9.  Vacuum Structure and Angular Momentum

Take everything out of a box and something is still left. This chapter works out what the substrate leaves behind when nothing is there, and finds it has structure: enough to explain why things are able to spin, and enough to matter for how the universe expands.

9.1.  The cosmological constant: substrate vs emergent energy

The Landau–Ginzburg valley energy of the modal vacuum is

$$\mathcal{F}\big{|}_{\mathcal{V}}=\int_{\mathcal{C}}\left[-\frac{(|T(C)|-\tau)^{2}}{4b}\right]d\mu(C)\;<\;0,$$ (75)

the value of $\mathcal{F}$ evaluated on the vacuum 6-sphere $\mathcal{V}\cong S^{6}\subset V_{7}$ at radius $r_{0}=\sqrt{\varepsilon/(2b)}$ rather than at the unstable symmetric maximum $\psi=0$. This is the substrate’s internal Landau–Ginzburg (LG) vacuum energy. The observed cosmological constant $\Lambda_{\mathrm{obs}}>0$ in the field equations of emergent spacetime is the energy density that gravity sees in its own vacuum after the substrate has been projected. These are not the same quantity, and the apparent sign mismatch between $\mathcal{F}\big{|}_{\mathcal{V}}<0$ and $\Lambda_{\mathrm{obs}}>0$ dissolves once that distinction is made precise. This section addresses the sign of $\Lambda$ through two complementary mechanisms: A, a calculable bare zero-point value whose sign is regulator-dependent, and B, a structural $w=-1$ contribution that carries the sign conditional on the substrate’s pre-spatial framing. The magnitude is the standard cosmological-constant fine-tuning problem of any quantum field theory coupled to gravity, which PST inherits from the Standard Model via the LG–Higgs structural identification, neither worsening nor solving it.

9.1.1.  What $\mathcal{F}\big{|}_{\mathcal{V}}$ is, and is not

The static valley depth $\mathcal{F}\big{|}_{\mathcal{V}}$ is the depth of the sombrero ring relative to the symmetric maximum $\psi=0$ in the substrate’s modal potential. It is internal to the precausal substrate. Coupling it directly to emergent gravity as if it were the cosmological constant would treat the substrate and the emergent geometry as a single thermodynamic system with a shared zero of energy. They are not: the substrate has no spacetime extent (cf. the Cosmological Principle theorem of §16.2); the projection $\Pi$ defines what couples to $g_{\mu\nu}$, not the bare LG potential. $\mathcal{F}\big{|}_{\mathcal{V}}<0$ is therefore a correct statement about the LG vacuum but not, on its own, a statement about $\Lambda$.

9.1.2.  Mechanism A: bare Goldstone zero-point on the slice $\mathcal{V}_{\hat{\eta}}\cong S^{1}$

The Goldstone phase $\theta$ of the broken $\mathrm{U}(1)$ on $\mathcal{V}$ is a massless field whose bare zero-point fluctuations at a fixed ultraviolet (UV) regulator $\Lambda_{\mathrm{UV}}$ contribute

$$\rho_{\mathrm{A}}^{\mathrm{bare}}\;=\;\frac{1}{16\pi^{2}}\,\Lambda_{\mathrm{UV}}^{4}\;+\;\text{counter-terms},$$ (76)

which is positive in absolute value at any naive cutoff and dominates $|\mathcal{F}\big{|}_{\mathcal{V}}|$. At PST’s modal cutoff $M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1.285$ TeV the bare contribution, taken at the un-suppressed naive-cutoff scale $\Lambda_{\mathrm{UV}}\sim M_{*}$ (dropping the loop factor of equation (76) for an order-of-magnitude estimate), is $\rho_{\mathrm{A}}^{\mathrm{bare}}\sim M_{*}^{4}\approx 10^{12}$ GeV⁴, larger than $|\mathcal{F}\big{|}_{\mathcal{V}}|\sim\varepsilon^{2}\sim 10^{8}$ GeV⁴ by four orders, and larger than the renormalised observed value $\Lambda_{\mathrm{obs}}/(8\pi G)\approx 10^{-47}$ GeV⁴ by some 60 orders. Matching $\Lambda_{\mathrm{obs}}$ requires the standard quantum field theory (QFT) with-gravity counter-term cancellation between $\rho_{\mathrm{A}}^{\mathrm{bare}}$, $\mathcal{F}\big{|}_{\mathcal{V}}$, and the bare cosmological constant. This is the standard cosmological-constant fine-tuning problem of any quantum field theory (QFT) coupled to gravity; PST inherits it through the LG–Higgs identification, and does not claim to resolve it.

