In quantum field theory, the kinetic energy term in the Lagrangian describes the dynamics of the field and its derivatives. It does not explicitly contain a mass term because the mass of a field is associated with the potential energy term.
To understand why this is the case, let's consider the simplest example of a scalar field, such as the Higgs field. The Lagrangian density for the scalar field can be written as:
L = (∂μφ)(∂μφ) - V(φ)
Here, (∂μφ)(∂μφ) represents the kinetic energy term, and V(φ) represents the potential energy term. The kinetic energy term involves the derivatives (∂μφ), which describe how the field changes in space and time.
The absence of a mass term in the kinetic energy arises from the requirement of Lorentz invariance, which is a fundamental symmetry of special relativity. Lorentz invariance dictates that the Lagrangian must remain the same under Lorentz transformations, which include boosts and rotations in spacetime.
Including a mass term in the kinetic energy would break this Lorentz invariance because the mass term introduces a preferred reference frame and violates the principle of relativity. Therefore, the mass term is incorporated into the potential energy term instead.
It's important to note that while the kinetic energy term does not contain a mass term, the presence of a non-zero mass for a field is reflected in the potential energy term of the Lagrangian. The potential energy term is responsible for the field acquiring a mass through the Higgs mechanism, for example.