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Size and dimension in Kaluza-Klein theories
Volume 150B, number 6
PHYSICS LETTERS
24 January 1985
SIZE AND DIMENSION IN KALUZA-KLEIN THEORIES
Enrico RODRIGO Center for Theoretical Physics, The University ofTexasatAustin, Austin, TX 78712, USA Received 12 October 1984
By freezing all degrees of freedom except the radius of the internal subspace of an M4 × Sn Kaluza-Klein geometry, a purely quantum mechanical method is used to determine how the radius of the internal subspace depends on n.
There have been several recent attempts to determine the size of the internal subspace of Kaluza-Klein theories [ 1 - 3 ] . These efforts fall into three categories: classical, semi-classical, and semi-quantum. The classical method [1] begins and ends with Finding a spontaneously compactified solution to Einstein's equations in 4 + n dimensions. The size of the internal subspace is then found to depend on a parameter (monopole strength) of the matter part of the solution. The semi-classical approach continues to treat gravity classically, but gravity has as source the vacuum expectation value of the stress-energy of quantum matter fields [2]. The size of the internal subspace depends on the number of quantum matter fields in the system. In the semi-quantum approach [3] both gravity and matter are given quantum properties. The linearized perturbation of gravity from a classical, spontaneously compactified solution is quantized to create a Casimir force, which is opposed by forces associated with the quantum matter fields [4,5]. The size of the internal subspace is taken to be that at which these forces cancel. These approaches each depend on spontaneous compactification to some degree. They are therefore essentially classical theories that undergo quantum perturbations. But such methods cannot adequately address the question of the size of the Kaluza-Klein internal subspace if, as we reasonably suppose, this size is of the scale o f a Planck length. A purely quantum approach is required. Recently, such an approach, which is based on the Wheeler-DeWitt equation, was proposed [6]. The 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
purpose o f this note is to use this idea to determine how the size of the Kaluza-Klein internal subspace depends on the subspace's dimensionality. Begin by considering the metric for the geometry of R 3 X Sn, =
,
(gab)
(1)
a2~/km
where i,] E {1, 2, 3), and ~/krn(k, m >~ 4) is the metric of the unit n-sphere. This can be inserted into an expression for the hamiltonian of (4 + n)-dimensional gravity coupled to matter. If the energy associated with the presence o f the matter is E, then the hamiltonian, which is constrained to be zero [7], may be written as [6] H -
1 f 2•4+n
[KabKa b _ (Kaa)2 _ (3+n)R]gl/2 d3+n x
+ E = ( - e Z K 4 + n / Z V ~ n ) p 2 - (V~n/ZK4+n)(½nr)2-4/n
+ E = 0,
(2)
where [3n -
Volume of S n a n
n(n - 1).
K is extrinsic curvature and R is the (3 + n)-dimensional intrinsic curvature, Kn+ 4 is the (4 + n)-dimensional gravitation constant, V is the volume of the R 3 subspace, and p is the momentum conjugate to r. The variable r, (which was introduced to circumvent the 425
Volume 150B, number 6
PttYSICS LETTERS
factor ordering problem by making the kinetic term of H a product o f identical factors) is related to the radius a o f the n-sphere by, r = ( 2 / n ) a n/2
(3)
One could replace the canonical m o m e n t u m p with ih d/dr, and thus obtain the Wheeler-DeWitt equation in one degree of freedom. But we are only interested in the ground state expectation value of r. Hence we replace p with h/r, minimize E with respect to r, and use eq. (3) to obtain (a)o = (n/[Jn)l/(n-1) [n/(4n - 8)] 1/(2n-2) X ( h c K 4 + n / V ) 1/(n -1)
= 2[(n - 2 ) / n ] l / 2 ( n - 1) V/hcK 4 ,
(4)
24 January 1985
n = 1, and n = 2 are special. When n = 0, the change of variable in eq. (3) is undefined (as is the notion of radius of a "0-sphere"). When n = 1, the potential term in (2) vanishes, because/31 = 0. When n = 2, the potential is constant, and there is no unique minimum. The principal advantage of this method is that it is nonperturbative. There are no references to classical solutions. Because we do not have to solve Einstein's equations, we can side step the problem of the cosmological constant. However the question of the existence of static classical solutions is easily answered by inspecting eq. (2). It is clear, for example, that in the absence o f matter (E = 0), there exists no solution to (2) unless the potential term vanishes. As we have noted, this only occurs for n = 1. Thus, with little effort, we reach a conclusion that normally requires a more extensive derivation [3].
where we have used the relation [8] n4+n = (Volume ofSn)t~4 = [(3nan/n(n - 1)] K4.
