Quantum mean field asymptotics and multiscale analysis

0.58 MB·

by  Zied Ammari

Quantum mean-field asymptotics and multiscale analysis
Z. Ammari, S. Breteaux and F. Nier
∗ † ‡
7 January 24, 2017
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0
2

n
a Abstract
J
We study, via multiscale analysis, some defect of compactness phenomena which occur in
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bosonic and fermionic quantum mean-field problems. The approach relies on a combination
2
of mean-field asymptotics and second microlocalized semiclassical measures. The phase space

] geometric description is illustrated by various examples.
h
p
-
h Keywords and phrases: Semiclassical and multiscale measures, reduced density matrices, second
t
a quantization, microlocal analysis.
m
2010 Mathematics subject classification: 81S30, 81Q20, 81V70, 81S05.
[

1
1 Introduction
v
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2 Motivations: Over thepasttwodecades, itbecomes clear thatmicrolocal analysis provides inter-
4
esting mathematical tools for the study of quantum field theories and quantum many-body theory,
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0 seefor instance [AmNi1,Rad]. In particular, in the analysis of general bosonicmean-field problems,
. as done in [AmNi1, AmNi2, AmNi3, AmNi4], the following defect of compactness problem arises.
1
0 If γ(p) denotes the p-particles reduced density matrix, one may have
ε
7
1
limTr[γ(p)˜b]= Tr[γ(p)˜b] (1)
: ε 0
v ε 0
→
i
X
for any p-particle compact observable ˜b, while it is not true for a general bounded ˜b, e.g.
r
a
limTr[γ(p)]> Tr[γ(p)].
ε 0
ε 0
→
In the fermionic case, it is even worse, because mean-field asymptotics cannot be described in terms
(p)
offinitelymanyquantumstates andtheright-handsideof (1)isusually0whilelim Tr[γ ] > 0.
ε 0 ε
→
From the analysis of finite dimensional partial differential equations, it is known that such defect of
compactness can be localized geometrically with accurate quantitative information by introducing
∗[e-mail], IRMAR, Universit´e de Rennes I, UMR-CNRS 6625, campus de Beaulieu, 35042
RennesCedex, France.
†[e-mail],BCAM-BasqueCenterforAppliedMathematics,AlamedadeMazarredo14,
48009 Bilbao, Spain.
‡[e-mail], LAGA, UMR-CNRS 9345, Universit´e de Paris 13, av. J.B. Cl´ement, 93430
Villetaneuse, France.
1
scales andsmallparameterswithinsemiclassical techniques (e.g. [Ger,GMMP]). We arethusled to
introducetwo small parameters ε > 0 for the mean-field asymptotics and h > 0 for the semiclassical
quantization of finite dimensional p-particles phase space. The small parameter ε stands for 1/n,
where n is the typical number of particles, while h is the rescaled Planck constant measuring
→ ∞
the proximity of quantum mechanics to classical mechanics. The combined analysis of this article
is concerned with the general situation when ε = ε(h) with lim ε(h) = 0. In order to keep
h 0
→
track of the information at the quantum level especially in the bosonic case we also introduce finite
dimensional multiscale observables in spirit of [Bon, FeGe, Fer, Nie].
Framework: The one particle space Z is a separable complex Hilbert space endowed with the
scalar product , (anti-linear in the left-hand side). For a Hilbert space h the set of bounded
h i
operator is denoted by (h), while the Schatten class is denoted by p(h), 1 p , the case
L L ≤ ≤ ∞
p = corresponding to the space of compact operators. Let Γ (Z) be the bosonic (+ sign) or
fermi∞onic ( sign) Fock-space built on the separable Hilbert spac±e Z :
−
⊥
Γ (Z)= nZ n,
⊗
± S±
n N
M∈
where tensor products and direct sum are Hilbert completed. The operator n is the orthogonal
projection given by S±
1
n(f f ) = s (σ)f f ,
1 n σ(1) σ(n)
S± ⊗···⊗ n! ± ⊗···⊗
σX∈Sn
where s (σ) equals 1 while s (σ) denotes the signature of the permutation σ and S is the n-
+ n
−
symmetric group.
The dense set of many-body states with a finite number of particles is
,alg
⊥
Γfin(Z) = nZ n,
⊗
± S±
n N
M∈
where the ,alg superscript stands for the algebraic orthogonal direct sum.
⊥
We shall use the notations [A,B] = [A,B] = ad B = AB BA for the commutator of two oper-
+ A
−
ators and the notation [A,B] = AB +BA for the anticommutator.
−
One way to investigate the mean-field asymptotics relies on parameter-dependent CCR (resp.
CAR). The small parameter ε > 0 has to be thought of as the inverse of the typical number
of particles and the Canonical Commutation (resp. Anticommutation) Relations are given by
[a (g),a (f)] = [a (g),a (f)] = 0 , [a (g),a (f)] = ε g, f .
∗ ∗ ∗
± ± ± ± ± ± ± ± ± h i
Let (̺ ) be a family of normal states (i.e., non-negative and normalized trace-class operators)
ε ε>0
on the Fock space Γ (Z), depending on ε > 0, we want to investigate the asymptotic behaviour
±
of reduced density matrices, defined below, as ε 0, by possibly introducing another scale h > 0
→
on the p-particles phase-space, with ε = ε(h) and lim ε(h) = 0.
h 0
→
2
Outline: In Section 2, we recall how Wick observables are used to define the reduced density
(p)
matrices γ . Notethatitismuchmoreconvenient here, inthegeneralgrandcanonical framework,
ε
to work with non normalized reduced density matrices. Some symmetrization formulas are also
recalled in this section. In Section 3, we present the geometry of the classical p-particles phase
space and introduce the formalism of double scale semiclassical measures, after [Fer, FeGe]. In
Section 4, we combine the mean-field asymptotics with semiclassical analysis, the two parameters
ε and h being related through ε = ε(h) with lim ε(h) = 0. Instead of studying the collection of
h 0
(p) →
non normalized reduced density matrices (γε(h))p N, it is more convenient to associate generating
∈
functionsz Tr ̺ ezdΓ±(aQ,h) ,andtouseholomorphyargumentspresentedthere. InSection5,
ε(h)
7→
some classical examples with various asymptotics illustrate the general framework: coherent states
(cid:2) (cid:3)
in the bosonic setting; simple Gibbs states in the fermionic case; more involved Gibbs states in
the bosonic case, which make explicit the separation of condensate and non condensate phases for
rather general non interacting steady Bose gases. The Appendices collect or revisit known things
about multiscale semiclassical measures, the (PI)-condition of bosonic mean-field problems, Wick
composition formulas, and traces of non self-adjoint second quantized contractions.
2 Wick observables and reduced density matrices
2.1 Wick observables
Notation: For n N, the operator n is an orthogonal projection in Z n so that ( n) = n.
⊗ ∗
However, we consid∈er n as a boundeSd±operator from Z n onto nZ n and its adjoiSn±t, denoSt±ed
⊗ ⊗
by n,∗ : nZ⊗n ZS⊗±n, is nothing but the natural embedding.S±
LetS˜b± S(±pZ p→; qZ q), the Wick quantization of ˜b is the operator on Γfin(Z) defined by
⊗ ⊗
∈ L S± S± ±
(n+p)!(n+q)!
˜bWick S±n+pZ⊗(n+p) = εp+2q p n! S±n+q(˜b⊗IdZ⊗n)S±n+p,∗.
(cid:12)
In the bosonic case,(cid:12)an element ˜b ( pZ p; qZ q) is determined by the symbol Z z
∈ L S+ ⊗ S+ ⊗ ∋ 7→
b(z) = z q, ˜bz p owing to the relation ˜b = 1 ∂q∂pb. Observe that b(z) admits higher Fr´echet
h ⊗ ⊗ i q!p! z z
derivatives with the natural identification of ∂kb(z) as a continuous form on kZ k and ∂kb(z) as
z S+ ⊗ z¯
a vector in kZ k. We shall use also the notation bWick =˜bWick.
