Helicity is not well-defined in the rest frame and it
depends on the reference frame. Helicity is determined by the momentum
and spin
orientation of a particle. We should be weary of two critical and subtle
points. First, there is no momentum when the particle is in the rest frame and
thus its helicity is not well-defined. This point is consistent with the
accurate solution of the Dirac equations for massive fermion, where no
left-handed or right-handed helicity is available. Second, the helicity of
massive fermion changes under the Lorentz transforamtions as its signs of
momentum reversed. These problems do not arise in massless particle as there is
no rest frame when massless. We may find a historical reason why these points
have been neglected from the fact that the concept of helicity has been
developed from the study of nuetrino, assumed massless then.
Because it limits the truth the approximation
indicates!. One may dismiss any
violation of principle in approximation, saying that the approximation is just
an
approximation. However there is a good reason for this and it concerns the
criteria of scientific logic. Let us say there is a person A walking on the
street and I observe that person A closely resembles person B I know. Without
any further assumptions or known facts, this approximate observation can tell either
A is B or A is similar to B. On the contrary, if I am with person B right now,
this approximation is limited to indicate that person A is a look-alike of
person B, not A is B. In other words, if we have a fact or assumption that
contradict any possible logical consequence of approximation observation, this
would limit its approximation for reveal what is true. In E158, we
approximately measure the helicity of electron or muon in the laboratory frame
and rely on the Standard Model for its calculations. However, we should be
cautious to conclude parity, a fundamental property of nature, is violated and
rather investigate any lack of rigor from this approximation of helicity. For
example, we can observe the Left-Right asymmetry in E158 as a Right-Right asymmetry
in any frame and thus the Left-Right asymmetry is not a Lorentz-invariant
observable.
No, the historical background to conclude parity
violation is not free of massless neutrino assumption. There are
different types of parity violation(atomic beta decay, neutral meson, neutrino:
note the first two are composite) and they are only assumed to be universal. The
conclusion
of parity violation is based on the assumption of massless neutrino,
which is quite essential since it is only evidence directly related to
point-like and fundamental particle. How come we keep insisting that the
logical consequence of massless neutrino still holds without any further
investigation when we find it massive?
Parity violation is Lorentz
violation for massive fermon. Consider a massive fermion with left-handed helicity.
Under parity, the massive fermion is supposed to be transformed into the
right-handed helicity. However, we can find the exactly same transformation
under the Lorentz transformations; the left-handed massive fermion can be
observed as a right-handed helicity one as its momentum becomes reversed in the
boosted frame. Therefore, what parity operator does to a massive fermion is the
same as Lorentz transformationa and the parity violation means Lorentz
violation for massive fermion.
Whether electromagnetic(EM) and weak interactions is
unified or not. The chirality of fermion can only be an observable
property when the fermion is massless as it becomes idential to helicity. For a
massive fermion, there is always two chirality to represent the fermion state
according to the Dirac equations. In other word, no massive particle can be
designated as, for example, a left-handed chirality fermion.
No. Accurately speaking, it is not
exactly free of massless fermion assumption as long as the fermion with
a massive term described with the property of massless fermion. In the Standard
Model, we start with massless fermion that can be labeled as left- and
right-handed chirality and helicity as they are the same for massless fermion.
After the Higgs mechanism of spontaneous symmetry breaking, the massless
fermion obtains a mass term. However, it treats these massivee fermion labeling
as left-handed or right-handed as if there were massless, while there is no
massive particle with only one chirality. There are two justification arguments
in general. First, one may claim now this left- and right-handedness is either
of helicity or of chirality which can be approximately the same in the high
energy limit or the massless limit. This means that the Standard Model is only
approximately true, but since this approximation violates Lorentz invariance,
the implication of this approximation reflecting a rigorous truth of
fundamental property of nature should be excluded and we must conclude that the
Standard Model is seemingly true, but not true. Second, one may say the
calculation would be the same whether massive or not as the chiral structure of
weak interaction picks up only one of two Weyl spinors. This statement is
inconsistent with the whole idea of the Standard Model of unification of
electroweak interactions, since the concept of chiral structure indicates that
the weak interactions structure is fundamentally different from electromagnetic
interactions.
No, it is not, exactly. Since the
Standard Model interaction is Lorentz-scalar, one may prove its Lorentz
invariance with an assumption of massless fermion, i.e. as long as we describe
a massive fermion with only one chirality. If then, this fermion with one
chirality would remain the same regardless of its reference frame. But what it
matters is whether it is Lorentz-scalar of a particle satisfying the Dirac
equations. And there is no such massive fermion according to the Dirac
equations; any massive fermion should have two nonzero Weyl spinors. One may
say one of the Weyl spinor is suppressed in the boosted frame. Yes, it is, but
it is only small, not zero. This is only approximation which violates Lorentz
violation. Then, this proof is inconsistent, since we have proved that a theory
is Lorentz invariance with an assumption of Lorentz vioation.
Because it does not have any specifics of physical
time and space. A physicsal event specifies its physical time and space. If an event does
not carry any concept of physics time and space that we can observed and
measure, it is not a physical event, but a hypothetical one.
No, though it is quite common to confuse with helicity and chirality. I believe what I have discussed in my article is more
subtle than people think and that is why my explanation deals with explicit
details.
Because the
actual calculation of lifetime is independent of the signs of coupling
constants as it takes the absolute value of interacting term. The lifetime
calculation is based on the assumption of local interactions. This assumption
treats particles as free separating its interactions as we solve the free Dirac
equation, rather than the interacting ones. Therefore, the Dirac field we use
for the lifetime calculation do not carry any information about the signs of
its coupling constants.
Yes and that is why we can say that the SM denies the
fundamentaliy of Dirac equaitons. My article is clearly based on the fundamentality
of Dirac equations while the Standard Model implies that the Dirac
equations are not completely fundamental equations to determine the properties
of fermions, but it requires some other explanation of Higgs field. If that
is where the SM stands at, then it should Cleary state that "the Dirac
equation is not fundamental, but the Higgs mechanism is more fundamental
one", but no where in the SM makes such a clear statement. Also,
our practical understanding of fermion is mainly based on the fundamentality of
the Dirac equations. At least, the Dirac equation is still commonly accepted as
fundamental in most references we can find including textbooks. If we would like
to deny the Dirac equations, we need a more cautious approach then the way the Higgs
mechanism has been upheld so far. After all, the Higgs mechanism has not been
verified yet and is still hypothetical as we do not have the specific space and
time information about the occurrence of symmetry breaking. That is why I think
it is a weak argument to deny that the Dirac equations is not fundamental and
at least it is worthy of our rigorous investigation.
Yes, an interacting field could be different from a free
field. However, in practice, we do NOT solve the interacting Dirac
equations, but rely on the free
Dirac equations assuming the locality of interactions, i.e. asymptotic completeness. Therefore, the
inconsistencies arise from the basic assumption of QFT and we cannot dismiss my
argument as inconsistent as long
as QFT is established on the same view. As a matter
of fact, this conventional approach of quantum field theory brings the
interacting field back to the free
field, which is at least practical way to evaluate interactions. This is
exactly where the fundamentality of symmetry fails and is limited by our
practical calculation.
Yes, quantum fields are quantum fields, but
the rigor of mathematics and any failure should be explicitly clarified. What
lies in this question is that quantum fields do not need to obey mathematical representation
rigorously. In my paper I take a point of view of investigating with the rigor
of mathematical reasoning. It could be reasonable to have flexible physical
interpretations, but it is not a good reason to claim its rigorousness or
dismiss a rigorous argument.
Last revised: 01/25/2006