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Dirac CAR, covariance, graded locality and positive Fock energy

Physical scope

Conditional conclusions for the selected noninteracting Dirac CAR model on flat affine Minkowski spacetime, with fixed (-+++) signature, length coordinates and μ≥0. No empirical premise, interaction, point-field observable or general spin-statistics claim is used.

Statement

Assume Countable Choice and the adopted Dirac branch of Specified bosonic scalar and CAR Dirac model adoption as specified by Exact Dirac particle/antiparticle free model. For all Schwartz spinor tests $F, G$ , the bounded smeared field $Ψ$ is antilinear in its test, satisfies ${Ψ (F), Ψ (G)} = 0, {Ψ (F), Ψ (G)^{†}} = ⟨ u_{F}, u_{G} ⟩ I,$ and solves the tested free Dirac equation. If $F$ and $G$ have compact spacelike-separated supports, the field-field and field-adjoint anticommutators vanish; consequently, finite even polynomials in the bounded smeared fields commute across spacelike-separated regions.

Let ${\tilde{G}}_{D}$ be the translation and explicit spin-lift group defined in Dirac Fock covariance, vacuum cyclicity and spectrum. Its strongly continuous unitary Fock representation $U$ satisfies $U (\tilde{g}) Ψ (F) U (\tilde{g})^{- 1} = Ψ (τ_{\tilde{g}} F), τ_{\tilde{g}} F (x) = (S^{- 1})^{†} F (Λ^{- 1} (x - a)),$ for $\tilde{g} = (a, Λ, S)$ . For the explicit spatial rotation lift, $S_{R} (2 π) = - I_{4}$ and $S_{R} (4 π) = I_{4}$ ; hence the $2 π$ test pullback is $F \mapsto - F$ and $U_{2 π} Ψ (F) U_{2 π}^{- 1} = - Ψ (F)$ , while the $4 π$ test pullback and field action are identities. The vacuum is invariant and cyclic for the global graded field algebra. On the $n$ -particle sector, $U_{2 π} = (- 1)^{n} I$ and $U_{4 π} = I$ .

The joint spectrum of translations lies in the closed future cone. If $K_{D}$ denotes the length-coordinate time-translation generator of Dirac Fock covariance, vacuum cyclicity and spectrum, its zero-energy eigenspace is $\ker K_{D} = C Ω$ , where $Ω$ is the normalized vacuum; thus normalized zero-energy vectors differ from $Ω$ only by a complex phase. For $μ > 0$ , the nonvacuum spectrum has gap at least $μ$ in inverse-length units, hence $m_{r e s t, S I} c^{2}$ in SI energy units. The physical energy is $H_{D}^{S I} = ℏ c K_{D} \geq 0$ on the same domain and has the same kernel. This Fock generator acts on a different Hilbert space from the first-quantized multiplier $h_{D}$ , which has both signs of energy. These are conditional deductions in the adopted model, not empirical claims.

Facts & Assumptions

Given: Countable Choice, $μ \geq 0$ , the fixed $(- + + +)$ and Fourier conventions, and the exact draft mathematical supplier interfaces stated below.

[A1]

Specified bosonic scalar and CAR Dirac model adoption adopts the free Dirac CAR representation, species/statistics assignment and vacuum. This is a physical model assumption, not a derived theorem or empirical conclusion.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) defines the Countable Choice axiom assumed by the Fock and Dirac mathematical suppliers; no stronger choice principle is used.

[F1]

Exact Dirac particle/antiparticle free model fixes the selected $μ \geq 0$ branch, signature, length coordinates, one-particle branches and bounded smeared field; Quantum field quantities and compatible SI normalizations specifies that inverse-length energy $ω$ converts to SI energy $ℏ c ω$ .

[F2]

Smeared free Dirac CAR fields and graded locality assumes Countable Choice and the stated Dirac projections, Fock/CAR and Fourier suppliers. Its complete local draft proof establishes the continuous bounded smeared field, the two CAR identities and the tested Dirac equation.

[F3]

The same Smeared free Dirac CAR fields and graded locality proof establishes causal support of the anticommutator kernel and vanishing anticommutators for compact spacelike-separated tests; its statement derives graded locality. Its current mathematical supplier chain remains conditionally escalated on the scalar-causal and upstream QFT-MF01/MF02 inputs.

[F4]

Dirac Fock covariance, vacuum cyclicity and spectrum assumes Countable Choice and the exact Dirac, CAR, mass-shell, Lorentz and translation-spectrum suppliers. Its complete local draft proof establishes unitary spin-lift covariance, an invariant cyclic vacuum, the $2 π$ and $4 π$ actions, forward-cone spectrum, zero-energy eigenspace $C Ω$ and positive-mass gap. Its current mathematical supplier chain remains conditionally escalated on the local CAR prerequisite and QFT-MF01/MF02 page inputs.

[F5]

Dirac matrix identities and orthogonal energy projections defines $S_{R} (θ) = \cos (θ / 2) I_{4} - i \sin (θ / 2) Σ_{n}$ for a unit rotation axis and proves it is the explicit spin lift used by [F4].

[F6]

Quarter-turn values and shifts by pi/2 and pi proves $\sin (x + π) = - \sin x$ and $\cos (x + π) = - \cos x$ , with $\sin π = 0$ and $\cos π = - 1$ ; setting $x = π$ gives $\sin 2 π = 0$ and $\cos 2 π = 1$ .

Proof

1.1A1A2F1F2F3givenalgebra

Under [A1], the physical field and conventions are those fixed in [F1]. The mathematical construction [F2], with the Countable-Choice hypothesis [A2] and its Fock/CAR hypotheses, establishes the displayed smeared CAR relations and tested Dirac equation. Its causal-support conclusion [F3] gives vanishing anticommutators between fields supported in compact spacelike-separated regions; taking adjoints gives the adjoint-adjoint case. Thus every generator from one region anticommutes with every generator from the other. Reordering monomials of degrees $r$ and $s$ contributes $(- 1)^{r s}$ ; for even $r, s$ this is $+ 1$ , so bilinearity gives commutation of finite even polynomials. This deduction does not establish that the adopted model describes nature.

2.1A2F1F4F5F6step 1.1

The covariance and Fock-representation conclusions follow from [F4] under [A2] for the same $μ$ , metric, projection and CAR conventions. For a $2 π$ spatial rotation, [F5] and [F6] give $S_{R} = \cos π I_{4} - i \sin π Σ_{n} = - I_{4}$ ; the spacetime rotation is the identity, so the displayed pullback is $- F$ and antilinearity gives $U_{2 π} Ψ (F) U_{2 π}^{- 1} = - Ψ (F)$ . Substituting $x = π$ into the pi-shift identities in [F6] gives $\cos 2 π = 1$ and $\sin 2 π = 0$ , hence $S_{R} (4 π) = I_{4}$ and both pullback and field action are identities. The Fock action on each sector is supplied by [F4]. The same covariance supplier gives a strongly continuous unitary representation, invariant vacuum and cyclic global field algebra.

3.1F1F4algebra∎

The covariance supplier in [F4] identifies its pure time-translation subgroup with the representation in its translation-spectrum supplier. That supplier defines the length-coordinate generator $H$ on the maximal multiplication domain by $U_{t r a n s} (a^{0}) = χ (a^{0} H)$ , where $χ (s) = \cos s + i \sin s$ ; its joint spectrum is contained in the closed future cone, its kernel is $C Ω$ , and for $μ > 0$ the nonvacuum spectrum has gap at least $μ$ . Write $K_{D} : = H$ for the generator named in the Statement. Since $x^{0} = c t$ , the SI time-evolution convention in [F1] gives $U_{e v o l} (t) = U_{t r a n s} (- c t) = χ (- c t K_{D}) = \exp (- i c t K_{D}) = \exp (- i t H_{D}^{S I} / ℏ),$ where $H_{D}^{S I} = ℏ c K_{D}$ as in the Statement. Thus the forward-cone spectral result gives $K_{D} \geq 0$ and hence $H_{D}^{S I} \geq 0$ . Since $ℏ c > 0$ , the domains and kernels of these two generators agree; a vector $z Ω$ has norm one exactly when $∣ z ∣ = 1$ . For $μ > 0$ the SI gap is $ℏ c μ = m_{r e s t, S I} c^{2}$ . The same supplier distinguishes this Fock generator from the two-signed first-quantized multiplier $h_{D}$ , which acts on a different Hilbert space.

Dirac CAR, covariance, graded locality and positive Fock energy

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources