Spacetime geometry of spin, polarization, and wavefunction collapse

To incorporate quantum nonlocality into general relativity, we propose that the preparation and measurement of a quantum system are simultaneous events. To make progress in realizing this proposal, we introduce a spacetime geometry that is endowed with particles which have no distinct points in their worldlines; we call these particles 'pointons'. This new geometry recently arose in nonnoetherian algebraic geometry. We show that on such a spacetime, metrics are degenerate and tangent spaces have variable dimension. This variability then implies that pointons are spin-$\tfrac 12$ fermions that satisfy the Born rule, where a projective measurement of spin corresponds to an actual projection of tangent spaces of different dimensions. Furthermore, the $4$-velocities of pointons are necessarily replaced by their Hodge duals, and this transfer from vector to pseudo-tensor introduces a free choice of orientation that we identify with electric charge. Finally, a simple composite model of electrons and photons results from the metric degeneracy, and from this we obtain a new ontological model of photon polarization.


Simultaneity of preparation and measurement
In this article we introduce the idea that the fundamental source of quantum phenomena is that clocks in the same inertial frame may tick at different rates.Specifically, we propose that time does not advance in a quantum system undergoing unitary evolution.Consequently, the preparation and measurement of a quantum system are simultaneous events.This simultaneity is then the origin of quantum nonlocality.Our aim here is to make this idea precise, both mathematically and physically, by modifying the geometry of classical general relativity.
The spacetime model we will introduce produces many of the standard features of quantum theory, but does not reproduce quantum theory exactly.In particular, we model quantum phenomena not with Hilbert spaces, but with degenerate spacetime metrics obtained from our new notion of simultaneity.We will show that such metrics yield a composite model of electrons and photons, as well as (ψ-epistemic ontological models of) spin, polarization, and their wavefunction collapse.In companion articles we reproduce photon path superposition and entanglement using this framework [B7], and extend our composite model of electrons and photons introduced here to all standard model particles [B1, B2].
We remark that the original motivation for the development of this new geometry, which we call 'nonnoetherian geometry', arose from brane tilings in string theory (e.g., [B6, F-W]). 1 However, here we apply nonnoetherian geometry directly to general relativity.
Notation: Tensors labeled with upper and lower indices a, b, . . .represent covector and vector slots respectively in Penrose's abstract index notation (so v a ∈ V and v a ∈ V * ), and tensors labeled with indices µ, ν, . . .denote components with respect to a coordinate basis.We denote the tangent space of a manifold M at p ∈ M by M p := T p M and its (vector space) dimension by dim M p .Given a curve β : I → M , we will often denote its image β(I) also by β.We use natural units, c = 1, and the signature (+, −, −, −) throughout.
1 A 'brane tiling' (or 'dimer model') is a directed graph that embeds in a torus with certain conditions.Associated to a brane tiling is a quiver gauge theory whose vacuum moduli space is either a toric algebraic variety (in which case the brane tiling is said to be 'consistent' and the gauge theory is superconformal; see e.g., [Br, D, DHP]) or a nonnoetherian scheme [B3] (in which case the brane tiling is 'inconsistent' and the gauge theory is not superconformal).The former gauge theories are stable, the latter are unstable, and both are physically allowed; in M-theory the two kinds of theories are on an equal footing.It had always been assumed, however, that nonnoetherian schemes do not admit concrete geometric descriptions-something that can be visualized-but rather are only abstract constructions.With the aim of a geometric picture for the nonnoetherian moduli spaces, the author introduced nonnoetherian algebraic geometry as the geometry of algebraic varieties with positive dimensional 'smeared-out' points [B4, B5].Under this new framework, the moduli space of an inconsistent brane tiling admits an honest geometric description as a toric algebraic variety with an embedded curve or surface that is identified as a single closed point [B6].

Spacetime with a degenerate metric
We would like to modify general relativity so that the preparation and measurement of a quantum system are simultaneous events.To this end, we introduce the following.
Definition 2.1.Let ( M , g) be an orientable Lorentzian 4-manifold.Consider a locally finite set of particles on M with worldlines β i ⊂ M .We call the set where each β j is a single point of M , an internal spacetime, or simply spacetime.We call M the external spacetime of M , and the particles pointons.
We thus construct spacetime M from M by replacing each β i in M with a single point.Although points are usually defined to be 0-dimensional objects (an exception being the points of a scheme in algebraic geometry), this is not the case in our construction of M .We call a pointon worldline β a 'point' because there are no (0-dimensional) points properly contained inside of it in M .However, we continue to regard β as a curve, that is, as a 1-dimensional object, since it appears to be a curve in a deleted neighborhood about it.In other words, in constructing M we do not contract β ⊂ M to a 0-dimensional point, as is done when taking geometric or topological quotients.Instead, we keep the embedding of β intact by using the metric on M to measure distances near β on M (made precise below).Informally, we do not 'throw away' M in forming M .The worldline β is then a continuum of distinct 0-dimensional points in M , and a single 1-dimensional point in M .In particular, time does not advance along β in spacetime M .
