Photon transversality at every coordinate depth

Current physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Particle patterns have different dimensional involvement in that space. The ten-dimensional matter domain is presently modeled by flat $\mathbb R^{10}$. Nested layers describe proposed spatial support, not additional manifolds. Depth counts a pattern's dimensional involvement, not a separate containing space. The familiar three dimensions describe the observer's view, not the ambient dimension. All spatial directions are of the same physical kind. Confinement and persistence of the layers remain dynamical questions.

Mathematical scope: Projective, round-sphere and prescribed global-nesting calculations below retain their stated additional premises. Their curvature, topology and finite-volume normalizations are conditional comparisons; they do not establish the current flat spatial geometry. Field components and mode spaces are distinct from physical position. IDWT has no compactified dimensions or mediator particles; interactions must come from $\Psi_\infty$. See spatial meaning and mathematical scope.

Current Reading and Scope — 1 November 2026

The current photon picture is one extended connection between material patterns, with phase advancing along a structure already spanning its free direction. No photon pellet or mediator is assumed. The proposed instantaneous completion of a process entirely in free directions must still be connected to observed phase-dependent timing, conservation and no-signalling. Transverse rotation weights alone do not derive that temporal mechanism.

The new particle-geometry and sector-depths case provides the current derivations, shared assumptions, glossary and reproduction instructions. The twenty-volume research map connects the complete account. Calculations below retain their own stated mathematical premises and do not inherit results from a different action.

1. The claim to explain

The photon is an excitation bound in two macroscopic spatial dimensions of the nested manifold:

$$\Xi_2\subset\Xi_3\subset\Xi_4\subset\cdots\subset\Xi_\infty.$$

The single time coordinate is shared at every depth. Ordinary observers occupy the $r=3$ level of this nesting.

The photon's two dimensions are ordinary spatial dimensions shared with every containing sector. For an electron there are six ordinary spatial dimensions, two of which contain the photon plane; for a tau there are ten. This comparison makes none of the containing particle's dimensions different in kind. Likewise, the electron's six are six of the tau's ten: the electron is lower-dimensional, or “flat,” relative to the tau. These are nested submanifolds, not particles made of separate dimensional compartments.

The transversality problem is therefore:

Why must the ambient momentum of a photon bound in two spatial dimensions be perpendicular to its two directions of oscillation at every depth that contains it?

Photon masslessness and the complete electromagnetic field equations are the neighbouring dynamical problems.

2. The physical spatial plane and its normal bundle

Let $x$ be the photon's centre and let

$$E_{\gamma,x}\subset T_x\Xi_\infty$$

be its oriented physical spatial two-plane. This is the local physical content of the $d=2$ assignment:

$$\dim E_{\gamma,x}=2.$$

Its orientation is a dynamical variable. At a containing depth $r\geq3$, write its image as

$$E_{\gamma,x}^{(r)}\subset T_x\Xi_r.$$

With the spatial metric restricted to that depth, there is an orthogonal decomposition

$$T_x\Xi_r = E_{\gamma,x}^{(r)} \oplus N_{\gamma,x}^{(r)},$$

where

$$N_{\gamma,x}^{(r)} = \left(E_{\gamma,x}^{(r)}\right)^\perp, \qquad \dim N_{\gamma,x}^{(r)}=r-2.$$

The second summand is the normal space available for propagation relative to the photon plane. At observer depth its dimension is

$$\dim N_{\gamma}^{(3)}=1.$$

At deeper levels the photon plane is still two-dimensional while its normal space has more possible directions.

Here “normal” is the geometric word for perpendicular to the chosen photon plane. It is a relationship to that plane, not a different kind of spatial dimension. There are four such directions within the electron's six and eight within the tau's ten. Neither containing sector acquires an inherent physical split from this description of the photon.

3. The depth-independent transversality theorem

This is a conditional geometric theorem: assigning polarization to the physical spatial plane and propagation momentum to its normal space already enforces their orthogonality. The depth-two assignment alone does not derive those assignments, select the separated mode below, or establish the electromagnetic constraint equations. Free outer dynamics also allow wavepackets; constant outer intensity describes the chosen momentum eigenmode, not every photon.

