Normal Dirac extension and the canonical mass coefficient

Summary: A full Dirac field on a conditional spatial half-cylinder induces the absolute tangential Dirac operator, while its kinetic norm cancels a constant bulk mass factor and local scalar contact excites complementary normal profiles.

Obtain a nonlocal operator from ordinary spatial continuation

The Killing-spinor transfer gives a Hermitian scalar-mass candidate when its polynomial is evaluated on the absolute spatial Dirac operator:

$$\widehat M_d=\frac{Q_d(R|\mathscr D|)}{d!},\qquad Q_d(x)=\prod_{j=1}^{d/2}(x-j)(x+j-1). \tag{1}$$

It has the desired simplex eigenvalues on both Dirac signs, under the specified round geometry. The absolute value makes it spatially nonlocal. A natural question is whether ordinary spatial continuation of the same field can supply that absolute value.

An exact product-space calculation does supply it as a map from a field's value to its perpendicular derivative. However, that same continuation induces a nontrivial time-kinetic norm. A constant bulk scalar coefficient remains constant after canonical normalization. A local nonlinear scalar source also drives normal profiles outside the initially selected decaying profile.

These statements concern a conditional geometry and boundary problem. Take a six-dimensional spatial manifold $Q_6$ and one ordinary perpendicular coordinate $y\ge0$, with metric $dy^2+g_{Q_6}$. The six tangential directions remain equivalent in kind. No dimension is compactified and no small geometric radius is assumed. This is a local normal-direction benchmark, not an assignment of a seven-dimensional particle or a derivation of the full six-to-ten nesting. Multiple normal directions require their complete Clifford structure and cannot be supplied by independent scalar copies.

A nested spatial submanifold does not itself impose a hard boundary, a one-sided domain or a decaying boundary condition. Those are explicit extra assumptions here. All retained and complementary profiles below are modes of the same Dirac field.

Keep the full parent spinor

Let the tangential operators satisfy

$$\mathscr D^\dagger=\mathscr D,\qquad \beta^\dagger=\beta,\qquad \beta^2=I,\qquad \{\mathscr D,\beta\}=0,\qquad A=|\mathscr D|>0. \tag{2}$$

Assume a positive spectral gap, as on the conditional round sphere, where $A$ has values $(k+3)/R$. Initially work on smooth finite spectral sums; the integrated formulas then extend on their indicated quadratic-form domains.

The parent scalar matrix must be distinct from the new spatial Clifford matrix. On the full doubled spinor bundle set

$$\mathbb D= \begin{pmatrix}\mathscr D&0\\0&\mathscr D\end{pmatrix},\quad \alpha_y= \begin{pmatrix}0&\beta\\\beta&0\end{pmatrix},\quad \beta_p= \begin{pmatrix}\beta&0\\0&-\beta\end{pmatrix}. \tag{3}$$

The two last matrices are Hermitian involutions. They anticommute with each other and with $\mathbb D$. For an ordinary rank-eight six-dimensional spinor, (3) retains all sixteen components of this parent Hamiltonian. These are field components, not additional spatial coordinates or particles.

The massless parent spatial operator is

$$H_0=\mathbb D-i\alpha_y\partial_y. \tag{4}$$

This product form belongs to the standard theory of Dirac Cauchy data and their spectral projectors; see Sharma, Introduction, pages 2–3. The particular lift, norm and scalar-action calculations needed here follow directly below.

Construct the complete decaying lift

Define the tangential involution and the parent isometry

$$T=i\beta\mathscr D A^{-1},\qquad T^\dagger=T,\qquad T^2=I,\qquad V=\frac1{\sqrt2}\begin{pmatrix}I\\T\end{pmatrix}. \tag{5}$$

Write $A_p=\operatorname{diag}(A,A)$ and $K=i\alpha_y\mathbb D$. Direct multiplication gives

$$\begin{gathered} K^\dagger=K,\qquad K^2=A_p^2,\qquad KV=VA,\\ V^\dagger V=I,\qquad VV^\dagger=\Pi_+=\frac12(I+KA_p^{-1}),\\ \beta_pV=V\beta,\qquad V^\dagger\alpha_yV=0. \end{gathered} \tag{6}$$

Thus $V$ spans the entire decaying half of the parent data and retains every tangential polarization. The scalar matrix preserves this subspace. The last identity makes its normal flux vanish for arbitrary complex data.

