IDOS Energy, Coherence, and Supporting Dynamics — Part 6

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For a real field with fixed orthogonal $J^2=-1$, define

$$\rho=|\psi|^2/2,\qquad e=\psi/|\psi|,\qquad a_i=\langle Je,\partial_i e\rangle,\qquad B_{ij}=\langle\partial_i e,\partial_j e\rangle-a_i a_j.$$

The positive-semidefinite matrix $B$ describes spatial orientation gradients. The underlying coordinates remain ordinary physical coordinates in $\mathbb R^{10}$. Real field-component pairs are not additional dimensions. Let $L_0>0$ be a supplied length and choose one positive scalar function $m(C)$ of $C=L_0^2B$. The action is

$$\mathcal L=\int\frac12\langle J\psi,\partial_t\psi\rangle\,dx-H, \qquad H=\int\rho w\,dx,\qquad w=\sqrt{m(C)^2+c^2|a|^2}.$$

Its canonical equation is $\psi_t=J\delta H/\delta\psi$ and its complete action and translation momentum are $S=\int\rho$ and $P_i=-\int\rho a_i$. Write $M_{ij}=\partial w/\partial B_{ij}$, $b_i=c^2a_i/w$, $q_i=\rho(b_i-2M_{ij}a_j)$ and $Q=1-ee^T$. The full functional derivative is

$$\frac{\delta H}{\delta\psi}=w\psi+\frac{QG}{|\psi|},\qquad G=-2q_iJ\partial_i e-(\partial_iq_i)Je -2\partial_i(\rho M_{ij}\partial_j e).$$

At a constant selected matrix with $Dm=0$, the exact equation reduces to

$$\psi_t=b\cdot\nabla\psi+\frac12(\nabla\cdot b)\psi +(w-b\cdot a)J\psi.$$

The tangent projector supplies the essential last subtraction. Let $e_0$ be a unit texture with $a_0=0$, constant $C=C_*$, $m(C_*)=\omega$ and $Dm(C_*)=0$. For any admissible smooth envelope $f$ and constant physical covector $p$, put $w_p=\sqrt{\omega^2+c^2|p|^2}$ and $b=c^2p/w_p$. Then the complete spatial field

$$\psi(t,x)=e^{J(p\cdot x+w_pt)}f(x+bt)e_0(x+bt)$$

solves this equation. Both the envelope and its internal texture translate. The complete quantities satisfy

$$E=Sw_p,\qquad P=-Sp,\qquad v=-b,\qquad E^2-c^2|P|^2=(S\omega)^2,\qquad P=Ev/c^2.$$

The comoving clock is $\omega^2/w_p$, and the rest inertia is $S\omega/c^2>0$. The speed parameter $c$ was inserted into the action; these branch relations do not establish full Lorentz covariance or a Lorentz-contracted profile. Fixing $S=A$ gives the common integrated $E_{\rm rest}/\omega=A$ without fitting a separate mass amplitude.

A Common Finite-Ladder Matrix Law

For $K_N=\{k\in\mathbb N_0^d:|k|_1\le N\}$ and $D=\binom{N+d}{d}$, take mutually orthogonal real component vectors $u_k,v_k$ whose $J$ images are also mutually orthogonal, and set

$$e_{0,N}=D^{-1/2}\sum_{k\in K_N} [u_k\cos(k\cdot x/L_0)+v_k\sin(k\cdot x/L_0)].$$

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