Already Everywhere — The Path Integral in IDWT

The usual story vs. the already-present field

In the textbook derivation a particle is imagined to explore every possible path from A to B; the amplitude for each path is \(e^{iS/\hbar}\) and the total amplitude is their coherent sum. This beautifully reproduces the double-slit pattern and the Schrödinger equation, but it leaves an ontological discomfort: how does the particle “know” about all the other paths?

In IDWT no particle explores every path. A particle follows one definite path — and it never needs to know about the others, because the thing that carries the alternatives is not the particle. There is one master Dirac spinor field, \(\Psi_\infty\), on the infinite-dimensional manifold \(M_\infty = \mathbb{R}_t \times \Xi_\infty\), and every possible configuration — every “path” in the usual language — is already an eigenmode of the sector Laplacian localized in one of the sectors \(d \in \{2,3,4,5,6,10\}\). The field is already everywhere; the particle is a localized excitation riding it.

The path integral is not a computational trick. It is the statement that the master field \(\Psi_\infty\) does not travel — it is already everywhere, and the observed amplitude selects the sector mode that satisfies the boundary conditions.

”Free without” — the \(d=2\) photon example

Consider a photon traveling from a laser to a screen. In ordinary language we say the photon takes all paths at once. In IDWT the \(d=2\) sector mode (the photon) is already present throughout the entire experimental region because its two coordinates are nested inside every higher sector. There is no packet that must “choose” a path. The field \(\Psi^{(2)}\) is co-located with every point in the apparatus.

The observed interference is the projection onto the detector of the already-present field configuration that satisfies the phase-matching condition at both source and detector. The kernel couples the different geometric contributions, but the field itself never “goes” anywhere — it is free without ever having to travel.

The double-slit in sector geometry

Both slits are open. The electron — a localized excitation of \(\Psi_\infty\) in the \(d=6\) sector — passes through one of them, and that is not where the interference comes from. The configuration of the one wave it rides is already present across the entire region, and the two slits impose boundary conditions on that configuration. The pattern on the screen is the natural beating between the two phase contributions the geometry supports, written into the landing statistics of electrons that each crossed a single opening.

This is why closing one slit instantly changes the pattern: it removes one of the already-present geometric constraints, not because information raced from the slit to the screen, but because the global mode structure of \(\Psi_\infty\) changed instantaneously across the whole space.

Path integral as mode selection

Formally, the Feynman path integral emerges in IDWT as the overlap between the already-present eigenmodes of the master field and the boundary projectors at initial and final times:

\[ \langle \phi_f | e^{-iHt/\hbar} | \phi_i \rangle = \int \mathcal{D}\phi \, e^{iS[\phi]/\hbar} \]

In IDWT this is realized through integration over the sector space:

\[ K = \int_{\Xi} \chi_n^*(\xi) \chi_n(\xi) e^{iS_{\rm eff}[x,\xi]/\hbar} \, d\mu_\xi \]

The functional integral is not over new trajectories in 3D space; it is the sum over the already-existing sector modes (eigenfunctions \(\chi_n(\xi)\) of the sector Laplacian) whose collective phase accumulation matches the effective action \(S_{\rm eff}\). The “sum over histories” is the statement that every history is already an excitation of \(\Psi_\infty\) somewhere in the infinite-dimensional sector space. The physical amplitude is the coefficient of the particular linear combination that satisfies the experimental boundaries.

This is “free without”: the field does not need to propagate information about alternative paths because all paths are already encoded in the global geometry of the master field.

Consistency with the rest of IDWT

This picture dovetails directly with refraction without slowing (the \(d=2\) field is already present in the medium) and with single-electron interference (the configuration the electron rides is already present across both slits; the electron crosses one). It also explains why the path integral works even for macroscopic objects: the higher-sector modes are still present, just with enormous phase density that makes classical paths dominate.

The measurement problem is likewise softened: collapse is not a physical process happening to a traveling packet, but the selection of one already-present sector-mode projection when the apparatus entangles with the system (see \(\rho(\mathbf{r},t) = \int |\Psi_\infty|^2 d\xi\)).