Communications
in
Commun.
Math.
Phys.
67,
1-16
(1979)
Mathematical
Physics
© by
Springer-Verlag
1979
The
Partition
Function
of a
Degenerate
Functional
A.
S.
Schwarz
Department
of
Theoretical Physics, Moscow Physical
Engineering
Institute,
Moscow
M
409,
USSR
Abstract.
The
partition
function
of a
degenerate quadratic functional
is
defined
and
studied.
It is
shown
that
Ray-Singer invariants
can be
interpreted
as
partition functions
of
quadratic
functionals.
In the
case
of a
degenerate
non-
quadratic
functional
the
semiclassical approximation
to the
partition
function
is
considered.
Section
1.
Introduction
The
degenerate Lagrangians
are
important
in
quantum
field
theory.
For
example
the
action
in
gauge theories
is
invariant with respect
to
infinite-dimensional group
of
local gauge transformations
and
therefore
the
corresponding Lagrangian
is
degenerate.
To
calculate
the
physical quantities
in
gauge theories
one
must impose
the
gauge conditions,
but
final
results must
be
independent
of the
gauge
conditions.
The
physical quantities
in the
gauge theories
and
other theories
described
by
degenerate Lagrangians were expressed through functional integrals
by
Faddeev
and
Popov
(see
[1]).
In the
present paper
we
give
a
rigorous treatment
of the
case when
the
action
is
a
degenerate quadratic
functional
(Sects.
2 and 3). Our
results
can be
useful
when
dealing
with various questions
on
quantum
field
theory.
For
example, they
are
connected with so-called anomalies. These results
can be
used
to
study
the
instanton contribution
in
Schwinger functions (Sect.
5). Our
assertions
can be
applied outside
of
quantum
field
theory
too.
They
are
closely related with
the
theorems proved
in [2,
3].
Namely,
we
show
that
the
Ray-Singer torsion
[2] can
be
considered
as a
partition
function
of
action
which
is
invariant
by
diffeo-
morphisms.
The
independence
of
Ray-Singer torsion
on the
choice
of
riemannian
metric
can be
interpreted
as
independence
of the
partition
function
on the
choice
of
gauge condition.
In a
similar
way one can get
invariants constructed
in
[37]
and
new
invariants.
One of the new
invariants
will
be
described
below.
Part
of our
results
was
formulated
in
[4].
A
short review
of
some mathematical
results
used
in
present paper
can be
found
in
[5].
0010-3616|79|0067|0085|$01.20
2 A. S.
Schwarz
The
functional
integrals
for
partition
functions
of
quadratic
functionals
are
gaussian
and
therefore they
can be
expressed through determinants.
We
must
define
therefore
the
determinant
of
infinite-dimensional
operator
and
some related
notions.
We say
that
the
non-negative
self-adjoint
operator
B in
Hubert space
is
regular
if
for
ί->+0
Sp(exp(-Bi)-ff(B))=
Σ*
k
(B)t~
k
+
0(f)
, (1)
where
ε
>0
and k run
over
finite
set K of
non-negative numbers.
The
symbol
Π(M)
denotes
here
and
later
the
projector
on the
kernel
of
operator
M
:
where
f
t
run
over zero modes
of M. The
trace
Sp M of
operator
M is
defined
as
where
e
t
run
over
orthonormal
basis. [Always when
we
consider
the
trace
of
operator
M we
suppose that
the
operator
M is of the
trace class,
i.e.
the sum of the
eigenvalues
of
(M*M)
1/2
converges.]
The
zeta
function
ζ(s\B)
of
operator
B for
large
Re (5) can be
defined
by the
formula
where
λ
j
run
over non-zero eigenvalues
of B. For
other
5 the
zeta
function
must
be
defined
by
means
of
analytic continuation.
It is
easy
to
check that
for
regular
operator
B the
analytic continuation
of
ζ(s\B)
in the
half-plane
Re(s)>
— ε,
ε>0
can be
written
in the
form
keK
+
}
Sp(exp(-Bί)-Π(B))-
£
o^Bf^dt
. (3)
0
\
keK
We
define
the
regularized determinant
D(B)
of the
regular operator
B by the
formula
This
definition
is
correct because
ζ(s\B)
is
analytic
at
point
s
=0.