For a tangential trace variable $f$, set

$$F(y)=V e^{-yA}f. \tag{7}$$

Since $H_0=-i\alpha_y(\partial_y+K)$, equations (6)–(7) give

$$\boxed{ H_0F=0,\qquad -\partial_yF(0)=VAf. } \tag{8}$$

The positive operator $A=|\mathscr D|$ is therefore the Dirichlet-to-Neumann operator: it maps the boundary field value to its outward derivative. A mode with $A=\lambda$ decays over length $1/\lambda$, or $R/(k+3)$ on the round six-space. This is a field profile length under the assumed geometry, not compactification.

The tangential and normal kinetic terms cancel in (8). Retaining only one of them would give a different reduced equation.

Normalize the entire action

The trace $f$ does not have the canonical norm of the parent field. Integration over the normal coordinate gives

$$\|F\|^2=\langle f,Zf\rangle,\qquad Z=\int_0^\infty e^{-2yA}dy=\frac1{2A}. \tag{9}$$

Consequently the time-kinetic term is $i\langle f,Z\partial_t f\rangle$. The canonically normalized tangential variable is

$$\chi=Z^{1/2}f,\qquad f=(2A)^{1/2}\chi. \tag{10}$$

As a linear control, add a constant real scalar coefficient $M$ to the parent equation, giving $H_M=H_0+M\beta_p$. Here $M$ is prescribed; the local self-interaction will be tested separately. Its bulk energy is

$$\begin{aligned} E_M(F) &=M\int_0^\infty\langle F(y),\beta_pF(y)\rangle\,dy\\ &=M\langle f,\beta Zf\rangle =M\langle\chi,\beta\chi\rangle. \end{aligned} \tag{11}$$

The last equality uses $[A,\beta]=0$. The resulting canonical equation and exact lifted solution are

$$i\partial_t\chi=M\beta\chi,\qquad F(t,y)=Ve^{-yA}(2A)^{1/2}e^{-itM\beta}\chi_0. \tag{12}$$

Substitution verifies (12) in every component of the full parent equation, with the specified decaying boundary data. The coefficient is $M$, independent of tangential level. The nonlocal operator in (8) has not become a mass enhancement. Its inverse also enters the kinetic norm and cancels from (11).

For comparison, a separately specified boundary quadratic form $b\langle f,\beta f\rangle$ compresses in the same canonical variables to

$$E_{\partial}=2b\langle\chi,A\beta\chi\rangle. \tag{13}$$

This gives a first-degree inverse-width factor, with a new boundary coefficient. It is the compression of an additional form onto (7). Its complete boundary condition and changed normal profile must still be solved before (13) is treated as an exact physical spectrum. The full polynomial and normalization in (1) have not been derived.

Why a free normal extension does not select a bound particle

On a complete normal line $y\in\mathbb R$, decay toward positive $y$ requires data in $\operatorname{Ran}\Pi_+$; decay toward negative $y$ requires data in $\operatorname{Ran}\Pi_-$, where $\Pi_-=I-\Pi_+$. Their intersection is zero.

Hence a smooth zero-frequency solution of $H_0F=0$ cannot use both free decays across $y=0$ with nonzero continuous data. The same obstruction applies to the lifted scalar eigensolutions of (12) at frequencies $\pm M$: after selecting their scalar eigenvalue, the remaining spatial equation is again $H_0F=0$. An interface, a boundary law or a nonlinear binding mechanism would have to supply different matching dynamics.

