In
physics, relativistic quantum mechanics (RQM) is any
Poincaré covariant formulation of
quantum mechanics (QM). This theory is applicable to
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 propagating at all
velocities up to those comparable to the
speed of light ''c'', and can accommodate
massless particle
In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, gl ...
s. The theory has application in
high energy 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) and ...
,
particle physics and
accelerator physics
Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beams ...
,
as well as
atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned w ...
,
chemistry and
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the ...
. ''Non-relativistic quantum mechanics'' refers to the
mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which ...
applied in the context of
Galilean relativity, more specifically quantizing the equations of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical m ...
by replacing dynamical variables by
operator
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another s ...
s. ''Relativistic quantum mechanics'' (RQM) is quantum mechanics applied with
special relativity. Although the earlier formulations, like the
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which ma ...
and
Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators ( observables and others) incorporate a dependency on time, bu ...
were originally formulated in a non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) also work with special relativity.
Key features common to all RQMs include: the prediction of
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 radioacti ...
,
spin magnetic moment
In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. For example, the electron is an elementary spin-1/2 fermion. Quantum electrodynamics gives the ...
s of
elementary spin fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s,
fine structure, and quantum dynamics of
charged particle
In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...
s in
electromagnetic fields.
The key result is the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
, from which these predictions emerge automatically. By contrast, in non-relativistic quantum mechanics, terms have to be introduced artificially into the
Hamiltonian operator to achieve agreement with experimental observations.
The most successful (and most widely used) RQM is ''relativistic
quantum field theory'' (QFT), in which elementary particles are interpreted as ''field quanta''. A unique consequence of QFT that has been tested against other RQMs is the failure of conservation of particle number, for example in
matter creation
Even restricting the discussion to physics, scientists do not have a unique definition of what matter is. In the currently known particle physics, summarised by the standard model of elementary particles and interactions, it is possible to disting ...
and
annihilation
In particle physics, annihilation is the process that occurs when a subatomic particle collides with its respective antiparticle to produce other particles, such as an electron colliding with a positron to produce two photons. The total ener ...
.
In this article, the equations are written in familiar 3D
vector calculus notation and use hats for
operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
(not necessarily in the literature), and where space and time components can be collected,
tensor index notation is shown also (frequently used in the literature), in addition the
Einstein summation convention is used.
SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
are used here;
Gaussian units
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
and
natural units are common alternatives. All equations are in the position representation; for the momentum representation the equations have to be
Fourier transformed – see
position and momentum space.
Combining special relativity and quantum mechanics
One approach is to modify the
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which ma ...
to be consistent with special relativity.
A
postulate of quantum mechanics is that the
time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
of any quantum system is given by the
Schrödinger equation:
:
using a suitable
Hamiltonian operator corresponding to the system. The solution is a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued
wavefunction , a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of the
3D position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
of the particle at time , describing the behavior of the system.
Every particle has a non-negative
spin quantum number
In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
. The number is an integer, odd for
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s and even for
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s. Each has ''z''-projection quantum numbers; .
[Other common notations include and etc., but this would clutter expressions with unnecessary subscripts. The subscripts labeling spin values are not to be confused for tensor indices nor 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 in ...
. This is an additional discrete variable the wavefunction requires; .
Historically, in the early 1920s
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 ...
,
Kronig,
Uhlenbeck and
Goudsmit were the first to propose the concept of spin. The inclusion of spin in the wavefunction incorporates the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
(1925) and the more general
spin–statistics theorem
In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles tha ...
(1939) due to
Fierz, rederived by Pauli a year later. This is the explanation for a diverse range of
subatomic particle
In physical sciences, a subatomic particle is a particle that composes an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a prot ...
behavior and phenomena: from the
electronic configuration
In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon atom ...
s of atoms, nuclei (and therefore all
elements on the
periodic table
The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of ch ...
and their
chemistry), to the quark configurations and
colour charge
Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD).
The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
(hence the properties of
baryon
In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classified ...
s and
mesons).
