Universal Bounds for Large Determinants from Non-Commutative H\"older Inequalities in Fermionic Constructive Quantum Field Theory
J.-B. Bru, W. de Siqueira Pedra
TL;DR
This paper establishes universal bounds for large fermionic determinants using non-commutative H"older inequalities, ensuring convergence of perturbation series in fermionic quantum field theories under certain decay conditions.
Contribution
It provides the sharpest known universal bounds for large fermionic determinants applicable to all one-particle Hamiltonians, including unbounded cases.
Findings
Smallest universal determinant bound is exactly 1.
Convergence of perturbation series is guaranteed under sufficient decay of matrix entries.
Bounds are derived using non-commutative H"older inequalities by Araki and Masuda.
Abstract
Efficiently bounding large determinants is an essential step in non-relativistic fermionic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength of the interparticle interaction. We provide, for large determinants of fermionic convariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly . In particular, the convergence of perturbation series at of any fermionic quantum field theory is ensured if the matrix entries, with respect to some fixed orthonormal basis, of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use H\"older inequalities for general non-commutative -spaces…
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Taxonomy
TopicsQuantum many-body systems · Advanced Operator Algebra Research · Noncommutative and Quantum Gravity Theories
Universal Bounds for Large Determinants from Non–Commutative Hölder Inequalities in Fermionic Constructive Quantum Field Theory
J.-B. Bru
W. de Siqueira Pedra
Abstract
Efficiently bounding large determinants is an essential step in non–relativistic fermionic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength of the interparticle interaction. We provide, for large determinants of fermionic convariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one–particle Hamiltonians. We find the smallest universal determinant bound to be exactly . In particular, the convergence of perturbation series at of any fermionic quantum field theory is ensured if the matrix entries, with respect to some fixed orthonormal basis, of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use Hölder inequalities for general non–commutative –spaces derived by Araki and Masuda [AM].
Keywords: determinant bounds; Hölder inequalities for non–commutative -spaces; interacting fermions, constructive quantum field theory.
AMS Subject Classification: 81T08, 82C22, 46L51, 81V70
Contents
-
2.1 Quasi–Free States Associated with the Determinants of the Discrete–time Covariance
-
2.2 Discrete–time Covariance and Bernoulli–Euler Approximations
-
2.3 Modular Objects Associated with Discrete–time Covariance
-
2.4 Determinant Bounds from Non–commutative Hölder Inequalities
-
2.5 Finite Dimensional Case and Hölder Inequalities for Schatten Norms
-
3.2 Representation of Discrete–time Covariance by Quasi–Free States
-
3.3 Correlation Functions and Tomita–Takesaki Modular Theory
1 Setup of the Problem
The convergence of perturbation expansions in non–relativistic fermionic constructive quantum field theory at weak coupling is ensured if the matrix entries, with respect to some fixed orthonormal basis, of the covariance and the interparticle interaction decay sufficiently fast and if certain determinants arising in the expansion can be bounded efficiently. For any one–particle Hamiltonian we show here how to get such bounds on determinants from non–commutative Hölder inequalities. To our knowledge, such estimates are unknown for the unbounded case, even for semibounded (one–particle) Hamiltonians. The unbounded case is important, for instance, in the context of fermionic theories in the continuum. See also Remarks 1.3 and 1.4.
The bounds on determinants (of fermionic covariances) obtained in this way turn out to be universal and sharp, in a sense to be made precise below (cf. (LABEL:universal_determinant_bound) and Corollary 2.4). A consequence of these estimates is that the convergence of perturbation expansions in non–relativistic fermionic quantum field theory is implied by decay properties of interaction and covariance alone. Similar to [dSPS], we give bounds which do not impose cutoffs on the Matsubara frequency, but the results obtained here are stronger than those of [dSPS] on determinants of fermionic covariances.
The paper is organized as follows: Definitions and notation are fixed in Sections 1.1–1.2. The problem of bounding large determinants and the importance of our results in the context of constructive quantum field theory are discussed in Section 1.3. Our main results are Theorem 2.2 and Corollaries 2.3–2.4 of Section 2. Our approach uses Hölder inequalities for general non–commutative –spaces. See, e.g., [AM]. The main lines of the proofs are explained in Section 2, while the technical details are postponed to Section 3.
Notation 1.1
A norm on a generic vector space is denoted by and the identity map of by . The space of all bounded linear operators on is denoted by . If is a Hilbert space, then denotes its scalar product. Units of –algebras are always denoted by .
1.1 Spaces of Antiperiodic Functions on Discrete Tori
We start by defining spaces of antiperiodic functions taking values in a fixed Hilbert space and next give the definition of the antiperiodic discrete delta function:
(i): Fix {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, an even integer n\in 2$$\mathbb{N} and let
[TABLE]
be the discrete torus of length 2{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}. This means that -{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\equiv{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}. Pick any Hilbert space and let be the Hilbert space of functions from to which are antiperiodic. That is here, for any ,
[TABLE]
The scalar product on is then defined to be
[TABLE]
The parameter is interpreted as being the inverse temperature in (fermionic and non–relativistic) quantum field theory, while refers to the so–called one–particle Hilbert space in the same context. The use of antiperiodic functions on the torus is related to the KMS property of equilibrium states and the canonical anticommutation relations (CAR). The discretization of the torus, leading to for n\in 2$$\mathbb{N}, arises from the use of the Trotter–Kato formula in the construction of correlation functions of such KMS states as Berezin–Grassmann integrals.
(ii): We see the Hilbert space as a subset of by using the discrete delta function {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}_{\mathrm{ap}}\in\ell_{\mathrm{ap}}^{2}(\mathbb{T}_{n};\mathbb{C}) defined by
[TABLE]
Vectors of are viewed as antiperiodic functions \hat{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}} of via the definition
[TABLE]
Note that this identification is isometric up to a constant, since
[TABLE]
The discrete delta function {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}_{\mathrm{ap}} is useful here because of the property
[TABLE]
with the convolution being defined by
[TABLE]
Indeed, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}_{\mathrm{ap}} is used below to construct the inverse of some discrete difference operator, see Equation (28).
1.2 Discrete–time Covariance
The discrete–time covariance is an operator defined from (i) a self–adjoint operator acting on the Hilbert space and (ii) the discrete derivative operator acting on the space of antiperiodic functions:
(i): Any (possibly unbounded) operator acting on with domain is viewed as an operator with domain
[TABLE]
by the definition
[TABLE]
If then is also self–adjoint on the Hilbert space of antiperiodic functions.
