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A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories.
Attempts to create a unified field theory based on classical physics are classical unified field theories. During the years between the two World Wars , the idea of unification of gravity with electromagnetism was actively pursued by several mathematicians and physicists like Einstein, Theodor Kaluza , [ 19 ] Hermann Weyl , [ 20 ] Arthur ...
The topic broadly splits into equations of classical field theory and quantum field theory. Classical field equations describe many physical properties like temperature of a substance, velocity of a fluid, stresses in an elastic material, electric and magnetic fields from a current, etc. [1] They also describe the fundamental forces of nature ...
Like many other classical unified field theorists, Eddington considered that in the Einstein field equations for general relativity the stress–energy tensor, which represents matter/energy, was merely provisional, and that in a truly unified theory the source term would automatically arise as some aspect of the free-space field equations.
When the canonical quantization procedure is applied to a field, such as the electromagnetic field, the classical field variables become quantum operators. Thus, the normal modes comprising the amplitude of the field are simple oscillators, each of which is quantized in standard first quantization, above, without ambiguity.
The most basic scalar field theory is the linear theory. Through the Fourier decomposition of the fields, it represents the normal modes of an infinity of coupled oscillators where the continuum limit of the oscillator index i is now denoted by x. The action for the free relativistic scalar field theory is then
In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that [citation needed] jet bundles and the variational bicomplex are the correct domain for such a description.
Covariant Hamiltonian field theory is the relativistic formulation of Hamiltonian field theory. Hamiltonian field theory usually means the symplectic Hamiltonian formalism when applied to classical field theory , that takes the form of the instantaneous Hamiltonian formalism on an infinite-dimensional phase space , and where canonical ...