3.1. -Conditions and Bosonic Second Quantization
Let
denote the Bose Fock space over a complex Hilbert space
, with Fock vacuum vector
. We denote the projection onto the one-particle space
by
. For
, we have the usual creation and annihilation operators
,
on (a dense domain in)
. Their sum
is essentially self-adjoint on this domain, and gives rise to unitary Weyl operators by
where the bar denotes the self-adjoint closure. The map
is only real linear, and defines a map from closed real linear subspaces
to von Neumann algebras
via
In the context of the Klein Gordon quantum field, the subspaces
H may be related to real Cauchy data with prescribed supports, see
Section 5.2.
We collect some well-known properties of this map in the following lemma. In its formulation, we make use of the symplectic complement of a closed real subspace H, taken w.r.t. the imaginary part of the scalar product of , which is again a closed real linear subspace.
Lemma 3.1. Let be a closed real subspace. Then- (a)
Ω is cyclic for if and only if is dense.
- (b)
Ω is separating for if and only if .
- (c)
The map preserves inclusions.
- (d)
.
For more detailed properties of this map, see [
5], or [
43, Thm. I.3.2] for a proof.
To discuss
- and
p-nuclearity-properties, we consider in addition to a closed real subspace
also a self-adjoint (possibly unbounded) linear operator
X of second quantized form on
. Later on,
X will be taken to be a modular operator of a von Neumann algebra containing
. In the context of energy nuclearity conditions (on ultrastatic spacetimes), one would take
for some inverse temperature parameter
and a second quantized Hamiltonian
L [
27]. In the present section, we keep
X abstract. When necessary, we will denote its restriction to
by
, and we will assume
and
. We then define the linear map
similar to the map (2.13) appearing in the modular
condition.
Let us next state a theorem of Buchholz and Jacobi [
28, Thm. 2.1] about nuclearity properties of
. Its formulation makes use of
conjugations Γ on a complex Hilbert space, which are here defined to be anti-unitary involutions. Given a conjugation Γ, we write
and note that these are real linear real self-adjoint projections with the obvious properties
,
,
,
.
Theorem 3.2. [
28]
Let be as above, satisfying the following two additional assumptions:- (a)
There exists a conjugation Γ
on which commutes with , and two closed complex linear subspaces , such that and - (b)
Denoting the (complex linear) projections onto by , the operators are trace class and satisfy .
Then is nuclear, and its nuclearity index can be estimated as Our following generalization of this result involves the real orthogonal projection onto H. To define it, we consider as a real Hilbert space, with scalar product Re〈⋅,⋅〉. This still induces the same norm on , and defines a notion of real adjoint of (real or complex) linear operators on . Since this real adjoint coincides with the usual adjoint for complex linear operators, we will denote it by a superscript * as usual. Then is a real linear (real) self-adjoint projection.
Theorem 3.3. In the notations above, the following hold true:- (a)
If is for some , and , then is p-nuclear and for . In particular, if the assumption holds for all , then is for all .
- (b)
for all .
There are three differences between Thm. 3.3 and Thm. 3.2. First, the assumption
of Thm. 3.2 is absent in Thm. 3.3; Second, different spectral density conditions (
p-nuclearity and
, for complex respectively real linear operators) are used; Third, we also demonstrate the necessity of one of our assumptions in part
. We did not try to derive a sharp bound on the
p-nuclearity index or
-quasi-norms of
. However, from the proof given later, one sees that in the situation of Thm. 3.3, one has a bound of the form
where Li is the polylogarithm and the
are the eigenvalues of a positive operator
constructed from
, satisfying
and
for some numerical constant
, cf. (3.17).
We begin with a discussion of assumption of Thm. 3.2. To this end, it is useful to characterize inclusions of the form (3.4) in a more invariant manner.
Lemma 3.4.- (a)
Let Γ be a conjugation on . Then a closed real subspace is of the form with two closed complex linear subspaces which are invariant under Γ if and only if .
- (b)
If H is a closed real subspace as in , the real orthogonal projection onto H is related to the complex orthogonal projections onto by
Proof. Suppose H has the form as described. Then, as , it follows immediately that .
For the other implication, assume that
, and define
. These are two complex linear subspaces which are both invariant under Γ, and we claim that they are also closed. In fact, if
is a Cauchy sequence in
, then so is
Hence and are Cauchy sequences, and the closedness of H implies the closedness of .
Using the same properties of Γ again, we also see that
But as and , we have , i.e. H is of the claimed form .
Since the are invariant under Γ, this conjugation commutes with the projections . Using this fact, it is straightforward to check that is a self-adjoint real linear projection. In view of , this space is pointwise invariant under Q. On the other hand, if for some , then . Thus, . This implies that Q and coincide. ☐
The situation described in part
of this lemma is generic: As we will show later, any closed real subspace
H admits a conjugation Γ such that
(Prop. 3.9). Furthermore, by virtue of the spectral theorem in its multiplication operator form [
44, Thm. VIII.4], any (complex linear) self-adjoint operator
is unitarily equivalent to an operator multiplying with a real-valued function on some
-space. Considering pointwise complex conjugation on that space, it follows that there exists a conjugation Γ commuting with
.
But in general, there does not exist a conjugation commuting with
and preserving
H at the same time, as it is assumed in Thm. 3.2. We will show later in
Section 3.2 that such a conjugation does also not always exist if
is taken to be the modular operator suggested from the modular
-condition. This complication of a missing suitable conjugation can be circumvented in our proof of Thm. 3.3 below, but results in less stringent bounds on the
-quasi-norms.
Before we can proceed to the proof of Thm. 3.3, we need a technical lemma.
Lemma 3.5. Let be two bounded complex linear operators, and Γ
a conjugation that commutes with both of them. Define the real linear operator- (a)
There holds the norm equality - (b)
Let . Then if and only if , and in this case, the corresponding -quasi-norms satisfy the boundswhere are numerical constants depending only on p.
Proof. The proof of (3.9) is based on the fact that for a conjugation Γ on
and two arbitrary vectors
, there always holds
because
are real orthogonal projections with
.
To begin with, note that it readily follows from our assumptions that
, and in particular
. But by complex linearity of
, we also have
and hence
. This implies
. Using these bounds and (3.12), we obtain,
,
Hence
. On the other hand,
which implies
. Together with
, this yields (3.9).
From Remark 2.2 we see that
implies
, with
. After renaming
, this shows (3.10). Furthermore, for
a quick calculation shows that in this case,
. Hence
which completes the proof of (3.11). ☐
Now we are ready for the proof of the main result of this section, Thm. 3.3.
Proof. As explained above, we first need to account for the possibility that there is no conjugation Γ such that
and
. We therefore start with a construction to introduce some additional structure. Let Γ be a conjugation on
, and consider
It is clear that is a conjugation on . Moreover, leaves the closed real subspace invariant and commutes with . The real linear projection onto is , which implies by our norm assumption on , and by the quasi-norm property (2.5).
We now use the natural unitary map implementing the equivalence , which carries the Fock vacuum of onto , and the von Neumann algebra onto .
Under this identification, we have
But clearly the maps , and , , are linear and bounded, with norm one, and . Hence and . It now suffices to prove the claim for the underlined objects.
Lemma 3.4 applies to
,
, so that we may write
and
with complex linear subspaces
and corresponding complex linear projections
, commuting with
. Thus,
has the form assumed in Lemma 3.5, with
, and we conclude
with some numerical constant
.
For
, the space
coincides with the trace class on
. In that situation, all assumptions of Thm. 3.2 are satisfied, and we can immediately conclude that
is (1-)nuclear, with the bound
For general
p, we need to re-examine the argument underlying Thm 3.2. One step in that proof is the construction of a certain joint least upper bound of
[
28, p. 316-317]. Going through the construction, it becomes apparent that it works for
-operators as well: If
, then there exists a positive operator
such that
and
.
To estimate the approximation numbers of
, we can then follow the argument in [
16]: Let
denote an orthonormal basis of
consisting of eigenvectors of
T,
i.e.,
, with
. Let
denote the corresponding “occupation number” orthonormal basis of
,
i.e.,
are summable functions. Then
cf. [
16, p. 338]. This implies
where Li denotes the polylogarithm. To show that this expression is finite, it is sufficient to estimate
for large enough
l. Recall that
for all
, and thus,
. Since
monotonically as
, we have
for
l larger than some
. Hence, for large enough
l, we have
. As
is summable in
l, this shows that the product (3.16) converges. Note that for
, (3.16) reduces to the familiar expression
underlying (3.15).
