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The Casimir effect in conformal field theories
Volume 190, number 1,2
PHYSICS LETTERS B
21 May 1987
THE CASIMIR EFFECT IN C O N F O R M A L FIELD THEORIES Bernardo BARBIELLINI-AMIDEI D~partement de Physique, Universit~de Gen~ve, CH-1211 Geneva4, Switzerland Received 13 October 1986
The aim of this letter is to give, by studying the zero-point energy of two-dimensional conformal tensor fields, a simple derivation of two well-known formulae occurring in two-dimensional conformal field theory (CFT): the spectrum of the conformal central charge c and the critical dimension in string theory. In order to find the formula for c < 1, we inspire ourself by the Reggepole model.
1. Conformal tensors. In C F T thinks a space as the complex plane, then the conformal transformations consist o f the analytic mapping z ~ f ( z ) , and the powerful machinery o f complex analysis can be brought into play, Let us consider a conformal tensor field on the unit circle transforming under the rotations according to
x'(o) =e-i:x(0), X'(2n) =X(2n)
~---e - i 2 1 r j
,
where j is the (conformal) spin (see fig. 1 ). One can expand X i n normal modes or oscillators:
X(O) = ~ Xn_j e 2nitn-j)O . --
oo
To avoid branch points, we require that j be a positive integer. The q u a n t u m zero-point energy (i.e. the Casimir effect) is given, after the (-function regulation (see appendix), by
1 ~ (n-j)=½((-1;-j) 2 n=O
=-,~+U(j+ l ) = - h c , o
o= 0 Fig. 1. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where c = 1 + 6 j 0 ' + 1 ) is called the conformal anomaly parameter or central charge. With the central charge, one can parametrise the representations of the conformal group. For a tensor repr8sentation o f spin j, c is an integer t> 1. We argue that the conformal tensors given all the unitary irreducible representations ( U I R ' s ) o f the conformal group having c i> 1.
2. UIR's with c< 1 [I]. Regge in 1959 proposed to treat the angular m o m e n t u m as a continuous complex variable - although, deafly, physically observable states must have integral or half-integral angular m o m e n t u m . Therefore inspired by Regge, let us consider the analytic continuation of c c(z)=l+6z,
z=j(j+ 1)~C,
which can be considered as a conformal field! A definition o f hermitean conjugation for an observable A ( z ) in CFT's is given by [A(z)] + = A ( - 1/z). In fact if the locations o f the asymptotic states are z = 0 and z = ~ , the conformal transformation f ( z ) = - 1/z maps 0 on ~ , hence we get the exchange o f the initial and final states. Note that in these theories z = 0 is a singularity, because of radial causality [ 1 ] ! Therefore the hermitian conjugate for c(z) is given by
c+(z)=l-6/z. 137
Volume 190, number 1,2
PHYSICS LETTERS B
21 May 1987
½
C
I
T discrete spectrum
continuous Spectrum
Fig. 2.
Unitarity hypothesis. Due to the unitarity, if c(z) parametrises a UIR, then also c + (z), except for the scalar field ( j = 0) which corresponds to the singularity z = 0 and the vector field ( j = 1 ) which can be associated with ghosts coming from the reparametrisations. Thus we can obtain all the U I R with c < 1 from the tensor ones by hermitian conjugation: c=l-6/m(m+l),
4. Concluding remark. The author hopes that these simple physical arguments provide a CTF treatment accessible to the non specialists without having to "go back to school to learn the formalism". Appendix ( a ) The Riemann zeta-function regularisation.
m>~2,
f f ( z ; a ) . ' = ~ ( n + a ) -z, Analytical
gives
((-1;a)=-~+
(b) A pedestrian way to get the same result. Start with n = ~ ne -n~, n=l
~0
n=l
0 ~, e_n, 0en=l 0 =-0--~ ( I - e - ' ) - ' 0
=-o-7 [1/e-'--he+O(d)] = 11~2 -- ~ • Zero point energy = ½ ~ n = J 4 ~ ~ - ~ . Now consider ½2(n+a) =-~
""° Z _ ,"-..
x
z=O
Fig. 3
138
continuation
½a(1-a).