S+ ⊗
Examples:
a) The annihilation operator a (f), f Z , is the Wick quantization of ˜b = f : Z 1 = Z
⊗
ϕ f , ϕ Z 0 = C. ± ∈ h | ∋
⊗
7→ h i ∈
b) The creation operator a (f), f Z , is the Wick quantization of ˜b = f : Z 0 = C λ
∗ ⊗
λf Z 1 = Z . ± ∈ | i ∋ 7→
⊗
∈
c) For ˜b (Z) its Wick quantization ˜bWick is nothing but
∈L
dΓ±(˜b) S±nZ⊗n = ε ˜b⊗IdZ ⊗···⊗IdZ +···+IdZ ⊗···⊗IdZ ⊗˜b .
(cid:12) h i
(cid:12)
3
When ˜b is self-adjoint one has
dΓ (˜b) = i∂ e itdΓ±(˜b) = i∂ Γ (e iεt˜b) ,
± t − t=0 t ± − t=0
while for a contraction C (Z;Z), (cid:12)(cid:12) (cid:12)(cid:12)
∈ L
Γ (C) = C C.
± S±nZ⊗n ⊗···⊗
(cid:12)
A particular case is ˜b = IdZ associated(cid:12)with the scaled number operator (N ,ε=1 stands for the
±
usual ε-independent number operator):
˜bWick = dΓ (IdZ)= N = εN ,ε=1.
± ± ±
From the definition of the Wick quantization one easily checks the following properties.
Proposition 2.1. For ˜b ( pZ p; qZ q):
⊗ ⊗
∈ L S± S±
˜bWick ∗ = [˜b ]Wick.
∗
•
h i
The operator (1+N ) m/2˜bWick(1+N ) m′/2 extends to a bounded operator on Γ (Z) as
− −
• ± ± ±
soon as m+m p+q with
′
≥
k(1+N±)−m/2˜bWick(1+N±)−m′/2kL(Γ±(Z)) ≤ Cm,m′k˜bkL(S±pZ;S±qZ), (2)
with C independent of ˜b and of ε (0,ε ).
m,m′ 0
∈
(˜b 0) (˜bWick 0), while this makes sense only for q = p.
• ≥ ⇔ ≥
TheWick quantized operators generally areunboundedoperators on Γ (Z)(e.g. N )butthey
are well defined on the dense set Γfin(Z) which is preserved by their act±ion. Hence ˜bW±ick ˜bWick
makes sense at least on Γfin(Z) an±d the following composition law holds true. 1 ◦ 2
±
Proposition 2.2 (Composition of Wick operators). Let ˜bj ( pjZ⊗pj; qjZ⊗qj), j = 1,2, then
∈ L S± S±
min{p1,q2} εk
˜bWick ˜bWick = ( 1)(p1 k)(p2+q2) (˜b ♯k˜b )Wick, (3)
1 ◦ 2 ± − k! 1 2
k=0
X
where ˜b1♯k˜b2 := (p1p−1!k)!(q2q−2!k)! S±q1+q2−k(˜b1⊗Id⊗q2−k)(Id⊗p1−k ⊗˜b2)S±p1+p2−k,∗.
For reader’s convenience, the proof of Prop. 2.2 is provided in Appendix C.
In the bosonic case the symbols b(z) = z q, ˜bz p are convenient for writing the composition
⊗ ⊗
of Wick quantized operators. If b ♯Wickb dehnotes the siymbol of ˜bWick ˜bWick, the composition law
1 2 1 ◦ 2
is summarized below (see [AmNi1]).
Proposition 2.3 (Composition of Wick symbols in the bosonic case).
min{p1,q2}εk
b1♯Wickb2(z) = eε∂z1·∂z2b1(z1)b2(z2) z1=z2=z = k!∂zkb1(z)·∂zkb2(z).
k=0
(cid:12) X
(cid:12)
4
The commutator of Wick operators in the bosonic case:
Wick
max{min{p1,q2},min{p2,q1}}εk
bWick, bWick = b ,b (k) ,
1 2  k! { 1 2} 
h i Xk=1
 
where the k-th order Poisson bracket is given by b ,b (k) = ∂kb (z) ∂kb (z) ∂kb (z) ∂kb (z).
{ 1 2} z 1 · z 2 − z 2 · z 1
Proposition 2.4. Let p, m, m N, such that m+m 2p 2. Then, there exist coefficients
′ ′
C 0 such that, for any ˜b ∈ ( ; ), ≥ −
j1,...,jk ≥ ∈ L Z Z
p 1
− Wick
dΓ±(˜b)p−(˜b⊗p)Wick = εp−k Cj1,...,jk S±k ˜bj1 ⊗···⊗˜bjkS±k,∗ (4)
Xk=1 0j1≤+j·1X·≤·+··j·k≤=jkp (cid:16) (cid:17)
and the estimate
(1+N ) m/2 dΓ (˜b)p (˜b p)Wick (1+N ) m′/2 εB ˜b p
k ± − ± − ⊗ ± − kL(Γ±(Z)) ≤ pk kL(Z)
(cid:0) (cid:1)
holds in both the bosonic case and the fermionic case, with B the p-th Bell number.
p
Remark 2.5. The p-th Bell number B can be defined as the number of partitions of a set with p
p
p
elements and satisfies B < 0.792p (see [BeTa]), and hence it grows much slower than p!.
p ln(p+1)
(cid:16) (cid:17)
Proof. We first prove Formula (4) by induction on p N .
∗
For p = 1, Formula (4) holds because dΓ (˜b)= (˜b∈)Wick.
We then set r (˜b) := dΓ (˜b)p (˜b p)Wick±. Assuming the result holds for some p N , one can
p ⊗ ∗
± − ∈
compute
dΓ (˜b)p+1 = (˜b p)Wick(˜b)Wick +r (˜b)Wick(˜b)Wick
⊗ p
±
using the composition formula (3) for
(˜b⊗p)Wick(˜b)Wick = (˜b⊗p+1)Wick +pε p ˜b⊗p−1 ˜b2 p,∗ Wick
S± ⊗ S±
(cid:0) (cid:1)
and for
εp−k k ˜bj1 ˜bjk k,∗ Wick(˜b)Wick
S± ⊗···⊗ S±
(cid:0) = εp(cid:1)+1−(k+1) k+1˜b ˜bj1 ˜bjk k+1,∗ Wick
S± ⊗ ⊗···⊗ S±
Wick
+kεp+1−k(cid:0) k ˜bj1 ˜bjk k,∗ k ˜b(cid:1) Id⊗j1+···+jk−1 k,∗ ,
S± ⊗···⊗ S± S± ⊗ Z S±
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
which yields the expected form for r (˜b), and achieves the induction.
p+1
We then remark that the sum of coefficients of order k,
S (p,k) = C ,
2 j1,...,jk
0j1≤+j·1X·≤·+··j·k≤=jkp
5
satisfiestherecurrencerelationS (p,k) = kS (p 1,k)+S (p 1,k 1), withS (p,1) = 1 = S (1,k)
2 2 2 2 2
− − −
for all p,k N , where the S (p,k) are the Stirling numbers of the second kind. Observe that, for
∗ 2
∈
M/2 k, and for any c˜ ( k k),
⊗
≥ ∈ L S±Z
c˜Wick(1+N ) M/2 c˜ .
k ± − kL(Γ±(Z)) ≤ k kL(S±kZ⊗k;S±kZ⊗k)
We thus get,
p 1
(1+N ) m/2 dΓ (˜b)p (˜b p)Wick (1+N ) m′/2 − εp kS (p,k) ˜b p
k ± − ± − ⊗ ± − kL(Γ±(Z)) ≤ − 2 k kL(Z)
(cid:16) (cid:17) Xk=1
and the estimate then follows from pk−=11εp−kS2(p,k) ≤ ε pk=1S2(p,k) = εBp with Bp the p-th
Bell number.