We have therefore found a
p-nuclear decomposition (2.7) of
, and conclude that this map is
p-nuclear, with
p-nuclearity index bounded by
Whereas up to this point, the value of was arbitrary, we now restrict to the case to apply Lemma 2.3, which then tells us that is also for any .
We remark that in the situation at hand, one can exploit the particular form of our p-nuclear decomposition in terms of an orthonormal basis of a Hilbert space to show that is even for any .
We now prove the second statement, so we may assume that is for some (otherwise the estimate is trivially true). We use the fact that a map has the same operator norm and rank as its dual , so if Ξ is for some , then so is . Combining this with Lemma 2.1 we see that .
Now let
be in the domain of
X. Assume for the moment that
is non-zero and define
. Writing
we may use
with
to estimate
where we used the fact that the projection
is real self-adjoint. It follows from this estimate that
. The same estimate holds when
, so it holds on a dense domain in
. Hence,
, which means that
is bounded and
by Lemma 2.1. Using the same argument as in Lemma 3.5, we also find
, from which the result follows. ☐
3.2. Second Quantization of Modular Operators
We now wish to apply the results of the previous subsection to the case where X is the modular operator of a second quantized von Neumann algebra, containing a subalgebra corresponding to the real subspace considered so far.
As before, we consider a closed real subspace H of a complex Hilbert space , and denote the symplectic complement of H by . Furthermore, we will need to work with two different orthogonal complements, a real and a complex one. The complex orthogonal complement of H refers to the scalar product of . It is denoted , and seen to coincide with by an elementary calculation. The real orthogonal complement of H, referring to the real scalar product Re〈⋅,⋅〉, was introduced before. We will write the real orthogonal complement of H as , and note that .
The natural setting of spatial modular theory is that of
standard subspaces (see [
45] for an overview). A closed real subspace
is called standard if
Thanks to these properties, any standard subspace
H has a well-defined densely defined Tomita operator
,
As usual, the polar decomposition of this anti-linear involution will be denoted , with an anti-unitary involution, and a complex linear positive operator, satisfying . As all our standard subspaces will be only real-linear, we drop the term “real” and refer to them simply as standard subspaces.
The second quantized von Neumann algebra
of a standard subspace
H has the Fock vacuum Ω as a cyclic separating vector (Lemma 3.1). The modular data of
are closely related to the modular data of
H [
46]:
Lemma 3.6. Let be a standard subspace. Then the modular data of are related to by second quantization: For an inclusion of standard subspaces , this shows that taking , , and the subspace , we are in the situation described in Thm. 3.3 for discussing nuclearity properties of (3.3).
In line with the situation described in
Section 2.1, we will however need to consider more general closed real subspaces
H, which do not necessarily satisfy (3.18) or (3.19). In that case,
H can be compressed to a standard subspace, as we describe now.
Note that
and
are closed complex subspaces that are orthogonal to each other. Hence there exists an orthogonal (complex linear) projection
such that
In this decomposition,
,
i.e.,
is the (complex) orthogonal complement of
in
H, and therefore separating. Considered as a subspace of
, the projected real space
is therefore standard [
43]. Analogously to
Section 2.1, we now define the Tomita operator
of a general closed real subspace by
referring to the decomposition (3.22).
We are now in the position to apply Thm. 3.3 to the modular setting.
Theorem 3.7. Let be an inclusion of closed real subspaces, and . If and is for all , then (3.3) is for all .
Proof. In view of the split (3.22), the Bose Fock space over
has the form
with Fock vacuum
in an obvious notation.
Furthermore, in this decomposition, the second quantized von Neumann algebra
and its commutant are [
43]
According to the definitions of
Section 2.1, the modular data of
are constructed by first projecting to the subspace generated by
, which is
. Then, in this subspace, we consider the projection onto the subspace generated by the commutant
, which is
. But on the last tensor factor,
is based on a standard subspace, with modular operator Δ the second quantization of
.
