Generalized Phase Space Representation of Operators

0.15 MB·English

by  John R. Klauder

Generalized Phase-Space Representation of Operators
John R. Klaudera,b,∗ and Bo-Sture K. Skagerstamc,†
aDepartment of Physics and Department of Mathematics,
University of Florida, Gainesville, FL 32611, U.S.A.‡
bDepartment of Physics, The Norwegian University of Science and Technology, N-7491 Trondheim, Norway
cComplex Systems and Soft Materials Research Group, Department of Physics,
The Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Introducing asymmetry into the Weyl representation of operators leads to a variety of phase-space
representationsandnewsymbols. SpecificgeneralizationsoftheHusimiandtheGlauber-Sudarshan
symbols are explicitly derived.
PACS numbers42.50.-p, 42.25.Kb, 03.65.-w February 1, 2008
7
0
0
I. INTRODUCTION nals and their transformations; in this connection, see
2
[3], Sec. 4.5 and Chap. 9.

n
a Phasespacerepresentationshaveproventheirvaluein
J classical mechanics and ever since the work of Wigner
A. Basics: Wigner, Weyl, and Moyal

2 [1] in quantum mechanics as well. The map introduced
2 byWignerfromstatefunctionstoquasi-probabilityfunc-
The basic phase space representation of quantum
tionsisnotablefortwofundamentalproperties. Thefirst
2 states and operators is due to Wigner [1], Weyl [4] and
reason it is notable relates to the fact that, in general,
v Moyal [5]. The basis of this representation involves the
the Wigner quasi-probability function is not everywhere
7 Heisenberg canonicaloperator pair, P and Q, subject to
nonnegative, and this keeps it from being a true proba-
3 the commutation rule [Q,P] = i (with ~ = 1 through-
0 bility function. The second reason it is notable relates
out), expressed in their exponential form
2 to the fact that the Wigner map is not unique, and this
1 property has led to a variety of alternative prescriptions U[p,q] ei(pQ−qP) , (1)
6 to define a quantum mechanical phase space distribu- ≡
0 andknownastheWeyloperator. Assuch,weassumethe
tion. The most widely known variation on the Wigner
/ operatorsQandP arebothselfadjoint,withcontinuous
h map is that due to Cohen [2]. Cohen’s generalization,
spectrum on the entire real line, and thus the operators
p which is typical of mostsuch efforts,normally steps out-
- sidethefamilyofoperatorimagesbytheWignermap;in U[p,q] are unitary and (weakly) continuous in the real
t parameters (p,q) R2. In terms of these operators the
n other words, generalizations are introduced that involve ∈
a commutation relation assumes the form
an auxiliary phase-space function that is generally not
u
obtained by means of an operator map. U[p,q]U[r,s] = ei(ps−qr)/2U[p+r,q+s]
q
: Two well known phase-space representations for op- = ei(ps−qr)U[r,s]U[p,q], (2)
v
erators, however, are defined by direct transcriptions
i
X of operators. In particular, we refer to the Husimi whichisthe versionweshalluse. Akeyrelationwe shall
employ is the distributional identity
r phase-space representation, and its dual, the Glauber-
a Sudarshanphase-spacerepresentation,bothofwhichare Tr(U[p,q]U[r,s]†) = Tr(U[r,s]†U[p,q])
recalled below. Both representations can be defined in
= 2πδ(p r)δ(q s), (3)
an entirely abstract operator setting with the only tran- − −
scriptionbetweenoperatorsandphase space being given and it is appropriate that we reestablishthis well known
byanexclusiveuseoftheWeyloperator. Thegoalofthe expression. To that end, consider the diagonal position
present paper is the generalizationof such schemes lead- space matrix elements
ingtoamultitudeofsuchdualpairsthataredefinedina y U[p,q]U[r,s]† y
strictly operator fashion, and which, thereby, transform h | | i
= y eipQ/2e−iqPei(p−r)Q/2eisPe−irQ/2 y
covariantly under coordinate transformations. h | | i
= ei(p−r)y/2 y e−iqPei(p−r)Q/2eisP y
While our discussion is presented in the notation and
h | | i
language of quantum mechanics, a completely parallel = ei(p−r)y/2 y+q ei(p−r)Q/2 y+s
h | | i
discussion applies to a time-frequency analysis of sig-
= ei(p−r)y/2ei(p−r)(y+s)/2δ(q s). (4)
−
Integration over y leads to
‡Permanentaddress Tr(U[p,q]U[r,s]†) = ei(p−r)yei(p−r)s/2δ(q s)dy
∗Electronicaddress: [email protected] Z −
†Electronicaddress: [email protected] = 2πδ(p r)δ(q s), (5)
− −
2
as required. We make occasional use of this identity, and, correspondingly,
which we call the basic identity.
dkdx
The Weyl representation of operators (limited to B[p,q] ei(kq−xp)B˜[k,x] . (16)
Hilbert-Schmidt operators for the time being), is given ≡Z 2π
by
In terms of these representatives we have
dkdx
A= U[k,x]A˜[k,x] , (6) dpdq
Z 2π Tr(A†B)= A[p,q]∗B[p,q] , (17)
Z 2π
where A˜[k,x] is a function to be determined. If we con-
as follows from Parseval’s Theorem.
sider Tr(U[k′,x′]†A) and use the basic identity Eq. (5),
WenotethatA[p,q]andB[p,q]asintroducedhere,are
we learn that
generally referred to as the Weyl symbols for the opera-
dkdx torsAandB,respectively. Hence,A˜[k,x]andB˜[k,x]are
Tr(U[k′,x′]†A) = Tr(U[k′,x′]†U[k,x])A˜[k,x] theFouriertransformofthecorrespondingWeylsymbols.
Z 2π
= A˜[k′,x′]. (7)
II. INITIAL INTRODUCTION OF
Therefore, we have the important completeness relation ASYMMETRY
dkdx
A= U[k,x]Tr(U[k,x]†A) . (8) It is clear from Eqs. (11) and (17) that the Weyl rep-
Z 2π
resentationhas led to a symmetric functional realization
inhowittreatsthe operatorsAandB. Thisisanatural
We take a second operator with an analogous represen-
representation choice for applications in which A and B
tation,
enter in a symmetric manner. However, there are cases
dkdx whenthatisnotappropriate,andinsuchsituationsitcan
B = U[k,x]Tr(U[k,x]†B) , (9)
Z 2π prove valuable to treat the representations for A and B
in an asymmetric manner. For example, it is well known
and we again introduce the notation that that in quantum optics observablesare commonly repre-
sented by normally ordered operators, which are readily
B˜[k,x] Tr(U[k,x]†B). (10) given a phase space representation via the Husimi rep-
≡
resentation [6]. In turn, the dual representation for this
Using the basic identity Eq.(5) again, we learn that important example is known as the Glauber-Sudarshan
representation [7]. We begin our analysis of asymmetric
Tr(A†B)
representations with this important conjugate pair.
dk′dx′dkdx To discuss these alternative representations it is con-
= A˜[k′,x′]∗Tr(U[k′,x′]†U[k,x])B˜[k,x]
Z (2π)2 venient to first recall canonical coherent states given by
dkdx
= A˜[k,x]∗B˜[k,x] . (11) p,q U[p,q] 0 , (18)
Z 2π | i≡ | i
where the fiducial vector 0 is the normalized ground
In summary, the Weyl representation for operators is | i
state of a suitable harmonic oscillator, or more simply
given by
that
dkdx
A= U[k,x]A˜[k,x] , (12) (Q+iP) 0 =0. (19)
Z 2π | i
From a notational point of view, it is also useful to
where
recall that
A˜[k,x]≡Tr(U[k,x]†A). (13) p,q z =eza†−z∗a 0 =e−|z|2/2eza†e−z∗a 0
| i ≡ | i | i | i
These relations are complemented by the expression = e−|z|2/2eza† 0 =e−|z|2/2 ∞ zna†n 0
| i n! | i
dkdx nX=0
Tr(A†B)= A˜[k,x]∗B˜[k,x] . (14)
Z 2π = e−|z|2/2 ∞ zn n . (20)
X √n!| i
ItwillalsobeimportanttointroducetheFouriertrans- n=0
forms of the Weyl representation elements as
In this expression we have introduced z = (q+ip)/√2,
dkdx a = (Q+iP)/√2, from which it follows that [a,a†] = 1
A[p,q] ei(kq−xp)A˜[k,x] , (15)
≡Z 2π and a 0 = 0, and finally the normalized states n
| i | i ≡
3
a†n 0 /√n! for all n 0. These are the familiar canoni- some time (see, e.g., Ref. [8], p. 185), and we do not
| i ≥
cal coherent states; however, since we will generalize ex- repeat a proof here; it will be implicitly reestablished in
pressionswellbeyondtheseparticularstates,weshallnot what follows.
focus on the complex parametrizationwhich is generally AdirectconnectionalsoexistsbetweentheHusimiand
limited to this special family of coherent states. the Glauber-Sudarshan representations as well. In par-
Returning to the phase space notation, the Husimi ticular, we can transform Eq. (22) to read
phase-space representation of B is given by
dpdq
A [r,s] = A [p,q] p,q r,s 2
BH[p,q] p,q B(P,Q) p,q H Z G−S |h | i| 2π
≡ h | | i
= h0|B(P +p,Q+q)|0i; (21) = A [p,q]e−[(r−p)2+(s−q)2]/2 dpdq .(27)
Z G−S 2π
and the operator representation given by
The functional form of this equation as a convolution
dpdq
A A [p,q] p,q p,q , (22) reflects the connection through multiplication in the
≡Z G−S | ih | 2π
Fourier space.
It is clear from the connections above that the Husimi
involves the Glauber-Sudarshan weight function
symbol B [p,q] is generally “smoother” than the cor-
A [p,q] in a weighted integral over the one dimen- H
G−S
responding Weyl symbol B[p,q], while the Glauber-
sionalprojectionoperators p,q p,q . Itisclearthatthe
| ih | SudarshansymbolA [p,q]isgenerally“rougher”than
two symbols A [p,q] and B [p,q] are related to their G−S
G−S H
the corresponding Weyl symbol A[p,q]. In any asym-
respective operators (i.e., A and B) in very different
metric treatment of the representations this dichotomy
ways. Nevertheless, one has the important relation
is inevitable. In particular, it is known (see Ref. [8],
dpdq p. 183) that for bounded operators, for example, the
Tr(A†B) = Z AG−S[p,q]∗Tr(|p,qihp,q|B) 2π Glauber-Sudarshan symbol AG−S[p,q] ′(R2), which
∈ Z
is the Fourier transform of the more familiar space of
dpdq
= A [p,q]∗B [p,q] , (23) distributionsintwodimensions, ′(R2). Theuseofsuit-
Z G−S H 2π D
abledistributionsforphase-spacesymbolsisperfectlyac-
and it is in this sense that we refer to these two distinct ceptable provided that they are always paired with dual
representations as dual to one another. symbols that lie in the appropriate test function space.
Despite their very different appearances, there is a
fairly close connection between the initial, symmetric
Weyl representations and the latter, asymmetric Husimi III. GENERALIZED ASYMMETRIC PHASE
– Glauber-Sudarshan representations. That relationship SPACE REPRESENTATIONS
is most easilyseenin the Fouriertransformsofthe given
symbols. First, letus startfromthe symmetric Weylex- We now come to the main topic of this paper, namely
pression(11)andmodifythatexpressioninthefollowing the introduction of a large class of asymmetric phase
way: space representations. In so doing, we shall see how this
analysis incorporates and generalizes the familiar asym-
Tr(A†B) metric example discussed in the previous section.
