Quantum Theory and Relativistic QFT

This chapter defines the quantum projection channel of the theory. It expands the typed channel map introduced in Chapter 3, using the notation and standards fixed in Chapter 4. For each record-capable realization support object $Y\in\mathsf{Real}_0$, the quantum channel returns a structured quantum readout object. The channel formalizes Hilbert-space state structure, the observable algebra, effects, POVMs, Born probability measures, dynamics, tensor-product composition, instruments, and record preforms, together with the locally covariant relativistic QFT structure on which the later sectors depend. The source-side derivation of the quantum readout, the source origin of Born weights, and the single-record realization selector remain active proof obligations of the companion programme.

The chapter supplies quantum-side structures to Chapter 6 for record deposition and objectivity, and to Chapter 8 for the quantum-to-geometry bridge. It does not define record deposition mechanics or the geometry bridge.

5.1 The Quantum Channel Map

The quantum projection channel expands the typed channel map

$$\boxed{ \Pi_Q:\mathsf{Real}_0\to\mathcal{O}_Q } \tag{5.1}$$

introduced in Chapter 3. For each record-capable realization support object $Y\in\mathsf{Real}_0$, the quantum channel returns a structured quantum readout object

$$\boxed{ Q_Y=\Pi_Q(Y)\in\mathcal{O}_Q. } \tag{5.2}$$

5.2 Quantum Theory and Locally Covariant QFT

5.2.1 Quantum readout

On a prepared realization $Y$, a quantum readout consists of a Hilbert space $\mathcal H_Y$, a normal state space, an observable algebra, a measurable outcome space and its effects. In the type-I reduction the state space is

$$\mathcal S(\mathcal H_Y) = \{\rho\in\mathcal T_1(\mathcal H_Y):\rho\ge0,\ \operatorname{Tr}\rho=1\}. \tag{5.3}$$

For a POVM $E:\Sigma\to\mathcal B(\mathcal H_Y)$,

$$E(B)\ge0, \qquad E(\Omega)=I, \qquad E\!\left(\bigcup_nB_n\right)=\sum_nE(B_n) \tag{5.4}$$

in the weak operator topology for disjoint $B_n$. The Born measure is

$$\boxed{ \mu_\rho(B)=\operatorname{Tr}[\rho E(B)]. } \tag{5.5}$$

The trace pairing with a positive trace-class state makes (5.5) a normalized countably additive probability measure. It assigns weights to alternatives; it does not by itself select or deposit a record.

Closed dynamics is generated by a self-adjoint Hamiltonian,

$$U(t)=e^{-it\mathsf H}, \qquad \rho(t)=U(t)\rho(0)U(t)^\dagger, \tag{5.6}$$

while open dynamics is completely positive and trace preserving,

$$\Phi(\rho)=\sum_aK_a\rho K_a^\dagger, \qquad \sum_aK_a^\dagger K_a=I. \tag{5.7}$$

For a composite nonrelativistic system,

$$\mathcal H_{AB}=\mathcal H_A\otimes\mathcal H_B, \qquad \rho_{AB}\in\mathcal S(\mathcal H_{AB}),$$

without any assumption that $\rho_{AB}$ factorizes. In relativistic QFT, composition is instead expressed by the net of local algebras and their causal commutation relations; a tensor-product factorization is not assumed across arbitrary local regions.

For general local QFT the algebraic formulation is primary. If $\mathcal A$ is a von Neumann algebra with predual $\mathcal A_*$, a Schrödinger-picture instrument is a countably additive map

$$\mathcal I:\Sigma\longrightarrow\operatorname{CP}(\mathcal A_*), \tag{5.8}$$

whose nonselective operation preserves normalization. Its effect is $E(B)=\mathcal I(B)^*(I)$, and the probability is $p_\omega(B)=\omega(E(B))$. Equation (5.5) is recovered in a density-operator representation.

5.2.2 Locally covariant gauge QFT

Let $\mathsf{Loc}_{\rm spin}$ be the category of oriented, time-oriented, globally hyperbolic spin spacetimes with admissible gauge-bundle data and causality-preserving embeddings. A locally covariant QFT is a functor

$$\mathcal A:\mathsf{Loc}_{\rm spin}\longrightarrow\mathsf{Alg}_*, \tag{5.9}$$

obeying covariance, Einstein causality and the time-slice property. Gauge symmetry is treated by the BV/BRST complex. For a background $\mathfrak b$, the perturbative physical algebra is

$$\boxed{ \mathcal A_{\rm phys}^{\rm form}(\mathfrak b) = H^0\!\left(\widehat s_{\rm BV}, \mathcal A_{\rm int}^{\rm ren}(\mathfrak b)[[\hbar,\mathbf g]]\right). } \tag{5.10}$$

The quantum master equation and the relevant local, global and mixed anomaly classes must vanish or be cancelled on an admitted branch. Renormalization is locally covariant and carries an explicit prescription and scale $\mu$; changes of scheme are related by the renormalization group and threshold matching.

At the classical level the BV action and antibracket obey

$$(S_{\rm BV},S_{\rm BV})=0, \qquad s_{\rm BV}=(S_{\rm BV},\,\cdot\,), \qquad s_{\rm BV}^{2}=0.$$

Gauge fixing is a choice of Lagrangian submanifold, equivalently a gauge-fixing fermion where that description applies; physical observables are degree-zero cohomology classes and do not depend on that choice.

For fermionic matter the background contains a tetrad $e^a{}_{\mu}$, a spin connection $\omega_\mu{}^{ab}$ and a spin structure compatible with the gauge bundle. The covariant derivative is

$$D_\mu = \partial_\mu +\frac14\omega_\mu{}^{ab}\gamma_{ab} -i\sum_Ag_AA_\mu^AT_A, \tag{5.11}$$

with the appropriate representation matrices $T_A$. The renormalized BV effective action $\Gamma_{\rm QFT}^{\rm ren}[g,e,A,\Phi;\mu]$ satisfies the quantum master equation up to the declared anomaly class. On a physical branch,

$$\frac12(\Gamma,\Gamma)-i\hbar\Delta\Gamma=0, \tag{5.12}$$

or the nonzero right-hand side is cancelled by the stated matter content and counterterms. The field equations are the stationary conditions $\delta\Gamma_{\rm QFT}^{\rm ren}/\delta\Phi=0$, interpreted perturbatively where appropriate.