This implies that the modular operator of
w.r.t. Ω is the second quantization of
, and thus given by
. From this we see that the map
has the form
Under the assumptions made, we know by Thm. 3.3 that the map , acting on the rightmost factor, is for all . Since the other two factor maps are of rank one, the claim follows. ☐
We wish to address two more topics: The norm bound appearing in the assumptions of Thm. 3.3, and the existence of a conjugation commuting with and preserving (cf. discussion after Lemma 3.4).
The norm bound required in Thm. 3.3 is almost a consequence of the -properties. As we will see in the applications to quantum field theory models, the appearing standard subspaces are typically “factors” in the sense that , in which case the norm bound is a consequence.
Lemma 3.8. Let be an inclusion of closed real subspaces and . Then , and whenever . If in addition is compact (or even for some ), then Proof. Let
and
. As
, it follows that the function
is analytic on the strip
and continuous on the closure of this strip. In view of
we see furthermore that
f is bounded. Moreover, on the boundary we have
, and
,
. Hence we may apply the three lines theorem [
47, Thm. 3.7] to the effect that
throughout the closed strip. Since
χ and
η are arbitrary, this entails
and hence
.
Now suppose that
satisfies
. Using the orthogonal decomposition
with
and
we then find from the first paragraph that
and hence
and
. Then note that for
the estimate
is strict, unless
f is constant (see, for example, [
47, Cor. 3.9]). Taking
,
f is the expectation value of a unitary one parameter group, which is constant if and only if
is constant. But that would imply
[
45, Prop. 2.1.14], and thus
. When this is assumed to be
, we find
,
i.e.,
has no eigenvectors with eigenvalue 1.
Now suppose that the real linear operator is compact. Similar to the complex linear case, a real linear compact operator can be represented as with two real orthonormal bases and w.r.t. Re〈⋅,⋅〉, and positive numbers such that for all n, and as . (Such a representation of T can be established by considering the complexification of . Then T gives rise to a complex linear compact operator on , which leaves invariant. The claimed representation of T then follows from the canonical form of complex linear compact operators, and restriction to .)
To show , it is therefore sufficient to show that any eigenvalue of the real linear operator is strictly less than 1, which follows from the first two paragraphs. ☐
We now come to our discussion of conjugations. In the proof of Thm. 3.3 we constructed a doubled Hilbert space with some complex conjugation Γ. It may happen that such a conjugation already exists without doubling the Hilbert space. In this case our estimates on the -quasi-norms can be improved significantly, essentially by taking a square root.
In Thm. 3.2, the existence of a conjugation Γ, commuting with and preserving the real subspace H, is assumed. This has its motivation in the theory of the Klein-Gordon field on Minkowski space in its vacuum representation. When formulated in terms of its time zero field and momentum, consider the von Neumann algebra generated by the time zero fields, with arbitrary support on the time zero surface . This is a maximally abelian second quantized von Neumann algebra, i.e., for some standard subspace in the single particle space. As a consequence, the modular operator is trivial, and hence is a conjugation, corresponding to complex conjugation for functions on . This conjugation preserves the time zero fields, smeared with real test functions in a given region , but changes the sign of the time zero momenta, smeared with real test functions with support in O. Therefore the standard subspace corresponding to has the structure assumed in Thm. 3.2.
In general, one can show given a closed real linear subspace , there always exists a conjugation preserving it. Note that this statement is non-trivial because H is only real linear.
Proposition 3.9. Let be a closed real subspace. Then there exists a conjugation Γ on such that .
Proof. We split as in (3.22), and have to construct a conjugation on each of the three summands. Since H has no components in the first summand, and the second (complex linear) summand is contained in H, we can pick arbitrary conjugations on the first two summands. In other words, it is sufficient to consider the case where H is standard.
Recall that standard subspaces H are in one-to-one correspondence with their Tomita operators via . Therefore a conjugation Γ preserves a standard subspace H if and only if it commutes with on the domain . Proceeding to the polar decomposition, this is also equivalent to Γ commuting with both, the modular conjugation , and the modular operator (on its domain).
We therefore need to construct a conjugation Γ commuting with both modular data,
J and Δ, of
H. (For brevity, we drop the index “
H” on these operators during this proof.) The proof is based on the relation
Let denote the spectral subspace of spectrum of Δ in (which is zero in the factor situation ). Furthermore, split into subspaces and , corresponding to spectrum of Δ in and , respectively. We then have , and Δ leaves all three subspaces invariant. In view of (3.28), we see that , , and .
The modular operator Δ restricts to a (complex linear, bounded) self-adjoint operator on , and thus we find a conjugation on that commutes with this restriction. Taking into account , the conjugation is seen to be well-defined on , and leave this space invariant. Furthermore, commutes with the restriction of Δ to because of (3.28).
Finally, on
, the modular operator restricts to the identity. Hence
is a conjugation on this space that commutes with
. Summarizing this discussion,
is a conjugation on
that commutes with Δ. By construction, it also commutes with
J. Thus,
. ☐
It has to be mentioned, however, that for a general inclusion of standard subspaces, a conjugation preserving both H and need not exist. Our counterexample is that of a half-sided modular inclusion, i.e. an inclusion satisfying for .
Lemma 3.10. Let be a non-trivial half-sided modular inclusion of standard subspaces. Then there exists no conjugation Γ with and .
Proof. Suppose Γ is a conjugation with and . Then Γ commutes with both modular operators, and . In view of the anti-linearity of Γ and , this implies in particular that for all .
But for a half-sided modular inclusion, there exists a unitary one-parameter group
T with positive generator such that
and
See [
45] for a proof of these facts in the standard subspace setting, and [
48,
49] for the original von Neumann algebraic situation.
Setting
, we therefore find from (3.30)
i.e.,
for all
. But this leads to a contradiction: Since
T is a one-parameter group, and
, we have,
,
But on the other hand, using (3.31), we also have
Hence for all , which is possible if and only if for all . Thus, , which contradicts the non-triviality of the inclusion. ☐
We leave it as an open problem to characterize inclusions of standard subspaces which allow for a conjugation preserving both H and .
3.3. -Conditions and Fermionic Second Quantization
To conclude this section, we also briefly consider the Fermionic case, where one again starts from a closed real subspace
as before, but proceeds to the Fermionic Fock space
, and the von Neumann algebra
generated by the Fermi field operators
,
. The structure of the map
is analogous to the one discussed in Lemma 3.1, with the commutant replaced by a twisted commutant (see, for example, [
50, Thm. 55]). We will denote the analogue of
(3.3) in the Fermi case by
, with identical assumptions on
as in
Section 3.1, but now on
instead of
.
There exists a result about second quantization of modular nuclearity conditions in the Fermionic case.
Theorem 3.11. [
29]
Let be a closed real subspace and a selfadjoint operator on , satisfying the same assumptions as in Thm. 3.2, with the exception of the norm bound , which is not assumed. Then is nuclear, and its nuclearity index can be estimated as The absence of the condition on the norm of
, and the sharper bound on
, are consequences of the Pauli principle [
29].
In our more abstract setting, we find
Theorem 3.12. Let be a closed real subspace.- (a)
If is for some , then is p-nuclear, and for .
- (b)
If is for some , then is .
- (c)
is for all if and only if is for all .
Proof. The proof is similar to that of Thm. 3.3, but simpler because we do not need to control the norm of . We again pick a conjugation Γ on , and consider the doubled system , (3.13), invariant under . Then, as in the Bose case, we can write with complex closed subspaces , and the corresponding operators lie in because . We again proceed to the joint upper bound , satisfying .
Nuclearity estimates on
are obtained by following [
29]: Denoting the eigenvalues of
T by
, it is shown there (p. 3051) that the corresponding Fermi second quantized orthonormal basis
of
satisfies
Note that the last expression is dominated by , and thus finite because T is an element of , , which is contained in the trace class.
If one estimates the
p-th powers of the expectation values instead, one gets in a similar manner
from which we read off
Hence is p-nuclear, and by Lemma 2.3, also for all .
The conversion of these estimates to corresponding ones for the system without the doubling now follows as in the Bose case.
The Fermi field operator
,
(sum of Fermionic creation and annihilation operator), is bounded with norm
, and an element of
satisfying
. Thus, the composition of bounded and
maps (where
denotes the projection
)
is
, meaning that
is in
. But this is equivalent to
.
This is a direct consequence of and . ☐
Also in the Fermi case, one can apply this general result to more concrete modular or energy nuclearity conditions. We refrain from giving the details here.