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Casimir effect in n dimensions
Volume 149B, number 4,5
PHYSICS LETTERS
20 December 1984
CASIMIR EFFECT IN n DIMENSIONS
H. VERSCHELDE x, L WILLE and P. PHARISEAU
Seminarie voor Theoretische Vaste Stof- en Lage Energie Kernfysica, R#ksuniversiteit Gent, Krf/gslaan281, Gebouw $9, 9000 Gent, Belgium Received 15 June 1984 Revised manuscript received 9 October 1984
We derive exact expressions for the Casimir interaction energy per unit area between two parallel, perfectly conducting, metal plates, in n (~ 2) dimensions.
In recent years, bag model calculations [1] have revived interest in the Casimir effect [2]. Progress, has been made in developing approximation techniques [3] as well as in calculating new geometries [4]. Nevertheless, the number of exact results is small and intuition on how geometry affects the qualitative aspects of the Casimir effect is completely lacking. An unanswered question in this respect is whether the attractive or repulsive character of the effect will be changed by going to higher dimensions [5]. In this letter, we will address this question for parallel plates and derive exact expressions for the interaction energy in N(~> 2) dimensions. A regularized sum over frequencies formula for the interaction energy per unit area (in two dimensions, this formula gives the interaction energy per unit length, in n dimensions it gives the interaction energy per unit of (n - 1)-dimensional area) between two parallel, perfectly conducting metal plates, located at x = -L/2, x = +L/2 is given by (we use units h = c = 1)
U(n'L'a)=-~-"
m=l
f
X exp { - a [(mlt/L) 2 + k 2] 1/2).
[(mTr/L)2+k211/2 (1)
We have omitted the m = 0 term in (1) because it is independent of L and therefore does not contribute to 1 Aspirant NFWO (Belgium). 396
the Casimir energy. If we let a go to zero, we recover the familiar sum-over-frequencies formula. The divergences will appear as singularities with respect to a, as a ~ 0. The finite, physically relevant, Casimir interaction energy per unit area will be given by the a independent part of U(n, L, a). The appearance of square roots in (1) makes the evaluation of (1)very difficult. However, they can be eliminated by using the following Laplace transform: f
e x p ( - p t ) exp(---a2/4t)
tl12
dt = (vc~'/VCp)e x p ( - a x/rp).
0
(2)
The interaction energy then becomes
U(n, L, a) = (02/~a2)f(n, L, a), with
F(n,L,a) = -n--g - -1, , N1 ×f
(3)
-/
~1 exp(-a2/4t) tl/2 = o dt
d/d~-I e x p { - t [ k 2+(m~lL)21}.
(2~)n-1
(4)
The integration over (n - 1)-dimensional k space is now trivial and yields
F(n, L, a) = ~(n - 1) 2 (1 -n)n-n/2 × ~ ? d t exp(-a2/4t)exp[-t(mn/L)2] m =1 ~) tn[ 2
(5)
0370-2693•84•$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 149B, number 4,5
PHYSICS LETTERS
If we use the following integral representation for the modified cylindrical Bessel function [6]
2(~)V/2Kv(2(z~)l/2) = f e -zt e -~/t t - v - 1 dt 0
20 December 1984
m • l K o ( m X )Ir/2x - 3'/2 + {ln(x/47r) +
=
(6) +rt ~ (Ix2 +(27r~)2] -1/2 - 1/2rr£), ~=1
(15)
(5) becomes where 3' = 0, 577215 .... Because of [6]
F(n, L, a) = {(n - 1) 2(2-n)/2rr-n/2a -(n-2) × f({(,
- 2), L, a),
(7)
where
[x 2 + (21rj~)2] -1/2 = S e - 2 7 r ~ t 0
Jo(xt) dt
(16)
and the expansion [6]
f(v, L, a) = ~
m=l
(m,a/L )VKv(mlra/L ).
(8)
Jo(xt) - 1 = ~ ( - 1 ) s (xt/2)2S/(s!)2,
(17)
s=l It is clear from (3) and (7) that the Casimir interaction energy per unit area Uc(n, L) which is the a independent part of U(n, L, a) will be given by
the last term in (15) becomes
Uc(n, L) = {(n - 1) 2 (4-n)/2~r-n/2b~n_2)/2 ,
rr S "[[x 2 + (2rr~) 2] -1/2 _ 1/2rr~) ~=1
(9)
where byu is the coefficient ofa u in a small a expansion for f(v, L, a). We now establish a recurrence relation for the coefficients b u. The modified cylindrical Bessel functions Kv(z ) satisfy the following recurrence relations [6] K _l(Z) -
=~ ~ S=1~=1 s=l
Kv+l(Z ) = -(2v/z)Kv(z ),
K v _l(Z) + Kv+l(Z ) = - 2 K v ( z ).
(10)
Appropriate combination of these relations and multiplication by z v+l yields
f(v + 1,L,a) = -aOf(v,L,a)]aa + 2vf(v,L,a),
(11)
(-l)Srr(x/2)2s f exp(-2rt£t)t 2s dt
=
(S!)2
24s+l(s!)2
0 (2S)! =
24s+l(s!)2
(18)
where ~'(z) is the Riemann zeta function. Therefore, (15) becomes oo
so that the recurrence relation for the b u's v becomes b"/) = [ - u +
- 1)] b"v - - l *
These recurrence relations can be solved for and give
b~n-2)~2 = (-l)(n-2)]22(n-2)]2(1n)! b~
(12) n b(n -2)/2
m=l
Ko(mX) = {(3' - In art) + -~ln x + 7r/2x
+ ~ (-1)S(2s)!f(2s + 1) (x/Ir)2S" s=l
(13)
for n even and b(n - 2 ) / 2 = (-1)(n-3)/22(n-3)/2 [-~(n - 1)1 ! b~/2(14 ) for n odd. What remains to be calculated is bg and
b7/2. To calculate b~, we use the following identity [7]
(19)
24s+l(s!) 2
Hence, because of (8), (15) and the definition of b u, we have that
n (-1)n/2n!~( n + 1) 1 b0 = 2 zn+l [(n/Z)!] 2 L--n-" To calculate
(20)
b~/2, we use the formula [6]
Kll 2(x) = X / ~ c-x/vrx
(21 ) 397
Volume 149B, number 4,5
PHYSICS LETTERS
to obtain
20 December 1984
which is the well known result [2], for the Casimir effect between two parallel, perfectly conducting metal plates in three dimensions. Because [6]
f(~, L, a) = ~ (mlra/L)ll 2K1 / 2(re,raiL ) m=l
Bn+ 1 = 2(-1)(n+3)/2(n + 1)! ~
=~
[expOra/L) 1] -1
(23)
(24)
One of us, (H.V.), would like to thank K. Debel for introducing him to the Casimir effect.
we obtain for bnl/2
bnl/2 = X / ~ [Bn+l/(n + 1)!] rrn/L n.
By combining (9), (13), (14) and (20), (24) we are able to obtain exact expressions for the Casimir interaction energy per unit area in n('~ 2) dimensions:
Uc(n,L ) = ±¼(n - 1) n!f(n + 1) 1 2'2nrtn/2(n/2)! L---ff
(25)
for n even and
Uc(n,L)=~(n - 1) (-1)(n+l)/2rr(n+l)12[{(n - 1)1 ! Bn+ 1 1 X
(n + 1)!
Ln
(26)
for n odd. For n = 2, (25) yields Uc(2, L ) = - ~(3)/16rrL 2,
(27)
which agrees with [8] up to a factor 1/2 (only one polarization). For n -- 3, (26) yields Uc(3, L) = -rr2/720L 3 ,
398
(29)
it follows from (26) that Uc(n, L) is negative for n odd. Therefore, the Casimir interaction energy is always negative. This means that the Casimir force between parallel plates is attractive, irrespective of the number of dimensions [5,9].
If we make use of the definition for the Bernoulli numbers [6]
0 ~Bn xn e x X- 1 = ~ =
(1/21rm) n÷l
m=l
(22)
-
(28)
References [1] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9 (1974) 3471. [2] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetenschap. 51 (1948) 793. [3 ] R. Balian and B. Duplantier, Ann. Phys. 104 (1977) 300; 112 (1978) 165. [4] J.H. Boyer, Phys. Rev. 174 (1968) 1764; K.A. Milton, L.L. DeRaad Jr. and J. Schwinger, Ann. Phys. 115 (1978) 388; D. Deutsch and P. Candelas, Phys, Rev. D20 (1979) 3063. [ 5 ] K. Debel, private communication. [6 ] W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and theorems for the special functions of mathematical physics (Springer, Berlin, 1966). [7] W. Magnus and F. Oberhettinger, Formein und Siitze ftir die speziellen Funktionen der mathematischen Physik (Springer, Berlin, 1948). [8] M. van den Berg, Phys. Lett. 81A (1981) 219. [91 J. Ambj#rn and S. Wolfram, Ann. Phys. (NY) 147 (1983) 1.
