Inertia of a moving cavity

Summary: A moving-well sum rule gives count-additive inertia for quadratic kinetics, while an exact Dirac cavity derives inertia from confined field motion and exposes the confinement stress needed for a closed particle.

The inertial quantity to calculate

The mass-frequency relationship is already available: $\omega_m=mc^2/\hbar$. The proposed spectrum is

$$\begin{aligned} m(n,d)&=\mu_dD_N,\\ D_N&=\binom{N+d}{d},\qquad N=n-1. \end{aligned} \tag{1}$$

A route to its dynamics is to calculate the momentum response of an oscillating spatial pattern when its overall velocity changes. Continuous internal oscillation is compatible with zero total momentum. In a translation-invariant setting, the target coefficient is

$$\left.\frac{\partial P_i}{\partial v_j}\right|_{v=0} =\mu_dD_N\delta_{ij}. \tag{2}$$

The following two controls isolate the required calculation. They are declared spatial wave equations, rather than derived IDWT particles. The quadratic comparison contains a kinetic mass parameter. The Dirac comparison admits zero bare mass and has explicit spatial confinement. Its confining background is prescribed, so its momentum must not be silently identified with that of a closed particle plus its well.

All coordinates used below are ordinary spatial coordinates. The controls do not change the particle depth assignments, impose compactification, or make one particle a constituent of another.

An exact translation sum rule

Consider a fixed Hermitian multiplication potential $V(x)$ and

$$H_X=\frac{p^2}{2m_{\rm kin}}+V(x-X),\qquad p=-i\hbar\nabla. \tag{3}$$

Assume its selected discrete states are separated from the complementary spectrum. Moving to coordinates $y=x-X(t)$ gives, at constant velocity, the generator

$$H_v=H_0-v\cdot p. \tag{4}$$

Complete the momentum square, keeping its phase transformation:

$$\begin{aligned} U_v&=\exp(i m_{\rm kin}v\cdot y/\hbar),\\ H_v&=U_vH_0U_v^\dagger-\frac12m_{\rm kin}|v|^2. \end{aligned} \tag{5}$$

Every selected eigenvalue therefore changes by the same amount, $\epsilon_a(v)=\epsilon_a(0)-m_{\rm kin}|v|^2/2$, and its full moving field is $U_vu_a$. This statement holds for an anharmonic or quasiperiodically perturbed well as well as for a harmonic one.

Suppose there are $D$ equally occupied orthogonal modes, each with canonical norm $q$. The occupied generator trace and the resulting translation term in the effective action are

$$\mathcal E(v)=q\sum_{a=1}^{D}\epsilon_a(0) -\frac12qDm_{\rm kin}|v|^2, \qquad L_{\rm trans}=\frac12qDm_{\rm kin}|v|^2. \tag{6}$$

Hence

$$\boxed{M_{ij}=qDm_{\rm kin}\delta_{ij}.} \tag{7}$$

The equal contribution follows from the kinetic translation symmetry; it does not require equal individual eigenfrequencies. For a cumulative oscillator band, $D=D_N$ gives the simplex multiplier exactly.

The two physical inputs in this comparison remain explicit. The single-mode inertia $m_{\rm kin}$ is present in (3), and equal occupation $q$ is a stated preparation. At fixed total norm $Q=qD$, (7) is $M=m_{\rm kin}Q$ and has no count enhancement. The calculation does not select $q$, identify canonical norm with electric charge, or derive a physical kinetic mass from the IDWT master field.