Since a pointon worldline β is a single point of spacetime M , we cannot define a tangent vector v ∈ Mβ(t) (or 4-velocity) along β in M .Furthermore, any 'metric' h on M , however it may be defined, should be equal on any vectors w 1 , w 2 ∈ Mp that map to points exp p (w 1 ), exp p (w 2 ) ∈ M which are identified in M and lie in a sufficiently small open neighborhood of p in M .
Lemma 2.2.Let β ⊂ M be a pointon worldline with tangent vector v.For p ∈ M , let h ab : Mp ⊗ Mp → R be a symmetric tensor such that h ab w a 1 = h ab w a 2 whenever the points exp p (w 1 ), exp p (w 2 ) ∈ M are identified in M but are not equal in M .Then h ab v a ≡ 0 along any geodesic segment of β.
Proof.Without loss of generality, we may suppose β(I) is a geodesic segment of β with t = 0 in the interval I ⊂ R. Then the points exp β(0) (λv) and exp β(0) (λ ′ v) lie in β(I) ⊂ M for all λ, λ ′ ∈ I, and are thus identified in M .Whence, for all w ∈ Mβ(0) , Therefore, by choosing λ ̸ = λ ′ we find h ab v a = h(v, −) = 0. □ Lemma 2.2 implies that to construct an internal metric h ab at p ∈ M from the metric g ab on M , it must project out each vector v tangent to a geodesic pointon worldline β ⊂ M at p. Let v be such a vector.If v is a timelike resp.spacelike unit vector, then the orthogonal projection of v is However, if v is null, then (1) does not project out v.
Lemma 2.3.Let v be a null vector.Any minimal orthogonal projection , a contradiction since [v] is an orthogonal projection.Therefore v ′ must also be null.□ We will determine the physical meaning of the null vector v ′ in Section 5.
Definition 2.4.Fix p ∈ M .We call the Lorentzian metric g ab : Mp ⊗ Mp → R an external metric at p.2 Let v 1 , . . ., v n ∈ Mp be the tangent vectors to the pointon worldlines β 1 , . . ., β n at p. We define the corresponding internal metric to be the degenerate symmetric rank-2 tensor given by the composition of projections ( 1) and (2), As is well known, for any Riemannian manifold M , there is a geodesic ball B p about each point p ∈ M for which the exponential map exp p : Mp → M is injective.By our definition of internal metric, this property continues to hold on internal spacetimes.
Definition 2.5.The internal tangent space at a point p ∈ M is the image of h at p, Mp , and thus is a subspace of the 4-dimensional tangent space Mp .We call a vector v ∈ Mp internal if v is in M p , and external otherwise.
We will find in the next sections that the variability of the dimensions of the internal tangent spaces M p is the source of spin, polarization, and their wavefunction collapse.

Internal 4-velocities: electric charge and spin
We first recall orientation and Hodge duality from Maxwell's equations to establish notation (e.g., [F, N]).
An orientation of a vector space V is given by fixing an ordered basis B of V , and declaring any ordered basis to be positive (resp.negative) if it can be obtained from B by a base change with a positive (resp.negative) determinant.
The Hodge dual ⋆ψ of an element ψ ∈ M * p is defined on the basis elements and extended R-multilinearly to M * p .For example, ⋆1 = vol( Mp ).Note that Hodge duals are pseudo-forms, that is, they depend on a choice of orientation, since they are defined using the volume form vol( Mp ).This fact will play a fundamental role in our framework.Now consider a pointon with timelike worldline β ⊂ M and 4-velocity v on M .Since time does not advance along β, v is orthogonal to spacetime M : h(v) = 0. We would therefore like to replace v with a new geometric object v that is intrinsic to spacetime M and independent of M .
Naively, we may replace v ∈ ker h ⊂ Mβ(t) at a point β(t) ∈ M with the Hodge dual ⋆vol(ker h) of the volume form of the kernel ker h ⊂ Mβ(t) .Indeed, each 1-form 3 The metric appears in the volume form (3) in arbitrary local coordinates x µ on M near p, vol( Mp ) = in ⋆vol(ker h) lies in the image im h = M β(t) , since im h is orthogonal to ker h in Mp .However, a Hodge dual is a pseudo-form, and thus depends on a choice of orientation of the subspace ker h.
To make our geometric replacement v of v fully intrinsic to M , we need to eliminate this dependency by allowing an orientation of ker h to be freely chosen, independent of any (non-physical) choice of orientation of Mβ(t) .Thus, by requiring v to be intrinsic to M we obtain a new Z 2 parameter.Fundamentally, this parameter arises because we are replacing a vector, which does not depend on any choice of orientation, with a pseudo-tensor that does.
Definition 3.1.We define the internal 4-velocity of a pointon at p = β(t) ∈ M with 4-velocity v ∈ ker h ⊂ Mp to be the pseudo-form where o ker h ∈ {±1} is a free parameter independent of any orientation of Mp .Note that the rank of v changes along β ⊂ M whenever the dimension of M β(t) changes.
(i) dim M p = 3: First suppose that β does not transversely intersect another pointon worldline at p. Then the internal 4-velocity at p is vabc = o 0 e 1 ∧ e 2 ∧ e 3 , where o 0 ∈ {±1} is a free choice of time orientation (in the rest frame of the pointon), independent of any orientation of Mp .We identify this timelike orientation with the electric charge of the pointon, ( 4) This is similar to the Stückelberg-Feynman interpretation of antimatter in quantum field theory, where positrons are regarded as electrons that travel backwards in time [S].We note, however, that in contrast to the Stückelberg-Feynman interpretation, time is stationary along the pointon's worldline β: since β is a single point of spacetime M , the pointon does not travel backwards along β just as it does not travel forwards.
(ii) dim M p = 1: Suppose β transversely intersects other pointon worldlines at p such that dim M p = 1.If we choose a tetrad for which ker h = span{e 0 , e 1 , e 2 } at p, then the internal 4-velocity is the vector where o P ∈ {±1} is a free choice of orientation of the plane P = span{e 1 , e 2 } orthogonal to va and independent of any orientation of Mp .We call the internal 4-velocity va the epistemic spin vector of the pointon at p, and identify the choice of orientation o P as spin, up ↑ or down ↓, in the e 3 direction, o P ∈ {↑, ↓}.
We then parallel transport va along β until it is projected under h onto a subsequent 1-dimensional subspace M β(t ′ ) .In the next section we will show how this projection yields a realist model of spin wavefunction collapse.
Remark 3.2.The identification of time orientation with electric charge yields a new physical interpretation of spinors, wherein spinor chirality becomes electric charge [B2,Proposition 2.2].This new interpretation in turns yields a derivation of the standard model particles from the free Dirac Lagrangian [B2,Theorem 7.1].In particular, it establishes a composite model (interpretation) of the standard model, where its particles are represented by bound states of pointons.This model has only one fundamental particle, namely the pointon, and only one fundamental interaction, namely pair creation and annihilation of pointons of opposite charge.We note, however, that the compositeness may be regarded as purely mathematical (similar to a direct sum decomposition of a representation into irreducibles), since the pointons in a bound state share the same wordline.In any case, this feature makes our model quite different from other composite (or preon) models and eliminates the preon problem of finite particle size.
The free Dirac Lagrangian specifies both the possible pointon bound states and, with two additional 'fusion rules', their possible interactions (the standard model trivalent vertices) [B1, Theorem B], [B2, Section 5; Theorem 5.9]. 4 In this framework, then, gauge theory is regarded as an approximate description of the Dirac Lagrangian.Furthermore, the neutral standard model particles (the photon, Z boson, Higgs boson, and neutrinos) are represented by bound states of an even number of pointons, half with positive charge and half with negative charge. 5In Section 5 we will introduce this model for electrons, positrons, and photons.
Finally, the possible free choices of orientation that may arise from vanishing subspaces are given in Table 1.In this article we only consider electric charge and spin.We note that the choice of orientation o 0123 ∈ {±1} for any external tangent Table 1.The pointon properties that emerge from vanishing subspaces of spacetime tangent spaces.free orientation from the vanishing subspace ⇝ physical identification

Tangent space projections: spin wavefunction collapse
Throughout, let β be the worldline of a pointon.Recall that β is a curve in the external spacetime M and a single point in the internal spacetime M .Thus, by β ⊂ M (resp.β ∈ M ), we mean we are viewing β as a curve in M (resp.a point in M ).Furthermore, by M β(t) we mean the subspace of the (4-dimensional) tangent space Mβ(t) given by the image of h at the point β(t) ∈ M ; in particular, M β(t) is not the same tangent space for each t.
• For w ∈ Mp , set • Finally, suppose the pointon has timelike 4-velocity v and let e 0 , . . ., e 3 ∈ Mβ(t) be an orthonormal tetrad along β ⊂ M for which e 0 = v.If w, w ′ are 4-vectors that lie in V := span{e 1 , e 2 , e 3 } ⊂ Mβ(t) , then we may write them as 3-vectors w, w ′ ∈ V with respect to e 1 , e 2 , e 3 , and denote their contraction by the dot product in V , w•w ′ = −w a w ′ a .Let β ⊂ M be a pointon worldline, and suppose that for t ∈ (0, ϵ], the dimensions of M β(−t) and M β(t) are constant and satisfy Let s be an internal unit vector parallel transported along β and set p := β(0) ∈ M .