Choose coordinates $u$ on the physical spatial plane and $y$ on its normal space:

$$u\in E_{\gamma,x}^{(r)}, \qquad y\in N_{\gamma,x}^{(r)}.$$

Bound within, free without gives the two coordinate sets different jobs. The photon's physical oscillation is the spatially bound mode $\chi_0(u)$ in its two bound spatial directions. Those same directions belong to the containing sectors. In the outer directions it has no localized envelope: a free momentum eigenmode has constant magnitude and a changing phase. The separated photon mode is

$$\Psi_\gamma(u,y,t) = \chi_0(u)\,\boldsymbol\varepsilon\, e^{i(\mathbf K\cdot y-\omega t)},$$

where

$$\boldsymbol\varepsilon \in E_{\gamma,x}^{(r)}\otimes\mathbb C, \qquad \mathbf K\in N_{\gamma,x}^{(r)}.$$

If this vector ansatz is identified directly with a source-free electric field, the full divergence constraint additionally requires $\boldsymbol\varepsilon\cdot\nabla_u\chi_0=0$. A real scalar envelope with constant circular polarization spanning the two bound directions must then have both transverse derivatives zero, so that particular factorization cannot provide transverse localization. Spatially varying polarization, additional components or a different reconstruction of the electric field can change the conclusion. Orthogonality alone does not solve these spatial constraints, and the Maxwell constraint must not be imposed on an unidentified master-field component without the stated field identification.

Its intensity is independent of the free coordinate:

$$|\Psi_\gamma(u,y,t)|^2 = |\chi_0(u)|^2|\boldsymbol\varepsilon|^2.$$

This is the precise meaning of “the photon is already there.” Its support is uniform in the free direction, while its phase gradient carries energy, momentum, and the sequence of possible interactions.

The polarization belongs to the physical spatial plane, $\boldsymbol\varepsilon\in E_\gamma$, while the phase gradient belongs to the outer free coordinates, $\mathbf K\in N_\gamma$. By orthogonality,

$$\boxed{\mathbf K\cdot\boldsymbol\varepsilon=0.}$$

This conclusion holds for every $r$ containing the photon plane. At depth $r=3$, the normal space is one-dimensional, producing the familiar picture of one propagation axis and a two-dimensional transverse plane. At depth $r>3$, the photon may propagate along any direction in an $(r-2)$-dimensional normal space, but every such direction remains perpendicular to both spatial photon directions.

Including the shared time coordinate, define

$$K^A=(\omega,\mathbf K), \qquad \varepsilon^A=(0,\boldsymbol\varepsilon).$$

Then

$$K_A\varepsilon^A=0.$$

If the photon is massless, its dispersion relation additionally gives

$$K_AK^A=0, \qquad \omega^2=\|\mathbf K\|^2.$$

Masslessness makes $K$ null. Transversality follows from the allocation of phase propagation and physical oscillation to orthogonal parts of the nested manifold.

4. Momentum without rest mass

Energy, momentum, and invariant rest mass obey

$$E^2=p^2c^2+m^2c^4.$$

For $m=0$ and $E>0$,

$$\boxed{p=\frac{E}{c}>0.}$$

The same result follows directly from the field stress tensor. Let $u$ be the wave's energy density and $\mathbf S$ its energy-flux vector. The field momentum density is

$$\mathbf g=\frac{\mathbf S}{c^2}.$$

For the source-free Maxwell control in three observed spatial dimensions, the general pointwise bound is $|\mathbf S|\le cu$. With real electric and magnetic fields $\mathbf E$ and $\mathbf B$,

$$u^2-\frac{|\mathbf S|^2}{c^2} =\frac{\epsilon_0^2}{4} \left[(|\mathbf E|^2-c^2|\mathbf B|^2)^2 +4c^2(\mathbf E\cdot\mathbf B)^2\right]\ge0.$$

Equality requires a null field. For a finite pulse of total energy $U$,

$$\mathbf P=\frac{1}{c^2}\int\mathbf S\,d^3x, \qquad |\mathbf P|\le\frac{U}{c}.$$

Integrated equality additionally requires aligned energy flux wherever the energy is nonzero. Two equal-energy, disjoint packets at angles $\pm\alpha$ to an axis have $|\mathbf P|=(U/c)\cos\alpha$. An individual massless excitation obeys $p=E/c$; a multidirectional classical pulse or an expectation four-vector need not saturate that relation. The exact plane-mode result does not establish equality for every free finite pulse.

