Weyl equation

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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[edit]

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

explicitly in SI units:

where

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[edit]

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

where

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:

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

where .

Helicity[edit]

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:

Here .

Derivation[edit]

The equations are obtained from the Lagrangian densities

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

See also[edit]

References[edit]

  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[edit]

External links[edit]