The (possibly unbounded) self–adjoint operator acting on the Hilbert space is viewed as the so–called one–particle Hamiltonian in (fermionic and non–relativistic) quantum field theory. Indeed, its second quantization refers to the free part of the full interaction of the fermion system.
(ii): The discrete derivative operator is the bounded operator defined by
[TABLE]
It is a normal invertible operator. Combining (7) and (8) we remark that
[TABLE]
for any operator acting on . Because the discrete derivative operator acts on a space of antiperiodic functions,
[TABLE]
Hence, if is any self–adjoint operator acting on , then is a (possibly unbounded) normal operator with bounded inverse. The discrete–time covariance is thus defined to be
[TABLE]
This type of operator appears as the covariance of Gaussian Berezin–Grassmann integrals used in the construction of correlation functions for systems of interacting fermions, see [S]. The discrete–time derivative is related to the corresponding Trotter–Kato product formula used to define such integrals, as already mentioned in Section 1.1.
1.3 Determinant Bounds in Constructive Quantum Field Theory
Correlation functions of interacting fermions can be constructed by perturbation series in the regime of weak couplings. In this context, the self–adjoint (possibly unbounded) operator acting on is the generator of the unperturbed dynamics of the fermion system.
Now, suppose, for simplicity, that is a separable Hilbert space with ONB \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{\mathfrak{i}}\}_{\mathfrak{i}\in\mathbb{I}}, being countable, and set
[TABLE]
for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, and measurable function from to . See (3), (7) and (9). We have in mind cutoff functions {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}:\mathbb{R\rightarrow}\left[0,1\right].
Another essential quantity in non–relativistic fermionic constructive quantum field theory is the so–called determinant bound of and defined as follows:
Definition 1.2** (Determinant bounds)**
**
The parameter {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 269\relax}}{\mbox{\boldmath\textstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 269\relax}}}_{H,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}}\in\mathbb{R}^{+} is a determinant bound of and the measurable function {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}:\mathbb{R\rightarrow R}_{0}^{+} if, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, , , with , and all parameters
[TABLE]
the following bound holds true:
[TABLE]
For we have in mind positive matrices appearing in the so–called Brydges–Kennedy tree expansions which have the following structure: For each non–oriented graph with vertices, all functions \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}}\in\left[0,1\right]^{\mathfrak{g}} and any parameter , we define the subgraph
[TABLE]
In fact, only minimally connected graphs (trees) are relevant for the Brydges–Kennedy tree expansions. Let \mathcal{R}_{\mathfrak{g}\left(\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}},s\right)}\subset\left\{1,\ldots,m\right\}^{2} denote the smallest equivalence relation for which one has (k,l)\in\mathcal{R}_{\mathfrak{g}\left(\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}},s\right)} for all such that the line belongs to the graph \mathfrak{g}\left(\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}},s\right). Then, for any , \mathfrak{M}=\mathfrak{M}\left(\mathfrak{g},\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}},t\right) is the symmetric positive real matrix defined by
[TABLE]
See for instance [AR, BK, SW].
Assume that the matrix entries, with respect to some fix orthonormal basis, of the interparticle interaction decay sufficiently fast, and let be the coupling constant of the considered interacting fermion system, i.e., quantifies the strength of the interparticle interaction. Then, it can be shown that, if the parameter \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}_{H,\mathbf{1}_{\mathbb{R}}}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 269\relax}}{\mbox{\boldmath\textstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 269\relax}}}_{H,\mathbf{1}_{\mathbb{R}}}^{2}\left|u\right| is small enough, the perturbation expansion of all correlation functions in terms of powers of converges absolutely. More precisely, all correlation functions are analytic functions of the coupling at with analyticity radius of order \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}_{H,\mathbf{1}_{\mathbb{R}}}^{-1}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 269\relax}}{\mbox{\boldmath\textstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 269\relax}}}_{H,\mathbf{1}_{\mathbb{R}}}^{-2}. See for instance [AR, SW].
The use of the cutoff function is important in multiscale analyses of correlation functions of interacting fermion systems. Indeed, even for couplings much larger than the convergence radius \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}_{H,\mathbf{1}_{\mathbb{R}}}^{-1}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 269\relax}}{\mbox{\boldmath\textstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 269\relax}}}_{H,\mathbf{1}_{\mathbb{R}}}^{-2} correlations functions can still be constructed via multiscale schemes related to the Wilson renormalization group: Take a family \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}_{L}\}_{L\in\mathbb{N}} of measurable functions from to such that
[TABLE]
(I.e., the family is a partition of unity.) If \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}_{H,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}_{L}}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 269\relax}}{\mbox{\boldmath\textstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 269\relax}}}_{H,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}_{L}}^{2}\left|u\right| is small enough for all , then, up to technical details, the perturbation series at scale in terms of powers of converges absolutely. In general, the smallness of the parameters \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}_{H,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}_{L}}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 269\relax}}{\mbox{\boldmath\textstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 269\relax}}}_{H,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}_{L}}^{2}\left|u\right| at all scales is a much weaker condition than the smallness of \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}_{H,\mathbf{1}_{\mathbb{R}}}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 269\relax}}{\mbox{\boldmath\textstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 269\relax}}}_{H,\mathbf{1}_{\mathbb{R}}}^{2}\left|u\right|. See for instance [dSP].
Note that the form of cutoff function we consider does not depend on the variables, that is, the dependency on the Matsubara frequency of covariance does not need to be regularized, in contrast to other approaches like for instance [GM, BGPS, GMP].
Indeed, coming back to the estimate of the form (11), one easily shows from the Gram bound for determinants that
[TABLE]
This kind of estimate gives no finite determinant bound of and because, in general, the norm of diverges, as . This problem appears already for bounded when , because in this case
[TABLE]
as . See (4). Nevertheless, similar to the multiscale analysis presented above, one can tackle this problem by using the Gram bound as previously for some regularized covariances C_{H}\hat{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}}_{L}(\hat{H},i\partial) at every . Here, for any , \hat{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}}_{L}:\mathbb{R}^{2}\rightarrow\left[0,1\right] is some measurable function of two variables in such a way that
[TABLE]
This decomposition can be chosen such that there are constants \hat{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 269\relax}}{\mbox{\boldmath\textstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptstyle\mathchar 269\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 269\relax}}}}_{L}\in\mathbb{R}^{+}, , which at least do not depend on and meanwhile satisfy
[TABLE]
As already mentioned, such a bound follows from the usual Gram bound for determinants. This kind of strategy is used for instance in [BGPS, Section 3], [GM, Section 3.2], (more recently) [GMP, Section 5.A.], and in many others works. [dSPS] shows that this multiscale analysis for the so–called the Matsubara UV problem is not necessary, by proving a new bound for determinants that generalizes the original Gram bound, see [dSPS, Theorem 1.3]. Note finally that using multiscale analysis to treat the Matsubara UV problem can, moreover, render useful properties of the full covariance less transparent. Hence, avoiding this kind of procedure brings various technical benefits.
