Derivation 4.5A — Coframe evolution and probability conservation

Keywords

theoretical physics, foundations of physics, momentum conservation, canonical momentum, special relativity, stationary gravity, gravitoelectromagnetism, quantum mechanics, FLRW cosmology

Purpose and assumptions

A varying Stage changes spatial derivatives, spin transport and the measure used to normalize a state. This derivation supports proposed §4.5, “Quantum motion in deformed space,” and the changing-Stage boundary in proposed §4.9. It derives a complete prescribed-background spinor control, including the terms that a shell calculation alone cannot fix.

Assume an oriented, torsion-free, metric-compatible spin geometry with a smooth prescribed Stage-adapted coframe, \(\phi^0{}_i=0\), \(N=\phi^0{}_0>0\). Retain the minimally coupled spinor equation and the Clifford convention of Derivation 4.4A. The background is not varied dynamically here. Define calculation aliases

\[ E^a{}_i=\phi^a{}_i,\quad w^i=(E^{-1})^i{}_a\phi^a{}_0,\quad h_{ij}=\delta_{ab}E^a{}_iE^b{}_j,\quad W=\det E>0. \]

Then

\[ d\ell^0=Ndx^0,\qquad d\ell^a=E^a{}_i(dx^i+w^idx^0), \] \[ ds^2=-N^2(dx^0)^2+h_{ij}(dx^i+w^idx^0)(dx^j+w^jdx^0). \]

Here \(h^{ij}\) means the inverse spatial metric, not the spatial block of the inverse spacetime metric. Physical spatial volume is \(W\,d^3x\). Coordinate momentum is \(P_i=p_{{\mathrm{coord}},i}\), and \(p_{{\mathrm{local}},a}=(E^{-1})^i{}_aP_i\).

Construct the connection before solving for evolution

Let \(\Gamma^\rho{}_{\mu\nu}\) be the Levi-Civita connection of \(g=\phi^{\mathsf T}\eta\phi\). In matrix notation \((\Gamma_\mu)^\rho{}_\nu=\Gamma^\rho{}_{\mu\nu}\). Coframe compatibility fixes the local frame connection:

\[ \mathcal A_\mu=(\phi\Gamma_\mu-\partial_\mu\phi)\phi^{-1}. \]

Its Lorentz-lowered pair is antisymmetric. For the convention \(\{\gamma^a,\gamma^b\}=-2\eta^{ab}I\), the compatible spin lift is

\[ \mathsf A_\mu=-\frac14\eta_{ac}\mathcal A_\mu{}^c{}_b\gamma^a\gamma^b, \qquad D_\mu=\partial_\mu+\mathsf A_\mu. \]

Indeed the Clifford commutator identity yields

\[ \partial_\mu\gamma^\nu+\Gamma^\nu{}_{\mu\lambda}\gamma^\lambda +[\mathsf A_\mu,\gamma^\nu]=0, \qquad \gamma^\mu=(\phi^{-1})^\mu{}_a\gamma^a. \]

This supplies an explicit construction and a check of every sign. It is not an independent gravitational field fitted after the generator is written.

Since \(\gamma^0=\beta/N\), define the Hermitian velocity matrices

\[ A^i=N\alpha^a(E^{-1})^i{}_a-w^i I=N\beta\gamma^i. \]

Multiplying the stipulated equation \((i\hbar\gamma^\mu D_\mu-p_f)\Psi=0\) by \(N\beta\) gives

\[ i\hbar\partial_0\Psi=\widehat M_{t,{\mathrm{raw}}}\Psi, \] \[ \widehat M_{t,{\mathrm{raw}}} =A^i\widehat P_i+\beta Np_f -i\hbar(\mathsf A_0+A^i\mathsf A_i), \qquad\widehat P_i=-i\hbar\partial_i. \]

The shift is an identity-valued drift \(-w^iP_i\), together with its connection effects. It is not generally a fixed momentum subtraction inside the \(\alpha\) term.

Put the changing norm into a fixed Hilbert space

The slice norm is \(\int W\Psi^\dagger\Psi\,d^3x\). Set \(\psi=W^{1/2}\Psi\). Differentiating both \(W^{-1/2}\psi\) in space and \(W^{1/2}\Psi\) in time gives

\[ \boxed{\widehat M_t=A^i\widehat P_i+\beta Np_f+Z,} \] \[ \boxed{Z=-i\hbar(\mathsf A_0+A^i\mathsf A_i) +\frac{i\hbar}{2} \bigl(A^i\partial_i\ln W+I\partial_0\ln W\bigr).} \]

Evolution is \(i\hbar\partial_0\psi=\widehat M_t\psi\), now with norm \(\int\psi^\dagger\psi\,d^3x\). The temporal derivative is essential whenever the spatial measure changes. This is the changing-inner-product issue treated in the curved-spinor literature, for example by Huang and Parker.