Uniform support and nonzero Noether momentum coexist because the plane-wave factor has unit magnitude and a nonzero phase gradient:

$$\left|e^{i(\mathbf K\cdot y-\omega t)}\right|=1, \qquad -i\nabla_y e^{i(\mathbf K\cdot y-\omega t)} = \mathbf K e^{i(\mathbf K\cdot y-\omega t)}.$$

If the one-quantum normalization is used, the same statement is

$$E=\hbar\omega, \qquad \mathbf p=\hbar\mathbf K, \qquad \omega=c|\mathbf K|.$$

The ratio $p/E=1/c$ is classical wave kinematics and is independent of the absolute value of $\hbar$. It is the wave's Noether momentum and energy flux.

Finally, $n=0$ labels the unexcited spatial mass mode. External frequency $\omega$, wave vector $\mathbf K$, energy $E$, and momentum $\mathbf p$ remain the freely specifiable wave data, allowing a zero-rest-mass sector mode to carry arbitrary photon energy.

5. Why there are two polarization modes

Choose an oriented orthonormal basis $\{\mathbf e_1,\mathbf e_2\}$ of the physical spatial plane. Any photon oscillation is

$$\boldsymbol\varepsilon = a_1\mathbf e_1+a_2\mathbf e_2.$$

The polarization vector has component count

$$\dim_{\mathbb C}\left(E_\gamma\otimes\mathbb C\right)=2,$$

at every containing depth. Increasing $r$ enlarges the possible normal propagation directions while preserving these two polarization modes.

The circular basis is

$$\mathbf e_\pm = \frac{\mathbf e_1\pm i\mathbf e_2}{\sqrt2}.$$

Under a rotation of the photon plane through angle $\alpha$,

$$\mathbf e_\pm \longmapsto e^{\mp i\alpha}\mathbf e_\pm.$$

These are the two orientation weights $\pm1$ of the physical spatial $SO(2)\simeq U(1)$ geometry. At depth $r=3$, where the plane is the complete spatial complement of the propagation direction, these weights are precisely the usual photon helicities $\pm1$. At higher depths the full massless little group is larger, and the same two circular polarization modes form the $d=2$ sector representation.

A parity-complete solution space permits both circular modes. A particular state can occupy one or both. Reality relates positive-frequency components to their negative-frequency conjugates; it does not require both independent positive-frequency helicities to be occupied. For example, $\operatorname{Re}[(\mathbf e_1+i\mathbf e_2)e^{i(kz-\omega t)}]$ is real and has one positive-frequency circular polarization.

6. Orientation at an arbitrary depth

The space of possible orientations of the photon plane at depth $r$ is the oriented Grassmannian

$$\operatorname{Gr}^{+}_2(T_x\Xi_r).$$

A photon configuration at that depth consists geometrically of a pair

$$(E_\gamma,\mathbf K), \qquad E_\gamma\in\operatorname{Gr}^{+}_2(T_x\Xi_r), \qquad \mathbf K\in E_\gamma^\perp.$$

This is the depth-independent state geometry. It reduces at $r=3$ to the familiar coincidence

$$\operatorname{Gr}^{+}_2(\mathbb R^3) \simeq S^2 \simeq \mathbb{CP}^1,$$

because an oriented plane in three dimensions is determined by its oriented unit normal. This realizes the physical photon plane in observer geometry.

For the $r=3$ realization, the right- and left-circular polarization line bundles over the direction sphere have

$$c_1(\mathcal L_\pm)=\pm2,$$

and can be written, up to orientation convention, as

$$\mathcal L_\pm\simeq\mathcal O(\pm2).$$

This is the spin-one topology visible at that depth. The helicity bundles are $\mathcal O(\pm2)$, while the basic amplitude Hopf line is $\mathcal O(1)$. Thus the physical spatial plane has the required observer-level realization.

7. Relation to a gauge field

The geometric condition

$$\mathbf K\cdot\boldsymbol\varepsilon=0$$

removes a longitudinal spatial oscillation once normal propagation has been established. A full account of electromagnetism must additionally explain the gauge connection and its dynamics.

