Relativistic wave equations

In <a href="/facts/Physics/a7aH2y08">physics</a>, specifically <a href="/facts/Relativistic_quantum_mechanics/DhkDNnHf">relativistic quantum mechanics</a> (RQM) and its applications to <a href="/facts/Particle_physics/YgIuyj2t">particle physics</a>, relativistic wave equations predict the behavior of <a href="/facts/Particles/JHKr2w6h">particles</a> at high <a href="/facts/Energy/wd8PrQM7">energies</a> and <a href="/facts/Velocity/Q3o1LVDe">velocities</a> comparable to the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a>. In the context of <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> (QFT), the equations determine the dynamics of <a href="/facts/Quantum_field/VoewsNJV">quantum fields</a>.
The solutions to the equations, universally denoted as ψ or Ψ (<a href="/facts/Greek_language/pzDd6eIz">Greek</a> <a href="/facts/Psi_(letter)/AUdz70Lq">psi</a>), are referred to as "<a href="/facts/Wave_function/KgM5ykjY">wave functions</a>" in the context of RQM, and "<a href="/facts/Field_(physics)/FJdAKeRA">fields</a>" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a> or are generated from a <a href="/facts/Lagrangian_density/p9BCUz57">Lagrangian density</a> and the field-theoretic <a href="/facts/Euler%25E2%2580%2593Lagrange_equation/Ymq3r3Dh">Euler–Lagrange equations</a> (see <a href="/facts/Classical_field_theory/7HuKY1qu">classical field theory</a> for background).
In the <a href="/facts/Schr%25C3%25B6dinger_picture/4VK89MbE">Schrödinger picture</a>, the wave function or field is the solution to the <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a>,

i
        ℏ
        
          
            ∂
            
              ∂
              t
            
          
        
        ψ
        =
        
          
            
              H
              ^
            
          
        
        ψ
        ,
      
    
    {\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi ={\hat {H}}\psi ,}

one of the <a href="/facts/Mathematical_formulation_of_quantum_mechanics/fRLw4sFU">postulates of quantum mechanics</a>. All relativistic wave equations can be constructed by specifying various forms of the <a href="/facts/Hamiltonian_(quantum_mechanics)/5XwK2HmD">Hamiltonian operator</a> Ĥ describing the <a href="/facts/Physical_system/tdSfxvIB">quantum system</a>. Alternatively, <a href="/facts/Richard_Feynman/32vGFrtB">Feynman</a>'s <a href="/facts/Path_integral_formulation/6lt2MHkW">path integral formulation</a> uses a Lagrangian rather than a Hamiltonian operator.
More generally – the modern formalism behind relativistic wave equations is <a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a> theory, wherein the spin of the particle has a correspondence with the <a href="/facts/Representations_of_the_Lorentz_group/XJRuPVkL">representations of the Lorentz group</a>.

Relativistic wave equations open-in-new

Relativistic wave equations