Commun. Math. Phys. 195, 1 – 14 (1998)
Communications in
Mathematical
Physics
© Springer-Verlag 1998
Anderson Localization for the Almost Mathieu Equation,
III. Semi-Uniform Localization, Continuity of Gaps, and
Measure of the Spectrum
Svetlana Ya. Jitomirskaya1,? , Yoram Last2
1
Department of Mathematics, University of California, Irvine, CA 92697, USA
Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena,
CA 91125, USA
2
Received: 7 July 1997 / Accepted: 15 September 1997
Abstract: We show that the almost Mathieu operator, (Hω,λ,θ 9)(n) = 9(n+1)+9(n−
1)+λ cos(πωn+θ)9(n), has semi-uniform (and thus dynamical) localization for λ > 15
and a.e. ω, θ. We also obtain a new estimate on gap continuity (in ω) for this operator
with λ > 29 (or λ < 4/29), and use it to prove that the measure of its spectrum is equal
to |4 − 2|λ|| for λ in this range and all irrational ω’s.
1. Introduction
In this paper we study localization for the almost Mathieu operator Hω,λ,θ acting on
`2 (Z):
(Hω,λ,θ 9)(n) = 9(n + 1) + 9(n − 1) + vn 9(n),
(1.1)
where the potential vn is given by
vn = λ cos(πωn + θ).
(1.2)
For background and some recent results on the almost Mathieu operator see [26, 18,
19, 15].
An often used notion of localization is that of pure point spectrum with exponentially
decaying eigenfunctions. It can be expressed by the following definition:
Definition 1. H exhibits localization if there exists a constant γ > 0 such that for any
eigenfunction ψs one can find a constant C(s) > 0 and a site n(s) ∈ Z (center of
localization) so that |ψs (k)| ≤ C(s)e−γ|n(s)−k| for any k ∈ Z.
? Alfred P. Sloan Research Fellow. The author was supported in part by NSF Grants DMS-9208029 and
DMS-9501265.
2
S. Ya. Jitomirskaya, Y. Last
Localization, in this sense, for the almost Mathieu operator has been proven for
λ > 15 and ω, θ satisfying certain arithmetic conditions [32, 13, 16].
At the same time, the physical understanding of localization is connected with what
is often called dynamical localization—the non-spreading of initially localized wavepackets under the Schrödinger time evolution. It can be expressed by, for example,
boundedness in the time t of kxeitH δ0 k2 = (e−itH δ0 , x2 e−itH δ0 ). An even stronger
condition that we prefer to adapt here is the following:
Definition 2. H exhibits dynamical localization if there exists a constant γ̃ > 0 such
that for any ` ∈ Z, there exists C̃(`) > 0 so that
sup |e−itH (n, `)| ≤ C̃(`)e−γ̃|n−`| .
(1.3)
t
Both of the above definitions (as well as Definitions 3 and 4 below), with Z replaced
by Zd , work for operators H acting on `2 (Zd ).
Dynamical localization has been established for the d-dimensional Anderson model
in [2, 10] (and, in a restricted form, [28]); it remained an open question for the almost
Mathieu operator.
While dynamical localization implies pure point spectrum [23, 5], the converse is
not true in general. There exist operators H with localization, but, nevertheless, with
lim supt→∞ kxeitH δ0 k2 /tα = ∞ for any α < 2 [7] (also see [27]). In fact, tα can be
replaced by an arbitrary function f (t) = o(t2 ). This example shows that Simon’s theorem
on the absence of ballistic motion for operators with pure point spectrum [31] is optimal,
and that mere “exponential localization” of eigenfunctions is not sufficient to determine
the dynamics.
Indeed, localization might not have much physical meaning if there is no control on
the dependence of C on n (or, equivalently, on the eigenenergy En ). In particular, if the
C(m)’s are allowed to grow arbitrarily fast with m, then eigenvectors may be “extended”
over arbitrarily large length-scales and one cannot effectively define a “localization
length” corresponding to a typical size of the “essential support" of the eigenfunction.
An appropriate level of control over the C(m)’s is given by the following definition,
introduced in [7]:
Definition 3. H has SULE (semi-uniformly localized eigenvectors) if there exists a
constant γ > 0 such that for any b > 0, there exists a constant C(b) > 0 such that for
any eigenfunction ψs one can find n(s) ∈ Z so that |ψs (k)| ≤ C(b)eb|n(s)|−γ|k−n(s)| for
any k ∈ Z.
The notion of SULE, which strengthens mere localization, also has a dynamical
counterpart, which strengthens mere dynamical localization:
Definition 4. H has SUDL (semi-uniform dynamical localization) if there exists a constant γ̃ > 0 such that for any b > 0, there exists a constant C̃(b) > 0 so that
sup |e−itH (n, `)| ≤ C̃(b)eb|`|−γ̃|n−`| .
(1.4)
t
It is shown in [7] that SULE implies SUDL (and thus dynamical localization), and
that SUDL, along with a simple spectrum, implies SULE.
There is another direction from which the notion of localization had been challenged
recently, when Gordon [14] and del Rio-Makarov-Simon [9] have shown that localization
can be destroyed by infinitesimally small localized rank-one perturbations. However,
Anderson Localization for Almost Mathieu Equation. III
3
SULE appears to be a condition that implies certain semistability (or physical stability)
of localization [7]. Specifically, if H has SULE and H 0 is obtained from H by adding a
localized rank-one perturbation, then all the spectral measures of H 0 are supported on
0
a set of zero Hausdorff dimension; and, more importantly, while kxe−itH δ0 k may be
2
unbounded, it never grows faster than C ln (t) [7].
That makes it particularly interesting to establish SULE for systems with localization. For the d-dimensional Anderson model, SULE had been derived in [7] from the
dynamical estimates of Aizenman [2] ([10] in the one-dimensional case). More precisely, the dynamical estimates imply SUDL, from which SULE follows by the SULE
⇔ SUDL relation. In the present paper, we obtain SULE for the almost Mathieu operator by direct analysis of eigenfunctions, and so we deduce SUDL (and, in particular,
dynamical localization) for this operator by using the SULE ⇔ SUDL relation in the
other direction. (It should be pointed out that we do not know any other way to prove
dynamical localization for the almost Mathieu operator.)
Throughout the paper we will often assume ω to be Diophantine; that is, that there
exist c(ω) > 0 and 1 < r(ω) < ∞ such that
| sin πjω| >
c(ω)
|j|r(ω)
(1.5)
for all j 6= 0. Our main result is the following:
Theorem 1.1. The almost Mathieu operator has SULE for any Diophantine ω, λ > 15,
and a.e. θ.
As discussed above this immediately implies:
Corollary 1.2. The almost Mathieu operator with ω, λ, θ as in Theorem 1 has dynamical
localization.
Remarks. 1. The set of parameters ω, λ in Theorem 1.1 is exactly the set for which
localization has been proven [16].
2. We will, in fact, obtain a polynomial bound on C(n(s)). See (2.2).
3. The set of θ’s for which we show SULE is smaller than the set of θ’s for which
localization has been proven. One can show that there is a zero measure set of θ’s
for which there is localization but not SULE.
4. Similar techniques can be applied to non-Diophantine ω’s with exponential rate of
approximation by rationals, for which localization is proven for large λ [21].
5. One can study local SULE, that is, semi-uniform localization of eigenfunctions with
corresponding eigenvalues belonging to a certain interval. Such local SULE can
be shown to imply dynamical localization for the spectral projection of H onto this
interval. Our method allows us to establish local SULE for all energy intervals where
localization has been shown so far. That includes certain intervals in the center of
the spectrum for λ > 5.6 [16].
6. One can think that a more natural control on the C(n)’s is given by uniform localization, UL, defined as localization with a uniform bound C(n) < c. However, a large
class of ergodic operators, including the almost Mathieu operator, has been shown
not to have UL [20, 7]. Indeed, for large families of ergodic potentials, vn = v(T n x),
UL implies phase-stability of pure point spectrum (see [20]), that is, localization
for all phases x [20, 7]. We would like to repeat a remark from [20] that for the
almost Mathieu operator, SULE is accompanied by certain phase-semistability of
pure point spectrum, as the spectral measures for all θ are zero-dimensional [19, 22].
4
S. Ya. Jitomirskaya, Y. Last
This suggests that the general relation between SULE and such semistability, or even
a stronger dynamical statement, could be similar to the general relation between UL
and the stability. While both phase-stability of pure point spectrum and UL, as in
the Maryland model [29, 12, 11, 30], are deeply connected with the lack of resonances (see [20]), both SULE and zero-dimensional spectral measures for the almost
Mathieu operator are implied by certain control over the resonances.
The estimates we obtain allow us, in addition, to answer some other almost Mathieu
questions. In particular, we establish a strong version of the continuity of gaps theorem
[3, 24], and use it to extend the result of [24] on the measure of the spectrum to the case
of an arbitrary irrational frequency ω. Let σ(ω, λ) denote the spectrum of Hω,λ,θ , which
is known to be completely θ-independent for any irrational ω [5]. We shall prove the
following:
Theorem 1.3. For all irrational ω’s and |λ| > 29 or |λ| < 4/29,
|σ(ω, λ)| = |4 − 2|λ||,
where | · | denotes Lebesgue measure.
Remark. The equality |σ(ω, λ)| = |4 − 2|λ|| was conjectured by Aubry and Andre [1] to
hold for every λ and irrational ω, and was studied by Thouless [33] and by Avron, van
Mouche, and Simon [3], who proved the lower bound. Last [24] obtained the equality
for every λ and a.e. ω.
In Sect. 2 we prove Theorem 1.1 up to some lemmas that we prove in Sections 3 and
4. In Sect. 5 we prove a result about continuity of gaps, which we use in Sect. 6 to prove
Theorem 1.3. The Appendix provides a proof for a somewhat technical lemma that we
use in Sect. 4.
2. Proof of Theorem 1.1
In this section we prove Theorem 1.1 up to some lemmas that will be proven later. Our
proof consists of two parts:
1. Obtaining uniform estimates in the proof of localization for the nonresonant regime.
2. Studying the statistics of resonances, and, particularly, the dependence of the number
of resonances on the position of the center of localization.
We introduce the sets of resonant phases:
j
2sj,k = θ : (k + 1)−s < sin 2π θ + ω < k −s , k ∈ N,
2
2sk = ∪k|j|=0 2sj,k ,
2s = lim 2sk ,
k→∞
and
2 = ∩s>r(ω) 2s .
Anderson Localization for Almost Mathieu Equation. III
5
Note that
2 = {θ : for every s > r(ω) the relation | sin 2π(θ + (j/2)ω)| < k−s for some |j| ≤ k
holds for infinitely many k 0 s}.
Since |2sk | = O(k −s ), it follows from the Borel-Cantelli lemma that every 2s , s > r(ω),
has zero Lebesgue measure and so does 2. For θ ∈
/ 2, we define the resonant rate as
s(θ) ≡ inf{s > r(ω) : θ ∈
/ 2s }+1 ≥ r(ω)+1. For s > s(θ)−1 we define the s-resonant
number of θ, k(θ, s) = #{m ∈ N : θ ∈ 2sm }. Let n1 (θ, s) < · · · < nk(θ,s) (θ, s) be the
positions of resonances: all m ∈ N such that θ ∈ 2sm . For a fixed θ, the numbers k(θ, s)
and nk(θ,s) (θ, s) decrease with s. Also, s(θ) is an invariant function: s(θ + ω) = s(θ), and
2 is an invariant set. Let k(θ) = k(θ, s(θ)). We put ni (θ) = ni (θ, s(θ)), i = 1, . . . , k(θ),
and will sometimes write nk(θ) for nk(θ) (θ).
We first obtain some elementary information on the sparseness of the sequence ni (θ).
Lemma 2.1. Suppose θ ∈ 2sn , ω satisfies (1.5), and s > r(ω). Then there exists a
s
positive constant c1 (ω) such that θ ∈
/ 2sm , n < m < c1 (ω)n r(ω) .
Proof. | sin 2π(θ + j/2ω)| ≤ n−s , some |j| ≤ n, and | sin 2π(θ + `/2ω)| ≤ m−s ,
some |`| ≤ m, |`| > |j|, imply | sin π(` − j)ω)| < 2n−s , and so, by (1.5), we have
1
m ≥ |`| ≥ |` − j| − |j| ≥ (c(ω)ns /2) r(ω) − n.
For θ ∈
/ 2, λ > 15 and ω satisfying (1.5) the localization has been proven in [16].
Thus every eigenfunction 9E with eigenvalue E, attains its maximal value at no more
than finitely many points. We define n(E) to be the position of the leftmost maximum
of 9E .
Our key technical result is
Lemma 2.2. Let θ ∈
/ 2, λ > 15 and ω satisfies (1.5). Then there exist C = C(θ, ω, λ) <
∞ and γ = γ(λ) > 0 such that for any eigenfunction 9E of Hθ , we have |9E (m)| <
2|9E (n(E))|e−γ(λ)|m−n(E)| for |m − n(E)| > C(θ, ω, λ) ln nk(θ+n(E)ω) .
Lemma 2.2 will be proven in Sect. 4. For the continuity of gaps and measure of the
spectrum parts, we will also need a similar, slightly more detailed statement, Lemma
5.3, asserting the exponential decay between the resonances.
In order to relate the number of resonances to the position of n(E), we will need the
following:
Lemma 2.3. Fix 0 < r < ∞. Then for a.e. θ ∈
/ 2s with s > r + 1, there exists
q(θ, r, s) < ∞ such that for every eigenvalue E of Hθ , we have n(E) > nrk(θ+n(E)ω) if
nk(θ+n(E)ω) > q(θ, r, s).
This lemma will be proven in Sect. 3.
Proof of Theorem 1.1. Suppose λ > 15 and ω satisfies (1.5). For a.e. θ ∈
/ 2, as in
Lemma 2.3, we obtain, using Lemma 2.2, that for any eigenfunction 9E of Hω,λ,θ and
any m ∈ Z,
|9E (m)| < 2|9E (nE )|(nk(θ+n(E)ω) )C(θ,ω,λ)γ(λ) e−γ(λ)|m−n(E)|) .
(2.1)
Thus, if we normalize 9 by 9E (n(E)) = 1 and fix 0 < r < s(θ) − 1, Lemma 2.3 yields
that
1
(2.2)
|9E (m)| < 2(max(n(E) r , q(θ, r, s(θ))))C(θ,ω,λ)γ(λ) e−γ(λ)|m−n(E)|)
which, in particular, proves the theorem.
6
S. Ya. Jitomirskaya, Y. Last
3. Resonant Sets: Proof of Lemma 2.3
Let psn (θ) = min{|m| : θ + 2πmω ∈ 2sn }.
Lemma 3.1. Fix 0 < r < s − 1. Then for a.e. θ, there exists q(θ, r, s) < ∞ such that
for n > q(θ, r, s), we have psn (θ) > nr .
Proof. Since |2sn | = O(n−s ), we have |{θ : psn (θ) < nr }| ≤ 2nr+1−s . Thus, the
Borel-Cantelli lemma implies the result.
Proof of Lemma 2.3. Since θ + n(E)ω ∈ 2snk(θ+n(E)ω) , we have by the definition of psn (θ)
that n(E) ≥ psnk(θ+n(E)ω) (θ). So, by Lemma 3.1, we obtain the needed statement.
4. Uniform Decay: Proof of Lemma 2.2
Throughout this section we assume that ω satisfies (1.5). We start with recalling the main
definitions and lemmas from the proof of localization in [16]. We will use the notation
G[x1 ,x2 ] (E) for the Green’s function (H − E)−1 of the operator Hω,λ,θ restricted to the
interval [x1 , x2 ] with zero boundary conditions at x1 − 1 and x2 + 1. Let us denote
#
"
.
Pk (θ, E) = det (H(θ) − E)
[0,k−1]
We now fix E ∈ R; 1 < m1 <
quantities:
where by
p
λ 1
2 , m1
p
1
M (E, λ) = √ |E + i + (E + i)2 − λ2 |,
3
(E + i)2 − λ2 we understand the value with positive imaginary part;
C(E, λ) =
and cλ, =
< m2 < 1, > 1. We will need the following
ln(m1 m2 )
ln(M (E,λ)) .
ln λ2
3
−
ln M (E, λ) 4
Given k > 0, let us denote
Ak (E) = {x ∈ Z : |Pk (θ + xω, E)| > mk1 }.
We will sometimes drop the E-dependence, assuming E is fixed.
Definition. Fix E ∈ R. A point y ∈ Z will be called (m2 , k)-regular if there exists an
interval [x1 , x2 ] containing y such that
|G[x1 ,x2 ] (y, xi )| < mk2 , and dist(y, xi ) ≤ k; i = 1, 2.
Otherwise y will be called (m2 , k)-singular.
The Poisson formula
9(y) = G[x1 ,x2 ] (y, x1 )9(x1 − 1) + G[x1 ,x2 ] (y, x2 )9(x2 + 1)
(4.1)
implies that if 9E is a generalized eigenfunction, then every point y with 9E (y) 6= 0 is
(m2 , k)-singular for k sufficiently large: k > k(E, m2 , θ, y). Of course, in general there
is no uniform bound on k(E, m2 , θ, y). It turns out though that if we pick Hω,λ,θ with
an eigenvalue E and take y to be n(E), such a bound becomes immediate.
Anderson Localization for Almost Mathieu Equation. III
7
2
Lemma 4.1. n(E) is (m, k)-singular for k > − lnlnm
.
Proof. Obvious from (4.1).
We will need to recall several statements from the proof of localization in [16].
Lemma 4.2 (Proposition 1 of [16]). For any > 1, there exists k(, E) such that for
k > k(, E) and all θ we have
|Pk (θ, E)| < (M (E, λ))k .
Lemma 4.3 (Proposition 2 of [16]). Suppose y ∈ Z is (m2 , k)-singular. Then for any
x such that k(1 − cλ, ) ≤ y − x ≤ kcλ, , we have that x does not belong to Ak .
The following lemma is proven in the proof of Lemma 3 in [16]:
1
Lemma 4.4. Let 2m
λ < b < 1. Then there exists k1 (b) < ∞ such that for k > k1 (b), if
the two points x1 , x2 ∈ Z are such that
/ Ak , i = 1, 2,
1) xi , xi + 1, ..., xi + [ k+1
2 ]∈
],
2) dist(x1 , x2 ) > [ k+1
2
then
cos 2π θ + k − 1 + x1 + j1 ω
2
(4.2)
k
k−1
4
− cos 2π θ +
+ x1 + j 2 ω
≤b
2
k+1
for some j1 , j2 ∈ [0, [ k+1
2 ]] ∪ [x2 − x1 , x2 − x1 + k − 1 − [ 2 ]].
Let E(θ) be a generalized eigenvalue of Hω,λ,θ , 9(x) the corresponding generalized
eigenfunction. To finish the proof of Lemma 2.2 we will need E-independent bounds
on how large the scale k should be in Lemmas 4.1–4.4. Since Lemmas 4.1, 4.3, 4.4 are
already E-independent, we will only have to take care of Lemma 4.2. It turns out that
Lemma 4.2, although formulated and proven in [16] in a non-uniform way, is, in fact, a
uniform statement:
Lemma 4.5. There exists k() < ∞ such that k(, E) ≤ k() for any E ∈ [−λ−2, λ+2].
Lemma 4.5 will be proven in the appendix.
Proof of Lemma 2.2. Let λ > 15. It was shown in [16] that in such a case, C(E, λ) >
C(λ + 2, λ) > 0 for any E ∈ [λ − 2, λ + 2]. Thus, there exist (E-independent) 1 <
1
< b < 1. Let
m1 < λ2 , m2 < 1 and > 1 such that 2cλ, − 1 > 21 . Fix 2m
λ
ln 2
k = |x − n(E)| > max[k(), k1 (b), − ln m2 ]. Suppose x is (m2 , k)-singular. Since, by
Lemma 4.1, n(E) is also (m2 , k)-singular, we can, by Lemma 4.3, apply Lemma 4.4
with x1 = x − [cλ, |x − n(E)|] and x2 = n(E) − [cλ, |x − n(E)|]. We then obtain, by
(1.5), (4.2),
bk/4 (3k/2)r(ω)
sin 2π θ + k − 1 + x1 ω + j1 + j2 ω
<
< bk/5 , k > k̂(ω, b)
2
2
2c(ω)
k+1
with some j1 , j2 ∈ [0, [ k+1
/
2 ]]∪[x2 −x1 , x2 −x1 +k −1−[ 2 ]]. Noting that θ+n(E)ω ∈
−5s(θ) ln nk(θ+n(E)ω)
s(θ)
or |k−1−2cλ, k+
2n for n > nk(θ+n(E)ω) , we obtain that either k <
ln b
−k
j1 + j2 | > b 5s(θ) . Since for any allowed j1 , j2 we have |k − 1 − 2cλ, k + j1 + j2 | < 5/2k,
the second inequality is contradictory for k > k̂1 (b, s(θ)). Thus any x with
8
S. Ya. Jitomirskaya, Y. Last
k = |x − n(E)| > k0 (, m2 , b, ω, θ)
ln 2
−5s(θ) ln nk(θ+n(E)ω)
= max k(), k1 (b), −
, k̂(ω, b), k̂1 (b, s(θ)),
ln m2
ln b
is (m2 , |x − n(E)|)-regular. Repeating the argument of [16], we have that there exists
an interval [y1 , y2 ] containing x such that
|x−n(E)|
|yi − x| ≤ |x − n(E)|, |G[y1 ,y2 ] (x, yi )| ≤ m2
, i = 1, 2.
Using (4.1), we obtain the estimate: |9(x)| ≤ 2|9(n(E))|e−γ(λ)|x−n(E)| , γ(λ) =
− ln m2 , for any x with |x − n(E)| > C1 (ω, θ, λ) ln nk(θ+n(E)ω) , since our choice of
, m2 , b was dependent on λ only.
5. Continuity of Gaps
Let σ(ω, λ, θ) denote the spectrum of Hω,λ,θ which depends on θ only if ω is a rational.
We denote S(ω, λ) = ∪θ σ(ω, λ, θ). In this section we establish the following continuity
property of the set S(ω, λ):
Theorem 5.1. For every λ > 29, there are constants C(λ), D(λ) > 0, such that if
ω, ω 0 ∈ R satisfy |ω − ω 0 | < C(λ), then for every E ∈ S(ω, λ) there is E 0 ∈ S(ω 0 , λ)
with
|E − E 0 | < F (|ω − ω 0 |),
where F (x) = −D(λ)x ln x.
Remarks. 1. This theorem with F (x) = 6(λx)1/2 was proven for any λ > 0 by Avron,
van Mouche, and Simon [3].
2. The constant D(λ) can be effectively estimated. Namely, as can be seen from the
λ. This estimate explodes for λ
proof, it can be shown that D(λ) < 3 ln 264λ
2M (λ+2,λ)5/6
approaching the root of λ = 2M (λ + 2, λ)5/6 which is slightly smaller than 29.
However, as λ grows, the estimate becomes increasingly better.
Theorem 5.1 immediately implies the following corollary:
Corollary 5.2. (i) For every λ > 29, there are constants C(λ), D(λ) > 0, such that
if |ω − ω 0 | < C(λ), then for every gap in S(ω, λ) with midpoint Eg , and measure |g|
larger than −2D(λ)|ω − ω 0 | ln |ω − ω 0 |, there is a corresponding (containing Eg ) gap
in S(ω 0 , λ) with measure larger than |g| + 2D(λ)|ω − ω 0 | ln |ω − ω 0 |.
(ii) The same continuity as in (i) also holds for the extreme edges of S(ω, λ) namely, for
|ω − ω 0 | < C(λ):
0
max
0
0
|max
min S(ω, λ) − min S(ω , λ)| < −D(λ)|ω − ω | ln |ω − ω |.
For the proof of Theorem 5.1, we will need a statement asserting decay between the
resonances. Fix θ. Let j(n) be the position of the resonance of order n : | sin 2π(θ +
j(n)
−s
, | sin 2π(θ + j2 ω)| > n−s , |j| < |j(n)|. By the proof of Lemma 2.1, we
2 ω)| < n
have, for m > n:
s
(5.1)
|j(m)| > c1 (ω)|j(n)| r(ω) .
Anderson Localization for Almost Mathieu Equation. III
9
Lemma 5.3. Let λ > 29, ω satisfies (1.5), θ ∈
/ 2, E is an eigenvalue of Hθ and
/ 2si , n < i < m. Then there exists
θ + n(E)ω ∈ 2sn ∩ 2sm , n < m; θ + n(E)ω ∈
k(λ, ω) = max(c2 (λ), −c3 (λ) ln(2c(ω))), such that for |l − n(E)| > k(λ, ω) we have
|9E (l)| < 2|9E (n(E))|e−γ(λ)|l−n(E)|
whenever 3j(n) < |l − n(E)| < 3/8j(m), where γ(λ) is the same as in Lemma 2.2.
For the proof of Theorem 5.1, we will need the following elementary lemma:
Lemma 5.4. For any Borel set S ⊂ [0, 1] with |S| > 0, there exist ω ∈ S for which
(1.5) holds with r(ω) = 3 and c(ω) > |S|
3 .
Proof. If not, then for every ω ∈ S with r(ω) = 3 (such ω’s form a set of full measure)
there would exist j 6= 0 such that
| sin πjω| <
|S|
.
3|j|3
(5.2)
If
Pwe denote by Aj the set of ω ∈ S for which (5.2) is satisfied, we obtain |S| ≤
|Aj | < |S|, and the contradiction proves the lemma.
Proof of Theorem 5.1. Fix λ > 29. By Lemma 5.4, we can find ω1 ∈ (ω, ω 0 ) satisfying
0
|
. Take E ∈ σ(ω1 , λ) and L > k(λ, ω1 )
(1.5) with r(ω1 ) = 3, such that c(ω1 ) > |ω−ω
3
(from Lemma 5.3). Pick θ ∈
/ 2 so that Hω1 ,λ,θ has pure point spectrum. Pick E1 ,
an eigenvalue of Hω1 ,λ,θ with |E1 − E| < e−3/8γ(λ)L . Let 9E1 be the corresponding
eigenfunction. Let j(ni ) be a sequence of resonance positions for the phase θ + n(E1 )ω1 .
Let i be such that j(ni ) < L < j(ni+1 ). Then by (5.1), one can find a constant 3/8 <
a < 3 such that 3j(ni ) < aL < 3/8j(ni+1 ). Let 8E1 be 9E1 restricted to [n(E1 ) −
aL, n(E1 ) + aL] and normalized by k8E1 k = 1. Then we have
||(Hω0 ,λ,θ − E)8E1 || ≤ ||(Hω1 ,λ,θ − Hω0 ,λ,θ )8E1 ||
+ ||(Hω1 ,λ,θ − E1 )8E1 || + |E − E1 |||8E1 ||
≤ 2aλ|ω − ω 0 |L + 4e−γ(λ)aL + e−3/8γ(λ)L
≤ 6λ|ω − ω 0 |L + 5e−3/8γ(λ)L .
Here we used Lemma 5.3 to estimate the second term. By taking L = −D1 (λ) ln |ω −ω 0 |,
where D1 (λ) = max(2c3 (λ), 8/3γ(λ)−1 ), we obtain the needed statement for λ > 29.
ln
λ
2
Proof of Lemma 5.3. The constant 29 was chosen so that ln M (31,29)
> 5/6. This implies
that for λ > 29, one can choose 1 < m1 < λ/2, m2 < 1, > 1 so that 2cλ, − 1 > 2/3.
That means, by Lemma 4.3, that every (m2 , k)-singular point “produces" [2k/3] points
not belonging to Ak . We will now formulate a version of Lemma 4.4:
1
Lemma 5.5. Let 2m
λ < b < 1. Then there exists k2 (b) < ∞ such that for k > k2 (b), if
the two points x1 , x2 ∈ Z are such that
/ Ak , i = 1, 2,
1) xi , xi + 1, ..., xi + 2[ k+1
3 ]∈
2) dist(x1 , x2 ) > [ k+1
],
2
then
10
S. Ya. Jitomirskaya, Y. Last
cos 2π θ + k − 1 + x1 + j1 ω
2
k−1
≤ b k4
− cos 2π θ +
+ x1 + j 2 ω
2
(5.3)
k+1
for some j1 , j2 ∈ [0, 2[ k+1
3 ]] ∪ [x2 − x1 , x2 − x1 + [ 3 ]].
The proof of this lemma is identical to that of Lemma 4.4 and is given in [16].
/ 2si , n < i < m. Assume x
Assume now that θ + n(E)ω ∈ 2sn ∩ 2sm , n < m; θ ∈
is (m2 , k)-singular, with k = |x − n(E)|. We will consider several cases depending on
the position of x with respect to the resonance.
• If x < n(E) < n(E) + j(n), we can apply Lemma 5.5 with x1 = x − 5k/6 and
x2 = n(E) − 5k/6. As in the proof of Lemma 2.2 this together with (1.5) will imply
a resonance condition
j
< bk/5 ,
sin 2π θ + n(E) + ω
2
(5.4)
k+1
for j = −8k/3−1+j1 +j2 , some j1 , j2 ∈ [0, 2[ k+1
3 ]]∪[k, k +[ 3 ]], and k satisfying
bk/4 (4k/3)r(ω)
< bk/5 .
2c(ω)
(5.5)
We get a contradiction provided j 6= j(n) or j(m) for all choices of j1 , j2 . Since for
any allowed j1 , j2 we have −8k/3 − 1 ≤ j ≤ 0, the contradiction follows from
k < 3/8j(m).
• If x > n(E) > n(E) + j(n), we apply Lemma 5.5 with x1 = x − 5/6k and
x2 = n(E) − k2 . We get a resonant condition (5.4) with j = 4k/3 − 1 + j1 + j2 , with
j1 + j2 ∈ [−4k/3, 4k/3]. By the same argument we obtain a contradiction from
k < 3/8j(m) if k obeys (5.5).
• If x > n(E) + j(n) > n(E), we apply Lemma 5.5 with x1 = x − 5/6k and x2 =
n(E)−5/6k. The possible values for j in (5.4) are now j ∈ [−2k/3, 0]∪[k/3, 8k/3].
For the contradiction, we need 3j(n) < k < 3/8j(m) and
bk/4 (5k/3)r(ω)
< bk/5 .
2c(ω)
(5.6)
• Similarly, if x < n(E) + j(n) < n(E), we apply Lemma 5.5 with x1 = x − 5/6k
and x2 = n(E) − k2 . Again, the contradiction follows if 3j(n) < k < 3/8j(m) and
k obeys (5.6).
In the same way as in the proof of Lemma 2.2, we obtain, using (5.5),(5.6), that for
ln 2 20 ln(2c(ω))
+ 3/2) and satisfying 3j(n) < |x −
|x − n(E)| > max(k(), k2 (b), −
ln m2 ,
ln b
n(E)| < 3/8j(m) we have that x is (m2 , |x − n(E)|)-regular which, as before, proves
the statement of Lemma 5.3. Here, in estimating how large k should be to satisfy (5.5),
(5.6), we used that c(ω) ≤ 1 for all ω obeying (1.5).
Anderson Localization for Almost Mathieu Equation. III
11
6. Measure of the Spectrum: Proof of Theorem 1.3
Once we have established the strong version of continuity of gaps, Theorem 5.1, the
proof of Theorem 1.3 simply follows the lines of the measure-of-the-spectrum theorem
in [24, 25]. We present the argument here for the reader’s convenience.
Let G(ω, λ) be the union of the gaps in S(ω, λ), so that
|S(ω, λ)| = max S(ω, λ) − min S(ω, λ) − |G(ω, λ)|.
(6.1)
If ω = p/q is a rational, then S(ω, λ) consists of no more than q bands, and G(ω, λ) of
no more than q − 1 intervals. It is well known that for any irrational ω, there exists a
sequence of rationals pn /qn → ω such that
|ω − pn /qn | <
1
.
qn2
(6.2)
Avron, van Mouche, and Simon [3] had proven that for every λ and every sequence
{pn /qn } with pn and qn relatively prime and qn → ∞,
lim |S(pn /qn , λ)| = |4 − 2|λ||.
n→∞
(6.3)
Equation (6.3) along with (any) gap continuity implies (see [3, 25]) the upper bound
|σ(ω, λ)| ≥ |4 − 2|λ||
(6.4)
for any irrational ω. We now obtain from (i) of Corollary 5.2:
|G(ω, λ)| > |G(pn /qn , λ)| − 2D(λ)(qn − 1)λ|ω − pn /qn | ln |ω − pn /qn |.
(6.5)
By (6.1) and (ii) of Corollary 5.2, this implies:
|S(ω, λ)| < |S(pn /qn , λ)| + 2D(λ)qn λ|ω − pn /qn | ln |ω − pn /qn |.
By (6.2) and (6.3), we obtain |σ(ω, λ)| = |S(ω, λ)| ≤ |4 − 2|λ||, which together with
(6.4), completes the proof of Theorem 1.3 for λ > 29. Since S(ω, λ) is independent of
the sign of λ, it is enough to have |λ| > 29. The result for |λ| < 4/29 follows from
duality: S(ω, λ) = (λ/2)S(ω, 4/λ).
7. Appendix: Proof of Lemma 4.5
We denote
B(θ, E) =
E − λ cos θ
−1
1
0
,
Bk (θ, E) = B(θ + kπω, E).
It was shown in the proof of Proposition 1 in [16] that for any k > 0, we have
|Pk (θ, E)| ≤
2
√
3
k+1 Y
k
kBj (θ, E)k,
i=0
√
√
a b
, we use ||A|| = max( a2 + c2 , b2 + d2 ). We now want to find
c d
k() (not dependent on E!) such that for any k ≥ k(),
where for A =
12
S. Ya. Jitomirskaya, Y. Last
2
√
3
k+1 Y
k
kBj (θ, E)k ≤ j=0
k
2
√
3
k
e
k
2π
R 2π
ln kB(θ,E)kdθ
0
,
which can be rewritten as:
X
Z 2π
k
k
2
+
ln kBj (θ, E)k ≤ k ln +
ln kB(θ, E)kdθ.
ln √
2π 0
3
j=0
(7.1)
(7.2)
√
R 2π
This will prove Lemma 4.5 since 0 ln kB(θ, E)kdθ = ln( 23 M (E, λ)).
Let pn /qn be the sequence of continued fraction approximants of ω. Let n(k) be
such that qn(k) ≤ k < qn(k)+1 . We will write r for r(ω) and c for c(ω).
Proposition 7.1. For any f ∈ C[0, 2π), k > 0 we have:
X
Z 2π
k−1
k
n(k) + 1
f (θ + jπω) −
f (θ)dθ ≤ k(c−1/r
)Var(f ).
2π 0
k 1/r
j=0
(7.3)
Proof. Writing k = bn qn + bn−1 qn−1 + · · · + b1 q1 + b0 and using the Denjoy-Koksma
inequality (see, e.g., Lemma 4.1, Ch. 3 [4]), we get
k−1
!
Z 2π
n X
X
k
q
i+1
Var(f ).
f (θ + jα) −
f (θ)kdθ ≤ (b0 + · · · + bn )Var(f ) ≤
2π 0
qi
j=0
i=0
(7.4)
qir
c ,
qi+1
qi
1−1/r
qi+1
c1/r
qi+1
(cqi+1 )1/r
we have
<
=
. The right-hand side of
Since (1.5) implies qi+1 <
(7.4) can now be estimated as
!
n(k)
X
k
k
1−1/r
1−1/r
−1/r
≤ c
Var(f ) < c−1/r n(k)qn(k) +
Var(f ).
qi
+
qn(k)
qn(k)
i=1
qr
1/r
, and
Since k < qn(k)+1 ≤ n(k)
c , we have qn(k) ≥ (ck)
can continue our estimate as
k
qn(k)
≤ (c−1/r (n(k) + 1)k 1−1/r )Var(f ).
≤ c−1/r k 1−1/r . Thus we
Proposition 7.1 implies that
X
Z 2π
k
k
ln kBk (θ, E)k −
ln kB(θ, E)kdθ
2π 0
j=0
c−1/r n(k)
max
Var(ln((E − λ cos θ)2 + 1)).
−λ−2<E<λ+2
k 1/r
√
Var(ln((E − λ cos θ)2 + 1)) = A(λ). Since qn ≥ ( 2)n , n ≥ 2, we
≤ k·
Denote
max
−λ−2<E<λ+2
that n(k) ≤ lnln√k2 ,
and we can find k() such that for any k > k() we have
√
ln(2/ 3)
c−1/r n(k)
≤ ln A(λ) +
k
k 1/r
which implies (7.2). This completes the proof of Lemma 4.5.
have
Anderson Localization for Almost Mathieu Equation. III
13
Acknowledgement. S.J. would like to thank J. Avron for the hospitality of the ITP at the Technion, and
B. Simon for the hospitality of Caltech, where parts of this work were done. We are also grateful to Ya. Sinai
and to B. Simon for enlightening conversations.
References
1. Aubry, G., Andre, G.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann.
Israel Phys. Soc. 3, 133–140 (1980)
2. Aizenman, M.: Localization at weak disorder: Some elementary bounds. Rev. Math. Phys. 6, 1163–1182
(1994)
3. Avron, J., van Mouche, P., Simon, B.: On the measure of the spectrum for the almost Mathieu operator.
Commun. Math. Phys. 132, 103–118 (1990)
4. Cornfeld, I., Fomin, S., Sinai, Ya.: Ergodic Theory. New York: Springer, 1982
5. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators. Berlin, Heidelberg, New
York: Springer 1987
6. del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: What is localization? Phys. Rev. Lett. 75, 117–119
(1995)
7. del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: Operators with singular continuous spectrum, IV.
Hausdorff dimensions, rank-one perturbations, and localization. J. d’Analyse Math. 69, 153–200 (1996)
8. del Rio, R., Jitomirskaya, S., Makarov, N., Simon, B.: Singular continuous spectrum is generic. Bull.
AMS 31, 208–212 (1994)
9. del Rio, R., Makarov, N., Simon, B.: Operators with singular continuous spectrum, II. Rank one operators.
Commun. Math. Phys. 165, 59–67 (1994)
10. Delyon, F., Kunz, H., Souillard, B.: One dimensional wave equations in disordered media. J. Phys. A
16, 25–42 (1983)
11. Figotin, A., Pastur, L.: An exactly solvable model of a multidimensional incommensurate structure.
Commun. Math. Phys. 95, 401–425 (1984)
12. Fishman, S., Grempel, D., Prange, R.: Localization in a d-dimensional incommensurate structure. Phys.
Rev. B 29, 4272–4276 (1984)
13. Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one dimensional quasi-periodic
Schrödinger operators. Commun. Math. Phys. 132, 5–25 (1990)
14. Gordon, A. Y.: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys. 164, 489–505 (1994)
15. Gordon, A., Jitomirskaya, S., Last, Y., Simon B.: Duality and singular continuous spectrum in the almost
Mathieu equation. Acta Math. 178, 169–183 (1997)
16. Jitomirskaya, S.: Anderson localization for the almost Mathieu equation; A nonperturbative proof.
Commun. Math. Phys. 165, 49–58 (1994)
17. Jitomirskaya, S., Simon, B.: Operators with singular continuous spectrum, III. Almost periodic
Schrödinger operators. Commun. Math. Phys. 165, 201–205 (1994)
18. Jitomirskaya, S.: Almost everything about the almost Mathieu operator II. Proceedings of XI International
Congress of Mathematical Physics, Paris 1994, Int. Press, 1995, pp. 373–382
19. Jitomirskaya, S., Last, Y.: Dimensional Hausdorff properties of singular continuous spectra. Phys. Rev.
Lett. 76, 1765–1769 (1996)
20. Jitomirskaya, S.: Continuous spectrum and uniform localization for ergodic Schrödinger operators. J.
Funct. Anal. 145, 312–322 (1997)
21. Jitomirskaya, S.: Almost Mathieu equations with weakly Liouville frequencies: Second critical constants.
In preparation
22. Jitomirskaya, S., Last, Y.: Power-Law subordinacy and singular spectra, II. Line operators. In preparation
23. Kunz, H., Souillard, B.: Sur le spectre des operateurs aux differences finies aleatoires. Commun. Math.
Phys. 78, 201–246 (1980)
24. Last, Y.: A relation between absolutely continuous spectrum of ergodic Jacobi matrices and the spectra
of periodic approximants. Commun. Math. Phys. 151, 183–192 (1993)
25. Last, Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys. 164, 421–432
(1994)
26. Last, Y.: Almost everything about the almost Mathieu operator I. Proceedings of XI International
Congress of Mathematical Physics, Paris 1994, Int. Press, 1995, pp. 366–372
14
S. Ya. Jitomirskaya, Y. Last
27. Last Y.: Quantum dynamics and decompositions of singular continuous spectra. J. Funct. Anal. 142,
406–445 (1996)
28. Martinelli, F., Scoppola, E.: Introduction to the mathematical theory of Anderson localization. Rivista
del Nuovo Cimento 10, N. 10 (1987)
29. Prange, R., Grempel, D., Fishman, S.: A solvable model of quantum motion in an incommensurate
potential. Phys. Rev. B 29, 6500–6512 (1984)
30. Simon, B.: Almost periodic Schrödinger operators, IV. The Maryland model. Ann. Phys. 159, 157–183
(1985)
31. Simon, B.: Absence of ballistic motion. Commun. Math. Phys. 134, 209–212 (1990)
32. Sinai, Ya.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Stat. Phys. 46, 861–909 (1987)
33. Thouless, D.J.: Bandwidth for a quasiperiodic tight binding model. Phys. Rev. B 28, 4272–4276 (1983)
Communicated by B. Simon
Commun. Math. Phys. 195, 15 – 28 (1998)
Communications in
Mathematical
Physics
© Springer-Verlag 1998
On Causal Compatibility of Quantum Field Theories and
Space-Times
Michael Keyl?
Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.
E-mail: [email protected]
Received: 30 October 1996 / Accepted: 7 November 1997
Abstract: In [Key96] the notion of causal compatibility is introduced as a method
to determine the conformal structure of space-time uniquely by the net of observable
algebras of a quantum field theory. In this work some new aspects of causal compatibility
are discussed. In particular it is shown that for a given poset of algebras there exists up to
conformal equivalence at most one Lorentzian manifold which is causally compatible.
This enhances previous results, where special properties of the index set of the net are
used.
1. Introduction
In the framework of algebraic quantum field theory it is interesting to ask, whether
the local observable algebras A(O), O ⊂ M of a quantum field theory determine
uniquely the space-time (M, g) in which they are located. Previous results considering
this question can be found in [Ban88, Ban94, Key93, Key96, Wol94] and the references
therein. The most satisfactory results concern the causal, and hence the conformal,
structure of space-time [Key93, Key96, Wol94], since there are several theorems stating
that the local algebras fix the conformal structure of the underlying space-time uniquely.
The easiest theorem of this kind assumes that the algebras A(O1 ), A(O2 ) commute
iff the space-time regions O1 , O2 are space-like separated, i.e. there is no causal curve
from O1 to O2 [BW92, Key93]. However this condition excludes physically significant
models, like free, massless fields on Minkowski space, where the algebras do not commute iff there is a null geodesic from O1 to O2 . To overcome this problem Wollenberg
uses algebras A(c) associated to curves c : (a, b) → M . The key observation is that at
least for the free scalar field those algebras are abelian if c is space-like and non-abelian,
if c is time-like (see [Wol94] for details).
? Present address: Inst. für Math. Phys., TU-Braunschweig, Mendelssohnstraße 3, D-38106 Braunschweig,
Germany.
16
M. Keyl
A different method to generalize the results of [Key93], called causal compatibility
(CC-compatibility in [Key96]) is proposed in [Key96]. The basic idea is, not to compare
the binary relation ⊥g , given by space-like separation of two regions, directly with the
relation ⊥max , which is defined by O1 ⊥max O2 : ⇐⇒ [A(O1 ), A(O2 )] = {0}, but to use
the causal closures operators associated to ⊥g and ⊥max (see e.g. [BW92] or [Key96]
for a definition of causal closures). It turns out that we get a uniqueness result similar to
the one given in [Key93] which is applicable to a wider class of models.
Another aspect of space-time structure which can be treated quite satisfactorily in
the outlined framework is the topology. However our initial question is somewhat ill
posed in this context. Since open, relatively compact subsets of space-time are used to
index the local algebras, the topology of space-time is needed before the algebras A(O)
can be defined. One way to circumvent this problem is the usage of order strucutures.
In [Key96] elements λ of an abstract partially ordered set (I, ≺) are used instead of
open relatively compact subsets of a manifold M to index the local algebras. In this
way we get an “abstract” net (A(λ))λ∈I which is defined without explicit references to
space-time. To derive space-time topology from this abstract net we have to search for
(or construct) a topological space M such that the elements of I can be realized by open,
relatively compact subsets of M and the relation ≺ becomes the usual inclusion relation
⊂. In other words we need an order isomorphism λ from I to the set B(M ) of all open,
relatively compact subsets of M . If this ansatz is combined with causal compatibility,
outlined in the last paragraph, it can be shown that up to conformal equivalence only
one space-time exists, such that the derived net B(M ) 3 O 7→ A(λ−1 (O)) on (M, g) is
causally compatible with this space-time (which implies that in addition to the causal
and topological structure the differentiable structure of space-time is determined by the
net B(M ) 3 O 7→ A(λ−1 (O))).
However this result depends highly on the correct choice of the index set (I, ≺). If we
assume that the order isomorphism λ from the last paragraph do not map I to B(M ) but
to the set of all causally closed sets in B(M ) or to the set of all double cones, the theorem
discussed in the last paragraph is, as it stands, not applicable. This is a substantial drawback, since there is in general much freedom to change the index set without changing
the quantum theory described by the corresponding net. It is therefore more appropriate
to consider, as it is done by Bannier in [Ban94], the set LB := {A(O) | O ∈ B} of all
local algebras from the net as a partially ordered set for itself and to use this structure
to determine space-time topology. The basic idea of this paper is to start with such a set
LB which consists of algebras and which is partially ordered by inclusion and to search
for a space-time (M, g) and a map B(M ) 3 O 7→ A(O) ∈ LB such that the corresponding net on (M, g) is causally compatible with this space-time. One main result is
that the uniqueness result of [Key96], just mentioned generalizes to this case (see Sect.
4), i.e. (M, g) is uniquely determined up to conformal equivalence. The second main
topic (Sect. 5) represents a first step towards generalized space-time concepts. Since
the conformal structure of a space-time (M, g) is uniquely determined by the set LB it
should be possible to reformulate statements about causality completely in terms of LB
without any reference to (M, g). Following this idea we will characterize “space-like
separation” in an algebraic way which makes sense even for sets of algebras LB to which
a “classical” space-time (M, g) can not be associated in the described way.
Causal Compatibility of Quantum Field Theories and Space-Times
17
2. Local Algebras
Before we start with this analysis let us recall some basic facts about local C*-algebras1
which we will need in the following. Hence we will consider a Lorentzian manifold2
(M, g) and a family of C*-algebras (A(O))O∈B indexed by the elements O ∈ B of a
base B for the topology of M . The latter we will denote with T (M ) in the following. The
selfadjoint elements of the algebra A(O) should describe physically local observables
which are measurable in the region O. Hence it is natural to assume that isotony
O1 ⊂ O2 ⇒ A(O1 ) ⊂ A(O2 )
(1)
holds, i.e. (A(O))O∈B is a net of C*-algebras. This implies especially that we can define
the algebra of quasi-local observables A by
[
A(O)
(2)
A :=
O∈B(M )
and more generally for each Q ∈ T (M ) the algebra


[
A(O) with T (Q) := {O |O ∈ T (M ), O ⊂ Q}.
Ã(Q) = C ∗ 
(3)
O∈B∩T (Q)
Since O ∈ T (O) ∩ B for each O ∈ B we get A(O) = Ã(O). In other words the family
(Ã(Q))Q∈T (M ) of C*-algebras extends the family (A(O))O∈B , therefore we will drop
the tilde in the following. Isotony holds as well for the extended family which defines
therefore an extended net which we will call the canonical extension of (A(O))O∈B .
If we consider now another base B1 of T (M ) we get another net (A(Õ))Õ∈B1 by
restricting the canonical extension (A(Q))Q∈T (M ) to the new base B1 . However if we
consider the canonical extension (A1 (Q))Q∈T (M ) of this new net (defined as in (3) with
B replaced by B1 ) we get A1 (O) 6= A(O) except the original net (A(O))O∈B is additive,
i.e.
!
!
[
[
∗
O =C
A(O) ,
(4)
A
O∈N
S
O∈N
holds for each subset N ⊂ B with O∈N O ∈ B. In this case the canonical extension
of (A(O))O∈B is additive as well and we get A(Q) = A1 (O) for all Q ∈ T (M ). In other
words an additive net does not depend on the set B used to index the algebras, as long
as B is a base for T (M ). From the physical point of view it is therefore convenient to
choose the set
B(M ) := {O ⊂ M | O open, O compact},
(5)
or an appropriate subset of B(M ) as the index set, because its elements represent bounded
space-time regions. Hence the observables in A(O) with O ∈ B(M ) are measurable in
a bounded region.
1 At many places in this paper associative algebras are sufficient. However this generalization does not
change the statements and the proofs substantially. Since the A(O) should describe algebras of local observables C*-algebras are the physically most reasonable choice.
2 We will assume throughout this paper that (M, g) is smooth, Hausdorff, second countable and timeoriented and that the strong causality condition holds on it.
18
M. Keyl
A second significant axiom, which links the structure of (A(O))O∈B to the causal
structure of (M, g) is locality:
O1 ⊥g O2 ⇒ [A(O1 ), A(O2 )] = {0},
(6)
where ⊥g denotes space-like separation:
O1 ⊥g O2 : ⇐⇒ O2 ⊂ M \ J + (O1 ) ∪ J − (O1 ) .
(7)
A net (A(O))O∈B satisfying locality is called a causal net. For a causal net locality holds
for the canonical extension and each subnet we can define by specifying another base
B1 .
Let us consider some properties of the binary relation ⊥g defined in (7). It is easy to
check that it satisfies the conditions:
O1 ⊥g O2 ⇐⇒ O2 ⊥g O1
i.e. ⊥g is symmetric,
(8)
O1 ⊂ O2 ∧ O2 ⊥g O3 ⇒ O1 ⊥g O3 ,
(9)
and
I ⊂ T (M ) ∧ O1 ⊥g O ∀O ∈ I ⇒ O1 ⊥g
[
!
O .