P P
2.2 Reduced density matrices
Reduced density matrices emerge naturally in the study of correlation functions of quantum gases
and in particular in the quantum mean-field theory they are the main quantities to be analysed.
We shall work with non normalized reduced density matrices, which are easier to handle. Going
back to the more natural reduced density matrices with trace equal to 1, requires attention when
normalizing and taking the limits.
Definition 2.6. Let ̺ 1(Γ (Z)) (ε > 0 is fixed here) be such that ̺ 0, Tr[̺ ] = 1 and
ε ε ε
Tr ̺ ecN± < for som∈eLc >±0. The non normalized reduced density ma≥trix of order p N,
ε
∞ ∈
γ(p) 1( pZ p), is defined by duality according to
ε (cid:0) (cid:1) ⊗
∈ L S±
˜b ( pZ p; pZ p), Tr γ(p)˜b = Tr ̺ ˜bWick .
∀ ∈ L S± ⊗ S± ⊗ ε ε
h i h i
The definition makes sense owing to the number estimate (2) and to (1+N )ke cN± (Γ (Z)).
−
When Tr γ(p) = 0, the normalized density matrix γ¯(p) is defined by γ¯(p) =± γε(p) ,∈thLat i±s
ε 6 ε ε Trhγε(p)i
h i
Tr ̺ ˜bWick
ε
˜b ( pZ p), Tr γ¯(p)˜b =
∀ ∈ L S± ⊗ h ε i Tr ̺ε(IhdS±pZ⊗p)iWick
h i
Tr ̺ ˜bWick
ε
= .
Tr[̺ N (N hε)...(Ni ε(p 1))]
ε
± ± − ±− −
Thesenormalizedreduceddensitymatricesγ¯(p)arecommonlyused,especiallywhen̺ 1( Z n),
ε ε ⊗
∈L S±
with nε 1 (see [BGM, BEGMY, BPS, KnPi, LNR]), for the following reason: When ̺
ε
∼ ∈
1( nZ n) 1(Z n) lies in the n-particles sector (with nε 1 for the mean-field regime)
⊗ ⊗
L S± ⊂ L →
it is simply given by the partial trace of ̺ when n> p. Actually
ε
n!
˜bWick S±nZ⊗n = εp(n p)!S±n(˜b⊗IdZ⊗(n−p))S±n,∗
−
(cid:12)
(cid:12)
6
and, as εpn(n 1) (n p+1) 1 if nε 1, it follows that
− ··· − → →
lim Tr γ(p)˜b = lim Tr ̺ (˜b Id ) .
nε 1 ε nε 1 ε ⊗ Z⊗(n−p))
ε ∼0 h i ε ∼0 h i
→ →
When Z = L2(M;dv) one thus often considers:
γ˜(p)(x ,...,x ;x ,...,x ) = ̺ (x ,...,x ,x;x ,...,x ,x) dv (n p)(x).
ε 1 p ′1 ′p ε 1 p ′1 ′p ⊗ −
ZMn−p
But, if the states ̺ are not localized on the n-particles sector, such an alternative definition does
ε
(p)
not coincide with γ , even asymptotically in the mean-field regime.
ε
(p+1)
As well there is no general relation between the non normalized density matrices γ and
ε
(p)
γ . Actually we have
ε
Wick (n+p+1)!
S±p+1(˜b⊗IdZ)S±p+1,∗ S±n+p+1Z = εp+1 n! S±n+p+1(˜b⊗IdZ⊗n+1)S±n+p+1,∗
(cid:16) (cid:17) (cid:12) = ε(n+1)˜bWick ,
(cid:12) nZn+p+1
S±
(cid:12)
from which we deduce (cid:12)
Tr γε(p+1)(˜b⊗IdZ) = Tr ̺εN±˜bWick +O(ε),
h i h i
while Tr γ(p)˜b = Tr ̺ ˜bWick ,
ε ε
h i h i
where we have again identified γ(p+1) as an element of 1(Z (p+1)). We thus conclude with the
ε ⊗
L
following important remark.
(p) (p′) (p)
Remark 2.7. Simple asymptotic relation between the γ and γ (or the normalized version γ¯
ε ε ε
(p′)
and γ¯ ) can be expected when ̺ = ̺ 1 (N ) with ν > 0 and lim δ(ε) = 0 but not
ε ε ε [ν δ(ε),ν+δ(ε)] ε 0
− ± →
otherwise (of course the condition above is sufficient but not necessary).
We shall use recurrently with variations the following lemma, with the following notations
1
˜b ˜b = ˜b ˜b ,
1⊙···⊙ p p! σ(1) ⊗···⊗ σ(p)
σX∈Sp
for ˜b ,...,˜b (Z).
1 p
∈ L
Wick
We also write shortly (˜b1 ... ˜bp)Wick and (˜b⊗p)Wick instead of p(˜b1 ... ˜bp) p,∗ and
⊙ ⊙ S± ⊙ ⊙ S±
p(˜b⊗p) p,∗ Wick. (cid:16) (cid:17)
S± S±
(cid:16) (cid:17)
Lemma 2.8. Quantum symmetrization lemma: In the bosonic and fermionic cases for any
p N, the equality
∈ p(˜b1 ˜bp) p,∗ = p(˜b1 ˜bσ(p)) p,∗, (5)
S± ⊗···⊗ S± S± ⊙···⊙ S±
holds in ( pZ p; pZ p) for all ˜b ,...,˜b (Z;Z).
⊗ ⊗ 1 p
AsacLonSs±equenceS,±undertheassumptions of∈DLefinition2.6, thenonnormalized (resp. normalized
if possible) reduced density matrix γ(p) (resp. γ¯(p)), p N, is completely determined by the set of
ε ε
∈
quantities Tr ̺ (˜b p)Wick ,˜b when is any dense subset of (Z;Z).
ε ⊗ ∞
∈ B B L
n h i o
7
(p) (p)
Remark 2.9. While computing Tr γε or studying γ¯ε one can simply add to the element IdZ
B
owing to pId⊗Zp p,∗ = Id pZ⊗p. Fhor εi> 0 fixed it is not necessary because compact observables
S± S± S±
(p) (p)
are sufficient to determine the total trace owing to Tr γε = supB ∞( pZ⊗p),0 B IdTr γε B .
∈L S± ≤ ≤
However, while considering weak∗-limits as ε 0, adhdingithe identity operator Id pZ⊗p tho the siet
of compact observables, or possibly replacing →by the Calkin algebra CId(Z) S±(Z), is useful
∞
B ⊕L
in order to control the asymptotic total mass.
Proof. For ˜b1,...,˜bp (Z), we decompose p ˜b1 ˜bp p,∗ p(ψ1 ψp) as
∈ L S± ⊗···⊗ S± S± ⊗···⊗
(cid:16) (cid:17)
1
p s (σ )(˜b ψ ) (˜b ψ )
′ 1 σ′(1) p σ′(p)
S±p! ± ⊗···⊗ 
σX′∈Sp
 
1
= s (σ)s (σ )(˜b ψ ) (˜b ψ ) .
p!p! ± ± ′ σ(1) σ◦σ′(1) ⊗···⊗ σ(p) σ◦σ′(p) 
σX∈SpσX′∈Sp
 
Setting σ = σ σ , with s (σ )= s (σ)s (σ ) yields the Eq. (5), after noting that ˜b ˜b =
′′ ′ ′′ ′ 1 p
1 ˜b ◦ ˜b ±commute±s with± p in both the bosonic case and the fermio⊙nic··c·a⊙se.
Np!owσt∈hSepnoσn(1)n⊗orm··a·l⊗izedσ(pr)educed density mSat±rix is determined by
P
Tr γ(p)B˜ = Tr ̺ B˜Wick
ε ε
h i h i
for B˜ ( pZ p) as 1( pZ p) is the dual of ( pZ p). But B˜ ( pZ p) means
∞ ⊗ ⊗ ∞ ⊗ ∞ ⊗
B˜ = ∈pBL˜′ pS,∗±with B˜′ L ∞S(±Z⊗p), while the algebLraicSt±ensor product ∈∞(LZ)⊗Sa±lgp is dense in
(ZS±p).S± ∈ L L
∞ ⊗
L
With the estimate
Tr ̺ B˜Wick = Tr ec/2N̺ ec/2Ne c/2NB˜Wicke c/2N
ε ε − −
(cid:12)(cid:12)(cid:12) h i(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)CTrh ̺εecN kB˜kL(S±pZ⊗p;S±pZ⊗p), i(cid:12)(cid:12)(cid:12)
(cid:2) (cid:3)
it suffices to consider B˜ = pB˜′ p,∗ with B˜′ ∞(Z)⊗algp. By linearity and density, γε(p) is
determined by the quantitiesST±r[̺SB˜±Wick] with B˜∈ =L˜b ˜b , ˜b . We conclude with
ε ′ 1 p i
⊗···⊗ ∈ B
p ˜b1 ˜bp p,∗ = p ˜b1 ˜bp p,∗,
S± ⊗···⊗ S± S± ⊙···⊙ S±
(cid:16) (cid:17) (cid:16) (cid:17)
and the polarization identity
p
1
˜b ˜b = ε ε ( ε b˜) p.
1⊙···⊙ p 2pp! 1··· p i i ⊗
εiX=±1 Xi=1
(p)
Remark 2.10. In the bosonic case, the non normalized reduced density matrices γ are also
ε
characterized by the values of Tr[γε(p)B] for B in = ψ⊗p ψ⊗p ,ψ Z Id⊗Zp . This does
B {| ih | ∈ }∪
not hold in the fermionic case. n o
8
(p)
The rest of the article is devoted to the asymptotic analysis of γ as ε 0. In particular we
ε
shall study their concentration at the quantum level while testing with fixed→observable ˜b (with ˜b
compact) and their semiclassical behaviour after taking semiclassically quantized observables, e.g.
a(x,hD ) with some relation ε = ε(h) between ε and h.
x
3 Classical phase-space and h-quantizations
When Z = L2(M1,dx), with M1 = M a manifold with volume measure dx, the classical one
particle phase-space is 1 = = T M1 and we will focus on the h-dependent quantizations which
∗
X X
associates with a symbol a(x,ξ) = a(X), X 1 an operator aQ,h = a(x,hD ) with the standard
x
∈ X
semiclassical quantization or when M1 = Rd, aQ,h = aW,h = aW(htx,h1 tD ) by using the Weyl
− x
quantization, t R being fixed.
∈
Note that in later sections the parameters ε and h will be linked through ε = ε(h) with
lim ε(h) = 0. In relation with the symmetrization Lemma 2.8, we introduce the adapted
h 0
→
p-particles phase-space which was also considered in [Der1], and the corresponding semiclassical
observables.
3.1 Classical p-particles phase-space
A fundamental principle of quantum mechanics is that identical particles are indistinguishable.
The classical description is thus concerned with indistiguishable classical particles. If one classical
particle is characterized by its position-momentum (x,ξ) 1 = T M1, x M being the position
∗
∈ X ∈
coordinate and ξ the momentum coordinates, p indistinguishable particles will be described by
their position-momentum coordinates (X ,...,X ) = (x ,ξ ,...,x ,ξ ) p/S = (T M)p/S =
1 p 1 1 p p p ∗ p
∈ X
T (Mp)/S , where the quotient by S simply implements the identification
∗ p p
σ S , (X ,...,X ) (X ,...,X ).
p σ(1) σ(p) 1 p
∀ ∈ ≡
Thegrand canonical description of a classical particles system then takes place in the disjoint union
p/S = (T M)p/S .
p ∗ p
p⊔NX p⊔N
∈ ∈
A p-particles classical observable will be a function on p/S and when the number of particles
p
X
is not fixed a collection of functions (a(p))p N each a(p) being a function on p/Sp. The situation
is presented in this way in [Der1]. A p-p∈articles observable is a functionXa(p) on p/S and a
p
X
p-particles classical state is a probability measure (and when the normalization is forgotten a non
negative measure) on p/S .
p
X
However while quantizing a classical observable, it is better to work in p which equals T (Mp), a
∗
X
function a(p) on p/S being nothing but a function on p which satisfies
p
X X
1
σ S , σ a(p) = a(p) = σ˜ a(p),
p ∗ ∗
∀ ∈ p!
