Classical Field Theory
Classical Mechanics is formulated for several particles. You can study systems constituted by a huge amount of particles, but they are all independent, there is no notion os "closeness". If we want our theory to be "local" we need to introduce fields. Is not the same as Continuum Mechanics?
Example.
Think in a particle
If we had lots of particles
representing a string. See also this.
To my knowledge (20/4/22) there are different approaches to Classical Field Theory:
- If we start from Lagrangian Mechanics (particles) we obtain Lagrangian field theory.
- If we start from Hamiltonian mechanics (particles) we obtain Hamiltonian field theory
- If we take into account that the change of the point of view of observers is not the Galilean group but the Lorentz group, that is, we consider special relativity, then we arrive to Relativistic Field Theory (I suppose it can be Lagrangian and Hamiltonian).
Why do we introduce fields?
In classical physics, the primary reason for introducing the concept of the field is to construct laws of Nature that are local. The old laws of Coulomb and Newton involve “action at a distance.” This means that the force felt by an electron (or planet) changes immediately if a distant proton (or star) moves. This situation is philosophically unsatisfactory. More importantly, it is also experimentally incorrect. The field theories of Maxwell and Einstein remedy the situation, with all interactions mediated locally by the field. The requirement of locality remains a strong motivation for studying field theories in the quantum world.
Extracted from here.
See also this in page 17, although I don't understand yet. Also in this video.
Mathematical formalism
According to this video.
A classical field theory is the following data:
- A
-dimensional manifold (space-time) , - A target space
, to construct the space of fields . - A Lagrangian, which is a polynomial
, for , where .
In most examples
Examples:
, , is a polynomial. , , where and ,
Type of fields
- Scalar fields. Section of a rank 1 vector bundle over spacetime. Related to spin 0 particles.
- Vector fields. Section of a rank
vector bundle over a -dimensional spacetime. Related to spin 1 particles. - Spinor fields (classical spinor field?). When they say a particle has spin 1/2 what they mean is that the particle is the quantum version of a classical spinor field, that is, for every point
in the (3+1)-spacetime we have a vector in such a way that if we consider a change of variables around then acts on by means of a spin representation of .