In the same spirit, we derive direct bounds of the type (11) that do not need the UV regularization of the Matsubara frequency. One technical advantage of the approach we present here is that the given covariance does not need to be decomposed as in [dSPS, Eq. (8)] in order to obtain determinant bounds. Moreover, our estimates are sharp (or optimal) and hold true for all (possibly unbounded, the latter not being limited to semibounded) one–particle Hamiltonians. Observe that [dSPS] gives sharp estimates up to a prefactor 2 for the class of bounded operators it applies, see [dSPS, Theorem 2.4 and discussions below it].
In this paper we show the (possibly infinite) general bound
[TABLE]
named here the universal determinant bound, is equal to . (Even if the class of all separable Hilbert spaces is not a set, the supremum is well–defined because of the separation axiom.) In particular, the convergence of perturbation series at of any non–relativistic fermionic quantum field theory (possibly in the continuum) is ensured by the smallness of the positive parameter \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}_{H,\mathbf{1}_{\mathbb{R}}}, i.e., if the interaction and the covariance are summable, only. To our knowledge, such estimates are unknown for unbounded self–adjoint operators , even for semibounded ones. Similar statements can also be derived while taking into account the (cutoff) function , see Corollary 2.3. Note that we consider separable Hilbert spaces in (LABEL:universal_determinant_bound) to avoid technical issues.
Remark 1.3** (Covariance in the continuum)**
**
In the continuous case, we would like to stress that, in contrast to the lattice case, we do not have in mind covariances of the form
[TABLE]
with (x_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}),(x_{2},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2})\in\mathbb{R}^{d}\times[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}) and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}\geq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}, i.e., Fourier transforms of the Fermi–Dirac distribution associated with dispersion relations . Indeed, such functions generally diverge for when {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}-{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2} tends to and, hence, cannot have a finite determinant bound. Formally, such covariances would correspond to use (Dirac) delta functions in (10), instead of the orthonormal vectors {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{\mathfrak{i}}.
Remark 1.4** (Determinant bounds in the continuum)**
**
For any fixed {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}\in L^{2}(\mathbb{R}^{d}), its Fourier transform has, of course, to decay at large frequencies. However, we cannot conclude from this that determinant bounds derived here are related to the boundedness of spacial frequencies, because the bounds are uniform with respect to the choice of the unit vectors {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{\mathfrak{i}}.
2 Main Results
The proofs are based on two consecutive transformations of the determinant of the left–hand side of Inequality (11):
- (a)
We first write this determinant as the limit {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\rightarrow\infty of correlation functions associated with quasi–free states {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}. This is reminiscent of [dSPS, Theorem 3.7], which represents determinants as time–ordered correlation functions of Fock states (a special case of quasi–free state). In contrast to the present work, [dSPS, Theorem 3.7] cannot be applied to the full covariance, but, rather, for each term of the decomposition [dSPS, Eq. (8)].
- (b)
For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, these correlation functions are represented as scalar products involving modular operators in the GNS representation of {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}. See Equation (76). As compared to [dSPS], the representation of the determinant of (11) obtained from this second transformation has the advantage of avoiding the decomposition [dSPS, Eq. (8)], which can be non–trivial to verify for general Hamiltonians and lead to artificial prefactors in the bounds.
These two transformations allow us to get bounds of the form (11) by using [AM, (A.2)], which can be viewed as Hölder inequalities for general non–commutative –spaces.
Sections 2.1 and 2.2 explain the main lines of (a). The details of this first transformation are postponed to Sections 3.1 and 3.2. In Section 2.3, we give a few key definitions and results on the Tomita–Takesaki modular theory used for the transformation (b), which is described in detail in Section 3.3. In particular, we explain the origin of modular objects appearing in our main theorem, that is, Theorem 2.2. This section is devoted to the readers who may not be acquainted with the Tomita–Takesaki modular theory. The main results of this paper, that is, Theorem 2.2 and Corollaries 2.3–2.4, are found in Section 2.4, while Section 2.5 illustrates the central arguments of the proofs in the finite dimensional case via Hölder inequalities for Schatten norms.
Recall that is an arbitrary separable Hilbert space. In all the section, we fix {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, , , with , while is any self–adjoint operator acting on . Note again that must not be bounded. To avoid triviality of assertions, we assume .
2.1 Quasi–Free States Associated with the Determinants of the
Discrete–time Covariance
The aim of this section is to represent the determinant of (11) in terms of quasi–free states. To this end, we first define CAR –algebras constructed from a fixed and some finite–dimensional Hilbert spaces , having in mind the positive matrices appearing in the Brydges–Kennedy tree expansions:
(i): The (generic) non–vanishing positive matrix gives rise to a positive sesquilinear form defined on by
[TABLE]
In general, this sesquilinear form is degenerated. The vector space is then defined to be the quotient
[TABLE]
Then, as usual, we introduce a scalar product on as
[TABLE]
and denotes the Hilbert space . Using the notation , where is the canonical basis of , note that
[TABLE]
(ii): The (extended) CAR –algebra associated with is the unital –algebra generated by the unit and the family of elements satisfying the canonical anticommutation relations (CAR), see (43)–(44) with . Notice that such a family always exists and two families satisfying these CAR are related to each other by a unique –automorphism on the –algebra . See, e.g., [BR2, Theorem 5.2.5].
The element is, in fermionic quantum field theory, the annihilation operator associated with whereas its adjoint
[TABLE]
is the corresponding creation operator.
Considering that represents the one–particle Hilbert space, is the –algebra that allows to represent the corresponding many–fermion system within the algebraic formulation of quantum mechanics. The extension of this –algebra to is pivotal to control the determinant of (11). Such determinants are naturally expressed through limits of quasi–free states on the –algebra : Quasi–free states are positive linear functionals {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\in\mathrm{CAR}(\mathfrak{h}\otimes\mathbb{M})^{\ast} such that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}(\mathbf{1})=1 and, for all and ,
[TABLE]
if , while in the case ,
[TABLE]
Remark 2.1** (Other definitions of quasi–free states in the literature)**
**
Some authors relax Condition (15) in the definition of quasi–free states. Within this more general framework (known as the self–dual formalism) quasi–free states fulfilling (15) are then referred as gauge invariant quasi–free states of the corresponding CAR –algebras. For instance, see [A, Definition 3.1]. Note indeed that [A, Definition 3.1, Condition (3.1)] only imposes on the quasi–free state to be even, but not necessarily gauge invariant.
The operator S^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}})}\in\mathcal{B}(\mathfrak{h}\otimes\mathbb{M}) defined from
[TABLE]
is named the symbol (or one–particle density matrix) of the quasi–free state . By the positivity and normalization of states, it follows that symbols are positive (self–adjoint) operators with spectrum lying on the unit interval . Conversely, any such positive operator on uniquely defines a quasi–free state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S} on such that
[TABLE]
The symbols allowing us to represent the determinant of (11) in terms of quasi–free states are defined as follows: For all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, define the function
[TABLE]
and let
[TABLE]
The relevant quasi–free states on the –algebra are those with symbol
[TABLE]
observing that 0<S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}\leq\mathbf{1}_{\mathfrak{h}\otimes\mathbb{M}}. The precise relationship between the quasi–free states {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, and the covariance appearing in the determinant of (11) is described below.
2.2 Discrete–time Covariance and Bernoulli–Euler Approximations
At fixed {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R} and large , note from (19) that
[TABLE]
is the well–known Bernoulli–Euler approximation of the exponential function \mathrm{e}^{\mp{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}. In particular, H_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}, as defined by (20), can be viewed as an approximation of the self–adjoint operator . The relevance of the function \digamma_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} results from the following observations:
(i): By the spectral theorem, there is a (–finite) measure space (\Omega_{H},\mathfrak{A}_{H},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{H}), a unitary map from to and a –measurable function {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}_{H}:\Omega_{H}\rightarrow\mathbb{R} such that
[TABLE]
where \mathrm{m}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}_{H}} is the multiplication operator on with the function {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}_{H}. Using the unitary we can identify with , i.e.,
[TABLE]
Recall that is the extension to of any operator acting on , as defined by (7). The latter, in turn, is canonically identified with
[TABLE]
In other words, by using , we identify with (24). Note that the above direct integral is well–defined because is finite dimensional and (\Omega_{H},\mathfrak{A}_{H},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{H}) is a –finite measure space, since is assumed to be separable.
(ii): With this convention,
[TABLE]
The discrete derivative defined by (8) is meanwhile written in the new Hilbert space as
[TABLE]
where is defined by
[TABLE]
In particular, the discrete–time covariance , defined by (9), can be represented as
[TABLE]
where R\left(\mathfrak{d},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\right)\in\mathcal{B}(\ell_{\mathrm{ap}}^{2}(\mathbb{T}_{n};\mathbb{C})) is the resolvent
[TABLE]
(iii): It is convenient to represent the last resolvent as a convolution (6) with an antiperiodic function. To this end, we solve the following equation
[TABLE]
in g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}\in\ell_{\mathrm{ap}}^{2}(\mathbb{T}_{n};\mathbb{C}) for any fixed {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in{\mathbb{R}}. (Compare with (25).)
(iii.a): For {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\neq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}^{-1}n and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, the antiperiodic function g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}\in\ell_{\mathrm{ap}}^{2}(\mathbb{T}_{n};\mathbb{C}) defined by
[TABLE]
is the unique solution on of the difference equation
[TABLE]
with the discrete delta function {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}_{\mathrm{ap}}\in\ell_{\mathrm{ap}}^{2}(\mathbb{T}_{n};\mathbb{C}) being defined by (2). In particular, g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}\in\ell_{\mathrm{ap}}^{2}(\mathbb{T}_{n};\mathbb{C}) solves (26) for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\neq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}^{-1}n.
Note that we take to ensure that
[TABLE]
and observe meanwhile that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}^{-1}n\in\mathbb{Z} if {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}\in\mathbb{T}_{n}. Therefore, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\neq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}^{-1}n and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+},
[TABLE]
Recall that is the sign function defined here as follows: for and otherwise.
(iii.b): For {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}^{-1}n, the (unique) solution on of the difference equation (28) is equal to
[TABLE]
We can write this function as the following limit:
[TABLE]
In particular, g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}\in\ell_{\mathrm{ap}}^{2}(\mathbb{T}_{n};\mathbb{C}) solves (26) for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}^{-1}n. Compare also (30) with (29).
(iv): The relationship between the function g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}\in\ell_{\mathrm{ap}}^{2}(\mathbb{T}_{n};\mathbb{C}) and the symbols S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (21) defining the quasi–free states {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, can be heuristically understood by considering the limit case :
(iv.a): The function g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}\in\ell_{\mathrm{ap}}^{2}(\mathbb{T}_{n};\mathbb{C}) plays the role, in the discrete case (), of the antiperiodic function g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}^{(\infty)}:\mathbb{R\rightarrow R} defined by
[TABLE]
which solves the differential equation
[TABLE]
Here, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}_{x} is the delta distribution at . Compare the last equation with (28). Up to the observation (22) and the special case {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}^{-1}n, the qualitative difference between (31) and (29) concerns the replacement of in (31) by {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}-n^{-1}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}} in (29) and the prefactor
[TABLE]
(iv.b): Using the symbol
[TABLE]
for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}\in\mathbb{T}_{\infty}\doteq\left(-{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\right] (seen as a torus) with {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}\leq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}, all entire analytic vectors {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{2} of and every ,
[TABLE]
with being the vectors of satisfying (14). The symbol is directly related to the the antiperiodic function g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}^{(\infty)} since
[TABLE]
Similar identities hold true in the discrete case for which and g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}}^{(\infty)} are replaced with S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (21) and g_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}} (29)–(30). In particular, the determinant of (11) can be represented in terms of a limit {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\rightarrow\infty (cf. (30)) of quasi-free states {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}} with symbol S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (21). See Lemma 3.2 and Corollary 3.3.
