Relativistic wave equations

In physics, specifically relativistic quantum mechanics and its applications to particle physics, relativistic wave equations are equations which describe particles as waves. They are also used in quantum field theory (QFT), but the particles are treated as quantum fields, rather than waves.

At the time quantum mechanics was developed, physicists attempted to extend its range of application by formulating versions of the Schrödinger equation that were compatible with special relativity.

The first such equation was discovered by Erwin Schrödinger himself; however, he realized that this equation, now called the Klein-Gordon equation, gave incorrect results when used to calculate the energy levels of hydrogen. Schrödinger discarded his relativistic wave equation, only to realize a few months later that its non-relativistic limit (what is now called the Schrödinger equation) was still of importance.

The following list of relativistic wave equations are categorised as linear and non-linear partial differential equations, therin the spin of the particles they describe.

Linear equations

Spin 0

• Klein-Gordon equation: describes a massless or massive spin-0 particle (such as Higgs bosons)

$(\hbar \partial_{\mu} + imc)(\hbar \partial^{\mu} -imc)\psi = 0$

Spin 1/2

• Weyl equation: describes massless spin-1/2 particles
• Dirac equation: describes massive spin-1/2 particles (such as electrons)

$\left( i \hbar \partial\!\!\!/ - m c \right) \psi = 0$

• Majorana equation: describes a massive Majorana particle

$i \hbar \partial\!\!\!/ \psi - m c \psi_c = 0$

• Breit equation: describes two massive spin-1/2 particles (such as electrons) interacting electromagnetically to first order in perturbation theory

Spin 1

• Maxwell equations: describe a photon (massless spin-1 particle)
• Proca equation: describes a massive spin-1 particle (such as W and Z bosons)

$\partial_\mu(\partial^\mu A^\nu - \partial^\nu A^\mu)+\left(\frac{mc}{\hbar}\right)^2 A^\nu=0$

Gauge fields

• Kemmer equation: an alternative equation for spin-0 and spin-1 particles

Spin 3/2

• Rarita-Schwinger equation: describes a massive spin-3/2 particle

$\epsilon^{\mu \nu \rho \sigma} \gamma^5 \gamma_\nu \partial_\rho \psi_\sigma + m\psi^\mu = 0$

Arbitrary spin

• Bargmann-Wigner equations: describe free particles of arbitrary integral or half-integral spin.

Non-linear equations

Gauge fields

• Yang-Mills equation: describes a non-abelian gauge field
• Yang–Mills–Higgs equations: describes a non-abelian gauge field coupled with a massive spin-0 particle

Spin 2

• Einstein field equations: describe interaction of matter with the gravitational field (massless spin-2 field).

$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}$

See also

• Special relativity
• Status of special relativity
• Lorentz transformations
• Quantum Field Theory
• Scalar field theory

Sources

• Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
• McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3
• The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
• Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0
• Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
• 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
• Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2
• Quantum Mechanics, E. Abers, Pearson Ed., Addision Wesley, Prentice Hall Inc, 2004, ISBN 9780131461000
• Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8

External links

• Bargmann-Wigner equations