(a) As s enters the lower dimensional internal space M p , s ∈ M β(−ϵ)→p is projected onto it by the normalized internal metric ĥp : Mp → M p .(b) As ĥp (s) exits M p , the time reversal of (a) occurs: a unit vector section If ĥp (s) ̸ = 0, then the probability that s ′ is chosen is given by what we call the Kochen-Specker conditional probability: Suppose dim M p = 1 and let ĥp (s) = va be the internal 4-velocity of the pointon at p = β(0).We call the parallel transports of ĥp (s) and s ′ along β to β(ϵ) the epistemic and ontic spin vectors of the pointon at β(ϵ), respectively.
In this case, we propose the following: (a ′ ) The projection s → h p (s) corresponds to a measurement of spin in the direction of the line M p ⊂ Mp : a spin wavefunction collapse arises from an actual projection between tangent spaces on spacetime.(b ′ ) The choice of unit vector s ′ corresponds to the randomness inherent in the outcome of a measurement of spin subsequent to p = β(0).Suppose an ontic spin vector s exits a 1-dimensional tangent space M p at p = β(0), is parallel transported along a pointon worldline β ⊂ M for a time ϵ > 0, and then enters another 1-dimensional tangent space M q at q = β(ϵ): We say the spin s of the pointon is prepared at p and measured at q. Furthermore, the parallel transport of h p (s) along β (which in general is not equal to s) is the pointon's epistemic spin vector, by Definition 4.2.Let e 0 , . . ., e 3 ∈ Mβ(t) be an orthonormal tetrad parallel transported along β such that e 0 = v.The 3-vector ĥβ(t) (s) ∈ span{e 1 , e 2 , e 3 } ⊂ Mβ(t) is a unit vector, and thus may be regarded as a point on the (parallel transported) unit sphere S 2 β(t) ⊂ Mβ(t) defined by the triad e 1 , e 2 , e 3 .
Recall the standard correspondence between the Bloch sphere S 2 and the spin Hilbert space H = C |↑⟩ ⊕ C |↓⟩ given by (7) w = (sin θ cos φ, sin θ sin φ, cos θ) ←→ |w⟩ = cos(θ/2) |↑⟩ + e iφ sin(θ/2) |↓⟩ , where 0 ≤ θ ≤ π and 0 ≤ φ < 2π.Under this correspondence, we may identify S 2 with the Bloch sphere, and the 3-vector ĥβ(t We thus obtain a spacetime geometric realization of an ontological model of spin called the Kochen-Specker model [KS], which we shall now briefly review. The Kochen-Specker model is a realist model where quantum spin states are epistemic: they specify our knowledge about ontic (i.e., physically real) states.The ontic state space of the model is the Bloch sphere S 2 .If the spin of a particle is measured to be |ψ⟩ ∈ H, then the particle's subsequent ontic state will be some vector λ ∈ S 2 that lies in the hemisphere 'above' the plane orthogonal to ψ ∈ S 2 , that is, For example, if the particle is measured to have spin up |(0, 0, 1)⟩ (resp.spin down |(0, 0, −1)⟩) in the z-direction, then its subsequent ontic state is some vector that lies in the northern hemisphere (resp.southern hemisphere) of S 2 .
In this model, the probability that a preparation procedure P ψ produces an ontic state λ represented by ψ is given by 7 , where H is the Heaviside function: H(x) = 1 if x > 0 and 0 otherwise.Since the probability p(λ|P ψ ) is independent of the preparation procedure P , for brevity we 7 Note that p(λ|P ψ ) is correctly normalized: it suffices to suppose ψ = (0, 0, 1) and λ = (sin θ cos φ, sin θ sin φ, cos θ).
denote it p(λ|ψ).Furthermore, the probability that a measurement |ϕ⟩⟨ϕ| of λ yields ϕ ∈ S 2 is given by ( 9) The conditional probabilities ( 8) and ( 9) together reproduce the Born rule, 8 Theorem 4.3.Our spacetime model of spin reproduces the Born rule for spin wavefunction collapse: Proof.The assumptions ( 5) and ( 6) of our model are equivalent to the assumptions ( 9) and ( 8) of the Kochen-Specker model, with Our model is therefore a spacetime geometric realization of the Kochen-Specker model, and the theorem follows.□

Photons
In this and the following two sections we will find that a consequence of internal spacetime geometry is that any particle with a null 4-velocity must be electrically neutral, have spin-1, and admit a trivalent interaction vertex similar to the electronphoton vertex.
Consider a pointon with geodesic worldline β ⊂ M and null 4-velocity v.By Lemma 2.3, a minimal orthogonal projection of v is of the form for some choice of null vector v ′ satisfying v a v ′ a = −1.The choice of v ′ is equivalent to a choice of unit timelike or spacelike vector Hence, a choice of v ′ is equivalent to a choice of inertial frame with time direction ẽ0 .The vectors ẽ0 , ẽ3 are orthogonal and satisfy 8 For completeness, we show (10) following the derivation of [L, Appendix B].It suffices to suppose ψ = (1, 0, 0) and ϕ = (cos φ ϕ , sin φ ϕ , 0).The Heaviside functions in (10) imply the restrictions where (i) holds by (7).A similar computation holds if φ ϕ is negative.
