Table Of ContentWHAT 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: [email protected]
1 Reminder of the Phenomena
Toestablishthesettingofthispresentation,considerasanexampletheclassic
problem of a scalar (cid:12)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 (cid:12)eld equation, H =−r2, several quantities are associated:
1.1 Vacuum energy
In the 1970s considerable attention was directed to de(cid:12)ning 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
(cid:12)nitesizeofthebox,tothediscretenessofthespectrumofnormalmodes,and
to the existence of a closed geodesic (or classical path) of length 2L. This is
the scalar analog of the classic Casimir e(cid:11)ect. It persists in a closed universe
without boundary. Second, there is a divergentdistribution of energyclinging
to the walls: T (x) / x−4, when x is the distance to the nearest wall. It
00
persists in an in(cid:12)nite space with only one wall. We may say that this e(cid:11)ect is
caused by the existence of the boundary, the spatial inhomogeneity near the
boundary of the set of mode functions, and the length (2x) 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
observingvacuumenergylocally, wecantellhowbig the worldis (L)andhow
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 operatorH, thRe heat kernelK(t;x;y)(cid:17)e−tH(x;y)
is the Green function such that u(x) = K(t;x;y)f(y)dy solves the initial-
value problem −@u = Hu, u(0;x) = f(x). It is well known that K has an
@t
asymptotic expansion
" #
X1
K(t;x;x)(cid:24)(4(cid:25)t)−m=2 1+ a (x)tn ; (1)
n
n=1
(m=spatialdimension)wherea (x)isalocal functionalofthecurvature,etc.
n
(covariantfunctionalsofthecoe(cid:14)cientfunctionsintheseond-orderdi(cid:11)erential
operator H) at x. In particular, a is identically zero inside our box (where
n
H = −r2). By studying the heat kernel expansion, you will never discover
the Casimir e(cid:11)ect!
1.3 Schr¨odinger and Schwinger{DeWitt kernels
The function U(t;x;y) (cid:17) 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 propagatorthat solves the time-dependent Schr¨odinger
equation. When H is a hyperbolic operator and t a (cid:12)ctitious proper time,
U is the Schwinger{DeWitt kernel used in renormalization of quantum (cid:12)eld
theories. TherotationofthetcoordinateinEq.(1)isalgebraicallytrivial,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:
h i
U(t;x;y)(cid:24)(4(cid:25)it)−1=2 eijx−yj2=4t−eijx+yj2=4t : (2)
Passing to the diagonal, we have
h i
U(t;x;x)(cid:24)(4(cid:25)it)−1=2 1−eix2=t ; (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
consequentlytheseriesisindeedvalidinvariousdistributionalsenses,asIshall
partially explain below.
1.4 The cylinder kernel
p
The Green function T(t;x;y) (cid:17) e−t H(x;y) solves the (elliptic) boundary-
value problem @2u = Hu, u(0;x) = f(x), u(t;x) ! 0 as t ! +1 (i.e.,
@t2
Rin a semi-in(cid:12)nite cylinder with our spatial manifold as its base) by u(x) =
T(t;x;y)f(y)dy. The cylinder kernel T shares many of the properties of
vacuum energy (hT i) and the latter’s progenitor, the Wightman two-point
00
function (W(t;x;y) (cid:17) h(cid:30)(t;x)(cid:30)(0;y)i), a Green function for the wave equa-
tion @2u = −Hu; however, T is technically simpler in several ways. Its study
@t2
thereforedeservesourattention,eventhoughithasnodirectphysicalinterpre-
tationwithtasatimecoordinate. Foranexampleweonceagainconsiderone
space dimension, whereH =− @2 (so we’re dealing with very classicalGreen
@x2
functions for thetwo-dimensionalLaplaceequation). Ifthe spatialmanifoldis
the entire real line, the kernel is
t 1
T (t;x;y)= ; (4)
0 (cid:25) (x−y)2+t2
whereas if the space is R , the problem can again be solved by images:
+
(cid:20) (cid:21)
t 1 1
T (t;x;y)= − : (5)
+ (cid:25) (x−y)2+t2 (x+y)2+t2
Passing to the diagonal and making a Taylor expansion, we get
(cid:20) (cid:21)
1 t2
T (t;x;x)= 1− +(cid:1)(cid:1)(cid:1) : (6)
+ (cid:25)t (2x)2
Thus we see that the asymptotic expansion of the cylinder kernel does probe
x (and also L, in a (cid:12)nite universe), as the vacuum energy does. On the other
hand, hT i and W share some of the delicate analytical complications of U,
00
whereas T is about as well-behaved as K.