2.3 Modular Objects Associated with Discrete–time Covariance
Our estimates are based on non–commutative Hölder inequalities [AM, (A.2)] (see also (75)), which requires the celebrated Tomita–Takesaki (modular) theory. Modular objects associated with discrete–time covariance are constructed, for any fixed {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, from the quasi–free state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}} with symbol S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (21) as follows:
(i): Let (\mathfrak{H}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},\varkappa_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}) be a cyclic representation of {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}. The weak closure of the –algebra is the von Neumann algebra
[TABLE]
As is usual, denotes the bicommutant of any subset of the space of bounded operators acting on a Hilbert space.
(ii): The vector {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} is, by assumption, a cyclic vector for \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}, i.e., is the closure of (the linear span of) the set
[TABLE]
Because the vector {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} represents a KMS state (see Section 3.3), it is also separating for \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}, i.e., for all A\in\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}, A{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}=0 iff .
(iii): We define two anti–linear operators and respectively by
[TABLE]
for any A\in\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} and B\in\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}^{\prime}. Since a cyclic and separating vector for \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} is also cyclic and separating for its commutant \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}^{\prime}, both operators are well–defined on the dense domains \mathrm{Dom}(\mathcal{S}_{0})=\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} and \mathrm{Dom}(\mathcal{F}_{0})=\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}^{\prime}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}. By [BR1, Proposition 2.5.9], and are closable and their closure are denoted by and , respectively. In fact, and .
(iv): The modular operator \mathbf{\Delta}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} and conjugation J_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} associated with the pair (\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}) are respectively the unique, positive, self–adjoint operator and the unique anti–unitary operator occurring in the polar decomposition of \mathcal{S}=J_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}\mathbf{\Delta}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}^{1/2}. The main result of the modular Tomita–Takesaki theory is the Tomita–Takesaki theorem [BR1, Theorem 2.5.14], which states in the current context that
[TABLE]
for all . The second assertion is related with the so–called modular automorphism group, as defined by (70) in its –rescaled version.
For more details on the theory of von Neumann algebras and modular objects, see for instance [BR1]. To make its key points more transparent, this theory is illustrated in the finite dimensional case in Section 2.5. In the same spirit, the non–commutative Hölder inequalities [AM, (A.2)], corresponding here to (75), are derived in the finite dimensional case from Hölder inequalities for Schatten norms. See (40)–(42).
2.4 Determinant Bounds from Non–commutative Hölder Inequalities
To prove our estimates, we rewrite the determinant of (11) by using cyclic representations of quasi–free states on the –algebra , as explained in Section 2.1. This allows us to use the bound [AM, (A.2)], which can be viewed as Hölder inequalities for general non–commutative –spaces. This yields the following assertions on determinants of fermionic covariances:
Theorem 2.2** (Representation of determinants of fermionic covariances)**
Let be any separable Hilbert space. Take {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, , , any self–adjoint operator acting on , and a non–vanishing with . Then there are von Neumann algebras \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}\subset\mathcal{B}(\mathfrak{H}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}), cyclic and separating unit vectors {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}\in\mathfrak{H}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (for \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}) and –homomorphisms \varkappa_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (from to \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}), where {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, such that for each bounded measurable positive function from to , all parameters
[TABLE]
and for any permutation of elements with sign \left(-1\right)^{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}} so that111The conditions on impose that it is a permutation of elements which orders the numbers {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{q}, , in the following way: {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(k)<{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(l) whenever {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{k}<{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{l} for while {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(k)<{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(N+l) whenever {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{k}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{N+l} for .
[TABLE]
where \tilde{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}}_{q}\doteq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{q} for and \tilde{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}}_{q}\doteq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{q}+n^{-1}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}} for , the following assertion holds true:
[TABLE]
The integer is defined to be the smallest element of so that \tilde{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(p)}\geq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2. \Delta_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} is the modular operator associated with the pair (\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}). For such that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}^{-1}(q)\in\{1,\ldots,N\},
[TABLE]
while for such that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}^{-1}(q)\in\{N+1,\ldots,2N\},
[TABLE]
Here, is the sign function defined as follows: for and otherwise.
Proof: Combining Lemma 3.1 and Corollary 3.3 with the construction done in Section 3.3, in particular Equation (76), one gets the assertion when all functions {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{1},\ldots,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{N}\in\mathfrak{D}\subset\mathfrak{h} belong the dense space (59). To extend it to all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{1},\ldots,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{N}\in\mathfrak{h}, by (75), note that both sides of Equation (33) are continuous with respect to {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{1},\ldots,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{N}.
For an explicit description of (\mathfrak{H}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},\varkappa_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}), which is a cyclic representation of the quasi–free state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}} for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, see Sections 2.1 and 2.3. Heuristic arguments can be found in Section 2.2.
Corollary 2.3** (Determinant bounds)**
Under the assumptions of Theorem 2.2,
[TABLE]
Compare with Definition 1.2.
Proof: This corollary is a direct consequence of Theorem 2.2 and Inequality (75). In fact, inequalities of the form [AM, (A.2)] (which generalize (75)) are intimately related to Hölder inequalities for non–commutative –spaces. In the finite dimensional case, the non–commutative –spaces correspond to spaces of Schatten class operators, as explained in Section 2.5.
Corollary 2.4** (Universal determinant bounds)**
The universal determinant bound defined by (LABEL:universal_determinant_bound) equals .
Proof: Invoking Corollary 2.3, we deduce , see (LABEL:universal_determinant_bound) and Definition 1.2. Now, let with canonical ONB denoted by . Take {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}=\mathbf{1}_{\mathbb{R}}, H={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\mathbf{1}_{\mathfrak{h}} with {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R} and with . Then, from Corollary 3.3 together with (21) and (16)–(17) , for each n\in 2$$\mathbb{N} and all , we directly compute that, for sufficiently large ,
[TABLE]
In particular, for every {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 290\relax}}{\mbox{\boldmath\textstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 290\relax}}}>0 and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, there are {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 290\relax}}{\mbox{\boldmath\textstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 290\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}}\in\mathbb{R} and n_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 290\relax}}{\mbox{\boldmath\textstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 290\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}}\in\mathbb{N} such that, for all n\geq n_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 290\relax}}{\mbox{\boldmath\textstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 290\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}} and ,
[TABLE]
Using this lower bound and Corollary 2.3, we then arrive at the equality .