Scope of Mechanism A. It might be supposed that Mechanism A delivers the sign of $\Lambda_{\mathrm{obs}}$. That would be too strong: the sign of the bare zero-point energy is in fact regulator-dependent (this is the heart of the cosmological-constant problem), and once a large counter-term cancellation is conceded for the magnitude, the sign of $\Lambda_{\mathrm{obs}}$ is set by the counter-term, not by $\rho_{\mathrm{A}}^{\mathrm{bare}}$. Mechanism A therefore contributes a calculable bare value at fixed regulator (positive in the naive $\Lambda_{\mathrm{UV}}^{4}/(16\pi^{2})$ form), but does not on its own pin the sign of the renormalised observed $\Lambda$. The sign of $\Lambda_{\mathrm{obs}}$ comes from Mechanism B below, subject to its own caveats.

A classical version of Mechanism A (a uniformly rotating phase $\psi=r_{0}e^{i\omega t}$ delivering positive kinetic energy $r_{0}^{2}\omega^{2}$) fails: the conjugate Noether charge $J^{0}=2r_{0}^{2}\omega$ is the global $\mathrm{U}(1)_{\mathrm{em}}$ charge of the universe, observationally constrained to zero.

9.1.3.  Mechanism B: the ongoing-sublimation source

A structurally independent contribution arises from the cosmological continuity equation derived in §16.5. The substrate maintains an equilibrium density $j_{\tau}$ of threshold-crossed configurations in its pre-geometric layer, a substrate-internal density measured against the configuration-space measure $\mu=\bigotimes_{a\in D}\mathrm{Bern}(\tfrac{1}{2})$ of P3, not against any emergent-spacetime proper volume (the substrate has no spatial extent against which a proper-volume rate could be defined). Each threshold crossing carries the energy quantum $\Delta\mathcal{F}=\mathcal{F}_{\mathrm{pre}}-\mathcal{F}\big{|}_{\mathcal{V}}>0$ (the configuration descends from the high-tension pre-threshold state to the valley). The modal-sublimation projection $\Pi:\mathcal{S}\to\mathcal{G}$ carries this substrate-internal saturation into emergent spacetime as a true energy density $\rho_{\mathrm{inst}}=j_{\tau}\,\Delta\mathcal{F}/c^{2}>0$ that neither dilutes nor redshifts, so the covariant continuity equation forces $w=-1$. This derivation (non-dilution from the substrate’s spacelessness, the continuity equation, and the positive sign from $\Delta\mathcal{F}>0$) is carried out in full in §16.5, its canonical home; here we note only what is distinctive to Mechanism B. The choice that makes the equation of state $w=-1$ rather than $w=0$ is the definition of $j_{\tau}$ as a saturation density in the pre-spatial substrate (measured against the configuration-space measure of P3, with no proper-volume rate to refer to) rather than as a per-emergent-volume creation rate: the latter would give Hoyle–Bondi–Gold-style steady-state ($w=0$, continuous matter creation compensating dilution), not a cosmological constant. This is forced by the pre-spatial framing of the substrate, and is what separates PST’s mechanism from the classic steady-state models.

Scope of Mechanism B. One might present the no-dilution property as a derived structural consequence of the substrate’s spacelessness. This is in fact close to circular: the assertion “the substrate has no spacetime extent, so its outputs do not dilute under expansion” is itself the substantive content of the pre-spatial framing of $\mathcal{S}$, not a derived corollary of it. Mechanism B is therefore best read as: conditional on the pre-spatial framing of the substrate (an integral part of P1–P3 and the modal-sublimation functor $\Phi:\mathcal{S}\to\mathcal{G}$, but not derived from any narrower fact), the ongoing-sublimation contribution to the energy budget of emergent spacetime carries $w=-1$ exactly. This places Mechanism B in the same family as the Hoyle–Narlikar steady-state C-field [147] and similar ongoing-creation proposals; that family is constrained by observation (cosmic microwave background [CMB] blackbody, nucleosynthesis), as discussed at the cosmology level in §16.5. The PST formulation differs by treating ongoing instantiation as a structural feature of the substrate-to-spacetime projection rather than a phenomenological addition to general relativity (GR).

The two mechanisms differ both on whether they deliver the sign and on which is dominant:

• Mechanism A contributes a calculable bare zero-point energy at fixed regulator. Its sign is regulator-dependent, and the observed sign of $\Lambda_{\mathrm{obs}}$ is set by the QFT counter-term, not by the bare Mechanism A value. PST inherits the Standard Model (SM) cosmological-constant magnitude problem through this mechanism.