PHYSICS LETTERS
20 December 1984
CASIMIR EFFECT IN n DIMENSIONS
H. VERSCHELDE x, L WILLE and P. PHARISEAU
Seminarie voor Theoretische Vaste Stof- en Lage Energie Kernfysica, R#ksuniversiteit Gent, Krf/gslaan281, Gebouw $9, 9000 Gent, Belgium Received 15 June 1984 Revised manuscript received 9 October 1984
We derive exact expressions for the Casimir interaction energy per unit area between two parallel, perfectly conducting, metal plates, in n (~ 2) dimensions.
In recent years, bag model calculations [1] have revived interest in the Casimir effect [2]. Progress, has been made in developing approximation techniques [3] as well as in calculating new geometries [4]. Nevertheless, the number of exact results is small and intuition on how geometry affects the qualitative aspects of the Casimir effect is completely lacking. An unanswered question in this respect is whether the attractive or repulsive character of the effect will be changed by going to higher dimensions [5]. In this letter, we will address this question for parallel plates and derive exact expressions for the interaction energy in N(~> 2) dimensions. A regularized sum over frequencies formula for the interaction energy per unit area (in two dimensions, this formula gives the interaction energy per unit length, in n dimensions it gives the interaction energy per unit of (n - 1)-dimensional area) between two parallel, perfectly conducting metal plates, located at x = -L/2, x = +L/2 is given by (we use units h = c = 1)
U(n'L'a)=-~-"
m=l
f
X exp { - a [(mlt/L) 2 + k 2] 1/2).
[(mTr/L)2+k211/2 (1)
We have omitted the m = 0 term in (1) because it is independent of L and therefore does not contribute to 1 Aspirant NFWO (Belgium). 396
the Casimir energy. If we let a go to zero, we recover the familiar sum-over-frequencies formula. The divergences will appear as singularities with respect to a, as a ~ 0. The finite, physically relevant, Casimir interaction energy per unit area will be given by the a independent part of U(n, L, a). The appearance of square roots in (1) makes the evaluation of (1)very difficult. However, they can be eliminated by using the following Laplace transform: f
e x p ( - p t ) exp(---a2/4t)
tl12
dt = (vc~'/VCp)e x p ( - a x/rp).
0
(2)
The interaction energy then becomes
U(n, L, a) = (02/~a2)f(n, L, a), with
F(n,L,a) = -n--g - -1, , N1 ×f
(3)
-/
~1 exp(-a2/4t) tl/2 = o dt
d/d~-I e x p { - t [ k 2+(m~lL)21}.
(2~)n-1
(4)
The integration over (n - 1)-dimensional k space is now trivial and yields
F(n, L, a) = ~(n - 1) 2 (1 -n)n-n/2 × ~ ? d t exp(-a2/4t)exp[-t(mn/L)2] m =1 ~) tn[ 2
(5)
0370-2693•84•$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 149B, number 4,5
PHYSICS LETTERS
If we use the following integral representation for the modified cylindrical Bessel function [6]
2(~)V/2Kv(2(z~)l/2) = f e -zt e -~/t t - v - 1 dt 0
20 December 1984
m • l K o ( m X )Ir/2x - 3'/2 + {ln(x/47r) +
=
(6) +rt ~ (Ix2 +(27r~)2] -1/2 - 1/2rr£), ~=1
(15)
(5) becomes where 3' = 0, 577215 .... Because of [6]
F(n, L, a) = {(n - 1) 2(2-n)/2rr-n/2a -(n-2) × f({(,
- 2), L, a),
(7)
where
[x 2 + (21rj~)2] -1/2 = S e - 2 7 r ~ t 0
Jo(xt) dt
(16)
and the expansion [6]
f(v, L, a) = ~
m=l
(m,a/L )VKv(mlra/L ).
(8)
Jo(xt) - 1 = ~ ( - 1 ) s (xt/2)2S/(s!)2,
(17)
s=l It is clear from (3) and (7) that the Casimir interaction energy per unit area Uc(n, L) which is the a independent part of U(n, L, a) will be given by
the last term in (15) becomes
Uc(n, L) = {(n - 1) 2 (4-n)/2~r-n/2b~n_2)/2 ,
rr S "[[x 2 + (2rr~) 2] -1/2 _ 1/2rr~) ~=1
(9)
where byu is the coefficient ofa u in a small a expansion for f(v, L, a). We now establish a recurrence relation for the coefficients b u. The modified cylindrical Bessel functions Kv(z ) satisfy the following recurrence relations [6] K _l(Z) -
=~ ~ S=1~=1 s=l
Kv+l(Z ) = -(2v/z)Kv(z ),
K v _l(Z) + Kv+l(Z ) = - 2 K v ( z ).