The spectral response includes every complementary mode

Differentiating (4) by ordinary eigenvalue perturbation theory gives another expression for the same inertia. Let $P$ denote the occupied spectral subspace and put $\Delta_{ba}=\epsilon_b-\epsilon_a$. Summing over the occupied modes cancels transitions internal to $P$:

$$M_{ij}(0)=2q\operatorname{Re} \sum_{a\in P,\ b\notin P} \frac{\langle a|p_i|b\rangle\langle b|p_j|a\rangle} {\Delta_{ba}}. \tag{8}$$

For a time-reversal-invariant real well, the symmetric response at angular frequency $\Omega$ is

$$M_{ij}(\Omega)=2q\operatorname{Re} \sum_{a\in P,\ b\notin P} \frac{\Delta_{ba}\langle a|p_i|b\rangle\langle b|p_j|a\rangle} {\Delta_{ba}^2-(\hbar\Omega)^2}. \tag{9}$$

The retarded continuation specifies how a pole is approached. Equation (7) constrains (9) at zero frequency; it does not make the entire response independent of the well. Eigenfunction overlaps and transition gaps remain in the finite-frequency result. No division of a response by a fitted particle mass has been used to obtain (7).

For an isolated excited level, rather than an occupied trace, the sum is over every other level and retains the signed denominators. Dropping lower or negative-energy levels changes that response.

A first-order Dirac cavity with no bare mass

Set $\hbar=c=1$ for this section. On one ordinary spatial coordinate, consider the local two-component operator

$$H=\sigma_xp+b\sigma_yx+m_0\sigma_z, \qquad b>0. \tag{10}$$

The linear pseudoscalar profile confines the wave. The parameter $m_0$ is a stated constant mass offset, and can be set to zero. The complete operator retains both Dirac signs; these signs receive no particle–antiparticle interpretation here.

With oscillator length $b^{-1/2}$, (10) couples the upper Hermite mode $h_{n-1}$ to the lower mode $h_n$. The two eigenvalues for $n\ge1$ are

$$E_{n,\pm}=\pm\sqrt{m_0^2+2bn}. \tag{11}$$

There is also an unpaired lowest state. The finite-basis calculations below explicitly check the spatial residual of each selected paired state so that an artificial cutoff state cannot replace it.

For the prescribed translated well $b(x-vt)$, let $r=\sqrt{1-v^2}$ and use the generator $H-vp$. Its exact paired spectrum is

$$\boxed{\mathcal E_{n,\pm}(v) =\pm r\sqrt{m_0^2+2bnr}.} \tag{12}$$

This formula follows from the full Dirac transformation, including the well. In coordinates $x'=\gamma(x-vt)$, its slope is $b'=b/\gamma$. The stationary frequency there is $E'=\pm\sqrt{m_0^2+2b'n}$. Writing $t'=t/\gamma-\gamma v(x-vt)$ gives (12). In terms of a normalized stationary spinor $u_{n,b'}$, the actual moving profile is

$$\phi_{n,v}(y) =e^{iE'\gamma vy} e^{\frac12\operatorname{artanh}(v)\sigma_x} u_{n,b'}(\gamma y),\qquad y=x-vt. \tag{13}$$

The boost phase, the two spinor components and the changed spatial width all matter. Replacing (13) by an unchanged real envelope loses part of the momentum response.

For the positive branch, differentiating the moving eigenvalue gives the momentum of this field:

$$P_n(v)=-\frac{d\mathcal E_n(v)}{dv} =v\left[ \frac{\sqrt{m_0^2+2bnr}}{r} +\frac{bn}{\sqrt{m_0^2+2bnr}} \right]. \tag{14}$$

Consequently its adiabatic translation coefficient is

$$\boxed{M_n=E_n+\frac{bn}{E_n}.} \tag{15}$$

The second term is precisely the integrated kinetic stress: $\langle\sigma_xp\rangle=bn/E_n$. It follows either from the two Hermite components or from the spatial virial identity. In particular,

$$\boxed{m_0=0\quad\Longrightarrow\quad M_n=\frac32E_n>0.} \tag{16}$$

This derives nonzero translational inertia from a confined, continuously oscillating Dirac field with no bare mass offset. It does not derive the inertia of a closed physical particle: the source maintaining the linear well has been held external to (10).

