In
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, the Dirac equation is a
relativistic wave equation derived by British physicist
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
in 1928. In its
free form, or including
electromagnetic interactions, it describes all
spin- massive particle
The physics technical term massive particle refers to a massful particle which has real non-zero rest mass (such as baryonic matter), the counter-part to the term massless particle. According to special relativity, the velocity of a massive particl ...
s, called "Dirac particles", such as
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
s and
quark
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
s for which
parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
is a
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
. It is consistent with both the principles of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and the theory of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the
hydrogen spectrum in a completely rigorous way.
The equation also implied the existence of a new form of matter, ''
antimatter
In modern physics, antimatter is defined as matter composed of the antiparticles (or "partners") of the corresponding particles in "ordinary" matter. Antimatter occurs in natural processes like cosmic ray collisions and some types of radioac ...
'', previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a ''theoretical'' justification for the introduction of several component wave functions in
Pauli
Pauli is a surname and also a Finnish male given name (variant of Paul) and may refer to:
* Arthur Pauli (born 1989), Austrian ski jumper
* Barbara Pauli (1752 or 1753 - fl. 1781), Swedish fashion trader
*Gabriele Pauli (born 1957), German politi ...
's
phenomenological theory of
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...
. The wave functions in the Dirac theory are vectors of four
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s (known as
bispinor
In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, spe ...
s), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the
Weyl equation.
Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the
positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collide ...
—represents one of the great triumphs of
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
. This accomplishment has been described as fully on a par with the works of
Newton,
Maxwell
Maxwell may refer to:
People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of ...
, and
Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
before him. In the context of
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin- particles.
The Dirac equation appears on the floor of
Westminster Abbey
Westminster Abbey, formally titled the Collegiate Church of Saint Peter at Westminster, is an historic, mainly Gothic church in the City of Westminster, London, England, just to the west of the Palace of Westminster. It is one of the United ...
on the plaque commemorating Paul Dirac's life, which was unveiled on 13 November 1995.
Mathematical formulation
In its modern formulation for field theory, the Dirac equation is written in terms of a
Dirac spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain co ...
field
taking values in a complex vector space described concretely as
, defined on flat spacetime (
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
)
. Its expression also contains
gamma matrices and a parameter
interpreted as the mass, as well as other physical constants.
In terms of a field
, the Dirac equation is then
and in
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
, with
Feynman slash notation
In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If ''A'' is a covariant vector (i.e., a 1-form),
: \ \stackrel\ \gamma^1 A_1 + ...
,
The gamma matrices are a set of four
complex matrices (elements of
) which satisfy the defining ''anti''-commutation relations:
These matrices can be realized explicitly under a choice of representation. Two common choices are the Dirac representation
where
are the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
, and the chiral representation: the
are the same, but
The slash notation is a compact notation for
where
is a four-vector (often it is the four-vector differential operator
). The summation over the index
is implied.
Dirac adjoint and the adjoint equation
The Dirac adjoint of the spinor field
is defined as
Using the property of gamma matrices (which follows straightforwardly from Hermicity properties of the
) that
one can derive the adjoint Dirac equation by taking the Hermitian conjugate of the Dirac equation and multiplying on the right by
:
where the partial derivative acts from the right on
: written in the usual way in terms of a left action of the derivative, we have
Klein–Gordon equation
Applying
to the Dirac equation gives
That is, each component of the Dirac spinor field satisfies the
Klein–Gordon equation
The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant ...
.
Conserved current
A conserved current of the theory is
Another approach to derive this expression is by variational methods, applying Noether's theorem for the global
symmetry to derive the conserved current
Solutions
Since the Dirac operator acts on 4-tuples of
square-integrable functions, its solutions should be members of the same
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The fact that the energies of the solutions do not have a lower bound is unexpected.
Plane-wave solutions
Plane-wave solutions are those arising from an ansatz
which models a particle with definite 4-momentum
where
For this ansatz, the Dirac equation becomes an equation for
:
After picking a representation for the gamma matrices
, solving this is a matter of solving a system of linear equations. It is a representation-free property of gamma matrices that the solution space is two-dimensional (see
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a ...
).
For example, in the chiral representation for
, the solution space is parametrised by a
vector
, with
where
and
is the Hermitian matrix square-root.
These plane-wave solutions provide a starting point for canonical quantization.
Lagrangian formulation
Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by:
If one varies this with respect to
one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to
one gets the Dirac equation.
In natural units and with the slash notation, the action is then
For this action, the conserved current
above arises as the conserved current corresponding to the global
symmetry through
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
for field theory. Gauging this field theory by changing the symmetry to a local, spacetime point dependent one gives gauge symmetry (really, gauge redundancy). The resultant theory is
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
or QED. See below for a more detailed discussion.
Lorentz invariance
The Dirac equation is invariant under Lorentz transformations, that is, under the action of the Lorentz group
or strictly
, the component connected to the identity.
For a Dirac spinor viewed concretely as taking values in
, the transformation under a Lorentz transformation
is given by a
complex matrix