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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 very important in physics, indeed the Standard Model is
a chiral theory since it differentiates 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 more 'fundamental' than 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 is 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 antipositron.
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.