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[ Pobierz całość w formacie PDF ]WHAT WE SHOULD HAVE LEARNED FROM G. H. HARDY
ABOUT QUANTUM FIELD THEORY UNDER EXTERNAL
CONDITIONS
S. A. FULLING
Mathematics Department, Texas A&M University,
College Station, TX 77843, USA
E-mail: fulling@math.tamu.edu
1
Reminder of the Phenomena
To establish the setting of this presentation, consider as an example the classic
problem of a scalar eld in a box. More precisely, consider a \Casimir slab"
of width
L
in 4-dimensional space-time, with the energy-momentum tensor
associated with minimal gravitational coupling. With the spatial operator in
the eld equation,
H
=
−r
2
, several quantities are associated:
1.1
Vacuum energy
In the 1970s considerable attention was directed to dening and calculating
the local density of energy in such situations.
2
It was found that there are
two contributions. First, there is a constant energy density throughout the
space, proportional to
L
−
4
. It is variously described as being related to the
nite size of the box, to the discreteness of the spectrum of normal modes, and
to the existence of a closed geodesic (or classical path) of length 2
L
.Thisis
the scalar analog of the classic Casimir eect. It persists in a closed universe
without boundary. Second, there is a divergent distribution of energy clinging
to the walls:
T
00
(
x
)
x
−
4
,when
x
is the distance to the nearest wall. It
persists in an innite space with only one wall. We may say that this eect is
caused by the existence of the boundary, the spatial inhomogeneity near the
boundary of the set of mode functions, and the length (2
x
) of a path that
reflects from the boundary and returns to the observation point. Therefore,
with some exaggeration in the eyes of an experimentalist, we can say: By
observing vacuum energy
locally
, we can tell how big the world is (
L
) and how
far we are from its edge (
x
).
/
1
1.2
The heat kernel expansion
Expanding heat kernels has long been a favorite industry of many, myself
included. For any positive operator
H
, the heat kernel
K
(
t;x;y
)
e
−tH
(
x;y
)
is the Green function such that
u
(
x
)=
R
K
(
t;x;y
)
f
(
y
)
dy
solves the initial-
value problem
@u
@t
=
Hu
,
u
(0
;x
)=
f
(
x
).
−
It is well known that
K
has an
asymptotic expansion
(4
t
)
−m=
2
"
1+
X
n
=1
a
n
(
x
)
t
n
#
;
K
(
t;x;x
)
(1)
(
m
= spatial dimension) where
a
n
(
x
)isa
local
functional of the curvature, etc.
(covariant functionals of the coecient functions in the seond-order dierential
operator
H
)at
x
. In particular,
a
n
is identically zero inside our box (where
H
=
2
). By studying the heat kernel expansion, you will never discover
the Casimir eect!
−r
1.3
Schrodinger and Schwinger{DeWitt kernels
e
−itH
(
x;y
) is obtained formally by replacing
t
in
the heat kernel by
it
.When
H
is an elliptic operator, a quantum-mechanical
Hamiltonian,
U
is the propagator that solves the time-dependent Schrodinger
equation. When
H
is a hyperbolic operator and
t
a ctitious proper time,
U
is the Schwinger{DeWitt kernel used in renormalization of quantum eld
theories. The rotation of the
t
coordinate in Eq. (1) is algebraically trivial, but
the resulting expansion is, in general, invalid if taken literally! This is most
easily seen by letting our space be the half-space
R
+
, for which the problem
can be solved exactly by the method of images:
The function
U
(
t;x;y
)
(4
it
)
−
1
=
2
h
e
ijx−yj
2
=
4
t
i
:
2
=
4
t
e
ijx
+
yj
U
(
t;x;y
)
−
(2)
Passing to the diagonal, we have
(4
it
)
−
1
=
2
h
1
e
ix
2
=t
i
;
U
(
t;x;x
)
−
(3)
and we see that the reflection term is exactly as large as the \main" term, in
blatant contradiction to the alleged asymptotic expansion (1) (which in this
case consists just of the main term and an implied error term vanishing faster
than any power of
t
). Does this mean that the Schwinger{DeWitt series, to
which so many graduate students have devoted their thesis years, is nothing
but a snare and a delusion?
Heaven forbid!
The information in that series
2
is meaningful when used correctly (for example, in renormalization theory).
In fact, the second term in Eq. (3), although large, is rapidly oscillatory, and
consequently the series is indeed valid in various distributional senses, as I shall
partially explain below.
1.4
The cylinder kernel
e
−t
p
H
(
x;y
) solves the (elliptic) boundary-
The Green function
T
(
t;x;y
)
value problem
@
2
u
@t
2
(i.e.,
in a semi-innite cylinder with our spatial manifold as its base) by
u
(
x
)=
=
Hu
,
u
(0
;x
)=
f
(
x
),
u
(
t;x
)
!
0as
t
!
+
1
R
T
(
t;x;y
)
f
(
y
)
dy
.