Admitted states are positive normalized algebraic states satisfying the Hadamard or corresponding microlocal spectrum condition required to define the renormalized stress tensor. For spacelike separated regions $O_1$ and $O_2$, Einstein causality gives

$$[\mathcal A(O_1),\mathcal A(O_2)]=0. \tag{5.13}$$

Renormalized time-ordered products obey locality, covariance, scaling and the Ward identities. Changing the scale $\mu$ is governed by the renormalization group, and integrating through a mass threshold requires matching of effective couplings and operators. These conditions state the QFT structure that a successful realization must recover; they are not placed in the source signature.

On an anomaly-free physical branch, perturbative unitarity holds on BRST cohomology order by order. A trace anomaly need not vanish; in four dimensions it has the renormalized form

$$\langle T^\mu{}_{\mu}\rangle = \sum_i\beta_i\mathcal O_i +aE_4+cW_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma} +b\Box R,$$

up to the chosen operator basis and scheme-dependent total-derivative terms. This contribution belongs to the same renormalized effective action and stress tensor used below; it is not added a second time in the gravitational equation.

Equation (5.10) is a formal $*$-algebra and is not automatically a $C^*$- or von Neumann algebra. In particular, there is no presumed $*$-homomorphism from the whole formal power-series algebra into a $C^*$-algebra. An operational completion in a relatively compact detector region $\mathfrak D$ is instead the datum

$$\mathfrak O_{\mathfrak D} = \bigl( \mathcal B_{\mathfrak D}, \mathcal B^0_{\mathfrak D}, \mathcal D^{\rm res}_{\mathfrak D}, \operatorname{Res}_{\mathfrak D}, \operatorname{Asym}_{\mathfrak D} \bigr). \tag{5.14}$$

Here $\mathcal B_{\mathfrak D}$ is a unital $C^*$-algebra, $\mathcal B^0_{\mathfrak D}$ is a norm-dense $*$-subalgebra, and $\mathcal D^{\rm res}_{\mathfrak D}$ is a $*$-algebra of coupling-dependent observables for which the chosen Borel–Écalle, analytic or nonperturbative reconstruction is defined. The partial resummation

$$\operatorname{Res}_{\mathfrak D}: \mathcal D^{\rm res}_{\mathfrak D}/\mathcal N_{\mathfrak D} \longrightarrow \mathcal B^0_{\mathfrak D} \tag{5.15}$$

is a $*$-isomorphism onto $\mathcal B^0_{\mathfrak D}$, with $\mathcal N_{\mathfrak D}$ the declared BRST- and detector-null ideal. The asymptotic map $\operatorname{Asym}_{\mathfrak D}$ sends elements of $\mathcal B^0_{\mathfrak D}$ to their formal BV/BRST expansions and satisfies

$$\operatorname{Asym}_{\mathfrak D} \circ\operatorname{Res}_{\mathfrak D}(F) \sim F \quad(F\in\mathcal D^{\rm res}_{\mathfrak D}) \tag{5.16}$$

to the declared asymptotic order, while respecting local covariance, the time-slice property and the physical cohomology. A genuinely nonperturbative construction may specify $\mathcal B_{\mathfrak D}$ directly, but it must still supply (5.16) on a dense comparison domain. Existence, covariance and uniqueness of $\mathfrak O_{\mathfrak D}$ for an interacting four-dimensional branch are companion-theorem obligations, not consequences of the formal algebra.

Only after such an operational completion has been established may a positive normalized state $\bar\omega$ on $\mathcal B_{\mathfrak D}$ be used in the GNS construction:

$$(\pi_{\bar\omega},\mathcal H_{\bar\omega},\Omega_{\bar\omega}) =\operatorname{GNS}(\mathcal B_{\mathfrak D},\bar\omega), \qquad \mathcal N_{\mathfrak D,\bar\omega} =\pi_{\bar\omega}(\mathcal B_{\mathfrak D})''. \tag{5.17}$$

Normal instruments act on $\mathcal N_{\mathfrak D,\bar\omega}$, never directly on (5.10):

$$\mathcal I_{\mathfrak D}:\Sigma_{\mathfrak D} \longrightarrow\operatorname{CP}_{\rm n}(\mathcal N_{\mathfrak D,\bar\omega}), \qquad p_{\bar\omega}(B) =\widetilde\omega\bigl(\mathcal I_{\mathfrak D,B}(1)\bigr). \tag{5.18}$$

For pairwise disjoint $B_n$ they satisfy

$$\mathcal I_{\mathfrak D,\cup_nB_n}(A) = \sum_n\mathcal I_{\mathfrak D,B_n}(A) \quad\text{ultraweakly}, \qquad \mathcal I_{\mathfrak D,\Omega_{\mathfrak D}}(1)=1. \tag{5.19}$$

For an admissible embedding $\chi:\mathfrak D\hookrightarrow\mathfrak D'$, covariance requires

$$\alpha_\chi\circ\mathcal I_{\mathfrak D,B} = \mathcal I_{\mathfrak D',\chi_*B}\circ\alpha_\chi. \tag{5.20}$$

For sequential instruments,

$$p(B_1,\ldots,B_n) = \widetilde\omega\!\left( \mathcal I^{(1)}_{B_1}\circ\cdots\circ \mathcal I^{(n)}_{B_n}(1)\right). \tag{5.21}$$

Spacelike local algebras commute. If a nonselective operation in $\mathfrak D_A$ acts identically on $\mathcal N_{\mathfrak D_B}$ for spacelike $\mathfrak D_B$, then the unconditioned $B$ marginal is independent of the choice at $A$. Selective conditioning may alter correlations but not enable signalling.

