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International Journal of Modern Physics: Conference Series

c ? World Scientific Publishing Company

SCALAR CASIMIR ENERGIES OF TETRAHEDRA

E. K. ABALO, K. A. MILTON

Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma

Norman, OK 73019, USA

abalo@nhn.ou.edu, milton@nhn.ou.edu

L. KAPLAN

Department of Physics, Tulane University

New Orleans, LA 70118, USA

lkaplan@tulane.edu

Received Day Month Year

Revised Day Month Year

New results for scalar Casimir self-energies arising from interior modes are presented for

the three integrable tetrahedral cavities. Since the eigenmodes are all known, the energies

can be directly evaluated by mode summation, with a point-splitting regulator, which

amounts to evaluation of the cylinder kernel. The correct Weyl divergences, depending

on the volume, surface area, and the corners, are obtained, which is strong evidence that

the counting of modes is correct. Because there is no curvature, the finite part of the

quantum energy may be unambiguously extracted. Dirichlet and Neumann boundary

conditions are considered and systematic behavior of the energy in terms of geometric

invariants is explored.

Keywords: Casimir energy; Casimir self-energy

PACS numbers: 13.70.+k,11.10.Gh,42.50.Lc,42.50.Pq

1. Introduction

The concept of a self-energy in the context of the Casimir effect remains elusive.

Since 1948, the year of H. B. G. Casimir’s seminal paper,1what is now called the

Casimir effect has captivated many. Yet, while Casimir later predicted an attractive

force for a spherical conducting shell,2Boyer proved the self-stress in that case to

be instead repulsive, which was an even more unexpected result.3Since Boyer’s

formidable calculation, many other configurations were examined: cylinders, boxes,

wedges, etc. The literature abounds with these results; for a review see Ref. 4. How-

ever, since there are other well-known cases of cavities where the interior modes are

known exactly, it is surprising that essentially no attention had been paid to these.

For example, recently we presented the first results for Casimir self-energies for cylin-

ders of equilateral, hemiequilateral, and right isosceles triangular cross sections,5

even though the spectrum is well-known and appears in general textbooks.6,7Pos-

1

arXiv:1112.0079v1 [hep-th] 1 Dec 2011

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2Abalo, Milton, and Kaplan

sibly, the reason for this neglect was that only interior modes could be included for

any of these cases, unlike the case of a circular cylinder, where both interior and

exterior modes must be included in order to obtain a finite self-energy. However,

the extensive attention to rectangular cavities puts the lie to this hypothesis.8–11

It seems not to have been generally appreciated that finite results can be obtained

in all these cases because there are no curvature divergences for boxes constructed

from plane surfaces.

In Ref. 5 we obtained exact, closed-form results for the three-mentioned inte-

grable triangles, both in a plane, and for cylinders with the corresponding cross

section. The expected Weyl divergences related to the volume, surface area, and

the corners of the triangle were obtained, going a long way toward verifying the

counting of modes, which is the most difficult aspect of these calculations. More-

over, we were able to successfully interpolate between the results for these triangles

by using an efficient numerical evaluation, and showed that the results, for Dirich-

let, Neumann, and perfect conducting boundary conditions, lie on a smooth curve,

which was reasonably well-approximated by the result of a proximity force calcula-

tion. In this paper we show that the same techniques can be applied to tetrahedral

boxes; again, there are exactly three integrable cases, where an explicit spectrum

can be written down. Again, it is surprising that the Casimir energies for these cases

are not well-known. The only treatment of a pyramidal box found in the Casimir

energy literature appears in a relatively unknown work of Ahmedov and Duru,12

which however, seems to contain a counting error.

In this paper we present Casimir energy calculations for three integrable tetra-

hedra. For each cavity we consider a massless scalar field subject to Dirichlet and

Neumann boundary conditions on the surfaces. We regulate the mode summation by

temporal point-splitting, which amounts to evaluation of what is called the cylin-

der kernel,13and extract both divergent (as the regulator goes to zero) and the

finite contributions to the energy. In the end, we explore the functional behavior of

the Casimir energies with respect to an appropriately chosen ratio of the cavities’

volumes and surface areas, including cubic and spherical geometries with the same

boundary conditions. Further details will appear in Ref. 14.

1.1. Point-splitting regularization

We regularize our results by temporal point-splitting. As explained in Ref. 5, after

a Euclidean rotation, we obtain

?

