Husimi Transform of an Operator Product

13 Pages·0.18 MB·

by  D. M. Appleby

HUSIMI TRANSFORM OF AN OPERATOR PRODUCT
D M APPLEBY
Department of Physics, Queen Mary and Westfield College, Mile End Rd, London
E1 4NS, UK
0
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(E-mail: [email protected])
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2

n
a
J

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2 Abstract

1 It is shown that the series derived by Mizrahi, giving the Husimi
v transform(orcovariantsymbol)ofanoperatorproduct,isabsolutely
3 convergentforalargeclassofoperators. Inparticular,thegeneralized
0
Liouville equation, describing the time evolutionof the Husimi func-
1
tion, is absolutely convergent for a large class of Hamiltonians. By
1
0 contrast,theseriesderivedbyGroenewold,givingtheWeyltransform
0 of an operator product, is often only asymptotic, or even undefined.
0 The result is used to derive an alternative way of expressing expec-
/
h tation values in terms of the Husimi function. The advantageof this
p formula is that it applies in many ofthe cases where the anti-Husimi
- transform(or contravariantsymbol)is so highly singular that it fails
t
n to exist as a tempered distribution.
a
u
q
:
v
i
X
r
a
1
1. Introduction
A particularly useful and illuminating way of studying the classical limit is to
formulate quantum mechanics in terms of phase space distributions [1, 2, 3]. The
advantage of such a formulation as compared with the standard Hilbert space for-
mulation is that it puts quantum mechanics into a form which is similar to the
probabilistic phase space formulation of classical mechanics. At least from a for-
mal, mathematical point of view it thus allows one to regard quantum mechanics
as a kind of generalized version of classical mechanics.
There are, of course, many different phase space formulations of quantum me-
chanics. TheonewhichwasdiscoveredfirstistheformulationbasedontheWigner
function [1, 2, 3, 4]. In the case of a system having one degree of freedom with
position xˆ, momentum pˆand density matrix ρˆthe Wigner function is defined by
1
W(x,p)= dyeipy x y ρˆ x+ y
2π − 2 2
Z
(cid:10) (cid:12) (cid:12) (cid:11)
(in units chosen such that ~ = 1). The Wigner fu(cid:12)nc(cid:12)tion continues to find many
important applications (in quantum tomography [3], for example). However, if
the aim is specifically to represent quantum mechanics in a manner which resem-
bles classical mechanics as closely as possible, then the Wigner function suffers
from the serious disadvantage that it is not strictly non-negative (except in spe-
cial cases [5])—which makes the analogywith the classicalphase space probability
distribution somewhat strained.
There has accordingly been some interest in the problem of constructing alter-
native distributions, which are strictly non-negative,and which can be interpreted
as probability density functions. There are, in fact, infinitely many such func-
tions [6, 7, 8,9, 10, 11]. The one which wasdiscoveredfirst, and whichis the focus
ofthispaper,istheHusimi,orQ-function[1,2,3,12,13,14,15],whichisobtained
from the Wigner function by smearing it with a Gaussian convolution:
1
Q(x,p)= dx′dp′ exp (x x′)2 (p p′)2 W(x′,p′)
π − − − −
Z
(cid:2) (cid:3)
(in units such that ~ = 1, and where we assume that x, p have been made di-
mensionless by choosing a suitable length scale λ, and making the replacements
x x/λ, p λp). It should be emphasised that it is not simply that the Husimi
→ →
function has the mathematical significance of a probability density function. It
also has this significance physically. It has been shown that the Husimi function
is the probability distribution describing the outcome of a joint measurement of
position and momentum in a number of particular cases [3, 8, 16, 17, 18]. More
generally it can be shown [19, 20] that the Husimi function has a universal signifi-
cance: namely, it is the probability density function describing the outcome of any
retrodictivelyoptimaljointmeasurementprocess. InAppleby [20]itisarguedthat
this means that the Husimi function may be regarded as the canonical quantum
mechanicalphase space probability distribution, which plays the same role in rela-
tion to joint measurements of x and p as does the function x ψ 2 in relation to
|h | i|
single measurements of x only.
If one wants to construct a systematic procedure for investigatingthe transition
fromquantaltoclassicalitisnotenoughsimplytofindananaloguefortheclassical
phase space probability distribution. One also needs an analogue of the classical
Liouville equation, giving the time evolution of the probability distribution. In
the formulation based on the Husimi function this is accomplished by means of
Mizrahi’s formula [15], giving the Husimi transform of an operator product (also
see Lee [2], Cohen [21], Prugoveˇcki[22] and O’Connell and Wigner [23]).
2
Let A denote the Weyl transform of the operator Aˆ, defined by [1, 2, 24]
W
A (x,p)= dyeipy x y Aˆ x+ y
W − 2 2
Z
(cid:10) (cid:12) (cid:12) (cid:11)
The Husimi transform (or covariant symbol) A i(cid:12)s th(cid:12)en given by
H
1
A (x,p)= dx′dp′ exp (x x′)2 (p p′)2 A (x′,p′) (1)
H π − − − − W
Z
(cid:2) (cid:3)
Mizrahi [15] has derived the following formula for the Husimi transform of the
product of two operators Aˆ, Bˆ:
←−−→
(AˆBˆ) =A e∂+∂−B (2)
H H H
where ∂ = 2−1/2(∂ i∂ ). Using this formula, and the fact that the Husimi
± x p
∓
function is just the Husimi transform of the density matrix scaled by a factor
1/(2π), it is straightforward to derive the following generalization of the Liouville
equation:
∂
Q= H ,Q (3)
∂t { H }H
where H is the Husimi transformofthe Hamiltonian, and H ,Q is the gener-
H { H }H
alized Poisson bracket
∞
2
H ,Q = Im ∂nH ∂nQ (4)
{ H }H n! + H −
n=0
X (cid:0) (cid:1)
Thefirstterminthesumontheright-handsideisjusttheordinaryPoissonbracket.
The remaining terms represent quantum mechanical corrections.
ItisnotapparentfromMizrahi’sderivation,whethertheseexpressionsareexact,
orwhethertheyareonlyasymptotic. InSection2wewillshowthatthereisalarge
class of operators for which the series in Eq. (2) [and consequently the series in
Eq. (4)] is absolutely convergent . This property is closely connected with the
complex analytic properties of the Husimi transform, as discussed by Mehta and
Sudarshan [25] and Appleby [26].
The significance of the result proved in Section 2 is best appreciated if one
compares Eqs. (2–4) with the corresponding formulae in the Wigner-Weyl formal-
ism [1, 2, 27, 28, 29]:
(AˆBˆ)W =AWexp 2i ←∂−x−∂→p −←∂−p−∂→x BW (5)
∂ h (cid:16) (cid:17)i
W = H ,W (6)
∂t { W }W
∞ ( 1)n 2n+1
{HW,W}W = (2n+−1)!22nHW ←∂−x−∂→p −←∂−p−∂→x W (7)
nX=0 (cid:16) (cid:17)
whereH istheWeyltransformoftheHamiltonian,andwhere H ,W denotes
W { W }W
the Moyal bracket [28].
Itcanbe seenthatEqs(2–4)andEqs.(5–7)areformallyverysimilar. However,
this formal resemblance is somewhat deceptive, for it turns out that the two sets
of equations have quite different convergence properties.
The formula for the Weyl transform of an operator product, Eq. (5), is exact if
either Aˆ or Bˆ is a polynomial in xˆ and pˆ(in which case the series terminates after
a finite number of terms). More generally, if A , B are C∞ functions satisfying
W W
appropriate conditions on their growth at infinity, then it can be shown that the
seriesisasymptotic[30]. However,therearemanyoperatorsofphysicalinterestfor
whichtheWeyltransformisonlydefinedinadistributionalsense,andforoperators
3
such as this the series can be highly singular. Consider, for example, the parity
operator Vˆ, whose action in the x-representation is given by
x Vˆ ψ = x ψ
−
We have (cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12)
V (x,p)=πδ(x)δ(p)
W
Substituting this expression into Eqs. (5) gives
(Vˆ2)W(x,p)=π2δ(x)δ(p)exp 2i ←∂−x−∂→p −←∂−p−∂→x δ(x)δ(p)
h (cid:16) (cid:17)i
The left-hand side of this equation = 1; whereas the expression on the right-hand
side is an infinite sum eachindividual term of which is ill-defined (being a product
of distributions concentrated at the origin).
