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Finite-regulator physical Gauss-sector existence and dynamics

Physical scope

Conditional exact existence and unitary evolution on the nonzero physical Gauss sector of the adopted fixed finite-graph rotor/CAR regulator, with its exact Hamiltonian domain and full Axiom of Choice. No continuum limit, continuum QED existence, or empirical validity is asserted.

Statement

Assume the fixed finite-graph model adopted by Specified finite spatial compact U(1) gauge regulator: a finite nonempty oriented graph, $s \geq 1$ finite CAR modes per vertex, fixed backgrounds $b_{x} \in {0, \dots, s}$ , link electric operators $E_{e}$ , the joint Gauss operators $G_{x}$ , and $H = H_{0} + V_{B} + V_{F}$ on the exact maximal domain $D (H_{0})$ of Exact compact U(1) rotor plus finite CAR data. Assume the Axiom of Choice and a constant $ℏ > 0$ in joule-seconds. Then the physical sector $H_{p h y s} = ⋂_{x \in V} \ker G_{x}$ is nonzero. The restriction $H_{p h y s} : = H ∣_{D (H) \cap H_{p h y s}}$ is self-adjoint, and $U_{p h y s} (t) = e^{- i t H_{p h y s} / ℏ}, t \in R,$ is a strongly continuous unitary group on $H_{p h y s}$ with the exact derivative domain $D (H) \cap H_{p h y s}$ . This is an exact fixed-regulator result conditional on the stated mathematical suppliers; it asserts no continuum dynamics or continuum QED construction.

Facts & Assumptions

Given: Full AC, the adopted finite-graph model, its CAR/Fourier conventions, the self-adjoint gauge-sector construction, and $ℏ > 0$ .

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function. In particular it supplies the Countable Choice assumption in the finite-graph definition.

[L1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice selects one element from each set in a sequence of nonempty sets. The finite-graph definition uses this weaker assumption for its conditional Fock/CAR construction.

[F1]

Specified finite spatial compact U(1) gauge regulator: the physical model adopts the finite oriented graph, compact rotor links, finite CAR matter, fixed backgrounds, and exact Hamiltonian/domain of Exact compact U(1) rotor plus finite CAR data. Its CAR inputs remain draft suppliers; the postulate does not call them independently accepted.

[F2]

Gauss operators, gauge group and nonempty physical projection: on the current draft interface the maximal diagonal Gauss operators have a nonzero joint kernel; for each allowed background, the zero-link Fourier vector tensored with an occupation vector having $b_{x}$ occupied modes at each vertex is a normalized vector in that kernel. The carrier and its complete conditional proof have a current Step-1 ready record; that record is not proof acceptance.

[F3]

Global unitary dynamics in the Gauss sector: for the same model data and full AC, $H$ is self-adjoint on $D (H_{0})$ , $e^{- i t H / ℏ}$ is strongly continuous, commutes with the gauge action and physical projection, and the restricted physical Hamiltonian $H_{p h y s} = H ∣_{D (H) \cap H_{p h y s}}$ is self-adjoint on $D (H) \cap H_{p h y s}$ . The restricted group has skew-adjoint infinitesimal generator $- i H_{p h y s} / ℏ$ and self-adjoint Stone operator $- H_{p h y s} / ℏ$ , both on that same domain. This current mathematical carrier is draft with a complete conditional proof and a Step-1 ready record, not independent proof acceptance.

Proof

Given: [A1], [L1], [F1]–[F3], and $ℏ > 0$ .

1.1A1L1givenconstruct

To check the weaker choice hypothesis in [L1], let $(X_{n})_{n \in N}$ be any sequence of nonempty sets. Apply [A1] to the family ${X_{n} : n \in N}$ and obtain a choice function $g$ on that family; then $f (n) : = g (X_{n})$ belongs to $X_{n}$ for every $n$ . Thus full AC supplies the Countable Choice input required by [F1]–[F3].

1.2F1F2F3given

The hypotheses in [F1] identify the adopted Hamiltonian, background range and maximal domain with the exact finite-graph data used by [F2] and [F3]; no continuum or spatial embedding data are needed for those mathematical suppliers.

1.3F1F2construct

For every vertex $x$ , the allowed range $0 \leq b_{x} \leq s$ lets us occupy exactly $b_{x}$ of its $s$ modes, including zero or all modes at the endpoints. Tensor the resulting normalized finite occupation vector with the Fourier basis vector having every link label $n_{e} = 0$ . Every link contribution to each Gauss multiplier then vanishes, including the cancelling outgoing and incoming terms of a self-loop; the matter term is $- (b_{x} - b_{x}) = 0$ . Thus $γ_{x} = 0$ at each vertex, so [F2] places this nonzero vector in $H_{p h y s}$ .

2.1A1F1F3step 1.2step 1.3algebra∎

By [F3], $H$ is self-adjoint on exactly $D (H_{0})$ and its unitary group preserves the physical projection. Restricting that group to the nonzero closed range $H_{p h y s}$ therefore gives the stated strongly continuous unitary group; [F3] gives its exact derivative domain and self-adjoint restricted Hamiltonian. The units in [F1] make $t H / ℏ$ dimensionless.

Finite-regulator physical Gauss-sector existence and dynamics

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources