Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space

22 Pages·0.3 MB·English

by  Cosmas K Zachos

ANL-HEP-PR-01-095
DEFORMATION QUANTIZATION:
QUANTUM MECHANICS LIVES AND WORKS IN PHASE-SPACE
2
0
0
2 COSMASZACHOS

n High Energy Physics Division, Argonne National Laboratory
a Argonne, IL 60439-4815, USA
J E-mail: [email protected]

9

3
v Wigner’s quasi-probabilitydistributionfunction inphase-space is a special (Weyl) rep-
4 resentation of the density matrix. It has been useful in describing quantum transport
1 in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum
1 chaos;“WelcherWeg”discussions;semiclassicallimits. Itisalsoofimportanceinsignal
0 processing. Nevertheless, a remarkableaspect of its internal logic, pioneered byMoyal,
1 has only emerged in the last quarter-century: It furnishes a third, alternative, formu-
1 lation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path
0 Integralformulations. Inthislogicallycompleteandself-standingformulation,oneneed
/ notchoose sides—coordinateormomentum space. It worksinfullphase-space, accom-
h
modatingtheuncertaintyprinciple. Thisisanintroductoryoverviewoftheformulation
t
- withsimpleillustrations.
p
e
1. Introduction
h
:
v There are three logically autonomous alternative paths to quantization. The first
i
X is the standard one utilizing operators in Hilbert space, developed by Heisenberg,
r Schr¨odinger,Dirac,andothersinthe Twentiesofthe pastcentury. The secondone
a reliesonPathIntegrals,andwasconceivedbyDirac1 andconstructedbyFeynman.
The last one (the bronze medal!) is the Phase-Space formulation, based on
Wigner’s (1932) quasi-distribution function2 and Weyl’s (1927) correspondence3
between quantum-mechanical operators and ordinary c-number phase-space func-
tions. The crucial composition structure of these functions, which relies on the
⋆-product,was fully understood by Groenewold(1946)4, who, together with Moyal
(1949)5,pulledthe entireformulationtogether. Still,insightsoninterpretationand
a full appreciation of its conceptual autonomy took some time to mature with the
workofTakabayasi6,Baker7,Fairlie8,andothers. (Abriefguidetosomelandmark
papers is provided in the Appendix.)
This complete formulation is based on the Wigner Function (WF), which is a
CK Zachos
quasi-probability distribution function in phase-space,
1 ~ ~
f(x,p)= dy ψ∗(x y) e−iypψ(x+ y). (1)
2π Z − 2 2
It is a special representation of the density matrix (in the Weyl correspondence,
as detailed in Section 8). Alternatively, it is a generating function for all spatial
autocorrelationfunctions of a given quantum-mechanical wavefunction ψ(x).
There are several outstanding reviews on the subject9,10,11. Nevertheless, the
centralconceitof this review is that the aboveinput wavefunctionsmay ultimately
be forfeited,since the WFs aredetermined, inprinciple,asthe solutionsofsuitable
(celebrated)functionalequations. ConnectionstotheHilbertspaceoperatorformu-
lation of quantum mechanics may thus be ignored, in principle—even though they
areprovidedinSection8forpedagogyandconfirmationoftheformulations’equiv-
alence. One might then envision an imaginary planet on which this formulation
of Quantum Mechanics preceded the conventional formulation, and its own tech-
niques andmethods aroseindependently, perhapsout ofgeneralizationsofclassical
mechanics and statistical mechanics.
It is not only wavefunctions that are missing in this formulation. Beyond an
all-important (noncommutative, associative, pseudodifferential) operation, the ⋆-
product, which encodes the entire quantum mechanical action, there are no opera-
tors. Observables and transition amplitudes are phase-space integrals of c-number
functions (which compose through the ⋆-product), weighted by the WF, as in sta-
tistical mechanics. Consequently, even though the WF is not positive-semidefinite
(it can, and usually does go negative in parts of phase-space), the computation of
observablesandtheassociatedconceptsareevocativeofclassicalprobabilitytheory.
