HEISENBERG PROOF OF THE BALIAN-LOW THEOREM+
Guy Battle*
Centre de recherches mathematiques
Universi1e de Montreal
C.P. 6128, A
Montreal, Quebec
H3C 3J7
+
*
Supported in part by the National Science Foundation under Grant No. OMS 8603795
On leave from the Mathematics Department, Texas A&M University, College Station,
Texas 77843.
2
Phase space Wannier
structure calculations
functions
functions
[1,2,3J have been of interest in electronic
[41 and signal analysis [5,6,7J.
Unlike classical Wannier
[8,9], which are defined in terms of the eigenfunctions
Hamiltonians,
of periodic
phase space Wannier functions have uniform tail lengths in both
position space and momentum space.
In this note we are concerned
with a
special class of such expansion functions, namely families of coherent states i.e., expansions based on windowed Fourier transforms [10].
Let T mn be the phase space translation operator in one dimension i.e. ,
(1 )
(Tmn<p)(x) = ei2nmx <p(x+n).
Definition.
Let f be a square-integrable function on IR such that {T mnf} is an
orthonormal basis. Then {Tmnf} is a basis of coherent states and f is said to
generate it.
Let X and P be the position and momentum operators, respectively.
Balian-Low Theorem. If f E L2 (fR) generates
a basis of coherent states then X f
and Pf cannot both be square-integrable.
This is a no-go theorem for localizing phase space with a windowed Fourier
transform.
It states that such functions must have infinitely long tails in either
position space or momentum space. Balian [2] and Low [3] independently
established this theorem with an ingenious topological argument showing that
if the function
Lt
f (k+.t )ei2nt k'
(2)
is continuous in both variables, then it must have a zero. Coifman and
Semmes (see [10]) have more recently closed a technical loophole concerning
what regularity properties for f will guarantee the continuity of (2).
We present here an elementary proof of the Balian-Low Theorem in which the
Heisenberg Uncertainty Principle enters directly.
3
New Proof. If Xf and Pf are both square-integrable, then
~ (Xf, T mnf) (Tmnf, Pf)
(Xf, Pf) = LJ
m,n
=
L
(3)
(T_m,_n
f, X ~ (P f,T_m,_nf)
m,n
because
([T-m,-n' X]f, f) = n(T_m,_n
f, f) = 0,
(4)
(f, [T_m-n'
, P]f) = 21tm( Tm,nf, f) = O.
(5)
On the other hand,
L
and so ([P, X]f, f) =
(6)
(T_m-nf, X ~ (Pf, T_m-nf) = (Pf, X f),
m,n'
,
This implies f == O .•
O.
Remark. Actually, one has to be a little careful here because the premise of our
argument does not guarantee the square-integrability
of XPf
and
PXf.
Let
{<pj} be a sequence of smooth, compactly supported functions such that <Pj-7 f,
X<Pj-~ Xf,'and P<Pj-7 Pf in the square integrable sense of convergence.
an approximating sequence certainly exists.) Then
i1lcpjll2
=
([X,P]cp.,cp.)
=
(Pcp.,
Xcp.)
- (X<p.,Pcp.)
J
J
J
J
J
J
-7 (Pf,Xf) - (Xf,Pf) =
But l/<pjl/2-7 0 implies
II
(Such
O.
(7)
f 112 = O.
One may believe that sacrificing orthogonality is a reasonable way to beat this
no-go theorem.
However, when one expands an arbitrary function with respect
to a basis that is not orthonormal, the coefficients are computed with respect to
the dual basis. The bad news is that if an L2-complete family of coherent states
has good phase space localization, then the dual basis cannot.
4
Theorem. Let {T mnf} and {T mng} be two families of coherent states which
are complete in L2(R) but are bi-orthonormal - Le.,
(T mnf, T m'n,g)
If Xf is square-integrable,
=
(8)
0mm,onn"
then Pg cannot be. If Pf is square-integrable,
then
Xg cannot be.
Proof.
Look at the expansions
(Xf, Pg) =
L
(Xf, Tmng) (Tmnf, Pg),
(9)
m,n
(10)
(Pf, Xg) = "£.J (Pf, T mn g) (T mn t, Xg)
m,n
and argue as before, using (8). Either of the two premises
(9) and Pt, Xg E L2(R) for (10)) will lead to the conclusion
(Xf, Pg E L 2(R) for
(f, g) = O.
+
In this context the van Neumann basis [3] easily comes to mind. However, this
basis, which has exponential
example
phase space localization,
because the dual basis functions
Further, t~e basis is "overcomplete".
is a rather extreme
are not even square-integrable.
It has been pointed out by Janssen [6] that
if f is the standard Gaussian generating the van Neumann basis, then f lies in
the closure of the linear span of {T mnf}(m,n)#(O,O)'
Acknowledgements.
It is a pleasure to thank I. Daubechies, F. Low, and
R. Balian for background references and technical comments.
to thank F.H. Clarke for his hospitality at CRM.
I would also like
5
References.
1.
K. Wilson, "Generalized Wannier Functions", Cornell University preprint.
2.
R. Balian, C. R. Acad. Sci. Paris 292 (1981), 1375.
3.
F. Low, in "A Passion for Physics", G.F. Chew Volume, World Scientific
Press, 1985.
4.
J.
Rehr, D. Sullivan,
Functions
J.
in Electronic
Wilkins, and K. Wilson, "Phase Space Wannier
Structure
Calculations",
Cornell
University
preprint.
J. Zak,
5.
H. Bacry, A. Grossmann, and
6.
A. Janssen, J. Math. Phys. 23 (1982), 720.
7.
I. Daubechies, A. Grossmann, and Y. Meyer, J. Math. Phys. 27 (1986),
1271.
8.
G. Wannier, Phys. Rev. 52 (1937), 191.
9.
J. des Cloizeaux, Phys. Rev. 135, 3A (1964), 698.
10.
I. Daubechies,
Phys. Rev. 812 (1975), 1118.
"The Wavelet Transform,
Time-~requency
and Signal Analysis", AT&T Bell Laboratories preprint.
Localization,