The complete energy of a retained spatial source

Mass is the selected particle frequency in equivalent units. A proposed same-field realization must also supply its corresponding inertial response. Once a sustaining mode is retained dynamically, its spatial kinetic energy and interactions belong in that calculation.

Stationary identity for a quartic action

Consider canonical Dirac fields $\Phi_j=e^{-i\omega_jt}\phi_j(x)$ with finite integrals and vanishing boundary terms. Write their complete energy as $E=T+U_2+U_4$, where $T$ is the spatial Dirac kinetic energy, $U_2$ is quadratic in the fields, and $U_4$ is quartic. Here the symbols denote integrated quantities. Let $Q_j=\int\phi_j^\dagger\phi_j$.

Multiplying each stationary field equation by its conjugate and summing gives Euler's homogeneous-function identity,

$$\sum_j\omega_jQ_j=T+U_2+2U_4, \qquad \boxed{E=\sum_j\omega_jQ_j-U_4.} \tag{1}$$

Additional binding terms must contribute their actual homogeneous variations; a smoothed or fractional binding term cannot silently be counted as quartic. Equation (1) is not a formula for particle mass from amplitudes. It is a consistency test on the energy of the specified stationary solution and its canonical normalization.

For a pure cross interaction between target $A$ and retained source $B$, let the target's other energy terms vanish on its solved branch. Then $\omega_aQ_a=U_\times$, while the normal equation gives $\omega_bQ_b=T_b+U_\times$ if it has no other terms. The complete energy is therefore

$$E=\omega_aQ_a+T_b. \tag{2}$$

The algebraic homogeneous comparison has $T_b=0$. A spatial source does not acquire this identity merely by reproducing the same target source matrix. Its kinetic expectation must be evaluated, along with any binding contributions omitted from the hypotheses of (2).

Spatial scaling supplies another necessary condition

Suppose the geometry is ordinary flat $\mathbb R^d$, the coefficients are spatially constant, and all currents and boundary terms allow the norm-preserving variation $\phi_j(x)\mapsto\lambda^{d/2}\phi_j(\lambda x)$. Then $T\mapsto\lambda T$, $U_2\mapsto U_2$, and $U_4\mapsto\lambda^dU_4$. Stationarity at fixed $Q_j$ implies

$$T+dU_4=0,\qquad E=U_2-(d-1)U_4, \quad \sum_j\omega_jQ_j=U_2-(d-2)U_4. \tag{3}$$

In particular, a massless canonical Dirac action with a nonnegative local quartic potential cannot have a nontrivial finite stationary solution of positive energy under these scaling hypotheses when $d>1$. This does not apply to the earlier inverse binding action, to a nonzero background without a justified subtraction, or to a different spatial geometry. It identifies which extra physical structure a proposed localized positive-frequency realization must account for.

Equations (1)–(3) follow directly from the stated action and admissible variations. A numerical stationary calculation should check the full equations, their energy identity and their spatial virial independently.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.