This formulation of Quantum Mechanics is useful in describing quantum trans-
port processes in phase space, of importance in quantum optics, nuclear physics,
condensed matter, and the study of semiclassical limits of mesoscopic systems12
and the transition to classical statistical mechanics13. It is the natural language
to study quantum chaos and decoherence14 (of utility in, eg, quantum computing),
and provides intuition in QM interference such as in “Welcher Weg” problems15,
probability flows as negative probability backflows16, and measurements of atomic
systems17. Theintriguingmathematicalstructureofthe formulationisofrelevance
to Lie Algebras18, and has recently been retrofitted into M-theory advances linked
to noncommutative geometry19, and matrix models20, which apply spacetime un-
certainty principles21 reliant on the ⋆-product. (Transverse spatial dimensions act
formally as momenta, and, analogouslyto quantum mechanics, their uncertainty is
increased or decreased inversely to the uncertainty of a given direction.)
As a significant aside, the WF has extensive practical applications in signal
processing and engineering (time-frequency analysis), since time and energy (fre-
quency)constituteapairofFourier-conjugatevariablesjustlikethexandpofphase
spacea.
aThus,timevaryingsignalsarebestrepresentedinaWFastimevaryingspectrograms,analogously
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hep-th/0110114 Deformation Quantization
For simplicity, the formulation will be mostly illustrated for one coordinate
and its conjugate momentum, but generalization to arbitrary-sizedphase spaces is
straightforward23,includinginfinite-dimensionalones,namelyscalarfieldtheory24,25,26,27.
2. The Wigner Function
As already indicated, the quasi-probability measure in phase space is the WF,
1 ~ ~
f(x,p)= dy ψ∗(x y) e−iypψ(x+ y). (2)
2π Z − 2 2
Itisobviouslynormalized, dpdxf(x,p)=1. Intheclassicallimit, ~ 0,itwould
→
reduce to the probability dRensity in coordinate space x, usually highly localized,
multiplied by δ-functions in momentum: the classical limit is “spiky”! This ex-
pression has more x p symmetry than is apparent, as Fourier transformation to
−
momentum-spacewavefunctionsyields acompletely symmetric expressionwiththe
rolesofxandpreversed,and,uponrescalingoftheargumentsxandp,asymmetric
classical limit.
It is alsomanifestly Realb. Itis alsoconstrainedby the SchwarzInequality to be
bounded, 2 f(x,p) 2. Again, this bound disappears in the spiky classical
−h ≤ ≤ h
limit.
p-orx-projectionleadstomarginal(bona-fide)probabilitydensities: aspacelike
shadow dpf(x,p)= ρ(x), or else a momentum-space shadow dxf(x,p) = σ(p),
respectivRely. Eitheris a(bona-fide)probabilitydensity,being poRsitivesemidefinite.
Butneither canbe conditionedonthe other,asthe uncertaintyprincipleisfighting
back: The WF f(x,p) itself can, and most often does go negative in some areas
of phase-space2,9, as is illustrated below, a hallmark of QM interference in this
language. (Infact,theonlypurestateWFwhichisnon-negativeistheGaussian28.)
The counter-intuitive “negative probability” aspects of this quasi-probability
distribution have been explored and interpreted29,30,16, and negative probability
flows amount to legitimate probability backflows in interesting settings16. Never-
theless, the WF for atomic systems can still be measured in the laboratory, albeit
indirectly17.