(10)
Appropriate combination of these relations and multiplication by z v+l yields
f(v + 1,L,a) = -aOf(v,L,a)]aa + 2vf(v,L,a),
(11)
(-l)Srr(x/2)2s f exp(-2rt£t)t 2s dt
=
(S!)2
24s+l(s!)2
0 (2S)! =
24s+l(s!)2
(18)
where ~'(z) is the Riemann zeta function. Therefore, (15) becomes oo
so that the recurrence relation for the b u's v becomes b"/) = [ - u +
- 1)] b"v - - l *
These recurrence relations can be solved for and give
b~n-2)~2 = (-l)(n-2)]22(n-2)]2(1n)! b~
(12) n b(n -2)/2
m=l
Ko(mX) = {(3' - In art) + -~ln x + 7r/2x
+ ~ (-1)S(2s)!f(2s + 1) (x/Ir)2S" s=l
(13)
for n even and b(n - 2 ) / 2 = (-1)(n-3)/22(n-3)/2 [-~(n - 1)1 ! b~/2(14 ) for n odd. What remains to be calculated is bg and
b7/2. To calculate b~, we use the following identity [7]
(19)
24s+l(s!) 2
Hence, because of (8), (15) and the definition of b u, we have that
n (-1)n/2n!~( n + 1) 1 b0 = 2 zn+l [(n/Z)!] 2 L--n-" To calculate
(20)
b~/2, we use the formula [6]
Kll 2(x) = X / ~ c-x/vrx
(21 ) 397
Volume 149B, number 4,5
PHYSICS LETTERS
to obtain
20 December 1984
which is the well known result [2], for the Casimir effect between two parallel, perfectly conducting metal plates in three dimensions. Because [6]
f(~, L, a) = ~ (mlra/L)ll 2K1 / 2(re,raiL ) m=l
Bn+ 1 = 2(-1)(n+3)/2(n + 1)! ~
=~
[expOra/L) 1] -1
(23)
(24)
One of us, (H.V.), would like to thank K. Debel for introducing him to the Casimir effect.
we obtain for bnl/2
bnl/2 = X / ~ [Bn+l/(n + 1)!] rrn/L n.
By combining (9), (13), (14) and (20), (24) we are able to obtain exact expressions for the Casimir interaction energy per unit area in n('~ 2) dimensions:
Uc(n,L ) = ±¼(n - 1) n!f(n + 1) 1 2'2nrtn/2(n/2)! L---ff
(25)
for n even and
Uc(n,L)=~(n - 1) (-1)(n+l)/2rr(n+l)12[{(n - 1)1 ! Bn+ 1 1 X
(n + 1)!
Ln
(26)
for n odd. For n = 2, (25) yields Uc(2, L ) = - ~(3)/16rrL 2,
(27)
which agrees with [8] up to a factor 1/2 (only one polarization). For n -- 3, (26) yields Uc(3, L) = -rr2/720L 3 ,
398
(29)
it follows from (26) that Uc(n, L) is negative for n odd. Therefore, the Casimir interaction energy is always negative. This means that the Casimir force between parallel plates is attractive, irrespective of the number of dimensions [5,9].
If we make use of the definition for the Bernoulli numbers [6]
0 ~Bn xn e x X- 1 = ~ =
(1/21rm) n÷l
m=l
(22)
-
(28)
References [1] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9 (1974) 3471. [2] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetenschap. 51 (1948) 793. [3 ] R. Balian and B. Duplantier, Ann. Phys. 104 (1977) 300; 112 (1978) 165. [4] J.H. Boyer, Phys. Rev. 174 (1968) 1764; K.A. Milton, L.L. DeRaad Jr. and J. Schwinger, Ann. Phys. 115 (1978) 388; D. Deutsch and P. Candelas, Phys, Rev. D20 (1979) 3063. [ 5 ] K. Debel, private communication. [6 ] W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and theorems for the special functions of mathematical physics (Springer, Berlin, 1966). [7] W. Magnus and F. Oberhettinger, Formein und Siitze ftir die speziellen Funktionen der mathematischen Physik (Springer, Berlin, 1948). [8] M. van den Berg, Phys. Lett. 81A (1981) 219. [91 J. Ambj#rn and S. Wolfram, Ann. Phys. (NY) 147 (1983) 1.