline{E}_{R} E_{L}
+ ...

But, both L_SM1 and L_SM2 fail to derive the massive Dirac equations
while the Lagrangian for the massive Dirac equations is

L = i \overline{E} \gamma^{\mu} \partial_{\mu} E + m E E[/quote]

All 3 of these Lagrangians are identical.


Last edited by Warren Siegel on Wed Apr 25, 2007 8:39 pm; edited 1 time in total
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jcyoon



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Posts: 213

PostPosted: Sat Oct 21, 2006 2:19 pm    Post subject: RE: Lorentz Violation of the Standard Model Reply with quote

Thu 2006-05-18 1:24 AM

Dear Professor Warren Siegel,

Thanks for you prompt reply, but
I would like to point out there are different.

m E E = m ( E_L + E_R ) ( E_L + E_R)
= m ( E_L E_L + E_R E_R + E_L E_R + E_R E_L)

It is not E_L E_R + E_R E_L and the term like E_L E_L might be assumed to be zero, but no good reason to be zero.

Thanks,
J.C. Yoon
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Warren Siegel



Joined: 28 Sep 2006
Posts: 14

PostPosted: Sat Oct 21, 2006 2:21 pm    Post subject: Re: Lorentz Violation of the Standard Model Reply with quote

Thu 2006-05-18 1:38 AM

[quote="On May 17, 2006, at 12:23 PM, J.C. Yoon"]m E E = m ( E_L + E_R ) ( E_L + E_R)
= m ( E_L E_L + E_R E_R + E_L E_R + E_R E_L)[/quote]

E is a 4-component spinor, E_L & E_R each 2-component.
In an appropriate basis, in matrix notation,

E = ( E_L while E-bar = ( E-bar_R , E-bar_L )
E_R )

E-bar is defined as the hermitian conjugate of E times gamma_0.
All the gamma matrices in this basis are off diagonal:

gamma = ( 0 , sigma
sigma-bar , 0 )

except for

gamma_5 = ( I , 0
0 , -I )

Thus, the gamma-derivative term connects LL & RR, while the mass term connects LR & RL.

See, e.g.,

http://en.wikipedia.org/wiki/Dirac_matrices#Weyl_basis


Last edited by Warren Siegel on Wed Apr 25, 2007 8:41 pm; edited 1 time in total
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jcyoon



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Posts: 213

PostPosted: Sat Oct 21, 2006 2:22 pm    Post subject: RE: Lorentz Violation of the Standard Model Reply with quote

Thu 2006-05-18 3:45 AM

Dear Professor Warren Siegel,

Thanks for your kind comment.
But we have to clarify the notation once again.

According to the notation I set before,
E_L & E_R each 4-component with zero part and e_L & e_R each 2-component.

[quote="On Sat 2006-05-13 5:36 AM
, J.C. Yoon"] Let us clarify our notation first. We have E = [ e_L ]
[ e_R ]
where both e_L, e_R should be nonzero to satisfy the massive Dirac
equations.

And for further reference in my argument, we can introduce E_L = [ e_L
]
[ 0 ]
which expresses the SM Lagrangian.[/quote]

Using my notation e_L & e_R as 2-component, your argument is that the Lagrangian for the massive Dirac equations

L = i \overline{E} \gamma^{\mu} \partial_{\mu} E + m E E

can be expressed as

L_{SMprime} = i \overline{e}_{R} {\partial_{\mu}_{0} - \sigma \nabla e_{L}
+ i \overline{e}_{L} {\partial_{\mu}_{0} + \sigma \nabla e_{L}
- m ( \overline{e}_{L} e_{R} + \overline{e}_{R} e_{L}
+ ...

not L_{SM1} or L_{SM2}. Therefore, L and L_{SMprime} are not identical with L_{SM1} or L_{SM2} expressed by 4-component E_L & E_R.

And there is a critical distinction between L and L_{SMprime} regarding the number of variables representing a field; we have to use L, not L_{SMprime}, for deriving the equation of motion for a fermion represented by E, not both of e_L and e_R. The equations of motion for a part of field e_L is meaningless.
Therefore, E is the right variable to represent a particle in the derivation of the equations of motion.

Before the Spontaneous Symmetry Breaking, we can express the massless electron with 2-component without this problem, but after SSB the Standard Model theory should have addressed this possible problem.

Experimentally, its momentum, energy and spin for free massive particle wave solutions are the same and the chirality differentiating these two is only measured by the approximation of the helicity.

Therefore, there is no theoretical and experimental reason to discard E and claim two independent variable e_{L,R} for the representation of one particle.

To avoid these problems, the ultrarelativistic approximation is inevitable.

