Relativistic wave equations
Encyclopedia
Before the creation of quantum field theory
, physicists attempted to formulate versions of the Schrödinger equation
which were compatible with special relativity
. Such equations are called relativistic wave equations.
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.
All the particle equations except the Breit, the Yang–Mills, Yang–Mills–Higgs and Einstein are
linear
.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, physicists attempted to formulate versions of the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
which were compatible with special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. Such equations are called relativistic wave equations.
The first such equation was discovered by Erwin Schrödinger
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
himself; however, he realized that this equation, now called the Klein-Gordon equation
Klein-Gordon equation
The Klein–Gordon equation is a relativistic version of the Schrödinger 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
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
) was still of importance.
List of relativistic wave equations
The following list of relativistic wave equations is categorised by the spin of the particles they describe.Spin 0
- Klein-Gordon equationKlein-Gordon equationThe Klein–Gordon equation is a relativistic version of the Schrödinger equation....
: describes a massless or massive spin-0 particle (such as Higgs bosonHiggs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
s)
Spin 1/2
- Weyl equationWeyl equation-Mathematical Formulation:-Derivation:The equations are obtained from the Lagrangian densitiesBy treating the spinor and its conjugate as independent variables, the relevant Weyl equation is obtained....
: describes massless spin-1/2 particles - Dirac equationDirac equationThe Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
: describes massive spin-1/2 particles (such as electronElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s)
-
- Majorana equation: describes a massive Majorana particle
- Breit equationBreit equationThe Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles interacting electromagnetically to the first order in perturbation theory. It accounts for magnetic interactions and...
: describes two massive spin-1/2 particles (such as electronElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s) interacting electromagnetically to first order in perturbation theory
- Breit equation
Spin 1
- Maxwell equations: describe a photonPhotonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
(massless spin-1 particle) - Proca equation: describes a massive spin-1 particle (such as W and Z bosonsW and Z bosonsThe W and Z bosons are the elementary particles that mediate the weak interaction; their symbols are , and . The W bosons have a positive and negative electric charge of 1 elementary charge respectively and are each other's antiparticle. The Z boson is electrically neutral and its own...
)
Gauge fields
- Yang-Mills equation: describes a non-abelian gauge field
- Yang–Mills–Higgs equationsYang–Mills–Higgs equationsIn mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle...
: describes a non-abelian gauge field coupled with a massive spin-0 particle - Kemmer equationDuffin–Kemmer–Petiau algebraIn mathematical physics, the Duffin–Kemmer–Petiau algebra , introduced by R.J. Duffin, Nicholas Kemmer and G. Petiau, is the algebra which is generated by the Duffin–Kemmer–Petiau matrices...
: an alternative equation for spin-0 and spin-1 particles
Spin 3/2
- Rarita-Schwinger equationRarita-Schwinger equationIn theoretical physics, the Rarita–Schwinger equation is therelativistic field equation of spin-3/2 fermions. It is similar to the Dirac equation for spin-1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinger in 1941...
: describes a massive spin-3/2 particle
Spin 2
- Einstein field equationsEinstein field equationsThe Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
: describe interaction of matter with the gravitational fieldGravitational fieldThe gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
(massless spin-2 field).
Arbitrary spin
- Bargmann-Wigner equations: describe free particles of arbitrary integral or half-integral spin
All the particle equations except the Breit, the Yang–Mills, Yang–Mills–Higgs and Einstein are
linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
.
See also
- Special relativitySpecial relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
- Status of special relativityStatus of special relativitySpecial relativity is a physical theory that plays a fundamental role in the description of all physical phenomena, as long as no considerable influence of gravitation occurs...
- Lorentz transformations
- Quantum Field TheoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
- Scalar field theoryScalar field theoryIn theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field...