Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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below. For the present, we simply note that for this
system the corresponding potential is the Weier-
strass }-function, A(x)A(x) = }() }(x), and the
resulting Hamiltonian system [1] is known as the
Calogero–Moser system. It is completely integrable
though, as already remarked, the ansatz did not
necessitate this. The Lax pair presented here and the
reduction of its consistency to a functional equation
and algebraic constraints follows Calogero (1976) in
which he discovered the elliptic generalization of the
model he had introduced in 1975.
A different ansatz for a Lax pair is
L
jk
¼
_
q
j
jk
þð1
jk
Þ
ffiffiffiffiffiffiffiffiffi
_
q
j
_
q
k
q
Aðq
j
q
k
Þ
M
jk
¼
jk
X
l 6¼j
_
q
l
Bðq
j
q
l
Þþð1
jk
Þ
ffiffiffiffiffiffiffiffiffi
_
q
j
_
q
k
q
Cðq
j
q
k
Þ
Now the consistency of the Lax pair yields equations
of motion of the form
€
q
j
¼
X
k 6¼j
_
q
j
_
q
k
Vðq
j
q
k
Þ
VðxÞ¼
AðxÞ AðxÞ
CðxÞ CðxÞ
¼VðxÞ
provided B(x) = B(x), C(x) = A
0
(x) A(x)G(x),
where we have defined G(x) = B(x) þ (1=2)V(x),
and the functions satisfy the functional equation
Aðx þ yÞ¼AðxÞAðyÞþ
AðxÞ AðyÞ
A
0
ðxÞ A
0
ðyÞ
GðxÞGðyÞ
¼
AðxÞ AðyÞ
CðxÞ CðyÞ
GðxÞGðyÞ
½3
Again we shall briefly defer describing the solution
of this equation and simply note that the general
solution for V(x) is again given in terms of the
Weierstrass }-function V(x) = }
0
(x)=(}() }(x))
and that the equations of motion follow from the
Hamiltonian
H ¼
X
j
e
p
j
Y
k 6¼j
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
}ðÞ}ðq
j
q
k
Þ
q
This is known as the Ruijsenaars–Schneider model
and it too is completely integrable. The Lax pair
here was constructed by Bruschi and Calogero.
In the two examples of Lax pairs just presented,
each particle interacts with every other pairwise. By
modifying the ansatz, it is possible to construct
models that interact with just their nearest neighbors
(which include the Toda systems). More generally,
an ansatz exists for a Lax pair associated with
equations of motion of the form
€
q
j
¼
X
k 6¼j
ða þ b
_
q
j
Þða þ b
_
q
k
ÞV
jk
ðq
j
q
k
Þ½4
which unifies, for example, the Calogero–Moser,
Ruijsenaars–Schneider, and Toda systems. The
functional equations now encountered are typically
(and whenever b 6¼0) of the form
1
ðx þ yÞ¼
2
ðxÞ
2
ðyÞ
3
ðxÞ
3
ðyÞ
4
ðxÞ
4
ðyÞ
5
ðxÞ
5
ðyÞ
½5
This functional equatio n, for five a priori unknown
functions, includes [2] and [3] as special cases.
The general analytic solution of [5] is, up to
symmetries, given by
1
ðxÞ¼
ðx;
1
Þ
ðx;
2
Þ
;
2
ðxÞ
3
ðxÞ
!
¼
ðx;
1
Þ
0
ðx;
1
Þ
!
4
ðxÞ
5
ðxÞ
!
¼
ðx;
2
Þ
0
ðx;
2
Þ
!
where
ðx; Þ
ð xÞ
ðÞðxÞ
e
ðÞx
½6
Here, (x) = (x)
0
=(x) is the Weierstrass -function.
The solution of [2] arises as the
2
!0 limit of [5].
The proof of the general solution just stated
is in fact constructive ( Braden and Buchstaber
1997). The parameters appearing in the solution
are determined as follows. Suppose x
0
is a generic
point for [5]. Then (for k = 1, 2), we have that
@
y
ln
2k
ðx þ x
0
Þ
2k
ðy þ x
0
Þ
2kþ1
ðx þ x
0
Þ
2kþ1
ðy þ x
0
Þ
y¼0
¼ ð
k
ÞðxÞð
k
xÞ
k
¼
1
x
k
þ
X
l¼0
F
l
x
lþ1
ðl þ 1Þ!
½7
The Laurent expansion determines the parameters
g
1
, g
2
(which are the same for both k = 1, 2)
characterizing the elliptic functions of [6] by
g
2
¼
5
3
F
2
þ 6F
2
0
; g
3
¼ 6F
3
0
F
2
1
þ
5
3
F
0
F
2
and the parameters
k
via F
0
= }(
k
). Here,
}(x) =
0
(x) is the Weierstrass elliptic }-function
426 Functional Equations and Integrable Systems
with periods 2!,2!
0
that satisfies the differential
equation
}
0
ðxÞ
2
= 4}ðxÞ
3
g
2
}ðxÞg
3
The constructive nature of the solutions of [5] means
that it is straightforward to construct solutions to
various specializations of the equation such as
1
ðx þ yÞ¼
4
ðxÞ
5
ðyÞþ
4
ðyÞ
5
ðxÞ
(obtained by requiring
2
(x) =
2
4
(x)and
3
(x) =
2
5
(x)). More complicated functional equations such as
1
ðx þ yÞ¼
2
ðx þ yÞ
2
ðxÞ
3
ðyÞ
þ
3
ðx þ yÞ
4
ðxÞ
5
ðyÞ½8
may be solved using the solutions of [5].
Finally, let us note that the general system [4] may
lead to functional equation s not just of the form [5],
for example,
6
ðx þ yÞ¼
1
ðx þ yÞ
4
ðxÞ
5
ðyÞðÞ
þ
2
ðxÞ
3
ðyÞ
0
2
ðxÞ
0
3
ðyÞ
½9
The genera l analytic solution to [9] has yet to be
determined although particular solutions are known.
As a final example of a functional equation
coming from an ansatz for a Lax pair, consider
L
jk
¼
ffiffiffiffiffiffiffiffiffi
p
j
p
k
p
Aðq
j
q
k
Þ; M
jk
¼
ffiffiffiffiffiffiffiffiffi
p
j
p
k
p
Cðq
j
q
k
Þ
where we now assume A(0) and C(0) regular. Then
the consistency of this Lax pair corresponds to the
equations of motion for the Hamiltonian
H ¼
X
n
j;k
p
j
p
k
f ðq
j
q
k
Þ½10
provided f is even and the functional equation
2A
0
ðx þ yÞ f ðxÞf ðy Þ½
Aðx þ y Þ f
0
ðxÞf
0
ðyÞ½¼
AðxÞ AðyÞ
CðxÞ CðyÞ
½11
is satisfied. The Hamiltonian system [10] corre-
sponds to geodesic motion. Nonanalytic solutions
are known to the functional equation [11].
An Algebraic Ansatz: Conserved
Quantities
Another way in which functional equations may
appear is by making an ansatz for an additional
conserved quantity beyond the Hamiltonian. For two
and three particles on the line, Hietarinta derived
functional equations by seeking a second quartic or
cubic integral (respectively). Here, a key ingredient is
the assumption of a further invariant polynomial in the
momenta. Polynomial invariance, together with sym-
metry, is quite constraining. Consider
Theorem 1 Let H and P be the (natural) Hamilto-
nian and center of mass momentum
H ¼
1
2
X
n
i¼1
p
2
i
þ V; P ¼
X
n
i¼1
p
i
Denote by Q an independent third-order quantity
Q ¼
X
n
i¼1
p
3
i
þ
1
6
X
i 6¼j 6¼k
d
ijk
p
i
p
j
p
k
þ
X
i 6¼j
d
ij
p
2
i
p
j
þ
1
2
X
ij
a
ij
p
i
p
j
þ
X
i
b
i
p
i
þ c
If these are S
n
-invariant and Poisson-commute,
P; H
fg
¼ P; Q
fg
¼ Q; H
fg
¼ 0
then
V ¼
1
6
X
i 6¼j
} q
i
q
j
þ const:
and we have the Calogero–Moser system.
Here, the symmetric group invariance means that
for any coefficient
ij
(q
1
, q
2
, ..., q
n
) in the expan-
sions above, we have
(i)(j)
(q
(1)
, q
(2)
, ..., q
(n)
) for
all 2 S
n
. In particular, V(q
1
, q
2
, ..., q
n
) = V(q
(1)
,
q
(2)
, ..., q
(n)
) for all 2 S
n
. We remark that had we
begun with particles of possibly different particle
masses, H = (1=2)
P
n
i = 1
m
i
p
2
i
þ V; the effect of
S
n
-invariance is such as to require these masses to
be the same. Thus, we are assuming the S
n
-invariant
Hamiltonian of the theorem. Finally, by ‘‘an
independent third-order quantity’’ Q, we mean one
functionally independent of H and P and for which
one cannot obtain an invariant of lower degree by
subtracting multiples of P
3
and PH. We are not
dealing with quadratic conserved quantities here.
The assumed polynomial behavior of the con-
served quantities means that when calculating
Poisson brackets, the coefficients of independent
monomials must vanish. This, together with sym-
metry, leads to the functional equation
111
FðxÞ FðyÞ FðzÞ
F
0
ðxÞ F
0
ðyÞ F
0
ðzÞ
¼ 0; x þ y þ z ¼ 0 ½12
Functional Equations and Integrable Systems 427
The result follows in light of
Theorem 2 Let f be a three-times differentiable
function satisfying the functional equation [12]. Up
to the manifest invariance
FðxÞ!FðxÞþ
the solutions of [12] are one of F(x) = }(x þ d),
F(x ) = e
x
or F(x) = x. Here, } is the Weierstrass
}-function and 3d is a lattice point of the }-function.
Again we note that the ansatz per se has not
established co mplete integrability: the ansatz leads
us to the Calogero–Moser model whose complete
integrability must be established by other means.
This result may be interpreted as a rigidity theorem
for the a
n
Calogero–Moser system and in part
explains this models’ ubiquity: demanding a cubic
invariant together with S
n
-invariance necessitates
the model. A natural generalization is to replace the
S
n
-invariance with the invariance of a general
Weyl group W and make connection with the
Calogero–Moser models associated to other root
systems (Perelomov 1990).
We shall encounter the functional equation [12]
again in this survey and now note that this may be
generalized to
111
FðxÞ GðyÞ HðzÞ
F
0
ðxÞ G
0
ðyÞ H
0
ðzÞ
¼ 0; x þ y þ z ¼ 0 ½13
If F, G, and H are three-times differentiable func-
tions satisfying the functional equation [13], then,
up to the manifest invariance,
FðxÞ!Fðx þ
1
Þþ
GðxÞ!Gðx þ
2
Þþ
HðxÞ!Hðx þ
3
Þþ
where
1
þ
2
þ
3
= 0, the nonconstant solutions of
[13] are given by F(x) = G(x) = H(x) = e
x
, x,or}(x).
If (say) H(z) is a constant then either
1. one of the functions F(x)orG(y) is the same
constant as H(z), in which case the remaining
function is arbitrary, or
2. F(x) = G(x) = e
x
.
We remark that in fact the exponential and linear
function solutions satisfy [12] and [13] without the
constraint x þ y þ z = 0. Further, the theorems
immediately give the general analytic solut ions to
the same functional equations viewed as functions of
a complex variable, showing that the solutions are
in fact meromorphi c. These theorems were estab-
lished in Braden and Byatt-Smith (1999) where
earlier results are described.
Quantum Calogero–Moser Systems
Quite a bit is known about the quantum general-
izations of the Calogero–Moser system. The poly-
nomial and Weyl group W-invariance of the
classical conserved quantities is replaced by a
commutative ring R of W-invariant, holomorphic,
differential operators, whose highest-order terms
generate W-invariant differential operators with
constant coefficients. The Poisson bracket is then
replaced by a commutator of operators. When this
is done functional equations again ensue and one
finds that the potential term for the Laplacian H
(the quantum Hamiltonian) has Calogero–Moser
potential appropriate to W (Oshima and Sekiguchi
1995). In this setting, it is known that the
commutativity of just a few low-order elements of
R dictate the form of the potential and the
commuting algebra (at least for the classical root
systems). In particular, Theorem 1 above is the classical
analog for the a
n
root system of a quantum result where
a functional equation equivalent to [12] was obtained
by requiring the commutativity of certain linear,
quadratic, and cubic holomorphic differential opera-
tors. Taniguchi’s results (Taniguchi 1997) are also
indicative of the rigidity of these quantum models: if H
is the quantum Hamiltonian just discussed, and Q
1, 2
are holomorphic (but not aprioriW-invariant),
differential operators of appropriate degrees for
which [Q
1, 2
,H] = 0, then Q
1, 2
2Rand consequently
[Q
1
, Q
2
] = 0.
An Algebraic Ansatz: The Poincare´
Algebra
We have earlier encountered the Ruijssenaars–
Schneider models when considering functional
equations ensuing from ansatz for Lax pairs. These
models were however discovere d by another route
(Ruijsenaars and Schneider 1986) in the course
of investigating mechanical models obeying the
Poincare´ algebra
fH; Bg¼P; fP; Bg¼H; fH; Pg¼0 ½14
Here, H will be the Hamiltonian of the system
generating time translations, P is a space-translation
428 Functional Equations and Integrable Systems
generator, and B the generator of boosts. Ruijsenaars
and Schneider began with the ansatz
H ¼
X
n
j¼1
cosh p
j
Y
k 6¼j
f ðx
j
x
k
Þ
P ¼
X
n
j¼1
sinh p
j
Y
k 6¼j
f ðx
j
x
k
Þ
B ¼
X
n
j¼1
x
j
With this ansatz and the canonical Poisson bracket
{p
i
, x
j
} =
ij
, the first two Poisson brackets of [14]
involving the boost operator B are automatically
satisfied. The remaining Poisson bracket is then
fH; Pg¼
X
n
j¼1
@
j
Y
k 6¼j
f
2
ðx
j
x
k
Þ
1
2
X
j 6¼k
coshðp
j
p
k
Þ
Y
l 6¼j
f ðx
j
x
l
Þ
Y
m 6¼k
f ðx
k
x
m
Þ @
j
ln f ðx
k
x
j
Þ
þ @
k
ln f ðx
j
x
k
ÞÞ
and for the independent terms proportional to
cosh (p
j
p
k
) to vanish we require that f
0
(x)=f (x)be
odd. This entails that f(x)iseitherevenorodd
(Ruijsenaars and Schneider assumed the function even)
and in either case F(x) = f
2
(x) is even. Supposing that
f (x) is so constrained, then the final Poisson bracket is
equivalent to the functional equation
fH; Pg¼0 ()
X
n
j¼1
@
j
Y
k 6¼j
f
2
ðx
j
x
k
Þ¼0 ½15
For n = 3, eqn [15] takes precisely the form [12]
with F(x) = f
2
(x). From Theorem 2, the even
solutions to this have the form F(x) = }(x) þ c.
This was found by Ruijsenaars and Schneider who
further showed this function satisfies [15] for all n.
The general solution to [15] has recently been
established.
Theorem 3 (Byatt-Smith and Braden 2003). The
general even solution of [15] amongst the class of
meromorphic functions whose only singularities on
the real axis are either a double pole at the origin, or
double poles at np (p real, n 2 Z) is:
(i) for all odd n given by the solution of Ruijsenaars
and Schneider while
(ii) for even n 4, there are in addition to the
Ruijsenaars–Schneider solutions the following:
F
i
ðzÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð}ðzÞe
j
Þð}ðzÞe
k
Þ
q
½16
where i, j, k are a cyclic permutation of 1, 2, 3.
These functions have simple expressions in terms
of Weierstrass elliptic functions, theta functions, and
the Jacobi elliptic functions (Whittaker and Watson
1927). For example,
F
1
ðzÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð}ðzÞe
2
Þð}ðzÞe
3
Þ
p
¼
2
ðzÞ
3
ðzÞ
2
ðzÞ
¼
3
ðvÞ
4
ðvÞ
2
1
ðvÞ
02
1
ð0Þ
4!
2
3
ð0Þ
4
ð0Þ
¼ b
dnðuÞ
sn
2
ðuÞ
where
ðzÞ¼
ðz þ !
Þ
ð!
Þ
e
zð!
Þ
u ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e
1
e
3
p
z
v = z=2!, b = e
1
e
3
with !
1
= !, !
2
= ! !
0
, and
!
3
= !
0
. For appropriate ranges of z the solutions
are real. Their degenerations yield all the even
solutions with only a double pole at x = 0on
the real axis. These degenerations may in fact
coincide with the degenerations of the Ruijsenaars–
Schneider solution.
Thus far, complete integrability has not been
mentioned. The models discovered by Ruijsenaars
and Schneider not only exhibited an action of the
Poincare´ algebra but were completely integrable as
well. In particular, Ruijsenaars and Schneider
demonstrated the Poisson commutativity for their
solutions of the light-cone quanti ties
S
k
¼
X
If1;2;... ;ng
jIj¼k
exp
X
i2I
p
i
!
Y
i2I
j62I
f ðx
i
x
j
Þ½17
Then, H = (S
1
þ S
1
)=2andP = (S
1
S
1
)=2. (Note
the even/oddness of the functions f (x) means that there
really are only n functionally independent quantities.)
It is an open problem whether the new solutions [16]
of Theorem 3 yield integrable systems. We know that
these new solutions do not always yield Poisson
commuting quantities using the ansatz of Ruijsenaars
and Schneider, but as yet one cannot rule out other
Poisson commuting conserved quantities.
Quantum Ruijsenaars–Schneider Models
Ruijsenaars later investigated the quantum version
of the classical models he and Schneider introduced.
Functional Equations and Integrable Systems 429
From the outset, he sought operator analogs of the
light-cone quantities [17]. He showed that (for
k = 1, ..., n)
^
S
k
¼
X
If1;2;...;n g
jIj¼k
Y
i2I
j62I
hðx
j
x
i
Þ
1=2
exp
ffiffiffiffiffiffiffi
1
p
X
i2I
@
i
!
Y
i2I
j62I
hðx
i
x
j
Þ
1=2
pairwise commute if and only if
X
If1;2;...;n g
jIj¼k
Y
i2I
j62I
hðx
j
x
i
Þhðx
i
x
j
iÞ
0
B
@
Y
i2I
j62I
hðx
i
x
j
Þhðx
j
x
i
iÞ
1
C
A
¼0 ½18
held for all k and n 1. Here, is an arbitrary
positive number and the sum is over all subsets with
k ele ments. Observe that upon dividing [18] by
and letting !0 this yields [15] with F(x) =
h(x)h(x) when k = 1.
Ruijsenaars found a solution to [18] which has
subsequently been shown to be unique. The general
solution of the functional equation [18] analytic in a
neighborhood of the real axis with either a simple
pole at the origin or an array of such poles at np on
the real axis (n 2 Z) is given by
hðxÞ¼b
ðx þ Þ
ðxÞðÞ
e
x
½19
This solution is related to the earlier Ruijsenaars–
Schneider solution via
ðx þ Þðx Þ
2
ðxÞ
2
ðÞ
¼ }ðÞ}ðxÞ
Geometric Ansatz
We have already encountered the Hamiltonian
system [10] corresponding to geodesic motion
while discussing Lax pairs. We shall now consider
various ansa¨ tze with a geometric flavor and their
attendant functional equations.
It is known that the Ruijsenaars–Schneider model
has the Calogero–Moser system as a scaling limit.
Other scaling limits also exist for the Ruijsenaars–
Schneider model. In particular, we may consider one
in which the Poincare´ algebra scales to either the
Galilean alge bra or a central extension of the
Galilean algebra.
Similar to our analysis of the Poincare´ algebra, we
find that the functions
H ¼
1
2
X
n
j¼1
p
2
j
Y
k 6¼j
f ðx
j
x
k
Þ;
P ¼
X
n
j¼1
p
j
Y
k 6¼j
f ðx
j
x
k
Þ; B ¼
X
n
j¼1
x
j
obey the algebra
fH; Bg¼P; fP; Bg¼; fH ; Pg¼0 ½20
if and only if f (x) is either an even or odd function
satisfying
X
n
j¼1
Y
k 6¼j
f ðx
j
x
k
Þ¼ ½21
where is a constant. When = 0 this is the
Galilean algebra, while 6¼0 is a central extension
of the Galilean algebra. Again we are encountering
models of the form H = (1=2)
P
n
j = 1
g
jj
p
2
j
and so
dealing with diagonal metrics. We note that if [21]
holds for n = 3 then it holds for all n; and if it
holds for n = 4 then it holds for all ‘‘even’’ n. This
type of behavior was already encountered in
Theorem 3.
Some particular solutions of [21] are known
although the general solution is not known as yet.
The odd functions f (x) = 1=x ( = 0), coth (x)( = 1
for n odd and = 0 for n even),
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
}(x) e
p
( = 0)
yield solutions for example. Interestingly, in the
case of an even number of particles, particular cases
of the elliptic Ruijsenaars–Schneider model are in
this list.
Diagonal metrics arise in many settings in integr-
able systems. By taking the ansatz
ds
2
¼
X
n
i¼1
Y
j 6¼i
ðx
i
x
j
Þ
!
ðdx
i
Þ
2
we may construct and solve a functional equation to
show that the potentially nonvanishing curvature
components R
i
jik
, R
i
jjk
(k 6¼i, j), and R
i
jij
have
1. R
i
jik
= R
i
jjk
= 0(k 6¼i, j ) if and only if
(x) = (e
2bx
1)
a
or x
a
. We may set = 1by
rescaling x.
2. R
i
jij
= (1)
n
b
2
when (x) = (e
2bx
1).
3. R
i
jij
= 0 when (x) = x.
Thus, (x) = x yields a solution of the Lame´
equations. These metrics are of Sta¨ ckel form. The
rational degenerations of the Galilean models above
are given by this theorem. They may be understood
as a parabolic limit of Jacobi elliptic coordinates.
430 Functional Equations and Integrable Systems
Similar techniques may be applied to the more
general metric
ds
2
¼
X
n
i¼1
i
ðx
i
Þ
Y
j 6¼i
ðx
i
x
j
Þ
!
ðdx
i
Þ
2
to show that R
i
jik
= R
i
jjk
= 0(k 6¼i, j) if and only
if (x) = (e
2bx
1)
a
or x
a
.
Ground-State Factorization
Some years ago, Sutherland and Calogero consid-
ered the problem as to when the ground-state wave
function of a one-dimensional n-body Schro¨ dinger
equation with pairwise interactions would factorize.
Thus, the problem is to determine those potentials
v(x) for which
h
2
2
X
n
i¼1
@
2
i
þ
1
2
X
i 6¼j
vðx
i
x
j
ÞE
()
ðx
1
; x
2
...x
n
Þ¼0 ½22
and where
ðx
1
; x
2
...x
n
Þ¼
Y
i<j
ðx
i
x
j
Þ
It is convenient to set
ðx
i
x
j
Þ¼exp
1
h
Z
x
i
x
j
f ðxÞdx
Substitution now shows
h
2
2
X
n
i¼1
@
2
i
¼ h
X
i<j
f
0
ðx
i
x
j
Þþ
X
i<j
f ðx
i
x
j
Þ
2
(
þ
X
i<j<k
f ðx
i
x
j
Þf ðx
i
x
k
Þ
f ðx
i
x
j
Þf ðx
j
x
k
Þ
þf ðx
j
x
k
Þf ðx
i
x
k
Þ
)
Comparison with [22] shows that this may be
expressed in terms of two-body potentials if and
only if we have the functional equation
f ðaÞf ðbÞf ðaÞf ðcÞþf ð cÞf ðbÞ
¼ G
1
ðaÞþG
1
ðcÞþG
2
ðbÞ; a þb þc ¼ 0 ½23
Now [23] is not quite the functional equation
studied by Sutherland and Caloger o. On physical
grounds, Sutherland implicitly, and Calogero expli-
citly, made the ‘‘assumption’’ that f is an odd
function. This ensured that the potential was even
and so bounded from below; equally it may be
imposed so that (x
i
x
j
) = (x
j
x
i
) and the
ground state describes bosons. With this assump-
tion, one arrives at the functional equation of
Sutherland:
f ðaÞf ðbÞþf ðbÞf ðcÞþf ðcÞf ðaÞ
¼ GðaÞþGðbÞþGðcÞ½24
Actually the assumption of f being odd is unneces-
sary. One can show that there is a bijection between
analytic solutions of [23] and analytic solutions of
[24] for which f
0
(x) is even. Upon requiring a
potential of the stated form then necessitates f
being odd. Whatever, we arrive at the functional
equation [24]. This is connected with [12] by
Lemma 4 If a þ b þ c = 0, then
f
00
ðaÞ f
00
ðbÞ f
00
ðcÞ
f
0
ðaÞ f
0
ðbÞ f
0
ðcÞ
111
¼ 0 ½25
() f ðaÞþf ðbÞþf ðcÞðÞ
2
¼ FðaÞþFðbÞþFðcÞ½26
() f ðaÞf ðbÞþf ðbÞf ðcÞþf ðcÞf ðaÞ
¼ GðaÞþGðbÞþGðcÞ½27
Now, we may use Theorem 2 to determine those
potentials with factorizable ground-state wave func-
tions. We remark that the -function potential a(x)
of many-body quantum mechanics on the line,
which also has a factorizable ground-state wave
function, can be viewed as the ! 0 limit of
b= sinh
2
(x= þ i=3) with a = 6b. Thus, all
of the known quantum mechanic al problems with
factorizable ground-state wave function are included
in [12].
Baker–Akiezer Functions
Baker–Akiezer functions are one of the foundations
of the algebro-geometric or finite-gap integration of
integrable systems. These functions may be viewed
as an extension of the exponential function to curves
of arbitrar y genus g. They have essential singula-
rities at various points on the curve and a prescribed
asymptotic expansion at these points. The functions
may be described in terms of theta functions on the
Jacobian of the curve, and suitable meromorphic
differentials on the curve. The functions [6] and [19]
may be viewed as the Baker–Akhiezer function for a
genus-1 curve. Now, just as the exponential function
satisfies Cauchy’s functional equation one may ask
what functional equations (if any) characterize the
Functional Equations and Integrable Systems 431
Baker–Akiezer function. This is an area of research
still ongoing. Theta functions of a general abelian
variety are known to satisfy addition formulas with
N = 2
g
terms. It appears Baker–Akiezer functions
satisfy a similar functional equation with far fewer
terms. Such a characterization of Baker–Akiezer
functions, if found, will provide an analogous
answer to that of the Riemann–Schottky problem
which seeks to describe the Jacobians of curves
amongst general abelian varieties.
The functional equations [5], and after suitably
symmetrizing [9], are particular cases of the func-
tional equation
X
N
i¼0
3i
ðx þ yÞ
3iþ1
ðxÞ
3iþ1
ðyÞ
3iþ2
ðxÞ
3iþ2
ðyÞ
¼ 0 ½28
with N = 1 in the former case and N = 2 in the
latter. In the case
3iþ2
=
0
3iþ1
, these may be viewed
as differentiated forms of
X
N
i¼0
3i
ðx þ yÞ
3iþ1
ðxÞ
3iþ1
ðyÞ¼1 ½29
For N = 0, this is Cauchy’s equation characterizing
the exponential function and for N = 2itis
equivalent to [8]. For N = 1 and N = 2, Buchstaber
and Krichever have shown that ‘‘all’’ the solutions to
this equation are the Baker–Akhiezer functions
corresponding to algebraic curves of genus 1 and
2, respectively. In general, the Baker–Akhiezer
functions for a genus-g curve are known to satisfy
[29] for N = g. Thus, many of the equations we have
encountered are related to Baker–Akhiezer func-
tions. Dubrovin, Fokas, and Santini have shown that
Baker–Akhiezer fun ctions for a genus- g curve are
related to the functional equatio n
qðx; yÞqðy ; zÞ
qðx; zÞ
¼ rðx; yÞrðz; yÞþ
X
g
k¼1
s
k
ðyÞp
k
ðx; zÞ
Multivariable generalizations of [29] have been sought
as a means of characterizing Baker–Akhiezer functions
but such a characterization remains unproved as yet.
Topology
Several of the functional equations we have encoun-
tered also arise in topology, where the German and
Russian schools have powerfully applied functional
equations to formal group laws and genera. It is still
unclear whether these common threads form part of
a greater fabric. A genus is a ring homomorphism
’ :Q ! R;’ð1Þ¼1
where is the cobordism ring and R an integral
domain over Q. To each even power series Q(x)with
Q(0) = 1, one can associate a genus ’
Q
and vice versa
(Hirzebruch et al. 1992). Defining the odd power
series f (x) = x=Q(x) with first term 1 and coefficients
in R, the inverse function g = f
1
is such that
g
0
ðyÞ¼
X
1
n¼0
’
Q
ðCP
n
Þy
n
The genus corresponding to Q(x) = x= tanh(x)is
known as the L-genus; it takes the value 1 on every
even complex projective space. The genus corre-
sponding to Q(x) = (x=2)=sinh( x=2) is known as the
^
A-genus. The so-called (string-inspired) Witten or
elliptic genus corresponds to Q(x) = x=(x). Certain
genera may be associated with the index of natural
differential operators on the manifold. Thus, the
signature of M, sign(M), is given in terms of the de
Rham differential d and its adjoint d
,
indðd þ d
Þ¼signðMÞ¼
Y
2n
j¼0
x
j
tanhðx
j
Þ
!
½M
with variants for the
^
A-genus and elliptic genus.
Further, when a compact topological group acts on
the manifold, Atiyah and Bott showed how these
indices may be determined from the fixed point sets
of the action.
Now, functional equations arise naturally in this
context when seeking genera with special properties.
Novikov’s school has shown, for example, that the
genera associated with the index theorems of known
elliptic operators arise as solutions of functional
equations which are particular examples of [5].
Similarly, one may seek the following property of
a genus ’: for the fiber bundle p : E
F
!
B with
smooth fiber and base, one has that
’ðEÞ¼’ðFÞ’ðBÞ
Such a genus is said to be strictly multiplicative. It
may be shown that a genus is strictly mul tiplicative
in bundles with fiber CP
n1
if and only if
X
n
j¼1
Y
k 6¼j
1
f ðx
j
x
k
Þ
¼ ½30
which is essentially [21]. Following the remarks of
that equation, a genus ’ is strictly multiplicative for
all fiber bundles with fibers CP
n1
if and only if it is
strictly multiplicative for all fiber bundles with fiber
CP
2
, in which case the genus is the L-genus. If, on the
other hand, we only demand strict multiplicativity for
all fiber bundles with fibers CP
2k1
, then this is
equivalent to requiring it to hold for all fiber bundles
with fiber CP
3
, in which case the genus is an elliptic
432 Functional Equations and Integrable Systems
genus. That the same functional equations arise in
both the integrable systems and topological settings
may reflect something deeper. String theory physics,
for example, allows some topology changes such as
flops, and physical quantities such as the partition
function should reflect this invariance; invariance
under classical flops characterizes the elliptic genus.
In addition, connections have been made between the
complex cobordism ring and conformal field theory.
Other Areas
The constraints placed on this review have meant that
several further applications of functional equations
and integrable systems can only be noted. Using an
ansatz together with functional equations, Wojcie-
chowski gives an analog of the Ba¨ cklund transforma-
tion for integrable many-body systems. Similarly,
Inozemtsev constructs generalizations of the
Calogero–Moser models, while this route was used to
construct new solutions to the Witten–Dijkgraaf–
Verlinde–Verlinde (WDVV) equations by Braden,
Marshakov, Mironov, and Morozov. In the quantum
regime, Gutkin derived and solved several functional
relations by requiring a nondiffractive potential, while
functional equations have been used to construct
R-operators, solutions of the quantum Yang–Baxter
equation on a function space.