• Mechanism B contributes $w=-1$ structurally, with the positive sign of $\rho_{\mathrm{inst}}$ following from the direction of threshold crossing. The non-dilution premise is the pre-spatial framing of the substrate, not an independent derivation; conditional on that framing, the equation of state is fixed.

The honest joint statement is therefore: PST does not derive $\Lambda_{\mathrm{obs}}$ from first principles. What it offers is (i) a calculable bare zero-point contribution (Mechanism A) whose sign is renormalisation-scheme-dependent, and (ii) a structural $w=-1$ equation-of-state contribution (Mechanism B) conditional on the substrate’s pre-spatial framing. Whether Mechanism B is the dominant contributor to $\Lambda_{\mathrm{obs}}$ is an open question; if it is, the small observed value would not require the SM fine-tuning, but a structural argument for the absolute magnitude $j_{\tau}\,\Delta\mathcal{F}$ would still be needed. GR’s metric and curvature emerge from the spectral action (Computation 5), while the cosmological-constant magnitude is renormalised just as in any QFT coupled to gravity.

One route to that magnitude from the theory’s own resources was asked about and is closed (Computation 188). The second edition derives no area law for $\Pi$ (Table LABEL:tab:framework-comparison, §19.9.3), and one written with the area quantum set by a length the theory owns would be false rather than weak: the energy bound it implies, that a region holds no more vacuum energy than a black hole of its size, is $10^{-32}$ of the critical density with the grain $\ell_{*}$ and $5\times 10^{-53}$ with $d_{0}$, against an observed $\rho_{\Lambda}$ of $0.7$ of it. The bound that lands is general relativity’s, reached through the emergent $G$ of (61); it is an inequality on the total, and since $\Delta\mathcal{F}$ enters only through the product $j_{\tau}\,\Delta\mathcal{F}$ it fixes neither factor. The finiteness of the internal structure sets a scale, $M_{*}^{4}$, fifty-nine orders above $\rho_{\Lambda}$, which is Mechanism A again. Nothing in the theory’s own equipment reaches $\Delta\mathcal{F}$, and nothing here needs a fourth postulate or an imported bound.

9.2.  The degenerate vacuum manifold and angular momentum

The order parameter $\psi$ is $V_{7}$-valued (P3, §1.7). For the angular-momentum analysis of this section we work in a 2-plane slice $\mathbb{C}_{\hat{\tau},\hat{\eta}}\subset V_{7}$ spanned by the substrate-determined direction $\hat{\tau}$ and a perpendicular unit direction $\hat{\eta}$. The slice projects $\psi$ to a complex scalar $\psi_{\mathbb{C}}=\langle\psi,\hat{\tau}\rangle+i\langle\psi,\hat{\eta}\rangle$. The two components of the slice (magnitude $r$ and slice-angle $\theta$) written in polar form as $\psi_{\mathbb{C}}=r\,e^{i\theta}$ with $r\geq 0$ and $\theta\in[0,2\pi)$ are the relevant degrees of freedom for the $\mathrm{U}(1)$ angular structure.

Restricted to the slice, the potential functional becomes:

$$\mathcal{F}(\psi_{\mathbb{C}},C)=\int_{\mathcal{C}}\Bigl{[}-\varepsilon\,|\psi_{\mathbb{C}}|^{2}+b\,|\psi_{\mathbb{C}}|^{4}+c\,|\nabla_{\mathcal{C}}\psi_{\mathbb{C}}|^{2}\Bigr{]}d\mu(C)$$ (77)

where $|\psi_{\mathbb{C}}|^{2}=r^{2}$. This functional is invariant under global phase rotations $\psi_{\mathbb{C}}\mapsto e^{i\alpha}\psi_{\mathbb{C}}$ for all $\alpha\in\mathbb{R}$: a continuous $\mathrm{U}(1)$ symmetry within the slice. For $\varepsilon(C)<0$, the unique minimum is $\psi_{\mathbb{C}}=0$, the symmetry is unbroken, and the potential is a paraboloid in the slice. For $\varepsilon(C)>0$, the symmetric minimum becomes an unstable saddle point, and the functional acquires a continuous ring of degenerate minima in the slice:

$$\mathcal{V}_{\hat{\eta}}=\bigl{\{}\,\psi_{\mathbb{C}}\in\mathbb{C}_{\hat{\tau},\hat{\eta}}\;:\;|\psi_{\mathbb{C}}|=r_{0}\,\bigr{\}},\qquad r_{0}=\sqrt{\frac{|T(C)|-\tau}{2b}}=\sqrt{\frac{\varepsilon}{2b}}$$ (78)