Equivalently,

\[ \widehat M_t=\frac12\{A^i,\widehat P_i\}+\beta Np_f+V_{\mathrm{spin}}, \qquad V_{\mathrm{spin}}=Z+\frac{i\hbar}{2}\partial_iA^i. \]

The spin potential is fixed by the compatible lift. Symmetrizing the derivative term by hand without this remainder is not the general coframe equation.

Continuity, adjointness and the boundary term

The spinor equation and its adjoint give \(\nabla_\mu j_{\mathrm{prob}}^\mu=0\), where \(j_{\mathrm{prob}}^\mu=\bar\Psi\gamma^\mu\Psi\) and \(\bar\Psi=\Psi^\dagger\beta\). Since \(\sqrt{-g}=NW\),

\[ \sqrt{-g}\,j_{\mathrm{prob}}^0=W\Psi^\dagger\Psi, \qquad \sqrt{-g}\,j_{\mathrm{prob}}^i=W\Psi^\dagger A^i\Psi. \]

Therefore

\[ \boxed{\partial_0(\psi^\dagger\psi) +\partial_i(\psi^\dagger A^i\psi)=0.} \]

In ordinary time the probability flux is \(c\psi^\dagger A^i\psi\). It is not the gravitational source tensor.

The same identity follows by subtracting the generator equation and its conjugate. The required coefficient relation is

\[ Z-Z^\dagger=-i\hbar\partial_iA^i. \]

For test spinors \(u,v\) in a spatial region \(\Omega\), integration by parts now gives the exact Green identity

\[ \langle u,\widehat M_t v\rangle- \langle\widehat M_t u,v\rangle =-i\hbar\int_{\partial\Omega}u^\dagger A^iv\,n_i\,dS. \]

The volume terms cancel; formal adjointness therefore requires a boundary domain, not merely real eigenvalues of the shell symbol. In the raw representation the corresponding coordinate-measure identity is instead

\[ \widehat M_{t,{\mathrm{raw}}}^\dagger W-W\widehat M_{t,{\mathrm{raw}}} =i\hbar\partial_0W. \]

To see the sign, differentiate \(\langle\Psi,W\Psi\rangle\): the two evolution terms contribute \((i/\hbar)\langle\Psi,(M_{t,{\mathrm{raw}}}^\dagger W-WM_{t,{\mathrm{raw}}})\Psi\rangle\), which cancels \(\langle\Psi,\partial_0W\Psi\rangle\). Setting the raw adjoint defect to zero while \(W\) changes would violate norm conservation.

Three explicit controls

Static isotropic space

Take \(E=b(\mathbf x)I\), \(w=0\) and \(F=N/b\). The scalar \(b\) is a spatial scale, distinct from the indexed four/six maps \(L\). For variation along \(x^1\), the nonzero spin coefficients are

\[ \mathsf A_0=\frac{N'}{2b}\alpha^1,\qquad \mathsf A_2=\frac{b'}{2b}\gamma^1\gamma^2,\qquad \mathsf A_3=\frac{b'}{2b}\gamma^1\gamma^3. \]

Thus \(\mathsf A_0+A^i\mathsf A_i=(N'/(2b)+Nb'/b^2)\alpha^1\). Combining this with \(\partial_1\ln W=3b'/b\) yields \(Z=-i\hbar\alpha^1F'/2\). The vector form is

\[ \boxed{\widehat M_t=\beta Np_f+ \frac12\{F,\boldsymbol\alpha\cdot\widehat{\mathbf P}\}.} \]

Its full semiclassical dispersion matrix, including the zeroth-order fermic term, squares to \(N^2(p_f^2+|\mathbf P|^2/b^2)I\). This is not the derivative-only principal symbol used to define characteristic cones. Its positive eigenvalue is \(M_t=NM=M_r\), with \(\mathbf p_r=F\mathbf P\) and \(p_{f,r}=Np_f\). Both slots use the same comparison. This control agrees with the Hermitian isotropic construction of Obukhov, Silenko and Teryaev.

A rotating flat-coordinate frame

Take \(N=1\), \(E=I\), \(w=\boldsymbol\Omega\times\mathbf x/c\), with constant \(\boldsymbol\Omega\). For rotation around \(z\), write \(\omega=\Omega/c\). The compatible spin lift gives \(\mathsf A_0=-i\omega\Sigma_z/2\) and \(\mathsf A_i=0\), where \(\Sigma_z=\operatorname{diag}(\sigma_z,\sigma_z)\). Hence

\[ \widehat M_t=\beta p_f+\boldsymbol\alpha\cdot\widehat{\mathbf P} -\frac1c\boldsymbol\Omega\cdot (\mathbf L_{\mathrm{orb}}+\mathbf S), \qquad \mathbf S=\frac\hbar2\boldsymbol\Sigma. \]

The shift supplies the orbital term and the connection supplies spin rotation. The geometry is flat: this checks frame transport, not a sourced rotating gravitational field. A timelike rotating laboratory lies inside its light cylinder; no global positivity claim for the rotating generator is required.

