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Helicity and Chirality
----------------------
Helicity corresponds to 'handedness' but is dependent
on the reference frame. For a massive particle it is
possible to get ahead of the particle and look back.
If one does, the handedness of the particle will be
reversed. Therefore, the notion of helicity is not
intrinsic to the particle.
Chirality, on the other hand, is intrinsic to the
particle and is independent of reference frames.
Particles with different chirality are truly different.
In the case of a massless particle that is moving at
the speed of light, the chirality and helicity are
the same - it is not possible to ever get ahead of
such a particle and look back.
While chirality and helicity are not the same, they
become approximately equivalent at high energies.
Chirality is extremely important in physics, indeed
the Standard Model is a chiral theory since it
distinguishes between left and right-handed particles.
The spin angular momentum of a particle can be
oriented in any direction. However, what if we want
to to find the component of spin in the direction of
motion. This leads us to the definition of helicity
as the projection of the spin onto the direction of
momentum. Therefore,
h = S.p where
----> right-handed helicity
<---- left-handed helicity
--------> p
This quantity is conserved.
Helicity can be visualized as follows:
The gray arrow represents the direction of motion.
The red and blue arrows represent the handedness
and NOT the actual spin of the particle which is an
intrinsic property unrelated to classical momentum.
Right handed particles follow the right hand rule.
If your thumb points in the direction of the gray
arrow, then your fingers wrap in the direction of
the red arrow.
Conversely, a particle is 'left-handed' because it
follows the left hand rule. If your thumb points
in the direction of the gray arrow, then your
fingers wrap in the direction of the blue arrow.
Weyl Spinors
------------
We can write the 4-component Dirac spinor in terms
of two 2-component WEYL SPINORS.
- - - -
ψ = | φ | ≡ | ψ_{L} |
| χ | | ψ_{R} |
- - - -
The reason it makes sense to decompose a Dirac
spinor this way is that, ψ_{L} and ψ_{R} transform
independently under Lorentz transformations (they
do not mix). The technical way to express this
is to say that a Dirac spinor forms a reducible
representation of the Lorentz group, whereas Weyl
spinors form irreducible representations. This
implies that, from a purely mathematical point of
view, Weyl spinors may be considered to be more
'fundamental' than the Dirac spinors.
Before we go any further we need to introduce the
5th GAMMA MATRIX. This is defined as:
^{ } - -
^{ } | 0 0 1 0 |
γ^{5} = iγ^{0}γ^{1}γ^{2}γ^{3} = | 0 0 0 1 |
^{ } | 1 0 0 0 |
^{ } | 0 1 0 0 |
^{ } - -
Where,
- - - -
γ^{i} = | 0 σ^{i} | and γ^{0} = | I_{2} 0 |
_{ }| -σ^{i} 0 | _{ } | 0 -I_{2} |
- - - -
A Dirac field, ψ, can be projected onto its left-
chiral and right-chiral components by:
ψ_{L} = (1/2)(I_{4} - γ^{5})ψ and ψ_{R} = (1/2)(I_{4} + γ^{5})ψ
There is another representation of the Gamma
matrices called the WEYL REPRESENTATION. In the
Weyl representation, the γ^{i} matrices are unchanged
but the γ^{0} matrix is different.
- - - -
γ^{i} = | 0 σ^{i} | and γ^{0} = | 0 I_{2} |
_{ }| -σ^{i} 0 | _{ } | I_{2} 0 |
- - - -
γ^{5} becomes:
^{ } - -
^{ } | -1 0 0 0 |
γ^{5} = iγ^{0}γ^{1}γ^{2}γ^{3} = | 0 -1 0 0 |
^{ } | 0 0 1 0 |
^{ } | 0 0 0 1 |
^{ } - -
and the chiral projections take a simple form:
_{ } - - _{ } - _{ } -
ψ_{L} = | I_{2} 0 |ψ and ψ_{R} = | 0 0_{ } |ψ
_{ } | 0 _{ }0 |_{ } | 0 I_{2} |
_{ } - - _{ } - _{ } -
Or,
- - -_{ } - - -
| I_{2} 0 || ψ_{L} | = | ψ_{L} |
| 0 _{ }0 || ψ_{R} | | 0 _{ }|
- - -_{ } - - -
and,
- - -_{ } - - -
| 0 0_{ } || ψ_{L} | = | 0_{ } |
| 0 I_{2} || ψ_{R} | | ψ_{R} |
- - -_{ } - - -
In the chiral representation, the eigenstates of
γ^{5} are:
- - - - _{ } - -
| -I_{2} 0 || ψ_{L} | = ±1| ψ_{L} |
| 0 I_{2} || ψ_{R} | | ψ_{R} |
- - - - _{ } - -
By a slight abuse of language, we will say ψ_{L} has
a chirality of -1 (left-chiral), whereas ψ_{R} has
a chirality of +1 (right-chiral). This is an abuse
of language because it is actually the 4-component
spinor with two components set to 0 that has a
definite chirality, not the 2-component spinors
themselves.