2.5 Finite Dimensional Case and Hölder Inequalities for Schatten
Norms
As already discussed, we use Hölder inequalities for non–commutative –spaces to derive determinant bounds (Definition 1.2). Here, we illustrate this approach in the finite dimensional case via Hölder inequalities for Schatten norms:
(i): Assume that is a finite dimensional Hilbert space. Then, the –algebra associated with can be identified with the space of all linear operators acting on the fermionic Fock space
[TABLE]
constructed from the one–particle Hilbert space .
(ii): Take any faithful state on with cyclic representation (\mathfrak{H},\varkappa,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}). By finite dimensionality, it follows that
[TABLE]
Because is faithful and is a matrix algebra, is separating for and the (Tomita–Takesaki) modular objects associated with it are well–defined. Denote by the modular operator associated with the pair \left(\varkappa\left(\mathrm{CAR}\left(\mathfrak{h}\otimes\mathbb{M}\right)\right),{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\right). See Section 2.3.
The cyclic representation (\mathfrak{H},\varkappa,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}) is uniquely defined, up to a unitary transformation. It is explicitly given, for instance, by the so–called standard (cyclic) representation [DF, Section 5.4]: The space corresponds to the linear space endowed with the Hilbert–Schmidt scalar product
[TABLE]
For any we define the left and right multiplication operators and acting on by
[TABLE]
respectively. The representation is the left multiplication, i.e.,
[TABLE]
The cyclic vector is defined by
[TABLE]
with being the unique positive operator such that
[TABLE]
I.e., is the density matrix of the state . In this representation, the modular operator associated with is equal to
[TABLE]
Note that if a state is faithful then its density matrix is invertible. The (–rescaled) modular group is the one–parameter group {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}\equiv\{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}_{t}\}_{t\in{\mathbb{R}}} defined by
[TABLE]
(iii): Now, we fix n\in 2$$\mathbb{N} and apply this last construction to the quasi–free states {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, which are defined from symbols S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (21). See Section 2.1. Denote their standard representations by (\mathfrak{H}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},\varkappa_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}), their density matrices by \mathrm{D}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} and the associated modular operators by \Delta_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}. We infer from (34), (35), (36), Corollary 3.3, the defining properties of Bogoliubov automorphisms (compare (37) with (69)–(71)), the cyclicity of traces, and the assumptions and definitions of Theorem 2.2 that
[TABLE]
that is, Equation (33).
(iv): Schatten norms on are defined by
[TABLE]
and
[TABLE]
Remark that the norm on the Hilbert space defined from the scalar product (34) is the Hilbert–Schmidt norm, i.e.,
[TABLE]
(v): Hölder inequalities for Schatten norms refer to the following bounds: For any n\in 2$$\mathbb{N}, such that , and all operators ,
[TABLE]
This type of inequality combined with (38) implies Corollary 2.3 in the finite dimensional case.
(vi): Indeed, for any integer and strictly positive parameter {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 272\relax}}{\mbox{\boldmath\textstyle\mathchar 272\relax}}{\mbox{\boldmath\scriptstyle\mathchar 272\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 272\relax}}}\in\mathbb{R}^{+}, define the tube
[TABLE]
Let be a faithful quasi–free state on and denote by H_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}}=H_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}}^{\ast}\in\mathcal{B}(\mathfrak{h}\otimes\mathbb{M}) the unique self–adjoint operator such that the symbol S^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}})} of equals
[TABLE]
See beginning of Section 2.1 for more explanations on quasi–free states in relation with their symbols.
Choose and pick a family of elements of , where the notation “” stands for either “” or “”. For any complex vector , we observe from (36) that
[TABLE]
By applying Hölder inequalities (39) and (40), we obtain from the last equality that
[TABLE]
which, combined with , in turn implies that
[TABLE]
This inequality corresponds to (75) in the finite dimensional case. Therefore, Equation (38) combined with Inequality (42) implies Corollary 2.3 when is a finite dimensional Hilbert space.
3 Technical Proofs
3.1 Quasi–Free States on General Monomials
Let be some Hilbert space and the associated CAR –algebra generated by the unit and the family \{a({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}})\}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}\in\mathcal{H}} of elements satisfying the canonical commutation relations (CAR): For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{2}\in\mathcal{H},
[TABLE]
Strictly speaking, the above conditions only define up to an isomorphism of –algebras. See, e.g., [BR2, Theorem 5.2.5]. As explained in Section 2.1 for the special case , the generator a({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}})\in\mathrm{CAR}(\mathcal{H}) is interpreted as the annihilation operator associated with {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}\in\mathcal{H} whereas its adjoint
[TABLE]
is the corresponding creation operator.
A monomial in the annihilation and creation operators is normally ordered if the creation operators appearing in the monomial are on the left side of all annihilation operators in the same monomial, like
[TABLE]
By the above definition, if is a quasi–free state and is a normally ordered monomial in the annihilation and creation operators, then {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}(\mathcal{M}) is the determinant of a matrix, the entries of which are given by acting on monomials of degree two. We show below that this pivotal property of quasi–free states remains valid even if is not normally ordered.
This is not surprising. For instance, [A, Definition 3.1, Condition (3.2)] also essentially says that if the state is quasi–free then expectation values (with respect to this state) of any monomial (not necessarily normally ordered) of arbitrary even degree is a determinant of a matrix, the entries of which are expectation values of monomials of degree two. However, beyond this fact, we would like to give the explicit behavior of such expectation values with respect to arbitrary permutations of creation and annihilation operators in large monomials. This point is crucial here and is given by Lemma 3.1.
To this end, we introduce some notation. If is a permutation of elements (i.e., a bijective function from to ) with sign (-1)^{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}, we define the monomial \mathbb{O}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}(A_{1},\ldots,A_{n})\in\mathrm{CAR}(\mathcal{H}) in by the product
[TABLE]
In other words, \mathbb{O}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}} places the operator at the {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(k)th position in the monomial (-1)^{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}A_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}^{-1}(1)}\cdots A_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}^{-1}(n)}. Further, for all , ,
[TABLE]
is the identity function if {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(k)<{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(l), otherwise {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}_{k,l} interchanges and . Then, the following property of quasi–free states holds true:
Lemma 3.1** (Quasi–free states on general monomials)**
**
Let be a quasi–free state on the –algebra , as defined by (15)–(16) for . For any , all permutations of elements and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{1},\ldots,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{N_{1}+N_{2}}\in\mathcal{H},
[TABLE]
if , while in the case ,
[TABLE]
Proof: By (43) and (44), if the monomial
[TABLE]
contains different numbers of annihilation and creation operators (i.e., ), then it can be written as a sum of normally ordered monomials with the same property. By (15) and the linearity of states, we thus deduce (47).
