In this work we get the formula (5) that it allows the explicit computation of the renormalized partition function associated with a physical system, whose states belong to a differentiable manifold of finite dimension, and whose energy function is invariant under the action of a compact Lie group. Furthermore, we extend the formula above to the infinite dimensional case using -regularization technique.
REFERENCES
1.
2.
M.
Bordag
, E.
Elizalde
, and K.
Kirsten
, “Heat kernel coefficients of the Laplace operator on the D-dimensional ball
,” J. Math. Phys.
37
, 895
–916
(1996
).3.
M.
Bordag
, E.
Elizalde
, K.
Kirsten
, and S.
Leseduarte
, “Casimir energies for massive fields in the bag
,” Phys. Rev. D
56
, 4896
–4904
(1997
).4.
M.
Bordag
, E.
Elizalde
, K.
Kirsten
, and S.
Leseduarte
, “Casimir energy for a massive fermionic quantum field with a spherical boundary
,” J. Phys. A
31
, 1743
–1759
(1998
).5.
J. B.
Bost
, “Fibres determinants, determinants regularises et mesures sur les espaces de modules de courbes complexes
,” Asterisque
152–153
, 113
–149
(1987
).6.
A. P.
Calderón
and A.
Zygmund
, “Singular integral operators and differential equations
,” Am. J. Math.
79
, 901
–921
(1957
).7.
G.
Cognola
, E.
Elizalde
, and S.
Zerbini
, “Heat-kernel expansion on noncompact domains and a generalized zeta-function regularization procedure
,” J. Math. Phys.
47
(8
), 083516
(2006
).8.
9.
E.
Elizalde
, “Explicit zeta functions for bosonic and fermionic fields on a non-commutative toroidal spacetime
,” J. Phys. A
34
, 3025
–3035
(2001
).10.
E.
Elizalde
, “Multidimensional Extension of the Generalized Chowla-Selberg Formula
,” Commun. Math. Phys.
198
, 83
–95
(1998
.11.
E.
Elizalde
, Ten Physical Applications of Spectral Zeta Functions
(World Scientific
, Singapore
, 1994
).12.
E.
Elizalde
, “Zeta functions: formulas and applications
,” J. Comput. Appl. Math.
118
, 125
–142
(2000
).13.
E.
Elizalde
and K.
Kirsten
, “Casimir energy of a massive field in a genus-1 surface
,” Phys. Lett. B
365
, 72
–78
(1995
).14.
E.
Elizalde
, L.
Vanzo
, and S.
Zerbini
, “Zeta-Function Regularization, the Multiplicative Anomaly and the Wodzicki Residue
,” Commun. Math. Phys.
194
, 613
–630
(1998
).15.
E.
Elizalde
, S. D.
Odintsov
, A.
Romeo
, A. A.
Bytsenko
, and S.
Zerbini
, Zeta Regularization Techniques with Applications
(World Scientific
, Singapore
, 1994
).16.
P.
Gómez
, R.
Mendoza
, and F.
Moraes
, “Fluctuating metrics in one-dimensional manifolds
,” J. Math. Phys.
38
(10
), 5293
–5300
(1997
).17.
R.
Hamilton
, “The inverse function Theorem of Nash and Moser
,” Bull., New Ser., Am. Math. Soc.
7
(1
), 65
–222
(1982
).18.
S. W.
Hawking
, “Zeta function regularization of path integrals in curved space-time
,” Commun. Math. Phys.
55
, 133
–148
(1977
).19.
S.
Helgason
, Differential Geometry, Lie Groups and Symmetric Spaces
(Academic
, New York
, 1978
).20.
R.
Mendoza
, J.
Rojas
, and F.
Moraes
, “Partition function for a nonlinear supersymmetric model
,” J. Phys. A
33
(48
), 8887
–8892
(2000
).21.
B.
Osgood
, R.
Phillips
, and P.
Sarnak
, “Moduli space, heights and isospectral sets of plane domains
,” Ann. Math.
129
, 293
–362
(1989
).22.
A. M.
Polyakov
, “Quantum geometry of bosonic strings
,” Phys. Lett.
103B
, 207
–210
(1981
).23.
A. M.
Polyakov
, “Quantum geometry of fermionic strings
,” Phys. Lett.
103B
, 211
–213
(1981
).24.
25.
D. B.
Ray
and I. M.
Singer
, “R-torsion and the Laplacian on Riemannian manifolds
,” Adv. Math.
7
, 145
–210
(1971
).26.
R. T.
Seeley
, “Complex powers of an elliptic operator
,” Proc. Symp. Pure Math.
10
, 288
–307
(1967
).27.
A.
Tromba
, Teichmüller Theory in Riemann Geometry
(Birkhauser
, Zurich
, 1992
).© 2007 American Institute of Physics.
2007
American Institute of Physics
You do not currently have access to this content.