Smoothing f by a filter of size larger than ~ (eg, convolving with a phase-space
Gaussian) results in a positive-semidefinite function, ie it may be thought to have
been coarsened to a classical distribution31c.
toamusicscore, iethe changing distributionof frequencies ismonitoredintime22: even though
thedescriptionisconstrainedandredundant,itgivesanintuitivepictureofthesignalthatamere
time profile or frequency spectrogram fails to convey. Applications abound in acoustics, speech
analysis, seismic data analysis, internal combustion engine-knocking analysis, failing helicopter
componentvibrations,etc.
bInonespace dimension,byvirtueofnon-degeneracy, itturnsout tobep-even, butthisisnota
propertyusedhere.
cThisoneiscalledtheHusimidistribution32,andsometimesinformationscientistsexamineiton
account of its non-negative feature. Nevertheless, it comes with a heavy price, as it needs to be
“dressed” back to the WF forall practical purposes when expectation values arecomputed with
it, ie it does not serve as an immediate quasi-probability distribution with no other measure32.
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CK Zachos
Among real functions, the WFs comprise a rather small, highly constrained,
set. When is a real function f(x,p) a bona-fide Wigner function of the form (2)?
Evidently, when its Fourier transform “left-right” factorizes,
f˜(x,y)= dp eipyf(x,p)=g∗(x ~y/2) g (x+~y/2), (3)
Z L − R
ie,
∂2lnf˜
=0, (4)
∂(x ~y/2) ∂(x+~y/2)
−
so g =g from reality.
L R
Nevertheless, as indicated, the WF is a distribution function: it provides the
integration measure in phase space to yield expectation values from phase-space
c-number functions. Such functions are often classical quantities, but, in general,
are associated to suitably ordered operators through Weyl’s correspondence rule3.
Given an operator ordered in this prescription,
1
G(x,p)= dτdσdxdp g(x,p)exp(iτ(p p)+iσ(x x)), (5)
(2π)2 Z − −
the corresponding phase-space function g(x,p) (the “classical kernel of the opera-
tor”) is obtained by
p p, x x. (6)
7→ 7→
That operator’s expectation value is then a “phase-space average”4,5,
G = dxdp f(x,p) g(x,p). (7)
h i Z
The classical kernel g(x,p) is often the unmodified classical expression, such as a
conventional hamiltonian, H = p2 + V(x), ie the transition from classical me-
2m
chanics is straightforward ; but it contains ~ when there are quantum-mechanical
ordering ambiguities, such as for the classical kernel of the square of the angular
momentum L L: this one contains O(~2) terms introduced by the Weyl ordering,
·
beyond the classical expression L2. In such cases, the classical kernels may still be
produced without direct consideration of operators. A more detailed discussion of
the correspondence is provided in Section 8.
Inthissense,expectationvaluesofthephysicalobservablesspecifiedbyclassical
kernels g(x,p) are computed through integration with the WF, in close analogy to
classicalprobabilitytheory, but for the non-positive-definitenessof the distribution
function. Thisoperationcorrespondstotracinganoperatorwiththedensitymatrix
(Sec 8).
Thenegative featureoftheWFis,inthelastanalysis,anasset,andnotaliability,andprovides
anefficientdescriptionof“beats”22.
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hep-th/0110114 Deformation Quantization
3. Solving for the Wigner Function
Given a specification of observables, the next step is to find the relevant WF for a
givenhamiltonian. Canthis be donewithoutsolvingforthe Schr¨odingerwavefunc-
tions ψ, i.e. not using Schr¨odinger’s equation directly? The functional equations
which f satisfies completely determine it.
Firstly, its dynamical evolution is specified by Moyal’s equation. This is the
extension of Liouville’s theorem of classical mechanics, for a classical hamiltonian
H(x,p), namely ∂ f + f,H =0, to quantum mechanics, in this language2,5:
t
{ }
∂f H⋆f f ⋆H
= − , (8)
∂t i~
where the ⋆-product4 is
⋆ ei2~(←∂x→∂p−←∂p→∂x). (9)
≡
Therighthandsideof(8),dubbedthe“MoyalBracket”(theWeyl-correspondent
of quantum commutators), is the essentially unique one-parameter (~) associative
deformationofthePoissonBracketsofclassicalmechanics33. Expansionin~around
0revealsthatitconsistsofthePoissonBracketcorrectedbytermsO(~). Theequa-
tion (8) also evokes Heisenberg’s equation of motion for operators, except H and
f here are classical functions, and it is the ⋆-product which introduces noncommu-
tativity. This language, thus, makes the link between quantum commutators and
PoissonBrackets more transparent.
Since the ⋆-product involves exponentials of derivative operators, it may be
evaluated in practice through translation of function arguments (“Bopp shifts”),
i~ → i~ →
f(x,p)⋆g(x,p)=f x+ ∂p, p ∂x g(x,p). (10)
(cid:18) 2 − 2 (cid:19)
The equivalent Fourier representation of the ⋆-product is7:
1
f ⋆g = dp′dp′′dx′dx′′ f(x′,p′) g(x′′,p′′)
~2π2 Z
2i
= exp − (p(x′ x′′)+p′(x′′ x)+p′′(x x′)) . (11)
(cid:18) ~ − − − (cid:19)
An alternate integral representation of this product is34
2i
f⋆g =(~π)−2 dudvdwdzf(x+u,p+v)g(x+w,p+z) exp (uz vw) , (12)
Z (cid:18)~ − (cid:19)
which readily displays noncommutativity and associativity.
⋆-multiplication of c-number phase-space functions is in complete isomorphism
to Hilbert-space operator multiplication4,
1
AB= dτdσdxdp (a⋆b) exp(iτ(p p)+iσ(x x)). (13)
(2π)2 Z − −
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CK Zachos
The cyclic phase-spacetrace is directly seen in the representation(12) to reduce to
a plain product, if there is only one ⋆ involved,
dpdx f ⋆g = dpdx fg = dpdx g⋆f . (14)
Z Z Z
Moyal’sequationisnecessary,butdoesnotsufficetospecifytheWFforasystem.
Intheconventionalformulationofquantummechanics,systematicsolutionoftime-
dependentequationsisusuallypredicatedonthespectrumofstationaryones. Time-
independent pure-state Wigner functions ⋆-commute with H, but clearly not every
function ⋆-commuting with H canbe a bona-fide WF (eg, any ⋆-function of H will
⋆-commute with H).
Static WFs obey more powerful functional ⋆-genvalue equations8 (also see 35),
i~ → i~ →
H(x,p)⋆f(x,p) = H x+ ∂p , p ∂x f(x,p)
(cid:18) 2 − 2 (cid:19)
= f(x,p)⋆H(x,p)=E f(x,p) , (15)
where E is the energy eigenvalue of Hψ = Eψ. These amount to a complete
characterizationof the WFs36.
Lemma: Forreal functions f(x,p), the Wigner form (2) for pure staticeigenstates
is equivalent to compliance with the ⋆-genvalue equations (15) ( and parts).
ℜ ℑ
Proof:
H(x,p)⋆f(x,p)=
= 1 (p i~ →∂x)2/2m+V(x) dy e−iy(p+i~2←∂x)ψ∗(x ~y) ψ(x+ ~y)
2π (cid:18) − 2 (cid:19)Z − 2 2
1 ~ → ~ ~ ~
= dy (p i ∂x)2/2m+V(x+ y) e−iypψ∗(x y) ψ(x+ y)
2π Z (cid:18) − 2 2 (cid:19) − 2 2
1 → ~ → ~ ~ ~
= dy e−iyp (i ∂y +i ∂x)2/2m+V(x+ y) ψ∗(x y) ψ(x+ y)
2π Z (cid:18) 2 2 (cid:19) − 2 2
1 ~ ~
= dy e−iypψ∗(x y) E ψ(x+ y)
2π Z − 2 2
= E f(x,p). (16)
Action of the effective differential operators on ψ∗ turns out to be null.
