Relativistic wave equations

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 context of quantum field theory (QFT), the equations determine the dynamics of quantum fields. − ℏ 2 ∂ 2 ψ ∂ t 2 + ( ℏ c ) 2 ∇ 2 ψ = ( m c 2 ) 2 ψ , {displaystyle -hbar ^{2}{frac {partial ^{2}psi }{partial t^{2}}}+(hbar c)^{2} abla ^{2}psi =(mc^{2})^{2}psi ,,}     (1) E 2 − ( p c ) 2 = ( m c 2 ) 2 , {displaystyle E^{2}-(pc)^{2}=(mc^{2})^{2},,}     (2) ( E c − α ⋅ p − β m c ) ( E c + α ⋅ p + β m c ) ψ = 0 , {displaystyle left({frac {E}{c}}-{oldsymbol {alpha }}cdot mathbf {p} -eta mc ight)left({frac {E}{c}}+{oldsymbol {alpha }}cdot mathbf {p} +eta mc ight)psi =0,,}     (3A) ( E c + α ⋅ p − β m c ) ψ = 0 , {displaystyle left({frac {E}{c}}+{oldsymbol {alpha }}cdot mathbf {p} -eta mc ight)psi =0,,}     (3B) p γ α ˙ A ϵ 1 ϵ 2 ⋯ ϵ n α ˙ β ˙ 1 β ˙ 2 ⋯ β ˙ n = m c B γ ϵ 1 ϵ 2 ⋯ ϵ n β ˙ 1 β ˙ 2 ⋯ β ˙ n {displaystyle p_{gamma {dot {alpha }}}A_{epsilon _{1}epsilon _{2}cdots epsilon _{n}}^{{dot {alpha }}{dot {eta }}_{1}{dot {eta }}_{2}cdots {dot {eta }}_{n}}=mcB_{gamma epsilon _{1}epsilon _{2}cdots epsilon _{n}}^{{dot {eta }}_{1}{dot {eta }}_{2}cdots {dot {eta }}_{n}}}     (4A) p γ α ˙ B γ ϵ 1 ϵ 2 ⋯ ϵ n β ˙ 1 β ˙ 2 ⋯ β ˙ n = m c A ϵ 1 ϵ 2 ⋯ ϵ n α ˙ β ˙ 1 β ˙ 2 ⋯ β ˙ n {displaystyle p^{gamma {dot {alpha }}}B_{gamma epsilon _{1}epsilon _{2}cdots epsilon _{n}}^{{dot {eta }}_{1}{dot {eta }}_{2}cdots {dot {eta }}_{n}}=mcA_{epsilon _{1}epsilon _{2}cdots epsilon _{n}}^{{dot {alpha }}{dot {eta }}_{1}{dot {eta }}_{2}cdots {dot {eta }}_{n}}}     (4B) [ ( γ 2 ) μ ( p 2 − A ~ 2 ) μ + m 2 + S ~ 2 ] Ψ = 0. {displaystyle Psi =0.} ( − i ℏ γ μ ∂ μ + m c ) α 2 α 2 ′ ψ α 1 α 2 ′ α 3 ⋯ α 2 s = 0 {displaystyle (-ihbar gamma ^{mu }partial _{mu }+mc)_{alpha _{2}alpha _{2}'}psi _{alpha _{1}alpha '_{2}alpha _{3}cdots alpha _{2s}}=0} where ψ is a rank-2s 4-component spinor. 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 context of quantum field theory (QFT), the equations determine the dynamics of quantum fields. The solutions to the equations, universally denoted as ψ or Ψ (Greek psi), 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 a wave equation or are generated from a Lagrangian density and the field-theoretic Euler–Lagrange equations (see classical field theory for background). In the Schrödinger picture, the wave function or field is the solution to the Schrödinger equation; one of the postulates of quantum mechanics. All relativistic wave equations can be constructed by specifying various forms of the Hamiltonian operator Ĥ describing the quantum system. Alternatively, Feynman's path integral formulation uses a Lagrangian rather than a Hamiltonian operator. More generally – the modern formalism behind relativistic wave equations is Lorentz group theory, wherein the spin of the particle has a correspondence with the representations of the Lorentz group. The failure of classical mechanics applied to molecular, atomic, and nuclear systems and smaller induced the need for a new mechanics: quantum mechanics. The mathematical formulation was led by De Broglie, Bohr, Schrödinger, Pauli, and Heisenberg, and others, around the mid-1920s, and at that time was analogous to that of classical mechanics. The Schrödinger equation and the Heisenberg picture resemble the classical equations of motion in the limit of large quantum numbers and as the reduced Planck constant ħ, the quantum of action, tends to zero. This is the correspondence 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 the speed of light, or when the number of each type of particle changes (this happens in real particle interactions; the numerous forms of particle decays, annihilation, matter creation, pair production, and so on). A description of quantum mechanical systems which could account for relativistic effects was sought for by many theoretical physicists; from the late 1920s to the mid-1940s. The first basis for relativistic quantum mechanics, i.e. special relativity applied with quantum mechanics together, was found by all those who discovered what is frequently called the Klein–Gordon equation: by inserting the energy operator and momentum operator into the relativistic energy–momentum relation: The solutions to (1) are scalar fields. The KG equation is undesirable due to its prediction of negative energies and probabilities, as a result of the quadratic 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 the Schrödinger equation) was still of importance. Nevertheless, – (1) is applicable to spin-0 bosons.