3. Strings and critical dimension D= 26 [2]. In string theory, the coordinate X u(O) is a set of conforreal scalar fields with cx=D, where D is the spacetime dimension. To get rid of the unphysical longitudinal and temporal modes, We need two ghosts with anticommuting modes and with conformal spin j = 1 (necessary for the propagation in other circle slices lying on the complex plane) (see fig. 3 ). Hence, Cghosts= -- 2 (1 + 6" 1"2) = -- 26, here, the minus sign is due to the anticommuting modes. If D = 26, we get Ctot= ex + cghost~= 0 and the quantum conformal invariance obtains.
•
Izl>l.
n=l
here m is no more a conformal spin! For m = 2, we obtain the trivial representation with c = 0. For m = 3, we obtain c = ½, which is the conforreal anomaly associated with the spinor field. These c-values give a sequence having c = 1 as an accumulation point. For a reducible representation, c is the sum of the c's of the building blocks hence c = 1 is an accumulation point, after c = 1 the c-spectrum is continuous, and before c = 1 it is discrete (see fig. 2).
X
+f(o0 ,
(0) string coordinate field
Volume 190, number 1,2
PHYSICS LETTERS B
21 May 1987
w h e r e f ( a ) must obey
References
f(0) =0
[ 1 ] D. Friedan, Z. Qiu and S. Shenker, in: Vertex operator in
(initial condition),
f ( a ) = ½a + f ( a + 1 )
(recurrent e q u a t i o n ) .
Mathematics and Physics, eds. M. Lepowsky et al. (Springer, Berlin); Phys. Rev. Lett. 52 (1984) 1575. [2] A.M. Polyakov, Phys. Lett. B 103 (1981) 207.
The solution is given by f ( c e ) = ~ a ( 1 - o 0 . (This simple derivation has been given by M. Peskin during a lecture. )
139
PHYSICS LETTERS B
21 May 1987
THE CASIMIR EFFECT IN C O N F O R M A L FIELD THEORIES Bernardo BARBIELLINI-AMIDEI D~partement de Physique, Universit~de Gen~ve, CH-1211 Geneva4, Switzerland Received 13 October 1986
The aim of this letter is to give, by studying the zero-point energy of two-dimensional conformal tensor fields, a simple derivation of two well-known formulae occurring in two-dimensional conformal field theory (CFT): the spectrum of the conformal central charge c and the critical dimension in string theory. In order to find the formula for c < 1, we inspire ourself by the Reggepole model.
1. Conformal tensors. In C F T thinks a space as the complex plane, then the conformal transformations consist o f the analytic mapping z ~ f ( z ) , and the powerful machinery o f complex analysis can be brought into play, Let us consider a conformal tensor field on the unit circle transforming under the rotations according to
x'(o) =e-i:x(0), X'(2n) =X(2n)
~---e - i 2 1 r j
,
where j is the (conformal) spin (see fig. 1 ). One can expand X i n normal modes or oscillators:
X(O) = ~ Xn_j e 2nitn-j)O . --
oo
To avoid branch points, we require that j be a positive integer. The q u a n t u m zero-point energy (i.e. the Casimir effect) is given, after the (-function regulation (see appendix), by
1 ~ (n-j)=½((-1;-j) 2 n=O
=-,~+U(j+ l ) = - h c , o
o= 0 Fig. 1. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where c = 1 + 6 j 0 ' + 1 ) is called the conformal anomaly parameter or central charge. With the central charge, one can parametrise the representations of the conformal group. For a tensor repr8sentation o f spin j, c is an integer t> 1. We argue that the conformal tensors given all the unitary irreducible representations ( U I R ' s ) o f the conformal group having c i> 1.