1.5 Summary and synopsis
ThefourGreenfunctions T,U,K,W demonstratetwodistinctions(Table1):
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
Table1: AsymptoticpropertiesofGreenfunctions.
Pointwise Distributional
Local K U
Global T W
The main points I wish to make are:
(cid:15) Asalreadyremarked,thesmall-texpansionofT containsnonlocalinfor-
mation not present in the corresponding expansion of K.
(cid:15) Nevertheless, both these expansions are determined by the high-energy
asymptoticbehaviorofthedensityofstates,or,moregenerally,thespec-
tral measures, of the operator H.
(cid:15) Thedetailedrelationshipsamongalltheseasymptoticdevelopmentsare,
at root, not a matter of quantum (cid:12)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 in(cid:12)nite 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 , the heat, cylinder, and Wightman
@x2
kernels are Z
1
1
K (t;x;y)= dke−tk2eikxe−iky; (7)
0
2(cid:25)
−1
Z
1
1
T (t;x;y)= dke−tjkjeikxe−iky; (8)
0
2(cid:25)
−1
Z
1 1 e−itjkj
W (t;x;y)= dk eikxe−iky: (9)
0 4(cid:25) jkj
−1
On the other hand, we can vary the space, or the operator; for instance, the
heat kernel for R is
+
Z
1
2
K (t;x;y)= d!e−t!2 sin!xsin!y; (10)
+
(cid:25)
0
4
while that for a box of length L is
X1
2 n(cid:25)x n(cid:25)y
K (t;x;y) = e−t(n(cid:25)=L)2 sin sin
L
L L L
n=1
Z 1 X1 (cid:16) (cid:17)
2 n(cid:25)
= d!e−t!2 (cid:14) !− sin!xsin!y: (11)
L L
0 n=1
In general, each object can be written in the schematic form
Z
1
g(t!)dE(!;x;y); (12)
0
where dE is integrationwith respect to the spectral measureof a givenH (on
a given manifold) and g is the kernel of a certain integral transform (Laplace,
Fourier,etc.) de(cid:12)ning the Green function in question. (To (cid:12)t someof the ker-
nels into the mpold (12), it is necessaryto rede(cid:12)ne some variables, for example
replacing t by 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{Ces(cid:18)aro Means
Eq. (12) is an instance of the general structure
Z
(cid:21)
f((cid:21))(cid:17) a((cid:27))d(cid:22)((cid:27)); (13)
0
where (cid:22) is some measure, such as E((cid:27);x;y). (For a totally continuous spec-
trum, d(cid:22)((cid:27)) equals (cid:22)0((cid:27))d(cid:27) where (cid:22)0 is a function, which wPe call the spectral
density. Foratotallydiscretespectrum,d(cid:22)((cid:27))isoftheform c (cid:14)((cid:27)−(cid:27) )d(cid:27).
n n n
If E((cid:27);x;x) is integrated over a compact manifold, then the corresponding
(cid:22)0((cid:27)) becomes the density of states, and (cid:22)((cid:21)) becomes the counting function
| the number of eigenvalues less than (cid:21).) If a = 1, then f is (cid:22) itself. If
a=g(t(cid:27)) and (cid:21)!1, then f is one of the kernels previously discussed.
Derivatives of negative order of f are de(cid:12)ned as iterated inde(cid:12)nite inte-
grals, which can be represented as single integrals:
Z Z
@−(cid:11)f((cid:21)) (cid:17) (cid:21)d(cid:27) (cid:1)(cid:1)(cid:1) (cid:27)(cid:11)−1d(cid:27) f((cid:27) )
(cid:21) 1 (cid:11) (cid:11)
0 Z 0
1 (cid:21)
= ((cid:21)−(cid:27))(cid:11)df((cid:27)): (14)
(cid:11)!
