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This is commonly taken as a general solution to the free Klein–Gordon equation. Note that because the initial Fourier transformation contained Lorentz invariant quantities like p ⋅ x = p μ x μ {\displaystyle p\cdot x=p_{\mu }x^{\mu }} only, the last expression is also a Lorentz invariant solution to the Klein–Gordon equation.
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 !
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions.
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 classical equations of motion of a field are typically identical in form to the (quantum) equations for the wave-function of one of its quanta. For example, the Klein–Gordon equation is the classical equation of motion for a free scalar field, but also the quantum equation for a scalar particle wave-function.
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.
Moreover, the free fields operators, i.e. when interactions are assumed not to exist, turn out to (formally) satisfy the same equation as do the fields (wave functions) in many cases. Thus the Klein–Gordon equation (spin 0) and the Dirac equation (spin 1 ⁄ 2) in this guise remain in the theory.