The cylinder kernel
T
shares many of the properties of
vacuum energy (
h
T
00
i
) and the latter's progenitor, the Wightman two-point
function (
W
(
t;x;y
)
h
(
t;x
)
(0
;y
)
i
), a Green function for the wave equa-
tion
@
2
u
Hu
; however,
T
is technically simpler in several ways. Its study
therefore deserves our attention, even though it has no direct physical interpre-
tation with
t
as a time coordinate. For an example we once again consider one
space dimension, where
H
=
@t
2
=
−
@
2
@x
2
(so we're dealing with very classical Green
functions for the two-dimensional Laplace equation). If the spatial manifold is
the entire real line, the kernel is
T
0
(
t;x;y
)=
t
−
1
y
)
2
+
t
2
;
(4)
(
x
−
whereas if the space is
R
+
, the problem can again be solved by images:
T
+
(
t;x;y
)=
t
(
x
+
y
)
2
+
t
2
:
1
1
y
)
2
+
t
2
−
(5)
(
x
−
Passing to the diagonal and making a Taylor expansion, we get
1
:
1
t
t
2
(2
x
)
2
T
+
(
t;x;x
)=
−
+
(6)
Thus we see that the asymptotic expansion of the cylinder kernel does probe
x
(and also
L
, in a nite universe), as the vacuum energy does. On the other
hand,
and
W
share some of the delicate analytical complications of
U
,
whereas
T
is about as well-behaved as
K
.
h
T
00
i
1.5
Summary and synopsis
The four Green functions
T
,
U
,
K
,
W
demonstrate two distinctions (Table 1):
that between local and global dependence on the geometry, and that between
pointwise and distributional validity of their asymptotic expansions (and, as
we'll see, of their eigenfunction expansions also).
3
Table 1: Asymptotic properties of Green functions.
Pointwise
Distributional
Local
K
U
Global
T
W
The main points I wish to make are:
As already remarked, the small-
t
expansion of
T
contains nonlocal infor-
mation not present in the corresponding expansion of
K
.
Nevertheless, both these expansions are determined by the high-energy
asymptotic behavior of the density of states, or, more generally, the spec-
tral measures, of the operator
H
.
The detailed relationships among all these asymptotic developments are,
at root, not a matter of quantum eld theory, nor even of partial dif-
ferential equations or operators in Hilbert space. They are instances of
some classical theory on the summability of innite series and integrals,
developed circa 1915.
9
;
10
2
Spectral Densities
The Green functions have spectral expansions in terms of the eigenfunctions
of
H
. If the manifold is
R
and
H
=
@
2
−
@x
2
, the heat, cylinder, and Wightman
kernels are
Z
1
1
2
dke
−tk
2
e
ikx
e
−iky
;
K
0
(
t;x;y
)=
(7)
−1
Z
1
1
2
dke
−tjkj
e
ikx
e
−iky
;
T
0
(
t;x;y
)=
(8)
−1
Z
1
dk
e
−itjkj
j
1
4
e
ikx
e
−iky
:
W
0
(
t;x;y
)=
(9)
k
j
−1
On the other hand, we can vary the space, or the operator; for instance, the
heat kernel for
R
+
is
Z
1
K
+
(
t;x;y
)=
2
d!e
−t!
2
sin
!x
sin
!y;
(10)
0
4
while that for a box of length
L
is
X
K
L
(
t;x;y
)=
2
L
sin
nx
L
sin
ny
L
e
−t
(
n=L
)
2
n
=1
Z
1
!
sin
!x
sin
!y:
d!e
−t!
2
X
n
=1
2
L
n
L
=
−
(11)
0
In general, each object can be written in the schematic form
Z
1
g
(
t!
)
dE
(
!;x;y
)
;
(12)
0
where
dE
is integration with respect to the spectral measure of a given
H
(on
a given manifold) and
g
is the kernel of a certain integral transform (Laplace,
Fourier, etc.) dening the Green function in question. (To t some of the ker-
nels into the mold (12), it is necessary to redene some variables, for example
replacing
t
by
p
t
.)
Spectral measures or densities are in a sense more fundamental and more
directly relevant than Green functions, since
all
functions of
H
can be ex-
pressed immediately in terms of them. However, the kernels (especially
K
and
T
) are more accessible to calculation and analytical investigation.
3
Riesz{Cesaro Means
Eq. (12) is an instance of the general structure
Z
f
(
)
a
(
)
d
(
)
;
(13)
0
where
is some measure, such as
E
(
;x;y
). (For a totally continuous spec-
trum,
d
(
) equals
0
(
)
d
where
0
is a function, which we call the spectral
density. For a totally discrete spectrum,
d
(
) is of the form
P
n
c
n
(
n
)
d
.
If
E
(
;x;x
) is integrated over a compact manifold, then the corresponding
0
(
) becomes the density of states, and
(
) becomes the counting function
| the number of eigenvalues less than
.)
−
If
a
=1,then
f
is
itself.
If
a
=
g
(
t
)and
,then
f
is one of the kernels previously discussed.
Derivatives of negative order
of
f
are dened as iterated indenite inte-
grals, which can be represented as single integrals:
!1
Z
Z
−
1
@
−
f
(
)
d
1
d
f
(
)
0
0
Z
1
!
)
df
(
)
:
=
(
−
(14)
0
5
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