σ˜X∈Sp
with X ...X p, σ a(p)(X ,...,X ) = a(p)(X ,...,X ).
1 p ∗ 1 p σ(1) σ(p)
∀ ∈ X
In the same way, we define for a Borel measure ν on p and σ S , the measure σ ν by
p
σ a(p)dν = a(p)d(σ ν) for all a(p) 0( p), or aXlternatively∈σ ν(E) = ν(σ 1E) f∗or all
X ∗ Xp ∗ ∈ Cc X ∗ −
R R
9
Borel subsetE of p. A non-negative measure on p/S is identified with a non-negative measure
p
X X
ν on p such that
X 1
σ S , σ ν = ν = σ˜ ν. (6)
p
∀ ∈ ∗ p! ∗
σ˜X∈Sp
Lemma 3.1 (Classical symmetrization lemma). Any Borel measure µ(p) on p/S is characterized
p
X
by the quantities a pdµ(p), a where the tensor power a p means a p(X ,...,X ) =
p a(X ) and isXapny⊗dense set in∈0C( 1) = f 0( 1),lim ⊗ f(X) =0⊗ . 1 p
i=1 i C(cid:8)R C∞(cid:9)X ∈C X X→∞
PQroof. By Stone-Weierstrass Theorem the subal(cid:8)gebra generated by the algebrai(cid:9)c tensor product
algp is dense in 0 ( p). Hence it suffices to consider
⊗
C C∞ X
1
a a = a a , a .
1 p σ(1) σ(p) i
⊙···⊙ p! ⊗···⊗ ∈ C
σX∈Sp
We conclude again with the polarization identity
p p
1 ⊗
a a = ε ...ε ε a .
1⊙···⊙ p 2pp! 1 p i i
!
εiX=±1 Xi=1
We will work essentially with M = Rd and = T Rd and therefore on p = T Rdp R2dp and
∗ ∗
X X ∼
recalltheinvarianceproperties,ifpossiblebyachangeofvariableinordertoextendittothegeneral
case. Remember that on Rdp, the standard and Weyl semiclassical quantization are asymptotically
equivalent a(x,hD ) aW(x,hD ) = O(h) when a S(1,dX2) (sup ∂αa(X) < for all
α N2d). Moreovxer−on Rdp, aWx(x,hD ) is unitar∈y equivalent to aXW∈(Th∗tRxd,ph|1XtD ) f|or a∞ny fixed
x − x
∈
t R so that result can be adapted to different scalings.
∈
3.2 Semiclassical and multiscale measures
We recall the notions of semiclassical (or Wigner) measures and multiscale measures in the finite
dimensionalcase. WestartwiththeresultsonM = RD (thinkofD = dp)andreviewtheinvariance
properties for applications to some more general manifolds M.
3.2.1 In the Euclidean Space
On RD the semiclassical Weyl quantization of a symbol a (R2D) will be written aW,h =
′
∈ S
aW(htx,h1 tD ) with t > 0 fixed and a kernel given by
− x
x+y dξ
[aW(x,D )](x,y) = eiξ(x y)a( ,ξ) .
x Rd · − 2 (2π)d
Z
Definition 3.2. Let (γ ) with 0 , (0,+ ), be a family of trace-class non-negative
h h
operators on L2(RD) such ∈thEat lim ∈TEr[γE] <⊂+ .∞The semiclassical quantization a aW,h =
h 0 h
aW(htx,h1 tD ) is said adapted to→the family (γ )∞ if 7→
− x h h
∈E
lim limsup Re Tr (1 χ(δ )W,h)γ = 0
h
δ 0+h ,h 0 − ·
→ ∈E → h i
10