Thus, ẽ3 is the direction of propagation of the pointon in the inertial frame with time direction ẽ0 .Furthermore, (11) implies that there are projections In particular, at points β(t) along β ⊂ M not transversely intersected by other pointon worldlines, the internal tangent space M β(t) ⊂ Mβ(t) is a plane orthogonal to ẽ0 and ẽ3 .Consequently, there is a free choice of plane orientation o 03 = o(ẽ 0 ∧ ẽ3 ), but no free choice of timelike orientation o 0 = o(ẽ 0 ), and therefore the pointon has no electric charge.
Theorem 5.1.A pointon with a geodesic worldline β ⊂ M and null 4-velocity v can only exist in a bound state with another pointon, and this state has nonzero spatial extent in the direction v + v ′ in M .Furthermore, the two pointons acquire opposite electric charges in the minimal projection where the pointons become timelike.
Proof.Recall the timelike and spacelike unit vectors ẽ0 and ẽ3 in (11).The vectors v and v ′ determine a unique minimal orthogonal projection of Mβ(t) for which the null pointon becomes timelike, namely

This projection yields the inclusions
In the smaller external space [ẽ 3 ] Mβ(t) , the null pointon has a timelike 3-velocity [ẽ 3 ]v = ẽ0 , by (12).Thus, at points β(t) ∈ M not transversely intersected by other pointon worldlines, the pointon's free choice of orientation that results from the vanishing of ẽ0 in [ẽ 3 ] Mβ(t) is electric charge o 0 , by (4).(Note that o 0 and o 03 are independent since any identification would not be well defined.)Therefore, although the pointon is neutral in Mβ(t) , the pointon does have an electric charge in the smaller external space [ẽ 3 ] Mβ(t) .But this subspace charge implies that the pointon has electric charge in the spatial plane orthogonal to ẽ3 in Mβ(t) , in contradiction to the pointon being neutral in Mβ(t) .This inconsistency is resolved if β is necessarily the worldline of two (or an even number of) pointons, both neutral in M and with opposite electric charges in [ẽ 3 ] Mβ(t) .
Finally, suppose that β is not transversely intersected by other pointon worldlines at β(t) ∈ M .Then the internal metric at β(t) is h = [v] = [ẽ 0 ][ẽ 3 ], by (13).Since h is obtained from the external metric g by projecting out tangent vectors to pointon worldlines, ẽ3 must be a tangent vector to the pointon bound state at β(t).Consequently, the bound state has nonzero spatial extent in the ẽ3 direction in M .More precisely, in the inertial frame with time direction ẽ0 , there is some ϵ > 0 for which the bound state contains the locus exp β(t) ((−ϵ, ϵ)ẽ 3 ) ⊂ M .□ We therefore propose that bound pairs of pointons of opposite electric charge are photons, and free pointons are electrons or positrons, depending on their charge.Furthermore, by allowing pairs of pointons of opposite charge to be created or annihilated, we obtain the electron-photon trivalent vertex.This is shown in Figure 1.9We will henceforth refer to bound pairs of pointons of opposite charge as 'photons' and free pointons as 'electrons'.
The constituent pointons of a photon each possess a spin 4-vector; we call these vectors s 1 and s 2 .In the next section we will use these vectors to model spin-1 polarization.
We defined a spin vector s exiting a 1-dimensional tangent space to have unit length, |s 2 | = 1, since spin vectors result from a free choice of plane orientation and an orientation does not specify a length.Similarly, we define photon spin vectors s 1 , s 2 exiting a 1-or 2-dimensional tangent space to span a parallelogram of unit area, since two plane orientations do not specify an area.But if two vectors have unit length and span a parallelogram with unit area, then they are orthogonal.Consequently, photon spin vectors s 1 , s 2 are orthogonal, ( 14) s a 1 s 2a = 0.For a photon with null 4-velocity v, a choice of null vector v ′ -which is required by Lemma 2.3-will serve two purposes in our model: (i) In Section 6 we will show that v ′ uniquely determines the photon's polarization type: linear, elliptical, or circular.(ii) Although the photon has a null 4-velocity, v ′ specifies an inertial frame from the relations (11).This inertial frame will be used to define the energy ω > 0 and 4-momentum k = ωv of the photon in [B7].

