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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: :i\hbar \frac\psi =\hat\psi 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 ...
p = \sqrt, it is: :E^2 = c^2\mathbf\cdot\mathbf + (mc^2)^2\,. 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: :\hat=i\hbar\frac\,,\quad \hat = -i\hbar\nabla\,, 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: :\fracA = \frac ,\hat\fracA\,, 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 ...
:;
;
:\psi_\sigma(\mathbf, t) \rightarrow D(\Lambda) \psi_\sigma(\Lambda^(\mathbf, t)) 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 ...
: :\hat = \frac + V(\mathbf,t) 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: :\hat = \hat = \sqrt \quad \Rightarrow \quad i\hbar\frac\psi = \sqrt \, \psi 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 ...
: :\hat_S = - \frac\hat\,,\quad \left, \boldsymbol_S\ = - g\mu_B \sigma\,, 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 :\hat_B = - \mathbf \cdot \hat_S 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: :\hat = \hat(\mathbf, t, \hat, \hat) 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 ...
: :\hat^2 \psi = c^2\hat\cdot\hat\psi + (mc^2)^2\psi \,, 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: : \left(\hat - c\boldsymbol\cdot\hat - \beta mc^2 \right)\left(\hat + c\boldsymbol\cdot\hat + \beta mc^2 \right)\psi=0 \,, 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 : :\alpha_i \beta = - \beta \alpha_i, \quad \alpha_i\alpha_j = - \alpha_j\alpha_i \,, and square to the identity matrix: : \alpha_i^2 = \beta^2 = I \,, so that terms with mixed second-order derivatives cancel while the second-order derivatives purely in space and time remain. The first factor: :\left(\hat - c\boldsymbol\cdot\hat - \beta mc^2 \right)\psi=0 \quad \Leftrightarrow \quad \hat = c\boldsymbol\cdot\hat + \beta mc^2 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: :\rho=\psi^\dagger \psi, \quad \mathbf = \psi^\dagger \gamma^0 \boldsymbol \psi \quad \rightleftharpoons \quad J^\mu = \psi^\dagger \gamma^0 \gamma^\mu \psi 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: :\rho = \frac\left(\psi^\frac - \psi \frac\right)\, ,\quad \mathbf = -\frac\left(\psi^* \nabla \psi - \psi \nabla \psi^*\right) \quad \rightleftharpoons \quad J^\mu = \frac(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*) 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 ...
: :\frac + \nabla\cdot\mathbf = 0 \quad \rightleftharpoons \quad \partial_\mu J^\mu = 0 \,, 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: :\hat \rightarrow \hat - q\phi \,, \quad \hat\rightarrow \hat - q \mathbf \quad \rightleftharpoons \quad \hat_\mu \rightarrow \hat_\mu -q A_\mu 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: :E - e\phi \approx mc^2\,,\quad \mathbf \approx m \mathbf\,, 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; :^2 \psi = c^2^2\psi + (mc^2)^2\psi \quad \rightleftharpoons \quad \left - ^2 \right\psi = 0. 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: :\left ( i\hbar \frac- q\phi\right) \psi = \frac^2 \psi \quad \Leftrightarrow \quad \hat = \frac^2 + q\phi.


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: :\left(i \hbar \frac - q \phi \right) \psi = \left \frac^2 \right\psi \quad \Leftrightarrow \quad \hat = \frac^2 + q \phi 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 ...
: :\psi=\begin\psi_ \\ \psi_ \end where the subscripts ↑ and ↓ refer to the "spin up" () and "spin down" () states.This spinor notation is not necessarily standard; the literature usually writes \psi=\begin u^1 \\ u^2 \end or \psi=\begin \chi \\ \eta \end 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; :\left(i \hbar \frac -q\phi \right)\psi = \gamma^0 \left c\boldsymbol\cdot - mc^2 \right\psi \quad \rightleftharpoons \quad \left gamma^\mu (\hat_\mu - q A_\mu) - mc^2 \rightpsi = 0 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 :\psi=\beginu \\ v \end = \begin u^1 \\ u^2 \\ v^1 \\ v^2 \end etc., but here we show informally the correspondence of energy, helicity, and spin states. :\psi=\begin\psi_ \\ \psi_ \end = \begin\psi_ \\ \psi_ \\ \psi_ \\ \psi_ \end 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: : \left(\frac + \boldsymbol\cdot \hat \right) \psi_ = 0 \,,\quad \left(\frac - \boldsymbol\cdot \hat \right) \psi_ = 0 \quad \rightleftharpoons \quad \sigma^\mu \hat_\mu \psi_ = 0\,,\quad \sigma_\mu \hat^\mu \psi_ = 0\,, 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: :\left sigma_a, \sigma_b \right= 2i \varepsilon_ \sigma_c \,, \quad \left sigma_a, \sigma_b \right = 2\delta_\sigma_0 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: :\left gamma^\alpha,\gamma^\beta\right = \gamma^\alpha\gamma^\beta + \gamma^\beta\gamma^\alpha = 2\eta^\,, (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; :\hat = \hat\cdot \frac = \hat \cdot \frac 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 \sqrt. For massless particles the helicity simplifies to: :\hat = \hat \cdot \frac