Mizrahi’s [15] derivationof the formula for the Husimi transformof an operator
product, Eq. (2), depends on the same kind of formal manipulation that is used
in Groenewold’s [27] derivation of Eq. (5), and so it might be supposed that the
validity of the formula is similarly restricted. However, it turns out that the sum
in Eq. (2) is actually much better behaved. In fact, it will be shown in Section 2
that, subject to certain not very restrictive conditions on the operators Aˆ and Bˆ,
the sum on the right-hand side of Eq. (2) is not only defined and asymptotic; it is
even absolutely convergent for all x, p. This is essentially because A is typically
H
a much less singular object than A [due to the Gaussian convolution in Eq. (1)].
W
In Section 3 we apply the result just described to the problem of expressing
expectation values in terms of the Husimi function.
TheexpectationvalueofanoperatorAˆcanbeobtainedfromtheWignerfunction
using the formula [1, 2]
Tr(ρˆAˆ)= dxdpA (x,p)W(x,p) (8)
W
Z
In certain cases we can also express the expectation value in terms of the Husimi
function using [1, 2, 15]
Tr(ρˆAˆ)= dxdpA (x,p)Q(x,p) (9)
H
Z
where A is the anti-Husimi transform (or contravariantsymbol) of Aˆ, defined by
H
A =e−∂+∂−A (10)
H H
[with ∂ =2−1/2(∂ i∂ ), as before]. Eq.(9)is valid (for example)whenever [31]
± x p
∓
A exists as a tempered distribution and Q belongs to the corresponding space
H
of test functions (i.e. the C∞ functions of rapid decrease). However, we have the
problem that A is often so highly singular that it is not defined as a tempered
H
distribution—whichmeansthattheusefulnessofEq.(9)issomewhatlimited. This
is often seen as a serious drawback of the Husimi formalism.
However,it turns out that it is often possible to circumvent this difficulty. Sup-
pose we substitute the series given by Eq. (10) into the right hand side of Eq. (9),
and suppose we then reverse the order of sum and integral. This gives
∞ ( 1)n
Tr(ρˆAˆ)= − dxdp ∂n∂nA (x,p) Q(x,p) (11)
n! + − H
n=0 Z
X (cid:0) (cid:1)
In Section 3 we show that it often happens that the sum on the right-hand side of
this equationis absolutely convergent,evenin many of the cases where A fails to
H
exist as a tempered distribution.
4
2. Convergence of the Product Formula
We will find it convenient to work in terms of coherent states. Define
1 1
aˆ= (xˆ+ipˆ) aˆ† = (xˆ ipˆ)
√2 √2 −
and let φ denote the nth (normalised) eigenstate of the number operator aˆ†aˆ:
n
1
aˆφ =0 φ = (aˆ†)nφ
0 n 0
√n!
Let Dˆ be the displacement operator
xp
Dˆ =ei(pxˆ−xpˆ)
xp
and define
φ =Dˆ φ φ =φ
n;xp xp n xp 0;xp
The φ are the coherent states. Let Aˆ be any operator (not necessarily bounded)
xp
with domain of definition D , and suppose that φ D for all x, p. It is then
Aˆ xp ∈ Aˆ
straightforwardto show that
A (x,p)= φ , Aˆφ
H h xp xpi
(wenolongerusetheDiracbra-ketnotation,becausetheexistenceof ψ, Aˆχ does
h i
not, in general, imply the existence of Aˆ†ψ, χ ).
h i
IfAˆ,Bˆ areboth bounded then the proofofEq.(2)is comparativelystraightfor-
ward. However, we want to make the proof as general as possible. We then have
the difficulty that the sum in Eq. (2) will only be defined if A and B are both
H H
C∞; whereas functions of the form Dˆ ψ, AˆDˆ ψ are, in general, not even once-
xp xp
h i
differentiable, let alone C∞. We are thus faced with the question: what conditions
must we impose onthe operatorAˆin order to ensure that the function A is C∞?
H
One answer to this question is given by the following theorem.
Theorem 1. Let D , D be the domains of definition of Aˆ, Aˆ† respectively.
Aˆ Aˆ†
Suppose that φ D D for all x, p. Suppose, also, that φ , Aˆφ ,
xp ∈ Aˆ ∩ Aˆ† h x1p1 x2p2i
φ , Aˆφ and φ , Aˆ†φ are continuous functions on R4. Then A
h 1;x1p1 x2p2i h 1;x1p1 x2p2i H
is an analytic function, which uniquely continues to a holomorphic function defined
on the whole of C2.
The continuation is given by
φ , Aˆφ
A (x,p)= h x−p− x+p+i (12)
H φ , φ
h x−p− x+p+i
where x,p are arbitrary complex, and where x ,p are the real variables defined by
± ±
1 i
x = (x+x∗) (p p∗) (13)
±
2 ± 2 −
1 i
p = (p+p∗) (x x∗) (14)
±
2 ∓ 2 −
This theorem is a strengthened version of results proved by Mehta and Sudar-
shan [25] and Appleby [26]. The proof is given in Appendix A.
It is worth noting that the condition in the statement of this theorem is quite
weak. If the three functions listed exist and are continuous then, without making
any explicit assumption regarding the differentiability of these functions, it auto-
matically follows that A must be complex analytic.
H
We also have the following lemma:
5
Lemma 2. Suppose that Aˆ satisfies the conditions of Theorem 1. Then
hφn;x−p−, Aˆφx+p+i = 1 n n (z z∗)n−r ∂r A (x,p) (15)
hφx−p−, φx+p+i √n!r=0(cid:18)r(cid:19) +− − ∂z−r H
X
hAˆ†φx−p−, φn;x+p+i = 1 n n (z z∗)n−r ∂r A (x,p) (16)
hφx−p−, φx+p+i √n!r=0(cid:18)r(cid:19) −− + ∂z+r H
X
where x , p are the variables defined by Eqs. (13) and (14), and where
± ±
1
z = (x ip)
±
√2 ±
The proof of this lemma is given in Appendix B.
If x, p are both real (so that z =z∗) Eqs. (15) and (16) become
− +
1
φ , Aˆφ = ∂nA (x,p) (17)
h n;xp xpi √n! − H
1
Aˆ†φ , φ = ∂nA (x,p) (18)
h xp n;xpi √n! + H
where
∂ 1 ∂ ∂
∂ = = i
±
∂z± √2(cid:18)∂x ∓ ∂p(cid:19)
Using these results the proof of the product formula becomes very straightfor-
ward. Let Aˆ, Bˆ be any pair of operators satisfying the conditions of Theorem 1.
Suppose, also, that φ D for all x,p R. Then, for real x, p,
xp ∈ AˆBˆ ∈
(AˆBˆ) (x,p)= φ , AˆBˆφ
H h xp xpi
= Aˆ†φ , Bˆφ
xp xp
h i
∞
= Aˆ†φ , φ φ , Bˆφ (19)
xp n;xp n;xp xp
h ih i
n=0
X
where we have used the fact that the φ constitute an orthonormal basis. The
n;xp
absoluteconvergenceofthissumisanimmediateconsequenceofbasicHilbertspace
theory. Using Eqs. (17) and (18) we deduce
∞
1 ←−−→
(AˆBˆ) (x,p)= ∂nA (x,p)∂nB (x,p)=A (x,p)e∂+∂−B (x,p)
H n! + H − H H H
n=0
X
which is the product formula. The absolute convergence of this sum follows from
the absolute convergence of the sum in Eq. (19).
3. Expectation Values
We now discuss the implications that the result just proved has for the con-
vergence of Eq. (11), giving the expectation value of Aˆ in terms of the Husimi
function.
Of course, one does not expect the right hand side of Eq. (11) to converge for
arbitrary Aˆ and ρˆ—since, apart from anything else, an unbounded operator does
not have a well-defined expectation value for every state ρˆ. We therefore need to
place some kind of restriction on the class of operators Aˆ and density matrices ρˆ
considered. Theresultweproveisprobablynotthemostgeneralpossible. However,
it will serve to illustrate the point, that the sum on the right hand side of Eq. (11)
is often absolutely convergent, even in many of the cases where the anti-Husimi
transform fails to exist as a tempered distribution.
6
We accordingly confine ourselves to the case of density matrices for which the
Husimi function Q I(R2), where I(R2) is the space of C∞ functions which are
∈
rapidly decreasing at infinity [31] (i.e. the space of test functions for the space of
tempered distributions). In other words, we assume that
sup (1+x2+p2)l∂m∂nQ(x,p) <
(x,p)∈R2 x p ∞
(cid:12) (cid:12)
for every triplet of non-negat(cid:12)ive integers l, m, n. (cid:12)
We assume that Aˆ has the properties
1. φ D D for all x,p R.
xp ∈ Aˆ∩ Aˆ† ∈
2. There exist positive constants K and non-negative integers N such that
± ±
Aˆφ K (1+x2+p2)N− (20)
xp −
k k≤
Aˆ†φ K (1+x2+p2)N+ (21)
xp +
k k≤
for all x, p R.
∈
We willsaythatanoperatorsatisfyingthese twoconditions is polynomial bounded.
The following lemma gives two properties of such operators which will be needed
in the sequel.
Lemma 3. Suppose that Aˆ is polynomial bounded. Then
1. Aˆ satisfies the conditions of Theorem 1. In particular, A is analytic.
H
2. For every pair of non-negative integers m, n there exists a positive constant
K and a non-negative integer N such that
mn mn
∂m∂nA (x,p) K (1+x2+p2)Nmn (22)
x p H ≤ mn
for all x, p R(cid:12). (cid:12)
∈ (cid:12) (cid:12)
The proof is given in Appendix C.
We now ready to prove the main result of this section.
Theorem 4. SupposethatAˆis polynomially bounded, andsupposethatthedensity
matrix ρˆ is such that the corresponding Husimi function is rapidly decreasing at
infinity. Suppose, also, that Aˆρˆis of trace-class. Then
∞ ( 1)n
Tr(Aˆρˆ)= − dxdp ∂n∂nA (x,p) Q(x,p) (23)
n! + − H
n=0 Z
X (cid:0) (cid:1)
where the sum on the right hand side is absolutely convergent.
Proof. We have
1
Tr(Aˆρˆ)= dxdp φ , Aˆρˆφ
xp xp
2π h i
Z
∞
1
= dxdp Aˆ†φ , φ φ , ρˆφ
xp n;xp n;xp xp
2π h ih i!
Z n=0
X
We now use Lebesgue’s dominated convergence theorem [31] to show that we may
reversetheorderofsumandintegral. Infact,itfollowsfromtheSchwartzinequality
that
m
Aˆ†φ , φ φ , ρˆφ
xp n;xp n;xp xp
(cid:12) h ih i(cid:12)
(cid:12)nX=0 (cid:12)
(cid:12) (cid:12) 1
(cid:12) m (cid:12) m 2
(cid:12) Aˆ†φ(cid:12) , φ 2 φ , ρˆφ 2
xp n;xp n;xp xp
≤ (cid:18)n=0 h i (cid:19)(cid:18)n=0 h i (cid:19)!
X(cid:12) (cid:12) X(cid:12) (cid:12)
Aˆ†φ (cid:12) ρˆφ (cid:12) (cid:12) (cid:12)
xp xp
≤k kk k
7
We have
1 1
ρˆφ = φ , ρˆ2φ 2 φ , ρˆφ 2 = 2πQ(x,p)
xp xp xp xp xp
k k h i ≤ h i
which, together with I(cid:0)nequality (21(cid:1)), imp(cid:0)lies (cid:1) p
Aˆ†φ ρˆφ √2πK (1+x2+p2)N+ Q(x,p)
xp xp +
k kk k≤
By assumption, Q(x,p) I(R2). It follows that Aˆ†φ pρˆφ is integrable. We
xp xp
∈ k kk k
may therefore use Lebesgue’s dominated convergence theorem [31] to deduce
∞
1
Tr(Aˆρˆ)= dxdp Aˆ†φ , φ φ , ρˆφ (24)
xp n;xp n;xp xp
2π h ih i
n=0Z
X
where the sum is absolutely convergent,since
∞
dxdp Aˆ†φ , φ φ , ρˆφ dxdp Aˆ†φ ρˆφ <
xp n;xp n;xp xp xp xp
h ih i ≤ k kk k ∞
n=0(cid:12)Z (cid:12) Z
X(cid:12) (cid:12)
We kn(cid:12)(cid:12)ow from Lemma 3 that Aˆ satisfies (cid:12)(cid:12)the conditions of Theorem 1. We may
therefore use the results proved in the last section to rewrite Eq. (24) in the form
∞
1
Tr(Aˆρˆ)= dxdp∂nA (x,p)∂nQ(x,p)
n! + H −
n=0 Z
X
Finally, it follows fromInequality (22),together with the fact that Q I(R2), that
∈
we may partially integrate term-by-term to obtain
∞ ( 1)n
Tr(Aˆρˆ)= − dxdp ∂n∂nA (x,p) Q(x,p)
n! + − H
n=0 Z
X (cid:0) (cid:1)
The right hand side of Eq. (8) (expressing Aˆ in terms of the Wigner function)
is defined whenever W I(R2) and A exihstsias a tempered distribution. On
W
the other hand, althoug∈h it is true that Q I(R2) whenever W I(R2) (see
∈ ∈
TheoremIX.3ofref.[31]),thefactthatA existsasatempereddistributionisnot
W
evidently sufficientto ensurethatAˆis polynomialbounded. So we havenotshown
that Eq. (23) has the same range of validity as Eq. (8). However, it can be shown
that Aˆ is polynomially bounded if φ D D , and if (Aˆ†Aˆ) and (AˆAˆ†)
xp ∈ Aˆ∩ Aˆ† W W
exist as tempered distributions (see Theorem IX.4 of ref. [31]). In the applications
onemeetswithoperatorssatisfyingtheseconditionsmuchmorecommonlythanone
meets withoperatorsforwhichA exists asa tempereddistribution. Forinstance,
H
everyboundedoperatorispolynomiallybounded,whereastherearemanybounded
operatorsofphysicalinterestforwhichA failstoexistasatempereddistribution.
H
The above result consequently represents a significant improvement on the results
that were previously known.
Of course, just from the fact that Eq. (23) is convergent, it does not necessar-
ily follow that the convergence is sufficiently rapid to make the formula useful in
practical, numerical work. This question requires further investigation.
4. Conclusion
AshasbeenstressedbyMizrahi[15],Lalovi´cetal [9],Davidovi´candLalovi´c[33]
and others, the Husimi formalism provides an especially perspicuous method for
studying the relationship between quantum and classical mechanics. It establishes
a one-to-one correspondence between the basic equations of the two theories, so
thatone canstartwith aclassicalformula,andthen turnit intothe corresponding
quantum formula by adding successive correction terms. Moreover, the fact that
8
Q(x,p)describestheoutcomeofaretrodictivelyoptimaljointmeasurementofxand
p [19, 20], means that one could reasonably argue that the Husimi function is the
mostnaturalchoicefor a quantummechanicalanalogueofthe classicalprobability
distribution.
In this paper we have investigated the convergence properties of two of the key
formulae in the Husimi formalism. We have shown that the formula giving the
Husimi transform of an operator product has much better convergence properties
than the corresponding formula in the Wigner function formalism. In particular,
theHusimiformalismleadstoaconvergentgeneralizationofthe Liouvilleequation
for a very large class of Hamiltonians. We have also shown that the convergence
properties of the formula expressing the expectation value Aˆ in terms of the
h i
Husimi function, although seemingly not as good as those of the corresponding
formula in the Wigner function formalism, are significantly better than the often
highly singular character of A would suggest.
H
These results lend additional support to the suggestion that, in so far as the
aimisspecifically toformulate quantummechanicsas akindofgeneralizedversion
of classical mechanics, then the formalism based on the Husimi function has some
significant advantages.
Appendix A. Proof of Theorem 1
For arbitrary complex x, p define
φ , Aˆφ
F(x,p)= h x−p− x+p+i
φ , φ
h x−p− x+p+i
where x , p are the (real) variables defined by Eqs. (13) and (14).
