Gaussian transform of the Weil representation

0.21 MB·English

by  A. V. Stoyanovsky

GAUSSIAN TRANSFORM OF THE WEIL
REPRESENTATION
A. V. STOYANOVSKY
6
0
0 Abstract. A description is given of the image of the Weil rep-
2
resentation of the symplectic group in the Schwartz space and in

n the space of tempered distributions under the Gaussian integral
a transform. We also discuss the problem of infinite dimensional
J
generalizationof the Weil representationin the Schwartzspace,in
2
order to construct appropriate quantization of free scalar field.
1

1
v
9 1. Introduction
2
0
This work arose during the study of the problem of quantization of
1
0 fields, i. e., mathematically and logically self-consistent construction
6 of quantum field theory. One of formulations of this problem is to
0
give a mathematical sense to the quantum field theory Schrodinger
/
h
variational differential equation [3,8] and its relativistically invariant
p
- generalization [13–15].
h
The Schrodinger equation for free scalar field reads
t
a
m ∂Ψ h2 δ2 1 m2
(1) ih = + (gradu(x))2 + u(x)2 Ψdx.
: ∂t − 2 δu(x)2 2 2
v Z (cid:18) (cid:19)
i Here Ψ is an unknown functional depending on a number t and on a
X
real function u(x), x = (x ,...,x ); δ is the variational derivative.
r 1 k δu(x)
a
Traditionally one solves equation (1) in the Fock Hilbert space con-
taining functionals of the form
1
Ψ(u) = Ψ (u)exp uˆ(p)uˆ( p)ω dp ,
0 p
−2h −
(cid:18) Z (cid:19)
where Ψ is a polynomial functional of u; p = (p ,...,p ), uˆ(p) =
0 1 k
1 eipxu(x)dx, ω = p2 +m2. It is easy to see that on these
(2π)k/2 p
functionals the right hand side of equation (1) equals infinity. To over-
R p
come this, one subtracts an “infinite constant” from the Hamiltonian
intherighthandsideof(1),andreducesthisHamiltoniantoanormally
orderedexpressionofcreationoperators 1 h δ +ω uˆ(p) and
√2ωp − δuˆ(−p) p
(cid:16) (cid:17)
Partially supported by the grant RFFI N 04-01-00640.
1
2 A.V. STOYANOVSKY
annihilation operators 1 h δ +ω uˆ(p) . This approach meets
√2ωp δuˆ(−p) p
big problems [8]. One of th(cid:16)em is that in the(cid:17)Fock space for k > 1
one cannot give a mathematical sense to relativistically invariant gen-
eralization of equation (1), i. e., one cannot perform quantization on
spacelike surfaces [16].
The idea of our approach to equation (1) was to try to define an
analog of the space of main functions and distributions on the infi-
nite dimensional space of functions u(x) (let us call them main and
distribution functionals), and to solve equation (1) in these distribu-
tion functionals. This approach is also valid from the physical point of
view (unlike the approach of considering dynamics in Hilbert space),
since quantum mechanical quantities like energy and momentum are
non-measurable in relativistic quantum dynamics, and the only mea-
surable quantities are the scattering sections. A negative result of the
present paper is that this approach turned out to be not fruitful for the
problem of giving mathematical sense to equation (1) and quantization
on spacelike surfaces (see Remark 7 at 6), but it led to some math-
§
ematical results which we also present in this paper. The problem of
quantization of a free field on spacelike surfaces is solved in the paper
[18].
The space of main functionals that we look for could satisfy the
following requirements.
1) It is a locally convex topological vector space with the action
of differential operators with polynomial coefficients, or rather with
the action of infinite dimensional analog of the Weyl algebra (cf., for
example, [17], 18.5).
§
2) It has an action of infinite dimensional symplectic group. (This
is needed, for example, for quantization on spacelike surfaces. Indeed,
in classical field theory the evolution operator of the Klein–Gordon
equationfromonespacelikesurfacetoanotheroneisalinearsymplectic
transformation of the phase space of a free field.)
3) It has the distribution functional 1.
4) Its finite dimensional analog is the Weil representation of the
symplectic group Sp(2n,R) in the Schwartz space S(Rn) of rapidly
decreasing smooth functions ([5,10]; [17], 18.5; see 2 below).
§ §
5) It contains Gaussian functionals, i. e., exponents of a quadratic
form.
Let us comment on the last requirement. In the finite dimensional
case Gaussian functions are transformed under the action of the sym-
plectic group in the most simple way. Hence one can expect that in
the infinite dimensional case main functionals will also contain analogs
GAUSSIAN TRANSFORM OF THE WEIL REPRESENTATION 3
of Gaussian functions. Besides that, the space of asymptotic states
as t should be identified with the Fock space, to make the
→ ±∞
S-matrix a unitary operator in this space. Hence one would like to
have an analog of the Gaussian integral, i. e., Gaussian distribution
functionals.
A direct generalization of the Schwartz space to infinite dimensions
meets difficulties. Indeed, if, for example, we define it as a space of
weakly smooth functionals on an infinite dimensional (say, nuclear)
space, with the norms of functionals analogous to the norms in the
space S(Rn), then a simplest functional
Ψ(u) = l(u)exp( B(u,u))
−
(l(u) is a linear functional, B(u,u) is a positive definite quadratic func-
tional) is in general unbounded, and hence it does not belong to this
space, which contradicts to requirement 5.
Since Gaussian functions are important for us, it is natural to try
to consider the finite dimensional case from the point of view of these
functions. ThisleadstotheGaussiantransform, which takesafunction
ψ(x ,...,x ) to the function
1 n
(2) u(Zjk) = ψ(x1,...,xn)e2i j,kZjkxjxkdx1...dxn.
Z P
The main purpose of the present paper is to describe the image of the
Weil representation under the Gaussian transform. It turns out that
it is rather easy to describe the image of the space S(Rn) and of the
space S′(Rn) of tempered distributions, and it is less easy to describe
the image of the space L (Rn) and other spaces, cf. Remarks in 6.
2
§
Thus, we obtain an explicit description of the Weil representation in
the spaces S(Rn) and S′(Rn), in which the Gaussian vectors and the
vector 1 play a distinguished role. This is the main result of the paper.
The paper consists of 6 sections. 2 contains preliminaries on the
§
Weil representation, 3 statements of main theorems, 4,5 proofs, 6
§ §§ §
concluding remarks.
A short exposition of the main theorems of the paper is contained in
[19].
The author is grateful to V. V. Dolotin and Yu. A. Neretin for nu-
merous helpful discussions.
4 A.V. STOYANOVSKY
2. The Weil representation
2.1. The Weil representation arises when one solves the Schrodinger
equation with a quadratic Hamiltonian:
(3)
∂ψ 1 ∂ h2 ∂ ∂ ih
ih = a x x +ihb x + c + b ψ.
jk j k jk j jk jj
∂t 2 ∂x 2 ∂x ∂x 2
k j k !
j,k j
X X
Hereψ(t,x)isanunknownfunctionofthevariablestandx = (x ,...,x );
1 n
a and c are real symmetric matrices; b is an arbitrary real matrix.
jk jk jk
The summand ih b is added to make the operator in the right
2 j jj
hand side self-adjoint, and to simplify the formulas below. Put the
P
initial condition
(4) ψ(0,x) = δ(x y).
−
It turns out that exact solution of the problem (3,4) is given by the
quasiclassical approximation. It reads [6]
(5) ψ(t,x) = a(t,h)exp(iS(t,x,y)/h),
where S(t,x,y) is the action of the corresponding Hamiltonian system
atthetimet,a(t,h)istheamplitude, whicharecomputedinthefollow-
ing way. The Hamiltonian flow at the time t gives a linear symplectic
p
transformation of the phase space , p = (p ,...,p )T. We will
x 1 n
(cid:18) (cid:19)
denote the points of the phase space by vector columns; the index T
denotes transposing (in the present case, transposing of a row). Denote
A B
the matrix of this transform by Sp(2n,R) (A,B,C,D are
C D ∈
(cid:18) (cid:19)
n n-matrices), i. e.,
×
q p p = Aq +By,
,
y 7→ x x = Cq +Dy.
(cid:18) (cid:19) (cid:18) (cid:19)
Then
(6) a(t,h) = 1/ (2πih)ndetC,
p
and the formula dS = pdx qdy H(x,p)dt (H(x,p) is the classical
− −
Hamiltonian) implies that
1 1
(7) S(t,x,y) = xTAC−1x yTC−1x+ yTC−1Dy.
2 − 2
GAUSSIAN TRANSFORM OF THE WEIL REPRESENTATION 5
Below we put h = 1. Thus, the evolution operator of equation (3) at
the time t has the form
(8)
1
(Uψ)(x) = ei(12xTAC−1x−yTC−1x+12yTC−1Dy)ψ(y)dy
(2πi)ndetC
Z
for detC = p0. The operator U is unitary and extends to a uni-
tary opera6tor on L (Rn). Indeed, this operator is the composition
2
of four operators: 1) the operator of multiplication by the function
exp iyTC−1Dy , 2) the linear change of coordinates with the matrix
2
(CT)−1, followed by multiplication by 1/√detiC, 3) the Fourier trans-
(cid:0) (cid:1)
form,4)theoperatorofmultiplicationbythefunctionexp ixTAC−1x .
2
All these operators are unitary. They also preserve the spaces S(Rn)
and S′(Rn). Besides that, they preserve the subspaces of e(cid:0)ven and odd(cid:1)
functions, which we will denote by the indices respectively + and ,
−
for example, S′ , (L ) .
+ 2 −
It turns out and it is not difficult to check that the set of operators U
A B
given by formula (8) for all matrices with detC = 0, can be
C D 6
(cid:18) (cid:19)
extended to a two-valued representation of the group Sp(2n,R) in the
spaces (L ) , S , S′ , i. e., to a representation of a two-fold covering of
2 ± ± ±
the group Sp(2n,R). The two-valuedness is related to non-uniqueness
of the square root from (2πi)ndetC. Let us call this covering by the
metaplectic group, denoted Mp(2n,R). An explicit description of this
group will be given in 2.2.
This representation of the groupMp(2n,R) is usually called theWeil
representation.
The Lie algebra of the group Sp(2n,R) acts by the differential oper-
ators
i ∂ i ∂ ∂ 1
(9) a x x +b x + c + b .
jk j k jk j jk jj
2 ∂x 2 ∂x ∂x 2
j,k (cid:18) k j k(cid:19) j
X X
These operators are identified with the matrices from the symplectic
Lie algebra by commuting with the operators
n
∂
(10) v x +w i .
j j j
∂x
j=1 (cid:18) j(cid:19)
X
The operators (10) form a 2n-dimensional symplectic vector space V ;
2n
the symplectic form on V is given by the commutator of operators.
2n
This implies (or one checks directly from (8)) that the operator U
(8) conjugates operators (10) by the standard action of the matrix
6 A.V. STOYANOVSKY
A B
onthespace V . Inother words, ifwe write thecoefficients
C D 2n
(cid:18) (cid:19)
v
of operator (10) as a column vector , then conjugation by U
w
(cid:18) (cid:19)
takesoperator(10)totheoperatoroftheform(10)withthecoefficients
A B v
.
C D w
(cid:18) (cid:19)(cid:18) (cid:19)
This property determines the operator U uniquely up to a scalar
factor. Indeed, if U′ is another operator with the same property, then
the operator U′U−1 commutes with the operators x and i ∂ , and
j ∂xj
hence it is multiplication by a constant, as it is not difficult to show.
This implies once more that the operators U form a projective repre-
E B
sentation of the group Sp(2n,R). In particular, the matrix
0 E
(cid:18) (cid:19)
acts by multiplication by the function exp i B x x , the ma-
2 j,k jk j k
0 E (cid:16) (cid:17)
trix acts (up to a constant factorP) by Fourier transform,
E 0
(cid:18) − (cid:19)
A 0
the matrix acts by composition of a linear change of
0 (AT)−1
(cid:18) (cid:19)
coordinates with the matrix A and multiplication by √detA. These
matrices generate the group Sp(2n,R), which gives one more proof of
existence of a projective action of Sp(2n,R) with the above described
commutation relations with operators (10).
2.2. Gaussian functions. These are functions
i
(11) ψ (x) = exp Z x x ,
Z jk j k
2
!
j,k
X
where Z is a symmetric complex matrix. We have ψ S′ iff the
Z
∈
imaginary part ImZ is nonnegative definite, ImZ 0. Further, ψ
Z
≥ ∈
S iff the matrix ImZ is positive definite, ImZ > 0.
The action of the metaplectic group on the Gaussian functions is
given by explicit formulas. This is seen from the fact that the function
ψ satisfies the system of equations
Z
∂
i + Z x ψ = 0, 1 k n.
jk j Z
∂x ≤ ≤
k !
j
X
GAUSSIAN TRANSFORM OF THE WEIL REPRESENTATION 7
A B
The action of the matrix takes these equations to the equa-
C D
(cid:18) (cid:19)
tions
∂
(12) v x +w i ψ = 0, 1 k n,
jk j jk
∂x ≤ ≤
j (cid:18) j(cid:19)
X
which, for det(CZ + D) = 0, are equivalent to the equations on the
6
Gaussian function ψ .