= e(k2+x2)/4A˜[k,x] ∗ e−(k2+x2)/4B˜[k,x] dkdx Let us introduce a nonvanishing operator σ which we
Z { } { } 2π require to be a trace-class operator, i.e., we require that
dkdx 0 < Tr(√σ†σ) < . Such operators have the generic
≡ Z A˜G−S[k,x]∗B˜H[k,x] 2π . (24) form given by ∞
∞
In (24) we have introduced
σ = c b a , (28)
l l l
| ih |
A˜ [k,x]=e(k2+x2)/4A˜[k,x] Xl=0
G−S
where a ∞ and b ∞ denote two, possibly iden-
=Z ei(px−qk)AG−S[p,q]dpdq/2π, (25) tical, c{o|mlpil}elt=e0ortho{n|orlmi}al=l0sets of vectors, and the co-
efficients c ∞ satisfy the condition
and { l}l=0
∞
B˜H[k,x] = e−(k2+x2)/4B˜[k,x] Tr(√σ†σ)= cl < . (29)
| | ∞
X
= ei(px−qk)B [p,q]dpdq/2π. (26) l=0
Z H
If σ† =σ, we may choose b = a , and the coefficients
l l
| i | i
This connection between the Weyl and the Husimi – c as real for all l; however, it is not required that σ be
l
Glauber-Sudarshan representations has been known for Hermitian.
4
dpdq
WeshallhaveneedofthefunctionTr(U[k,x]σ)defined = A [p,q]∗Tr(U[p,q]σU[p,q]†B) , (35)
for all (k,x) in phase space. Observe that because σ is Z −σ 2π
traceclass,thisfunctioniscontinuous. Fornow,weinsist
an equation which, thanks to its validity for all B of the
that the expression
form B = φ ψ for arbitrary φ and ψ in the Hilbert
| ih | | i | i
Tr(U[k,x]σ)=0, (30) space, carries the important implication that
6
forall(k,x) R2;later,webrieflydiscussarelaxationof dpdq
∈ A† A [p,q]∗U[p,q]σU[p,q]† , (36)
thiscondition,butwewillalwaysrequirethatTr(σ)=0. ≡Z −σ 2π
6
The operator σ will allow us to generalize the discussion
of the previous section. As an advance notice we point or if we take the Hermitian adjoint that
out that if we make the special choice that
dpdq
A A [p,q]U[p,q]σ†U[p,q]† . (37)
σ = 0 0 , (31) ≡Z −σ 2π
| ih |
thenthegeneraldiscussionthatfollowsreferstothecase
Observe that this equation implies a very general op-
of the Husimi – Glauber-Sudarshan dual pairs.
erator representation as a linear superposition of basic
We begin again with the symmetric expression for operatorsgivenby U[p,q]σ†U[p,q]†, for a generalchoice
Tr(A†B) given by (11) which we modify so that
of σ that satisfies the conditions given initially.
Tr(A†B) Equation (37) for A is the sought for generalization
of the Glauber-Sudarshan representation; indeed, if σ =
A˜[k,x]∗ dkdx
= Tr(U[k,x]σ)B˜[k,x] 0 0, it follows immediately that
Z Tr(U[k,x]σ){ } 2π | ih |
= A˜[k,x] ∗ Tr(U[k,x]σ)B˜[k,x] dkdx A = Z A−σ[p,q]U[p,q]|0ih0|U[p,q]† d2pπdq
Z {Tr(U[k,x]†σ†)} { } 2π
dpdq
≡Z A˜−σ[k,x]∗B˜σ[k,x]dk2πdx = Z A−σ[p,q]|p,qihp,q| 2π (38)
dpdq
dpdq = A [p,q] p,q p,q . (39)
≡Z A−σ[p,q]∗Bσ[p,q] 2π . (32) Z G−S | ih | 2π
InthefinallinewehaveintroducedtheFouriertransform Once again there is a direct connection between the
of the symbols in the line above. Our task is to find al- generalizationoftheHusimirepresentation,Aσ[p,q],and
ternativeexpressionsinvolvingthesymbolsA [p,q]and the generalization of the Glauber-Sudarshan representa-
−σ
Bσ[p,q] directly in their own space of definition rather tion, A−σ[p,q]. In particular, it follows from (37) that
than implicitly through a Fourier transformation.