See also: Calogero–Moser–Sutherland Systems of
Nonrelativistic and Relativistic Type; Classical r-matrices,
Lie Bialgebras, and Poisson Lie Groups; Cohomology
Theories; Eigenfunctions of Quantum Completely
Integrable Systems; Integrability and Quantum Field
Theory; Integrable Systems and Algebraic Geometry;
Integrable Systems: Overview; Lie Groups: General
Theory; Quantum Calogero–Moser Systems; Toda
Lattices; WDVV Equations and Frobenius Manifolds.
Further Reading
Aczel J (1996) Lectures on Functional Equations and Their
Applications. New York: Academic Press.
Braden HW (2001) Rigidity, functional equations and the
Calogero–Moser model. Journal of Physics A 34: 2197–2204.
Braden HW and Buchstaber VM (1997) The general analytic
solution of a functional equation of addition type. SIAM
Journal on Mathematical Analysis 28: 903–923.
Braden HW and Byatt-Smith JGB (1999) On a functional
differential equation of determinantal type. Bulletin of the
London Mathematical Society 31: 463–470.
Bruschi M and Calogero F (1987) The Lax representation for an
integrable class of relativistic dynamical systems. Communica-
tions in Mathematical Physics 109: 481–492.
Buchstaber VM and Krichever IM (1993) Vector addition
theorems and Baker–Akhiezer functions. Teoreticheskaya i
Matematicheskaya Fizika 94: 200–212.
Byatt-Smith JGB and Braden HW (2003) Functional equations
and Poincare´ invariant mechanical systems. SIAM Journal on
Mathematical Analysis 34: 736–758.
Calogero F (1975) Exactly solvable one-dimensional many-body
problems. Lettere al Nuovo Cimento 13(2): 411–416.
Calogero F (1976) On a functional equation connected with
integrable many-body problems. Lettere al Nuovo Cimento
16: 77–80.
Calogero F (2001) Classical Many-Body Problems Amenable to
Exact Treatments, Lecture Notes in Physics, New Series m:
Monographs, vol. 66. Berlin: Springer.
Dubrovin BA, Fokas AS, and Santini PM (1994) Integrable
functional equations and algebraic geometry. Duke Mathema-
tical Journal 76: 645–668.
Hietarinta J (1987) Direct methods for the search of the second
invariant. Physics Reports 147: 87–154.
Hirzebruch F, Berger T, and Jung R (1992) Manifolds
and Modular Forms. Aspects of Mathematics, vol. E20.
Braunschweig: Vieweg.
Oshima T and Sekiguchi H (1995) Commuting families of
differential operators invariant under the action of a Weyl
group. Journal of Mathematical Sciences, The University of
Tokyo 2: 1–75.
Perelomov AM (1990) Integrable Systems of Classical Mechanics
and Lie Algebras, vol. I. Basel: Birkha¨user.
Ruijsenaars SNM (1987) Complete integrability of relativistic
Calogero–Moser systems and elliptic function identities.
Communications in Mathematical Physics 110: 191–213.
Ruijsenaars SNM and Schneider H (1986) A new class of
integrable systems and its relation to solitons. Annals of
Physics (NY) 170: 370–405.
Sutherland B (1975) Exact ground-state wave function for a
one-dimensional plasma. Physical Review Letters 34:
1083–1085.
Taniguchi K (1997) On uniqueness of commutative rings of
Weyl group invariant differential operators. Publicati ons of
the Research Institute for Mathematical Sciences, Kyoto
University 33: 257–276.
Whittaker ET and Watson GN (1927) A Course of Modern
Analysis. Cambridge: Cambridge University Press.
Functional Equations and Integrable Systems 433
Functional Integration in Quantum Physics
C DeWitt-Morette, The University of Texas at Austin,
Austin, TX, USA
ª 2006 Elsevier Ltd. All rights reserved.
The Domain of Integration
Functional integration is integration over function
spaces, that is, the variable of integration is a
function f with values in a D-dimensional manifold:
f : U ! M
D
½1
Generically a space of functions is an infinite-
dimensional space. Our understanding of infinite-
dimensional spaces has progressed significantly
during the twentieth century, and we can formulate
functional integration in its proper setting.
Let F be the domain of integration, and f 2F
the variable of integration. If the domain of f is
a subset U of R, the functional integral is called
apathintegral;ifU is of dimension higher than 1
(e.g., spacetime), F isoftencalledaspaceof
histories.
The information necessary for defining a domain
of integration includes
the domain and the range of the variable of
integration f,
the analytical properties of f, and
possibly additional information, such as require-
ments on the values of f on its boundary.
Examples of variables of integration f in [1]
The domain U of f may be a time interval, a scale
range, or any parameter. The range M
D
of f may be
a group manifold, a Riemannian manifold, a
symplectic manifold, a multiply-connected space,
etc., or simply R
D
. The domain of integration F
may be a space of pointed paths, for example,
x :¼ T ! M
D
; T ¼ t
a
; t
b
½ ½2
xt
b
ðÞ¼x
b
2 M
D
for all x 2F
The paths x may be continuous (e.g., Brownian
paths), or may have square integrable derivatives;
F is then an L
2, 1
space (e.g., quantum physics).
Z
T
dt
_
xðtÞ
jj
2
< 1; x 2F ½3
Given a domain of integration F, one needs to
select a volume element appropriate to F. This is a
challenge which has been met in a number of cases
(Cartier and DeWitt-Morette 2006). Examples are
given below. Given a volume element, one can then
characterize the functional s F on F integrable with
respect to the chosen volume element.