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: :\psi(\mathbf,t) = \begin \psi_(\mathbf,t) \\ \psi_(\mathbf,t) \\ \vdots \\ \psi_(\mathbf,t) \\ \psi_(\mathbf,t) \end\quad\rightleftharpoons\quad ^\dagger = \begin ^\star & ^\star & \cdots & ^\star & ^\star \end 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: :\psi(\mathbf,t) = \begin \psi_(\mathbf,t) \\ \psi_(\mathbf,t) \\ \vdots \\ \psi_(\mathbf,t) \\ \psi_(\mathbf,t) \\ \psi_(\mathbf,t) \\ \psi_(\mathbf,t) \\ \vdots \\ \psi_(\mathbf,t) \\ \psi_(\mathbf,t) \end\quad\rightleftharpoons\quad ^\dagger\begin ^\star & ^\star & \cdots & ^\star \end 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'': :\psi(\mathbf,t) = \begin \psi_(\mathbf,t) \\ \psi_(\mathbf,t) \end 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: :\hat = \frac\hat which has eigenvalues that take ''any'' value. In RQM, the Dirac theory, it is: :\hat = \frac\left hat,\hat\right/math> which must have eigenvalues between ±''c''. See Foldy–Wouthuysen transformation for more theoretical background.


Relativistic quantum Lagrangians

The Hamiltonian operators in the Schrödinger picture are one approach to forming the differential equations for . An equivalent alternative is to determine a Lagrangian (really meaning ''
Lagrangian density Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
''), then generate the differential equation by the field-theoretic Euler–Lagrange equation: : \partial_\mu \left( \frac \right) - \frac = 0 \, For some RWEs, a Lagrangian can be found by inspection. For example, the Dirac Lagrangian is: :\mathcal = \overline(\gamma^\mu P_\mu - mc)\psi and Klein–Gordon Lagrangian is: :\mathcal = - \frac \eta^ \partial_\psi^ \partial_\psi - m c^2 \psi^ \psi\,. This is not possible for all RWEs; and is one reason the Lorentz group theoretic approach is important and appealing: fundamental invariance and symmetries in space and time can be used to derive RWEs using appropriate group representations. The Lagrangian approach with field interpretation of is the subject of QFT rather than RQM: Feynman's path integral formulation uses invariant Lagrangians rather than Hamiltonian operators, since the latter can become extremely complicated, see (for example) Weinberg (1995).


Relativistic quantum angular momentum

In non-relativistic QM, the
angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
is formed from the classical
pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...
definition . In RQM, the position and momentum operators are inserted directly where they appear in the orbital
relativistic angular momentum In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the thr ...
tensor defined from the four-dimensional position and momentum of the particle, equivalently a
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
in the exterior algebra formalism:Some authors, including Penrose, use ''Latin'' letters in this definition, even though it is conventional to use Greek indices for vectors and tensors in spacetime. :M^ = X^\alpha P^\beta - X^\beta P^\alpha = 2 X^ P^ \quad \rightleftharpoons \quad \mathbf = \mathbf\wedge\mathbf\,, which are six components altogether: three are the non-relativistic 3-orbital angular momenta; , , , and the other three , , are boosts of the
centre of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the rotating object. An additional relativistic-quantum term has to be added for particles with spin. For a particle of rest mass , the ''total'' angular momentum tensor is: :J^ = 2X^ P^ + \frac\varepsilon^ W_\gamma p_\delta \quad \rightleftharpoons \quad \mathbf = \mathbf\wedge\mathbf + \frac\star(\mathbf\wedge\mathbf) where the star denotes the Hodge dual, and :W_\alpha =\frac\varepsilon_M^p^\delta \quad \rightleftharpoons \quad \mathbf = \star(\mathbf\wedge\mathbf) is the
Pauli–Lubanski pseudovector In physics, the Pauli–Lubanski pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang Pauli and Józef Lubański, It describ ...
. For more on relativistic spin, see (for example) Troshin & Tyurin (1994).


