Weyl equation

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

Equation

The general equation can be written: ^[1]^[2]^[3]

\sigma ^{\mu }\partial _{\mu }\psi =0

explicitly in SI units:

I_{2}{\frac {1}{c}}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0

where

\sigma ^{\mu }=(\sigma ^{0},\sigma ^{1},\sigma ^{2},\sigma ^{3})=(I_{2},\sigma _{x},\sigma _{y},\sigma _{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

There are left and right handed Weyl spinors, each with two components. Both have the form

\psi ={\begin{pmatrix}\psi _{1}\\\psi _{2}\\\end{pmatrix}}=\chi e^{-i(\mathbf {k} \cdot \mathbf {r} -\omega t)}=\chi e^{-i(\mathbf {p} \cdot \mathbf {r} -Et)/\hbar }

,

where

\chi ={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}

is a constant two-component spinor which satisfies

\sigma ^{\mu }p_{\mu }\chi =(I_{2}E-{\vec {\sigma }}\cdot {\vec {p}})\chi =0

.

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:

|\mathbf {p} |=\hbar |\mathbf {k} |=\hbar \omega /c\,\rightarrow \,|\mathbf {k} |=\omega /c

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

{\begin{aligned}&\sigma ^{\mu }\partial _{\mu }\psi _{R}=0\\&{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=0\end{aligned}}

where ${\bar {\sigma }}^{\mu }=(I_{2},-\sigma _{x},-\sigma _{y},-\sigma _{z})$ .

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:

\mathbf {p} \cdot \mathbf {J} \left|\mathbf {p} ,\lambda \right\rangle =\lambda |\mathbf {p} |\left|\mathbf {p} ,\lambda \right\rangle

Here $\lambda =\pm 1/2$ .

Derivation

The equations are obtained from the Lagrangian densities

{\mathcal {L}}=i\psi _{R}^{\dagger }\sigma ^{\mu }\partial _{\mu }\psi _{R}

{\mathcal {L}}=i\psi _{L}^{\dagger }{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}

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

References

↑ Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
↑ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
↑ An Introduction to Quantum Field Theory, M.E. Peskin, D.V. Schroeder, Addison-Wesley, 1995, ISBN 0-201-50397-2

External links

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[1] Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0

[2] The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.

[3] An Introduction to Quantum Field Theory, M.E. Peskin, D.V. Schroeder, Addison-Wesley, 1995, ISBN 0-201-50397-2