Thanks,
J.C. Yoon
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Warren Siegel



Joined: 28 Sep 2006
Posts: 14

PostPosted: Sat Oct 21, 2006 2:24 pm    Post subject: Re: Lorentz Violation of the Standard Model Reply with quote

Thu 2006-05-18 3:56 AM

[quote="On May 17, 2006, at 2:45 PM, J.C. Yoon"] According to the notation I set before, E_L & E_R each 4-component
with zero part and e_L & e_R each 2-component.[/quote]

That makes no difference.
In either case the mass term for the Dirac Lagrangian involves only LR crossterms.
That is the only mass term that is both Lorentz & U(1) invariant.

[quote="J.C. Yoon"] And there is a critical distinction between L and L_{SMprime}
regarding the number of variables representing a field; we have to use
L, not L_{SMprime}, for deriving the equation of motion for a fermion
represented by E, not both of e_L and e_R. The equations of motion for
a part of field e_L is meaningless.[/quote]

E has 4 components, so varying it gives 4 component field equations.
2x2=4, so it doesn't matter whether you look @ 1 4-component equation, or 2 2-component equations, or 4 1-component equations.


Last edited by Warren Siegel on Wed Apr 25, 2007 8:42 pm; edited 1 time in total
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jcyoon



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Posts: 213

PostPosted: Sat Oct 21, 2006 2:26 pm    Post subject: RE: Lorentz Violation of the Standard Model Reply with quote

Thu 2006-05-18 6:29 AM

Dear Professor Warren Siegel,

I would like to appreciate your patience and effort.

I was wrong about the Lagrangians stating they all are different. At least, I should have mentioned that their mathematical representations are the same.

And it left us very elusive aspects to argue.

[quote="On Wednesday, May 17, 2006 11:56 AM, Warren Siegel"] E has 4 components, so varying it gives 4 component field equations.
2x2=4, so it doesn't matter whether you look @ 1 4-component equation,
or 2 2-component equations, or 4 1-component equations.
[/quote]

If we consider the whole set of equation to interpret one
particle, then it would not matter. But, just considering a
subset of equations makes things different, especially
when they are coupled due to the mass term; e_L and
e_R is not independent any more and therefore it is not
rigorously correct to derive the equations of motions
treating them as independent variables.

When we say a theory is Lorentz covariant, we should
verify the Lorentz invariance of field itself, as well as that
of Lagrangian. E_L 4-component for massive particle is
not exact solution to the massive Dirac equations.

In order to have such a Lagrangian, the only way to express is

L = i \overline{E} \gamma^{\mu} \partial_{\mu} E + m E E

It may be seemingly identical to the following Lagrangian

L_{SM3} = i \overline{E} \gamma^{\mu} ( 1 - \gamma5) \partial_{\mu} E
+ i \overline{E} \gamma^{\mu} ( 1 + \gamma5) \partial_{\mu} E
+ m EE

But, splitting the kinetic part into (1 +- \gamma5) as in L_{SM3}
is physically meaningless, since the practical equation is solved
without these projections and there is no corresponding
measurement of free particle to these separate terms.

For example, we do not impose any significant interpretation
to the following two terms separately
1/2 (1 - 2) f(x) + 1/2 ( 1 + 2 ) f(x) = 0
and we just solve for f(x) = 0.

And one of two term is neglected with an approximation; Just
looking at the first term and saying we are picking up one part
is ignoring the second term in the whole equations.

Also, the Standard Model has been described by
E_L = 1/2 (1 - \gamma_5) E
as a 4-component and in its experimental tests, E_L is
approximately identified as an electron with left-handed helicity.

Thanks,
J.C. Yoon
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Warren Siegel



Joined: 28 Sep 2006
Posts: 14

PostPosted: Sat Oct 21, 2006 2:28 pm    Post subject: Re: Lorentz Violation of the Standard Model Reply with quote

Thu 2006-05-18 8:15 PM

As I said before, neglecting the mass is just an approximation.
It is verified by treating the mass as a perturbation, & checking the corrections are small.
So we look @ an amplitude: E.g., for elastic scattering of electrons that are initially polarized, & whose final polarizations are
measured: A(s,t,h_i,m), where s & t are the momentum invariants, h_i are the 4 helicities, as measured with respect to the laboratory frame, & m is the electron mass. Then in the limit of high energies,
|s|>>m^2, |t|>>m^2, we find

A(s,t,h_i,m) = A(s,t,h_i,0) + O(m/s)

where A(s,t,h_i,0) is the corresponding massless amplitude (with the helicities h_i now invariants), & O(m/s) is a term smaller, by order m/s (or m/t) than the largest of the amplitudes A(s,t,h_i,0) (some of which vanish for certain helicities).
These amplitudes, @ lowest order in (electromagnetic) coupling, can be found in the literature explicitly & exactly for both the massive & massless cases, and one can easily see that this approximation works.
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jcyoon



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Posts: 213

PostPosted: Sat Oct 21, 2006 2:29 pm    Post subject: RE: Lorentz Violation of the Standard Model Reply with quote

Thu 2006-05-18 11:54 PM

Dear Professor Warren Siegel,

Thanks for your comments and reply.
I am very grateful for your effort and time in this matter, which have been a great opportunity for me to learn.