Pairs of spin vectors: photon polarization
In Section 5 we introduced a composite photon-electron model where a photon consists of two pointons, one with positive timelike orientation (an electron) and one with negative timelike orientation (a positron).Attached to each pointon is a spin vector, and thus a photon has two spin vectors.In this section we will introduce an extension of the Kochen-Specker model to spin-1 polarization and its wavefunction collapse, by using pairs of spin vectors in place of single spin vectors.We assume that photons travel with speed |v| equal to c in a vacuum and less than c in a medium, as is the case for electromagnetic waves (e.g., [BW,Chapt. 1]). 10 6.1.Photon polarization in a medium.Consider a photon with worldline β ⊂ M , timelike 4-velocity v, and spin 4-vectors s 1 , s 2 .We would like to identify the pair s 1 , s 2 with the polarization of the photon.
We briefly recall the classical and quantum descriptions of polarization.In the reference frame with time direction e 0 , a monochromatic electromagnetic plane wave with Poynting vector S = |S|e 3 has the form where E 1 , E 2 ∈ C are complex amplitudes and Re denotes the real part.The ratio of the amplitudes characterizes the type of polarization of E: Furthermore, E 2 /E 1 has negative (resp.positive) imaginary part if and only if E has right-handed (resp.left-handed) elliptical polarization.In general, the polarization of E may be represented by its Jones vector In a homogeneous isotropic medium with dielectric constant ϵ and magnetic permeability µ, we assume |v| = c/ √ ϵµ, in accordance with Snell's law.Of course, the phase velocity or group velocity of a superposition of monochromatic electromagnetic waves, all with speeds less than c, may exceed c or be negative (as in anomalous dispersion).
in the (quantum) Hilbert space H. Set To associate a polarization state to the pair s 1 , s 2 , decompose each s j = [e 0 ]s j into 3-vectors parallel and orthogonal to the photon's direction of propagation e 3 , where s ∥ j := (s j •e 3 )e 3 .Note that by our choice of tetrad, namely v = e 0 + ae 3 , s j has e 0 component a(s j •e 3 ), since s a j v a = 0 by assumption.Since We map the spin vectors s 1 , s 2 to a monochromatic electromagnetic plane wave E with complex amplitudes for j = 1, 2. Equivalently, setting Under these identifications we recover exactly the classical and quantum descriptions of linear and circular polarization: Lemma 6.1.Consider a timelike photon with spin vectors s 1 , s 2 .
(1) Suppose and thus the photon has circular polarization.
(2) Suppose , and thus the photon has linear polarization.
(2) Now suppose s 1 = s ⊥ 1 and s 2 = s ⊥ 2 , that is, s 1 and s 2 both lie in the plane orthogonal to the photon's direction of propagation e 3 .Then the amplitudes E j reduce to

□
Similarly, letting |H⟩ := |e 1 ⟩ and |V ⟩ := |e 2 ⟩ we obtain exactly the Jones vectors for the horizontal-vertical, diagonal-antidiagonal, and circular polarization bases of H: Note that the factor of i in the circular polarizations |L⟩, |R⟩ corresponds to a spin vector parallel to the photon's direction of propagation, as in Lemma 6.1.
Recall that a waveplate is an optical device that may be used to change linear polarization to circular polarization.For a photon with spin vectors s 1 , s 2 incident on a waveplate, we propose that the components s j •e slow of the spin vectors s j along the slow axis e slow remain unchanged, whereas the components s j •e fast of the spin vectors s j along the fast axis e fast are rotated in the plane orthogonal to the slow axis e slow .In order for our pointon model to agree with the electromagnetic wave model, we take the rotation angles to be the relative phase shifts of the corresponding electromagnetic wave E.
Specifically, consider a photon propagating in the e 3 direction with initial spin vectors s 1 = e 1 and s 2 = e 2 , that is, with diagonal polarization |D⟩, incident on a waveplate with slow axis e 1 and fast axis e 2 .
(i) If the waveplate is a quarter plate, then s 2 will be rotated 90 • in the e 2 ∧ e 3 plane, whence s 2 will exit pointing in the e 3 direction, while s 1 will exit unchanged.The photon will thus exit with circular polarization.(ii) If the waveplate is a half plate, then s 2 will be rotated 180 • in the e 2 ∧ e 3 plane, so s 2 will exit pointing in the −e 2 direction, while again s 1 will exit unchanged.The photon will therefore exit with antidiagonal polarization |A⟩.
Theorem 6.2.Let {s 1 , s 2 } and {s ′ 1 , s ′ 2 } be two polarizations.Then, by possibly relabeling subscripts, That is, the quantum state vectors of two polarizations are orthogonal in the Hilbert space H ∼ = C 2 if and only if the pairs of spin 4-vectors are parallel and antiparallel in spacetime.
