Fermionic Dirac translation generators and positive spectrum

Realize the one-particle space as the closed subspace of the direct sum $L^2({\mathbb{R}}^3;{\mathbb{C}}^4)\oplus L^2({\mathbb{R}}^3;\overline{{\mathbb{C}}^4})$ determined by $P_{+}$ and $\overline{P_{-}}$. In each fiber, $\mathrm{\mid }\chi (s){\mathrm{\mid }}^2={\cos }^{2}s+{\sin }^{2}s=1$ by [F6], so the phase multiplier preserves norms and both projected ranges. The addition formulas give $\chi (s+t)=\chi (s)\chi (t)$, hence $V(a)V(b)=V(a+b)$.

If $a_m\to a$, the multiplier phases converge pointwise by continuity of sine and cosine, and the squared difference applied to any $u\in \mathcal{K}$ is bounded by $4\mathrm{\mid }u{\mathrm{\mid }}^2$. Dominated convergence [F5] gives $\mathrm{\parallel }V(a_m)u-V(a)u\mathrm{\parallel }\to 0$. On an elementary $n$-fold tensor, the difference of $V(a{)}^{\otimes n}$ and $V(b{)}^{\otimes n}$ is a sum of $n$ terms, each with one factor $(V(a)-V(b))u_r$ and all other factors of unchanged norm. Thus strong continuity holds on elementary tensors, then on each completed antisymmetric sector by density and the unit norm bound. Finite-sector truncations are dense in Fock space, so the direct-sum lift is strongly continuous on all of ${\mathcal{F}}_a(\mathcal{K})$.

The ambient one-particle space has two species and four complex fiber coordinates per species, so it is the finite orthogonal sum of eight scalar $L^2({\mathbb{R}}^3)$ spaces. Distributing a finite $n$-fold Hilbert tensor over this sum gives a finite orthogonal sum indexed by the $n$ species/fiber-coordinate choices; applying the scalar product-tensor identification [F3] successively identifies each summand with scalar $L^2(({\mathbb{R}}^3{)}^n)$. The projected space ${\mathcal{K}}^{\widehat{\otimes }n}$ is a closed subspace of this product space, and its antisymmetric sector is the range of the signed permutation average. The function $s_n$ is measurable and permutation-invariant. Therefore multiplication by ${\mathbf{1}}_{s_n^{-1}(B)}$ preserves that sector and is an orthogonal projection. The direct sum over $n$, with vacuum multiplier ${\mathbf{1}}_B(0)$, satisfies $E(B)E(C)=E(B\cap C)$ and $E({\mathbb{R}}^4)=I$. For disjoint $B_m$ with union $B$, the squared norm of the difference between $E(B)\mathrm{\Psi }$ and ${\sum }_{m=0}^{N}E(B_m)\mathrm{\Psi }$ is the integral of $\mathrm{\mid }\mathrm{\Psi }{\mathrm{\mid }}^2$ over the decreasing remainder preimages; the indicators tend pointwise to zero and are bounded by one, so [F5] gives norm convergence. Hence $E$ is a strongly countably additive projection-valued measure.

Fix one real coordinate multiplier $s_{n,j}$ on an antisymmetric sector. It is finite, measurable, real, and permutation-invariant. Its maximal domain is dense because the cutoffs ${\mathrm{\Psi }}_n{\mathbf{1}}_{\{\mathrm{\mid }{s}_{n,j}\mathrm{\mid }\le N\}}$ are antisymmetric domain vectors converging in norm by [F5]; multiplication is symmetric because $s_{n,j}$ is real. If ${\mathrm{\Psi }}_n$ is in the adjoint domain with adjoint value $Q_n$, then $v_N=(s_{n,j}{\mathrm{\Psi }}_n-Q_n){\mathbf{1}}_{\{\mathrm{\mid }{s}_{n,j}\mathrm{\mid }\le N\}}$ is antisymmetric and in the maximal domain. The adjoint identity gives $0=⟨s_{n,j}{v}_N,{\mathrm{\Psi }}_n⟩-⟨v_N,Q_n⟩={\int }_{\{\mathrm{\mid }{s}_{n,j}\mathrm{\mid }\le N\}}\mathrm{\mid }{s}_{n,j}{\mathrm{\Psi }}_n-Q_n{\mathrm{\mid }}^2$. Letting $N\to \mathrm{\infty }$ proves $Q_n=s_{n,j}{\mathrm{\Psi }}_n\in L^2$; the converse adjoint inclusion follows from the same real-multiplier identity. Thus each sector multiplier is self-adjoint on its maximal domain.

The direct-sum multiplier $A_j={⨁}_n{M}_{s_{n,j}}$ is symmetric on the displayed domain. If $\mathrm{\Psi }$ belongs to $D(A_j^{\ast })$ with adjoint value $Q$, testing the adjoint identity against vectors supported in one sector shows ${\mathrm{\Psi }}_n\in D(M_{s_{n,j}}^{\ast })$ and $(A_j^{\ast }\mathrm{\Psi }{)}_n=M_{s_{n,j}}^{\ast }{\mathrm{\Psi }}_n=s_{n,j}{\mathrm{\Psi }}_n$, by step 3.1. Since $Q$ is in the Fock direct sum, ${\sum }_n\mathrm{\parallel }{s}_{n,j}{\mathrm{\Psi }}_n{\mathrm{\parallel }}^2={\sum }_n\mathrm{\parallel }{Q}_n{\mathrm{\parallel }}^2<\mathrm{\infty }$, so $\mathrm{\Psi }\in D(A_j)$. Conversely, if $\mathrm{\Psi }\in D(A_j)$, sectorwise symmetry and Cauchy--Schwarz for the square-summable sector sequences give $⟨A_j\mathrm{\Phi },\mathrm{\Psi }⟩=⟨\mathrm{\Phi },A_j\mathrm{\Psi }⟩$ for every $\mathrm{\Phi }\in D(A_j)$; hence $\mathrm{\Psi }\in D(A_j^{\ast })$ with $A_j^{\ast }\mathrm{\Psi }=A_j\mathrm{\Psi }$. Therefore $A_j=A_j^{\ast }$ with exactly the stated domain. Truncating both particle number and the multiplier to bounded level sets approximates each domain vector in graph norm by [F5], so finite-sector domain vectors form a graph core.

For every point of $X$, $q^0=\omega \ge \mathrm{\mid }\mathbf{q}\mathrm{\mid }$ by [F1]. For any finite list $q_1,\dots ,q_n$, the time component of their sum is nonnegative and $\mathrm{\mid }{\sum }_r{\mathbf{q}}_r\mathrm{\mid }\le {\sum }_r\mathrm{\mid }{\mathbf{q}}_r\mathrm{\mid }\le {\sum }_r{q}_r^0$ by [F7]. Thus every $s_n$ lies in the closed future cone, including $s_0=0$, so the spectral measure from step 2.2 is supported there. If $\mu >0$, then $s_{n,0}\ge n\mu \ge \mu$ for $n\ge 1$. If $\mu =0$, $\omega (\mathbf{k})>0$ except at the Lebesgue-null point $\mathbf{k}=0$; consequently $s_{n,0}>0$ almost everywhere on every $n\ge 1$ product sector and the zero-energy projection has only the vacuum range.

If $\mu =0$ and $\epsilon >0$, choose ${\mathbf{k}}_0$ with $0<\mathrm{\mid }{\mathbf{k}}_0\mathrm{\mid }<\epsilon \mathrm{/}4$. Since $P_{+}({\mathbf{k}}_0)$ has rank two, choose $e\in {\mathbb{C}}^4$ with $P_{+}({\mathbf{k}}_0)e\mathrm{\ne }0$. Continuity of $P_{+}$ away from the origin gives a bounded open ball $B$ about ${\mathbf{k}}_0$, with closure avoiding zero and contained in $\{\mathrm{\mid }\mathbf{k}\mathrm{\mid }<\epsilon \mathrm{/}2\}$, on which $P_{+}(\mathbf{k})e\mathrm{\ne }0$. The measurable function $u={\mathbf{1}}_B{P}_{+}e$ is square-integrable, nonzero, and supported where $0<\omega <\epsilon$. Hence the one-particle spectral measure has nonzero mass below every $\epsilon >0$, so there is no positive massless gap.

For real $r$, the mean value theorem and $\mathrm{\mid }\cos s\mathrm{\mid }\le 1$ from [F6] give $\mathrm{\mid }\sin r\mathrm{\mid }\le \mathrm{\mid }r\mathrm{\mid }$, hence $\mathrm{\mid }\chi (r)-1\mathrm{\mid }=2\mathrm{\mid }\sin (r\mathrm{/}2)\mathrm{\mid }\le \mathrm{\mid }r\mathrm{\mid }$. Also ${\chi }^{\mathrm{\prime }}(r)=i\chi (r)$ by the sine and cosine derivative formulas. For a coordinate translation set ${\epsilon }_0=1$ and ${\epsilon }_j=-1$ for $j=1,2,3$, and write $U_{\alpha }(t)=U(t{e}_{\alpha })$; on each sector this is multiplication by $\chi ({\epsilon }_{\alpha }t{s}_{n,\alpha })$ by the definition of $V(a)$. If $\mathrm{\Psi }$ is in the maximal multiplication domain, then $\mathrm{\mid }(\chi ({\epsilon }_{\alpha }t{s}_{n,\alpha })-1)\mathrm{/}t\mathrm{\mid }\le \mathrm{\mid }{s}_{n,\alpha }\mathrm{\mid }$; dominated convergence [F5] makes the norm derivative equal $i{\epsilon }_{\alpha }{s}_{n,\alpha }\mathrm{\Psi }$. Conversely, if the difference quotient has a norm limit, take $t_m=1\mathrm{/}m$. Pointwise the quotient tends to $i{\epsilon }_{\alpha }{s}_{n,\alpha }{\mathrm{\Psi }}_n$, so Fatou's lemma [F6] gives ${\sum }_n\mathrm{\parallel }{s}_{n,\alpha }{\mathrm{\Psi }}_n{\mathrm{\parallel }}^2\le {\mathrm{lim\hspace{0.17em}inf}}_m\mathrm{\parallel }(U_{\alpha }(t_m)-I)\mathrm{\Psi }\mathrm{/}{t}_m{\mathrm{\parallel }}^2<\mathrm{\infty }$. Thus the derivative domain is exactly the displayed maximal domain; the self-adjoint generator with convention $U(t{e}_0)=\chi (tH)$ or $U(t{e}_j)=\chi (-t{P}_j)$ is the multiplier $s_{n,\alpha }$. Applying the same argument to $a\mapsto U(ta)$ gives the maximal multiplier by $a^0{s}_{n,0}-{\sum }_j{a}^j{s}_{n,j}$ for any real $a$.