The spinors transform in the same way under
rotations, but oppositely under boosts.
The Weyl Equations
------------------
From the discussion of the The Dirac equation,
the Dirac spinor, ψ, can be written as a linear
combination of left-handed and right-handed
spinors. Therefore,
ψ = ψ_{L} + ψ_{R} and ψ^{†} = ψ_{L}^{†} + ψ_{R}^{†}
Note: In Group Theory language this means that
the Dirac spinor representation of the Lorentz
group is reducible. It decomposes into two
irreducible representations, acting only on
2-component Weyl spinors. ψ_{R} is in the (1/2,0)
representation of the Lorentz group while ψ_{L} is
in the (0,1/2) representation. The Dirac spinor
lies in the (1/2,0) ⊕ (0,1/2) representation.
_
We can use the relationship ψ = ψ^{†}γ^{0}, we get:
_
ψψ = ψ^{†}γ^{0}ψ
- ^{ } - - - - -
= | ψ^{†}_{L} ψ^{†}_{R} || 0 1 || ψ_{L} |
- ^{ } - | 1 0 || ψ_{R} |
^{ } - - - -
= ψ^{†}_{L}ψ_{R} + ψ^{†}_{R}ψ_{L}
In the Weyl representaion we can use the more
compact form:
- -
^{ } | 0 ^{ }σ^{μ} |
γ^{μ} = | _ ^{ } |
^{ } | σ^{μ} 0^{ } |
- -
^{ } _
Where by analogy with γ^{0} and γ^{μ}, σ^{μ} and σ^{μ} are
defined as:
σ^{μ} = (σ^{0},σ^{x},σ^{y},σ^{z})
σ^{0} is sometimes considered as the Identity
matrix. Thus,
σ^{μ} = (I,σ^{x},σ^{y},σ^{z})
^{ } = (1,σ^{i})
and,
_
σ^{μ} = (1,-σ^{i})
_
Note: The here has nothing to do with the Dirac
adjoint. It is purely convention.
We can now write the Dirac Lagrangian as:
_{ } _
L = iψ_{L}^{†}σ^{μ}∂_{μ}ψ_{L} + iψ_{R}^{†}σ^{μ}∂_{μ}ψ_{R} - m(ψ_{L}^{†}ψ_{R} + ψ_{R}^{†}ψ_{L})
For a massless particle the equations of motion
derived from the Euler-Lagrange equations are:
^{ }_
iσ^{μ}∂_{μ}ψ_{L} + iσ^{μ}∂_{μ}ψ_{R} = 0
Therefore,
iσ^{μ}∂_{μ}ψ_{L} = 0
and,
_
iσ^{μ}∂_{μ}ψ_{R} = 0
These are the WEYL EQUATIONS for a massless spin
1/2 particle where ψ_{L} and ψ_{R} are again the Weyl
Spinors.
It is important to note that massive fermions like
electrons are not Weyl spinors, they are Dirac
spinors. The subtlety is that the Dirac spinors
consist of 2 massless Weyl spinors that are mixed
by the mass term. For example, an 'electron'
propagating in space and interacting with a mass
due to the Higgs field connects a left-chiral
electron with a right-chiral anti-positron. This
means that the two types of particles are exhibiting
quantum mixing. The 'physical electron' which is
propagating through space is a combination of e_{L}
and e_{R} particles that swap back and forth. At a
particular point in time the particle may be e_{L},
but if you observe it a moment later, the very same
particle might manifest itself as e_{R}.
For a more details about this, please see the
discussion regarding the Higgs Field.