[quote="On Thursday, May 18, 2006 4:15 AM, Warren Siegel"] As I said before, neglecting the mass is just an approximation.[/quote]

I agree with you that an approximation itself does not bear any problem.

But, my point reaches out to whether we can conclusively claim the symmetry violation based on this approximation; at least, we have to postpone our final conclusion on the symmetry violation as long as it is based on the approximation.

For example, parity violation in SLAC E158 implies that an Lorentz-invariant amplitude is not invariant under parity. But, this observation itself is based on the approximation making the amplitude Lorentz-variant:
A(s,t,++,m) can be observed as A(s,t,-+,m) in other frame.

We may see this as a good approximation in the lab frame, but it is insufficient to claim the violation of fundamental symmetry which requires the measurement of exact Lorentz-invariant amplitude.

Thanks,
J.C. Yoon
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Warren Siegel



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PostPosted: Sat Oct 21, 2006 2:31 pm    Post subject: Re: Lorentz Violation of the Standard Model Reply with quote

Fri 2006-05-19 12:03 AM

[quote="On May 18, 2006, at 10:53 AM, J.C. Yoon"] For example, parity violation in SLAC E158 implies that an
Lorentz-invariant amplitude is not invariant under parity. But, this
observation itself is based on the approximation making the amplitude
Lorentz-variant:
A(s,t,++,m) can be observed as A(s,t,-+,m) in other frame.

We may see this as a good approximation in the lab frame, but it is
insufficient to claim the violation of fundamental symmetry which
requires the measurement of exact Lorentz-invariant amplitude.[/quote]

Parity violation is a frame-independent statement.
Parity is independent of Lorentz invariance.


Last edited by Warren Siegel on Wed Apr 25, 2007 8:43 pm; edited 1 time in total
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jcyoon



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PostPosted: Sat Oct 21, 2006 2:32 pm    Post subject: RE: Lorentz Violation of the Standard Model Reply with quote

Fri 2006-05-19 1:57 AM

Dear Professor Warren Siegel,

I am afraid that my argument lacks elaboration.
For more information, you may refer to my preprint "The Origin of CP Violation", if you haven't already
http://uk.arxiv.org/abs/hep-ph/0211005

[quote="On Thursday, May 18, 2006 8:03 AM, Warren Siegel"] Parity violation is a frame-independent statement.
Parity is independent of Lorentz invariance.
[/quote]

I agree that pairty violation is a frame-independent statement.
But, parity of massive fermion implies Lorentz invariance.

Parity was introduced with the assumption of massless fermion, and this assumption implicitly remains through when this concept was extended to massive fermion. However, parity of massive fermion is distinctively different from that of massless one.

For example, an electron with left-handed helicity should have the same lifetime as an electron with right-handed helicity.
Since under Lorentz transformation the left-handed helicity electron can be observed as a right-handed helicity one, parity of massive particle implies Lorentz invariance.

Thanks,
J.C. Yoon
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Warren Siegel



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PostPosted: Sat Oct 21, 2006 2:33 pm    Post subject: Re: Lorentz Violation of the Standard Model Reply with quote

Fri 2006-05-19 2:04 AM

[quote="On May 18, 2006, at 12:56 PM, J.C. Yoon"] I agree that pairty violation is a frame-independent statement.
But, parity of massive fermion implies Lorentz invariance.[/quote]

On the contrary, nonrelativistic fermions (e.g., Schroedinger equation for spin 1/2, in external electromagnetic field) are quite parity invariant, but not Lorentz covariant.


Last edited by Warren Siegel on Wed Apr 25, 2007 8:45 pm; edited 1 time in total
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jcyoon



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PostPosted: Sat Oct 21, 2006 2:35 pm    Post subject: RE: Lorentz Violation of the Standard Model Reply with quote

Fri 2006-05-19 2:22 AM

Dear Professor Warren Siegel,

I see your point, but my parity argument lies within the framework assuming free particle excluding external fields.

In my opinion, your argument in a specific case of norelativisic fermions under external field does not contradict mine, though my statement fails in its accuracy.

Thanks,
J.C. Yoon
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