Re: Lorentz violation of the Standard Model
- From: strangerep@xxxxxxxxxxxx
- Date: Wed, 15 Feb 2006 20:18:49 +0000 (UTC)
I wrote:
>> The SM is not saying that the original
>> massless fermion fields are "really" massive.
>> It is saying that fermion mass is an emergent
>> phenomenon, arising from the way the
>> unavoidable non-zero Higgs vev couples the
>> massless left- and right-handed fermion
>> fields together, in such a way that they
>> appear physically to be one massive Dirac
>> fermion field.
J.C.Yoon wrote:
I agree that the SM is not saying that the
massless fermion is really massive. Rather,
it says a massive fermion is indeed massless,
at least for its chiral property
I'm having difficulty extracting meaning from
your above paragraph. But let's continue...
and it behaves as if massive due to the
Higgs mechanism.
In my opinion, the scientific logic of SM
sounds like the following analogy. We measure
1 kg of mass for an object, but a theory
suggests that it is indeed 0.8 kg and the
rest 20% is from some other mechanism, which
we can not measure its time and location of
occurrences. Maybe, this theory is true, but
I find it unbalanced to discard the direct
measurement account.
People only discarded the "direct" approach
because they were forced to do so by experiment.
The history goes something like this:
Fermi's theory of the weak interaction involved a
point interaction which agreed with experiment
well enough at low energy, but not so well at
higher energy. People then suspected this was
because the weak interaction is mediated by
massive vector boson(s), which looks like a point
interaction at low energy. But no one knew how to
construct a _renormalizable_ quantum field theory
of a massive vector boson. Introducing the Higgs
mechanism is a way around that. The SM QFT is
renormalizable.
There was also another headache: it's difficult to
write down a Dirac mass term coupling the left-
and right-handed components of a fermion field
without violating SU(2)xU(1) electroweak gauge
invariance. But generating fermion mass by
inserting the Higgs field in between the left- and
right-handed parts solves this problem.
So unless you can construct a renormalizable QFT
of a massive vector boson which is based on a
Lagrangian containing all the observed fields and
is both Poincare- and gauge-invariant, it's hard
to avoid resorting to mass generation via the
Higgs mechanism.
Besides this elusive argument of SM causes a
real problem in measurement. If the
chiarility of massive fermion is that of
massless one, then the chiarlity of massive
fermion is the helicity of massive fermion.
[...]
Let's review the important point that "Max"
mentioned earlier. (I hope Max is not offended if
I try to re-explain his point in different words...)
The wave function of a free massive fermion (i.e:
obeying the Dirac equation) can be written as the
sum of two 2-component Weyl spinors of opposite
chirality. For these Weyl spinors in isolation,
the notions of chirality and helicity coincide.
However, because the massive 4-component Dirac
spinor solution is a superposition of these Weyl
spinors, neither chirality nor helicity is a
determinate property of such a Dirac spinor. Max
used the phrase: "they are not well-defined",
but I prefer the term "indeterminate". For a
massive Dirac spinor at rest, there's a 50% chance
that a chirality measurement will say
"left-handed" as the answer, and 50% chance that
it will say "right-handed". If we now
Lorentz-boost the Dirac spinor, one of the Weyl
spinors gets a larger factor, but the other's
factor is reduced. This changes the measurement
probabilities such that the likelihoods of each
result are no longer equal. For an extreme Lorentz
boost, there is almost a 100% chance of one
answer, and almost zero chance of the other.
The key point here is that it doesn't make sense
to speak of "the chirality of a massive fermion"
as if chirality were a definite (determinate)
property of the massive fermion. This is because
the Dirac spinor representing the massive fermion
is not an eigenstate of the chirality operator
(gamma5).
.
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