(AZ+B)(CZ+D)−1
The matrices Z with ImZ > 0 form the so called Siegel upper half-
plane ([4], 50), denoted . This is a homogeneous space of the group
§ SG
A B
Sp(2n,R); the matrix Sp(2n,R) acts by a fractional lin-
C D ∈
(cid:18) (cid:19)
ear transform:
Z (AZ +B)(CZ +D)−1.
7→
The stabilizer of the matrix Z = iE is the unitary group U(n).
An explicit calculation shows that
1
(13) Uψ = ψ .
Z (AZ+B)(CZ+D)−1
det(CZ +D)
Define the metaplecticpgroup Mp(2n,R) as the set of pairs
A B
(14) , det(CZ +D) ,
C D
(cid:18)(cid:18) (cid:19) (cid:19)
p
A B
where Sp(2n,R), and det(CZ +D) is one of the two
C D ∈
(cid:18) (cid:19)
continuous branches of the square ropot from det(CZ + D) = 0, Z
. Then the multiplication law in Mp(2n,R) can be define6d so tha∈t
SG
formula (13) will give a correctly defined single-valued action of this
group on the functions ψ .
Z
Denote matrices Z with ImZ = 0 by the letter a:
i
(15) ψ (x) = exp a x x .
a jk j k
2
(cid:18) (cid:19)
X
The Mp(2n,R)-orbit of the function ψ consists, in general, of distri-
a
butions. This orbit can be found as follows. The action of an element
of Mp(2n,R) on the function ψ takes it to the function satisfying the
a
system of equations (12) with real coefficients v , w . The operators
jk jk
in the left hand side of the system are linearly independent and pair-
wise commute, i. e., form a basis of a Lagrangian subspace L in V .
2n
Conversely, for any real Lagrangian subspace L V there exists a
2n
⊂
unique, up to proportionality, distribution solution ψ of the system
L
(12), in which the operators in the left hand side form a basis in L. It
8 A.V. STOYANOVSKY
is clear that the distribution ψ depends only on the subspace L but
L
not on its basis. An example of ψ is the delta-function δ(x) satisfying
L
the system of equations x δ(x) = 0, 1 k n. It is not difficult to
k
≤ ≤
see that in the general case ψ is the product of a Gaussian function
L
(15)on certain subspace in Rn andthe delta-function in the transversal
direction. Let us emphasize once more that ψ is defined only up to a
L
scalar factor.
Thus, we have an embedding of the real Lagrangian Grassmannian
Λ into the projectivization of the space S′, which takes L to Cψ .
n L
The image of this embedding is the Sp(2n,R)-orbit of the function 1.
(Here one can see what is the form of the operator (8) for detC = 0:
this is an integral operator with the Gaussian kernel supported on cer-
tain subspace of the space (x,y) and corresponding to the Lagrangian
subspace in R2n R2n which is the graph of the symplectic trans-
⊕
A B
form .) There arises an Mp(2n,R)-equivariant complex line
C D
bundl(cid:18)e µ on Λ(cid:19), whose fiber at the point L is the line Cψ . This
n L
bundle is trivialized by the functions ψ (15) on the open dense chart
a
of the Grassmannian parameterized by symmetric matrices a. Under
change of chart and trivialization, given by the action of element (14)
of Mp(2n,R), the corresponding transition functions can be found us-
ing Maslov’s method of canonical operator [6]. Indeed, the function
ψ corresponds to the Lagrangian subspace L in the method of canon-
L
ical operator. Under evolution of the Schrodinger equation (3), the
function ψ goes to
a
ψ det(Ca+D) −1/2 eiπk/2
(Aa+B)(Ca+D)−1
·| | ·
for some integer k (the Maslov index). The same formula is obtained
in another way in the book [17], 21.6. For a third way of calculation
§
of transition functions see below.
The orbits of other functions ψ with ImZ 0 also, in general,
Z
≥
consist of distributions. These orbits form the closure of the Siegel
SG
upper half-plane in the complex Lagrangian Grassmannian. For L
∈
the function ψ given by corresponding system (12) with complex
L
SG
coefficients, is in general the product of the Gaussian function ψ (11)
Z
oncertain subspace inRn, with ImZ 0, andthe delta-function in the
transversal direction. Sp(2n,R)-orbit≥s are parameterized by the rank
of the matrix ImZ on the subspace in Rn which is the support of the
function ψ .
L
The line bundle µ extends to an Mp(2n,R)-equivariant line bundle
µ on the closure , trivialized by the functions ψ over . This
1 Z
SG SG
and formula (13) imply a formula for the transition functions of the
GAUSSIAN TRANSFORM OF THE WEIL REPRESENTATION 9
bundle µ: the action of the element (14) corresponds to the transition
function
(16) lim 1/ det(CZ +D).
ImZ→0
(Cf. with the method of complepx germ [7,8].)
2.3. Gaussian transform. For a function ψ(x) its Gaussian trans-
form u(Z) = u (Z) is given by formula (2). Under the action of the
ψ
group Mp(2n,R) on the function ψ(x) its Gaussian transform u (Z)
ψ
transformsbyanexplicit formula, seeTheorem 5below. Themainpur-
pose of this paper is to describe the image of the Weil representation
in the space S under the Gaussian transform.
First let us find what obvious necessary conditions the function
u (Z) satisfies for ψ S. The function u (Z) is well defined as a
ψ ψ
∈
holomorphic function on the Siegel upper half-plane, which satisfies
the following system of partial differential equations:
∂ ∂ ∂ ∂
(17) u = u,
∂Z ∂Z ∂Z ∂Z
jl km jm kl
where the operators ∂ , for any j,k, are defined from the equalities
∂Zjk
∂ ∂ ∂
(18) du = u dZ = 2 u dZ + u dZ .
jk jk jj
∂Z ∂Z ∂Z
jk jk jj
j,k j<k j
X X X
This system and the holomorphness condition are obtained by differen-
tiation under the sign of the integral (2). For an odd function ψ S ,
−
∈
the function u (Z) vanishes.
ψ
Further, the function u extends on the boundary of the Siegel upper
half-plane as the section
(19) u(L) = u (L) = (ψ,ψ )
ψ L
of the line bundle µ∗ dual to µ . This section is continuous together
1 1
with all its derivatives with respect to the action of the Lie algebra of
Sp(2n,R). Inparticular,onedefinesaninfinitelydifferentiablefunction
(20) u(a) = uψ(a) = ψ(x)e2i ajkxjxkdx,
Z P
satisfying the system of equations
∂ ∂ ∂ ∂
(21) u = u.
∂a ∂a ∂a ∂a
jl km jm kl
It will be shown below that this system is invariant under the change
of a chart and a trivialization by action of an element of Mp(2n,R).
10 A.V. STOYANOVSKY
Let us also define an analog of the Gaussian transform for odd func-
tions ψ(x) S . This transform is given by the formula
−
∈
(22) ul(Z) = ul,ψ(Z) = ψ(x)xle2i Zjkxjxkdx, 1 l n.
≤ ≤
Z P
The holomorphic vector valued function (u (Z)) on the Siegel upper
l,ψ
half-plane satisfies the system of equations
∂ ∂
(23) u (Z) = u (Z).
l k
∂Z ∂Z
jk jl
The vector valued function (u (Z)) extends to a section
l,ψ
n
∂
(24) u (L) = u (L) = ψ, v x +w i ψ
v,w v,w,ψ j j j L
∂x
j=1 (cid:18) j(cid:19) !
X
of the vector bundle W µ∗ on the closure of the Siegel upper half-
⊗ 1
plane. HereW isthetautologicaln-dimensionalvector bundleover ,
SG
whose fiber over a point L is the space L itself or the isomorphic
∈ SG
space (V /L)∗. The right hand side of (24) gives a linear functional
2n
on the space V of vectors (v,w), vanishing on the subspace L, i. e.,
2n
an element of (V /L)∗.
2n
The section u (L) is also continuous together with all derivatives
v,w,ψ
withrespect to theactionof theLie algebraof Sp(2n,R). Inparticular,
one defines an infinitely differentiable vector valued function
(25) ul(a) = ul,ψ(a) = ψ(x)xle2i ajkxjxkdx, 1 l n,
≤ ≤
Z P
satisfying the system of equations
∂ ∂
(26) u (a) = u (a).
l k
∂a ∂a
jk jl
It will be shown below that this system is also invariant under change
of a chart and a trivialization by action of an element of Mp(2n,R).
3. Theorems
The main theorem for even functions.
Theorem 1. The transform (20) identifies the space S with the
+
space of smooth functions u(a) satisfying equations (21) and extending
to sections of the bundle µ∗ on the closure of the Siegel upper half-
1
plane, which are holomorphic on the upper half-plane and continuous
togetherwithallderivativeswithrespecttotheactionoftheLiealgebra
of Sp(2n,R). The corresponding holomorphic function u(Z) on the
Siegel upper half-plane automatically satisfies equations (17).