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Weyl equation

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Weyl equation

In physics, particularly quantum field theory, the Weyl Equation is a relativistic wave equation for describing massless spin-1/2 particles. It is named after the German physicist Hermann Weyl.

Contents

Equation

The general equation can be written:

σ μ ∂ μ ψ = 0

explicitly in SI units:

I 2 1 c ∂ ψ ∂ t + σ x ∂ ψ ∂ x + σ y ∂ ψ ∂ y + σ z ∂ ψ ∂ z = 0

where

σ μ = ( σ 0 , σ 1 , σ 2 , σ 3 ) = ( I 2 , σ x , σ y , σ z )

is a vector whose components are the 2 × 2 identity matrix for μ = 0 and the Pauli matrices for μ = 1,2,3, and ψ is the wavefunction - one of the Weyl spinors.

Weyl spinors

The elements ψL and ψR are respectively the left and right handed Weyl spinors, each with two components. Both have the form

ψ = ( ψ 1 ψ 2 ) = χ e − i ( k ⋅ r − ω t ) = χ e − i ( p ⋅ r − E t ) / ℏ

where

χ = ( χ 1 χ 2 )

is a constant two-component spinor.

Since the particles are massless, i.e. m = 0, the magnitude of momentum p relates directly to the wave-vector k by the De Broglie relations as:

| p | = ℏ | k | = ℏ ω / c → | k | = ω / c

The equation can be written in terms of left and right handed spinors as:

σ μ ∂ μ ψ R = 0 σ ¯ μ ∂ μ ψ L = 0

where σ ¯ μ = ( I 2 , − σ x , − σ y , − σ z ) .

Helicity

The left and right components correspond to the helicity λ of the particles, the projection of angular momentum operator J onto the linear momentum p:

p ⋅ J | p , λ ⟩ = λ | p | | p , λ ⟩

Here λ = ± 1 / 2 .

Derivation

The equations are obtained from the Lagrangian densities

L = i ψ R † σ μ ∂ μ ψ R L = i ψ L † σ ¯ μ ∂ μ ψ L

By treating the spinor and its conjugate (denoted by † ) as independent variables, the relevant Weyl equation is obtained.

References

Weyl equation Wikipedia
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