The Weyl equation of particle physics , named after Hermann Weyl , is the Dirac equation for massless particles with spin 1/2. It is used in describing the weak interaction . Correspondingly, fermions that satisfy this equation are called Weyl fermions .
Derivation
The representation of the Lorentz group on Dirac spinors is reducible . In a suitable representation of the Dirac matrices , the Weyl representation , the first two and the last two components of the 4 spinors transform separately, which is why they are also referred to as bispinors :
Ψ
=
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Ψ
L.
Ψ
R.
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{\ displaystyle \ Psi = {\ begin {pmatrix} \ Psi _ {L} \\\ Psi _ {R} \ end {pmatrix}}}
The 2 spinors and are the left and right handed Weyl spinors . They are the eigenstates of the chirality operator when it is written in the Weyl representation.
Ψ
L.
{\ displaystyle \ Psi _ {L}}
Ψ
R.
{\ displaystyle \ Psi _ {R}}
γ
5
{\ displaystyle \ gamma ^ {5}}
γ
5
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Ψ
L.
0
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=
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Ψ
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γ
5
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{\ displaystyle {\ begin {aligned} \ gamma ^ {5} {\ begin {pmatrix} \ Psi _ {L} \\ 0 \ end {pmatrix}} & = - {\ begin {pmatrix} \ Psi _ {L } \\ 0 \ end {pmatrix}} \\\ gamma ^ {5} {\ begin {pmatrix} 0 \\\ Psi _ {R} \ end {pmatrix}} & = {\ begin {pmatrix} 0 \\ \ Psi _ {R} \ end {pmatrix}} \ end {aligned}}}
.
In the Dirac equation for a free spin 1/2 particle they are coupled by the mass :
m
{\ displaystyle m}
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i
γ
μ
∂
μ
-
m
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-
m
i
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∂
μ
i
σ
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μ
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m
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Ψ
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{\ displaystyle \ left (\ mathrm {i} \ gamma ^ {\ mu} \ partial _ {\ mu} -m \ right) \ Psi = {\ begin {pmatrix} -m & \ mathrm {i} {\ bar { \ sigma}} ^ {\ mu} \ partial _ {\ mu} \\\ mathrm {i} \ sigma ^ {\ mu} \ partial _ {\ mu} & - m \ end {pmatrix}} {\ begin { pmatrix} \ Psi _ {L} \\\ Psi _ {R} \ end {pmatrix}} = 0}
Here is and , where the three are Pauli matrices and the two-dimensional identity matrix .
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μ
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{\ displaystyle \ sigma ^ {\ mu} = {\ begin {pmatrix} \ sigma ^ {0} & {\ vec {\ sigma}} \ end {pmatrix}}}
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{\ displaystyle {\ bar {\ sigma}} ^ {\ mu} = {\ begin {pmatrix} \ sigma ^ {0} & - {\ vec {\ sigma}} \ end {pmatrix}}}
σ
→
{\ displaystyle {\ vec {\ sigma}}}
σ
0
{\ displaystyle \ sigma ^ {0}}
If the mass ( ) disappears , the four-dimensional Dirac equation decouples into two two-dimensional equations for the left- and right-handed spinor:
m
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{\ displaystyle m = 0}
i
σ
μ
∂
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Ψ
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{\ displaystyle {\ begin {aligned} \ mathrm {i} \ sigma ^ {\ mu} \ partial _ {\ mu} \ Psi _ {L} & = 0 \\\ mathrm {i} {\ bar {\ sigma }} ^ {\ mu} \ partial _ {\ mu} \ Psi _ {R} & = 0 \ end {aligned}}}
Chiral coupling
To describe the electroweak interaction, it is important that the left- and right-handed spinors can couple differently, but Lorentz covariant , to vector fields ( chiral coupling). The coupling is created by replacing the derivatives with covariant derivatives :
D.
μ
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∂
μ
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T
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{\ displaystyle D _ {\ mu} = \ partial _ {\ mu} - \ mathrm {i} gT ^ {a} W _ {\ mu} ^ {a}}
Designate
The calibration group can be chosen differently for left- and right-handed particles without the Lorenz covariance being impaired.
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