masssquared

Mass squared, denoted m^2, is the square of a particle’s invariant mass. In special relativity, energy, momentum and mass satisfy the relation E^2 = p^2 c^2 + m^2 c^4. Equivalently, m^2 = (E^2/c^4) − (p^2/c^2). In natural units (c = 1) this becomes E^2 = p^2 + m^2 and m^2 = E^2 − p^2. The invariant mass is the same in all inertial frames, unlike energy and momentum, which transform with the frame.

If m^2 > 0 the particle is massive and has a time-like four-momentum; if m^2 = 0 the particle is massless (such as the photon); if m^2 < 0 the four-momentum is space-like, a tachyonic possibility that implies imaginary mass and is not observed in known physics.

In quantum field theory the parameter m^2 appears in the Lagrangian as a mass term, and in the Klein-Gordon equation (□ + m^2) φ = 0. Mass terms are central to spontaneous symmetry breaking: a negative m^2 in the potential V(φ) ∝ −m^2 φ^2 can drive symmetry breaking, with the physical mass set by the curvature of the potential at its minimum.

Experimentally and phenomenologically, one often reconstructs the invariant mass of a system from its constituents: m^2 = (Σ E_i)^2/c^4 − (Σ p_i)^2/c^2. In quantum field theory, bare masses receive renormalization, shifting m^2 from its classical value. The squared mass also appears in discussions of neutrino mass differences, Δm^2, relevant to oscillations and related phenomena.