A [r,s] =
We begin first with the symbol B [p,q]. In particular, σ
σ
dpdq
we note that = A [p,q]Tr(U[r,s]σU[r,s]†U[p,q]σ†U[p,q]†)
Z −σ 2π
dkdx
B [p,q]= ei(kq−xp)Tr(U[k,x]σ)B˜[k,x]
σ Z 2π = A [p,q]
Z −σ
dkdx
= Tr(U[p,q]†U[k,x]U[p,q]σ)Tr(U[k,x]†B) dpdq
Z 2π [Tr(U[r p,q s]σU[r p,q s]†σ†) . (40)
× − − − − 2π
dkdx
= Tr(U[k,x]U[p,q]σU[p,q]†)Tr(U[k,x]†B)
Z 2π Again, this equation is a convolution, which just reflects
=Tr(U[p,q]σU[p,q]†B), (33) the multiplicativeconnectionbetweenthesetwosymbols
inthe Fourierspace. Note thatthe convolutionkernelin
where in the middle line we have used the Weyl form of (40) is generally complex unless σ† =σ; c.f., Eq. (27).
the commutation relations Eq. (2), and in the last line
we have used Eq. (5), which has led us to the desired
expression for B [p,q]. This expression is the sought for IV. EXAMPLES OF GENERALIZED PHASE
σ
generalization of the Husimi representation; indeed, if SPACE OPERATOR REPRESENTATIONS
σ = 0 0 it follows immediately that
| ih |
B [p,q] = Tr(U[p,q] 0 0 U[p,q]†B) In this section we offer a sample of the generalization
σ
| ih | offered by our formalism. In particular, let us choose a
= p,q B p,q =B [p,q]. (34)
h | | i H thermal density matrix
For general σ, to find the expression for A [p,q] we
−σ σ =Ze−βN , (41)
appeal to the relation
dpdq where β > 0, N = a†a is the number operator, with
Tr(A†B)= A [p,q]∗B [p,q]
Z −σ σ 2π spectrum 0,1,2,3,... , and Z = 1 e−β normalizes
{ } −
5
σ so that Tr(σ) = 1. Observe that if we take a limit which is particularly useful if the Weyl symbolis a poly-
in which β , then σ 0 0 appropriate to the nomial.
→ ∞ → | ih |
standard case. For convenience, we label this example We now turn our attention to a different but closely
simply by the parameter β. related example, which we distinguish by a prime (′). In
It follows first that particular, we choose
Tr(U[k,x]σ)=ZTr(U[k,x]e−βN)=e−θ(k2+x2)/4,(42) σ′ Z′( 1)Ne−βN =(1+e−β)e−(β+iπ)N . (51)
≡ −
where This example also corresponds to a Hermitian choice
for σ, but it is not positive definite as was our previ-
1+e−β
ous choice. The results of interest are easily calculated
θ . (43)
≡ 1 e−β simply by an analytic extension of the parameter β to
−
β′ β+iπ. In particular, it follows that
Clearly, θ >1, and θ 1 as β . Furthermore, ≡
→ →∞
B [p,q]=ZTr(U[p,q]e−βNU[p,q]†B) Tr(U[k,x]σ′)=e−θ′(k2+x2)/4 , (52)
β
∞
where
=(1 e−β) e−βn nU[p,q]†B(P,Q)U[p,q] n
− nX=0 h | | i 1+e−β′ 1 e−β
θ′ = = − . (53)
∞ 1 e−β′ 1+e−β
=(1 e−β) e−βn n B(P +p,Q+q) n . (44) −
− h | | i
X Thus, in the present case, we have a similar rescaling
n=0
factor, but this time, the parameter θ′ < 1 , with again
This expression admits alternative forms as well. the limit that θ′ 1 as β .
