Constraint-Induced Effective Mass in Massless Field Propagation

TL;DR

The paper addresses how constraints on massless field propagation modify the low-energy dispersion by modeling restrictions with a linear operator $C$ acting on the mode space, which yields a constrained subspace $\ker C$ and a positive semidefinite operator $M=C^\dagger C$ encoding the spectral effect. It introduces an enforcement framework $\mathcal{D}_{\mathrm{eff}}(\alpha)=\mathcal{D}_0+\alpha\mathcal{M}$ with $\mathcal{M}=C^\dagger C$, showing that in the limit $\alpha\to\infty$ a lowest-lying spectral branch governed by $\ker C$ emerges and the leading gap is controlled by the smallest nonzero eigenvalue $\lambda_1$ of $\mathcal{M}$, via $\omega_0^2=\beta\lambda_1$. The Mass Induction Principle formalizes this as: if the constrained spectrum has a gapped lowest branch with $\omega_0>0$, the dispersion takes a Proca-like form with $m_{\mathrm{eff}}^2 c^4 = \hbar^2 \omega_0^2 = \hbar^2 \beta\lambda_1$, and the rank of $C$ bounds the number of penalised directions (with the actual observable branches depending on symmetry). Canonical realisations in plasmas, superconductors, and periodic media illustrate the mechanism as a structural, environment-dependent effect that does not modify the underlying massless field equations or introduce new dynamics.

Abstract

Constrained propagation of massless fields is ubiquitous in physical systems, arising from boundaries, material structure, or other restrictions on admissible modes. This paper shows that such constraints generically induce mass-like terms in the effective dispersion relation, without modifying the underlying field equations or introducing new degrees of freedom. Working at an abstract level, constraints are represented as linear operators acting on the field's mode space. Restriction of the admissible mode manifold produces a spectral gap whose magnitude is set by the smallest non-zero eigenvalue of an associated positive semidefinite operator. This gap may be identified with an effective mass parameter, yielding a Proca-like dispersion relation in the long-wavelength limit. The resulting Mass Induction Principle identifies rank reduction of the accessible mode space as the structural mechanism responsible for effective mass generation in constrained massless fields. Familiar systems such as plasmas, superconductors, and periodic media realise this structure as special cases, without introducing new dynamics. The analysis is deliberately dispersion-level and non-phenomenological: it does not assert a field-theoretic mass term, does not address vacuum propagation, and does not make claims about bounds on intrinsic particle masses.