(10)
O∈I
Following [BW92] Def. 7.1.1 we call each relation ⊥ ⊂ T (M ) × T (M ) satisfying
(8)–(10) a “causal disjointness relation”.
It is quite easy to see that the relation ⊥g contains all information about the conformal
structure of (M, g) (see [Key96]). More precisely if ⊥g1 = ⊥g2 holds, then the two
metrics g1 , g2 are conformally equivalent. Hence, if we want to determine the conformal
equivalence class of g by algebraic properties of the net we have to describe ⊥g by
relations between algebras. The easiest way to do this is to define the “maximal3 ”
relation ⊥max (see [BW92], Prop. 7.2.7):
O1 ⊥max O2 : ⇐⇒ [A(O1 ), A(O2 )] = {0}.
(11)
The condition ⊥max = ⊥g fixes now the conformal structure uniquely (and the differentiable structure as well; see [Key93]). The drawback of this idea is that it excludes
physically reasonable cases, e.g. free massless fields on Minkowski space.
To get an improved ansatz, we have to introduce the notions of causal complement
and causal closure first. The causal complement of an arbitrary open subset Q ⊂ M
with respect to a causal disjointness relation ⊥ is (see [BW92], Def. 7.1.6):
[
O with Qc := {O ∈ T (M ) | O⊥Q}.
(12)
Q⊥ :=
O∈Qc
Note that we can replace in this definition the set Qc by {O ∈ B | O⊥O1 ∀O1 ∈
B with O1 ⊂ Q} if B is a base for T (M ). The causal closure of an O ∈ T (M ) is
simply given by O⊥⊥ and O ∈ T (M ) is called causally closed if O = O⊥⊥ . Instead
of comparing ⊥g and ⊥max directly the idea is now to compare the closure operators
3 ⊥
max is the maximal element of the set of all causal disjointness relations with which a given net of
C*-algebras is a causal net (see [Wol94]).
Causal Compatibility of Quantum Field Theories and Space-Times
19
associated to ⊥g and ⊥max . Hence we call a net (A(O))O∈B causally compatible with
the space-time (M, g), if (A(O))O∈B is a causal net4 and if
O⊥g ⊥g = O⊥max ⊥max
∀O ∈ B
(13)
holds. Note that causal compatibility of the net (A(O))O∈B do not imply causal compatibility of the canonical extension even if additivity holds (in general causal compatibility
of (A(O))O∈B only implies Q⊥max ⊥max ⊂ Q⊥g ⊥g for a general open set Q ⊂ M ).
Therefore it is necessary frequently to assume that not only (A(O))O∈B is causally
compatible with (M, g) but as well its canonical extension (see Def. 3.6). However this
stronger condition is still more general than ⊥max = ⊥g .
A particular example for a causally compatible net arises from the free scalar field
on a globally hyperbolic space-time (see the [Dim80] for details). Due to Cor. 14.4 of
[Key96] the following proposition holds:
Proposition 2.1. Consider an analytic5 globally hyperbolic space-time (M, g) and the
net (A(O))O∈B(M ) constructed from the Klein-Gordon equation according to [Dim80].
For each convex open set U ⊂ M we can construct now the net (A(O))O∈B(U ) , which
is additive and causally compatible with the space-time (U, g|U ).
Note that it is not clear whether the whole net is causally compatible with (M, g).
Therefore causal compatibility might be a local property. This would imply that the
structures we are going to develop reflect only local properties of space-time.
3. Posets of C*-algebras
As we have indicated already in the introduction the basic idea of this paper is to consider
instead of a net (A(O))O∈B only the set
L(A(O), O ∈ B) := {A(O) | O ∈ B}
(14)
of local algebras and to ask whether this structure is sufficient to determine the spacetime (up to conformal equivalence). The purpose of this section is to provide some
technical tools which are necessary to make this rough idea more precise. Note that our
approach is closely related (and partly inspired) by a previous work of Bannier [Ban94].
We will discuss at the end of this section the differences and similarities between both
approaches.
The basic objects we have to deal with are sets of C*-algebras or more precisely sets
of C*-subalgebras of a C*-algebra as in the following definition:
Definition 3.1. A set LB is called a poset of C*-algebras if the elements of LB are
C*-subalgebras of a C*-algebra6 I and if I is the C*-algebra generated by the union
of all A ∈ LB . We will call I the maximal algebra of LB . (Note that we have I 6∈ LB
in general.)
4 Condition (13) does not imply locality. In [Key96] a not necessarily causal net satisfying this equation is
therefore called “CC-compatible” (causally closed compatible).
5 It is very likely that analyticity is only a restriction of the methods used in the proof and not of the
statement itself.
6 To distinguish local algebras A(O) and the quasilocal algebra A on the one hand from elements of L
B
which are not a priori related to space-time regions on the other hand, we will use caligraphic letters in the
first case and gothic ones in the latter.
20
M. Keyl
In terms of posets of algebras we can look at a net (A(O))O∈B on a space-time (M, g)
as a monotone map A : B → LB from a base B of T (M ) onto a poset of C*-algebras.
Hence if we want to consider the canonical extension of the net (A(O))O∈B we need an
extension L of the poset LB which contains for each subset N ⊂ L the supremum
_
N := C
[
∗
!
A .
(15)
A∈N
Therefore we define
Definition 3.2. A poset L of C*-algebras is called a semi-lattice of C*-algebras,
W
(i) if for each subset N ⊂ L the
W supremum L given in (15) is an element of L. (Hence
the maximal element I = L is in L.)
C*-algebras there is a smallest semi-lattice L containing LB
(ii) For each posetW
LB of W
and satisfying L = LB . It will be called the semi-lattice generated by LB .
If we consider now a net B 3 O 7→ A(O) ∈ LB ⊂ L we can interpret its canonical
extension as the map
T (M ) 3 Q 7→ A(Q) :=
_
{A(O) | O ∈ B, O ⊂ Q} ∈ L.
If
S the net is additive and if we consider T (M ) as a complete semi-lattice (with
Q∈N Q) this map is a homomorphism of semi-lattices, i.e. we have
A
[
Q∈N
!
Q
=
_
A(Q)
(16)
W
N =
(17)
Q∈N
for each subset N ⊂ T (M ). Furthermore it is the unique homomorphism of semi-lattices
which extends the original map from B to LB .
Although we have not introduced new structure, but reinterpreted definitions already
given in the last section, we have changed our point of view significantly. We do not treat
the space-time (M, g) as something given, but as something which has to be associated
to a given set of local algebras. Hence we will record the ideas just introduced in the
following definition.
Definition 3.3. Consider a poset LB of C*-algebras and the semi-lattice L it generates.
A realization of LB on a space-time (M, g) is a homomorphism T (M ) 3 Q 7→ A(Q) ∈
L of semi-lattices, which maps a base B ⊂ B(M ) of T (M ) to LB .
In terms of this definition our aim is to find conditions which rule out up to conformal
equivalence all but one realization of a given poset LB . Before we consider this question
let us specify what “conformal equivalence” means in this context.
Definition 3.4. Consider a poset LB of C*-algebras. Two realizations A1 : T (M1 ) → L
and A2 : T (M2 ) → L of LB on the space-times (M1 , g1 ) and (M2 , g2 ) are called
conformally equivalent if there is a (smooth) conformal transformation f : (M1 , g1 ) →
(M2 , g2 ) with A1 (Q) = A2 (f (Q)) for all Q ∈ T (M1 ).