± ±
It is easily seen that, if x, p are both real, then
F(x,p)= φ , Aˆφ =A (x,p)
h xp xpi H
Theproblemthusreducestothatofshowingthat,ifAˆhasthepropertiesstipulated,
then F is holomorphic. We will do this by showing that F satisfies the Cauchy-
Riemann equations with respect to the complex variables
1 1
z = (x ip )= (x ip) (25)
± ± ±
√2 ± √2 ±
In fact, it is straightforwardto show that φ , regardedas a vector-valuedfunc-
xp
tion of two real variables, is differentiable in the norm topology; the derivatives
being given by
∂ 1 i
φ = φ pφ
xp 1;xp xp
∂x √2 − 2
∂ i i
φ = φ + xφ
xp 1;xp xp
∂p √2 2
Also,
1 1 i
φ , φ =exp (x x )2 (p p )2+ (p x p x )
h x−p− x+p+i −4 +− − − 4 +− − 2 + −− − +
Consequently, F is differ(cid:2)entiable with respect to the variables x , p . More(cid:3)over
− −
∂ 1 φ , Aˆφ 1
F(x,p)= h 1;x−p− x+p+i + (z∗ z )F(x,p)
∂x− √2 hφx−p−, φx+p+i √2 −− +
∂
=i F(x,p) (26)
∂p
−
from which it follows that F satisfies the Cauchy-Riemann equations with respect
to the complex variable z =(x ip )/√2.
− − −
−
9
We can alternatively write
Aˆ†φ , φ
F(x,p)= h x−p− x+p+i
φ , φ
h x−p− x+p+i
Consequently, F is also differentiable with respect to the real variables x , p .
+ +
Moreover
∂ 1 Aˆ†φ , φ 1
F(x,p)= h x−p− 1;x+p+i + (z∗ z )F(x,p)
∂x+ √2 hφx−p−, φx+p+i √2 +− −
∂
= i F(x,p) (27)
− ∂p
+
from which it follows that F satisfies the Cauchy-Riemann equations with respect
to the complex variable z =(x +ip )/√2.
+ + +
If Aˆ has the properties specified in the statement of theorem, then we see from
Eqs. (26) and (27) that the partial derivatives ∂F/∂x , ∂F/∂p , are continuous
± ±
functions on R4. It follows [32] that F is a holomorphic function of the complex
variables z . Referring to Eq. (25) it can be seen that the variables z are linear
± ±
combinations of x, p. We conclude that F is a holomorphic function of x, p.
Appendix B. Proof of Lemma 2
It is straightforwardto show that φ , regardedas a vector valued function
n;x−p−
of two real variables, is differentiable in the norm topology. Moreover
1 ∂ ∂ 1
i φ =√n+1φ + z φ
√2(cid:18)∂x− − ∂p−(cid:19) n;x−p− (n+1);x−p− 2 − n;x−p−
Hence
1 ∂ ∂ φ , Aˆφ
+i h n;x−p− x+p+i
√2(cid:18)∂x− ∂p−(cid:19) hφx−p−, φx+p+i
φ , Aˆφ φ , Aˆφ
=√n+1h (n+1);x−p− x+p+i + z∗ z h n;x−p− x+p+i
φ , φ −− + φ , φ
h x−p− x+p+i h x−p− x+p+i
(cid:0) (cid:1)
Iterating this result, and using
1 ∂ ∂ r ∂r
+i A (x,p)= A (x,p)
2r/2 ∂x ∂p H ∂zr H
(cid:18) − −(cid:19) −
we obtain Eq. (15).
The proof of Eq. (16) is similar.
Appendix C. Proof of Lemma 3
Proofof(1). Weneedtoshowthat,ifAˆispolynomialbounded,thenthefunctions
φ , Aˆφ , φ , Aˆφ and φ , Aˆ†φ are continuous.
h x1p1 x2p2i h 1;x1p1 x2p2i h 1;x1p1 x2p2i
Consider the function φ , Aˆφ . We have
h 1;x1p1 x2p2i
φ , Aˆφ φ , Aˆφ
1;x′1p′1 x′2p′2 − 1;x1p1 x2p2
(cid:12) (cid:12)
(cid:12)(cid:10) (cid:11)(φ (cid:10) φ ), A(cid:11)ˆ(cid:12)φ + φ , Aˆ(φ φ )
(cid:12) ≤ 1;x′1p′1 − 1;x1p1 (cid:12) x′2p′2 1;x1p1 x′2p′2 − x2p2
(cid:12) (cid:12) (cid:12) (cid:12)
In view of Eq. (20(cid:12))(cid:10)we have (cid:11)(cid:12) (cid:12)(cid:10) (cid:11)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(φ1;x′1p′1 −φ1;x1p1), Aˆφx′2p′2 ≤K−(1+x′22+p′22)N−kφ1;x′1p′1 −φ1;x1p1k
(cid:12) (cid:12)
(cid:12)(cid:10) (cid:11)(cid:12)
(cid:12) (cid:12)