A homogeneous changing Stage

For \(N=1\), \(w=0\), \(E=\operatorname{diag}(a_1,a_2,a_3)\), let \(H_i=\partial_0\ln a_i\). The compatible lift has \(\mathsf A_0=0\), \(\mathsf A_i=(\partial_0a_i)\alpha^i/2\). Consequently \(A^i\mathsf A_i=\sum_iH_i I/2\), and its contribution to \(Z\) cancels \(i\hbar\partial_0\ln W/2\) exactly. The result is

\[ \widehat M_t=\beta p_f+\sum_i\alpha^i\widehat P_i/a_i. \]

For \(a_i=a\), identify Expansion’s supplied yield \(\chi=1/a>0\):

\[ \widehat M_\chi=\beta p_f+\chi\boldsymbol\alpha\cdot\widehat{\mathbf P}, \qquad M_\chi=\sqrt{p_f^2+\chi^2|\mathbf P|^2}. \]

Spatial homogeneity conserves the canonical labels \(P_i\). A narrow packet following an instantaneous branch has \(\dot x^i=c\chi^2P_i/M_\chi\). The factor \(\chi^2\) follows by differentiating the branch magnitude, not from an additional law. The background history remains supplied, not solved here.

Bounded unitary evolution and the positive-sector boundary

A clean existence statement uses a finite time interval and a compact spatial slice without boundary, for example a periodic control. Assume smooth coefficients with bounded derivatives, \(N\) bounded away from zero, and uniformly nondegenerate \(E\). The rescaled equation is a linear symmetric hyperbolic system with time matrix \(I\). Sobolev Cauchy existence, uniqueness and continuous dependence supply forward and backward evolution for smooth data. The proved continuity law preserves its \(L^2\) norm. Extension by density and backward evolution give an onto isometry, hence a unitary propagator of norm one. “Bounded” here describes the propagator and time interval, not the generally unbounded differential generator.

For stationary coefficients this is a strongly continuous unitary group. If also \(|w|_h<N\), the derivative symbol has no zero eigenvalue at nonzero covector, so it is elliptic; on the smooth compact slice its self-adjoint realization has domain \(H^1\). With a physical boundary one must specify a maximal admissible boundary domain. Requiring zero flux of each individual state alone is not such a specification. MIT-type reflecting boundary problems are an established example under their own hypotheses; see Große and Murro. Noncompact falloff, singular charts and nonlinear backreaction are outside this bounded statement.

Unitarity does not imply preservation of an instantaneous positive spectral subspace. For one homogeneous momentum mode and a polarization block, write

\[ h_j=p_f\sigma_z+\chi_jP\sigma_x,\qquad \epsilon_j=\sqrt{p_f^2+\chi_j^2P^2}. \]

At a sudden change the spinor is continuous. Projecting the old positive state onto the new negative subspace gives

\[ \begin{aligned} P_{+\to-} &=\operatorname{tr}\left[ \frac{I-h_2/\epsilon_2}{2}\frac{I+h_1/\epsilon_1}{2}\right]\\ &=\frac12\left[1- \frac{p_f^2+\chi_1\chi_2P^2}{\epsilon_1\epsilon_2}\right]. \end{aligned} \]

The quench is a separate piecewise-constant control, not a smoothness premise of the Cauchy theorem. For smooth change set \(\tan\vartheta=\chi P/p_f\). Rotation to the instantaneous basis supplies an off-diagonal term of magnitude \(\hbar|\partial_0\vartheta|/2\), against gap \(2\epsilon\). Thus the local adiabatic diagnostic is

\[ \partial_0\vartheta=\frac{p_fP\partial_0\chi}{p_f^2+\chi^2P^2}, \qquad \frac{\hbar|\partial_0\vartheta|}{4\epsilon}\ll1. \]

A controlled branch approximation also needs a nonclosing gap, sufficiently regular slow history and control of accumulated transitions over the interval; a small instantaneous ratio alone is not a global error theorem. For \(p_f=0\), fixed nonzero \(P\) and positive \(\chi\), the eigendirection is constant and this mixing vanishes. Neither example identifies a massless spinor with a photon or derives particle creation without a field-state prescription.

Result and return to the chapter

The prescribed minimal spinor control has an explicit generator, conserved probability flux and a bounded unitary Cauchy evolution under stated domain hypotheses. It matches the static local/reference shell and Expansion’s homogeneous operator. Proposed §§4.5 and 4.9 can use those results while distinguishing full-spinor evolution from branch-following approximations. The construction does not select native microscopic coupling or a self-consistent quantum Stage law.