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Klein–Gordon and Dirac equations

Attempts to combine quantum physics with special relativity began with building relativistic wave equations from the relativistic energy–momentum relation E 2 = (p c) 2 + (m 0 c 2) 2, {\displaystyle E^{2}=(pc)^{2}+\left(m_{0}c^{2}\right)^{2},} instead of nonrelativistic energy equations. The Klein–Gordon equation and the Dirac equation are two such equations. The Klein–Gordon equation, − 1 c 2 ∂ 2 ∂ t 2 ψ + ∇ 2 ψ = m 2 c 2 ℏ 2 ψ, {\displaystyle -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi +\nabla ^{2}\psi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi,} was the first such equation to be obtained, even before the nonrelativistic one-particle Schrödinger equation, and applies to massive spinless particles. Historically, Dirac obtained the Dirac equation by seeking a differential equation that would be first-order in both time and space, a desirable property for a relativistic theory. Taking the "square root" of the left-hand side of the Klein–Gordon equation in this way required factorizing it into a product of two operators, which Dirac wrote using 4 × 4 matrices α 1, α 2, α 3, β {\displaystyle \alpha _{1},\alpha _{2},\alpha _{3},\beta }. Consequently, the wave function also became a four-component function, governed by the Dirac equation that, in free space, read (β m c 2 + c (∑ n = 1 3 α n p n)) ψ = i ℏ ∂ ψ ∂ t. {\displaystyle \left(\beta mc^{2}+c\left(\sum _{n\mathop {=} 1}^{3}\alpha _{n}p_{n}\right)\right)\psi =i\hbar {\frac {\partial \psi }{\partial t}}.}

This has again the form of the Schrödinger equation, with the time derivative of the wave function being given by a Hamiltonian operator acting upon the wave function. Including influences upon the particle requires modifying the Hamiltonian operator. For example, the Dirac Hamiltonian for a particle of mass m and electric charge q in an electromagnetic field (described by the electromagnetic potentials φ and A) is: H ^ Dirac = γ 0, {\displaystyle {\hat {H}}_{\text{Dirac}}=\gamma ^{0}\left,} in which the γ = (γ, γ, γ) and γ are the Dirac gamma matrices related to the spin of the particle. The Dirac equation is true for all spin- 1 ⁄ 2 particles, and the solutions to the equation are 4-component spinor fields with two components corresponding to the particle and the other two for the antiparticle.

For the Klein–Gordon equation, the general form of the Schrödinger equation is inconvenient to use, and in practice the Hamiltonian is not expressed in an analogous way to the Dirac Hamiltonian. The equations for relativistic quantum fields, of which the Klein–Gordon and Dirac equations are two examples, can be obtained in other ways, such as starting from a Lagrangian density and using the Euler–Lagrange equations for fields, or using the representation theory of the Lorentz group in which certain representations can be used to fix the equation for a free particle of given spin (and mass).

In general, the Hamiltonian to be substituted in the general Schrödinger equation is not just a function of the position and momentum operators (and possibly time), but also of spin matrices. Also, the solutions to a relativistic wave equation, for a massive particle of spin s, are complex-valued 2(2 s + 1) -component spinor fields.