Relativistic wave equations
In physics, specifically relativistic quantum mechanics 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, the equations determine the dynamics of quantum fields.
The solutions to the equations, universally denoted as or , 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.
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.
History
Early 1920s: Classical and quantum mechanics
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, 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.Late 1920s: Relativistic quantum mechanics of spin-0 and spin- particles
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 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 – 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 was still of importance. Nevertheless, – is applicable to spin-0 bosons.
Neither the non-relativistic nor relativistic equations found by Schrödinger could predict the fine structure in the Hydrogen spectral series. The mysterious underlying property was spin. The first two-dimensional spin matrices were introduced by Pauli in the Pauli equation; the Schrödinger equation with a non-relativistic Hamiltonian including an extra term for particles in magnetic fields, but this was phenomenological. Weyl found a relativistic equation in terms of the Pauli matrices; the Weyl equation, for massless spin- fermions. The problem was resolved by Dirac in the late 1920s, when he furthered the application of equation to the electron – by various manipulations he factorized the equation into the form:
and one of these factors is the Dirac equation, upon inserting the energy and momentum operators. For the first time, this introduced new four-dimensional spin matrices and in a relativistic wave equation, and explained the fine structure of hydrogen. The solutions to are multi-component spinor fields, and each component satisfies. A remarkable result of spinor solutions is that half of the components describe a particle while the other half describe an antiparticle; in this case the electron and positron. The Dirac equation is now known to apply for all massive spin- fermions. 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- fermions, and still predicts negative energy solutions, which caused controversy at the time.
1930s–1960s: Relativistic quantum mechanics of higher-spin particles
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, and ideally with positive energy solutions.This was introduced and solved by Majorana in 1932, by a deviated approach to Dirac. Majorana considered one "root" of :
where is a spinor field now with infinitely many components, irreducible to a finite number of tensors or spinors, to remove the indeterminacy in sign. The matrices and are infinite-dimensional matrices, related to infinitesimal Lorentz transformations. He did not demand that each component of to satisfy equation, instead he regenerated the equation using a Lorentz-invariant action, via the principle of least action, and application of Lorentz group theory.
Majorana produced other important contributions that were unpublished, including wave equations of various dimensions. They were anticipated later by de Broglie, and Duffin, Kemmer, and Petiau 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.
Dirac in 1936, and Fierz and Pauli in 1939, built equations from irreducible spinors and, symmetric in all indices, for a massive particle of spin for integer :
where is the momentum as a covariant spinor operator. For, the equations reduce to the coupled Dirac equations and and together transform as the original Dirac spinor. Eliminating either or shows that and each fulfill.
In 1941, Rarita and Schwinger focussed on spin- particles and derived the Rarita–Schwinger equation, including a Lagrangian to generate it, and later generalized the equations analogous to spin for integer. 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 and by an arbitrary constant, subject to a set of conditions which the wave functions must obey.
Finally, in the year 1948, 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 : the Bargmann–Wigner equations. In the early 1960s, a reformulation of the Bargmann–Wigner equations was made by H. Joos and Steven Weinberg, the Joos–Weinberg equation. Various theorists at this time did further research in relativistic Hamiltonians for higher spin particles.
1960s–present
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 are still present.Linear equations
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 wave functions are denoted ', and are the components of the four-gradient operator.
In matrix equations, the Pauli matrices are denoted by ' in which, where is the identity matrix:
and the other matrices have their usual representations. The expression
is a matrix operator which acts on 2-component spinor fields.
The gamma matrices are denoted by ', in which again, and there are a number of representations to select from. The matrix is not necessarily the identity matrix. The expression
is a matrix operator which acts on 4-component spinor fields.
Note that terms such as "" scalar multiply an identity matrix of the relevant dimension, the common sizes are or, and are conventionally not written for simplicity.