The classical field theory Lan70 describes certain physical
systems as infinite-dimensional classical objects whose states need as
many classical coordinates as there are points in the 3D space.
Therefore, index that two minutes ago was used to enumerate the
classical coordinates, now changes into the space point
the coordinate itself - into the value of a certain function
, called the field, at point
In the physical world, the classical fields described above replace forces that act between particles. The whole Universe is thus composed of (classical) particles and (classical) fields. Particles are sources of fields, and fields exert forces on particles. The novelty here is the notion that a particle does not ``feel'' other particles; it only feels the fields generated by other particles. The so-called action at a distance disappeared from the theory, and a change of position of one particle, influences other particles only after the field it generates propagates to the rest of the world.
It is clear that the classical field theory is tailored to address
the question of time propagation of fields, and makes the full sense
within the relativistic approach where all fields propagate with
one common and unchangeable velocity. Classical electrodynamics and
classical gravity are theories of this type. Relativistic invariance
takes then the place of a basic ingredient of the theory, and, e.g.,
action corresponding to
Lagrangian (11) is manifestly relativistically invariant,