This concerns this product-space construction. It does not exclude different normal geometries or a fully solved nonlinear profile. It also does not replace IDWT's cylindrical free continuation by an already established localized normal profile.

Vary the actual local scalar contact

The round local-gap construction provides the exact spatial pair $\varepsilon,\xi=\beta\varepsilon$. Its Dirac eigenvalues are $+\lambda$ and $-\lambda$, respectively, with $\lambda=3/R$. The pair is pointwise orthonormal. Its scalar-current matrix is the ordinary Pauli $X$, and arbitrary complex coefficients give a spatially constant scalar current.

On this pair, $V$ in (5) is represented pointwise by the same isometry as in the operator calculation. Choose a tangential field $\chi=a\varepsilon+b\xi$ and define

$$w_\lambda(y)=\sqrt{2\lambda}\,e^{-\lambda y},\qquad F(y,q)=w_\lambda(y)V\chi(q),\qquad j=\chi^\dagger\beta\chi,\quad \rho=\chi^\dagger\chi. \tag{14}$$

Both $j$ and $\rho$ are constant on the spatial sphere. Tangential spatial measure is held fixed throughout; unit spatial measure may be used to isolate the normal normalization. The elementary normal integrals are

$$\int_0^\infty w_\lambda^2dy=1,\qquad \int_0^\infty w_\lambda^4dy=\lambda,\qquad \int_0^\infty w_\lambda^6dy=\frac43\lambda^2. \tag{15}$$

Consider the written local parent scalar energy $\kappa\int(F^\dagger\beta_pF)^2/2$. Its compression onto (14) gives

$$E_{\rm int}^{\rm comp} =\frac{\kappa\lambda}{2}\int_{Q_6}j^2\,d\mu,\qquad N_{\rm comp}(\chi)=\kappa\lambda j\beta\chi. \tag{16}$$

This inverse-width coefficient is obtained from the actual local quartic interaction with its canonical normal norm. But compression alone is not the complete nonlinear equation.

The full local source is

$$N(F)=\kappa w_\lambda^3 j\,\beta_pV\chi. \tag{17}$$

Its normal profile is $w_\lambda^3$, not $w_\lambda$. Let $W\chi=w_\lambda V\chi$ be the canonical lift on the pair. Then its projection is $W^\dagger N=\kappa\lambda j\beta\chi$, and the complementary normal source obeys

$$\boxed{ \|(I-WW^\dagger)N\|_y^2 =\frac{\kappa^2\lambda^2j^2\rho}{3}. } \tag{18}$$

Here the norm integrates the normal coordinate at a spatial point; integrating over $Q_6$ gives the full spatial statement. Indeed, (15) gives $\|N\|_y^2=4\kappa^2\lambda^2j^2\rho/3$, whereas $\|W^\dagger N\|^2=\kappa^2\lambda^2j^2\rho$. Their difference proves (18). For nonzero current the omitted force is one half of the full force norm, or $1/\sqrt3$ times its retained part.

The source stays within the spatial Killing pair, so this residual already lies in genuine complementary normal profiles; it is not an artifact of omitting angular spinor modes. The fixed decaying lift is therefore not an exact nonlinear invariant family for nonzero local scalar current. Solving those profiles can change both the effective interaction and its normalization, as the nonlinear mass reduction explains.

Solve the full-line binding question within the spatial Killing channel

For this spatial pair, the complete nonlinear normal problem admits a stronger conclusion than the residual (18). Retain four arbitrary complex functions of $y$: the two spatial Killing coefficients in each parent block. Write their vector as $z(y)\in\mathbb C^4$. This includes growing, decaying and oscillatory normal profiles; the lift (14) is no longer imposed.