The slice $\mathcal{V}_{\hat{\eta}}$ of the vacuum manifold is topologically $S^{1}$: the full $\mathcal{V}\cong S^{6}\subset V_{7}$ of §9.1 meets the complex slice $\mathbb{C}_{\hat{\tau},\hat{\eta}}$ in a circle. In the two-component representation $(\psi_{1},\psi_{2})$, the potential surface takes the form of a sombrero potential: a local maximum at the origin surrounded by a continuous circular valley of degenerate minima at radius $r_{0}$. It is this structure, not the paraboloid, that constitutes the correct vacuum of any instantiated configuration.

The spontaneous breaking of the $\mathrm{U}(1)$ symmetry, when the system selects a specific phase $\theta_{0}$ and settles into one point on $\mathcal{V}$, is accompanied by a massless excitation in the $\theta$ direction, the Goldstone mode of the broken symmetry [43]. In the precausal context, this massless mode corresponds physically to motion along $\mathcal{V}$ at zero energetic cost: the phase of the order parameter can vary without climbing the potential, because all points on the ring are degenerate. After projection through $\Pi$, this precausal excitation manifests as a propagating mode in the instantiated geometry.

One qualification, and the photon does not depend on it. The degeneracy described in this paragraph is the $\mathrm{U}(1)$ slice of a picture in which the order parameter’s direction in $V_{7}$ is determined by P2 rather than spontaneously chosen. Those angular directions therefore supply no Goldstone, and the symmetry that survives fixes the vacuum and so supplies none either (Computation 170). The mode described here is the phase of the complexified order parameter, which determination cannot touch because $\hat{\tau}$ is a real direction and carries no phase (Computation 171).

And it is not the photon, which is worth saying plainly because the sentence above reads as though it were. Both alternatives fail: left as a free massless scalar below $M_{*}$ it would break F4 on its own, the Standard Model having no massless scalar, and a $\mathrm{U}(1)$ that is both gauged and broken gives a massive boson, which is a superconductor rather than electromagnetism. The consistent reading is the Standard Model’s own: $\theta$ is eaten in the electroweak Higgs mechanism, among the would-be Goldstones taken up by $W^{\pm}$ and $Z$, and the photon is the unbroken combination, massless because that generator is unbroken. So $\theta$ is the mode whose wave equation §4.12 argues from, described before it is eaten, and the photon is what the gauging step below delivers.

Which component it is, concretely. With $\psi=H^{0}$ the substrate’s complex line $\mathbb{C}\hat{\tau}$ carries two real degrees of freedom and they are the neutral component’s: the modulus is the Higgs and the phase is the would-be Goldstone the $Z$ takes up (Computation 172). The doublet’s other two, $H^{\pm}$, come from $\mathrm{SU}(2)_{L}$ rather than from the complex line, which is consistent, since $\hat{\tau}$ is a direction and a phase and carries no charge. Two and two make the doublet’s four.

Before making the identification with the photon, a structural caveat is required. The $\mathrm{U}(1)$ symmetry broken here is a global symmetry of a configuration-space scalar. The $\mathrm{U}(1)$ gauge symmetry of electromagnetism is a local symmetry of a spacetime vector field. The mathematical move from global to local symmetry is gauging: introducing a connection on a principal $\mathrm{U}(1)$ bundle over $M$, and promoting ordinary derivatives to gauge-covariant ones. That step is carried out constructively in §13.3: the projection $\Pi$ maps the continuum of substrate configurations to the full manifold $M$, so independent configurations at different loci of $M$ select independent vacuum phases $\theta(x)$; the symmetry parameter is therefore an arbitrary smooth function $\alpha(x)$ and the symmetry is local, with $\theta(x)$ a section of a $\mathrm{U}(1)$ principal bundle and $D_{\mu}\theta=\partial_{\mu}\theta-A_{\mu}$ its covariant derivative. What that step delivers is worth stating exactly, since an earlier version of this paragraph had it backwards. The connection it introduces couples to a phase whose $\mathrm{U}(1)$ is broken, $|\psi|=r_{0}\neq 0$, and a gauged broken $\mathrm{U}(1)$ gives a massive vector: in unitary gauge $\theta$ is eaten and the Stueckelberg term $r_{0}^{2}A_{\mu}A^{\mu}$ is what remains, a superconductor rather than electromagnetism. Once the electroweak embedding of §13.4 is in place that vector is the $Z$ (Computation 172). Nor is the Maxwell term (155) obtained by integrating out $\theta$; it is supplied by the inner fluctuations of the spectral action (Computation 6). The photon is the connection of the unbroken combination $Q=T_{3}+Y/2$, massless because $Q$ annihilates the vacuum, which is the Standard Model’s own argument and the one this book inherits (§13.3); masslessness does not follow from the inhomogeneous transformation law of $A_{\mu}$ on its own, since gauge invariance permits exactly the Stueckelberg mass above. So the route from the phase is constructive and it ends at the $Z$, not at the photon: $\theta$ is not the photon (Computation 171, Computation 172), and the photon has one route, as the unbroken connection, with minimal coupling forced by gauge covariance (Computation 166).

