<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
branch	O
of	O
mathematics	O
,	O
it	O
is	O
sometimes	O
possible	O
to	O
generalize	O
the	O
notion	O
of	O
the	O
determinant	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
of	O
finite	O
order	O
(	O
representing	O
a	O
linear	B-Architecture
transformation	I-Architecture
from	O
a	O
finite-dimensional	O
vector	O
space	O
to	O
itself	O
)	O
to	O
the	O
infinite-dimensional	O
case	O
of	O
a	O
linear	B-Architecture
operator	I-Architecture
S	O
mapping	O
a	O
function	B-Algorithm
space	I-Algorithm
V	O
to	O
itself	O
.	O
</s>
<s>
The	O
corresponding	O
quantity	O
det(S )	O
is	O
called	O
the	O
functional	B-Algorithm
determinant	I-Algorithm
of	O
S	O
.	O
</s>
<s>
There	O
are	O
several	O
formulas	O
for	O
the	O
functional	B-Algorithm
determinant	I-Algorithm
.	O
</s>
<s>
They	O
are	O
all	O
based	O
on	O
the	O
fact	O
that	O
the	O
determinant	O
of	O
a	O
finite	O
matrix	B-Architecture
is	O
equal	O
to	O
the	O
product	O
of	O
the	O
eigenvalues	O
of	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
where	O
the	O
zeta	O
function	O
in	O
the	O
point	O
s	O
=	O
0	O
is	O
defined	O
by	O
analytic	B-Algorithm
continuation	I-Algorithm
.	O
</s>
<s>
To	O
give	O
it	O
a	O
rigorous	O
meaning	O
it	O
must	O
be	O
divided	O
by	O
another	O
functional	B-Algorithm
determinant	I-Algorithm
,	O
thus	O
effectively	O
cancelling	O
the	O
problematic	O
'	O
constants	O
 '	O
.	O
</s>
<s>
These	O
are	O
now	O
,	O
ostensibly	O
,	O
two	O
different	O
definitions	O
for	O
the	O
functional	B-Algorithm
determinant	I-Algorithm
,	O
one	O
coming	O
from	O
quantum	O
field	O
theory	O
and	O
one	O
coming	O
from	O
spectral	O
theory	O
.	O
</s>
<s>
Each	O
involves	O
some	O
kind	O
of	O
regularization	B-Algorithm
:	O
in	O
the	O
definition	O
popular	O
in	O
physics	O
,	O
two	O
determinants	O
can	O
only	O
be	O
compared	O
with	O
one	O
another	O
;	O
in	O
mathematics	O
,	O
the	O
zeta	O
function	O
was	O
used	O
.	O
</s>
<s>
have	O
shown	O
that	O
the	O
results	O
obtained	O
by	O
comparing	O
two	O
functional	B-Algorithm
determinants	I-Algorithm
in	O
the	O
QFT	O
formalism	O
agree	O
with	O
the	O
results	O
obtained	O
by	O
the	O
zeta	O
functional	B-Algorithm
determinant	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
is	O
to	O
find	O
a	O
way	O
to	O
make	O
sense	O
of	O
the	O
determinant	O
of	O
an	O
operator	O
S	O
on	O
an	O
infinite	O
dimensional	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
where	O
V	O
is	O
the	O
function	B-Algorithm
space	I-Algorithm
and	O
the	O
L2	O
inner	O
product	O
,	O
and	O
the	O
Wiener	O
measure	O
.	O
</s>
<s>
The	O
basic	O
assumption	O
on	O
S	O
is	O
that	O
it	O
should	O
be	O
self-adjoint	O
,	O
and	O
have	O
discrete	O
spectrum	O
λ1	O
,	O
λ2	O
,	O
λ3	O
,	O
…	O
with	O
a	O
corresponding	O
set	O
of	O
eigenfunctions	B-Algorithm
f1	O
,	O
f2	O
,	O
f3	O
,	O
…	O
which	O
are	O
complete	O
in	O
L2	O
(	O
as	O
would	O
,	O
for	O
example	O
,	O
be	O
the	O
case	O
for	O
the	O
second	O
derivative	O
operator	O
on	O
a	O
compact	O
interval	O
Ω	O
)	O
.	O
</s>
<s>
where	O
N	O
is	O
an	O
infinite	O
constant	O
that	O
needs	O
to	O
be	O
dealt	O
with	O
by	O
some	O
regularization	B-Algorithm
procedure	O
.	O
</s>
<s>
If	O
all	O
quantities	O
converge	O
in	O
an	O
appropriate	O
sense	O
,	O
then	O
the	O
functional	B-Algorithm
determinant	I-Algorithm
can	O
be	O
described	O
as	O
a	O
classical	O
limit	O
(	O
Watson	O
and	O
Whittaker	O
)	O
.	O
</s>
<s>
Otherwise	O
,	O
it	O
is	O
necessary	O
to	O
perform	O
some	O
kind	O
of	O
regularization	B-Algorithm
.	O
</s>
<s>
The	O
most	O
popular	O
of	O
which	O
for	O
computing	O
functional	B-Algorithm
determinants	I-Algorithm
is	O
the	O
zeta	O
function	O
regularization	B-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
this	O
allows	O
for	O
the	O
computation	O
of	O
the	O
determinant	O
of	O
the	O
Laplace	O
and	O
Dirac	O
operators	O
on	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
,	O
using	O
the	O
Minakshisundaram	O
–	O
Pleijel	O
zeta	O
function	O
.	O
</s>
<s>
Since	O
the	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
the	O
zeta	O
function	O
is	O
regular	O
at	O
zero	O
,	O
this	O
can	O
be	O
rigorously	O
adopted	O
as	O
a	O
definition	O
of	O
the	O
determinant	O
.	O
</s>
<s>
This	O
kind	O
of	O
Zeta-regularized	O
functional	B-Algorithm
determinant	I-Algorithm
also	O
appears	O
when	O
evaluating	O
sums	O
of	O
the	O
form	O
.	O
</s>
<s>
For	O
one-dimensional	O
potentials	O
,	O
a	O
short-cut	O
yielding	O
the	O
functional	B-Algorithm
determinant	I-Algorithm
exists	O
.	O
</s>
<s>
which	O
is	O
also	O
a	O
meromorphic	O
function	O
of	O
m	O
,	O
we	O
see	O
that	O
it	O
has	O
exactly	O
the	O
same	O
poles	O
and	O
zeroes	O
as	O
the	O
quotient	O
of	O
determinants	O
we	O
are	O
trying	O
to	O
compute	O
:	O
if	O
m	O
is	O
an	O
eigenvalue	O
of	O
the	O
operator	O
number	O
one	O
,	O
then	O
will	O
be	O
an	O
eigenfunction	B-Algorithm
thereof	O
,	O
meaning	O
;	O
and	O
analogously	O
for	O
the	O
denominator	O
.	O
</s>
