In quantum field theory (QFT), the action integral for a free particle is expressed using the Lagrangian density. The Lagrangian density describes the dynamics of a field, and for a free particle, it simplifies to a term involving the field and its derivatives. Here's the general form of the action integral for a free particle in QFT:
S = ∫ d^4x L
where S is the action, d^4x represents integration over all four spacetime coordinates (three spatial dimensions and time), and L is the Lagrangian density.
For a free particle, the Lagrangian density is given by:
L = ψ̄(x)(iγμ∂μ - m)ψ(x)
In this expression, ψ(x) represents the quantum field associated with the particle, ψ̄(x) is the adjoint field (complex conjugate transpose), γμ are the Dirac matrices, ∂μ represents the partial derivative with respect to the spacetime coordinate μ, and m is the mass of the particle.
Plugging this Lagrangian density into the action integral, we have:
S = ∫ d^4x ψ̄(x)(iγμ∂μ - m)ψ(x)
The action integral describes the total variation of the fields and their derivatives in the system, and it is a fundamental concept in the path integral formulation of QFT. By varying the action with respect to the fields, one can derive the equations of motion for the quantum field and study the behavior of the free particle within the framework of quantum field theory.