A fundamental prediction of special relativity is the relativistic
energy–momentum relation; for a particle of
rest mass , and in a particular
frame of reference with
energy and 3-
momentum with
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
in terms of the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
, it is:
:
These equations are used together with the
energy and
momentum operator
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another s ...
s, which are respectively:
:
to construct a
relativistic wave equation
In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the conte ...
(RWE): a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
consistent with the energy–momentum relation, and is solved for to predict the quantum dynamics of the particle. For space and time to be placed on equal footing, as in relativity, the orders of space and time
partial derivatives should be equal, and ideally as low as possible, so that no initial values of the derivatives need to be specified. This is important for probability interpretations, exemplified below. The lowest possible order of any differential equation is the first (zeroth order derivatives would not form a differential equation).
The
Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators ( observables and others) incorporate a dependency on time, bu ...
is another formulation of QM, in which case the wavefunction is ''time-independent'', and the operators contain the time dependence, governed by the equation of motion:
:
This equation is also true in RQM, provided the Heisenberg operators are modified to be consistent with SR.
Historically, around 1926,
Schrödinger and
Heisenberg
Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent serie ...
show that wave mechanics and
matrix mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
are equivalent, later furthered by Dirac using
transformation theory.
A more modern approach to RWEs, first introduced during the time RWEs were developing for particles of any spin, is to apply
representations of the Lorentz group
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representatio ...
.
Space and time
In
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical m ...
and non-relativistic QM, time is an absolute quantity all observers and particles can always agree on, "ticking away" in the background independent of space. Thus in non-relativistic QM one has for a
many particle system .
In
relativistic mechanics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
, the
spatial coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
and
coordinate time
In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spat ...
are ''not'' absolute; any two observers moving relative to each other can measure different locations and times of
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of ev ...
s. The position and time coordinates combine naturally into a
four-dimensional spacetime position corresponding to events, and the energy and 3-momentum combine naturally into the
four momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentu ...
of a dynamic particle, as measured in ''some''
reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
, change according to a
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
as one measures in a different frame boosted and/or rotated relative the original frame in consideration. The derivative operators, and hence the energy and 3-momentum operators, are also non-invariant and change under Lorentz transformations.
Under a proper
orthochronous Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
in
Minkowski space, all one-particle quantum states locally transform under some
representation of the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physic ...
:
[;]
;
:
where is a finite-dimensional representation, in other words a
square matrix . Again, is thought of as a
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
containing components with the allowed values of . The
quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be ...
s and as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of may occur more than once depending on the representation.
Non-relativistic and relativistic Hamiltonians
The
classical Hamiltonian for a particle in a
potential is the
kinetic energy plus the
potential energy , with the corresponding quantum operator in the
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which ma ...
:
:
and substituting this into the above Schrödinger equation gives a non-relativistic QM equation for the wavefunction: the procedure is a straightforward substitution of a simple expression. By contrast this is not as easy in RQM; the energy–momentum equation is quadratic in energy ''and'' momentum leading to difficulties. Naively setting:
:
is not helpful for several reasons. The square root of the operators cannot be used as it stands; it would have to be expanded in a
power series before the momentum operator, raised to a power in each term, could act on . As a result of the power series, the space and time
derivatives are ''completely asymmetric'': infinite-order in space derivatives but only first order in the time derivative, which is inelegant and unwieldy. Again, there is the problem of the non-invariance of the energy operator, equated to the square root which is also not invariant. Another problem, less obvious and more severe, is that it can be shown to be
nonlocal and can even ''violate
causality
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
'': if the particle is initially localized at a point so that is finite and zero elsewhere, then at any later time the equation predicts delocalization everywhere, even for which means the particle could arrive at a point before a pulse of light could. This would have to be remedied by the additional constraint .
There is also the problem of incorporating spin in the Hamiltonian, which isn't a prediction of the non-relativistic Schrödinger theory. Particles with spin have a corresponding spin magnetic moment quantized in units of , the
Bohr magneton
In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.
The Bohr magneton, in SI units is defined as
\mu_\mathrm ...
:
:
where is the (spin)
g-factor for the particle, and the
spin operator
Spin is a conserved quantity carried by elementary particles, and thus by composite particles (hadrons) and atomic nuclei.
Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. The orbita ...
, so they interact with
electromagnetic fields. For a particle in an externally applied
magnetic field , the interaction term
:
has to be added to the above non-relativistic Hamiltonian. On the contrary; a relativistic Hamiltonian introduces spin ''automatically'' as a requirement of enforcing the relativistic energy-momentum relation.
Relativistic Hamiltonians are analogous to those of non-relativistic QM in the following respect; there are terms including
rest mass and interaction terms with externally applied fields, similar to the classical potential energy term, as well as momentum terms like the classical kinetic energy term. A key difference is that relativistic Hamiltonians contain spin operators in the form of
matrices, in which the
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
runs over the spin index , so in general a relativistic Hamiltonian:
:
is a function of space, time, and the momentum and spin operators.
The Klein–Gordon and Dirac equations for free particles
Substituting the energy and momentum operators directly into the energy–momentum relation may at first sight seem appealing, to obtain 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 ...
:
:
and was discovered by many people because of the straightforward way of obtaining it, notably by Schrödinger in 1925 before he found the non-relativistic equation named after him, and by Klein and Gordon in 1927, who included electromagnetic interactions in the equation. This ''is''
relativistically invariant, yet this equation alone isn't a sufficient foundation for RQM for a at least two reasons: one is that negative-energy states are solutions,
another is the density (given below), and this equation as it stands is only applicable to spinless particles. This equation can be factored into the form:
:
where and are not simply numbers or vectors, but 4 × 4
Hermitian matrices
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the - ...
that are required to
anticommute
In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswappe ...
for :
:
and square to the
identity matrix:
:
so that terms with mixed second-order derivatives cancel while the second-order derivatives purely in space and time remain. The first factor:
:
is the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
. The other factor is also the Dirac equation, but for a particle of
negative mass
In theoretical physics, negative mass is a type of exotic matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg. Such matter would violate one or more energy conditions and show some strange properties such as t ...
.
Each factor is relativistically invariant. The reasoning can be done the other way round: propose the Hamiltonian in the above form, as Dirac did in 1928, then pre-multiply the equation by the other factor of operators , and comparison with the KG equation determines the constraints on and . The positive mass equation can continue to be used without loss of continuity. The matrices multiplying suggest it isn't a scalar wavefunction as permitted in the KG equation, but must instead be a four-component entity. The Dirac equation still predicts negative energy solutions,
so Dirac postulated that negative energy states are always occupied, because according to the
Pauli principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
,
electronic transitions from positive to negative energy levels in
atoms would be forbidden. See
Dirac sea
The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the ...
for details.
Densities and currents
In non-relativistic quantum mechanics, the
square modulus of the
wavefunction gives the
probability density function . This is the
Copenhagen interpretation
The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as featu ...
, circa 1927. In RQM, while is a wavefunction, the probability interpretation is not the same as in non-relativistic QM. Some RWEs do not predict a probability density or
probability current (really meaning ''probability current density'') because they are ''not''
positive-definite functions of space and time. The
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
does:
:
where the dagger denotes the
Hermitian adjoint (authors usually write for the
Dirac adjoint) and is the
probability four-current, while 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 ...
does not:
:
where is the
four-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus.
In special relativity and in quantum mechanics, the four-gradient is used to define the properties ...
. Since the initial values of both and may be freely chosen, the density can be negative.
Instead, what appears look at first sight a "probability density" and "probability current" has to be reinterpreted as
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
and
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional are ...
when multiplied by
electric charge. Then, the wavefunction is not a wavefunction at all, but reinterpreted as a ''field''.
The density and current of electric charge always satisfy a
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
:
:
as charge is a
conserved quantity
In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system.
Not all systems have conserved quantities, and conserved quantities are ...
. Probability density and current also satisfy a continuity equation because probability is conserved, however this is only possible in the absence of interactions.
Spin and electromagnetically interacting particles
Including interactions in RWEs is generally difficult.
Minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution. This minimal coupling is in contrast to, ...
is a simple way to include the electromagnetic interaction. For one charged particle of
electric charge in an electromagnetic field, given by the
magnetic vector potential defined by the magnetic field , and
electric scalar potential , this is:
:
where is the
four-momentum that has a corresponding
4-momentum operator
In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, ...
