Multicomponent WKB and Quantization

10 Pages·0.12 MB·English

by  C. Emmrich

Multicomponent WKB and Quantization
C. Emmrich, H. R¨omer
6
9 Fakult¨at fu¨r Physik
9 Universit¨at Freiburg
1
Hermann-Herder-Straße 3

n
79104 Freiburg
a
J

5
2 1 Introduction

1
Hamiltonians whose symbols are not simply real valued, but matrix or, more
v
8 generally, endomorphism valued functions appear in many places in physics,
3 examples being the Dirac equation, multicomponent wave equations like elec-
1 trodynamics in media, and Yang-Mills theories, and the Born-Oppenheimer
1
approximation in molecular physics.
0
Whereasthe semiclassicalandWKBapproximationofscalarsystemsiswell
6
9 understood,thisisnotthecasetothesameextentforHamiltonianswithmatrix
/ valuedsymbols. In particular,semiclassicalstatesin the scalarcasehaveanice
h
geometricinterpretationashalfdensitiesonLagrangeansubmanifoldsinvariant
t
-
under the Hamiltonian flow, and discrete spectra may be computed using the
p
e Bohr-Sommerfeldcondition,takingintoaccounttheMaslovcorrection[14,1,7,
h 10]. In the multicomponent case, the analogous structures are not known. The
:
v usual WKB ansatz
i Ψ(x)=a(x,~)exp(iS(x)/~)
X
r with a(x,~)=a0(x)+a1(x)~+... being a power series in ~ with vector valued
a coefficients is still possible [9, 12, 14, 5] in this situation, but it is in general
defined only locally and leads to equations which don’t have an obvious geo-
metrical interpretation.
The aim of this paper is to give a completely geometric approach to this
problems, and to reduce the problem “as far as possible” to the scalar case. As
observed in [13] the use of a star-product formulation of quantum mechanics
proves to be particularly useful in this context. However, whereas [13] restrict
themselves to the use of the Moyal product and thus to the study of trivial
bundles(orlocaltrivializations)overR2n,wewillconsidergeneralbundlesover
arbitrary symplectic manifolds. Here, Fedosov’s construction [8] will be the
adequate tool, since it gives an explicit construction for star products in this
general setting.
1
2 WKB on R2n
In this section, we give a short overview of some well known results on WKB
for multicomponent systems:
Multicomponent WKB deals with equations of the form
(Hˆ E)ψ =0
−
where Hˆ is an (N N)-matrix-valued differential operator, or equivalently, an
×
(N N)-matrix of differential operators. The WKB-ansatz for a solution is:
×
ψ =ae~iS, a=a0+~a1+...
At zeroth order, this yields:
(H (q,dS) E)a =0,
0 0
−
where H is the principal symbol of Hˆ, i.e. the zerothorder part of the symbol
0
of Hˆ. In the scalar case (N = 1) this simply is the Hamilton-Jacobi-equation
for the scalar Hamiltonian H (q,p).
0
ForN >1,thisequationhastwoimplications: Firstly,S hastobeasolution
of a Hamilton-Jacobi-equation
λ (q,dS)=E,
α
where λ (q,p) is an eigenvalue of H (q,p). (Here, we need the fact that—at
α 0
leastlocally—thedegeneraciesoftheeigenvaluesdonotchangeinordertodefine
the function λ .) Secondly, a (q,p) has to be an eigenvector of H (q,p) with
α 0 0
eigenvalue E.
Solutions of the Hamilton-Jacobi-equation define Lagrangean submanifold
Λ of R2n and we may interpret
α
ae~iS dµ
p
as a half-density on Λ with values in RN. (strictly speaking, it has to be ten-
sorized with a section of the Maslov bundle).
To study the next order in the WKB approximation, we introduce the pro-
jectorπ0 onthe αtheigenspaceofH . We makethe assumptionsthatthe mul-
α 0
tiplicity ofλ is constantona neighborhoodofΛ, andthat kernelandrangeof
α
H λ are complementary (which is in particular fulfilled for hermitian H).
α
−
Then, the first order equation, which is a “transport equation” for u =
a √dµ, may be written as [15]:
⊗
a dµ+D a dµ
⊗LXλα Xλα ⊗
p p
1
+ π0 π0,H λ I π0 +iπ0H π0 a dµ = 0
(cid:20)−2 α{ α 0− α } α α 1 α(cid:21) ⊗
p
As compared to the scalar case, there are two additional terms for N >1:
2
1. “Berry term” for Berry-connection [3, 13, 15]
Da= π0dπ0a
α α
Xα
2. “Curvatureterm”,involvesthecurvatureF andsecondfundamentalforms
S,S∗ of the Berry-connection. With
Sξ d=ef(1 π0)(dπ0)ξ, S∗η d=ef π0(d(1 π0))η
− α α − α − α
this term may be rewritten as [15]
λ <Π,F > <Π,S∗ (H λ π0)S >,
α − ∧ 0− α α
where Π is the Poisson tensor.
