Quantum Field Theory/Classical Lagrangian field theory

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Special relativity[edit | edit source]

Special relativity was proposed by Albert Einstein in the beginning of the 20th century. The Special Theory of Relativity is a successor of Classical Mechanics, which is based on Newtonian mechanics, which was developed by Isaac Newton (as the name suggests). Classical mechanics is valid to a good accuracy in day-to-day phenomena involving speeds much less than the speed of light. However, at speeds comparable to the speed of light, classical mechanics breaks down. Classical mechanics is mainly based on invariance under Galilean transformations. This tells us how a phenomenon oberseved in one reference frame would appear in another reference frame which has a different velocity relative to the original reference frame . According to the Galilean transformation, the coordinates transform as follows:

On the other hand, the special theory of relativity is based on invariance under the Lorentz transformation,

where Here, it is assumed that the reference frame had a velocity with respect to in direction.

Note that under the Lorentz transformation, the interval remains unchanged. Or, in other words, the interval transforms like a scalar under the Lorentz transformation. The time and space coordinates together form a four vector . Any quantity which transforms like the space-time coordinates under Lorentz transformation is defined as a four-vector. An example of a four-vector other than itself is the energy-momentum or the momentum four vector . The dual of a fourvector is denoted by . The dual vector is related to as . A product of a vector with a dual vector transforms like a scalar. Such a product is called as the inner product.

Variational principle[edit | edit source]

Action and Lagrangian[edit | edit source]

In classical mechanics, the action and the Lagrangian are related as follows:

These two quantities are defined similarly in quantum field theory. However, in quantum field theory it is often convenient to introduce a Lagrangian density . Hence the action can also be defined as:

Variational principle[edit | edit source]

One of the most important principles in physics which is also often called "Stationary Action Principle" or "Least Action Principle". Can be formulated in several ways:

  1. Of all possible fields with a given boundary condition the one that provides an extremum (often minimum, cf. Least Action) of the action is The Solution.
  2. The field for which the variation of the action vanishes is The Solution.

In other words if is The Solution and we add an arbitrary small variation to it then the (linear part of the) variation of the action vanishes, .

Note that the variation must not change the boundary condition of and must therefore vanish at the boundary.

Note also that the action must be real (just to talk about minima) and must be a 4-scalar (Lorentz invariant).

Euler-Lagrange equation[edit | edit source]

In classical mechanics, the Lagrangian is a function of the canonical coordinates and the canonical momenta . The Euler-Lagrange Equation is as follows:

In quantum field theory, however, the two variables of the Lagrangian are the fields and the corresponding derivatives and . Furthermore, quantum field theory treats time and spatial derivatives at equal footing. Thus, the Euler-Lagrange Equation reads:

where is the Lagrangian density.

Translation invariance, energy and momentum[edit | edit source]

Energy-momentum tensor[edit | edit source]

Conservation of energy and momentum[edit | edit source]

Hamiltonian[edit | edit source]

Conserved current[edit | edit source]