Scalar-tensor Theory - Tensor Fields and Field Theory

Tensor Fields and Field Theory

Modern physics tries to derive all physical theories from as few principles as possible. In this way, Newtonian mechanics as well as quantum mechanics are derived from Hamilton's principle of least action. In this approach, the behavior of a system is not described via forces, but by functions which describe the energy of the system. Most important are the energetic quantities known as the Hamilton function (or Hamiltonian) and the Lagrange function (or Lagrangian). Their derivatives in space are known as Hamiltonian or Hamilton density and Lagrangian or Lagrange density. Going to these quantities leads to the field theories.

Modern physics uses field theories to explain reality. These fields can be scalar, vectorial or tensorial. For them, there is:

• Scalars are tensors of rank zero.
• Vectors are tensors of rank one.
• Matrices are tensors of rank two.

Scalars are numbers, quantities of the form f(x), like the temperature. Vectors are more general and show a direction. In them, every component of the direction is a scalar. Tensors (degree 2) are a wider generalization, the most well known example of which are matrices (that can give equation systems). Higher order tensors are found for example in the deformation theory and in General Relativity.