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.
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.
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.
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.
[2] 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.
[3] Neither the non-relativistic nor relativistic equations found by Schrödinger could predict the fine structure in the Hydrogen spectral series.
For the first time, this introduced new four-dimensional spin matrices α and β in a relativistic wave equation, and explained the fine structure of hydrogen.
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 with any spin; both fermions and bosons, and in the same equations their antiparticles (possible because of the spinor formalism introduced by Dirac in his equation, and then-recent developments in spinor calculus by van der Waerden in 1929), and ideally with positive energy solutions.
[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) see Duffin–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 spinors A and B, symmetric in all indices, for a massive particle of spin n + 1⁄2 for integer n (see Van der Waerden notation for the meaning of the dotted indices): where p is the momentum as a covariant spinor operator.
[6] In 1941, Rarita and Schwinger focussed on spin-3⁄2 particles and derived the Rarita–Schwinger equation, including a Lagrangian to generate it, and later generalized the equations analogous to spin n + 1⁄2 for integer n. In 1945, Pauli suggested Majorana's 1932 paper to Bhabha, 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 as Feynman's path integral formulation was cast), Bargmann and Wigner 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): the Bargmann–Wigner equations.
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.
[5] The following equations have solutions which satisfy the superposition principle, that is, the wave functions are additive.
Throughout, the standard conventions of tensor index notation and Feynman 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 gamma matrices are denoted by γμ, in which again μ = 0, 1, 2, 3, and there are a number of representations to select from.
Note that terms such as "mc" scalar multiply an identity matrix of the relevant dimension, the common sizes are 2 × 2 or 4 × 4, and are conventionally not written for simplicity.
If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation (in Lorenz gauge)
Under a proper orthochronous Lorentz transformation x → Λx in Minkowski space, all one-particle quantum states ψjσ of spin j with spin z-component σ locally transform under some representation D of the Lorentz group:[12][13]
Here ψ is thought of as a column vector containing components with the allowed values of σ.
From these all other representations can be built up using a variety of standard methods, like taking tensor products and direct sums.
In general, the (A, B) representation space has subspaces that under the subgroup of spatial rotations, SO(3), transform irreducibly like objects of spin j, where each allowed value: