Notes on phase space quantization

0.31 MB·English

by  J. Kiukas

NOTES ON PHASE SPACE QUANTIZATION
J. KIUKAS, P. LAHTI, AND K. YLINEN
fAb:sΩtra
Rt. We
onsiderquestΩionsrelatedtoaquantizations
hemeinwhi
hEa
flassi
alvariable
→ on a phaseEsfpa
e is asso
iated with a semisΓpe(f
tkr)al measuΓre , su
h that the
moment operators of are required to be of the form , with a suitable mapping
from the set of
lassi
al variables to the set of (not ne
essarily bounded) operators in some
Γ
Hilbert spa
e. In parti
ular, we investigate the situation where the map is implemented by
6 the operator integral with respe
t to some (cid:28)xed positive operator measure. The phase spa
e
Ω
00 is (cid:28)rst taRke2n to be an abstra
tmeasurable spa
e, then a lo
ally
oΓm(fpka)
t unimodular group,
and (cid:28)nally , wherewedetermineexpli
itlythe relevantoperators for
ertainvariables
2 f Γ
, in the
asewhere the quantizationmap isimplemented by atranslation
ovariantpositive

n
operator measure. In addition, we
onsider the question under what
onditions a positive
a
operator measure is proje
tion valued.
J

2

1
v 1. Introdu
tion
9
0
Quantization
an be any pro
edure whi
h asso
iates a quantum me
hani
al observable to a
0
given
lassi
al dynami
al variable. The traditional way to realize a quantization is to assign to
1
0 ea
h
lassi
al variable a Hermitean (symmetri
, or even essentially selfadjoint) operator whi
h
6
should des
ribe the quantum observable. In the modern view of a quantum observable as a
0
/ semispe
tral measure, this kind of quantization is no longer su(cid:30)
ient. The question is how to
h
p modify the traditional s
heme in order to (cid:28)t it into the
ontext of modern quantum me
hani
s.
- Classi
al variables
an be represented by real valued measurable fun
tions de(cid:28)ned on some
t (Ω, )
n measurable spa
e RA2n, whi
h is the phase spa
e of the
lassi
al system. The phase spa
e
an
a
u be taken to be e.g. , in whi
h
ase the variaΓbles are Borel fun
tions. In the
onventional
q approa
h to quantization, we would have a map from the set of real measurable fun
tions to
Γ(f)
:
v the set of alllinear(not ne
essarily bounded) operators in H, and would be the observable
f
i
X
orresponding to the
lassi
al variable (see e.g [9, 15, 28, 32, 33℄). We would like to modify
f
r this s
heme so that we
ould assign a semispe
trΓa(lfmk)easkure tNo the fun
tion , instead of an
a
operator. To do this, we use all the operators , ∈ , instead of just the (cid:28)rst one
oEfft:hem(R,)andLth(en)
onsider the moment proxbkledmEfo=f (cid:28)Γn(dfink)g the unkiqueNsemispe
txrkadlEmfeasure
B → H with the property that for all ∈ . Here is the
L(xk,Ef) Rx xk Ef R
operator integral of the fun
tion 7→ with respe
t to ; see the next Se
tion for
Γ f Ef
its de(cid:28)nition. So if and are su
h that there existsΓa(fukn)iquke soNlution of the des
ribed
moment problem, then the
olle
tion of the operators { | ∈ } is eligible to represent a
Ef f
quantization of .
We mention that the approa
h des
ribed above is used, for instan
e, in [14℄ in the des
ription
of a quantum measurement. In a typi
al measurement situation, where one aims to measure
a (traditional) quantum observable represented by a selfadjoint operator, one is a
tually mea-
suring a noisy or unsharp version of that observable. The noisy version, represented by a
semispe
tral measure, may agree with the observable intended to be measured, on the statisti-

al level of expe
tations. The unsharpness of the measurement is re(cid:29)e
ted in the fa
t that the
1
dispersion of the measurement statisti
s is a
tually greater than what would be obtained in the
noiseless
ase. The observable, as represented by a semispe
tral measure,
annot be des
ribed
using a single operator. A

ordingly, in the approa
h of [14℄, ea
h moment of the measurement
out
ome distribution is
onsidered to be an average of
ertain operator,
alled (cid:17)operational
observable(cid:17), and the
olle
tion of these operators then represents the measured observable. We
want to point out that these (cid:17)operational observables(cid:17) are nothing else but the moment oper-
ators of the semispe
tral measure representing the observable. Note that these moments need
not determine the semispe
tral measure uniquely. In
ertain spe
ial
ases, however, they do
[12, 13℄, and sometimes even the (cid:28)rst moment is enough [8, 25℄.
Γ
One way to obtain a quantization map is to use the operator integral with respe
t to
E Γ f L(f,E)
some given positive operator measure , i.e. de(cid:28)ne to be the map 7→ . In this
Ef B E(f 1(B))
−

ase, if we de(cid:28)ne on the Borel sets of the real line by 7→ , we have simply
L(xk,Ef) = L(fk,E)
, as is easily seen by using the de(cid:28)nition of the next Se
tion and the usual
Ef

hange of variables in the integral with respe
t to a
omplex measure. So then is a solution
to the moment problem des
ribed above, whi
h leaves us with the uniqueness question. Note
E
that if we
hoose to be a spe
tral measure, we end up getting only spe
tral measures as the
L(fk,E)
quantized observables, with all the operators mutually
ommuting. We also remark
that, in any
ase, all the observables obtained via this type of quantization are fun
tionally

oexistent [26℄, so that they
an be measured together in the sense of Ludwig [29, D.3.1, p.
153℄.
The stru
ture of this note is as follows. In the Se
tion 2 we give some results on the theory
of operator integrals and give a simple
hara
terization of the quantization maps whi
h
an be
represented by an operator integral with respe
t to a positive operator measure. In Se
tion
3, we
onsider the operator integral with respe
t to
ovariant positive operator measures on a
lo
ally
ompa
t topologi
al group. Se
tioRn24 is devoted to the quantizations obtained by using
the
ovariant phase spa
e observables in , and in Se
tion 5, we dis
uss the (cid:17)optimal(cid:17)
hoi
e
for su
h a
ovariant observable, in view of quantization. In the last Se
tion, we
onsider the
question under what
onditions a positive operator measure is a spe
tral one.
2. The operator integral
The basi
tool in our quantization pro
edure is the operator integral. It asso
iates a linear
(notne
essarily bounded)operatorintheHilbertspa
e Hofaquantum system toea
h
omplex
Ω
measurablefun
tiononthephase spa
e . Inthisse
tion we review thebasi
results
on
erning
the theory of operator integrals and give some additional remarks in relation to quantization.
Ω σ Ω
Let be a nonempty set and A a -algebra of subsets of . Let H be a
omplex Hilbert
L( ) E : L( )
spa
e and H the set of bounded operators on H. Let A → H be a positive operator
σ
measure, i.e. a positive operator valued set fun
tion whi
h is -additive with respe
t to weak
ϕ,ψ E B ψ E(B)ϕ
ψ,ϕ
operator topology. For ea
h ∈ H, let denote the
omplex measure 7→ h | i.
E
We re
all that a normalized positive operator measure is
alled a semispe
tral measure, and
E E(A B) = E(A)E(B)
that a semispe
tral measure is a spe
tral measure exa
tly when ∩
A,B
for all ∈ A. As mentioned in the Introdu
tion, semispe
tral measures are also
alled
observables.
(Ω, ) (Ω)
Let F A , or F in brief, denote the set of all
omplex A-measurable fun
tions de(cid:28)ned
Ω ( ) f (Ω)
on , and O H the set of all (not ne
essarily bounded) linear operators in H. For ∈ F ,
2
de(cid:28)ne
D(f,E) = ϕ f E ψ ,
ψ,ϕ
{ ∈ H | is -integrable for ea
h ∈ H}
and
D(f,E) = ϕ f 2 E .
ϕ,ϕ
{ ∈ H | | | is -integrable}
L(f,E)
e
The following result was proved in [22℄. The operator appearing in it is
alled the
f
operator integral of with respe
t to E.
D(f,E)
Theorem 1. (a) The set is a linear (not ne
essarily dense) subspa
e of H, and
L(f,E) = fdE D(f,E)
there is a unique linear operator on the domain satisfying
R
ψ L(f,E)ϕ = fdE
ψ,ϕ
h | i Z
ψ ϕ D(f,E)
for all ∈ H and ∈ .
D(f,E) D(f,E)
(b) The set is a subspa
e of .
f L(f,E)
(
) If is reeal valued, is a symmetri
operator.
D(f,E) D(f,E) D(f,E) = D(f,E)
(d) While the in
lusion ⊂ may in general be proper, in
E
the
ase where is aespe
tral measure. e
E σ
Remark. Sin
e the operator measure is also strongly -additive, ea
h set fun
tion A ∋
B E := E(B)ϕ ϕ
ϕ
7→ ∈ H, for ∈ H, is an H-valued ve
tor measure. The de(cid:28)nition of the
D(f,E) ϕ f
operator integral states that is the set of those ∈ H for whi
h is integrable with
E L(f,E)ϕ = fdE
ϕ ϕ
respe
t to the ve
tor measure in the sense of [20, p. 21℄, and for ea
h
ϕ D(f,E) R
∈ . As pointed out in [20, p. 37℄, this de(cid:28)nition of integrability is equivalent to that
of [10, p. 323℄ (for proof, see [41, Corollary 3.6℄).
L(Tf,hEe)ve
tor measure appfroa
h provides an easy way to
hafra
ter(izΩe)the opneraNtor intfe˜gral
n
by˜approximating with bounded˜fun
tions. For ea
h ∈ F , and ∈ , let be
f (x) = f(x) f(x) n f (x) = 0
n n
su
h that if | | ≤ , and otherwise. It is a well-known fa
t that in
E
the
ase where is a spe
tral measure, we have
D(f,E) = D(f,E) = Df,E := ϕ lim L(f˜ ,E)ϕ .
n
{ ∈ H | n exists}
−→∞
L(f,E)ϕ = lime L(f˜ ,E)ϕ ϕ Df,E
n n
In addition, −→∞ for all ∈ (see e.g. [11, p. 1196℄). In the
ase
E
of a general positive operator measurRe , this need not be true. For example, take a probability
measure de(cid:28)ned on the Borel sets of , su
h that ithas a density whi
h is an even fun
tion, and
∞xdµ(x) = E B µ(B)I f(x) = x f
0 ∞. Let be the positive operator measure 7→ , and . Now is
R µ D(f,E) = 0 ϕ ψ L(f˜ ,E)ϕ = ψ ϕ n xdµ(x) =
not -integrable, so { }. But if ∈ H, we have h | n i h | i n
0 n N ψ µ Df,E = R−
for all ∈ and ∈ H be
ause the density of is even, so that˜ H.
lim L(f ,E)ϕ
n n
Thus in the general
ase, the existen
e of the limit −→∞ does not guarantee
ϕ D(f,E)
that ∈ . However, the following result holds.
E : L( ) f (Ω)
Proposition 1. Let A → H be a positive operator measure, and ∈ F . Then
˜
D(f,E) = ϕ lim L(χ f ,E)ϕ B ,
B n
{ ∈ H | n exists for ea
h ∈ A}
−→∞
˜
L(f,E)ϕ = lim L(f ,E)ϕ ϕ D(f,E)
n n
and −→∞ for all ∈ .
3
D
Proof. Denote by the right hand side of the set˜equality appearing i˜n the statement. Let
ϕ D B η = lim L(χ f ,E)ϕ f
B n B n n
∈ , and let ∈ A, with −g→∞ g (.x)Sin
f˜e(exa)
h 1 is boundend, wNe
an

hoose a sequen
e of A-simple fun
tions n, su
h that | n − n | ≤ n for ea
h ∈ and
x Ω (g ) f
n
∈ . Clearly the sequen
e
onverges to pointwise. Now
1 4
˜ ˜
L(χ g χ f ,E)ϕ = sup (g f )dE sup E (Ω) E(Ω) ϕ ,
B n B n n n ψ,ϕ ψ,ϕ
k − k (cid:12)Z − (cid:12) ≤ n | | ≤ nk kk k
ψ 1(cid:12) B (cid:12) ψ 1
k k≤ k k≤
(cid:12) (cid:12)
(cid:12) (cid:12)
so that 4
˜
g dE η E(Ω) ϕ + L(χ f ,E)ϕ η .
n ϕ B B n B
kZ − k ≤ nk kk k k − k
B
( g dE ) B η
It follows that the sequen
e B n ϕ of ve
tors
onverges for ea
h ∈ A (to B), so by
R f E
ϕ
the de(cid:28)nition of [10, p. 323℄, is integrable with respe
t to the ve
tor measure , i.e.
ϕ D(f,E) ϕ D(f,E)
∈ (see the Remark following Theorem 1). Conversely, let ∈ . Sin
e
˜
D(f,E) = D( f ,E) B (χ f ) χ f
B n B
| | ˜by de(cid:28)nition, and for ea
h ∈ A the sequen
e
onverges to
χ f χ f
B n B
pointwise, with | | ≤ | |, it follows e.g. from the dominated
onvergen
e theorem for
E
ϕ
the ve
tor measure (see [10, p. 328℄) that
˜ ˜
L(χ f,E)ϕ = χ fdE = lim χ f dE = lim L(χ f ,E)ϕ
B B ϕ B n ϕ B n
Z n Z n
−→∞ −→∞
B ϕ D L(f,E)ϕ = lim L(f˜ ,E)ϕ (cid:3)
n n
for ea
h ∈ A. Thus ∈ , and −→∞ .
Remark.
D(f,E) D(f,E) Df,E
(a) We now have the subspa
e in
lusions ⊂ ⊂ , with ea
h of them
E
possiblyproper. Inthe
asewhere isaespe
tralmeasure, bothin
lusionsareequalities.
E D(f,E)
(b) It is well known that in the
ase where is a spe
tral measure, the domain is
dense. As seen before Proposition 1, this need not be the
ase in general, so a question
E f D(f,E)
arises what is required for and to make dense.
f (Ω)
(
) Another di(cid:27)eren
e to the spe
tral
ase is that for real valued ∈ F , the symmetri
L(f,E)
operator is not ne
essarily selfadjoint (the adjoint need not even exist), and
L(f,E)
it seems to be di(cid:30)
ult, in general, to determine when might have selfadjoint
f D(f,E)
extensions. In the
ase where is positive and dense, the positive symmetri
L(f,E)
operator of
ourse has its selfadjoint Friedri
hs extension.
Γ
As noted in the Introdu
tion, the starting point of our quantization s
heme is a map from
( )
real valued measurable fun
tions to the set O H . The following theorem
hara
terizes those
Γ : (Ω) ( )
maps F → O H whi
h are implemented by an operator integral. The
orresponding
˜
f f
n
result involving bounded fun
tions is well known. Here the fun
tions are de(cid:28)ned for ea
h
as in Proposition 1.
Γ : (Ω) ( ) f L(f,E)
Theorem 2. A map F → O H
oin
ides with the map 7→ for a (
learly
E
unique) positive operator measure if and only if the following
onditions are satis(cid:28)ed.
Γ L( )
(i) restri
ted to bounded fun
tions is a positive linear map with values in H ;
(f )
n
(ii) if is an in
reasing sequen
e of positive A-measurable fun
tions
onverging pointwise
f (Ω) sup ϕ Γ(f )ϕ = ϕ Γ(f)ϕ ϕ
to a bounfded ∈(ΩF) , then nD∈(NΓh(f|)) n Γi(f)h | i for ea
h ∈ Hϕ;
(iii) for ea
h ∈ F , the domain of
onsists of those ve
tors ∈ H for
˜
(Γ(χ f )ϕ) B
B n
whi
h the sequen
e of ve
tors
onverges for ea
h ∈ A.
4
˜
ϕ D(Γ(f)) (Γ(f )ϕ) Γ(f)ϕ
n
(iv) for ea
h ∈ , the sequen
e
onverges to .
E Γ(f) = L(f,E)
Proof. Assume (cid:28)rst that there is a positive operator measure , su
h that
f
for ea
h . The above properties follow easily: The property (i) is well known (see e.g. [2,
pp. 26-28℄), and (ii) follows from the monotone
onvergen
e theorem. Proposition 1 gives
Γ : (Ω) ( )
(iii) and (iv). Next assume that (i)-(iii) hold for a map F → O H . By (i) the
B EΓ(B) := Γ(χ ) ( )
B
map A ∋ 7→ ∈ O H is a positive operator valued additive set fun
tion.
Γ sup
n N
Sin
liem is positive for bounded fun
tions by (i), B ∈ ϕinE
Γo(Bnd)iϕtion of (ii)
an be repla
ed
n
by EΓ−→∞. This implies that ea
h set funf
tion(Ω)7→ h | i is a positive measure, so
that is a positive operator measure. If ∈ F is a bounded positive fun
tion, we have
ϕ Γ(f)ϕ = ϕ L(f,EΓ)ϕ ϕ f
h | i h | i for all ∈ H, as is seen by approximating with an in
reasing
sequen
e of simple fun
tions and using linearity, (ii), and the monotone
onvergen
e theorem.
f (Ω) Γ(f) = L(f,EΓ)
Hen
e, if ∈ F is bounded, it follows by linearity and polarization that .
f (Ω) D(Γ(f)) = D(f,EΓ)
Now let ∈ F ˜ be arbitrary. It follows by (iii) and Proposition 1 that ,
f
n
and (sin
e ea
h is bounded), also
L(f,EΓ)ϕ = lim L(f˜ ,EΓ)ϕ = lim Γ(f˜ )ϕ = Γ(f)ϕ
n n
n n
−→∞ −→∞
ϕ D(Γ(f)) L(f,EΓ) = Γ(f) (cid:3)
for all ∈ (where (iv) is used), so .
Remark. Inthequantizations
hemedes
ribedintheIntrodu
tionthe
lassi
alvariableswere
(Ω)
thought to be real valued. Obviously, the pre
eding Theorem holds also with F repla
ed
by the set of real A-measurable fun
tions.
Weendthisse
tionbydis
ussingbrie(cid:29)yasimplewayofobtainingquantizationmapswithout
the use of positive operator measures. This approa
h is essentially the one frequently used in
the
onventional qfua:nRtiz2nationR(e.g. Weyl quantization): the operator
orresponding to a given

lassi
al variable → is obtained by integrating the variable with respe
t to some
operator valued fun
tion de(cid:28)ned on the phase spa
e. The quantization of the variablLe2t(hRe)n
be
omes a
ontinuous distribution valued operator de(cid:28)ned on some dense subspa
e of ,
whi
h does not depend on the variable itself. In the
ase of Weyl quantization, for example,
all the quantized operators are de(cid:28)ned on a
ommon domain (see e.g. [9, 36℄). However, to
make the situation similar to that of the operator integral map
onsidered above, we de(cid:28)ne the
quantization map in the following simple way. The proof is a dire
t adaptation of the proof of
Theorem1 (a)(see [22℄). Note that itfollowsfrom(i)that theintegrandin(ii)isA-measurable.
ϕ = sup ψ ϕ ψ , ψ = 1 ϕ
This is so be
ause k k {|h | i| | ∈ M k k } for any ve
tor ∈ H, where M is
a (cid:28)xed
ountable dense set in the separable Hilbert spa
e H. Hen
e the integral in (ii) is well
de(cid:28)ned.
(Ω, ,µ) Λ : Ω
Proposition 2. Let A be a measure spa
e, H a separable Hilbert spa
e, and →
L( )
H a map with the following properties:
ω ψ Λ(ω)ϕ ψ,ϕ
(i) 7→ h | i is A-measurable for all ∈ H;
Λ(ω)ϕ dµ(ω) < ϕ B µ(B) <
(ii) Bk k ∞ for ea
h ∈ H and ∈ A with ∞.
R f : Ω C Γ (f)
Λ
Then for ea
h A-measurable fun
tion → , there exists a linear operator in H,
su
h that
D(Γ (f)) = ϕ f ψ Λ( )ϕ µ ψ ,
Λ
{ ∈ H | h | · i is -integrable for ea
h ∈ H}
5
and
ψ Γ (f)ϕ = f(x) ψ Λ(ω)ϕ dµ(ω) ψ ,ϕ D(Γ (f)).
Λ Λ
h | i Z h | i ∈ H ∈
D(Γ (f)) (f )
Λ n
Proof. It is
lear that ⊂fH is a vef
tor subfspa
e. Lent Nbe a sequenψ
eΛ(oωf)sϕimple
n
fuψn
tΛio(nωs)ϕ
onverging ψpo,iϕntwise,ωto Ω, with | | ≤ | | for all ∈ . Sin
e n|h |N i| ≤
k kk k for ea
h ∈ H ∈ , it follows from (ii) and (i) that for ea
h ∈ , we have
D(Γ (f )) = ψ f (ω) ψ Λ(ω)ϕ dµ(ω)
Λ n n
ϕ H and thenlineNar fun
ϕtional 7→ R ηϕ h | i is
ontinuous for ea
h
∈ H. Hen
e, for ea
h ∈ and ∈ H, there is n ∈ H, su
h that
ψ ηϕ = f (ω) ψ Λ(ω)ϕ dµ(ω)
h | ni Z n h | i
ψ ϕ D(Γ (f)) f f n
Λ n
for all ∈ H. Now, let ∈ . Sin
e | | ≤ | | for all , the dominated
onvergen
e
ψ ηϕ ψ f(ω) ψ Λ(ω)ϕ dµ(ω)
theorem implies that the sequen
e h | ni
onverges for ea
h ∈ H to h | i ,
R Γ (f)ϕ
Λ
so by the uniform boundedness theorem and the re(cid:29)exivity of H, there is ∈ H, su
h
that
ψ Γ (f)ϕ = f(ω) ψ Λ(ω)ϕ dµ(ω)
Λ
h | i Z h | i
ψ D(Γ (f)) ϕ Γ (f)ϕ
Λ Λ
for all ∈ H. Clearly, the map ∋ 7→ ∈ H is linear, so the proof (cid:3)is

omplete.
Remark. Consider the situation where we have a lo
ally
ompa
t unimodular topologi
al
G µ
group , with a left Haar measure , and a strongly
ontinuous proje
tive unitary repre-
U : G ( ) A L( ) Λ : G L( )
sentation → U H . Then, for any ∈ H , the map → H , de(cid:28)ned by
Λ(g) = U(g)AU(g)
∗
, satis(cid:28)es the
onditions of the pre
eding Proposition, so we get the
or-
Γ
Λ
responding quantization map . Noti
e that in the
ase where the representation is square
A Γ
Λ
integrable and positive, the quantization map
an be represented by an operator integral
A Tr[A] = D(Γ (χ )) = 0
Λ G
if and only if has (cid:28)nite tra
e. Namely, the
ase ∞ gives { } (see
Tr[A] <
e.g. [19, Lemma 2℄), while the
ase ∞ leads to the usual quantization map given by
the operator integral with respe
t to a
ovariant positive operator measure (multipliedby some

onstant). G = R2 U(g) = U((q,p)) A
ConsidLer2(tRhe)
asψe wheψr(e ) L2,(R) aretheWeyloperators, andΓ istheparity
Λ
operator ∋ 7→ −· ∈ (mulftipliLed1(bRy2)a sLu2it(aRb2l)e
onstant). NoΓw(f) is the Weyl
Λ
quantizationmfap.LIt2(iRsw2)ellknowΓn t(hfa)tfor ∈ ∪ ,theoperator f isbounded.
Λ
Moreover, if ∈ , then is a Hilbert-S
hmidt operator, and if is a S
hwartz
Γ (f)
Λ
fun
tion,then isatra
e
lassoperator(seee.g. [36℄). Itisawellknownfa
tthattheWeyl
quantizationsofthe
lassi
alpositionandmomentumvariablesarethe positionandmomentum
Q P Q
operators and , in the (cid:17)distributional(cid:17) sense. Note, however, that the a
tual domains of
P
and are not given by the formula of the pre
eding theorem. For example, the
hara
teristi
χ Q (q,p) q χ Λ(q,p)χ
[ 1,1] [ 1,1] [ 1,1]
fun
tion − is in the domain of , but the fun
tion 7→ h − | − i is not
(Lebesgue)-integrable, as is easily seen by
al
ulating the expli
it form of the fun
tion.
3. Covariant quantization
Ω
We now take the set of the pre
eding se
tion to be a lo
ally
ompa
t se
ond
ountable
G (G) σ
unimodular topologi
al group, hen
eforth denoted as , and let B denote the Borel -
G λ G ( )
algebra of . Fix to be a Haar measure in . Let T H denote the Bana
h spa
e of tra
e
Aut( ( ))

lass operators on the Hilbert spa
e H, and let T H denote the set of linear, positive,
6
( )
tra
e norm preserving bije
tions frAoumt(T(H))ontβo itseTlfr.[AWβ(eT
)o]nsidCer it equiAppedLw(it)h the
topology given by the fun
tionals T H ∋ 7→ ∈ , where ∈ H and
T ( )
∈ T H .
β : G Aut( ( ))
Assume further that there is a
ontinuous group homomorphism → T H and a
d > 0

onstant , satisfying
Tr[P β(g)(P )]dλ = d P ,P .
1 2 1 2
(1) Z for all one-dimensional proje
tions on H
G
We now
onsider quantizations
onne
ted to the stru
ture of given by the homomorphism
β Γ : (G) ( ) Γ(f) L( )
, in the following sense: A map F → O H with the property that ∈ H for all
f (G) β β(g) (Γ(f)) = Γ(f(g )) g G
∗
bounded fun
tions ∈ F is said to be -
ovariant, if · for all ∈
f (G)
and all bounded fun
tions ∈ F .
Γ
If is su
h that it
an be represented by the operator integral with respe
t to an observable
E Γ Γ
, (i.e. satis(cid:28)es the
onditions of Theorem 2), then it is straightforward to verify that is
β E β
-
ovariant if and only if the observable is -
ovariant in the following sense: An observable
E : (G) L( ) β β(g) (E(B)) = E(g 1B) g G B (G)
∗ −
B → H is said to be -
ovariant if forall ∈ , ∈ B .
Covariant observables are essential in quantum me
hani
s, and hen
e they have been studied
quite extensively. The
anoni
al examples of
ovariant observables are
onstru
ted e.g. in [7℄,
and there are (at least) two
ompletely di(cid:27)erent ways to obtain their
hara
terization: a dire
t
approa
h [16, 38, 19℄, whi
h uses the theory of integration with respe
t to ve
tor measures, and
a group theoreti
al approa
h [6℄. The most general of these
hara
terizations is in [6℄. In our

ontext of a unimodular group, the
hara
terization is given by the following theorem.
T β
Theorem 3. Let be a positive operator of tra
e one. Then there is a -
ovariant observable
ET : (G) L( )
B → H , su
h that
ET(B) = d 1 β(g)(T)dλ(g)
−
(2) Z
B
B (G) E : (G) L( )
in the ultraweak sense for ea
h ∈ B . Conversely, assume that B → H is
β T
a -
ovariant observable. Then there is a unique positive operator of tra
e one, su
h that
E = ET
.
E β T
If is a -
ovariant observable, we
all the
orresponding tra
e-one positive operator the
E β E
generating operator for . Thus, for a -
ovariant observable , we have
D(f,E) = ϕ g f(g) ψ β(g)(T)ϕ λ ψ
{ ∈ H | 7→ h | i is -integrable for ea
h ∈ H}
and
ψ L(f,E)ϕ = d 1 f(g) ψ β(g)(T)ϕ dλ(g)
−
h | i Z h | i
ϕ D(f,E) ψ T E
for all ∈ and ∈ H, where is the generating operator for .
β(g) β(g)(T) = U(g)TU(g)
∗
A

ording to the Wigner theorem, ea
h has the form for some
U(g) β(g)
unitary or antiunitary operator , whi
h is unique up to a phase fa
tor, so that
or-
responds to the asso
iated equivalen
e
lass of unitary operators (see e.g. [5, p. 19℄ or [16,
G g U(g)
p. 22℄). It follows that, in the
ase where is
onne
ted, the map 7→ is a weakly
G U(g)
(Borel) measurable proje
tive unitary representation of , where ea
h is
hosen from the
β(g)
equivalen
e
lass
orresponding to by means of some (measurable) se
tion (see [16, p. 23℄
7
and [5, pp. 30, 100℄). The relation (1) gives the so-
alled square integrability
ondition
ψ U(g)ϕ 2dλ(g) = d
Z |h | i|
ψ,ϕ ϕ,ψ g G ψ U(g)ϕ =
for allunit ve
tors ∈ H. Clearly, for ea
h ∈ H there is a ∈ su
h that h | i 6
0 U(g)ϕ g G ϕ
. This implies that the
losed linear span of { | ∈ } is dense in H for ea
h ∈ H,
g U(g) β
whi
h means that the proje
tive representation 7→ is irredu
ible. The -
ovarian
e
E U(g) E(B)U(g) = E(g 1B) g G
∗ −

ondition for an observable takes the form for all ∈ ,
B (G)
∈ B .
Γ : (G) ( )
Hen
e, we know that ea
h
ovariant quantization map F → O H , whi
h
an be
Γ = ΓT := L( ,ET)
represented by an operator integral, is of the form · for some generating
T
operator . Itisworthnotingthatin
ertain
asestheobservablesprodu
edbythequantization
ΓT U
s
heme asso
iated with a map are never spe
tral measures. Namely, the irredu
ibility of
implies the following, perhaps well-known result.
G U
Proposition3. Assume that is
onne
ted, andthat theproje
tiverepresentation asso
iated
β T ( )
with is strongly
ontinuous. Let ∈ T H be positive and of tra
e one. Then the only
ET O I
proje
tions in the range of are and .
ET(X) X (G)
Proof. First we noti
e that if is a proje
tion for some ∈ B and positive operator
T ϕ E ϕ ϕ (X)
| ih |
of tra
e one, then there is a nonzero ∈ H, su
h that is a proje
tion. Indeed, let
T λ > 0 T λ 1 ϕ
be a positive operator of tra
e one, an eigenvalue of (so that ≤ ) and ∈ H an
T T = λ ϕ ϕ +(1 λ)T ϕ ϕ
′
asso
Tiated eigenve
tor. Then we
an de
ompose asET(X)|=ihλE| ϕ ϕ(X−)+(1, whλe)rEe T|′(iXh )|
′ | ih |
and are positive and of tra
e one, so we
an write − .
A L( ) 0 A I
Sin
e any proje
tion is an extreme point of the
onvex set { ∈ H | ≤ ≤ } (see e.g.
ET(X) ET(X) = E ϕ ϕ(X)
| ih |
[7, p. 19℄), it follows that if is a proje
tion, then .
η
Hen
e, it su(cid:30)
es to show that for ea
h unit ve
tor ∈ H, the only proje
tions in the range
E η η O I T = η η P
| ih |
of are and . Denote | ih |, and assume that there is a proje
tion in the range
ET PET(B) = ET(B)P B (G) ϕ
of . Then for all ∈ B [27℄. Let ∈ H. Now
ϕ Pβ(g)(T)ϕ dλ(g)= ϕ β(g)(T)Pϕ dλ(g)
Z h | i Z h | i
B B
B (G) g ϕ (Pβ(g)(T) β(g)(T)P)ϕ
for all ∈ B . Sin
e 7→ h | − i is
ontinuous, it is thus zero for all
g G Pβ(g)(T) = β(g)(T)P g G
∈ . Hen
e, for all ∈ .
U g U(g)η
η
Let denote the map 7→ . We then have
P U (g) U (g) = U (g) U (g) P
η η η η
| ih | | ih |
g G
for all ∈ .
g G U (g) P( ) U (g) P( ) f : G 0,1
η η ⊥
It follows that for ea
h ∈ , either ∈ H or ∈ H . Let → { }
f(g) = 0 U (g) P( ) f(g) = 1 U (g) P( ) f
η η ⊥
be the fun
tion su
h that if ∈ H and if ∈ H . Then
0,1 g G
0
is
ontinuous, when the set { } is equipped with the dis
rete topology. Indeed, let ∈ .
U W = U 1( ϕ U (g ) ϕ < √2 ) G
Sin
e η is
ontinuous, η− { ∈ H | k η 0 − k } is an open set in
ontaining
g f(g ) = 0 U (g)
0 0 η
. Assume (cid:28)rst that . Sin
e all ve
tors are of unit length, it follows that
U (g) U (g ) = √2 f(g) = 1 f(W) 0
η η 0
k − k whenever . Hen
e, ⊂ { }. Similarly, if we assume that
f(g ) = 1 f(W) 1 f G
0
, it follows that ⊂ { }. This implies that is
ontinuous. Sin
e is
onne
ted,
f U (g) P( ) g G U (g) P( ) g G
η η ⊥

