# Theories Of The Lagrangian And Its Symmetries

Decent Essays
2.1. The Lagrangian and its symmetries
In classical dynamics, a system’s equations of motion can be derived from its Lagrangian, L; the difference between the kinetic and potential energies of the system, here denoted by T and V respectively. This is often given as a function of a set of generalized coordinates, qi
, and their time derivatives, ˙qi
. This treatment is sufficient to describe a discrete system of particles, however in order to properly describe a continuous system, the Lagrangian must be replaced by a Lagrangian density: L(qi
, q˙i) → L(φi
, ∂µφi). (1)
Whereby the generalized coordinates qi are replaced by the fields φi(x µ ), and the time derivatives are replaced by the derivative of these fields with respect to each of the four space-time coordinates,
(2)
(It should be noted that henceforth a system’s Lagrangian Density will be referred to only as its
Lagrangian and no reference will be made to a system’s Lagrangian.)
To formulate a Quantum Field Theory, one must replace the single particle wavefunctions of quantum mechanics with multi-particle excitations of a quantum field that must satisfy the field equation derived from the system’s Lagrangian. The Lagrangian for a spinor field ψ(x) satisfying the free-particle Dirac equation is given by [9]
LD = iψγµ
∂µψ − mψψ (3) where γ µ represents the four Dirac gamma matrices, and ψ(x) the four component complex spinor; ψ(x) =

 ψ1 ψ2 ψ3 ψ4


, γ0 =

1 0
0