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Any solution of the free Dirac equation is, for each of its four components, a solution of the free Klein–Gordon equation. Despite historically it was invented as a single particle equation the Klein–Gordon equation cannot form the basis of a consistent quantum relativistic one-particle theory, any relativistic theory implies creation and ...
The Klein paradox is an unexpected consequence of relativity on the interaction of quantum particles with electrostatic potentials. The quantum mechanical problem of free particles striking an electrostatic step potential has two solutions when relativity is ignored.
The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important in quantum field theory. These functions are most simply defined in terms of the vacuum expectation value of products of field operators.
A classical free scalar field satisfies the Klein–Gordon equation. If a scalar field is denoted φ {\displaystyle \varphi } , a quartic interaction is represented by adding a potential energy term ( λ / 4 !
However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes.
The Higgs field φ satisfies the Klein–Gordon equation. The weak interaction fields Z, W ± satisfy the Proca equation. These equations can be solved exactly. One usually does so by considering first solutions that are periodic with some period L along each spatial axis; later taking the limit: L → ∞ will lift this periodicity restriction.
The equations for stationary configurations of the FK model reduce to those of the standard map or Chirikov–Taylor map of stochastic theory. [1] In the continuum-limit approximation the FK model reduces to the exactly integrable sine-Gordon (SG) equation, which allows for soliton solutions.
The above matrix operator contracts with one bispinor index of ψ at a time (see matrix multiplication), so some properties of the Dirac equation also apply to the BW equations: the equations are Lorentz covariant, all components of the solutions ψ also satisfy the Klein–Gordon equation, and hence fulfill the relativistic energy–momentum ...