annot be a surje
tion, so either ∈ H for all ∈ , or ∈ H for all ∈ .
8
U
But, due to the irredu
ibility of the proje
tive representation , the
losed linear span of the
U (g) g G P = I P = O
η
set { | ∈ } is dense in H. This is
learly possible only if either or . Th(cid:3)e
proof is
omplete.
The following observation is another
onsequen
e of the irredu
ibility of the proje
tive rep-
β
resentation asso
iated with . It uses a
al
ulation similar to that appearing e.g. in [40, p.40℄
in a di(cid:27)erent
ontext. Part (a) is mentioned also in [39℄.
G U
Proposition 4. Assume that is
onne
ted and let be the proje
tive representation asso
i-
β E : (G) L( ) β
ated with . Let B → H be a -
ovariant observable.
f (G) D(f,E) D(f(g ),E) g G
(a) Assume that ∈ F is su
h that ⊂ · for all ∈ . Then
U(g)D(f,E) = D(f,E) g G D(f,E) = 0 D(f,E)
for all ∈ , aend either e { } or is dense.
f (G) D(f,E) D(f(g ),E) g G
(b) Assumee that e∈ F is su
h that ⊂e · for eall ∈ . Then
U(g)D(f,E) = D(f,E) g G D(f,E) = 0 D(f,E)
for all ∈ , and either { } or is dense.
Moreover,
U(g) L(f,E)U(g) L(f(g ),E)
∗
(3) ⊂ ·
g G
for all ∈ .
T E
Proof. Let be the positive tra
e one operator asso
iated with , so that
D(f,E) = ϕ g f(g) 2 ϕ U(g)TU(g) ϕ λ ;
∗
{ ∈ H | 7→ | | h | i is -integrable }
D(f,E) = ϕ g f(g) ψ U(g)TU(g) ϕ λ ψ .
e ∗
{ ∈ H | 7→ | ||h | i| is -integrable for all ∈ H}
h,g G U(h) U(g) = c(g,h)U(h 1g) (h,g) c(h,g)
∗ −
For all ∈ , we have , where 7→ is some torus
U(g) U(h) = [U(h) U(g)] = c(g,h) 1U(h 1g)
∗ ∗ ∗ − − ∗
valued fun
tion, so that , and hen
e
U(h) U(g)TU(g) U(h) = U(h 1g)TU(h 1g) .
∗ ∗ − − ∗
(4)
ϕ D(f,E) h G
(a) Let ∈ , and ∈ . By the left invarian
e of the Haar measure and (4), we
have e
f(g) 2 U(h)ϕ U(g)TU(g) U(h)ϕ dλ(g) = f(g) 2 ϕ U(h 1g)TU(h 1g) ϕ dλ(g)
∗ − − ∗
Z | | h | i Z | | h | i
= f(hg) 2 ϕ U(g)TU(g) ϕ dλ(g),
∗
Z | | h | i
ϕ D(f(h ),E)
withallthe integrands positive. Sin
e ∈ · by assumption, the last integralis(cid:28)nite,
U(h)ϕ D(f,E) D(f,E)
so ∈ . Thus is an einvariant subspa
e of the proje
tive representation
U D(f,E) U
e e
, implying that the
losure is a
losed invariant subspa
e of . It follows from the
U D(f,E) U(h)D(f,E) = D(f,E)
irredu
ibilityof that eis either trivialor dense. The fa
t that
U(h 1) = c(h)U(h) c
− ′ ∗ ′
follows be
ause we havee for some torus valued fun
tioen . The peroof of
(a) is
omplete.
h G ϕ D(f,E) ψ
(b) Let ∈ , ∈ , and ∈ H. Then by using (4) and the assumption, we get
f(g) ψ U(g)TU(g) U(h)ϕ dλ(g) = f(g) U(h)U(h) ψ U(g)TU(g) U(h)ϕ dλ(g)
∗ ∗ ∗
Z | ||h | i| Z | ||h | i|
= f(g) U(h) ψ U(h 1g)TU(h 1g) ϕ dλ(g)
∗ − − ∗
Z | ||h | i|
= f(hg) U(h) ψ U(g)TU(g) ϕ dλ(g) < ,
∗ ∗
Z | ||h | i| ∞
9
U(h)ϕ D(f,E) D(f,E) U D(f,E)
so that ∈ . Thus is an invariant subspa
e for , and hen
e is
U(h)D(f,E) = D(f,E)
either trivial or dense. The fa
t that follows for the same reason as
the
orresponding one in (a).
h G
Let ∈ . By repeating the pre
eding
al
ulation without the absolute value signs, we get
ψ L(f,E)U(h)ϕ = U(h) ψ L(f(h ),E)ϕ = ψ U(h)L(f(h ),E)ϕ
∗
h | i h | · i h | · i
ψ ϕ U(h) D(f,E) = D(f,E) D(f(h ),E) (cid:3)
∗
for ea
h ∈ H and ∈ ⊂ · , so that (3) holds.
Remark.
D(f,E) = D(f(g ),E) g G
(a) If we assume · for all ∈ , then Proposition4 (b) gives the stri
t
U(g) L(f,E)U(g) = L(f(g ),E) g G
∗
operator equality · for ea
h ∈ , with dense domain
D(f,E) E
. This resembles the
ovarian
e
ondition for the observable .
E
ψ,ϕ
(b) Sin
e ea
h is a (cid:28)nite measure, it is
lear that the
onditions of Proposition 4 (a)
f (G)
and (b) are satis(cid:28)ed e.g. by all fun
tions ∈ F with the property that for ea
h
h G K M f(hg) K f(g) +M
h h h h
∈ there agre noGnnegative
onstantsG =aRnd2n , su
h that | | ≤ | |
for almost all ∈ . In the
aRs2enwhe(rxe ,...,x )(see pth(xe b)eginRning of thpe :neRxt Se
Rtion),
1 2n i
all polynomials of the form ∋ 7→ ∈ , where → is a
i = 1,...,2n
polynomial and , are like this.
R2
4. Phase spa
e quantization on
G = R2 λ n
Consider the spe
ial
ase where U : L,2(wRi)th the Lebesgue measure. Fix {| i} to be
an orthonormal basis of H, and let → H be the unitary operator whi
h maps
n h n W(q,p) = UW (q,p)U 1 (W (q,p)f)(t) =
n 0 − 0
the th Hermite fun
tion to | i. De(cid:28)ne , where
ei1qpeiptf(t+q) β : R2 Aut( ( )) β(q,p)(T) = W( q,p)TW( q,p) β
2 ∗
, and → T H by − − . Now is a
d = 2π λ

Ro2ntinuoAus qroup homomorphism, satisfying (1), with , annd the LebesgQue meaPsure of
. Let ± be the ladder oper1at(oArs a+ssAo
ia)ted wit1hi(tAhe basAis {)| i}, and de(cid:28)ne anAd =toAbe
the
QlosuresPof the operators √2 + − and √2 + − − , respe
tively. Then L+2(R) ∗−,
and and are unitarily equivalent to the position and momentum operators in via
U N A A
+
. Let denote the selfadjoint operator −.
T
A

ording to the general result des
ribed above, ea
h positive operator of tra
e one gen-
f L(f,ET)
erates the map 7→ , where
ET(B) = d 1 β(q,p)(T)dλ(q,p).
−
Z
B
T T = w n n w = 1 w 0
The generating operators of the form n n| ih |, where n n , and n ≥ for
n P EPT
ea
h , have a spe
ial signi(cid:28)
an
e, as they are the ones for whi
h is
ovariant with respe
t
to the phase shifts also, i.e.
eiθNET([0, ) B)e iθN = ET([0, ) (B +θ))
−
∞ × ∞ ×
θ [0,2π) B ([0,2π)) R2 = [0, ) [0,2π) B + θ
for all ∈ and ∈ B , where ∞ × and the sum is
2π
understo(oqd,pm) oduWlo(q,p.)(
f. [23℄). R2
Sin
e 7→ isastrongly
ontinuousproje
tiverepresentation, and is
onne
ted,
ET
Proposition3tellsusthattherangeof doesnot
ontainnontrivialproje
tions. Inparti
ular,
the
orresponding quantization s
heme
annot then produ
e spe
tral measures.
10