Based on the relation → →∞
Turning attention to the relevant symbols,we first see
B˜ [k,x]=e−θ(k2+x2)/4B˜[k,x], (45) that
β
B [p,q]=Z′Tr(U[p,q]e−β′NU[p,q]†B)
it follows that β′
∞
B [p,q]= e−(1/θ)[(p−p′)2+(q−q′)2]B[p′,q′]dp′dq′ .(46) =(1+e−β) (−1)ne−βnhn|U[p,q]†B(P,Q)U[p,q]|ni
β Z πθ 2π nX=0
∞
Still another connection is given by =(1+e−β) ( 1)ne−βn nB(P +p,Q+q)n . (54)
− h | | i
X
n=0
B [p,q]= ei(kq−xp)e−θ(k2+x2)/4B˜[k,x]dkdx
β Z 2π This expression also admits alternative forms as well.
Based on the relation
=e(θ/4)(∂2/∂p2+∂2/∂q2) ei(kq−xp)B˜[k,x]dkdx
Z 2π B˜ [k,x]=e−θ′(k2+x2)/4B˜[k,x], (55)
β′
=e(θ/4)(∂2/∂p2+∂2/∂q2)B[p,q]. (47)
it follows that
assLoectiautesdtuwrinthouAr−βat[pte,nq]t.ioInnttohitshecadseu,awl erefporceussenotnattiohne Bβ′[p,q]=Z e−(1/θ′)[(p−p′)2+(q−q′)2]B[πp′θ,′q′]dp2′πdq′ .
operator representation given by
(56)
dpdq
A=(1−e−β) Z A−β[p,q]U[p,q]e−βNU[p,q]† 2π Still another connection is given by
∞ B [p,q]=e(θ′/4)(∂2/∂p2+∂2/∂q2)B[p,q]. (57)
= (1 e−β) e−βn A [p,q] β′
− Z −β
X
n=0 Let us again turn our attention to the dual represen-
dpdq
tation associated with A [p,q]. In this case, we focus
p,q;n p,q;n , (48) −β′
×| ih | 2π on the operator representation given by
where we have introduced the notation [9] dpdq
A=(1−e−β′) Z A−β′[p,q]U[p,q]e−β′NU[p,q]† 2π
p,q;n U[p,q] n (49)
| i≡ | i ∞
=(1+e−β) ( 1)ne−βn A [p,q]
for those coherent states for which the fiducial vector is − Z −β′
X
n . Thereisanalternativerepresentationforthesymbol n=0
|unider discussion which is given by dpdq
p,q;n p,q;n .
×| ih | 2π
A [p,q]=e−(θ/4)(∂2/∂p2+∂2/∂q2)A[p,q], (50) (58)
−β
6
Thereisanotherrepresentationforthesymbolunderdis- σ = n n, where n denotes the n particle state intro-
| ih | | i
cussion given by duced earlier. For such a choice we learn that (see, e.g.,
[10])
A [p,q]=e−(θ′/4)(∂2/∂p2+∂2/∂q2)A[p,q], (59)
−β′
Tr(U[k,x] n n)= n U[k,x] n
| ih | h | | i
whichisparticularlyuseful, onceagain,ifthe Weylsym- =L (1(k2+x2))e−(k2+x2)/4 , (64)
n 2
bol is a polynomial.
It is of interest to observe, for the thermal choice of σ where Ln(y) denotes a Laguerre polynomial defined, as
(oronewithacomplextemperaturesuchasσ′)discussed usual, by
in this section, that every bounded operator A may be
ey dn
arerpereesleemnteendtsforofany′(cRho2i)c,ejoufstβ absy wweaisghtthseAc−aβs[ep,fqo]rththaet Ln(y)≡ n! dyn(yne−y). (65)
Z
Glauber-Sudarshan representation. This holds because
Each Laguerre polynomial L (y) is real and has n dis-
if we multiply a distribution in ′(R2) by an expression n
tinct, real zeros. Thus the choice σ = n n violates the
oftheformexp[θ(k2+x2)/4],itrDemainsadistributionin basic postulate that Tr(U[k,x]σ) = 0|foirhal|l (k,x) R2
′(R2) because multiplication with such a factor leaves 6 ∈
whenevern>0. However,wecanovercomethisproblem
tDhe test function space (R2) invariant.
by choosing instead a non-Hermitian example, such as
D
The examplediscussedabovehassomesimilaritywith
one treated by Cahill and Glauber [10]. In their work, σ =R n n +iT m m , (66)
| ih | | ih |
they interpret the additional factor as arising from a re-
ordering of the basic Weyl operator, i.e., a smooth tran- where m=n, and R and T are two real, nonzerofactors
6
sition from normal ordered to anti-normal ordered. On that reflect the relative weight of the two contributions.
the other hand, we interpret a similar factor as coming As a consequence, we are led to
from an alternative definition of the associated symbol.