A phase-gradient proposal would write

$$A_A=\partial_A\theta, \qquad F_{AB}=\partial_AA_B-\partial_BA_A.$$

For a globally smooth single-valued phase this gives

$$F=d^2\theta=0.$$

This is a pure-gauge configuration. A propagating photon uses a genuine $U(1)$ connection, defined independently or patchwise, whose physical components are restricted to the rank-two photon bundle $E_\gamma$.

The projected electromagnetic action must derive:

1. the connection's propagating sector is restricted to $E_\gamma$;
2. its momentum lies in $E_\gamma^\perp$.

With an effective curvature action

$$S_{\rm EM} = -\frac14\int F_{AB}F^{AB}\,d\mu,$$

a plane wave obeys

$$K^2\varepsilon^B-K^B(K\cdot\varepsilon)=0.$$

For a nonzero null mode this is consistent with $K\cdot\varepsilon=0$, while gauge equivalence

$$\varepsilon^A\sim\varepsilon^A+\alpha K^A$$

removes the pure-gauge direction. The sector projector restricts the physical polarization and bound oscillation to the spatial $d=2$ photon plane. The phase momentum remains in the perpendicular directions described above; restricting polarization is not restricting that momentum to the photon plane.

8. The Hopf sector connection and its extension

The physical $\mathbb{CP}^1$ sector geometry offers a nontrivial local connection. For a normalized representative $z\in\mathbb C^2$, with $z^\dagger z=1$ and $z\sim e^{i\alpha}z$, define

$$A_A=-iz^\dagger\partial_Az.$$

Then

$$A_A\longmapsto A_A+\partial_A\alpha$$

and

$$F_{AB} = -i\left( \partial_Az^\dagger\partial_Bz - \partial_Bz^\dagger\partial_Az \right),$$

which can be nonzero. This is a genuine improvement over $A=d\theta$.

This sector curvature is the pullback of the Fubini--Study area form $\omega$ on the two-real-dimensional target $\mathbb{CP}^1$. Since $\omega\wedge\omega=0$ there,

$$F\wedge F=0.$$

9. The dynamical masslessness condition

The convention $S(0,2)=0$ labels the photon zero mode. Exact masslessness requires the projected action to possess a gauge symmetry or dynamical identity forbidding a scalar mass endomorphism on this mode:

$$\boxed{P_{\gamma}\mathcal M P_{\gamma}=0},$$

where $\mathcal M$ is the full radiatively corrected mass operator.

10. Exact result and open bridge

For every containing depth $r\geq3$, the IDWT premises are:

1. the photon has an oriented rank-two spatial plane $E_\gamma^{(r)}\subset T\Xi_r$;
2. the nesting metric supplies the orthogonal normal bundle $N_\gamma^{(r)}=(E_\gamma^{(r)})^\perp$;
3. photon oscillations lie in $E_\gamma^{(r)}$;
4. bound-within/free-without factorizes the mode into spatial $u$ dependence and a constant-magnitude free phase in the outer coordinates $y$;
5. the solution space permits both $SO(2)$ orientation weights; a state need not occupy both.

The phase gradient then lies in $N_\gamma^{(r)}$, and

$$\boxed{ \boldsymbol\varepsilon\in E_\gamma^{(r)}, \qquad \mathbf K\in N_\gamma^{(r)}, \qquad \mathbf K\cdot\boldsymbol\varepsilon=0, \qquad \dim E_\gamma^{(r)}=2.}$$

Thus transversality and the two polarization modes are invariant with observer depth. At $r=3$, the normal space has dimension one and the two spatial rotation weights are the familiar helicities $\pm1$.

The projector equation follows from this factorization:

$$\boxed{P_{E_\gamma}\mathbf K=0.}$$

The action-level closure targets are the separated solution, the dynamics and source of the full-manifold $U(1)$ connection, its relation to the sector Hopf curvatures, and protection of the $n=0$ mode's exact masslessness.

Observer-level cross-checks

• E. Palmerduca and H. Qin, Photon topology: the two photon helicity line bundles over nonzero three-momentum have first Chern numbers $\mp2$.
• P. Caban and J. Rembieliński, Photon polarization and Wigner's little group: the massless little group and helicity phase at observer depth $r=3$.