3 Star Products and Geometrization
As mentioned earlier, the star product approachto quantization is particularly
adapted to our problem: Firstly, its structure allowsus to dealwith the expan-
sionin~inasimpleway,andsecondly,itistheonlyknowngeneralquantization
scheme which allows the quantization of any symplectic manifold, and which is
notrestrictedtojustofasmallsubclassofobservables(as,e.g. geometricquan-
tization). The idea of star product quantization is to pull back the operator
product, which is defined on operators on Hilbert space , to an associative,
H
non-commutativeproductonthefunctionsonphasespace,viaasuitablesymbol
calculus [2]: Assuming that we we are given a quantization procedure
Q: ∞(M) C[[~]] End
C ⊗ → H
we may define a star product on ∞(M) C[[~]] by:
C ⊗
a b=Q−1(Q(a)Q(b)).
∗
The simplest example for such a star product is the case M = R2n and Q
being Weyl-ordering, which leads to the well known Weyl-Moyal product. It is
explictly given by:
(a b)(x)=(ei2~ωij∂∂xi⊗∂∂yja(x)b(y))x=y,
⋄ |
where ω are the components of the standard symplectic form on R2n. Al-
ij
though we have only considered scalar valued observables so far, the matrix
generalization is obvious in this simple example: In the matrix product, one
simply has to replace the ordinarypointwise product ofthe matrix elements by
the Moyal product.
The formal definition is as follows:
3
Definition 3.1 A star product on a symplectic manifold M is an associatiative
product on ∞(M) C[[~]] with
C ⊗
1. a b=ab+O(~)
∗
2. a b b a=i~ a,b
∗ − ∗ { }
3. a 1=1 a=a
∗ ∗
4. supp(a b) supp(a) supp(b)
∗ ⊂ ∩
5. a b=¯b a¯
∗ ∗
Star products exist for every symplectic manifold M [6, 8]. Given a star
product and a differential operator W = W +~W +... on ∞(M) C[[~]]
0 1
C ⊗
with W1=0, we may define a new star product:
a b=e−~W(e~Wa e~Wb).
W
∗ ∗
and are called equivalent. The importance of this notion is that “physics
W
∗ ∗
remainsunchanged”,ifthechangeof to isaccompaniedbytheapplication
W
of the isomorphisme~W to the observ∗able∗s: Heuristically,this means that if we
change the ordering prescription and at the same time the symbol calculus in
suchawaythatthe operatorsonHilbertspaceremainunchanged,thenphysics
remains unchanged. We shall exploit this freedom later on.
4 Fedosov’s star product
In this section, we give a short overview of Fedosov’s explicit construction of
a star product on endomorphism bundles of vector bundles over a symplectic
manifold. The arena is as follows: We consider an Hermitean vector bundle V
overasymplecticmanifoldM. DenotingbyC[[~]]thesetofformalpowerseries
in ~ with coefficients in C, we construct from these data the bundles V[[~]] d=ef
V C[[~], E[[~]]= End(V ) C[[~]],andthealgebra =Γ(E[[~]] W),
m∈M m
⊗ ∪ ⊗ A ⊗
where W denotes the Weyl-bundle overM, the bundle whose fibres W are the
q
Weyl algebras over T M
q
Finally we consider the algebra: Ω(M) , with Ω(M) being the algebra
⊗A
of differential forms on M. On this algebra, we define two globally defined
operators δ,δ−1 : Ω(M) Ω(M) whose coordinate expressions are
⊗A → ⊗A
given by
∂ 1
δaˆ=dxi aˆ, δ−1aˆ= i aˆ
∧ ∂yi p+q yi∂∂xi
for a q-form aˆ with values in the homogeneous polynomials in y of degree p.
Here, xi are coordinates on M, and yi the induced coordinates on T M.
q
To construct a unique star product on E[[~]] two additional data are re-
quired: a Hermitian connection on V (inducing a connection on E[[~]]) and
∇
4
a symplectic connection ∂ on M. From these we can construct a connection
s
∂ = 1 ∂ + 1 on E[[~]] W. Now, these data given, the main step of
s
⊗ ∇⊗ ⊗
Fedosov’s construction consists in finding by a recursive procedure a covariant
exterior derivative
i
D = δ+∂+[ r, ]
− ~ ·
with r Ω1(M) such that
∈ ⊗A
1. D2 =0, i.e., D is a flat connection.
2. Covariantlyconstantelements in Ω (M) are in linear one-to-one cor-
0
respondence to Γ(E[[~]]) ⊗A
3. D is a derivationonΩ(M) (hence, covariantlyconstantsections form
⊗A
a subalgebra)
Ifwedenotetheisomorphismin2.,whichmapsa ΓE[[~]]toitscovariantly
∈
constant continuation in Ω (M) , by Q, it is given by the solution of
0
⊗A
i
Q(a)=a+δ−1(∂Q(a)+[ r,Q(a)]),
~
which may be solved iteratively.
With these notions, Fedosov’s star product is simply given by
a b=Q−1(Q(a) Q(b)),
∗ ⋄
where is the fibrewise Moyal product.
⋄
Differentchoicesof (and∂ )leadto differentstarproducts. However,the
s
∇
star products obtained in this way are equivalent:
Theorem 4.1 LetD(1,2) betheFedosovconnectionsassociatedto (1,2),∂(1,2).
∇
Then there is A and
(1,2)
∈A
i
U =exp ( A )
(1,2) ⋄ ~ (1,2)
such that
D(1)aˆ=0 D(2)(U aˆ U−1 )=0
⇔ (1,2)⋄ ⋄ (1,2)
and
φ(a)d=efQ(2)−1(U Q(1)(a) U−1 )
(1,2)⋄ ⋄ (1,2)
is an isomorphism of the products (1) and (2)
∗ ∗
Lemma 4.2
(1) (2) =O(~m)
∇ −∇
Q(2)−1(U aˆ U−1 ) Q(1)−1(aˆ)=O(~m+1)
⇒ (1,2)⋄ ⋄ (1,2) −
for every aˆ with D(1)aˆ=0.
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5 Application to WKB
We arenow readyto applythe abovetechniquesto multicomponentWKB.Let
H =H +~H +...
0 1
beasectionofE[[~]],π0(x)theprojectiononthe(regular)eigenspacebelonging
α
to λ (x). Now, the main problem in WKB, which prevents us from naively
α
reducing the problem to a scalar problem on each bundle of eigenspaces, is
the fact that, due to quantum corrections, WKB states are not sections in
these bundles, but have higher order corrections. In the star product approach
this is related to the fact that the star commutator of the Hamiltonian with
an observable commuting with π0(x) does not commute with π0(x), hence a
α α
reduction to the bundles of eigenspacesis not possible. Thus, we have to find a
kind of “quantum diagonalization procedure”.
The formal reason for this problem in Fedosov’s approach to star products
isthefactthattheprojectionsarenotcovariantlyconstantunderthehermitian
connectionwhichdefinesthestarproduct. Ourstrategyforsolvingthisproblem
istousethefreedomtochange whenapplyingthecorrespondingstarproduct
∇
isomorphism at the same time to preserve physics. This is possible indeed, as
expressed by the following theorem:
Theorem 5.1 ThereexistsaformallyorthogonaldecompositionV[[~]]= V ,
α α
⊕
dimV = m with corresponding quantum projections π(∞) and a hermitian
α α α
connection (∞) with
∇ (∞)π(∞) =0
∇ α
such that the corrected Hamiltonian (obtained by applying the corresponding
isomorphism φ(∞)):
H(∞) =φ(∞)(H)
preserves the decomposition:
[H(∞),π(∞)] =0 α,
α ∗∞ ∀
where is the Fedosov star product defined by (∞)
∞
∗ ∇
Proof:
The proof is by induction: To start we set π(1) = π(0) and define (1) by
α α
∇
(1)ψ = π(1) π(1)ψ. With H(1) =φ(1)(H), we have
∇ α α ∇ α
P
(1)π(1) =0, [π(1),H(1)] =O(~).
∇ α ∗1
Nowassumewehavefoundπ(k), (k) = π(k) π(k),H(k) =φ(k)(H)such
α ∇ α α ∇ α
that P
(k)π(k) =0, [π(k),H(k)] =O(~k)
∇ α ∗k
6
We construct π(k+1) with
α
[π(k+1),H(k)] =O(~(k+1))
α ∗k
using the ansatz
π(k+1) =ei~kA π(k) e−i~kA,
α ∗k α ∗k
which has a solution
π(k)Wπ(k)
α β
A=
λ λ
αX6=β α− β
where W is defined by
H(k) = π(k) H(k) π(k)+~kW.
α ∗k ∗k α
Xα
Now, setting (k+1) = π(k+1) π(k+1)ψ and using lemma 4.2, the theo-
∇ α α ∇ α
rem follows. P
2
6 Compatibility
Now, we are ready to address the issue of compatibility of observables, i.e., the
question which observables preserve the quantum decomposition found above.
The results may be considered generalizations of results found in [4] for the
Dirac equation. In physics language, we find an answer to the question which
observables are “slow” in the sense that their time evolution is analytic in ~,
and hence does not depend on inverse powers of ~.
Take a,b Γ(E[[~]]) such that
∈
[π(∞),a] =[π(∞),b] =0
α ∗∞ α ∗∞
then
[π(∞),a b] =0
α ∗∞ ∗∞
Compatible elements form a -subalgebra ˜. In particular, for b = H(∞) we
∞
seethatthestarcommutatoro∗fobservablesinO˜withthecorrectedHamiltonian
is in ˜ again. O
O
Theorem 6.1 Letφ(∞) betheisomorphism betweenthestarproducts and .
∞
For every a in the subalgebra =φ∞−1(˜) the Heisenberg evolution∗equat∗ion
O O
1
a˙ = [H,a]
i~ ∗
is a well defined differential equation in (i.e., there are no inverse powers of
~ on the right hand side). O
7
7 O(h)-correction
Inthe formalismdeveloppedso far,itis straightforwardto compute the correc-
tion of order ~: We have
H(∞) = π(2) H(1) π(2)+O(~2)
α ∗1 ∗1 α
Xα
=ei~A π(0) H(1) π(0) e−i~A+O(~2)
∗ α ∗ ∗ α ∗
A straightforwardcalculation, using the explicit formula for U leads to [16]
i~
π(0) H(1) π(0) =π(0)H(0)π(0)+ ωil π(0)( π0) π(0)
α ∗1 ∗1 α α α 2 α ∇i γ ∇l α
Xγ
With the curvature of the “Berry connection”
FBψ = π(0) π(0) π(0)ψ = π(0)F0π(0)+ π(0)( π0)( π(0))
α ◦∇◦ α ∇ α α α α ∇ γ ∇ α
Xα Xα Xα
and the “Second fundamental form”
Sβα(X)ψ =(π(0) π(0))ψ (β =α)
β ◦∇X ◦ α 6
we finally get:
i~ i~
π(0) H(1) π(0) = λ <Π,π0(FB F0)π0 >+ λ <Π,SγαSαγ >.
α ∗ ∗ α 2 α α − α 2 γ
γX6=α
This result is a generalization of our results in [15]: There, the starting
point was the Moyalproduct on R2n, which is a Fedosov star product for a flat
connection. Hence, the curvature F0 of was missing there.
∇
8 Change of compatible connection
IntheconstructionsabovewehavemadeexplicituseofBerrytypeconnections.
Nevertheless,wearenotreallyforcedtodoso,butitis justaconvenientchoice
fortheproof. Hence,thequestionariseswhetherthecorrectiontermscomputed
above are just a consequence of the choice of our connection, or whether they
are really “physical”.
To answer this question, take another compatible connection ˜. With
∇
(1) ˜(1) =∆Γ
∇ −∇
we obtain
π0 H˜(1) π0 =π0 H(1) π0 +i~π0 ∆Γ(X ) π0
α∗ ∗ α α∗ ∗ α α λα α
“Berry phase” term
| {z }
8
Hence, the only difference is the appearence of an additional Berry phase,
which was so far hidden in the explicit use of a non-flat connection. However,
the curvature term remains and it still is constructed from the curvature of the
Berry connection, not from the curvature of the chosen connection.
The correction terms involving the curvature and the second fundamental
form of the Berry connection have intrisic meaning.
9 Problems
In this section, we just mention a few open problems which are discussed in
more detail in [16].
1. Nontrivial holonomy of compatible connections may give an obstruction
to the existence of WKB states: only projectors exist because of a U(n)
holonomy. Solutionareknownonlyform =1(nodegeneracy)ordimM =
α
2 [11, 16]
2. AMaslov-correctionhastobeimplemented. Thisisstraightforwardifthe
problem can really be reduced to a scalar problem, e.g. for m =1.
α
3. Level crossings appear in many physical applications. These have been
excluded by our regularity conditions. Nevertheless, our approach is still
applicable to the open submanifold where the degeneracies are constant.
The level crossing points have to be studied in a second step.
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