9.3.  The incomplete vacuum of standard physics

The vacuum formulation underlying General Relativity and its semiclassical extensions assumes a unique, symmetric ground state. The cosmological constant $\Lambda$ enters Einstein’s field equations [20] as a constant energy density associated with the symmetric minimum $\psi=0$. This is the $\varepsilon<0$ regime of the PST potential, the pre-threshold paraboloid, and it correctly describes configurations that have not yet crossed the modal threshold. Once sublimation has occurred and $\varepsilon>0$, the symmetric minimum is no longer a stable state; it is the local maximum at the centre of the sombrero. The energy assigned to $\psi=0$ in the standard treatment is the energy of an unstable critical point, not a ground state.

The assumption of a unique symmetric vacuum is exact within its domain: for pre-threshold configurations, the paraboloid is the correct potential. The error is one of domain. The cosmological vacuum is an instantiated vacuum, and instantiated configurations inhabit the $\varepsilon>0$ regime. The physical ground state is not $\psi=0$ but any point on the slice $\mathcal{V}_{\hat{\eta}}\cong S^{1}$, selected by spontaneous breaking of the $\mathrm{U}(1)$ symmetry. What no theory built on the assumption of a unique symmetric vacuum can account for, without additional postulate, is why every gravitationally bound system in the observable universe, from the orbits of planets to the rotation of galaxies, exhibits stable circular structure. PST requires no additional postulate.

9.4.  Angular momentum as Noether charge

The global $\mathrm{U}(1)$ symmetry of $\mathcal{F}$ under $\psi\mapsto e^{i\alpha}\psi$ is inherited by the instantiated action $S$ through the projection $\Pi$: the Goldstone mode $\theta(\mathbf{x})$ enters $S$ as a massless spacetime scalar with a standard kinetic term, and the phase rotation is a global symmetry of that term. Noether’s theorem [21] applies to $S$, not to $\mathcal{F}$ directly; $\mathcal{F}$ is a free energy whose minima $\mathcal{V}$ determine which symmetry is spontaneously broken, while $S$ is the Lorentzian action whose dynamics generate the conserved charge. This distinction matters: Noether charges require a time for the charge to be conserved along; time is coinstantiated, so of course conservation laws are instantiated-domain structures.

With $\psi=r\,e^{i\theta}$ and the Goldstone kinetic term $r_{0}^{2}\,\partial_{\mu}\theta\,\partial^{\mu}\theta$ in $\mathcal{L}$, the Noether current associated with the phase rotation is:

$$j^{\mu}=2r^{2}\,\partial^{\mu}\theta$$ (79)

where $\partial^{\mu}$ are spacetime derivatives in $\mathcal{G}$. The conserved Noether charge is:

$$L=\int j^{0}\,d^{3}x=2r_{0}^{2}\,\dot{\theta}$$ (80)

where $\dot{\theta}$ is the time derivative of the phase in the instantiated domain. This charge $L$ is angular momentum. It is not postulated; it is the Noether charge of the projected instantiated action corresponding to the $\mathrm{U}(1)$ symmetry that $\mathcal{F}$ breaks spontaneously at threshold.

Before making the identification with physical angular momentum, a distinction must be drawn between two related but separate claims. The first is that the spontaneous breaking of $\mathrm{U}(1)$ symmetry generates a conserved Noether charge $L$ in configuration space. This is established above and is rigorous. The second is that this charge projects via $\Pi$ into the macroscopic orbital angular momentum of gravitationally bound physical systems. These claims are related but not identical. In standard field theory, the Goldstone mode of a broken $\mathrm{U}(1)$ is a massless field excitation propagating through the geometry; it is not automatically identified with the macroscopic orbital motion of massive bodies. The identification of $L$ with physical orbital angular momentum therefore requires the explicit action of $\Pi$: the projection of the configuration-space Noether charge onto the instantiated geometry must be shown to recover the orbital angular momentum of bodies moving on geodesics. The structural argument for this identification is clear, as $\Pi$ is by construction the structure-preserving map between the precausal and instantiated domains, and any conserved charge in configuration space must project to a conserved quantity in the instantiated geometry. The explicit computation of this projection, showing in detail how $L=2r_{0}^{2}\dot{\theta}$ maps to the orbital angular momentum of a planetary body or an electron in an atomic orbital, is carried out in the next subsection (§9.5), where the identification $L_{z}=mr^{2}\omega$ is obtained as a structural consequence of the Poincaré invariance of the projected Goldstone action.

With that qualification in place: in the instantiated domain, $L$ projects via $\Pi$ into physical angular momentum. Motion along the vacuum manifold, the order parameter tracing the ring of degenerate minima, is, after projection, the circular orbital motion of any instantiated system. Orbital mechanics is not an initial condition imposed from outside; it is the Noether charge of the spontaneously broken $\mathrm{U}(1)$ symmetry of the vacuum, conserved by construction. The stable circular orbits of gravitationally bound systems, which General Relativity describes correctly but does not derive, are, in PST, the structural projection of $\mathrm{U}(1)$ symmetry conservation across the sublimation threshold, as derived in the next subsection (§9.5).

The quantisation of $L$ in the near-threshold regime deserves comment. Near the modal threshold, where $\varepsilon(C)\approx 0$, the order parameter $\psi$ is not settled at a definite point on $\mathcal{V}$ but fluctuates across the ring. The charge $L$ is then not a classical continuous quantity but takes discrete values, the Noether charge computed over a fluctuating path on $S^{1}$. The quantisation condition $L=n\hbar$ for integer $n$ is the precausal statement that the phase of the order parameter can only wind an integer number of times around the ring without discontinuity. Spin is the near-threshold manifestation of the same $\mathrm{U}(1)$ charge that produces orbital angular momentum in the far-threshold classical regime.

9.5.  Angular momentum and the Goldstone–geodesic connection

The previous subsection established the Noether charge $L=2r_{0}^{2}\dot{\theta}$ and announced that the explicit computation of its projection to physical orbital angular momentum is carried out here. This section delivers it by deriving the angular momentum tensor from the Goldstone action and identifying the precise condition under which it reproduces the orbital angular momentum of a body in geodesic motion.

Angular momentum tensor from the Goldstone kinetic term. The Goldstone phase $\theta:M\to\mathbb{R}/(2\pi\mathbb{Z})$ enters the projected action with kinetic term $\mathcal{L}_{\theta}=r_{0}^{2}\,\partial_{\mu}\theta\,\partial^{\mu}\theta$. The canonical stress-energy tensor is

$$T_{\mu\nu}^{\theta}\;=\;2r_{0}^{2}\,\partial_{\mu}\theta\,\partial_{\nu}\theta\;-\;g_{\mu\nu}\,r_{0}^{2}\,\partial_{\lambda}\theta\,\partial^{\lambda}\theta,$$ (81)

and the angular momentum density 3-tensor is the standard $M_{\mu\nu}{}^{\rho}=x_{\mu}T_{\nu}{}^{\rho}-x_{\nu}T_{\mu}{}^{\rho}$ with conserved charges

$$L_{ij}=\int\left(x_{i}T_{j0}^{\theta}-x_{j}T_{i0}^{\theta}\right)\mathrm{d}^{3}x=2r_{0}^{2}\int\left(x_{i}\,\partial_{j}\theta-x_{j}\,\partial_{i}\theta\right)\dot{\theta}\,\mathrm{d}^{3}x.$$ (82)

These are conserved by the Bianchi identity [10] $\nabla^{\mu}T_{\mu\nu}^{\theta}=0$ (equation (71)), which follows from the diffeomorphism invariance derived in Chapter 4. The tensor structure is identical to that of standard angular momentum in a scalar field theory: PST derives this structure from the $\mathrm{U}(1)$ Noether symmetry rather than postulating it.

Vortex identification: localised bodies as topological defects. A spatially localised body moving along a circular geodesic of radius $r$ with angular frequency $\omega$ corresponds, in the Goldstone field, to a vortex-like configuration in which $\theta$ winds by $2\pi n$ ($n\in\mathbb{Z}$) around the body’s worldtube. In the far-field ($|\mathbf{x}-\mathbf{x}_{\mathrm{body}}|\gg\xi$, where $\xi\sim r_{0}^{-1}$ is the vortex core size), the gradient of $\theta$ is azimuthal: $\nabla\theta\approx n\hat{\phi}/|\mathbf{x}-\mathbf{x}_{\mathrm{body}}|$. The $z$-component of $L_{ij}$ integrated over a shell $[r_{1},r_{2}]$ around the vortex is

$$L_{z}^{\mathrm{shell}}\;=\;2r_{0}^{2}\int_{r_{1}}^{r_{2}}\!n\,\omega\,4\pi r^{2}\,\mathrm{d}r\;=\;\frac{8\pi}{3}\,n\,r_{0}^{2}\,\omega\,(r_{2}^{3}-r_{1}^{3}),$$ (83)

the integrand being $2r_{0}^{2}n\dot{\theta}$ because $x\partial_{y}-y\partial_{x}$ is $\partial_{\varphi}$ and $\partial_{\varphi}(n\varphi)=n$ is dimensionless. This scales as $r^{3}$, so the far-field Goldstone contribution is not localised: it grows without bound with the outer radius. That divergence is not removed by the short-distance cutoff, being infrared rather than ultraviolet, and it is the classic global-vortex behaviour. The winding label is an integer,

$$n\;\in\;\mathbb{Z},$$ (84)

because $\pi_{1}(\mathcal{V}_{\hat{\eta}})=\pi_{1}(S^{1})=\mathbb{Z}$ on the slice. The topology fixes the integer and not the unit. For $\theta=n\varphi$ a spatial rotation by $\alpha$ acts as an internal phase shift by $n\alpha$, so $L_{z}=nQ$ with $Q$ the internal charge, and single-valuedness under a $2\pi$ rotation then gives $L_{z}\in\hbar\mathbb{Z}$. But that last step is canonical quantisation, and $\hbar$ enters through equation (106) of §10.6, where it is supplied as the unit of momentum rather than derived. PST supplies the vacuum topology that makes the spectrum discrete; it does not supply $\hbar$ (Computation 161).

Classical limit and the condition for $L=mvr$. For a macroscopic body of mass $m$ at orbital radius $r$ with angular velocity $\omega$, the physical angular momentum is $L_{\mathrm{phys}}=mr^{2}\omega$. Matching $L_{z}=n\hbar$, the winding integer of equation (84) times the canonical unit, in the classical ($\hbar\to 0$, $n\to\infty$) limit requires

$$m\;=\;\lim_{n\to\infty}\frac{n\hbar}{r^{2}\omega}\;=\;\frac{2r_{0}^{2}}{r^{2}}\int\dot{\theta}\,\mathrm{d}^{3}x_{\perp}$$ (85)

where the integral is over the cross-section of the body perpendicular to the orbital plane. This is consistent with the stress-energy identification $m=\int T_{00}\,\mathrm{d}^{3}x$ derived from the projected action, confirming that the same Goldstone field that produces the angular momentum also carries the inertial mass. The explicit computation of the cross-sectional integral for a specific matter configuration (connecting $m$ and $L$ to the microscopic PST parameters $r_{0}$, $c$, and $d_{0}$) is treated below.

Closure via Lorentz covariance. The form $L_{z}=mr^{2}\omega$ follows without evaluating the cross-sectional integral. The projected Goldstone action

$$S_{\theta}\;=\;r_{0}^{2}\!\int_{M}\partial_{\mu}\theta\,\partial^{\mu}\theta\,\sqrt{-g}\,\mathrm{d}^{4}x$$ (86)

is diffeomorphism-covariant; in the local inertial frame along any geodesic it reduces to a Poincaré-invariant scalar field theory. For any Poincaré-invariant field theory, the total 4-momentum $P^{\mu}=\int T^{0\mu}\,\mathrm{d}^{3}x$ is a Lorentz 4-vector [97]. A global Goldstone vortex is not a finite-energy configuration: with $|\nabla\theta|=n/\rho$ the energy per unit length is $2\pi r_{0}^{2}n^{2}\ln(R/d_{0})$, logarithmically divergent in the outer scale $R$, and $d_{0}$ regulates only the core. The soliton momentum theorem needs finite total energy, so it applies to the regulated configuration, with $R$ set by the inter-defect spacing, and the mass it delivers is correspondingly cutoff-dependent. With that understood, and only with it [97]: $\mathbf{P}=\gamma m\mathbf{v}$, where $m=E_{\mathrm{rest}}/c^{2}$ is the rest mass. In the non-relativistic limit $v=r\omega\ll c$:

$$\mathbf{P}\;=\;m\mathbf{v},\qquad v=r\omega.$$ (87)

For a body in a circular geodesic of radius $r$ with angular velocity $\omega$, the tangential momentum is $P_{\phi}=mr\omega$ and the $z$-component of orbital angular momentum is

$$L_{z}\;=\;r\,P_{\phi}\;=\;mr^{2}\omega.$$ (88)

Including the intrinsic topological contribution $n\hbar$, the winding integer of equation (84) times the canonical unit:

$$L_{z}^{\mathrm{total}}\;=\;mr^{2}\omega\;+\;n\hbar.$$ (89)

In the macroscopic classical limit $n\hbar\ll mr^{2}\omega$, the winding term is negligible and $L_{z}=mr^{2}\omega$. The macroscopic orbital angular-momentum identification is thereby resolved as a structural consequence of the Poincaré invariance of the projected Goldstone action: the formula $L_{z}=mr^{2}\omega$ holds for any localised Goldstone vortex in geodesic motion, independent of the specific matter configuration.

PST elementary mass scale. It remains to express $m=E_{\mathrm{rest}}/c^{2}$ in terms of the PST microscopic parameters. In the rest frame of the vortex, $\partial_{t}\theta=0$ and the stress-energy density reduces to $T_{00}^{\theta}=r_{0}^{2}|\nabla\theta|^{2}$, so

$$mc^{2}\;=\;r_{0}^{2}\!\int_{M}|\nabla\theta|^{2}\,\mathrm{d}^{3}x.$$ (90)

For a localised phase lump of width $d_{0}$ with linear profile $\theta(\rho)=\pi\bigl{(}1-\rho/d_{0}\bigr{)}$ for $\rho\leq d_{0}$ (and $\theta=0$ beyond), the gradient is uniform inside the core: $|\nabla\theta|=\pi/d_{0}$. The integral evaluates to

$$\int_{|\mathbf{x}|\leq d_{0}}|\nabla\theta|^{2}\,\mathrm{d}^{3}x\;=\;\frac{\pi^{2}}{d_{0}^{2}}\cdot\frac{4\pi d_{0}^{3}}{3}\;=\;\frac{4\pi^{3}d_{0}}{3},$$ (91)

giving the PST elementary mass scale

$$m_{0}\;=\;\frac{4\pi^{3}}{3}\cdot\frac{r_{0}^{2}\,d_{0}}{c^{2}}.$$ (92)

What this is, and what it is not. It is a dimensional scale: any phase configuration localised on a width $d_{0}$ with an $O(1)$ excursion has rest energy $\propto r_{0}^{2}d_{0}$, and $m_{0}\propto r_{0}^{2}d_{0}/c^{2}$ is robust. It is not the mass of a vortex quantum, and the profile above is not a vortex: $\theta(\rho)$ depends on the radius alone, so it is constant on every closed loop and its winding number is zero. Nor could it be, since $\pi_{2}(S^{1})=0$ and a $\mathrm{U}(1)$ order parameter admits no point or spherical defect in three dimensions; the stable defects are lines, classified by $\pi_{1}(S^{1})=\mathbb{Z}$, and those are exactly the configurations whose energy diverges above. The coefficient $4\pi^{3}/3\approx 41.3$ is exact for this profile and carries no further significance. The orbital angular-momentum treatment is therefore closed on the terms it can be: it supplies the tensor, the integer winding label and a mass scale, $L_{z}=nQ$ is an identity for every axisymmetric configuration with $Q$ the Noether charge, and the unit is the absolute normalisation of the Goldstone stiffness $r_{0}^{2}$, which is the gradient-term normalisation Appendix C carries as the canonical-normalisation premise (Computations 194, 201); $\hbar$ is inherited at (106), and the treatment is carried there rather than as an item of its own.

9.6.  Summary

This chapter works out what the substrate leaves behind when nothing is there. The modal vacuum has negative Landau–Ginzburg valley energy, which fixes the sign of the cosmological constant on the substrate side and separates it from the emergent energy the instantiated sector sees (§9.1); its magnitude is configuration content, not structure. The vacuum manifold is the degenerate 6-sphere $\mathcal{V}\cong S^{6}\subset V_{7}$, read in the 2-plane slice spanned by $\hat{\tau}$ and $\hat{\eta}$ as the ring of minima (§9.2), which standard physics, with its unique symmetric ground state, lacks. The global phase symmetry of the Goldstone mode survives projection, and its Noether charge $L=2r_{0}^{2}\dot{\theta}$ is orbital angular momentum (§9.4), delivered explicitly through the Goldstone action and the geodesic condition (§9.5); its unit is inherited at (106) and is carried with the normalisation premise.