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${\displaystyle i\hslash \frac{d}{dt}|\mathrm{\Psi }⟩=\hat{H}|\mathrm{\Psi }⟩}$ Schrödinger equation
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Inphysics, specificallyrelativistic quantum mechanics (RQM) and its applications toparticle physics,relativistic wave equations predict the behavior ofparticles at highenergies andvelocities comparable to thespeed of light. In the context ofquantum field theory (QFT), the equations determine the dynamics ofquantum fields.The solutions to the equations, universally denoted asψ orΨ (Greekpsi), are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of awave equation or are generated from aLagrangian density and the field-theoreticEuler–Lagrange equations (seeclassical field theory for background).

In theSchrödinger picture, the wave function or field is the solution to theSchrödinger equation,$${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi =\hat{H}\psi ,}$$one of thepostulates of quantum mechanics. All relativistic wave equations can be constructed by specifying various forms of theHamiltonian operatorĤ describing thequantum system. Alternatively,Feynman'spath integral formulation uses a Lagrangian rather than a Hamiltonian operator.

More generally – the modern formalism behind relativistic wave equations isLorentz group theory, wherein the spin of the particle has a correspondence with therepresentations of the Lorentz group.^[1]

The failure ofclassical mechanics applied tomolecular,atomic, andnuclear systems and smaller induced the need for a new mechanics:quantum mechanics. The mathematical formulation was led byDe Broglie,Bohr,Schrödinger,Pauli, andHeisenberg, and others, around the mid-1920s, and at that time was analogous to that of classical mechanics. The Schrödinger equation and theHeisenberg picture resemble the classicalequations of motion in the limit of largequantum numbers(higher energy levels) and as the reducedPlanck constantħ (the quantum ofaction) tends to zero. This is thecorrespondence principle. At this point,special relativity was not fully combined with quantum mechanics, so the Schrödinger and Heisenberg formulations, as originally proposed, could not be used in situations where the particles travel near thespeed of light, or when the number of each type of particle changes (this happens in realparticle interactions; the numerous forms ofparticle decays,annihilation,matter creation,pair production, and so on).

A description of quantum mechanical systems which could account forrelativistic effects was sought for by many theoretical physicists from the late 1920s to the mid-1940s.^[2] The first basis forrelativistic quantum mechanics, i.e. special relativity applied with quantum mechanics together, was found by all those who discovered what is frequently called theKlein–Gordon equation:

$${\displaystyle -{\hslash }^2\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{t}^2}+(\hslash c{)}^2{\mathrm{\nabla }}^2\psi =(m{c}^2{)}^2\psi ,}$$

1

by inserting theenergy operator andmomentum operator into the relativisticenergy–momentum relation:

$${\displaystyle E^2-(pc{)}^2=(m{c}^2{)}^2.}$$

2

The solutions to (1) arescalar fields. The KG equation is undesirable due to its prediction ofnegativeenergies andprobabilities, as a result of thequadratic nature of (2) – inevitable in a relativistic theory. This equation was initially proposed by Schrödinger, and he discarded it for such reasons, only to realize a few months later that its non-relativistic limit (what is now called theSchrödinger equation) was still of importance. Nevertheless, (1) is applicable to spin-0bosons.^[3]

Neither the non-relativistic nor relativistic equations found by Schrödinger could predict thefine structure in theHydrogen spectral series. The mysterious underlying property wasspin. The first two-dimensionalspin matrices (better known as thePauli matrices) were introduced by Pauli in thePauli equation; the Schrödinger equation with a non-relativistic Hamiltonian including an extra term for particles inmagnetic fields, but this wasphenomenological.Weyl found a relativistic equation in terms of the Pauli matrices; theWeyl equation, formassless spin-1/2 fermions. The problem was resolved byDirac in the late 1920s, when he furthered the application of equation (2) to theelectron – by various manipulations he factorized the equation into the form

$${\displaystyle \left(\frac{E}{c}-\mathit{\alpha }\cdot \mathbf{p}-\beta mc\right)\left(\frac{E}{c}+\mathit{\alpha }\cdot \mathbf{p}+\beta mc\right)\psi =0,}$$