Tr(U[k,x]σ)=[RL (1(k2+x2))
n 2
+ iTL (1(k2+x2))]e−(k2+x2)/4 , (67)
m 2
V. FURTHER EXAMPLES AND
GENERALIZATIONS which is never zero for any choice of (k,x). Further ex-
tensions along these lines are evident.
Associated with each choice of σ above is the corre-
A. Additional examples
sponding dual representationgiven, as usual, by
For a fixed, nonzero point (r,s) in phase space, con- dpdq
A A [p,q]U[p,q]σ†U[p,q]† , (68)
sider the non-Hermitian example for which ≡Z −σ 2π
σ =(1 e−β)U[r,s]†e−βN . (60) which we do not need to spell out in detail again.
−
In this case
B. Relaxation of nonvanishing criterion
Tr(U[k,x]σ)=(1 e−β)Tr(U[k,x]U[r,s]†e−βN)
−
=(1 e−β)e−i(ks−xr)/2Tr(U[k r,x s]e−βN) Returning to a general choice of σ, we observe that so
=e−i−(ks−xr)/2e−θ[(k−r)2+(x−s)2]/−4 , − (61) lBon[gp,aqs]T=r(0Uf[okr,xa]llσ)(pn,eqv)ervRa2n,isthheesn,wtheecoapnearassteorrtBth=at0if.
σ
∈
This property follows from the fact that when B˜[k,x] =
where
0,itfollowsthatTr(B†B)= B˜[k,x]2dkdx=0,which
| |
1+e−β can only happen if B =0. HoRwever, if
θ (62)
≡ 1 e−β
− Tr(U[ko,xo]σ)=0 (69)
as before. With this example, we find that
for some phase space point (k ,x ), for example, then
o o
B˜ [k,x]= the very operator B = U[ko,xo] would lead to the fact
σ that B˜[k,x] = 0, for all (k,x) in phase space, with the
e−i(ks−xr)/2e−θ[(k−r)2+(x−s)2]/4Tr(U[k,x]†B), (63)
consequence that B [p,q] 0 despite the fact that B =
σ
≡ 6
0.
which has the feature that phase space points at and
Turning to the dual symbols, we observe that so long
near (r,s) are “emphasized”in comparisonwith the rest
asTr(U[k,x]σ)nevervanishes,wecanformallyrepresent
of phase space. For applications in the time-frequency
every operator in the form
domain, for example, such an emphasis may of value.
Another class of examples is offered by the following dpdq
A= A [p,q]U[p,q]σ†U[p,q]† , (70)
observation. First, let us make a preliminary choice of Z −σ 2π
7
where the symbol – specifically, F(0,x) = 1 = F(k,0) – this is the proce-
dureusedbyCohen[2]. However,ageneralfactorF[k,x]
A [p,q]= ei(px−qk) A˜[k,x] dkdx (71) cannotnormallyberepresentedasTr(U[k,x]σ)forsome
−σ Z Tr(U[k,x]†σ†) 2π σ. For example, the function Tr(U[k,x]σ) is necessarily
continuousandboundedby Tr(U[k,x]σ) Tr(√σ†σ),
is a distribution appropriate to the situation under con- | |≤
the trace class norm of σ. However, more to the point,
sideration. More precisely, we can represent all bounded
the introduction of a general expression such as F[k,x]
operators in such a fashion, and every operator can be
would normally introduce elements outside the Hilbert
obtained as a suitable limit from the set of bounded op-
space and its operators that we have so far consistently
erators. On the other hand, if we relax the condition
stayed within. Such an extension may of course be con-
that Tr(U[k,x]σ) never vanishes, we loose this general-
sidered,butthenon-operatornatureintheextensionbe-
ity. Suppose again that Tr(U[k ,x ]σ) = 0 for some
o o
ing made should be appreciated and accepted.
particular point in phase space (k ,x ). In that case, it
o o
is strictly speaking not possible to generate the operator Ofcourse,the investigationof quasi-distributionscon-
U[k ,x ]σ†U[k ,x ]† . tinues unabated. As one comparatively recent example
o o o o
Ontheotherhand,ifoneisonlyinterestedinoperators of such investigations,we cite the work of [12].
that are polynomial in P and Q, then the support of the
Weyl symbol A˜[k,x] or B˜[k,x] is strictly at the origin in
phase space, e.g., for C˜ equal either A˜ or B˜,
D. Non-canonical generalizations
I,J di dj
C˜[k,x]= c δ(k)δ(x), (72) RepresentationsofHilbertspaceoperatorsintheman-
X i,jdki dxj ner of the Weyl representation may be carried out for
i,j=0
a great variety of groups. In addition, since coherent
where I < and J < . In that case, the fact that states may be defined for other groups, analogs of the
∞ ∞
Tr(U[k,x]σ) may vanish away from the origin has no Husimi and dual Glauber-Sudarshan representations ex-
influence on the matter, and thus so long as Tr(σ) = 0, ist in such cases as well, and these have often been dis-
6
andthusbycontinuityTr(U[k,x]σ)isnonzeroinanopen cussed in the literature. Consequently, it follows that
neighborhood of the origin, all polynomials in P and Q asymmetric representations of various forms, analogous
can be represented in the form to those presented in this paper for the Weyl group, can
be introduced for other groups, e.g., the groups SU(2)
dpdq
A=Z A−σ[p,q]U[p,q]σ†U[p,q]† 2π , (73) and SU(1,1), as well.
and, additionally, the symbol
B [p,q]=Tr(U[p,q]σU[p,q]†B) (74)
σ VI. CONCLUSIONS
uniquely determines the operatorB providedthat it is a
polynomial. This result generalizes a result established Quite naturally, an increase in the family of represen-
some time ago [11]. tations of various systems offers new ways to study such
systems. Ageneralformulationofstatesandobservables
must lead to a symmetric, abstract description; but for
C. Arbitrary weight factors specificsystems,wherethestatesofinterestmaybe“bet-
ter behaved” than the observables, it may be useful to
It is of course possible to reweight the Weyl represen- take advantage of that distinction with an asymmetric
tation by quite arbitrary factors in the manner representation pair. Likewise, within the realm of time-
frequency analysis, signals and their analyzers may have
dkdx
Tr(A†B)= A˜[k,x]∗B˜[k,x] very different characteristics that also suggest the util-
Z 2π ity of an asymmetric pair. Moreover, to minimize any
= (F[k,x]∗)−1A˜[k,x] ∗ F[k,x]B˜[k,x] dkdx extraneouselementsinsuchasymmetricrepresentations,
Z { } { } 2π it is appropriate that the representations in question be
dkdx formulatedwithin the initial abstractformalism, a point
≡Z A˜−F[k,x]∗B˜F[k,x] 2π of view developed in the present paper.
dpdq Althoughitisdifficulttopredictinadvancejustwhich
≡Z A−F[p,q]BF[p,q] 2π , (75) asymmetric representation pairs may prove useful, there
may well arise special situations for which certain asym-
where F[k,x] is an arbitrary nonvanishing factor, and, metricrepresentationsproveuseful–justaswasthecase
as usual, we have introduced the Fourier transform ele- in quantum optics for the Husimi – Glauber-Sudarshan
ments in the last line. Apart from additional conditions representation pair.
8
ACKNOWLEDGEMENTS of Science and Technology, in Trondheim, as well as the
support provided through the 2006 Lars Onsager Pro-
JRK is pleased to acknowledge the hospitality of the fessorship for an extended visit during which time this
Complex Systems and Soft Materials ResearchGroup of paper was prepared.
the Department of Physics, The Norwegian University
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