2. UIR's with c< 1 [I]. Regge in 1959 proposed to treat the angular m o m e n t u m as a continuous complex variable - although, deafly, physically observable states must have integral or half-integral angular m o m e n t u m . Therefore inspired by Regge, let us consider the analytic continuation of c c(z)=l+6z,
z=j(j+ 1)~C,
which can be considered as a conformal field! A definition o f hermitean conjugation for an observable A ( z ) in CFT's is given by [A(z)] + = A ( - 1/z). In fact if the locations o f the asymptotic states are z = 0 and z = ~ , the conformal transformation f ( z ) = - 1/z maps 0 on ~ , hence we get the exchange o f the initial and final states. Note that in these theories z = 0 is a singularity, because of radial causality [ 1 ] ! Therefore the hermitian conjugate for c(z) is given by
c+(z)=l-6/z. 137
Volume 190, number 1,2
PHYSICS LETTERS B
21 May 1987
½
C
I
T discrete spectrum
continuous Spectrum
Fig. 2.
Unitarity hypothesis. Due to the unitarity, if c(z) parametrises a UIR, then also c + (z), except for the scalar field ( j = 0) which corresponds to the singularity z = 0 and the vector field ( j = 1 ) which can be associated with ghosts coming from the reparametrisations. Thus we can obtain all the U I R with c < 1 from the tensor ones by hermitian conjugation: c=l-6/m(m+l),
4. Concluding remark. The author hopes that these simple physical arguments provide a CTF treatment accessible to the non specialists without having to "go back to school to learn the formalism". Appendix ( a ) The Riemann zeta-function regularisation.
m>~2,
f f ( z ; a ) . ' = ~ ( n + a ) -z, Analytical
gives
((-1;a)=-~+
(b) A pedestrian way to get the same result. Start with n = ~ ne -n~, n=l
~0
n=l
0 ~, e_n, 0en=l 0 =-0--~ ( I - e - ' ) - ' 0
=-o-7 [1/e-'--he+O(d)] = 11~2 -- ~ • Zero point energy = ½ ~ n = J 4 ~ ~ - ~ . Now consider ½2(n+a) =-~
""° Z _ ,"-..
x
z=O
Fig. 3
138
continuation
½a(1-a).
3. Strings and critical dimension D= 26 [2]. In string theory, the coordinate X u(O) is a set of conforreal scalar fields with cx=D, where D is the spacetime dimension. To get rid of the unphysical longitudinal and temporal modes, We need two ghosts with anticommuting modes and with conformal spin j = 1 (necessary for the propagation in other circle slices lying on the complex plane) (see fig. 3 ). Hence, Cghosts= -- 2 (1 + 6" 1"2) = -- 26, here, the minus sign is due to the anticommuting modes. If D = 26, we get Ctot= ex + cghost~= 0 and the quantum conformal invariance obtains.
•
Izl>l.
n=l
here m is no more a conformal spin! For m = 2, we obtain the trivial representation with c = 0. For m = 3, we obtain c = ½, which is the conforreal anomaly associated with the spinor field. These c-values give a sequence having c = 1 as an accumulation point. For a reducible representation, c is the sum of the c's of the building blocks hence c = 1 is an accumulation point, after c = 1 the c-spectrum is continuous, and before c = 1 it is discrete (see fig. 2).
X
+f(o0 ,
(0) string coordinate field
Volume 190, number 1,2
PHYSICS LETTERS B
21 May 1987
w h e r e f ( a ) must obey
References
f(0) =0
[ 1 ] D. Friedan, Z. Qiu and S. Shenker, in: Vertex operator in
(initial condition),
f ( a ) = ½a + f ( a + 1 )
(recurrent e q u a t i o n ) .
Mathematics and Physics, eds. M. Lepowsky et al. (Springer, Berlin); Phys. Rev. Lett. 52 (1984) 1575. [2] A.M. Polyakov, Phys. Lett. B 103 (1981) 207.
The solution is given by f ( c e ) = ~ a ( 1 - o 0 . (This simple derivation has been given by M. Peskin during a lecture. )
139