Two Basic Techniques
The tw o most useful techniques for computing
integrals are change of variable of integration and
integration by parts. They follow from fundamental
properties that apply to functional integrals as well
as to ordinary integrals. Let us recall them in the
context of ordinary integrals.
Let f and g be functions on R of compact support.
Let I stand for integration
Iðf Þ¼
Z
R
dxfðxÞ; x 2 R
and D for derivation of f with respect to x,
ðDf ÞðxÞ¼
d
dx
f ðxÞ
The fundamental rule
DI ¼ 0 ¼) 0 ¼
d
dx
Z
R
dxfðxÞ½4
The functional I(f ) is invariant under a change of
variable of integration.
Another fundamental rule is ID = 0:
ID ¼ 0 ¼) 0 ¼
Z
d f ðxÞg ðxÞðÞ
¼
Z
df ðxÞgðxÞþ
Z
dgðxÞf ðxÞ½5
The fundamental rules [4] and [5] apply to
functional integration. The derivation D can be
either a functional derivative or a Lie derivative
defined as follows. Let K be the reals R or the
complex C, let f be a differentiable functional on a
Banach space X
f : U X ! K ½6
The functional derivative Df j
x
0
of f at x
0
is defined
by the equation
fx
0
þ hðÞfx
0
ðÞ¼Df j
x
0
h þ RðhÞ½7
with the norm kR(h)k of order less than the
norm khk.
The Lie derivative L
V
along the vector field V is
conceptually intuitive and of practical interest: an
infinite-dimensional space X of paths x is not an
intuitive concept, but a one-parameter family of
434 Functional Integration in Quantum Physics
paths {x()} 2 X, with 2 [0, 1], is a convenient
tool for dealing with X:
xðÞ : T ! M
D
xð; tÞ :¼ xðÞðÞðtÞ2M
D
½8
Set x(0) = x
0
. A differentiable family {x()} defines a
vector field V(x
0
) along the path x
0
(see Figure 1):
Vðx
0
Þ :¼
d
d
xðÞ
¼0
Vx
0
ðtÞðÞ¼
@
@
xð; tÞ
¼0
½9
The functional vector field V on the tangent bundle
of X defines a group of transformations on X, and a
Lie derivative L
V
of tensor fields on X.
The Lie derivative L
V
obeys the Cartan (Elie and
Henri) equation
L
V
¼ d{
V
þ {
V
d ½10
where d is the exterior differential and {
V
is the
interior product, defined as usual on Banach spaces.
Remark (Berezin integrals). To show the power of
the rules [4] and [5], we can mention that they
provide Berezin rules of integration over Grassmann
variables (Cartier and DeWitt-Morette 2006).
Path Integrals and Quantum Dynamics
The hist ory of path integrals in quantum physics did
not begin with the definitions of domain of integra-
tion, volume elements, etc. It began with the Ph.D.
thesis of R P Feynman in 1942. Feynman expressed
the time evolution of a system as the limit N = 1 of
the following N-tuple integral:
x
t
jx
a
ðÞ¼lim
N ¼1
ZZ
Z
x
t
jx
N
ðÞdx
N
x
N
jx
N 1
ðÞ
dx
N1
x
2
jx
1
ðÞdx
1
x
1
jx
0
ðÞ ½11
where the time interval T = [t, 0] has been replaced
by N of its points {t
i
}, 1 i N:
t
0
< t
1
< < t
N
< t ½12
and the path x : T ! R
D
is replaced by N of its values
x
i
:¼ xt
i
ðÞ ½13
Dirac (1933) had shown that (x
t
jx
0
) defines the
exponential of a quantum function S
Q
,by
exp iS
Q
x
t
; x
0
; tðÞ=h
:¼ x
t
jx
0
ðÞ ½14
such that the real part of S
Q
is the classical action
function (a.k.a. Hamilton’s principal function;
further studies have shown that the correct state-
ment is: the real part is the classical action, up to
order h), and the imaginary part of S
Q
is of order h,
the normalized Planck constant
h ¼ h=2 ½15
Feynman remarked that for a system with
Lagrangian L the short-time probability amplitude
(x
tþt
jx
t
) is ‘‘often equal to
A
1
exp i tL
x
tþt
x
t
t
; x
tþt
.
h
½16
within a normalizing constant A as the limit t
approaches zero.’’ The absolute value of A can be
obtained from a unitary requirement (Morette 1951).
Feynman expressed the finite probability ampli-
tude as a path integral, limit of the discretized
expression [11]
x
t
jx
0
ðÞ¼
Z
Dx exp iSðxÞ=hðÞ½17
where S(x) is the action functional
SðxÞ¼
Z
t
t
0
dsL
_
xðsÞ; xð sÞðÞ ½18
The undefined symbol Dx is a ‘‘volume element’’ on
the space of paths, corresponding to the infinite
product of the normalization constant A
1
.
The issues raised by the path integral [17] are
the definition of the volume element D x; and
a method for computing [17] for a given action
functional S.
The explicit calculation of the limit [11] of an
N-tuple integral when N = 1 is a Herculean task of
very limited use. But two other methods of wide
applications, leaving the volume element Dx as a
heuristic symbol, have vindicated the power of
functional integration: the diagram technique and
the semiclassical expansions.
x(0)
x(
α)
x(
α, t )
t
b
x(0, t )
t
a
v
D
=
0
Figure 1 A one-parameter family of paths with fixed endpoints.
Functional Integration in Quantum Physics 435