Symmetrically,
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hep-th/0110114 Deformation Quantization
f ⋆H =
1 1 → ~ → ~ ~ ~
= dy e−iyp (∂y ∂x)2+V(x y) ψ∗(x y) ψ(x+ y)
2π Z (cid:18)−2m −2 − 2 (cid:19) − 2 2
= E f(x,p), (17)
where the actionon ψ is now trivial. Conversely,the pair of ⋆-eigenvalueequations
dictate, for f(x,p)= dy e−iypf˜(x,y) ,
R
1 → ~ → ~
dy e−iyp (∂y ∂x)2+V(x y) E f˜(x,y)=0. (18)
Z (cid:18)−2m ±2 ± 2 − (cid:19)
Hence, Real solutions of (15) must be of the form
f = dy e−iypψ∗(x ~y)ψ(x+ ~y)/2π, such that Hψ =Eψ.
− 2 2
TRhe eqs (15) lead to spectral properties for WFs8,36, as in the Hilbert space
formulation. For instance, projective orthogonality of the ⋆-genfunctions follows
from associativity, which allows evaluation in two alternate groupings:
f ⋆H⋆g =E f ⋆g =E f ⋆g. (19)
f g
Thus, for E =E , it is necessary that
g f
6
f ⋆g =0. (20)
Moreover,precludingdegeneracy(whichcanbe treatedseparately),choosingf =g
above yields,
f ⋆H⋆f =E f ⋆f =H⋆f ⋆f, (21)
f
and hence f ⋆f must be the stargenfunction in question,
f ⋆f f. (22)
∝
Pure state fs then ⋆-project onto their space. In general, it can be shown6,36 that
1
f ⋆f = δ f . (23)
a b a,b a
h
The normalization matters6: despite linearity of the equations, it prevents super-
position of solutions. (Quantum mechanical interference works differently here, in
comportance with density matrix formalism.)
By virtue of (14), for different ⋆-genfunctions, the above dictates that
dpdx fg =0 . (24)
Z
Consequently, unless there is zero overlap for all such WFs, at least one of the two
must go negative someplace to offset the positive overlap9—an illustration of the
negative values’ feature, quite far from being a liability.
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CK Zachos
Further note that integrating (15) yields the expectation of the energy,
H(x,p)f(x,p) dxdp=E f dxdp=E. (25)
Z Z
NB Likewised, integrating the above projective condition yields
1
f2 dxdp= , (26)
Z h
that is the overlap increases to a divergent result in the classical limit, as the WFs
grow increasingly spiky.
In classical (non-negative) probability distribution theory, expectation values
of non-negative functions are likewise non-negative, and thus result in standard
constraint inequalities for the constituent pieces of such functions, such as, e.g.,
moments of the variables. But it was just seen that for WFs which go negative for
an arbitrary function g, g 2 need not 0: this can be easily seen by choosing
h| | i ≥
the support of g to lie mostly in those regions of phase-space where the WF f is
negative.
Still, such constraints are not lost for WFs. It turns out they are replaced by
g∗⋆g 0, (27)
h i≥
and lead to the uncertainty principle37. In Hilbert space operator formalism, this
relation would correspond to the positivity of the norm. This expression is non-
negative because it involves a real non-negative integrand for a pure state WF
satisfying the above projective conditione,
dpdx(g∗⋆g)f =h dxdp(g∗⋆g)(f⋆f)=h dxdp(f⋆g∗)⋆(g⋆f)=h dxdpg⋆f 2.