It
follows
from
(3)
that
- J
Sp
(exp
(
-
Bt)
-
Π(B))Γ
l
i
k
l
-
J
(Sp
(exp
(-Bt)-
77(5))-
Σ
«
k
(B)Γ
k
}Γ
l
dt
.
(4)
0\
keK
Partition
Functions
of
Degenerate Functionals
3
If
B is an
operator acting
from
Hubert space
^
into Hubert space
34f
2
and the
operator
B*B is
regular, then
one can
define
the
regularized determinant D(B)
as
-\
^-ζ(s\B*Bη
=D(B*B)
112
. (5)
If
B is a
self-adjoint
regular operator this
definition
of
D(B) concide with
the
preceding one.
We
define
smooth regular
family
of
operators
as a
family
of
regular operators
B(u\
0
^
u
^
1,
satisfying
-B(u)t-Π(B))-
£
a
k
(B(u))Γ
k
}\
^
keK
for
0<ί<l
and
j
V^.f
V'
x
l^
V
-t-'v""/''/
"V
*-V//
=
~./V"
\'/
αw
for
ί^l.
(Here
K
denotes
finite
set of
non-negative numbers,
N
is an
arbitrary
number,
the
positive constants
ε, C,
C
N
do not
depend
on u.)
We say
that
the
operators
A,
B
acting
in
Hubert space
ffl
form
a
regular pair
if
B is a
non-negative
self-adjoint
operator
SpA(exp(-Bt)-Π(B))=
Σ
β
k
(A\B)Γ
k
+
0(t
ε
)
(8)
keK
for
ί->+0
and
Sp
A(QXp
(-Bt)-
Π(B))
=
O(t
~
N
)
(9)
for
f-»
oo.
[Here
as in (3)
K
denotes
a
finite
set of
non-negative numbers,
ε>0,
N as
an
arbitrary
integer.]
If A = 1
then
the
pair
(A,
B}
is
regular
if and
only
if the
operator
B is
regular.
Let us
consider
differential
operators
on a
compact manifold
M
(i.e.
differential
operators acting
in the
spaces
of
sections
of
vector bundles with
the
base
M). The
coefficient
functions
of
differential
operators
(as
well other
functions
under
consideration) will
be
always supposed smooth.
The
family
of
differential
oper-
ators depending
on
parameter
u,
O^u^
1,
will
be
called smooth
if the
coefficient
functions
are
smooth with respect
of all
arguments (including
u).
Further
in
present
section
we use the
notations
A or
A
{
for
differential
operators
and the
notation
B for
self-adjoint
non-negative elliptic
differential
operator.
Lemma
1. The
operator
A
λ
exp(
—
Bt)A
2
is of the
trace class
for t >0 and
Sp
A!
exp
(-Bt)A
2
= Sp
A
2
A
l
exp
(-
Bή
= Sp exp
(-
Bt)A
2
A
ί
.
Lemma
2. The
function
Sp
^4(exp
(—
Bt) —
Π(B))
decreases faster than
any
power
of
t at
infinity.
Lemma
3. For
t-+
+0
4 A. S.
Schwarz
where
ε
>0
and k
—
, / is an
integer. (Here
n
denotes
the
dimension
of
manifold
M
and m
denotes
the
order
of
operator
B.)
Lemma
4. // M is an
odd-dimensional
manifold,
then
Ψ
0
(A\B)
=
0.
The
coefficients
Ψ
k
(A\B)
can be
calculated
by
semiclassical method. They
are
given
by
local formulae.
In
other words
the
following assertion
is
correct.
Lemma
5. // the
coefficient
functions
of
operators
B
ί
and
B
2
coincide
in the
domain
GcM
and the
coefficient
functions
of A
vanish
in M\G
then
Ψ
0
(A\B
ί
)=Ψ
0
(A\B
2
)
.
It
follows
from
Lemma
2 and
Lemma
3
that
the
pair
(A,
B) is
regular
and the
coefficients
β
k
(A\B)
are
given
by
β
0
(A\B)=Ψ
0
(A\B)-SpAΠ(B)
9
β
k
(A\B)=Ψ
k
(A\B)
for
/c>0.
For
arbitrary elliptic operator
C the
operator
B = C*C is a
non-negative
self-adjoint
elliptic
operator;
one can
therefore
define
the
regularized determinant
D(C)
for
arbitrary elliptic operator.
Let
A(u)
be a
smooth
family
of
differential
operators
and
B(u)
a
smooth
family
of
self-adjoint
non-negative elliptic operators
(O^w^l).
We
assume that
Sp
Π(B(u)}
— dim ker
B(u)
does
not
depend
on u.
Lemma
6. The
function
Ψ
k
(A(u)\B(u))
is
smooth
with
respect
to u and
Sp
A(u)
exp
(-
B(u)t)
-
£
Ψ
k
(A(u)\B(u)t
-k
<Ct
ε
for
0<ί<l
|Sp^(u)(exp
(
-
B(u)t)
-
Π(B))\
^
for
ί^l.
(Here
ε>0,
N is
arbitrary,
C and
C
N
do not
depend
on u.)
Lemma
7.
It
follows
from
Lemma
6 and
Lemma
7
that B(u)
is a
smooth regular
family.
The
lemmas above
can be
derived
from
well known results.
In
particular
Lemmas
3-6 can be
deduced
from
the
results
of [6] and
[7].
Section
2. The
Partition Function
of
Quadratic Functional
Let
<9"
be a
quadratic
functional
on a
pre-Hilbert
space
Γ
0
i.e.
(1)
where
S is a
self-adjoint
operator acting
in
Γ
0
.
If the
functional
£f
is
non-
degenerate (i.e.
Sf
=
Q
if and
only
if
/
= 0) and
S
2
is a
regular operator
one can
define
the
partition
function
Z of
¥
as
D(S)~
1/2
=D(S
2
)~
1/4
.
[Formally
we can
define
Z as the
functional
integral
of exp ( —
5^)
over
Γ
0
.
The
formal calculation
of
this gaussian integral leads
to the
answer
(detS)~
1/2
.]
Partition Functions
of
Degenerate Functionals
Let
us
consider
now the
quadratic functional
^
on
pre-Hilbert
space
Γ
0
and a
linear
map T of
pre-Hilbert space
Γ^
into
Γ
0
satisfying
&(f+Ώi)
=
&(f)
(2)
for
every
/zeΓ
r
[It is
easy
to
check that
the
requirement
(2) is
satisfied
if and
only
if
ST =
Q.~]
It
follows from
(2)
that every element
ΛejΓi
generates
a
symmetry
transformation
of
£f.
If
TΦO
then
the
functional
^
is
degenerate.
We
assume that
there exists adjoint operator
T*
defined
on
Γ
0
and
taking values
in
Γ
0
;
the
operators
S
2
and
T*T
will
be
supposed regular. Then
we can
define
the
partition
function
of
Sf
as
Z
=
D(S)-
1/2
D(T).
(3)
(This
definition will
be
justified
in
Appendix
by
means
of
Faddeev-Popov trick.)
By
definition
D(S)
=
D(S
2
)
ί/2
,
D(T)
=
D(T*T)
1/2
and
therefore
It
is
easy
to
check that
S
2
+ TT* is
regular operator
and
D(S
2
+
TT*)
=
D(S
2
)D(TT*)
(this
equality
can be
deduced from relations
S
2
-
TT*
=0,
TT*S
2
-0
which follows
from
ST
= 0). The
non-zero eigenvalues
of TT*
coincide with non-zero eigenvalues
of
T*T and
therefore
D(TT*)
=
D(T*T).
Hence
we can
represent
the
partition
function
in the
form
3/4
,
(4)
where
Π
0
=
S
2
+
TT*,
(5)
D!
=
Γ*T.
(6)
If
S is a
regular operator then
the
partition
function
can be
represented
in the
form
Z
=
D(S+TT*Γ
1/2
D(T*T)
. (7)
It
is
important
to
note that
not
only
the
functional
y
but
also
the map T and
the
scalar products
in the
spaces
Γ
0
,
Γ^
are
used
in the
definition
of
partition
function.
We
will study
now the
variation
of
partition
function
by
variation
of
scalar products
in
Γ
0
,
T^
and by
variation
of T Our
proofs will
be
based
on the
following
Lemma.
Lemma
8. Let us
suppose that
-
Σ
λ
q
Sp(
Q
χp(-tA
q
(u))-Π(A
q
(u)))
Σ
Sp/yu)(exp(-tT
r
(M))-Π(T
Γ
(w))),
(8)