(5)
We have assumed that their are no quantum fluctuations over the entire R 3 subspace, which has volume V. But if V is macroscopic, this assumption is clearly false. Quantum fluctuations in volume are presumably of the scale of the cube o f a Planck length, therefore the largest volume over which the above assumption holds is this Planck volume. Replacing V with the Planck volume simulates the effect of giving the Sn submetric an arbitrary number o f quantum degrees of freedom [9]. We may now rewrite eq. (4): (a) 0 = 2 [(n - 2)/n] 1/2(n - 1)(heK4) 1/2 .
(6)
The n-dependent coefficient o f the Planck length X/h--C-Kin eq. (6) is the result we seek. For large n the radius o f the n-sphere (the "size" of the internal subspace) grows linearly with 17. The cases for which n = 0,
426
References
[ 1 ] E. Cremmer and J. Scherk, Nucl. Phys. B127 (1977) 57; P. Freund and M. Rubin, Phys. Lett. 97B (1980) 233; M.J. Duff, B. Nilsson and C. Pope, Phys. Rev. Lett. 50 (1983) 2043. [21 P. Candelas and S. Weinberg, Nucl. Phys. B237 (1984) 397. [3] T. Appelquist and A. Chodos, Phys. Rev. D28 (1983) 772; T. Appelquist and A. Chodos, Phys. Rev. Lett. 50 (1983) 141. [4] M.A. Rubin and B.D. Roth, Nucl. Phys. B226 (1983) 444; G. Gilbert, B. McClean and M.A. Rubin, Dynamical quantum effects in Kaluza-Klein theories, University of Texas preprint UTTG-15-83 (1983). [5] M.A. Rubin and B.D. Roth, Phys. Lett. 127B (1983) 55. [6] E. Rodrigo, Phys. Lett. 105A (1984) 196. [7] B.S. DeWitt, Phys. Rev. 160 (1967) 1113. [8] S. Weinberg, Phys. Lett. 125B (1983) 265. [9] E. Rodrigo, unpublished (1984).
PHYSICS LETTERS
24 January 1985
SIZE AND DIMENSION IN KALUZA-KLEIN THEORIES
Enrico RODRIGO Center for Theoretical Physics, The University ofTexasatAustin, Austin, TX 78712, USA Received 12 October 1984
By freezing all degrees of freedom except the radius of the internal subspace of an M4 × Sn Kaluza-Klein geometry, a purely quantum mechanical method is used to determine how the radius of the internal subspace depends on n.
There have been several recent attempts to determine the size of the internal subspace of Kaluza-Klein theories [ 1 - 3 ] . These efforts fall into three categories: classical, semi-classical, and semi-quantum. The classical method [1] begins and ends with Finding a spontaneously compactified solution to Einstein's equations in 4 + n dimensions. The size of the internal subspace is then found to depend on a parameter (monopole strength) of the matter part of the solution. The semi-classical approach continues to treat gravity classically, but gravity has as source the vacuum expectation value of the stress-energy of quantum matter fields [2]. The size of the internal subspace depends on the number of quantum matter fields in the system. In the semi-quantum approach [3] both gravity and matter are given quantum properties. The linearized perturbation of gravity from a classical, spontaneously compactified solution is quantized to create a Casimir force, which is opposed by forces associated with the quantum matter fields [4,5]. The size of the internal subspace is taken to be that at which these forces cancel. These approaches each depend on spontaneous compactification to some degree. They are therefore essentially classical theories that undergo quantum perturbations. But such methods cannot adequately address the question of the size of the Kaluza-Klein internal subspace if, as we reasonably suppose, this size is of the scale o f a Planck length. A purely quantum approach is required. Recently, such an approach, which is based on the Wheeler-DeWitt equation, was proposed [6]. The 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
purpose o f this note is to use this idea to determine how the size of the Kaluza-Klein internal subspace depends on the subspace's dimensionality. Begin by considering the metric for the geometry of R 3 X Sn, =
,
(gab)
(1)
a2~/km
where i,] E {1, 2, 3), and ~/krn(k, m >~ 4) is the metric of the unit n-sphere. This can be inserted into an expression for the hamiltonian of (4 + n)-dimensional gravity coupled to matter. If the energy associated with the presence o f the matter is E, then the hamiltonian, which is constrained to be zero [7], may be written as [6] H -
1 f 2•4+n
[KabKa b _ (Kaa)2 _ (3+n)R]gl/2 d3+n x
+ E = ( - e Z K 4 + n / Z V ~ n ) p 2 - (V~n/ZK4+n)(½nr)2-4/n
+ E = 0,
(2)
where [3n -
Volume of S n a n
n(n - 1).
K is extrinsic curvature and R is the (3 + n)-dimensional intrinsic curvature, Kn+ 4 is the (4 + n)-dimensional gravitation constant, V is the volume of the R 3 subspace, and p is the momentum conjugate to r. The variable r, (which was introduced to circumvent the 425
Volume 150B, number 6
PttYSICS LETTERS
factor ordering problem by making the kinetic term of H a product o f identical factors) is related to the radius a o f the n-sphere by, r = ( 2 / n ) a n/2
(3)
One could replace the canonical m o m e n t u m p with ih d/dr, and thus obtain the Wheeler-DeWitt equation in one degree of freedom. But we are only interested in the ground state expectation value of r. Hence we replace p with h/r, minimize E with respect to r, and use eq. (3) to obtain (a)o = (n/[Jn)l/(n-1) [n/(4n - 8)] 1/(2n-2) X ( h c K 4 + n / V ) 1/(n -1)
= 2[(n - 2 ) / n ] l / 2 ( n - 1) V/hcK 4 ,
(4)
24 January 1985
n = 1, and n = 2 are special. When n = 0, the change of variable in eq. (3) is undefined (as is the notion of radius of a "0-sphere"). When n = 1, the potential term in (2) vanishes, because/31 = 0. When n = 2, the potential is constant, and there is no unique minimum. The principal advantage of this method is that it is nonperturbative. There are no references to classical solutions. Because we do not have to solve Einstein's equations, we can side step the problem of the cosmological constant. However the question of the existence of static classical solutions is easily answered by inspecting eq. (2). It is clear, for example, that in the absence o f matter (E = 0), there exists no solution to (2) unless the potential term vanishes. As we have noted, this only occurs for n = 1. Thus, with little effort, we reach a conclusion that normally requires a more extensive derivation [3].
where we have used the relation [8] n4+n = (Volume ofSn)t~4 = [(3nan/n(n - 1)] K4.
(5)
We have assumed that their are no quantum fluctuations over the entire R 3 subspace, which has volume V. But if V is macroscopic, this assumption is clearly false. Quantum fluctuations in volume are presumably of the scale of the cube o f a Planck length, therefore the largest volume over which the above assumption holds is this Planck volume. Replacing V with the Planck volume simulates the effect of giving the Sn submetric an arbitrary number o f quantum degrees of freedom [9]. We may now rewrite eq. (4): (a) 0 = 2 [(n - 2)/n] 1/2(n - 1)(heK4) 1/2 .
(6)
The n-dependent coefficient o f the Planck length X/h--C-Kin eq. (6) is the result we seek. For large n the radius o f the n-sphere (the "size" of the internal subspace) grows linearly with 17. The cases for which n = 0,
426
References
[ 1 ] E. Cremmer and J. Scherk, Nucl. Phys. B127 (1977) 57; P. Freund and M. Rubin, Phys. Lett. 97B (1980) 233; M.J. Duff, B. Nilsson and C. Pope, Phys. Rev. Lett. 50 (1983) 2043. [21 P. Candelas and S. Weinberg, Nucl. Phys. B237 (1984) 397. [3] T. Appelquist and A. Chodos, Phys. Rev. D28 (1983) 772; T. Appelquist and A. Chodos, Phys. Rev. Lett. 50 (1983) 141. [4] M.A. Rubin and B.D. Roth, Nucl. Phys. B226 (1983) 444; G. Gilbert, B. McClean and M.A. Rubin, Dynamical quantum effects in Kaluza-Klein theories, University of Texas preprint UTTG-15-83 (1983). [5] M.A. Rubin and B.D. Roth, Phys. Lett. 127B (1983) 55. [6] E. Rodrigo, Phys. Lett. 105A (1984) 196. [7] B.S. DeWitt, Phys. Rev. 160 (1967) 1113. [8] S. Weinberg, Phys. Lett. 125B (1983) 265. [9] E. Rodrigo, unpublished (1984).