Z Z Z Z | |
(28)
To produce Heisenberg’s uncertainty relation, one only need choose
g =a+bx+cp, (29)
forarbitrarycomplexcoefficientsa,b,c. Theresultingpositivesemi-definitequadratic
form is then
a∗a+b∗b x⋆x +c∗c p⋆p +(a∗b+b∗a) x +(a∗c+c∗a) p +c∗b p⋆x +b∗c x⋆p 0,
h i h i h i h i h i h i≥
(30)
dThis discussion applies to proper WFs, corresponding to pure states’ density matrices. E.g., a
sum of two WFs is not a pure state in general, and does not satisfy the condition (4). For such
generalizations, the “impurity” is4 dxdp(f −hf2)≥0, where the inequality is only saturated
intoanequalityforapurestate. ForRinstance,forw≡(fa+fb)/2withfa⋆fb=0,theimpurity
isnonvanishing, dxdp(w−hw2)=1/2.
eSimilarly, if f1Rand f2 are pure state WFs, the transition probability between the respec-
tive states is also non-negative38, manifestly by the same argument37, namely, dpdxf1f2 =
(2π~)2 dxdp|f1⋆f2|2≥0. R
R
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hep-th/0110114 Deformation Quantization
for any a,b,c. The eigenvalues of the corresponding matrix are then non-negative,
and thus so must be its determinant. Given
x⋆x=x2, p⋆p=p2, p⋆x=px i~/2, x⋆p=px+i~/2, (31)
−
and the usual
(∆x)2 (x x )2 , (∆p)2 (p p )2 , (32)
≡h −h i i ≡h −h i i
this condition on the 3 3 matrix determinant amounts to
×
2
(∆x)2 (∆p)2 ~2/4+ (x x )(p p ) , (33)
≥ (cid:16)h −h i −h i i(cid:17)
and hence
∆x ∆p ~/2. (34)
≥
The ~ enteredinto the moments’constraintthrough the action of the ⋆-product.
4. Illustration: the Harmonic Oscillator
To illustrate the formalism on a simple prototype problem, one may look at the
harmonic oscillator. In the spirit of this picture, one can, in fact, eschew solving
theSchr¨odingerproblemandpluggingthe wavefunctionsinto(2); instead,onemay
solve (15) directly for H =(p2+x2)/2 (with m=1, ω =1),
i~ i~
(x+ ∂ )2+(p ∂ )2 2E f(x,p)=0. (35)
p x
(cid:18) 2 − 2 − (cid:19)
Forthishamiltonian,theequationhascollapsedtotwosimplePDEs. Thefirstone,
the Imaginary part,
(x∂ p∂ )f =0, (36)
p x
−
restricts f to depend on only one variable, the scalar in phase space, z = 4H/~ =
2(x2+p2)/~. Thus the second one, the Real part, is a simple ODE,
z E
z∂2 ∂ f(z)=0. (37)
(cid:18)4 − z − z− ~(cid:19)
Setting f(z)=exp( z/2)L(z)yields Laguerre’s equation,
−
E 1
z∂2+(1 z)∂ + L(z)=0. (38)
(cid:18) z − z ~ − 2(cid:19)
It is solved by Laguerre polynomials,
ez∂n(e−zzn)
L = , (39)
n
n!
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CK Zachos
for n=E/~ 1/2=0,1,2,.., so the ⋆-gen-Wigner-functions are 4
−
( 1)n
f = − e−2H/~ L (4H/~);
n n
π
L =1, L =1 4H/~, L =8H2/~2 8H/~+1,... (40)
0 1 2
− −
But for the Gaussian ground state, they all have zeros and go negative. These
functions become spiky inthe classicallimit ~ 0; e.g.,the groundstate Gaussian
→
f goes to a δ-function.
0
Their sum provides a resolution of the identity5,
f =1/h. (41)
n
Xn
(For the rest of this section, for algebraic simplicity, set ~=1.)
Oscillator Wigner Function, n=3
f
p
xx
Fig.1. WFforthe3rdexcitedstate. Notethenegativevalues.
Dirac’s hamiltonian factorizationmethod for the alternate algebraic solution of
thesameproblemcarriesthroughintact,with⋆-multiplicationsupplantingoperator
multiplication. That is to say,
1 1
H = (x ip)⋆(x+ip)+ . (42)
2 − 2
10