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.

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

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

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

The elements ψ_L and ψ_R are respectively the 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.

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

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.

See also

• Dirac equation (which describes massive spin-1/2 particles)
• Angular momentum operator
• Momentum operator
• Spin (physics)

References

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

Further reading

• Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
• Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7
• Supersymmetry P. Labelle, Demystified, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
• The Road to Reality, Roger Penrose, Vintage books, 2007, ISBN 0-679-77631-1

External links