The confinement stress belongs in the particle calculation

For a stationary localized stress-energy distribution, a small boost gives a translational momentum coefficient

$$\frac{\partial P_i}{\partial v_j} =\frac1{c^2}\left(E\delta_{ij} +\int T^{ij}\,d^dx\right). \tag{17}$$

For a complete conserved tensor in a flat translation-invariant domain, the integrated stationary stresses vanish under the required decay conditions. To see the condition, integrate $\partial_k(x^iT^{kj})=T^{ij}+x^i\partial_kT^{kj}$. Conservation and a vanishing boundary term give $\int T^{ij}=0$ for a stationary tensor. A periodic localized tensor gives the corresponding cycle-averaged identity. The covariant total momentum of a closed system then gives $M=E/c^2$. The geometric hypotheses and background-field qualifications are developed by Giulini.

The positive stress in (15) is therefore a specific term that a self-contained confinement calculation must balance. Within IDWT, the balancing contribution must come from the same master field and its spatial geometry. An uncounted external wall is not that derivation. On a curved sector, the available translation or Killing symmetries and the appropriate momentum definition must be stated separately.

Numerical and full-field checks

The scalar control uses a two-dimensional oscillator with a bounded quasiperiodic perturbation of operator norm at most $0.3$. Its cumulative degree-four band has fifteen states and remains separated by a gap. Two choices of the three phases change the spatial eigenfunctions and occupied energy sums, but give

$$M(0)=15I \tag{18}$$

to below $5\times10^{-13}$ in the final spatial basis. At $\Omega=0.6$, their $xx$ responses are approximately $23.86076$ and $23.69556$, respectively. Thus count-additive inertia coexists with different finite-frequency response in this specified action.

The driven calculation evolves all fifteen complex occupied columns under $H-v(t)p_x$, with $v(t)=10^{-3}\sin(0.4t)$ through one period. Halving the drive amplitude reduces the leading response error by approximately four, as expected for the first nonlinear correction. The linear prediction differs from the refined full momentum by $7.24\times10^{-8}$. Time refinement changes the complete fields by about $1.00\times10^{-9}$. Increasing the spatial cutoff from 24 to 32 changes the complete fields, after a fixed initial spectator-basis alignment, by $5.47\times10^{-7}$ in the reported norm. The actual complex trajectories and bases are retained with the report.

The Dirac checks use slopes $b=0.8$, offsets $m_0=0,0.7$, levels $n=1,4,12$, and velocities through $|v|=0.5$. Complete signed spectral sums reproduce (15). Independent coordinate quadrature verifies the norm, momentum, uncompressed spatial Dirac equation and the analytic complex spinor (13). The final residuals and profile errors are of order $10^{-14}$ to $10^{-13}$, with spatial cutoffs 64 and 96 agreeing.

For example, at $m_0=0.7,n=1$, the energy is $1.44568323$ and the inertia is $1.99905480$. Negative-energy intermediate modes contribute $-0.09890517$ to that inertia. Omitting them would give a different answer even though the occupied branch is positive.

The implementations are cavity_inertia.py and cavity_inertia_checks.py.

Consequence for the inverse mass problem

The count in (1) can enter a collective inertial trace without being the eigenvalue of a rank-one point-contact operator. Equation (7) establishes that possibility for a concrete kinetic law. Equations (10)–(17) identify the extra calculation required when inertia comes from the actual Dirac field motion: its full spatial spectrum and confinement stress must be included.

The route remains conditional on deriving the common per-mode inertia, the occupied mode content, the physical cavity and its same-field confinement. A count of available states alone supplies none of these dynamical inputs. The complementary Dirac calculation provides a related normal-mode energy and response construction; the homogeneous inverse action instead supplies prescribed simplex frequencies with a different mass mechanism.

The exact oscillator level moments are relevant when computing the energy of an occupied cavity. Summing oscillator energies and counting equally weighted modes are distinct operations. Finally, the spatial localization conditions remain necessary when promoting a finite mode or external-well control to an actual localized same-field particle.