, and the
four-potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
. In the following, the non-relativistic limit refers to the limiting cases:
:
that is, the total energy of the particle is approximately the rest energy for small electric potentials, and the momentum is approximately the classical momentum.
Spin 0
In RQM, the KG equation admits the minimal coupling prescription;
:
In the case where the charge is zero, the equation reduces trivially to the free KG equation so nonzero charge is assumed below. This is a scalar equation that is invariant under the ''irreducible'' one-dimensional scalar representation of the Lorentz group. This means that all of its solutions will belong to a direct sum of representations. Solutions that do not belong to the irreducible representation will have two or more ''independent'' components. Such solutions cannot in general describe particles with nonzero spin since spin components are not independent. Other constraint will have to be imposed for that, e.g. the Dirac equation for spin , see below. Thus if a system satisfies the KG equation ''only'', it can only be interpreted as a system with zero spin.
The electromagnetic field is treated classically according to
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
and the particle is described by a wavefunction, the solution to the KG equation. The equation is, as it stands, not always very useful, because massive spinless particles, such as the ''π''-mesons, experience the much stronger strong interaction in addition to the electromagnetic interaction. It does, however, correctly describe charged spinless bosons in the absence of other interactions.
The KG equation is applicable to spinless charged
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s in an external electromagnetic potential.
As such, the equation cannot be applied to the description of atoms, since the electron is a spin particle. In the non-relativistic limit the equation reduces to the Schrödinger equation for a spinless charged particle in an electromagnetic field:
:
Spin
Non relativistically, spin was ''
phenomenologically
Phenomenology may refer to:
Art
* Phenomenology (architecture), based on the experience of building materials and their sensory properties
Philosophy
* Phenomenology (philosophy), a branch of philosophy which studies subjective experiences and a ...
'' introduced in the
Pauli equation by
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 ...
in 1927 for particles in an
electromagnetic field:
:
by means of the 2 × 2
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 in ...
, and is not just a scalar wavefunction as in the non-relativistic Schrödinger equation, but a two-component
spinor field In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
:
:
where the subscripts ↑ and ↓ refer to the "spin up" () and "spin down" () states.
[This spinor notation is not necessarily standard; the literature usually writes or etc., but in the context of spin , this informal identification is commonly made.]
In RQM, the Dirac equation can also incorporate minimal coupling, rewritten from above;
:
and was the first equation to accurately ''predict'' spin, a consequence of the 4 × 4
gamma matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
. There is a 4 × 4
identity matrix pre-multiplying the energy operator (including the potential energy term), conventionally not written for simplicity and clarity (i.e. treated like the number 1). Here is a four-component spinor field, which is conventionally split into two two-component spinors in the form:
[Again this notation is not necessarily standard, the more advanced literature usually writes
: etc.,
but here we show informally the correspondence of energy, helicity, and spin states.]
:
The 2-spinor corresponds to a particle with 4-momentum and charge and two spin states (, as before). The other 2-spinor corresponds to a similar particle with the same mass and spin states, but ''negative'' 4-momentum and ''negative'' charge , that is, negative energy states,
time-reversed momentum, and
negated charge. This was the first interpretation and prediction of a particle and ''corresponding
antiparticle
In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
''. See
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 ...
and
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 ...
for further description of these spinors. In the non-relativistic limit the Dirac equation reduces to the Pauli equation (see
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
for how). When applied a one-electron atom or ion, setting and to the appropriate electrostatic potential, additional relativistic terms include the
spin–orbit interaction
In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orb ...
, electron
gyromagnetic ratio
In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol Gamma, , gamma. Its ...
, and
Darwin term. In ordinary QM these terms have to be put in by hand and treated using
perturbation theory. The positive energies do account accurately for the fine structure.
Within RQM, for massless particles the Dirac equation reduces to:
:
the first of which is the
Weyl equation
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
, a considerable simplification applicable for massless
neutrinos.