The spatial pair has constant Gram and scalar matrices, so its complete local source is still independent of the tangential point. On a full normal line the stationary equation is

$$\begin{gathered} (D_\lambda+K_m)z=\omega z,\qquad K_m=-i\alpha_y\partial_y+m(y)\beta_p,\\ D_\lambda=I_2\otimes \begin{pmatrix}\lambda&0\\0&-\lambda\end{pmatrix}, \qquad m(y)=\kappa z^\dagger\beta_p z. \end{gathered} \tag{19}$$

In this equation $\alpha_y$ and $\beta_p$ are the four-component restrictions of (3). The normal direction is free at both infinities; there is no boundary or external normal potential. Assume a bounded smooth field of finite canonical normal norm. Then $m$ is bounded and integrable because $|m(y)|\le|\kappa|\,|z(y)|^2$. The stationary equation puts $z$ in $H^1$.

Since $m$ depends only on $y$, the exact identities are

$$\{D_\lambda,K_m\}=0,\qquad D_\lambda^2=\lambda^2I. \tag{20}$$

Integration by parts on $H^1$ gives the norm identity

$$\boxed{ (\omega^2-\lambda^2)\|z\|^2=\|K_mz\|^2\ge0. } \tag{21}$$

Thus no nonzero bound state can have $|\omega|<\lambda$, even with a completely relaxed nonlinear normal profile. At $|\omega|=\lambda$, equation (21) requires $K_mz=0$. Its exact first-order solution has the form

$$z(y)=\exp\!\left[-i\alpha_y\beta_p \int_{y_0}^{y}m(s)\,ds\right]z(y_0). \tag{22}$$

The integral stays finite at infinity. This matrix and its inverse therefore stay bounded, so nonzero initial data cannot approach zero in norm and cannot be square integrable.

Finally suppose $|\omega|>\lambda$. The stationary equation is

$$z'=B_0z+m(y)Cz,\qquad B_0=i\alpha_y(\omega-D_\lambda),\qquad C=-i\alpha_y\beta_p. \tag{23}$$

Here $B_0^2=-(\omega^2-\lambda^2)I$, so both $e^{yB_0}$ and $e^{-yB_0}$ are uniformly bounded. In the interaction variable $a(y)=e^{-yB_0}z(y)$,

$$a'=m(y)e^{-yB_0}Ce^{yB_0}a. \tag{24}$$

Its coefficient has integrable operator norm. Gronwall's inequality applied in both directions gives a strictly positive lower bound on $|a(y)|$ for every nonzero initial value. The bounded inverse free evolution gives the same lower bound on $|z(y)|$. Square integrability is again impossible.

Equations (21)–(24) exhaust real stationary frequencies. Hence the written local scalar interaction has no nonzero bounded smooth stationary field of finite normal norm on the full line within this complete spatial Killing channel, for either sign of $\kappa$. This result retains all four complex normal amplitudes and their self-consistent current.

Its scope is the selected spatial pair and product geometry. Tangentially varying currents, coupling to other spatial modes, different normal geometry and boundary dynamics change the problem. Normal localization itself is not an established requirement of IDWT's cylindrical free continuation. The theorem settles this particular attempted mechanism without selecting a physical geometry.

The calculation identifies an explicit spatial origin for $|\mathscr D|$ and follows it through the full spinor and action normalization. Deriving the complete simplex mass operator still requires a normal geometry and boundary or binding law inherited from the master field, together with its consistent nonlinear spatial solution.

Verification

normal_dirac_extension_checks.py checks every Clifford anticommutator of the rank-sixteen parent, the complete decaying projector and scalar intertwining, and arbitrary complex paired tangential spectra under unitary changes of basis. These paired matrices test operator identities; they are not a replacement for curved spatial multiplicities.

Independent normal quadrature verifies the induced kinetic metric, bulk scalar energy and profile moments. Full complex action variations check both bulk and boundary forms and the local quartic source. The last tests differentiate the actual six-dimensional Killing fields in a spatial chart, retain both parent blocks, and measure the complementary normal force in (18). Independent normal Fourier matrices test the full anticommutation and positive square in (20)–(21) with spatially varying scalar profiles and all four complex amplitudes. Those finite periodic matrices verify operator algebra; the full-line nonexistence statement follows from the analytic argument (21)–(24).