We consider the case . Assertion (48) trivially holds if and we can assume from now on that .
For convenience, the notation “” stands for either “” or “”. In particular, we write the monomial
[TABLE]
Let
[TABLE]
The parameter k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}} is the position the first annihilation operator appearing in the monomial while k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}^{+} is the position of the last creation operator appearing in . In other words,
[TABLE]
Note that k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}=N+1 iff the monomial is normally ordered. The same holds true if k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}^{+}=N. In particular, k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}=N+1 iff k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}^{+}=N. We will prove Assertion (48) by induction in the parameter
[TABLE]
Observe that N_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}=0 iff the monomial is normally ordered and Assertion (48) holds in this case because of (43), (16) and the antisymmetry of the determinant under permutations of its lines or rows.
Assume now that N_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}\geq 1. Thus, k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}\leq N and k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}^{+}\geq N+1. If k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}>2 and 2N-k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}>3 then
[TABLE]
with . Mutatis mutandis if k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}=1,2 or 2N-k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}=2,3. It is convenient to use the definition
[TABLE]
which implies a_{k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}}=a({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{N+q_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}}). By combining (3.1) with the CAR (43) and (44), we deduce the equality
[TABLE]
when k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}>2 and 2N-k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}>3. Mutatis mutandis if k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}=1,2 or 2N-k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}=2,3. For any , we fix a permutation {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}^{(k)} of elements such that
[TABLE]
(Recall that is assumed without loss of generality.) Similarly, \tilde{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}} is a permutation of elements such that
[TABLE]
By using this notation, we rewrite (3.1) as
[TABLE]
For all , note that
[TABLE]
As a consequence, for any , the induction parameter N_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}^{(k)}} associated with the permutation {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}^{(k)} satisfies:
[TABLE]
Similarly,
[TABLE]
which in turn imply
[TABLE]
Observe furthermore that, for any such that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(k)>k_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}},
[TABLE]
Therefore, by using (54) together with (3.1), we arrive at the equality
[TABLE]
We use now the following definitions: For any , the coefficients
[TABLE]
, are the entries of two matrices and , respectively. Let
[TABLE]
be the k,l\in\{1,\ldots,N\}\minor of , that is, the determinant of the matrix that results from deleting the th row and the th column of . From the Laplace expansion for determinants (sometimes called cofactor expansion),
[TABLE]
To derive the equality (57) we also use that \tilde{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}_{k,N+l}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}_{k,N+l} whenever l\neq q_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}, whereas it is the identity of the set for l=q_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}. On the other hand, using (52)–(53) and the induction hypothesis for all \tilde{N}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}\geq 0 with \tilde{N}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}<N_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}}, we deduce that
[TABLE]
and
[TABLE]
Thus, by induction, it follows from (44), (3.1), (56) and (57) that
[TABLE]
3.2 Representation of Discrete–time Covariance by Quasi–Free
States
(i): We pick a (possibly unbounded) self–adjoint operator acting on and fix from now on . Then, because of (25), (26), (29) and (30), for any fixed {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+} we introduce the unitary operator
[TABLE]
and the (possibly unbounded) operator H_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}\doteq\digamma_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}\left(H\right), see (19) and (20). For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}, the Hamiltonian H_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} gives rise to the symbol S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (21), which, as explained in Section 2.1, in turn yields a quasi–free state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}, with symbol S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}>0, on the CAR –algebra .
Let
[TABLE]
By the spectral theorem, it is a dense subspace of entire analytic vectors of H_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}. Note additionally that does not depend on {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}.
(ii): Similar to the permutation (46), for all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}\in\mathbb{T}_{n}\cap[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}), we define the permutation
[TABLE]
as the identity map if {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}\leq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}, while {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}} interchanges and when {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}>{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}.
Then, quasi–free states {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, give rise to the following representation of the discrete–time covariance:
Lemma 3.2** (Representation of the covariance by a quasi–free state)**
**
Let be any separable Hilbert space. Fix {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, a self–adjoint operator acting on , and . Then, for each bounded measurable positive function from to , all , non–vanishing with , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}\in\mathbb{T}_{n}\cap[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}), {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{2}\in\mathfrak{D} and ,
[TABLE]
with being the vectors of satisfying (14) and where \mathbb{O}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}}} is defined by (45) for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}}.
Proof: Fix all the parameters of the lemma. Note that
[TABLE]
Therefore, we can assume without loss of generality that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}=\mathbf{1}_{\mathbb{R}}. We deduce from Equations (25) and (26) that, for any
[TABLE]
with
[TABLE]
In the right–hand side of (3.2) observe that \hat{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 288\relax}}{\mbox{\boldmath\textstyle\mathchar 288\relax}}{\mbox{\boldmath\scriptstyle\mathchar 288\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 288\relax}}}}_{1} is seen as an element of , see (24). By (3), observe that
[TABLE]
which, combined with Equations (5) and (3.2), yields
[TABLE]
Therefore, by using the explicit expressions (20), (29)–(30) and (58), we deduce from (23) and (3.2) the equality
[TABLE]
for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}\leq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2} while, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}>{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2},
[TABLE]
using . On the other hand, if {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}\leq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2} then {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}}=\mathbf{1}_{\{1,2\}} and we infer from Equations (13), (14), (18), (21) and (45) that
[TABLE]
and the assertion holds true when {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}\leq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2}. If {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1}>{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{2} then
[TABLE]
which, combined with (44), implies that
[TABLE]
Using again (13), (14) and (21), we thus arrive from the last equality at
[TABLE]
By combining (62) and (3.2) with (64) and (65), we arrive at the assertion with {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 276\relax}}{\mbox{\boldmath\textstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptstyle\mathchar 276\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 276\relax}}}=\mathbf{1}_{\mathbb{R}}.
Corollary 3.3** (Determinants of the covariance and quasi–free states)**
Let be any separable Hilbert space. Fix {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, a self–adjoint operator acting on , and . Then, for each bounded measurable positive function from to , all , non–vanishing with and
[TABLE]
the following identity holds true:
[TABLE]
for any permutation of elements such that
[TABLE]
where \tilde{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}}_{q}\doteq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{q} for and \tilde{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}}_{q}\doteq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{q}+n^{-1}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}} for .