E =1

2lim

τ→0

−d

dτ

??

k,m,n

e−τ√

λ2

kmn,(1)

where the sum is over the quantum numbers that characterize the eigenvalues, and

τ is the Euclidean time-splitting parameter, supposed to tend to zero at the end of

the calculation. One recognizes the sum as the integrated cylinder kernel.13Next,

we proceed to re-express the sum with Poisson’s summation formula.

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Scalar Casimir Energies of Tetrahedra3

1.2. Poisson resummation

Poisson’s summation formula allows one to recast a slowly convergent sum into a

more rapidly convergent sum of its Fourier transform,

??∞

By point-splitting and resumming, we are able to isolate the finite parts, which are

Casimir self-energies, and the corresponding divergent parts, which are the the Weyl

terms. The resummed expressions below are obtained by evaluating the Fourier

transforms in spherical coordinates.5,8They are used in regularizing expressions

throughout the paper.

?

p,q,r

∞

?

m=−∞

f(m) =

∞

?

n=−∞

−∞

e2πimnf(m)dm

?

.(2)

−

d

dτ

??

e−τ√

α(p+a)2+β(q+b)2+γ(r+c)2=

24π

√αβγ τ4−

1

2π3√αβγ

(3)

×

??

p,q,r

?

e−2πi(pa+qb+rc)

(p2/α + q2/β + r2/γ)2

?

,

?

−

d

dτ

??

p,q

e−τ√

α(p+a)2+β(q+b)2=

4π

√αβ τ3−

1

4π2√αβ

??

p,q

e−2πi(pa+qb)

(p2/α + q2/β)3/2,

(4)

?

−

d

dτ

??

p

e−τ√

α(p+a)2=

2

√ατ2−

√α

2π2

??

p

e−2πi(pa)

p2

.(5)

Here the prime means that all positive and negative integers are included in the

sum, but not the case when all are zero.

2. Casimir Energies of Tetrahedra

The three integrable tetrahedra mentioned above are not recent discoveries. They

have, in fact, been the subject of a few articles.15–17However, there appears to be

only one Casimir energy article concerning one of these tetrahedra, which we denote

as the “small” tetrahedron.12These tetrahedra are integrable in the sense that their

eigenvalue spectra are known explicitly, and there are no other such tetrahedra.

We will successively look at the “large,” “medium,” and “small” tetrahedra, as

defined below, and obtain interior scalar Casimir energies for Dirichlet and Neumann

boundary conditions. Although the exterior problems cannot be solved in these

cases, the finite part of the interior energies are well defined because the curvature

is zero, and hence the second heat kernel coefficient vanishes.

2.1. Large tetrahedron

The first tetrahedron, sketched in Fig. 1, which we denote “large,” is comparatively

the largest or rather the most symmetrical. One can obtain a medium tetrahedron

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4Abalo, Milton, and Kaplan

z

x

y

A

B

D

C

Fig. 1.Large tetrahedron: −x < z < x and x < y < 2a − x.

by bisecting a large tetradron and idem for the small and medium tetrahedra. One

should note that the terms “large,” “medium,” and “small” are merely labels, since

one can always rescale each tetrahedron independently of the others. The spectrum

and complete eigenfunction set for the large tetrahedron, as well as those of the

other tetrahedra, are known and appear in Ref. 15,

λ2

kmn=

π2

4a2

?3(k2+ m2+ n2) − 2(km + kn + mn)?.(6)

With Dirichlet and Neumann boundary conditions, different constraints are imposed

on the spectrum, that is, on the ranges of the integers k, m, and n.

2.1.1. Dirichlet BC

The complete set of modes for Dirichlet boundary conditions is given by the re-

strictions 0 < k < m < n. After extending the sums to all of (k,m,n)-space and

removing unphysical terms, the Dirichlet Casimir energy for the large tetrahedron

can be defined in terms of the function

g(p,q,r) = e−τ√

(π/a)2(p2+q2+r2),(7)

and written as

E =

1

48lim

τ→0

?

−d

dτ

??

p,q,r

?

g(p,q,r) + g(p + 1/2,q + 1/2,r + 1/2) − 6g(p,q,q) (8)

− 6g(p + 1/2,q + 1/2,q + 1/2) + 8g(√3p/2,0,0)

+ 3g(p,0,0),

?

where the sums extend over all positive and negative integers including zero. (In the

third and fourth terms only p and q are summed over, while in the last two terms

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Scalar Casimir Energies of Tetrahedra5

only p is summed.) Note that the time-splitting has automatically regularized the

sums, and it is easy to extract the finite part,

?

1

16πZ2b(3/2;2,1) −π

E(D)

L

=1

a

−

1

96π2

?Z3(2;1,1,1) + Z3b(2;1,1,1)?+

96−π√3

1

8πζ(3/2)L−8(3/2) (9)

+

72

?

,

where18

??

m,n

(m2+ 2n2)−s= 2ζ(s)L−8(s).(10)

The energy then evaluates numerically to

E(D)

L

= −0.0468804266

a

.(11)

The function L−8 is a Dirichlet L-series, which are defined as Lk(s)

?∞

??

=

n=1χk(n)n−swhere χk is the number-theoretic character.18The Epstein zeta

functions Z3, Z3b, and Z2bare defined respectively as

Z3(s;a,b,c) =

k,m,n

(ak2+ bm2+ cn2)−s,(12)

Z3b(s;a,b,c) =

??

k,m,n

(−1)k+m+n(ak2+ bm2+ cn2)−s,(13)

and

Z2b(s;a,b) =

??

m,n

(−1)m+n(am2+ bn2)−s.(14)

The divergent parts, also extracted from the regularization procedure, follow

the expected form of Weyl’s law with the quartic divergence associated with the

volume V , the cubic divergence associated with the surface area S, and the quadratic

divergence matched with the corner coefficient

E(D)

div=

3V

2π2τ4−

S

8πτ3+

C

48πτ2. (15)

Here and subsequently, the corner coefficient C for a polyhedron is defined as19

?π

C =

?

j

αj

−αj

π

?

Lj,(16)

where the αj are dihedral angles and the Lj are the corresponding edge lengths.

The above expression for the divergences will be the same for all subsequent cavities

with Dirichlet boundary conditions.

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6Abalo, Milton, and Kaplan

2.1.2. Neumann BC

In the case of Neumann boundary conditions, the complete set of mode numbers

must satisfy 0 ≤ k ≤ m ≤ n, excluding the case when all mode numbers are zero.

The Neumann Casimir energy can be defined in terms of the preceding Dirichlet

result as

?

which gives us a numerical value of

E(N)

L

= E(D)

L

−

1

8πa

2ζ(3/2)L−8(3/2) + Z2b(3/2;2,1)

?

,(17)

E(N)

L

= −0.1964621484

a

.(18)

The divergent parts also match the expected Weyl terms for Neumann boundary

conditions. We note that the cubic divergence’s coefficient changes sign when com-

paring Dirichlet and Neumann divergent parts:

E(N)

div=

3V

2π2τ4+

S

8πτ3+

C

48πτ2.(19)

This form is also obtained for all the following calculations involving Neumann

boundary conditions.

2.2. Medium tetrahedron

z

x

y

A

B

D

C

Fig. 2. Medium tetrahedron: 0 < z < x and x < y < 2a − x.

The eigenvalue spectrum of the medium tetrahedron, shown in Fig. 2, is of the

same form as that of the large tetrahedron [Eq. (6)] with different constraints.

2.2.1. Dirichlet BC

The complete set of mode numbers for the Dirichlet case satisfies the constraints

0 < m < n < k < m + n. Following the same regularization procedure used in the

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Scalar Casimir Energies of Tetrahedra7

preceding cases, we obtain the Dirichlet Casimir energy in terms of the Dirichlet

result for the large tetrahedron,

?

where we used18

??

The Casimir energy evaluates to

E(D)

M

=1

2E(D)

L

+

1

96πa

3ζ(3/2)β(3/2) − (1 +

√2)π2?

,(20)

m,n

(m2+ n2)−s= 4ζ(s)β(s). (21)

E(D)

M

= −0.0799803933

a

.(22)

Here the function β is the Dirichlet beta function, also known as L−4, defined as

β(s) =?∞

2.2.2. Neumann BC

n=0(−1)n(2n + 1)−s.

With Neumann boundary conditions, the complete set of mode numbers is restricted

to 0 ≤ m ≤ n ≤ k ≤ m + n, excluding the all-null case. As with the Dirichlet case,

the Neumann Casimir energy for the medium tetrahedron can be expressed in terms

of the Neumann result for the large tetrahedron:

?

= −0.1997008024

a

E(N)

M

=1

2E(N)

L

−

1

96πa

3ζ(3/2)β(3/2) + (1 +

√2)π2?