3A

and one of these factors is theDirac equation (see below), upon inserting the energy and momentum operators. For the first time, this introduced new four-dimensional spin matricesα andβ in a relativistic wave equation, and explained the fine structure of hydrogen. The solutions to (3A) are multi-componentspinor fields, and each component satisfies (1). A remarkable result of spinor solutions is that half of the components describe a particle while the other half describe anantiparticle; in this case the electron andpositron. The Dirac equation is now known to apply for all massivespin-1/2fermions. In the non-relativistic limit, the Pauli equation is recovered, while the massless case results in the Weyl equation.

Although a landmark in quantum theory, the Dirac equation is only true for spin-1/2 fermions, and still predicts negative energy solutions, which caused controversy at the time (in particular – not all physicists were comfortable with the "Dirac sea" of negative energy states).

The natural problem became clear: to generalize the Dirac equation to particles withany spin; both fermions and bosons, and in the same equations theirantiparticles (possible because of thespinor formalism introduced by Dirac in his equation, and then-recent developments in spinor calculus byvan der Waerden in 1929), and ideally with positive energy solutions.^[2]

This was introduced and solved by Majorana in 1932, by a deviated approach to Dirac. Majorana considered one "root" of (3A):

$${\displaystyle \left(\frac{E}{c}+\mathit{\alpha }\cdot \mathbf{p}-\beta mc\right)\psi =0,}$$

3B

whereψ is a spinor field, now with infinitely many components, irreducible to a finite number oftensors or spinors, to remove the indeterminacy in sign. Thematricesα andβ are infinite-dimensional matrices, related to infinitesimalLorentz transformations. He did not demand that each component of3B satisfy equation (2); instead he regenerated the equation using aLorentz-invariantaction, via theprinciple of least action, and application of Lorentz group theory.^[4][5]

Majorana produced other important contributions that were unpublished, including wave equations of various dimensions (5, 6, and 16). They were anticipated later (in a more involved way) by de Broglie (1934), and Duffin, Kemmer, and Petiau (around 1938–1939) seeDuffin–Kemmer–Petiau algebra. The Dirac–Fierz–Pauli formalism was more sophisticated than Majorana's, as spinors were new mathematical tools in the early twentieth century, although Majorana's paper of 1932 was difficult to fully understand; it took Pauli and Wigner some time to understand it, around 1940.^[2]

Dirac in 1936, and Fierz and Pauli in 1939, built equations from irreducible spinorsA andB, symmetric in all indices, for a massive particle of spinn + 1/2 for integern (seeVan der Waerden notation for the meaning of the dotted indices):

$${\displaystyle p_{\gamma \dot{\alpha }}{A}_{{ϵ}_1{ϵ}_2\cdots {ϵ}_n}^{\dot{\alpha }{\dot{\beta }}_1{\dot{\beta }}_2\cdots {\dot{\beta }}_n}=mc{B}_{\gamma {ϵ}_1{ϵ}_2\cdots {ϵ}_n}^{{\dot{\beta }}_1{\dot{\beta }}_2\cdots {\dot{\beta }}_n},}$$

4A

$${\displaystyle p^{\gamma \dot{\alpha }}{B}_{\gamma {ϵ}_1{ϵ}_2\cdots {ϵ}_n}^{{\dot{\beta }}_1{\dot{\beta }}_2\cdots {\dot{\beta }}_n}=mc{A}_{{ϵ}_1{ϵ}_2\cdots {ϵ}_n}^{\dot{\alpha }{\dot{\beta }}_1{\dot{\beta }}_2\cdots {\dot{\beta }}_n},}$$

4B

wherep is the momentum as a covariant spinor operator. Forn = 0, the equations reduce to the coupled Dirac equations, andA andB together transform as the originalDirac spinor. Eliminating eitherA orB shows thatA andB each fulfill (1).^[2] The direct derivation of the Dirac–Pauli–Fierz equations using the Bargmann–Wigner operators is given by Isaev and Podoinitsyn.^[6]