Proof.This is readily verified from the basis vectors ( 16).□ Remark 6.3.There are differences between our composite model of photon polarization and the electromagnetic wave and Jones vector models.(i) In the electromagnetic wave model, the electric vector E is always transverse to the direction of wave propagation, S = E × H.In contrast, the spin vectors s 1 , s 2 need not be orthogonal to the direction of propagation of the photon: by Lemma 6.1, circular polarization is obtained by s 1 or s 2 being parallel to the photon's direction of propagation.Nevertheless, the spin 4-vectors s 1 , s 2 are always orthogonal to the photon's 4-velocity v: s a 1 v a = s a 2 v a = 0.This is compatible with the Lorenz gauge ∂ a A a = 0 together with Maxwell's equation ∂ a F ab = 0, where (ii) In Section 4 we showed that the Bloch sphere for electron spin is a geometric feature of internal spacetime geometry.In contrast, the Poincaré sphere for photon polarization does not correspond to physical spacetime geometry.Indeed, orthogonal polarization states are represented by antipodal points on the Poincaré sphere (similar to the Bloch sphere), but Lemma 6.2 implies that this is not the case for the (parallel transported) unit sphere that parameterizes the possible vectors 1 √ 2 (s 1 + s 2 ).6.2.Photon polarization in a vacuum.Consider a photon with geodesic worldline β ⊂ M , null 4-velocity v, and spin 4-vectors s 1 , s 2 .Since v 2 = 0, Lemma 2.3 implies that h β(t) projects out a 2-dimensional subspace of Mβ(t) at points β(t) along β where the photon is isolated: with v ′ some null 4-vector satisfying v a v ′ a = −1.In the following, we describe the constraints that the null condition v 2 = 0 imposes on s 1 , s 2 , and thus on the photon's polarization.
Corollary 6.6.A null photon cannot possess exact circular polarization, but only an approximate circular polarization where one of its two spin 3-vectors s j := [e 0 ]s j points arbitrarily close to the direction of the photon's propagation, up to sign.
Proof.Follows from Proposition 6.4.2, since b is nonzero but can be chosen to be arbitrarily close to zero.□

Polarization wavefunction collapse
Consider a photon with worldline β ⊂ M , timelike or null 4-velocity v, and spin 4-vectors s 1 , s 2 .Suppose the photon meets an electron at a point p ∈ M .In our composite model of the standard model [B1, B2], we declare there to be an electron-  1].Note the similarity between ( 19) and ( 5).
Lemma 7.1.Suppose there is an electron-photon vertex at p ∈ M which conserves 4-momentum.Let v be the photon's 4-velocity.Then generically, Proof.Let v 1 , v 2 , v 3 ∈ Mp be the respective 4-velocities of the photon and two electrons at p, and denote by k i = ω i v i , ω i > 0, their 4-momenta.By 4-momentum conservation we have The set v 1 , v 2 , v 3 is thus linearly dependent.The internal metric h is therefore given by the composition of projections Furthermore, the rank of the projection (1) of a timelike vector is 3, and the rank of the projection (2) of a null vector is 2. Consequently, if v 1 , v 2 are generic, v 1 ̸ = −v 2 , then (20) holds. □ We now show that our model yields a partial ontological spacetime description of linear polarizers.
Consider a photon with normal incidence on a linear polarizer.Suppose the polarizer is at rest (with 4-velocity e 0 ) and the photon has initial 4-velocity for some a ∈ (0, 1].Further suppose that, at a point p ∈ M in the polarizer, the photon meets an electron that is confined to a straight wire (or molecule) which runs along the ±e 1 direction.The wire thus acts as an external constraint to the photon-electron interaction at p.In particular, Lemma 7.1 does not hold in this case.Consequently, for some b, b ′ ∈ (0, 1), the electron has ideal initial 4-velocity If there is an electron-photon vertex at p, then we say the photon is 'absorbed' by the electron; otherwise the photon is 'transmitted' at p.
(i) If the photon has circular polarization, then one of its spin vectors s j is ±e 3 . Whence, Thus, (19) trivially holds.Therefore a photon with circular polarization will be absorbed by the electron.(ii) If the photon has linear polarization, then ( 14) and ( 19) together imply that the photon is transmitted at p if and only if 4 , then the photon has diagonal or antidiagonal polarization, or s 1 + s 2 = ±(e 1 − e 2 ).( 14) then implies that one of the spin vectors s j is ±e 1 .Thus, h p (s j ) = h p (e 1 ) = 0.
But then (19) trivially holds.Therefore the photon is absorbed by the electron.(iii) If the electron emits a new photon, then the photon's spin vectors will satisfy ( 21) by (b) in Section 4.
A photon may propagate between the wires (or stretched molecules) in a linear polarizer, yet still exit with linear polarization.We expect that this may be dealt with using pointon spinors, introduced in [B2, Section 3], though this requires further investigation.