0
5
The Riesz{Ces(cid:18)aro means of f are de(cid:12)ned from these by a change of normal-
ization: Z (cid:16) (cid:17)
(cid:21) (cid:27) (cid:11)
R(cid:11)f((cid:21))(cid:17)(cid:11)!(cid:21)−(cid:11)@−(cid:11)f((cid:21))= 1− df((cid:27)): (15)
(cid:21) (cid:21) (cid:21)
0
If the limit of f((cid:21)) as (cid:21) approaches 1 exists, then R(cid:11)f((cid:21)) also approaches
(cid:21)
that limit (though perhaps more slowly). On the other hand, R(cid:11)f((cid:21)) may
(cid:21)
convergeatin(cid:12)nitywhenf((cid:21))doesnot. Inthatcase,R(cid:11)f(1)servestode(cid:12)ne
(cid:21)
f(1). This extended notion of summability of an in(cid:12)nite integral generalizes
the summation of classically divergent Fourier series by Cesa(cid:18)ro means (the
averagesof the (cid:12)rst N partial sums).
4 Riesz Means with Respect to Di(cid:11)erent Variables
Tothispointournotationhasbeenratherloose,soastodiscussallthevarious
kernels in a uni(cid:12)ed, schematic framework without propounding cumbersome
de(cid:12)nitions. Now it is necessary to tighten up. Henceforth (cid:21) will denote the
eigenvalueparameterofoursecond-orderellipticdi(cid:11)erentialoperatorH,and!
will denote its square root, the frequency parameter. (On the real line, where
the prototype eigenfunction is (2(cid:25))−1=2eikx, we have (cid:21) = k2 and ! = jkj.)
Thus the basic relations are
(cid:21)=!2; d(cid:21)=2!d!: (16)
This seeming triviality of calculus has surprising impact. For (cid:11) > 0 the
Riesz means R(cid:11)(cid:22) andR(cid:11)(cid:22)arenot the same things. In fact, one can calculate
(cid:21) !
R(cid:11)(cid:22) in terms of the R(cid:12)(cid:22) with (cid:12) (cid:20) (cid:11), and vice versa.9;11 The striking result
(cid:21) !
is that the (cid:21) ! 1 behavior of the R ((cid:22)) is completely determined by that of
(cid:21)
the R ((cid:22)), but the converse is false; the asymptotics of the R ((cid:22)) depend on
! !
integrals of the R ((cid:22)) over all (cid:21), not just on their asymptotic values. That
(cid:21)
is, in the passage from (cid:21) to !, new terms in the asymptotic development of
R(cid:11)(cid:22) arise as undeterminable constants of integration; in going from ! to (cid:21),
!
thereare\magicalcancellations"thatcausethesetermstodisappearfromthe
asymptotics of R(cid:11)(cid:22).
(cid:21)
In more detail: The rigorous asymptotic approximation
X(cid:11)
R(cid:11)(cid:22)((cid:21))= a (cid:21)(m−s)=2+O((cid:21)(m−(cid:11)−1)=2) (17)
(cid:21) (cid:11)s
s=0
can be shown, as can the corresponding asymptotic approximation
X(cid:11) X(cid:11)
R(cid:11)(cid:22)(!)= c !m−s+ d !m−sln!+O(!m−(cid:11)−1ln!); (18)
! (cid:11)s (cid:11)s
s=0 s=m+1
s−m odd
6
where
(cid:15) c =constant(cid:2)a if s(cid:20)m or s−m is even;
(cid:11)s (cid:11)s
(cid:15) c is undetermined by a if s>m and s−m is odd;
(cid:11)s (cid:11)s
(cid:15) d =constant(cid:2)a if s>m and s−m is odd.
(cid:11)s (cid:11)s
The constants are combinatorial structures (involving mostly gamma func-
tions) whose detailed form is not important now.
5 Some Conclusions
We can now relate these facts to the spectral theory and physics discussed
earlier.
1. From the foregoing it is clear that the ! expansion coe(cid:14)cients of (cid:22)
(Eq. (18)) contain more information than the (cid:21) expansion coe(cid:14)cients
(Eq. (17)). In the application to quantum (cid:12)eld theory it turns out that
the new data are \global" and the old ones are \local". Indeed, the (cid:21)
coe(cid:14)cients are in one-to-one correspondence with the terms in the heat
kernelexpansion, while the whole list of ! coe(cid:14)cients correspondto the
terms in the expansion of the cylinder kernel (see below).
2. TheRieszmeansof(cid:22)((cid:21))(cid:17)E((cid:21);x;y)giverigorousmeaningtotheformal
high-frequency expansions obtained by formally inverting the asymp-
totics of K and T (i.e., applying the inverse of the appropriate inte-
gral transform (12) to the asymptotic expansion term-by-term). The
existence of discrete spectra (which makes the exact (cid:22) a step function)
shows that those high-frequency expansions cannot be literally asymp-
totic. That they neverthelesscanbe givena precisemeaning in terms of
someaveragingprocedurewas pointed out by Brownell1 in the ’50s; the
connection with Riesz means was drawn by Ho¨rmander.11
3. Similarly, the concepts of Riesz{Cesa(cid:18)ro summation tighten up the con-
vergence of the spectral expansions of U and W, and also the t ! 0
asymptotics of those distributions.
7
6 Kernel Expansions from Riesz Means
When f and (cid:22) are related by Eq. (13), a Riesz mean of f can be expressed in
terms of the corresponding Riesz mean of (cid:22):
R(cid:11)f((cid:21)) =a((cid:21))R(cid:11)(cid:22)((cid:21))
(cid:21) (cid:21)
(cid:18) (cid:19)Z
(cid:11)X+1 (−1)j (cid:11)+1 (cid:21)
+(cid:21)−(cid:11) d(cid:27)((cid:21)−(cid:27))j−1(cid:27)(cid:11)@ja((cid:27))R(cid:11)(cid:22)((cid:27)): (19)
(j−1)! j (cid:27) (cid:27)
j=1 0
InthecaseofEq.(12),Eq.(19)relatestheasymptoticsofthespectrumtothe
asymptotics of the kernels. Eq. (17) corresponds to
Z
1 X1
K(t)(cid:17) e−(cid:21)td(cid:22)((cid:21))(cid:24) b t(−m+s)=2; (20)
s
0 s=0
where
Γ((m+s)=2+1)
b = a : (21)
s ss
Γ(s+1)
Eq. (18) corresponds to
Z
1 X1 X1
T(t)(cid:17) e−!td(cid:22)(cid:24) e t−m+s+ f t−m+slnt; (22)
s s
0 s=0 s=m+1
s−m odd
wheree andf arerelatedtoc andd inmuchthesamewayasb isrelated
s s ss ss s
to a .7 (One canconcentrateonthe spectralcoe(cid:14)cients with (cid:11)=s,because
ss
those with (cid:11)6=s contain no additional information.)
Onehastowonderwhether thereissomethingmoreprofoundandgeneral
in the Riesz{Hardy theory of spectral asymptotics with respect to di(cid:11)erent
variables, waiting to be discovered and applied. We have concentrated on the
counterpoint of (cid:21)1=2 versus (cid:21), where the extra data associated with the for-
mer is geometrically global in a spectral problem, that associated with the
latter strictly local. This geometrical signi(cid:12)cance, of course, we could not
have learned from Hardy and Riesz; it has to be observed in the application.
The entire development (with di(cid:11)erent exponents in the series) could be re-
peatedfor(cid:21)versus(cid:21)2 andappliedtothespectralasymptoticsofH versusH2
(a fourth-order di(cid:11)erential operator). Examinationof the formulas of Gilkey8
for the heat kernel expansion of a fourth-order operator verify that, indeed,
there are terms in the asymptotics of H that disappear from the asymptotics
ofH2(becausethecombinatorialcoe(cid:14)cientsmultiplyingthemvanish). Inthis
case, however, both series are totally local, so there does not appear to be a
8
qualitativedi(cid:11)erence between the two sets of spectral invariants. Is the e(cid:11)ect,
therefore, a mere curiosity in that case, or does it have a deeper signi(cid:12)cance?
In the other direction, one can expect that the Riesz means with respect to
(cid:21)1=3 (and hence, the Green function for the operator e−tH1=3) contain new
information di(cid:11)erent from that contained in T; and that (cid:21)1=6 subsumes them
both; and so on. Should we care about this new spectral data? Finally, as
a wild speculation, might the investigation of these Riesz means with respect
to arbitrarilyextreme fractionalpowersoftheeigenparameterprovideaneasy
way to get access to at least a part of the detailed spectral information that
is encoded in the lengths of the closed geodesics (or periodic classical orbits)
of H?
7 Distributional/Ces(cid:18)aro Theory of Spectral Expansions
Finally, I shall summarize some more technical mathematics surrounding the
Green function integrals (12):4
(cid:15) The locality of the asymptotics of G hinges on the regularity of g at 0.
Forexample,thedi(cid:11)erentbehaviorofK andT arisesfromthedistinction
p
betweene−x (whichhasaTaylorseriesat0)ande− x (whichdoesnot).
(cid:15) The pointwise versus distributional/Ces(cid:19)aro validity of the asymptotics
ofGhingesonthebehaviorofg atin(cid:12)nity. Thus,thedi(cid:11)erencebetween
K and U arises from the contrast between e−x and e−ix.
(cid:15) Thelarge-(cid:21)asymptoticsofEisvalid(andlocal)inadistributional/Cesa(cid:18)-
ro sense (but usually not pointwise).
It remains to explain the term \distributional/Ces(cid:18)aro". The point is that
Riesz{Ces(cid:18)aro limits (de(cid:12)ned after Eq. (15)) are equivalent to distributional
limits in t, or in (cid:21) or !, under scaling.3 If f belongs to the distribution space
D0(R ), thenfor(cid:11)in theinterval(−k−2;−k−1),f(t)=O(t(cid:11))intheCes(cid:18)aro
+
sense if and only if there exist \moments" (cid:22) ;:::;(cid:22) such that
0 k
Xk (−1)j(cid:22) (cid:14)(j)(t)
f((cid:27)t)= j +O((cid:27)(cid:11)) (23)
j!(cid:27)j+1
j=0
in the topology of D0, as (cid:27) !+1 (with a similar but more complicated state-
ment when (cid:11) is an integer). This theorem has applications to both G(t) and
E((cid:21)) in the role of f.4;5
9
Acknowledgments
The distributional/Cesa(cid:18)ro theory of spectral expansions reported here is pri-
marilytheworkofmycollaboratorRicardoEstrada4;6;3;5. RobertGustafson7
combinatorially veri(cid:12)ed the \magical cancellations". Michael Taylor recom-
mendedthe studyofthecylinder kernelto me, andStuartDowkeracquainted
mewiththeworkofBrownellmanyyearsago. Inoralpresentationsofthisma-
terial I show a graph demonstrating Cesa(cid:18)ro summation of Fourier series that
was prepared by an undergraduate student, Philip Scales. Finally, I thank
Michael Bordag for the hospitality extended to me at the Workshop and the
opportunity to disseminate this material in a conference presentation.
References
1. F. H. Brownell, Extended asymptotic eigenvalue distributions for
bounded domains in n-space, J. Math. Mech. 6, 119 (1957).
2. B. S. DeWitt, Quantum (cid:12)eld theory in curvedspacetime, Phys. Reports
19, 295 (1975).
3. R. Estrada, The Cesa(cid:18)ro behavior of distributions, Proc. Roy. Soc. Lon-
don A, to appear.
4. R. Estrada and S. A. Fulling, Distributional asymptotic expansions of
spectralfunctionsandoftheassociatedGreenkernels,submittedtoElec-
tronic Journal of Di(cid:11)erential Equations (funct-an/9710003).
5. R. Estrada, J. M. Gracia-Bond(cid:19)(cid:16)a, and J. C. Va(cid:19)rilly, On summability of
distributions and spectral geometry, Commun. Math. Phys. 191, 219
(1998)
6. R. Estrada and R. P. Kanwal, Asymptotic Analysis: A Distributional
Approach (Birkha¨user, Boston, 1994).
7. S. A. Fulling, with appendix by R. A. Gustafson, Some properties of
Rieszmeansandspectralexpansions,submittedtoElectronic Journal of
Di(cid:11)erential Equations (physics/9710006).
8. P. B. Gilkey, The spectral geometryof the higher order Laplacian,Duke
Math. J. 47, 511 (1980).
9. G. H. Hardy, The second theorem of consistency for summable series,
Proc. London Math. Soc. 15, 72 (1916).
10. G. H. Hardy and M. Riesz, The General Theory of Dirichlet’s Series
(Cambridge U. Press, Cambridge, 1915).
11. L. Ho¨rmander, The spectral function of an elliptic operator, Acta Math.
121, 193 (1968).
10