Thomas precession and spin–orbit interactions

In 1926, the
Thomas precession In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a part ...
is discovered: relativistic corrections to the spin of elementary particles with application in 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 ...
of atoms and rotation of macroscopic objects. In 1939 Wigner derived the Thomas precession. In
classical electromagnetism and special relativity The theory of special relativity plays an important role in the modern theory of classical electromagnetism. It gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transfor ...
, an electron moving with a velocity through an electric field but not a magnetic field , will in its own frame of reference experience a Lorentz-transformed magnetic field : :\mathbf' = \frac \,. In the non-relativistic limit : :\mathbf' = \frac \,, so the non-relativistic spin interaction Hamiltonian becomes: :\hat = - \mathbf'\cdot \hat_S = -\left(\mathbf + \frac \right) \cdot \hat_S \,, where the first term is already the non-relativistic magnetic moment interaction, and the second term the relativistic correction of order , but this disagrees with experimental atomic spectra by a factor of . It was pointed out by L. Thomas that there is a second relativistic effect: An electric field component perpendicular to the electron velocity causes an additional acceleration of the electron perpendicular to its instantaneous velocity, so the electron moves in a curved path. The electron moves in a rotating frame of reference, and this additional precession of the electron is called the ''Thomas precession''. It can be shown that the net result of this effect is that the spin–orbit interaction is reduced by half, as if the magnetic field experienced by the electron has only one-half the value, and the relativistic correction in the Hamiltonian is: :\hat = - \mathbf'\cdot \hat_S = -\left(\mathbf + \frac \right) \cdot \hat_S \,. In the case of RQM, the factor of is predicted by the Dirac equation.


History

The events which led to and established RQM, and the continuation beyond into quantum electrodynamics (QED), are summarized below P.W_Atkins_(1974).html" ;"title="Peter_Atkins.html" ;"title="ee, for example, R. Resnick and R. Eisberg (1985), and Peter Atkins">P.W Atkins (1974)">Peter_Atkins.html" ;"title="ee, for example, R. Resnick and R. Eisberg (1985), and Peter Atkins">P.W Atkins (1974) More than half a century of experimental and theoretical research from the 1890s through to the 1950s in the new and mysterious quantum theory as it was up and coming revealed that a number of phenomena cannot be explained by QM alone. SR, found at the turn of the 20th century, was found to be a ''necessary'' component, leading to unification: RQM. Theoretical predictions and experiments mainly focused on the newly found
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 ...
, nuclear physics, and particle physics; by considering spectroscopy, diffraction and scattering of particles, and the electrons and nuclei within atoms and molecules. Numerous results are attributed to the effects of spin.


Relativistic description of particles in quantum phenomena

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
in 1905 explained of the
photoelectric effect The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid sta ...
; a particle description of light as
photons A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
. In 1916, Sommerfeld explains fine structure; the splitting of the
spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to identi ...
s of
atoms Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, and ...
due to first order relativistic corrections. The
Compton effect Compton scattering, discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a charged particle, usually an electron. If it results in a decrease in energy (increase in wavelength) of the photon ...
of 1923 provided more evidence that special relativity does apply; in this case to a particle description of photon–electron scattering.
de Broglie Louis Victor Pierre Raymond, 7th Duc de Broglie (, also , or ; 15 August 1892 – 19 March 1987) was a French physicist and aristocrat who made groundbreaking contributions to quantum theory. In his 1924 PhD thesis, he postulated the wave nat ...
extends
wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the ...
to
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic parti ...
: the de Broglie relations, which are consistent with special relativity and quantum mechanics. By 1927, Davisson and Germer and separately G. Thomson successfully diffract electrons, providing experimental evidence of wave-particle duality.


Experiments

* 1897
J. J. Thomson Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered. In 1897, Thomson showed that ...
discovers the electron and measures its
mass-to-charge ratio The mass-to-charge ratio (''m''/''Q'') is a physical quantity relating the ''mass'' (quantity of matter) and the ''electric charge'' of a given particle, expressed in units of kilograms per coulomb (kg/C). It is most widely used in the electrody ...
. Discovery of the Zeeman effect: the splitting a
spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to identi ...
into several components in the presence of a static magnetic field. * 1908 Millikan measures the charge on the electron and finds experimental evidence of its quantization, in the oil drop experiment. * 1911
Alpha particle Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay, but may also be produce ...
scattering in the Geiger–Marsden experiment, led by Rutherford, showed that atoms possess an internal structure: the atomic nucleus. * 1913 The
Stark effect The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several compo ...
is discovered: splitting of spectral lines due to a static electric field (compare with the Zeeman effect). * 1922
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throug ...
: experimental evidence of spin and its quantization. * 1924 Stoner studies splitting of energy levels in magnetic fields. * 1932 Experimental discovery of the neutron by Chadwick, and
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 collides ...
s by Anderson, confirming the theoretical prediction of positrons. * 1958 Discovery of the
Mössbauer effect The Mössbauer effect, or recoilless nuclear resonance fluorescence, is a physical phenomenon discovered by Rudolf Mössbauer in 1958. It involves the resonant and recoil-free emission and absorption of gamma radiation by atomic nuclei bound in a ...
: resonant and recoil-free emission and absorption of gamma radiation by atomic nuclei bound in a solid, useful for accurate measurements of
gravitational redshift In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well (seem to) lose energy. This loss of energy ...
and
time dilation In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
, and in the analysis of nuclear electromagnetic moments in
hyperfine interaction In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the ...
s.