The problem with applying these definitions to an internal spacetime M is that y : U → M in ( 22) is not injective.Thus, we cannot replace x −1 in (23) with y −1 .Morally, the tangent vector v = β ′ (0) to a pointon worldline β should be zero on M because β : (−ϵ, ϵ) → M is a constant map, and so any directional derivative along β should vanish.However, without a parametrization of M at β ∈ M we cannot make this precise using (23).To formulate a suitable definition of tangent space M p of M at p ∈ M , we require the following:11 (i) M p should be a vector space,12 and in particular a subspace of Mp ; and (ii) M p should not contain tangent vectors to pointon worldlines at p.These two conditions naturally lead to orthogonal projections.Indeed, suppose that there is a single timelike pointon worldline β in M at p = β(0).Fix an orthonormal basis of Mp that contains v = β ′ (0), say e 0 = v, e 1 , e 2 , e 3 .Then e 0 vanishes in M p .Consequently, an arbitrary vector 3 j=0 a j e j in Mp , with a j ∈ R, is mapped to 3 j=1 a j e j in M p .We therefore define M p to be the image of the degenerate metric h at p ∈ M (Definition 2.5): We call M p a 'tangent space' because it is a vector space that reduces to the tangent space Mp of the manifold M whenever M = M (that is, whenever the set of pointon worldlines is empty).Moreover, since M p is a subspace of Mp that does not contain v, the (vector space) dimension of M p can be at most three: the dimension of a vector space is the number of elements in any basis, and there is a basis for Mp which contains v.
We would like to compare this tangent space dimension with the dimension of M itself.M is not a manifold, however, and so we cannot apply the definition of manifold dimension given above.Nevertheless, we define the dimension of M to be the dimension of its external manifold M since M is obtained from M by removing a locally finite set of curves in M , and such a set of curves has measure zero in M . 13n this sense, then, the dimension of M is four.
Recall that the dimension of a tangent space on a manifold is always equal to the dimension of the manifold itself.In contrast, if singularities are allowed-as in algebraic geometry-then this equality of dimensions necessarily fails.In fact, a singular point of a variety is precisely a point where the tangent space dimension is larger than the dimension of the variety. 14For example, if X is a plane algebraic curve, then its tangent spaces are 1-dimensional at all smooth points of X and 2-dimensional at all singular points.In the case of an internal spacetime M , we have found the opposite to hold: the tangent space dimensions dim M β(t) along a pointon worldline β ⊂ M are smaller than the dimension of M , rather than larger.Consequently, pointon worldlines are a novel type of singularity whose tangent spaces have smaller dimension, rather than larger dimension, to that of the underlying geometric space.

Appendix B. Ontological models
We briefly review the classification of ontological (hidden variable) models given by Harrigan and Spekkens [HS].Consider a quantum system with Hilbert space H, and suppose the system possesses an underlying ontic state space Λ.Let p(λ|P ) be the probability distribution that an ontic state λ ∈ Λ results from the preparation procedure P ; and let p(k|M, λ) be the probability distribution that the outcome k results from the measurement M of λ.Let ρ be the density operator associated to P , and let E k be the POVM associated to the outcome k of M .In order for the model to reproduce quantum statistics, 15 it must satisfy Born's rule [HS,  The relationship between Λ and H specifies the following classes of ontological models [HS,Definitions 4 and 5].
There are two subclasses of ψ-ontic models: (i) ψ-complete: Each quantum state ψ represents a unique ontic state λ.
(ii) ψ-supplemented: There is a quantum state ψ that represents more than one ontic state in Λ.(b) ψ-epistemic: There is an ontic state λ that is represented by more than one quantum state in H; thus (24) does not hold for λ.A quantum state ψ is therefore ψ-ontic if and only if a variation of ψ implies a variation of reality, and ψ-epistemic if and only if a variation of ψ does not imply a variation of reality.In terms of mappings, a model is ψ-complete if the correspondence Λ → PH is bijective; ψ-supplemented if the correspondence Λ → PH is surjective but 14 Specifically, a point m of a variety Max R is singular if and only if the vector space dimension over k = R m /m of the (Zariski) cotangent space m/m 2 at m, or equivalently, the dimension of the tangent space (m/m 2 ) * = Hom k (m/m 2 , k), is larger than the Krull dimension of the local ring R m . 15For our purposes, P and M produce the pure states ψ and ϕ in H respectively; whence ρ = |ψ⟩⟨ψ|, E k = |ϕ⟩⟨ϕ|, and tr(ρE k ) = |⟨ϕ|ψ⟩| 2 .
not injective; and ψ-epistemic if the opposite correspondence PH → Λ is surjective but not injective.

Figure 1 .
Figure 1.From left to right: an electron; a positron; a photon; an electron and positron join to form a photon; a photon splits into an electron and positron; an electron absorbs a photon; and an electron emits a photon.The arrowheads indicate time orientations.
photon trivalent vertex at p if and only if (19) h p (s 1 ) |h p (s 2 )| = −h p (s 2 ) |h p (s 1 )| .This simplifies to ĥp (s 1 ) = − ĥp (s 2 ) whenever h p (s 1 ) and h p (s 2 ) are nonzero.If we omit the minus sign in (19), then the bound state of the two pointons represents the Z boson instead of the photon; see [B2, Table