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

$\sigma^\mu = (\sigma^0,\sigma^1,\sigma^2,\sigma^3)= (I_2,\sigma_x,\sigma_y,\sigma_z)$

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:

$\begin{align} & \sigma^\mu \partial_\mu \psi_R = 0 \\ & \bar{\sigma}^\mu \partial_\mu \psi_L = 0 \end{align}$

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.

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