Quantum non-locality and relativistic locality

In 1935; Einstein, Rosen, Podolsky published a paper concerning
quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
of particles, questioning quantum nonlocality and the apparent violation of causality upheld in SR: particles can appear to interact instantaneously at arbitrary distances. This was a misconception since information is not and cannot be transferred in the entangled states; rather the information transmission is in the process of measurement by two observers (one observer has to send a signal to the other, which cannot exceed ''c''). QM does ''not'' violate SR. In 1959, Bohm and Aharonov publish a paper on the
Aharonov–Bohm effect The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confine ...
, questioning the status of electromagnetic potentials in QM. The EM field tensor and EM 4-potential formulations are both applicable in SR, but in QM the potentials enter the Hamiltonian (see above) and influence the motion of charged particles even in regions where the fields are zero. In 1964, Bell's theorem was published in a paper on the EPR paradox, showing that QM cannot be derived from local hidden-variable theories if locality is to be maintained.


The Lamb shift

In 1947 the Lamb shift was discovered: a small difference in the 2''S'' and 2''P'' levels of hydrogen, due to the interaction between the electron and vacuum.
Lamb Lamb or The Lamb may refer to: * A young sheep * Lamb and mutton, the meat of sheep Arts and media Film, television, and theatre * ''The Lamb'' (1915 film), a silent film starring Douglas Fairbanks Sr. in his screen debut * ''The Lamb'' (1918 ...
and Retherford experimentally measure stimulated radio-frequency transitions the 2''S'' and 2''P'' hydrogen levels by
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency rang ...
radiation. An explanation of the Lamb shift is presented by Bethe. Papers on the effect were published in the early 1950s.



Development of quantum electrodynamics

* 1943 Tomonaga begins work on renormalization, influential in QED. * 1947 Schwinger calculates the anomalous magnetic moment of the electron. Kusch measures of the anomalous magnetic electron moment, confirming one of QED's great predictions.


See also


Atomic physics and chemistry

*
Relativistic quantum chemistry Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of ...
*
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 ...
*
Electron spin resonance Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spi ...
*
Fine structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between ele ...


Mathematical physics

* Quantum spacetime * Spin connection * Spinor bundle *
Dirac equation in the algebra of physical space In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-di ...
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Casimir invariant In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operat ...
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Casimir operator In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
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Wigner D-matrix The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjug ...


Particle physics and quantum field theory

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Zitterbewegung In physics, the zitterbewegung ("jittery motion" in German, ) is the predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first discussed by Gregory Breit in 1928 and la ...
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Two-body Dirac equations In quantum field theory, and in the significant subfields of quantum electrodynamics (QED) and quantum chromodynamics (QCD), the two-body Dirac equations (TBDE) of constraint dynamics provide a three-dimensional yet manifestly covariant reformulat ...
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Relativistic Heavy Ion Collider The Relativistic Heavy Ion Collider (RHIC ) is the first and one of only two operating heavy-ion colliders, and the only spin-polarized proton collider ever built. Located at Brookhaven National Laboratory (BNL) in Upton, New York, and used by a ...
* Symmetry (physics) * Parity * CPT invariance *
Chirality (physics) A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle ...
* Standard model * Gauge theory *
Tachyon A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
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Modern searches for Lorentz violation Modern may refer to: History * Modern history ** Early Modern period ** Late Modern period *** 18th century *** 19th century *** 20th century ** Contemporary history * Moderns, a faction of Freemasonry that existed in the 18th century Philosoph ...


Footnotes


References


Selected books

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Group theory in quantum physics

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Selected papers

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Further reading


Relativistic quantum mechanics and field theory

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Quantum theory and applications in general

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External links

* * * * * * * {{Quantum mechanics topics Quantum mechanics Mathematical physics Electromagnetism Particle physics Atomic physics Theory of relativity