Proof: Fix all the parameters of the corollary. Take any permutation of elements such that
[TABLE]
See, respectively, (iv) before Lemma 3.2 and Equation (46) for the definitions of the permutations {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{k},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{N+l}} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}_{k,N+l} of two elements. Then, (66) follows from Lemmata 3.1 and 3.2. To conclude the proof observe that a permutation of elements satisfying (67) exists and also satisfies (68), keeping in mind Equation (1).
3.3 Correlation Functions and Tomita–Takesaki Modular Theory
(i): As above, fix {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, a self–adjoint operator acting on , , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}^{+}, and a non–vanishing positive real matrix with . Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}\equiv\{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t}\}_{t\in{\mathbb{R}}} be the unique –group (that is, strongly continuous group) of automorphisms on the –algebra satisfying
[TABLE]
See (20). It is well—known that the quasi–free state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}, which is defined from the symbol S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (21), is the unique ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS state on .
(ii): Recall that (\mathfrak{H}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},\varkappa_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}) is a cyclic representation of {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}} (Section 2.3). The weak closure of the –algebra is the von Neumann algebra \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} (32). The state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}}\circ\varkappa_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} on \varkappa_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}(\mathrm{CAR}(\mathfrak{h}\otimes\mathbb{M})) extends uniquely to a normal state on the von Neumann algebra \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} and the –group \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t}\circ\varkappa_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}\}_{t\in\mathbb{R}} also uniquely extends to a –weakly continuous –automorphism group on \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}. Both extensions are again denoted by {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}} and \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t}\}_{t\in\mathbb{R}}, respectively. By [BR2, Corollary 5.3.4], {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{S_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}} is again a ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS state on \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}.
(iii): By [BR2, Corollary 5.3.9], the cyclic vector {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} is separating for \mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}, i.e., A{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}=0 implies for all A\in\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}. Denote by \Delta_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} the (possibly unbounded) Tomita–Takesaki modular operator of the pair \left(\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}\right). The (–rescaled) modular group is the –weakly continuous one-parameter group {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}\equiv\{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}_{t}\}_{t\in{\mathbb{R}}} defined by
[TABLE]
(If {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}=-1 then is the well–known modular automorphism group associated with the pair \left(\mathcal{X}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}\right), see [BR1, Definition 2.5.15].) By Takesaki’s theorem [BR2, Theorem 5.3.10], we deduce that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}. In particular, using (69) we arrive at the equality
[TABLE]
(iv): Recall that (59) is a dense subspace of entire analytic vectors for H_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}}, while for any and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 272\relax}}{\mbox{\boldmath\textstyle\mathchar 272\relax}}{\mbox{\boldmath\scriptstyle\mathchar 272\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 272\relax}}}\in\mathbb{R}^{+}, \mathfrak{T}_{N}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 272\relax}}{\mbox{\boldmath\textstyle\mathchar 272\relax}}{\mbox{\boldmath\scriptstyle\mathchar 272\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 272\relax}}})} is the tube defined by (41). For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}\in\mathfrak{D} and , the maps
[TABLE]
from to the –algebra are entire analytic functions. Fix , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{1},\ldots,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}_{N}\in\mathfrak{D}, , and pick a family
[TABLE]
where the notation “” stands for either “” or “”. For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}\in\mathfrak{D}, and , we also use the convention
[TABLE]
with
[TABLE]
Then, for any fixed integer , the map from to defined by
[TABLE]
is an entire analytic function.
(v): For , define a family of elements of the –algebra , where, as before, “” or “”. Then, by applying [AM, Lemma A.1], we obtain the following assertions:
- (A)
The (cyclic and separating) vector {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} belongs to the domain of definition of the possibly unbounded operator
[TABLE]
for any (z_{1},\ldots,z_{N})\in\mathfrak{T}_{N}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)} with
[TABLE]
This inequality is a special case of [AM, (A.2)], which is intimately related to Hölder inequalities for non–commutative –spaces.
- (B)
The map from \mathfrak{T}_{N}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)} to \mathfrak{H}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}} defined by
[TABLE]
is norm continuous on the whole tube \mathfrak{T}_{N}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)} and analytic on its interior.
Using the notation
[TABLE]
we consider now the map from \mathfrak{T}_{p}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)}\times\mathfrak{T}_{N-p+1}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)} to defined by
[TABLE]
Compare for instance with [AM, Lemma A], which explains the properties of . (Notice that [AM] uses a different convention for sesquilinear forms.) By (75), this function is uniformly bounded for all n\in 2$$\mathbb{N}. The same is trivially true for the map (74) on
[TABLE]
Moreover, by using (71) we deduce that and are equal to each other on . For each fixed imaginary vector , the maps and are both continuous as functions of (\tilde{z}_{p},z_{p+1},\ldots,z_{N})\in\mathfrak{T}_{N-p+1}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)} and analytic in the interior of \mathfrak{T}_{N-p+1}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)}, by (B) [AM, Lemma A.1]. Hence, from Hadamard’s three line theorem (see, e.g., [RS2, Appendix to IX.4]), and are equal to each other for any fixed imaginary vector and all complex vectors (\tilde{z}_{p},z_{p+1},\ldots,z_{N})\in\mathfrak{T}_{N-p+1}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)}. Applying this argument again at fixed (\tilde{z}_{p},z_{p+1},\ldots,z_{N})\in\mathfrak{T}_{N-p+1}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)} for and viewed as functions of (z_{1},\ldots,z_{p})\in\mathfrak{T}_{p}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)}, we conclude that on \mathfrak{T}_{p}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)}\times\mathfrak{T}_{N-p+1}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2)}.
In particular, for , any family (73) of elements of the –algebra , and all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{1},\ldots,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{N}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}] such that
[TABLE]
the following equality holds true:
[TABLE]
with defined to be the smallest element of such that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}_{p}\geq{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}/2.
Acknowledgments: This research is supported by the FAPESP, the CNPq, the Basque Government through the grant IT641-13 and the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323 and MTM2014-53850. We are very grateful to the BCAM and its management, which supported this project via the visiting researcher program. Finally, we thank the referee for his thorough revision work, Zosza Lefevre for linguistic hints and Christian Jäkel for the nice lectures he gave in 2015 on non–commutative –spaces.
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