5.2.3 State-dependent stress tensor and the record interface

Backreaction is governed by the in-in, or closed-time-path, effective action [70, 71] rather than by a state-independent in-out functional. For a Hadamard initial state $\omega$, let

$$\Gamma_{\rm QFT,CTP}^{\rm ren} [g^+,\Phi^+;g^-,\Phi^-\mid\omega] = \Gamma_{\rm QFT,dyn}^{\rm ren} [g^+,\Phi^+;g^-,\Phi^-\mid\omega] +\Gamma_{\rm grav,loc}^{\rm ren}[g^+] -\Gamma_{\rm grav,loc}^{\rm ren}[g^-]+C_\omega . \tag{5.22}$$

The dynamical influence functional is normalized at a fixed reference configuration by

$$\Gamma_{\rm QFT,dyn}^{\rm ren} [g_{\rm ref},\Phi_{\rm ref};g_{\rm ref},\Phi_{\rm ref}\mid\omega]=0. \tag{5.23}$$

The subtraction changes no functional derivative. The local pure-metric counterterms $\Gamma_{\rm grav,loc}^{\rm ren}$ renormalize the gravitational couplings and are included once, on the gravitational side of (7.54); they are excluded from the dynamical QFT stress. The state-dependent renormalized stress is

$$\boxed{ \langle T_{\mu\nu}^{\rm QFT,dyn}\rangle_\omega = -\frac{2}{\sqrt{-g}} \left. \frac{\delta\Gamma_{\rm QFT,dyn}^{\rm ren}} {\delta g^{+\mu\nu}} \right|_{g^+=g^-=g,\,\Phi^+=\Phi^-=\Phi} . } \tag{5.24}$$

On backgrounds with external fields $b_A$, its Ward identity has the form

$$\nabla^\mu\langle T_{\mu\nu}^{\rm QFT,dyn}\rangle_\omega = \sum_A\frac{1}{\sqrt{-g}} \left. \frac{\delta\Gamma_{\rm QFT,dyn}^{\rm ren}}{\delta b_A^+} \right|_{+=-} \nabla_\nu b_A +\mathcal A_\nu^{\rm diff}, \tag{5.25}$$

where the diffeomorphism anomaly term must vanish or be cancelled for ordinary covariant coupling to gravity.

For a finite-resolution detector cell $B$ with positive measure, or for an outcome in a standard-Borel outcome space on which a regular conditional instrument is defined almost everywhere, the QFT instrument supplies a record preform

$$D_B^{\rm pre} = (B,\mathfrak D,\mathcal I_B,w_B,\widetilde\omega_B, \text{context and realization lineage}). \tag{5.26}$$

Here $w_B=\mu(B)$ for a finite-resolution cell and is the registered Radon–Nikodym density at $B$ for an almost-everywhere regular conditional instrument. In a continuous POVM a singleton may have zero measure without being physically excluded: its conditional object is understood through the regular conditional instrument on its almost-everywhere domain or through a registered finite-resolution cell. Only events contained in a measurable null set on which no such conditional instrument is defined are excluded from actionable record preparation. Crucially, the instrument produces alternatives and their preforms. It neither selects an outcome nor deposits an objective record.

The geometry $g$ entering (5.9)–(5.25) is itself reconstructed and dynamically constrained below. Conversely, (5.24) enters the gravitational equation. The QFT algebra, its operational completion, the instruments, the objective archive and the gravitational solution must therefore belong to one simultaneous branch:

$$\bigl(g,\Phi,\omega,\mathcal I,u, \mathcal A^{\rm obj},\mathcal M_{\rm dim},\mathfrak m_R\bigr) \in\mathcal S_{\rm QFT-R-G}(Y,\mathcal N,B), \tag{5.27}$$

where the set on the right is defined explicitly by (8.10)–(8.11). Its existence and stability are part of the R3/R9 programme, not assumed by writing the sector equations separately.

5.3 Hilbert Space and Quantum State Space

5.3.1 Definition 5.3 — Hilbert Space

For each $Y\in\mathsf{Real}_0$, $\mathcal{H}_Y$ is a complex separable Hilbert space with inner product

$$\langle\cdot,\cdot\rangle_Y.$$

Unless a later finite-dimensional specialization is declared, $\mathcal{H}_Y$ is allowed to be infinite-dimensional and separable. Let

$$\mathcal{B}(\mathcal{H}_Y)$$

denote the bounded operators on $\mathcal{H}_Y$, and

$$\mathcal{T}_1(\mathcal{H}_Y)$$

denote the trace-class operators on $\mathcal{H}_Y$.

5.3.2 Definition 5.4 — State Space

The quantum state space is

$$\boxed{ \mathcal{S}(\mathcal{H}_Y) = \left\{ \rho\in\mathcal{T}_1(\mathcal{H}_Y): \rho\ge0,\ \mathrm{Tr}(\rho)=1 \right\}. } \tag{5.28}$$

A pure state is the special case

$$\boxed{ \rho=|\psi\rangle\langle\psi|, \qquad \|\psi\|_Y=1. } \tag{5.29}$$

The conditions in (5.28) define Tier-1 quantum readout structure. Their source derivation is routed through the companion programmes R1 and R2 of Chapter 17.

5.4 Observable Algebra, Effects, and POVMs

5.4.1 Definition 5.5 — Observable Algebra

The observable algebra $\mathcal{A}_{obs,Y}$ is taken to be a von Neumann algebra

$$\boxed{ \mathcal{A}_{obs,Y}\subseteq\mathcal{B}(\mathcal{H}_Y). } \tag{5.30}$$

This convention fixes the operator setting of the present chapter. Later sector chapters may specialize to finite-dimensional matrix algebras or to concrete field-theoretic algebras, but the general channel convention is von Neumann-algebraic.

5.4.2 Definition 5.6 — Effects

An effect is an operator $E\in\mathcal{A}_{obs,Y}$ satisfying

$$\boxed{ 0\le E\le I. } \tag{5.31}$$

The order means that $E$ is positive and $I-E$ is positive.

5.4.3 Definition 5.7 — POVM

Let $(\Omega_Y,\Sigma_Y)$ be a measurable outcome space. A POVM is a map

$$\boxed{ E_Y:\Sigma_Y\to\mathcal{B}(\mathcal{H}_Y) } \tag{5.32}$$

such that

$$\boxed{ E_Y(B)\ge0 \quad\text{for all }B\in\Sigma_Y, } \tag{5.33}$$ $$\boxed{ E_Y(\Omega_Y)=I, } \tag{5.34}$$

and $E_Y$ is countably additive in the weak operator topology:

$$\boxed{ \left\langle\psi,E_Y\!\left(\bigcup_{n=1}^{\infty}B_n\right)\phi\right\rangle = \sum_{n=1}^{\infty} \langle\psi,E_Y(B_n)\phi\rangle } \tag{5.35}$$

for pairwise disjoint $B_n\in\Sigma_Y$ and all $\psi,\phi\in\mathcal{H}_Y$.

For a discrete outcome set,

$$\boxed{ E_i=E_Y(\{i\}), \qquad E_i\ge0, \qquad \sum_iE_i=I. } \tag{5.36}$$

The set of all POVMs satisfying these conditions is denoted

$$\boxed{ \mathsf{POVM}_Y. } \tag{5.37}$$

POVM effects may be taken in $\mathcal A_{obs,Y}$ when the measurement is internal to the observable algebra; otherwise the POVM is an allowed bounded-operator measurement interface in $\mathcal B(\mathcal H_Y)$.

5.5 Born Measure

Given $\rho\in\mathcal{S}(\mathcal{H}_Y)$ and $E_Y\in\mathsf{POVM}_Y$, define

$$\boxed{ \mu_{\rho,Y}(B)=\mathrm{Tr}(\rho E_Y(B)), \qquad B\in\Sigma_Y. } \tag{5.38}$$

For a discrete outcome $i$,

$$\boxed{ p_i=\mu_{\rho,Y}(\{i\})=\mathrm{Tr}(\rho E_i). } \tag{5.39}$$

The Born measure supplies probability weights over measurement alternatives. It does not select a single realized deposited record. The selector variable $\chi_i$ is reserved for Chapter 6 and for the selection-depth programme R7 of Chapter 17.

5.6 Elementary Probability Lemmas

5.6.1 Lemma 5.1 — State-Effect Pairing

For any state $\rho\in\mathcal{S}(\mathcal{H}_Y)$ and any effect $E$ with $0\le E\le I$,

$$\boxed{ 0\le\mathrm{Tr}(\rho E)\le1. } \tag{5.40}$$

Proof. Since $\rho\ge0$ is trace-class and $E\ge0$ is bounded, $\mathrm{Tr}(\rho E)\ge0$. Since $E\le I$, the operator $I-E\ge0$. Therefore

$$\mathrm{Tr}(\rho(I-E))\ge0.$$

Using linearity of trace,

$$\mathrm{Tr}(\rho)-\mathrm{Tr}(\rho E)\ge0.$$

Because $\mathrm{Tr}(\rho)=1$, it follows that

$$\mathrm{Tr}(\rho E)\le1.$$

Thus $0\le\mathrm{Tr}(\rho E)\le1$. $\square$

5.6.2 Lemma 5.2 — Born Measure Is a Probability Measure

If $\rho\in\mathcal{S}(\mathcal{H}_Y)$ and $E_Y$ is a POVM on $(\Omega_Y,\Sigma_Y)$, then

$$\boxed{ \mu_{\rho,Y}(B)=\mathrm{Tr}(\rho E_Y(B)) } \tag{5.41}$$

defines a probability measure on $(\Omega_Y,\Sigma_Y)$.

Proof. Non-negativity. For every $B\in\Sigma_Y$, $E_Y(B)\ge0$. By Lemma 5.1 applied to $E_Y(B)$, $\mathrm{Tr}(\rho E_Y(B))\ge0$. Hence $\mu_{\rho,Y}(B)\ge0$.

Normalization. Since $E_Y(\Omega_Y)=I$,

$$\mu_{\rho,Y}(\Omega_Y) = \mathrm{Tr}(\rho I) = \mathrm{Tr}(\rho) = 1.$$

Countable additivity. Let $\{B_n\}_{n=1}^{\infty}$ be pairwise disjoint measurable sets. By weak operator countable additivity of $E_Y$,

$$E_Y\!\left(\bigcup_{n=1}^{\infty}B_n\right) = \sum_{n=1}^{\infty}E_Y(B_n)$$

in the weak operator sense. Since $\rho$ is trace-class, the trace pairing with bounded operators is normal, so it preserves countable increasing sums of positive operators. Therefore

$$\mu_{\rho,Y}\!\left(\bigcup_{n=1}^{\infty}B_n\right) = \mathrm{Tr}\!\left(\rho E_Y\!\left(\bigcup_{n=1}^{\infty}B_n\right)\right) = \sum_{n=1}^{\infty}\mathrm{Tr}(\rho E_Y(B_n)) = \sum_{n=1}^{\infty}\mu_{\rho,Y}(B_n).$$

Thus $\mu_{\rho,Y}$ is a normalized, nonnegative, countably additive measure, hence a probability measure. $\square$

5.7 Quantum Dynamics

5.7.1 Definition 5.8 — Closed Dynamics

For closed Tier-1 quantum readout dynamics, let $\mathsf H_Y$ be a self-adjoint Hamiltonian on $\mathcal{H}_Y$. The sans-serif symbol $\mathsf H_Y$ is the Chapter 4 Hamiltonian convention, distinguishing the Hamiltonian from the Higgs doublet $H$ (Chapter 9) and the Hubble readout $H(a)$ (Chapter 12). The unitary evolution is

$$\boxed{ U_Y(t)=e^{-it\mathsf H_Y} } \tag{5.42}$$

in units $c=\hbar=1$. The state evolves as

$$\boxed{ \rho(t)=U_Y(t)\rho(0)U_Y(t)^\dagger. } \tag{5.43}$$

When $\hbar$ is restored,

$$U_Y(t)=e^{-it\mathsf H_Y/\hbar}.$$

5.7.2 Definition 5.9 — Open Dynamics

An open quantum dynamics map is a completely positive trace-preserving map

$$\boxed{ \Phi:\mathcal{T}_1(\mathcal{H}_Y)\to\mathcal{T}_1(\mathcal{H}_Y). } \tag{5.44}$$

A Kraus representation has the form

$$\boxed{ \Phi(\rho)=\sum_a K_a\rho K_a^\dagger, \qquad \sum_aK_a^\dagger K_a=I. } \tag{5.45}$$

The map $\Phi$ is part of Tier-1 quantum-readout structure. The source derivation of the admissible dynamics class is routed through the companion programmes R1 and R2 of Chapter 17.

5.8 Tensor-Product Composition

For a bipartite quantum readout system,

$$\boxed{ \mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B. } \tag{5.46}$$

A product state has the form

$$\boxed{ \rho_{AB}=\rho_A\otimes\rho_B. } \tag{5.47}$$

The state space also allows entangled states, i.e. states $\rho_{AB}\in\mathcal{S}(\mathcal{H}_{AB})$ not expressible as convex combinations of product states.

The marginal state on $A$ is obtained by partial trace:

$$\boxed{ \rho_A=\mathrm{Tr}_B(\rho_{AB}). } \tag{5.48}$$

This composition block is required by Chapter 6 for decoherence and objectivity, and by Chapter 8 for the quantum-record-geometry bridge constructions.

5.9 Measurement Contexts and Instruments

Let $\mathcal A$ be a Tier-1 von Neumann algebra and let $\mathcal A_*$ be its normal predual. A normal state is a positive normalized element $\omega\in\mathcal A_*$. In a Hilbert-space representation with density operator $\rho$, one has $\omega(A)=\operatorname{Tr}(\rho A)$.

A complete measurement context is

$$\boxed{ C_{\rm meas} = (\Omega,\Sigma,E,\mathcal I,o_C). }$$

Here $(\Omega,\Sigma)$ is a standard Borel outcome space, $E:\Sigma\to\mathcal A$ is a POVM, $\mathcal I$ is a normal quantum instrument, and $o_C$ is a measurable outcome code. The code is part of the physical/source context. It is fixed before the source selector coordinate is evaluated, transforms under context isomorphisms, and never depends on the outcome subsequently selected.

In Schrödinger form, the instrument is

$$\boxed{ \mathcal I:\Sigma\to \operatorname{CP}(\mathcal A_*). }$$

For every $B\in\Sigma$, $\mathcal I(B)$ is completely positive and trace-nonincreasing. For pairwise disjoint $B_n$,

$$\boxed{ \mathcal I\!\left(\bigcup_{n=1}^{\infty}B_n\right)(\omega) = \sum_{n=1}^{\infty}\mathcal I(B_n)(\omega) }$$

with convergence in the predual norm. The nonselective map is normalization preserving:

$$\boxed{ \mathcal I(\Omega)(\omega)(I)=\omega(I)=1. }$$

The associated POVM is obtained from the Heisenberg dual:

$$\boxed{ E(B)=\mathcal I(B)^*(I), \qquad p_\omega(B)=\omega(E(B)). }$$

In the density-operator representation this reduces to

$$p_\rho(B)=\operatorname{Tr}[\rho E(B)].$$

This algebraic form is primary for relativistic QFT. The trace formula is the normal-representation and ordinary-QM limit.

Instruments and operational probabilities follow the operational approach of [19]; continuous-outcome measuring processes follow [20].

5.10 Record Preforms and Candidate Deposition

For a discrete outcome $i$, the quantum-side record preform is

$$\boxed{ R_i=(E_i,p_i,\sigma_i,\lambda_i^{\rm lab}), \qquad p_i=\omega(E_i). }$$

The effect $E_i$ and probability $p_i$ do not themselves constitute a deposited record. The signature $\sigma_i$ contains the physical and informational interface data required by the record channel, while $\lambda_i^{\rm lab}$ identifies the outcome within its transported context code.

The candidate deposition preparation map is

$$\boxed{ \operatorname{PrepDep}_Y: \mathcal P_Y^{\rm rec}\longrightarrow\mathcal D_Y^{\rm cand}. }$$

For each preform,

$$\boxed{ D_i^{\rm cand} = \operatorname{PrepDep}_Y(R_i) = (R_i,\tau_i^D,\ell_i^{\rm rel},\eta_i,\kappa_i^{\rm lin}). }$$

The component $\tau_i^D$ is a dependency/order marker and is not primitive Tier-1 time. The component $\ell_i^{\rm rel}$ is relational support identity and is not a coordinate or pre-existing location. The data $\eta_i$ encode environmental and redundancy support. The lineage link $\kappa_i^{\rm lin}$ traces the candidate to its source realization, quantum context, and record preform.

The object $D_i^{\rm cand}$ is a deposition blueprint or potential. Constructing it does not mean that outcome $i$ has actually been deposited.

5.11 Record Deposition Boundary and the Selection Residue

Chapter 6 receives the record-deposition interface expression

$$\boxed{ D_i^{\rm cand}=\operatorname{PrepDep}(R_i), \qquad D_{i_\ast}^{\rm sel}\ \text{with}\ \chi_{i_\ast}=1, \qquad \mathcal R_{T1} := D_{i_\ast}^{\rm obj} = \operatorname{ObjPers}\bigl(\operatorname{Dep}^{\rm act}(D_{i_\ast}^{\rm sel})\bigr). } \tag{5.49}$$

The selector variables satisfy

$$\boxed{ \chi_i\in\{0,1\}, \qquad \sum_i\chi_i=1. } \tag{5.50}$$

This chapter does not derive $\chi_i$. The compatibility condition between Born weights and selector statistics is the Chapter 6 / R7 proof obligation

$$\boxed{ \mathbb{E}[\chi_i]=p_i. } \tag{5.51}$$

Equation (5.51) is a compatibility target linking Born weights to single-record realization statistics. It is not discharged in this chapter.

Chapter 5 produces record preforms and candidate depositions only, through $\operatorname{PrepDep}$. Selection, actual deposition $\operatorname{Dep}^{\rm act}$, objectivity, and persistence are Chapter 6 operations; no Chapter 5 object selects an outcome.

5.12 Sequential Instruments and History Measures

For a history $h_n=(i_1,\ldots,i_n)$, define

$$\boxed{ W_\omega(h_n) = \left( \mathcal I_{i_n}^{(n,h_{n-1})} \circ\cdots\circ \mathcal I_{i_1}^{(1)} (\omega) \right)(I). }$$

In a density-operator representation this is

$$W_\rho(h_n) = \operatorname{Tr} \left[ \mathcal I_{i_n}^{(n,h_{n-1})} \circ\cdots\circ \mathcal I_{i_1}^{(1)}(\rho) \right].$$

For $W_\omega(h_{n-1})>0$, set

$$\boxed{ p(i_n\mid h_{n-1}) = \frac{W_\omega(h_n)}{W_\omega(h_{n-1})}. }$$

The coordinate $u_{v_n}$ is applied to this conditional distribution. Induction yields

$$\boxed{ \Pr_{\mu_X} [\operatorname{Sel}_{1:n}=h_n] = W_\omega(h_n). }$$

The history-dependent contexts and conditional kernels are required to be measurable. Their consistent cylinder distributions determine a unique countable-history measure by the Ionescu–Tulcea construction. A zero-weight history is reached only on a null set; an attempted transition from one returns the null-history outcome.

For repeated independent trials with identical context, the corresponding source coordinates are independent and identically distributed. Hence

$$\boxed{ \frac1N\sum_{n=1}^N\chi_i^{(n)} \longrightarrow p_i \quad\text{almost surely}. }$$

Expectation compatibility and frequency compatibility are therefore distinct, explicit results.

The Born-weight statement of this channel is therefore the pushforward/history-measure theorem above, with its exact hypotheses; no single-event frequency claim is made, and the selector-Born compatibility obligation of Chapter 6 remains active under R7.

5.13 Locally Covariant Gauge QFT in Detail

The compact construction above is now developed in detail: the admissible backgrounds, spin structure, field–antifield geometry, BV/BRST quantization, renormalization and thresholds, physical states, unitarity, stress tensor, Ward identities and trace anomaly of the relativistic sector, followed by its recovery limits.

5.13.1 QFT Background and Field Bundle

Candidate background category

Let

$$\boxed{ \mathsf{Bkg}^{\rm spin,G}_{gh} } \tag{5.52}$$

be the category whose objects are

$$\boxed{ \mathfrak b = ( \mathcal M, g, o, t, \mathfrak s, \mathcal P_G, \mathbf b ). } \tag{5.53}$$

Here:

• $(\mathcal M,g)$ is a four-dimensional globally hyperbolic Lorentzian Tier-1 candidate;

• $o$ and $t$ are orientation and time-orientation;

• $\mathfrak s$ is a spin structure;

• $\mathcal P_G\to\mathcal M$ is the candidate gauge principal bundle;

• $\mathbf b$ is the declared family of external background fields and spacetime-dependent couplings.

The metric and spin geometry in (5.53) are downstream geometry candidates. They are not source inputs.

A morphism

$$\chi:\mathfrak b\to\mathfrak b' \tag{5.54}$$

is a causally convex isometric embedding preserving orientation, time-orientation, spin lift, principal-bundle structure, backgrounds, and coupling labels.

Vierbein and spin connection

For fermionic fields, choose a vierbein representative

$$g_{\mu\nu} = \eta_{ab} e^a{}_\mu e^b{}_\nu \tag{5.55}$$

with spin connection

$$\omega_\mu{}^{ab}(e). \tag{5.56}$$

For a Standard Model branch, the spin/gauge covariant derivative is

$$\boxed{ D_\mu\psi = \left( \partial_\mu +\frac14\omega_{\mu ab}\gamma^{ab} -ig_3G_\mu^AT^A -ig_2W_\mu^at^a -ig_YYB_\mu \right)\psi. } \tag{5.57}$$

The ordinary hypercharge coupling is $g_Y$. If the parameter table uses the SU(5)-normalized coupling,

$$g_1 = \sqrt{\frac53}\,g_Y. \tag{5.58}$$

Field and antifield bundle

Let

$$\boldsymbol\varphi_{\rm BV} = ( A_\mu, \psi, H, c, \bar c, b; \boldsymbol\varphi^\ddagger ) \tag{5.59}$$

denote the graded field multiplet. It includes:

• gauge connections;

• chiral fermions;

• the Higgs field;

• ghosts and antighosts;

• Nakanishi–Lautrup auxiliaries;

• antifields.

The exact gauge group, representations, chirality, global quotient, and neutrino branch are inputs from the admitted matter branch. QFT quantization does not silently select them.

Classical BV structure

Define

$$\boxed{ S_{\rm BV} = S_M +S_{\rm antifield} +S_{\rm gf} +S_{\rm gh}. } \tag{5.60}$$

The BV antibracket satisfies the classical master equation

$$\boxed{ (S_{\rm BV},S_{\rm BV})=0. } \tag{5.61}$$

The classical BRST differential is

$$s_{\rm BV} = (S_{\rm BV},\,\cdot\,). \tag{5.62}$$

The BV gate requires:

1. properness of the BV action;

2. Green hyperbolicity of the gauge-fixed linearized operator;

3. existence of retarded and advanced propagators;

4. well-defined wavefront-set products for the microcausal algebra.

Gauge fixing may use a fermion $\Psi$, but physical algebras for admissible choices of $\Psi$ must be related by a specified quasi-isomorphism.

The field–antifield construction is the Batalin–Vilkovisky formalism [13] with BRST physicality [14], in the perturbative algebraic form of [15, 16].

5.13.2 BV/BRST Physical Algebra

Locally covariant algebra functor

The raw locally covariant QFT construction is a functor

$$\boxed{ \mathcal A_{\rm QFT}: \mathsf{Bkg}^{\rm spin,G}_{gh} \to \mathsf{Alg}_\ast, } \tag{5.63}$$

where $\mathsf{Alg}_\ast$ contains unital $*$-algebras and injective unital $*$-homomorphisms.

Functoriality requires

$$\mathcal A_{\rm QFT}(\operatorname{id}_{\mathfrak b}) = \operatorname{id}_{\mathcal A_{\rm QFT}(\mathfrak b)}, \tag{5.64}$$ $$\boxed{ \mathcal A_{\rm QFT}(\chi_2\circ\chi_1) = \mathcal A_{\rm QFT}(\chi_2) \circ \mathcal A_{\rm QFT}(\chi_1). } \tag{5.65}$$

The time-slice property requires a morphism containing a Cauchy surface to induce an algebra isomorphism.

Einstein causality requires

$$\boxed{ [ \mathcal A_{\rm QFT}(O_1), \mathcal A_{\rm QFT}(O_2) ] =0 } \tag{5.66}$$

when $O_1$ and $O_2$ are spacelike separated.

The locally covariant functorial formulation follows the generally covariant locality principle [7], extending the algebraic approach of [6]; perturbative renormalization on curved backgrounds follows [8, 9, 10].

Renormalized interacting physical algebra

Let

$$\mathcal A_{\rm int}^{\rm ren} (\mathfrak b) [[\hbar,\mathbf g]] \tag{5.67}$$

be the renormalized interacting microcausal algebra as a formal power series in $\hbar$ and couplings $\mathbf g$. Let $\widehat s_{\rm BV}$ be the quantum BV differential.

The physical algebra is

$$\boxed{ \mathcal A_{\rm phys}(\mathfrak b) = H^0 \left( \widehat s_{\rm BV}, \mathcal A_{\rm int}^{\rm ren} (\mathfrak b) [[\hbar,\mathbf g]] \right). } \tag{5.68}$$

BRST-closed representatives differing by a BRST-exact term determine the same physical observable.

A level separation is essential here. The physical algebra above is a formal unital $*$-algebra in $[[\hbar,\mathbf g]]$; it does not automatically carry a $C^*$-norm, weak topology, normal state, or ultraweakly additive instrument, and no such structure may be inferred from it alone. Every exact operational claim proceeds through the canonical operational-completion relation $\operatorname{OpComp}$ of Chapter 5: an admitted completion supplies the $C^*$-algebra $\mathcal B_{\mathfrak D,\tau}$, its GNS representation, and the detector von Neumann algebra $\mathcal N_{\mathfrak D,\bar\omega}$ on which normal instruments act. Formal positivity never substitutes for an admitted operational completion.

5.13.3 QFT Renormalization, Anomalies, and States

Quantum master equation and anomaly gate

At renormalization order $n$, let

$$\mathcal A_{\rm BV}^{(n)} \tag{5.69}$$

be the local ghost-number-one quantum-master anomaly. The recursive renormalization step is admitted only when

$$\boxed{ [ \mathcal A_{\rm BV}^{(n)} ] = 0 \quad \text{in} \quad H^1_{\rm loc}(s_{\rm BV}|d). } \tag{5.70}$$

Vanishing of (5.70) means the anomaly can be removed by an allowed local counterterm. The complete anomaly gate consumes separate certificates for:

1. perturbative gauge anomalies;

2. mixed gauge–gravitational anomalies;

3. pure gravitational anomalies where applicable;

4. global gauge anomalies and determinant-line consistency;

5. global gauge-group quotient and spin/$\mathrm{Spin}^G$ compatibility;

6. BRST gauge-fixing independence.

The matter and flavour chapters own the anomaly arithmetic and global-group analysis; the QFT sector consumes their conclusions.

A trace anomaly is not a QFT gate failure.

Renormalization prescription

Every QFT output carries

$$\boxed{ ( \mathfrak R, \mu, \theta_{\rm ren}(\mu), \mathcal B_{\rm EFT}, \mathcal P_{\rm prov} ), } \tag{5.71}$$

where:

• $\mathfrak R$ is the renormalization scheme;

• $\mu$ is the scale;

• $\theta_{\rm ren}(\mu)$ is the renormalized parameter vector;

• $\mathcal B_{\rm EFT}$ is the operator basis and truncation;

• $\mathcal P_{\rm prov}$ records parameter provenance.

Renormalized time-ordered products satisfy:

• local covariance;

• causal factorization;

• microlocal spectrum conditions;

• unitarity and reality;

• field independence;

• scaling;

• smooth background dependence;

• Ward identities modulo explicitly computed anomalies.

Running parameters obey

$$\boxed{ \mu \frac{d\theta_i}{d\mu} = \beta_i(\theta). } \tag{5.72}$$

At a heavy threshold $M_h$,

$$\boxed{ \theta_\lt (M_h) = \mathcal M_h \left[ \theta_>(M_h); M_h, \mathcal D_h \right], } \tag{5.73}$$

where $\mathcal M_h$ is the declared matching map, $\mathcal D_h$ the threshold data, and omitted terms are bounded by the stated EFT order.

Local covariant time-ordered products and their renormalization freedom follow [9, 10].

Physical states and positivity levels

The reference-state space contains positive normalized Hadamard states:

$$\boxed{ \mathsf{St}_0(\mathfrak b) = \left\{ \omega_0: \omega_0(1)=1, \ \omega_0(A^\ast A)\ge0, \ \omega_0\text{ is Hadamard} \right\}. } \tag{5.74}$$

The interacting formal state is obtained through the perturbative Møller map $R_V$:

$$\omega_V = \omega_0\circ R_V. \tag{5.75}$$

Use three distinct state statuses:

1. Reference positivity: (5.74) is proved on the free/reference algebra.

2. Perturbative positivity: (5.75) is normalized and Hermitian, has a positive leading coefficient, and satisfies the declared order-by-order positivity condition.

3. Exact positivity: an actual completed algebra and positive state have been constructed.

Formal perturbative positivity must not be relabelled exact positivity.

The Hadamard/microlocal spectrum condition follows [12]; the curved-spacetime state framework follows [21].

Perturbative unitarity

For Hermitian interaction $V$, perturbative unitarity requires

$$\boxed{ S(V)^\ast S(V)=1 } \tag{5.76}$$

order by order, together with physical unitarity on BRST cohomology. It is not an exact nonperturbative $S$-matrix theorem.

5.13.4 QFT Operational, Record, and Stress Interfaces

Renormalized stress tensor

For the inter-sector ownership assignment, split the full matter 1PI functional as

$$\Gamma_{\rm matter}^{\rm 1PI} = S_M^{\rm cl}+\Gamma_{\rm QFT}^{\rm ren}.$$

The symbol $\Gamma_{\rm QFT}^{\rm ren}$ in the stress interface denotes the renormalized quantum/loop, counterterm, and state-dependent remainder after the classical/tree-level action $S_M^{\rm cl}$ has been assigned to $T_{\mu\nu}^{\rm cl}$. A convention using a full effective action must subtract the separately owned classical term before both stresses enter $\operatorname{Sol}_G$.

The canonical QFT stress tensor is (6.8):

$$\boxed{ \left\langle T_{\mu\nu}^{\rm ren}\right\rangle_\omega = -\frac{2}{\sqrt{-g}} \frac{\delta\Gamma_{\rm QFT}^{\rm ren}} {\delta g^{\mu\nu}}. } \tag{5.77}$$

Its variation convention is

$$\delta\Gamma_{\rm QFT}^{\rm ren} = -\frac12 \int_{\mathcal M} \operatorname{vol}_g \left\langle T_{\mu\nu}^{\rm ren}\right\rangle_\omega \delta g^{\mu\nu}. \tag{5.78}$$

Finite local renormalization ambiguity is recorded as

$$\boxed{ T'_{\mu\nu}-T_{\mu\nu} = c_0m^4g_{\mu\nu} +c_1m^2G_{\mu\nu} +c_2H_{\mu\nu}^{(1)} +c_3H_{\mu\nu}^{(2)} +\cdots. } \tag{5.79}$$

The coefficients and curvature tensors in (5.79) are part of the scheme and provenance record.

Renormalized stress-tensor construction and semiclassical consistency follow [22, 21]; perturbative stress conservation follows [11].

Ward identity and exchange

For external backgrounds $b_A$, the diffeomorphism Ward identity is

$$\boxed{ \nabla^\mu \left\langle T_{\mu\nu}^{\rm ren}\right\rangle_\omega = \sum_A \frac{1}{\sqrt{-g}} \frac{\delta\Gamma_{\rm QFT}^{\rm ren}} {\delta b_A} \nabla_\nu b_A + \mathcal A_\nu^{\rm diff}. } \tag{5.80}$$

For a closed, anomaly-free, on-shell QFT sector with covariantly constant couplings,

$$\boxed{ \nabla^\mu \left\langle T_{\mu\nu}^{\rm ren}\right\rangle_\omega =0. } \tag{5.81}$$

For a dynamical QFT–record coupling, define

$$J_\nu^{\rm QFT\leftrightarrow rec} = \nabla^\mu \left\langle T_{\mu\nu}^{\rm ren}\right\rangle_\omega, \tag{5.82}$$ $$\nabla^\mu T_{\mu\nu}^{\rm rec} = -J_\nu^{\rm QFT\leftrightarrow rec}, \tag{5.83}$$

only when (5.82)–(5.83) follow from the total diffeomorphism-invariant action and its equations of motion. The exchange current is not an arbitrary source.

Anomalous Ward-identity consistency follows [17].

Trace anomaly

The trace takes the form

$$\boxed{ \left\langle T^\mu{}_\mu\right\rangle = \sum_i \beta_i\mathcal O_i +aE_4 +cW^2 +b\Box R +\text{mass terms}. } \tag{5.84}$$

For fixed field content:

• $a$ and $c$ are physical anomaly coefficients;

• $b$ is scheme-dependent;

• $\beta_i\mathcal O_i$ records running couplings;

• mass terms are classified separately.

The QFT gate requires the trace anomaly to be computed and routed, not to vanish.

Matter-to-QFT construction

Let

$$\mathsf{MBranch}_{\rm adm} \tag{5.85}$$

be the admitted set-valued matter branch produced by the Standard Model selector. The quantization relation is

$$\boxed{ \operatorname{Quant}_{SM}: \mathsf{MBranch}_{\rm adm} \times_{\rm bg} \mathsf{Bkg}^{\rm spin,G}_{gh} \times \Theta_{\rm ren} \times \mathsf{RenSch} \to \mathsf{QFT}_{\rm raw}^{\bot}. } \tag{5.86}$$

It performs:

1. field/antifield bundle construction;

2. classical BV action construction;

3. hyperbolicity check;

4. recursive locally covariant renormalization;

5. anomaly computation and counterterm test;

6. physical algebra cohomology;

7. state-space construction;

8. stress, instrument, and RG interface extraction.

If the matter branch is imported rather than source-selected, the QFT output is labelled a conditional recovery branch and does not discharge R4.

The detector-relative operational reduction $\operatorname{OpRead}_{\mathfrak D}$ and the QFT instrument construction $\operatorname{Instr}_{\rm QFT}$, together with their record handoff, are stated in Chapter 5 (quantum channel) and are consumed here without restatement.

QFT-to-geometry and reciprocal geometry dependence

The stress map is

$$\boxed{ \operatorname{Stress}_{\rm QFT}: \mathsf{QFT}_{\rm raw} \to \mathsf T_{\rm ren}, \qquad \mathfrak q \mapsto \left\langle T_{\mu\nu}^{\rm ren}\right\rangle_\omega. } \tag{5.87}$$

The reciprocal map sends a candidate geometry to its QFT background:

$$\boxed{ \operatorname{Bg}_{\rm QFT}: \mathsf G_{\rm cand} \to \mathsf{Bkg}^{\rm spin,G}_{gh}, \qquad g\mapsto\mathfrak b(g). } \tag{5.88}$$

Relative Cauchy evolution records the response of the algebra to compactly supported metric variation. The QFT sector supplies (5.87)–(5.88). The gravity sector owns the effective-action sum and $\operatorname{Sol}_G$; the realization construction owns joint fixed-point existence.

5.13.5 Recovery Limits

Ordinary quantum mechanics

If the detector algebra is a finite type-I factor,

$$\mathcal N_{\mathfrak D} \cong M_n(\mathbb C), \tag{5.89}$$

and the selected scaling regime is isolated and nonrelativistic, the operational reduction (5.14)–(5.18) yields the Hilbert-space state/effect/instrument structure of Chapter 5.

Flat-spacetime QFT

Restrict the functor to Minkowski objects and a selected Poincaré-covariant state:

$$(\mathcal M,g) = (\mathbb R^4,\eta). \tag{5.90}$$

This is a restriction of the locally covariant functor, not the bare substitution $g\to\eta$.

Classical field theory

The $\hbar^0$ coefficient of the formal algebra and of the full matter functional $S_M^{\rm cl}+\Gamma_{\rm QFT}^{\rm ren}$ recovers the classical BV/Peierls structure; it is not counted again inside the loop remainder.

Standard Model

Conditional on the selected Standard Model branch and its parameter solution, the QFT sector recovers the perturbative Standard Model on its admitted background. The condition does not discharge source selection or parameter fixation.

Semiclassical gravity

Variation of

$$\Gamma_{\rm EH} + \Gamma_{\rm QFT}^{\rm ren} \tag{5.91}$$

under the Ward and state-renormalization gates produces the semiclassical stress source consumed by $\operatorname{Sol}_G$.

Effective field theory

For $E/\Lambda_{\rm UV}\ll1$, operators of dimension $\Delta>4$ are retained with declared suppression

$$\left( \frac{E}{\Lambda_{\rm UV}} \right)^{\Delta-4} \tag{5.92}$$

and an explicit truncation-error estimate.

5.14 From Quantum Alternatives to Records

The quantum and field-theoretic structures of this chapter supply states, effects, instruments and record preforms. Chapter 6 develops what happens next: contextual selection among the weighted alternatives, actual deposition, and the conditions under which a deposited record becomes objective. Chapter 7 then constructs geometry from those objective records, and Chapter 8 assembles the full quantum–record–geometry route.