The Lagrangian for the standard model of particle physics uses Lagrangian densities since particle physics is based on field theories using space-time ψ(x,t).
In the simplest case for gravity the Lagrangian density ₤ is defined where
L = T - V = ∫ ₤d³x
T and V are the kinetic and potential energies respectively.
∫ ₤d³x is integrating over a volume.
The action S is defined as;
S = ∫ Ldt = ∫ ₤d⁴x
∫ ₤d⁴x is integrating over space-time.
Lagrangians in quantum field theories depend only on the fields and their first derivatives.
₤ → ₤(ψ, ∂ₙψ).
The Euler-Lagrange equation can formulated in terms of a quantum field theory.
δs = ∫ (δ₤)d⁴x
The terms in the square brackets are the
total derivative of δ₤.
The second term in the right hand most bracket can be integrated by parts and since the endpoints are fixed (refer
post#55);
Combining with the previous equation;
The Euler-Lagrange equation for the field is;
The Lagrangian density of a field is more complicated as the potential of the field and the density of the central mass needs to be considered.
Whereas the Lagrangian L of Newtonian gravity is simply T - V, the Lagrangian density ₤ of a gravitational field is;
▽ is the del operator
i(∂/∂x) +
j(∂/∂y) +
k(∂/∂z)
Φ is the gravitational potential.
ρ is the mass density.
G is the gravitational constant.
A property of the Lagrangian density is that it must remain invariant under mathematical transformations.
A typical transformation is a unitary transformation U = exp(iθ) which is a rotation.
If ₤ ≡ ₤(ψ) then ₤(U(ψ)) = ₤(ψ) = ₤.
Rotations can be global or local space-time.
For local transformations it is found the condition ₤(U(ψ)) = ₤(ψ) = ₤ is not met.
Physicists borrowed a technique from classical physics in electrodynamics; the electromagnetic field can be described by a potential Aₙ.
The corresponding space-time dependent field Aₙ = Aₙ(x) for every point x in the field is known as the gauge potential.
In a global transformation both the space-time ψ(x,t) and the first derivative ∂ₙψ(x,t) transform as
ψ → Uψ and ∂ₙψ → U∂ₙψ respectively.
The introduction of a gauge field allows the use of the
covariant derivative where;
Dₙψ = ∂ₙψ - iAₙψₙ
Locally the transformations are ψ → Uψ and Dₙψ → UDₙψ.
The covariant derivative preserves the invariance of ₤ in local space-time.
There also a physical significance of the invariance of ₤ locally under transformations which results in the creation of
gauge bosons or force carriers from the gauge field for the electromagnetic, weak and strong forces and have been experimentally confirmed.