.(23)

2.3. Small tetrahedron

z

x

y

A

B

D

C

Fig. 3. Small tetrahedron: 0 < z < x and 0 < y < a.

The small tetrahedron (Fig. 3) may be visualized as the result of a bisection of a

medium tetrahedron. The form of the eigenvalue spectrum for the small tetrahedron

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8 Abalo, Milton, and Kaplan

is different from that for the previous two tetrahedra but the same as the cube’s:

λ2

kmn=π2

a2

?k2+ m2+ n2?.(24)

The Dirichlet case is the aforementioned “pyramidal cavity” considered in Ref. 12.

2.3.1. Dirichlet BC

The modal restriction for the complete set is 0 < k < m < n. The finite part

obtained is thus

?

−π

7296

E(D)

S

=1

a

−

1

192π2Z3(2;1,1,1) +

64−π√3

1

16πζ(3/2)L−8(3/2) +

1

32πζ(3/2)β(3/2)

−π√2

?

,(25)

which evaluates to

E(D)

S

= −0.1005414622

a

.(26)

This result differs from that of Ref. 12. The discrepancy appears to stem from a

mode-counting error in Ref. 12, and the result found there is likely wrong.

2.3.2. Neumann BC

For the Neumann case, we again find the same condition that the mode numbers

must satisfy 0 ≤ k ≤ m ≤ n excluding the origin. The Neumann Casimir energy is

derived to be

?

with a numerical value of

E(N)

S

= E(D)

S

−ζ(3/2)

16πa

2L−8(3/2) + β(3/2)

?

,(27)

E(N)

S

= −0.2587920021

a

.(28)

3. Analysis

Seeking to gain a better understanding of self-energies, at least their correlation

to the system’s geometry, we add the well-known cases of cubic and spherical

geometries.8,20,21The scaled Casimir energies, ESc= E×V/S, are tabulated in Ta-

ble 1, and are plotted against the corresponding isoareal quotients, Q = 36πV2/S3

in Fig. 4. A noteworthy difference between the calculations of the energies for tetra-

hedra as compared to a sphere is that in the polyhedral cases only the interior

modes are considered (the exterior modes are unknown) whereas in the spherical

cases both interior and exterior are (necessarily) included to cancel the curvature

divergences.

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Scalar Casimir Energies of Tetrahedra9

Table 1. Scaled energies and isoareal quotients.

Q

E(D)

Sc

E(N)

Sc

Small T.

Medium T.

Large T.

Cube

Spherical Shell

0.22327

0.22395

0.27768

0.52359

1

−0.00694

−0.00696

−0.00552

−0.00261

0.00093

−0.01787

−0.01739

−0.02315

−0.04755

−0.07459

0.20.40.60.81.0

Q

?0.07

?0.06

?0.05

?0.04

?0.03

?0.02

?0.01

E?V?S

Fig. 4.

connects Dirichlet (Neumann) results. From left to right, the circular markers indicate small,

medium (which cannot be resolved on this graph), large tetrahedra, cube, and sphere data points.

It is interesting to note the relative order of the scaled energies of the small and medium tetrahedra:

E(D)

MS

M

S

.

Scaled Dirichlet and Neumann energies vs. isoareal quotients. The upper (lower) curve

< E(D)

and E(N)

> E(N)

4. Conclusions

In this paper, we have extended the work of Ref. 5 from infinite cylinders to in-

tegrable tetrahedra. The previous work was essentially two dimensional, so it was

possible to give closed form results for the Casimir self energy. This is apparently

not possible, at least not currently, for the cases considered in this paper.

The emerging systematics are intriguing, but not yet conclusive, since the cases

we can evaluate are limited. Numerical work will have to be done to explore the

geometrical dependence of the Casimir energy of cavities composed of flat surfaces

of arbitrary shape.

Work on other boundary conditions, in particular electromagnetic boundary

conditions, is currently under way. Unlike for cylinders, the electromagnetic energy

of a tetrahedron is not merely the sum of Dirichlet and Neumann parts; there is

no break-up into TE and TM modes in general. So this is a formidable task. Finite

triangular prisms and polytopes are also currently the subject of ongoing work.

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10Abalo, Milton, and Kaplan

Acknowledgments

We thank the US National Science Foundation and the US Department of Energy

for partial support of this work. We further thank Nima Pourtolami and Prachi

Parashar for collaborative assistance.

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