[.] This time there is a 2 × 2
identity matrix pre-multiplying the energy operator conventionally not written. In RQM it is useful to take this as the zeroth Pauli matrix which couples to the energy operator (time derivative), just as the other three matrices couple to the momentum operator (spatial derivatives).
The Pauli and gamma matrices were introduced here, in theoretical physics, rather than
pure mathematics itself. They have applications to
quaternions and to the
SO(2) and
SO(3) Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s, because they satisfy the important
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
nbsp;, and
anticommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
nbsp;, sub>+ relations respectively:
:
where is the
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
Levi-Civita symbol. The gamma matrices form
bases in
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercom ...
, and have a connection to the components of the flat spacetime
Minkowski metric in the anticommutation relation:
:
(This can be extended to
curved space
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
time by introducing
vierbein
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independen ...
s, but is not the subject of special relativity).
In 1929, the
Breit equation
The 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 (electrons, for example) interacting electromagnetically to the first ...
was found to describe two or more electromagnetically interacting massive spin fermions to first-order relativistic corrections; one of the first attempts to describe such a relativistic quantum
many-particle system
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
. This is, however, still only an approximation, and the Hamiltonian includes numerous long and complicated sums.
Helicity and chirality
The
helicity operator is defined by;
:
where p is the momentum operator, S the spin operator for a particle of spin ''s'', ''E'' is the total energy of the particle, and ''m''
0 its rest mass. Helicity indicates the orientations of the spin and translational momentum vectors. Helicity is frame-dependent because of the 3-momentum in the definition, and is quantized due to spin quantization, which has discrete positive values for parallel alignment, and negative values for antiparallel alignment.
An automatic occurrence in the Dirac equation (and the Weyl equation) is the projection of the spin operator on the 3-momentum (times ''c''), , which is the helicity (for the spin case) times
.
For massless particles the helicity simplifies to:
:
Higher spins
The Dirac equation can only describe particles of spin . Beyond the Dirac equation, RWEs have been applied to
free particles of various spins. In 1936, Dirac extended his equation to all fermions, three years later
Fierz and Pauli rederived the same equation. The
Bargmann–Wigner equations
:''This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators.
In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles with non ...
were found in 1948 using Lorentz group theory, applicable for all free particles with any spin. Considering the factorization of the KG equation above, and more rigorously by
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physic ...
theory, it becomes apparent to introduce spin in the form of matrices.
The wavefunctions are multicomponent
spinor field In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
s, which can be represented as
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
s of
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s of space and time:
:
where the expression on the right is the
Hermitian conjugate. For a ''massive'' particle of spin , there are components for the particle, and another for the corresponding
antiparticle
In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
(there are possible values in each case), altogether forming a -component spinor field:
:
with the + subscript indicating the particle and − subscript for the antiparticle. However, for ''massless'' particles of spin ''s'', there are only ever two-component spinor fields; one is for the particle in one helicity state corresponding to +''s'' and the other for the antiparticle in the opposite helicity state corresponding to −''s'':
:
According to the relativistic energy-momentum relation, all massless particles travel at the speed of light, so particles traveling at the speed of light are also described by two-component spinors. Historically,
Élie Cartan found the most general form of
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s in 1913, prior to the spinors revealed in the RWEs following the year 1927.
For equations describing higher-spin particles, the inclusion of interactions is nowhere near as simple minimal coupling, they lead to incorrect predictions and self-inconsistencies. For spin greater than , the RWE is not fixed by the particle's mass, spin, and electric charge; the electromagnetic moments (
electric dipole moments and
magnetic dipole moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
s) allowed by the
spin quantum number
In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
are arbitrary. (Theoretically,
magnetic charge
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
would contribute also). For example, the spin case only allows a magnetic dipole, but for spin 1 particles magnetic quadrupoles and electric dipoles are also possible.
For more on this topic, see
multipole expansion and (for example) Cédric Lorcé (2009).
Velocity operator
The Schrödinger/Pauli velocity operator can be defined for a massive particle using the classical definition , and substituting quantum operators in the usual way:
:
which has eigenvalues that take ''any'' value. In RQM, the Dirac theory, it is:
: