Complete generator and scalar inputs for Bogoliubov innerness

Written by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Original programme text: CC0 to the extent of rights held. Human-source notices remain with their sources.

The Bogoliubov lesson concerns the tracial representation of the CAR over a separable infinite-dimensional complex Hilbert space $K$. Its six results and twelve solved exercises use the entire Hilbert space, the whole normal factor and, in Lemma 3.1, an arbitrary self-adjoint generator. This component supplies the generator and scalar proofs and identifies the complete algebraic inputs.

For the bounded continuous calculus, Hilbert completion, adjoints, orthogonal projections and positive square roots, use CF1–10. For all normal functionals, rather than a selected family of vector states, use the complete CP1–6 coefficient predual. Continuous vector and operator integrals are constructed by tagged sums in Elementary calculus R03–R04.

BG0. The complete CAR and product inputs

The concrete construction in CO3 starts with the tracial product of full $2$-by-$2$ matrices. With $Z_j=1-2n_j$, its fields are

CO3 proves the CAR signs and $\|a(f)\|=\|f\|$, extends the fields in norm to all $K$, proves the full matrix algebra for every finite subspace $E$, and constructs the isometric map from that algebra to the one for $uE$. No fixed finite-mode exhaustion is required to be invariant under $u$. The compatible maps extend to the completed CAR. Uniqueness of each normalized finite matrix trace gives preservation of the complete tracial state.

CO3 then constructs the trace GNS unitary

and the normal automorphism $x\mapsto W_uxW_u^*$ of the whole factor. Its inverse is the extension for $u^*$. Strong convergence of $u$ gives convergence on every field and polynomial and then on every CAR element. Strong convergence of both GNS unitaries and their adjoints gives norm convergence of the pullback of every square-summable coefficient series in CP1–6. This proves the exact point-predual topology used in Theorem 1.1, for nets as well as sequences.

For an eigenbasis $uf_j=e^{is_j}f_j$, the commuting matrix legs are generated by $b_j=Z_1\cdots Z_{j-1}a(f_j)$, not by the raw anticommuting fields. The even unitary

sends $b_j$ to $e^{-is_j}b_j$ and has trace $\cos(s_j/2)$. The negative phase agrees with the conjugate-linear annihilator convention.

The complete diagonal necessity and sufficiency argument is CO4, using the whole matrix-tail expectation FH3 and finite trace products SS2. Necessity writes $E_m(U)=a_m\prod_{j\le m}w_j$, with $|a_m|\to1$, and bounds every later product of overlap moduli from below. Sufficiency adjusts phases after a finite zero-overlap prefix and proves strong convergence of the products and their adjoints on every finite-support vector. Thus the limit is unitary in the factor, not merely an isometry on the trace vector. The zero-prefix cases are part of this proof.

The five complete analytic arguments in Sections 6.1–6.5 of the lesson prove the whole Hilbert–Schmidt ideal, arbitrary compact-normal diagonalization including its kernel, interval projections with every boundary eigenspace, a logarithm including the $-1$ eigenspace, and the dyadic diagonal perturbation. Its full squared-error budget is $3M^2\,2^{-N}$, with the initial level counted separately from the later parent-band increments. The proof works on every reducing cyclic component and distributes an arbitrary prescribed tolerance over their countable orthogonal sum. BG1 supplies the unbounded input; using only a bounded exponential does not cover Lemma 3.1.

BG1. Resolvents, the common domain and every real time

Let $h=h^*$ be a densely defined self-adjoint operator on a complex Hilbert space. The adjoint is defined by its pairing against all vectors in $\operatorname{Dom}h$. That definition makes the adjoint graph closed: if $y_n\to y$ and $h^*y_n\to z$, the pairing passes to the limit against each domain vector and gives $y\in\operatorname{Dom}h^*$, $h^*y=z$. Consequently $h$ is closed.

For $r>0$ and $x\in\operatorname{Dom}h$, symmetry makes the cross terms cancel:

If $(h\pm ir)x_n$ converges, this identity makes both $x_n$ and $hx_n$ Cauchy. Completeness and closedness show that its limit is again in the range. The range is therefore closed. Its orthogonal complement is $\ker(h\mp ir)$, which is zero by (G1). Thus $h\pm ir$ is onto, with inverse norm at most $1/r$.

Conversely, let $S$ be closed, densely defined and symmetric, and suppose both $S-ir$ and $S+ir$ are onto. For $y\in\operatorname{Dom}S^*$, choose $x\in\operatorname{Dom}S$ with

Then $y-x\in\ker(S^*-ir)=\operatorname{Ran}(S+ir)^\perp=0$. Since symmetry already gives $S\subset S^*$, this proves equality of the domains and $S=S^*$. This is the surjective-resolvent criterion, including its domain conclusion.

Now let $k=k^*$ be bounded. The operator $h+k$ on exactly $\operatorname{Dom}h$ is symmetric. It is closed: convergence of $x_n$ and $(h+k)x_n$ implies convergence of $hx_n=(h+k)x_n-kx_n$, so closedness of $h$ applies. For $r>\|k\|$,

The first factor is invertible by its norm-convergent geometric inverse series. Both ranges are all of $K$. The criterion just proved makes $h+k$ self-adjoint on the original domain, with no extension of that domain.

We construct its groups for an arbitrary $h$, before differentiating a product. Set

Pairings give $R_+^*=R_-$. To verify the resolvent identity, apply $h+i$ to $R_-x-R_+x$: the result is $2iR_-x$, so $R_--R_+=2iR_+R_-$. Applying $h-i$ instead gives the same identity with the two inverses reversed. They commute. Hence

In particular $\ker(1-c)=0$ and $\operatorname{Ran}(1-c)=\operatorname{Dom}h$.

The bounded continuous calculus gives

It commutes with $c,c^*$, has zero kernel and dense range. Let $P_n$ be the support projection of $(D-1/n)_+$. The support is obtained as the strong limit of $B(B+\delta)^{-1}$ for $B=(D-1/n)_+$. The proof of the limit and the interval order bounds is the full Section 6.3 argument. Thus the $P_n$ increase, commute with $c,c^*$, satisfy $D\ge1/n$ on $P_nK$ and $D\le1/n$ on its complement. The bound

and density of $\operatorname{Ran}D$ show $P_n\to1$ strongly.

On $P_nK$, both $1-c$ and its adjoint are bounded below by $1/n$. Its range is closed and dense there, hence onto. Define the bounded operator

Every vector in this subspace belongs to $\operatorname{Ran}(1-c)=\operatorname{Dom}h$. Substitution using $1-c=2iR_+$ gives $H_nx=hx$. It is bounded symmetric on its whole Hilbert space, so is self-adjoint. Also $P_n$ preserves $\operatorname{Dom}h$ and commutes there with $h$: commute it through $R_+$, whose range is precisely that domain. These restrictions agree on nested subspaces.

The exponential series for the bounded $H_n$ gives $e^{itH_n}$. Extend it by the identity on $(1-P_n)K$ and denote that unitary by $U_n(t)$. For $n\ge m$, the restrictions agree on $P_mK$, so for every real $t$,

Completeness gives a strong limit $U(t)$, uniformly in $t$ on each fixed vector. The same estimate for $-t$ gives the adjoint limit. Products of uniformly bounded strongly convergent operators converge strongly, by adding and subtracting one mixed product. Thus $U(t)$ is unitary, $U(t)^*=U(-t)$, and $U(t+s)=U(t)U(s)$. Approximation by $P_mx$, followed by the norm-continuous bounded group on that subspace, proves strong continuity.

For $x\in\operatorname{Dom}h$, $P_nx\to x$ and $hP_nx=P_nhx\to hx$. On $P_nK$ we have $hU_n(t)P_nx=U_n(t)P_nhx$. Taking the two limits in the closed graph of $h$ gives

Using $-t$ shows that $U(t)\operatorname{Dom}h=\operatorname{Dom}h$. On each bounded subspace the exponential series and the continuous fundamental theorem give

By (G2) the integrands converge uniformly on compact time intervals. The complete tagged-sum integral therefore gives

The integral is oriented when $t<0$. These properties determine the group: for another group $V$ preserving the domain with this derivative, differentiation of $V(-t)U(t)x$ gives zero for every domain vector. The difference quotient is legitimate because (G3) makes the vector path continuous in the graph norm. The vector fundamental theorem makes that product constant, and density gives $V=U$. We write this constructed group as $e^{ith}$.

Suppose now that $k=k^*$ is Hilbert–Schmidt. Section 6.1 proves it bounded, the ideal inequalities

its norm completeness, finite-rank density and unitary invariance. The group for $h+k$ preserves the same domain. For $\xi$ in that domain, set $y(s)=e^{is(h+k)}\xi$. In addition to ordinary continuity,

is continuous. Thus $y$ is graph-continuous for $h$.

For clarity, the product derivative requires only the following split at time $s$:

The first term tends to $i e^{-ish}(h+k)y(s)$, by the derivative of $y$ and strong continuity. The second tends to $-i e^{-ish}hy(s)$, by (G4) at the fixed domain vector $y(s)$. Their sum is $i e^{-ish}k e^{is(h+k)}\xi$.

The entire integrand is continuous in Hilbert–Schmidt norm. For arbitrary $\xi,\eta\in K$, with no generator-domain assumption, put $\theta_{\xi,\eta}z=\xi\langle\eta,z\rangle$. Parseval gives $\|\theta_{\xi,\eta}\|_{\mathrm{HS}}=\|\xi\|\|\eta\|$. For fixed unitary groups $U,V$,

Adding and subtracting the intermediate rank-one operator bounds its difference at $s,t$ by

This tends to zero. Finite sums prove the finite-rank case. Approximating $k$ in Hilbert–Schmidt norm, while keeping both groups fixed, makes the error uniformly at most that approximation error, for every $s$. This proves continuity for general $k$, with constant norm $\|k\|_{\mathrm{HS}}$.

Completeness and tagged sums now supply the Hilbert–Schmidt integral. Integrating the domain derivative, then using density of $\operatorname{Dom}h$ and $\|T\|\le\|T\|_{\mathrm{HS}}$, proves the whole bounded-operator identity

This holds for every real $t$ and every self-adjoint $h$, possibly unbounded, on the exact common domain used above. In the commuting bounded case the integrand is $ke^{isk}$, giving $e^{itk}-1$; its first-order term $itk$ fixes the positive sign.

BG2. Scalar chords, logarithms and complete diagonal singular sums

The exponential and logarithm construction gives the absolutely convergent exponential series, its addition law, derivative, positive real range, inverse logarithm and continuity. CAR Section 17.1 proves Rolle and the mean-value theorem from real completeness, and defines $\pi$ from the first positive cosine zero. Its quadrant calculation gives

Here is the chord step used in the solved exercises. If $f''\le0$ on an interval, the mean-value theorem applied to $f'$ shows that $f'$ is nonincreasing. For $a<t<b$, applying that theorem on $[a,t]$ and $[t,b]$ gives

Multiplication by the positive denominators and rearrangement shows that $f(t)$ lies above its endpoint chord. Endpoints follow by equality. Taking $f=\sin$, $a=0$, $b=\pi/2$ proves

The exponential derivative and oriented scalar integral also give

The addition and conjugation laws yield

Consequently

To check the interval for the examples explicitly, the first positive cosine zero $q=\pi/2$ satisfies $e^{iq}=i$. The integral bound gives $\sqrt2=|i-1|\le q$, so $\pi\ge2\sqrt2>1$. Thus both $x=1/j$ and $x=1/\sqrt j$ lie in the range of (S2) for every $j\ge1$.

The logarithmic estimate in Exercise 5 also follows from the entire inverse construction. If $q>0$, continuity of $\log$ and the exponential derivative show

Indeed set $\epsilon=\log(q+\delta)-\log q$; then $\epsilon\to0$ and $\delta=q(e^\epsilon-1)$, so the quotient is $q^{-1}\epsilon/(e^\epsilon-1)$. For $0<r\le1$, apply the mean-value theorem to $\log$ on $[r,1]$. It gives

For finitely many $r_j\in(0,1]$, the addition law for the logarithm thus gives

A uniform lower bound $c>0$ on these products bounds every finite defect sum by $-\log c$. Alternatively the algebraic reciprocal-product bound in CO4 gives the same finiteness conclusion without logarithms. A zero overlap is excluded only after the finite prefix has been handled.

Let $T e_j=d_j e_j$ on a whole orthonormal basis, with bounded $d_j$. The operator and its adjoint act diagonally, and

The last diagonal operator is positive and squares to $T^*T$. Uniqueness of the positive square root, proved by the continuous calculus, identifies it with $|T|$. Its nonzero eigenvalues, counted with multiplicity, are the singular values; the kernel contributes zeros. In this diagonal compact setting, trace class means their sum is finite, equivalently

and Hilbert–Schmidt class means $\sum_j|d_j|^2<\infty$. The positive trace sum is independent of the chosen basis: Parseval and interchange of nonnegative double sums give

For the nonnegative sum, each equality means the supremum over finite partial sums; infinite values are allowed. Hence no unproved trace-class spectral theorem is needed for these exercises.

Compactness of this diagonal $T$ is exactly $d_j\to0$ (with finitely many coordinates treated automatically). If this holds, finite diagonal truncations approximate $T$ with norm $\sup_{j>m}|d_j|$, so compactness follows from Section 6.1. Conversely, if infinitely many $|d_j|$ are at least $\epsilon>0$, the vectors $Te_j=d_je_j$ have pairwise distances at least $\sqrt2\epsilon$. They have no convergent subsequence, contradicting compactness.

For $d_j=e^{i/j}-1$, (S2) gives $|d_j|\le1/j$ and $|d_j|\ge2/(\pi j)$. The square sum is at most $2$: on $2^n\le j<2^{n+1}$ its contribution is at most $2^{-n}$, and the sum of these geometric block bounds is $2$. The singular sum diverges, since each such harmonic block contributes at least $1/2$, before multiplication by $2/\pi$.

For $d_j=e^{i/\sqrt j}-1$, the norm of the omitted diagonal tail is at most $(m+1)^{-1/2}$, proving compactness. But $|d_j|^2\ge4/(\pi^2 j)$, so its Hilbert–Schmidt sum diverges. Its singular sum diverges too, since $j^{-1/2}\ge j^{-1}$. These are the complete inner-but-not-trace-class and compact-but-outer examples.

For the inner example, the positive overlaps $r_j=\cos(1/(2j))$ satisfy $1-r_j\le1/(8j^2)$, from the twice-integrated bound $1-\cos t\le t^2/2$. This bound follows by integrating $\sin s=\int_0^s\cos v\,dv$, using $|\cos v|\le1$, and integrating again. For $n>m$,

It is the entire trace-vector tail estimate in Exercise 4. BG0/CO4 extends it to all finite-support vectors and both adjoints, giving the asserted strong* unitary implementer.

BG3. All solution inputs and the full automorphism group

The solution inputs can be read in the following order.

The extra density conclusion in Exercise 11 concerns every normal automorphism $\beta$ of $R$, not just Bogoliubov automorphisms. The complete proof is CT1, with the entire projection comparison FCT0–7 and finite approximations AF1.

Here is its application. Uniqueness of the normal trace gives $\tau\beta=\tau$. For the matrix units of a finite head $A_n$, equal-trace comparison supplies a partial isometry $v$ with $v^*v=e_{11}$ and $vv^*=\beta(e_{11})$. Set

Matrix multiplication gives $Z_n^*Z_n=Z_nZ_n^*=1$ and $Z_ne_{ij}Z_n^*=\beta(e_{ij})$. The finite trace expectation $E_n$ recovers every trace GNS vector. Therefore, with $\gamma_n=\operatorname{Ad}Z_n$,

The GNS unitaries converge strongly. Their inverses converge strongly as well: if $V_n\to V$ strongly and all are unitary with unitary limit, then $\|(V_n^*-V^*)\xi\|=\|\xi-V_nV^*\xi\|\to0$. CP1–6 and its finite-list/square-summable-tail estimate now give convergence on every normal functional in norm. Thus the full point-predual norm density follows. The $j^{-1/2}$ example and the constant gauge $-1$ are outer, so this dense subgroup is proper.

BG4. Exact source comparison and distinct implementation questions

Carey's freely accessible Numdam article, printed pp.141–149, treats real Clifford spaces and more general quasifree representations. The definitions on p.142, Theorem 1.1 and Corollary 1.2 on p.143, and the whole parity argument on p.148 distinguish the $R-I$ Hilbert–Schmidt branch with even finite $-1$ multiplicity from the $R+I$ Hilbert–Schmidt branch with odd finite $+1$ multiplicity. Its proofs invoke Araki's implementation criteria and Ruijsenaars' phase and determinant formulas. Those source dependencies are not premises of BG0–3.

For a complex-linear unitary, a finite complex $\pm1$ eigenspace has twice that dimension as a real space. Thus an odd finite real $+1$ eigenspace cannot occur. The real Hilbert–Schmidt square norm of $u-1$ is twice its complex square norm, by the real basis $e_j,ie_j$. The complex tracial-factor theorem consequently has the single criterion $u-1$ Hilbert–Schmidt; this does not import the general real converse.

Araki's freely accessible EMS article, printed pp.385–442, supplies the state-preserving GNS comparison in Lemma 4.2, pp.389–390, and Corollary 6.2, p.401. Its Theorem 5 and whole proof, pp.421–424, concern innerness in the norm-complete CAR, with trace-class and parity conditions. Theorem 7, p.432, concerns implementation in a Fock representation and a Hilbert–Schmidt off-diagonal relative to its basis projection; its converse line prints “trace class” where the stated Hilbert–Schmidt hypothesis is needed. Its earlier representation-equivalence proof chain is separate from the present factor proof.

An implementer in $R$, an implementer in the norm CAR and an implementer on the Fock Hilbert space answer different questions. The numbered lesson results and all twelve solutions use the tracial-factor argument with BG0–3. Human article access grants no new reuse licence, and no human proof text is relabelled CC0 here.