In 1941, Rarita and Schwinger focussed on spin-3/2 particles and derived theRarita–Schwinger equation, including aLagrangian to generate it, and later generalized the equations analogous to spinn + 1/2 for integern. In 1945, Pauli suggested Majorana's 1932 paper toBhabha, who returned to the general ideas introduced by Majorana in 1932. Bhabha and Lubanski proposed a completely general set of equations by replacing the mass terms in (3A) and (3B) by an arbitrary constant, subject to a set of conditions which the wave functions must obey.^[7]

Finally, in the year 1948 (the same year asFeynman'spath integral formulation was cast),Bargmann andWigner formulated the general equation for massive particles which could have any spin, by considering the Dirac equation with a totally symmetric finite-component spinor, and using Lorentz group theory (as Majorana did): theBargmann–Wigner equations.^[2][8] In the early 1960s, a reformulation of the Bargmann–Wigner equations was made byH. Joos andSteven Weinberg, theJoos–Weinberg equation. Various theorists at this time did further research in relativistic Hamiltonians for higher spin particles.^[1][9][10]

The relativistic description of spin particles has been a difficult problem in quantum theory. It is still an area of the present-day research because the problem is only partially solved; including interactions in the equations is problematic, and paradoxical predictions (even from the Dirac equation) are still present.^[5]

The following equations have solutions which satisfy thesuperposition principle, that is, the wave functions areadditive.

Throughout, the standard conventions oftensor index notation andFeynman slash notation are used, including Greek indices which take the values 1, 2, 3 for the spatial components and 0 for the timelike component of the indexed quantities. The wave functions are denotedψ, and∂_μ are the components of thefour-gradient operator.

Inmatrix equations, thePauli matrices are denoted byσ^μ in whichμ = 0, 1, 2, 3, whereσ⁰ is the2 × 2identity matrix:and the other matrices have their usual representations. The expression$${\displaystyle {\sigma }^{\mu }{\mathrm{\partial }}_{\mu }\equiv {\sigma }^0{\mathrm{\partial }}_0+{\sigma }^1{\mathrm{\partial }}_1+{\sigma }^2{\mathrm{\partial }}_2+{\sigma }^3{\mathrm{\partial }}_3}$$is a2 × 2 matrixoperator which acts on 2-component spinor fields.

Thegamma matrices are denoted byγ^μ, in which againμ = 0, 1, 2, 3, and there are a number of representations to select from. The matrixγ⁰ isnot necessarily the4 × 4 identity matrix. The expressionis a4 × 4matrixoperator which acts on 4-componentspinor fields.

Note that terms such as "mc"scalar multiply anidentity matrix of the relevantdimension, the common sizes are2 × 2 or4 × 4, and areconventionally not written for simplicity.

Particlespin quantum numbers	Name	Equation	Typical particles the equation describes
0	Klein–Gordon equation	${\displaystyle (\hslash {\mathrm{\partial }}_{\mu }+imc)(\hslash {\mathrm{\partial }}^{\mu }-imc)\psi =0}$	Massless or massive spin-0 particle (such asHiggs bosons).
1/2	Weyl equation	${\displaystyle {\sigma }^{\mu }{\mathrm{\partial }}_{\mu }\psi =0}$	Massless spin-1/2 particles.
	Dirac equation	${\displaystyle \left(i\hslash \mathrm{\partial }/-mc\right)\psi =0}$	Massive spin-1/2 particles (such as electrons).
	Two-body Dirac equations	${\displaystyle [({\gamma }_1{)}_{\mu }(p_1-{\tilde{A}}_1{)}^{\mu }+m_1+{\tilde{S}}_1]\mathrm{\Psi }=0,}$ ${\displaystyle [({\gamma }_2{)}_{\mu }(p_2-{\tilde{A}}_2{)}^{\mu }+m_2+{\tilde{S}}_2]\mathrm{\Psi }=0.}$	Massive spin-1/2 particles (such as electrons).
	Majorana equation	${\displaystyle i\hslash \mathrm{\partial }/\psi -mc{\psi }_c=0}$	MassiveMajorana particles.
	Breit equation	${\displaystyle i\hslash \frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }t}=\left(\sum _i{\hat{H}}_D(i)+\sum _{i>j}\frac{1}{r_{ij}}-\sum _{i>j}{\hat{B}}_{ij}\right)\mathrm{\Psi }}$	Two massive spin-1/2 particles (such aselectrons) interacting electromagnetically to first order in perturbation theory.
1	Maxwell's equations (inQED using theLorenz gauge)	${\displaystyle {\mathrm{\partial }}_{\mu }{\mathrm{\partial }}^{\mu }{A}^{\nu }=e\overline{\psi }{\gamma }^{\nu }\psi }$	Photons, massless spin-1 particles.
	Proca equation	${\displaystyle {\mathrm{\partial }}_{\mu }({\mathrm{\partial }}^{\mu }{A}^{\nu }-{\mathrm{\partial }}^{\nu }{A}^{\mu })+{\left(\frac{mc}{\hslash }\right)}^2{A}^{\nu }=0}$	Massive spin-1 particle (such asW and Z bosons).
3/2	Rarita–Schwinger equation	${\displaystyle {ϵ}^{\mu \nu \rho \sigma }{\gamma }^5{\gamma }_{\nu }{\mathrm{\partial }}_{\rho }{\psi }_{\sigma }+m{\psi }^{\mu }=0}$	Massive spin-3/2 particles.
s	Bargmann–Wigner equations	whereψ is a rank-2s 4-componentspinor.	Free particles of arbitrary spin (bosons and fermions).[9][11]
	Joos–Weinberg equation	${\displaystyle [(i\hslash {)}^{2s}{\gamma }^{{\mu }_1{\mu }_2\cdots {\mu }_{2s}}{\mathrm{\partial }}_{{\mu }_1}{\mathrm{\partial }}_{{\mu }_2}\cdots {\mathrm{\partial }}_{{\mu }_{2s}}+(mc{)}^{2s}]\psi =0}$	Free particles of arbitrary spin (bosons and fermions).

TheDuffin–Kemmer–Petiau equation is an alternative equation for spin-0 and spin-1 particles:$${\displaystyle (i\hslash {\beta }^a{\mathrm{\partial }}_a-mc)\psi =0}$$

Start with the standardspecial relativity (SR) 4-vectors

• 4-position${\displaystyle X^{\mu }=\mathbf{X}=(ct,\overrightarrow{\mathbf{x}})}$
• 4-velocity${\displaystyle U^{\mu }=\mathbf{U}=\gamma (c,\overrightarrow{\mathbf{u}})}$
• 4-momentum${\displaystyle P^{\mu }=\mathbf{P}=\left(\frac{E}{c},\overrightarrow{\mathbf{p}}\right)}$
• 4-wavevector${\displaystyle K^{\mu }=\mathbf{K}=\left(\frac{\omega }{c},\overrightarrow{\mathbf{k}}\right)}$
• 4-gradient${\displaystyle {\mathrm{\partial }}^{\mu }=\mathrm{\partial }=\left(\frac{{\mathrm{\partial }}_t}{c},-\overrightarrow{\mathrm{\nabla }}\right)}$

Note that each 4-vector is related to another by aLorentz scalar:

• ${\displaystyle \mathbf{U}=\frac{d}{d\tau }\mathbf{X}}$, where${\displaystyle \tau }$ is theproper time
• ${\displaystyle \mathbf{P}=m_0\mathbf{U}}$, where${\displaystyle m_0}$ is therest mass
• ${\displaystyle \mathbf{K}=(1/\hslash )\mathbf{P}}$, which is the4-vector version of thePlanck–Einstein relation & thede Brogliematter wave relation
• ${\displaystyle \mathrm{\partial }=-i\mathbf{K}}$, which is the4-gradient version ofcomplex-valuedplane waves

Now, just apply the standard Lorentz scalar product rule to each one:

• ${\displaystyle \mathbf{U}\cdot \mathbf{U}=(c{)}^2}$
• ${\displaystyle \mathbf{P}\cdot \mathbf{P}=(m_{0}c{)}^2}$
• ${\displaystyle \mathbf{K}\cdot \mathbf{K}={\left(\frac{m_{0}c}{\hslash }\right)}^2}$
• ${\displaystyle \mathrm{\partial }\cdot \mathrm{\partial }={\left(\frac{-i{m}_{0}c}{\hslash }\right)}^2=-{\left(\frac{m_{0}c}{\hslash }\right)}^2}$

The last equation is a fundamental quantum relation.

When applied to a Lorentz scalar field${\displaystyle \psi }$, one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations.

• ${\displaystyle \left[\mathrm{\partial }\cdot \mathrm{\partial }+{\left(\frac{m_{0}c}{\hslash }\right)}^2\right]\psi =0}$: in 4-vector format
• ${\displaystyle \left[{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}^{\mu }+{\left(\frac{m_{0}c}{\hslash }\right)}^2\right]\psi =0}$: in tensor format
• ${\displaystyle \left[(\hslash {\mathrm{\partial }}_{\mu }+i{m}_{0}c)(\hslash {\mathrm{\partial }}^{\mu }-i{m}_{0}c)\right]\psi =0}$: in factored tensor format

TheSchrödinger equation is the low-velocitylimiting case (v ≪c) of the Klein–Gordon equation.

When the relation is applied to a four-vector field${\displaystyle A^{\mu }}$ instead of a Lorentz scalar field${\displaystyle \psi }$, then one gets theProca equation (inLorenz gauge):$${\displaystyle \left[\mathrm{\partial }\cdot \mathrm{\partial }+{\left(\frac{m_{0}c}{\hslash }\right)}^2\right]A^{\mu }=0}$$

If the rest mass term is set to zero (light-like particles), then this gives the freeMaxwell equation (inLorenz gauge)$${\displaystyle [\mathrm{\partial }\cdot \mathrm{\partial }]A^{\mu }=0}$$

Under a properorthochronous Lorentz transformationx → Λx inMinkowski space, all one-particle quantum statesψ^j_σ of spinj with spin z-componentσ locally transform under somerepresentationD of theLorentz group:^[12][13]$${\displaystyle \psi (x)\to D(\mathrm{\Lambda })\psi ({\mathrm{\Lambda }}^{-1}x)}$$whereD(Λ) is some finite-dimensional representation, i.e. a matrix. Hereψ is thought of as acolumn vector containing components with the allowed values ofσ. Thequantum numbersj andσ as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value ofσ may occur more than once depending on the representation. Representations with several possible values forj are considered below.

Theirreducible representations are labeled by a pair of half-integers or integers(A,B). From these all other representations can be built up using a variety of standard methods, like takingtensor products anddirect sums. In particular,space-time itself constitutes a4-vector representation(⁠1/2⁠,⁠1/2⁠) so thatΛ ∈D^{(1/2, 1/2)}. To put this into context;Dirac spinors transform under the(⁠1/2⁠, 0) ⊕ (0,⁠1/2⁠) representation. In general, the(A,B) representation space hassubspaces that under thesubgroup of spatialrotations,SO(3), transform irreducibly like objects of spinj, where each allowed value:$${\displaystyle j=A+B,A+B-1,\dots ,|A-B|,}$$occurs exactly once.^[14] In general,tensor products of irreducible representations are reducible; they decompose as direct sums of irreducible representations.

The representationsD^{(j, 0)} andD^(0,j) can each separately represent particles of spinj. A state or quantum field in such a representation would satisfy no field equation except the Klein–Gordon equation.

There are equations which have solutions that do not satisfy the superposition principle.

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

• Einstein field equations: describe interaction of matter with thegravitational field (massless spin-2 field):$${\displaystyle R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R+g_{\mu \nu }\mathrm{\Lambda }=\frac{8\pi G}{c^4}{T}_{\mu \nu }}$$ The solution is ametrictensor field, rather than a wave function.

• List of equations in nuclear and particle physics
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• Lorentz transformation
• Mathematical descriptions of the electromagnetic field
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• Minimal coupling
• Scalar field theory
• Status of special relativity

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