Already Everywhere

In the textbook path-integral picture a particle is imagined to explore every possible path from A to B, and the observed pattern is the coherent sum of a phase carried along each one. This reproduces the double-slit pattern, but it leaves an ontological discomfort: how does the particle “know” about all the other paths?

IDWT does not have a path integral, and does not need the question. 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.

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 simply the intensity of the already-present field configuration that satisfies the phase-matching condition at both source and detector, read directly by the detector's density coupling. The kernel couples the different geometric contributions, but the field itself never “goes” anywhere — it is free without ever having to travel.

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.

IDWT does not need a sum-over-histories construction to reach the observed pattern, because it does not need an amplitude at all. The conserved Noether density \(|\Psi_\infty|^2\) is the physical intensity of the wave at every point (P1), and every interaction — including detection — is the kernel's density–density coupling (P4), so a detector registers a point at a rate proportional to the local intensity there. Probability is the relative rate across outcomes, \(\rho(\mathbf r) = |\Psi_\infty(\mathbf r)|^2 / \int |\Psi_\infty|^2\) — a ratio, so the global amplitude cancels and never needs to be known (Part 1 §2.3). No separate sum-over-paths axiom is required to get here: because the field is already present with the correct phase relationship at every point the boundary conditions permit, the interference pattern is simply what the intensity \(|\Psi_\infty|^2\) looks like once the sector modes consistent with both slits are added together geometrically — not summed as alternative histories, but coexisting as one configuration of the one field.

This is “free without”: the field does not need to propagate information about alternative paths because all the geometry the pattern depends on is already present in \(\Psi_\infty\), everywhere the boundary conditions reach, at once.

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 macroscopic objects behave classically: the higher-sector modes are still present, just with enormous phase density, so a single dominant configuration overwhelms the rest.

The measurement problem is likewise softened: there is no collapse of a traveling packet, because there was never a superposition of separate packets to begin with — only one field, everywhere present, whose observable marginal at \(d=3\) is \(\rho(\mathbf{r},t) = \int |\Psi_\infty|^2 d\xi\). The apparatus is itself a mode of \(\Psi_\infty\) coupling through the coordinates it shares with the system; a measurement reads that shared-coordinate density directly, with no separate projection step.