Avoiding degenerate coframes in an affine gauge approach to quantum gravity

Eckehard W. Mielke; J. Dermott McCrea; Yuval Ne’eman; Friedrich W. Hehl

doi:10.1103/PhysRevD.48.673

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Avoiding degenerate coframes in an affine gauge approach to quantum gravity

Institute for Theoretical Physics, University of Cologne, D-50923 Köln, Germany

Department of Mathematical Physics, University College, Dublin 4
Dublin Institute for Advanced Studies, Dublin 4, Ireland

Raymond and Beverley Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

Institute for Theoretical Physics, University of Cologne, D-50923 Köln, Germany

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Phys. Rev. D 48, 673 – Published 15 July, 1993

DOI: https://doi.org/10.1103/PhysRevD.48.673

Abstract

In quantum models of gravity, it is surmised that configurations with degenerate coframes could occur during toplogy change of the underlying spacetime structure. However, the coframe is not the true Yang-Mills-type gauge field of the translations, since it lacks the inhomogeneous gradient term under gauge transformations. By explicitly restoring this ‘‘hidden’’ piece within the framework of the affine gauge approach to gravity, one can avoid the metric or coframe degeneracy which would otherwise interfere with the integrations within the path integral. This is an important advantage for quantization.

References (47)

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F.W. Hehl, P. von der Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48, 393 (1976); see also F.W. Hehl, in Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity, Proceedings of the International School, Erice, Italy, 1979, edited by P.G. Bergmann and V. de Sabbata,NATO Advanced Study Institute Series B: Physics, Vol. 58 (Plenum, New York, 1980), p. 5.
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C.J. Isham, in Recent Aspects of Quantum Fields, Proceedings of the XXXth International Univer sitäts wo chen für Kernphysik, Schladming, Austria, 1991, edited by H. Mitter and H. Gausterer,Lecture Notes in Physics, Vol. 396 (Springer, Berlin, 1991), p. 123.
E.W. Mielke, Ann. Phys. (N.Y.) 219, 78 (1992).
P. Baekler, E.W. Mielke and F.W. Hehl, Nuovo Cimento B 107, 91 (1992).
This is analogous to the connection representation in which the ``triad density" is represented by the functional derivative ${}^{underline {*}}{underline}$ { vartheta $}^{B}$ := δ / δ { buildrel pm over { underline{A $}_{B}$ }} with respect to the canonically conjugate Ashtekar variable buildrel pm over{ underline{A $}_{B}$ }. This provides a mapping from the Hamiltonian constraint of gravity with a cosmological term to the Chern Simons three form [7] of the Ashtekar Sen connection, see B. Brügmann, R. Gambini and J. Pullin, Nucl. Phys. B385, 587 (1992).
E. Witten, Nucl. Phys. B311, 46 (1988/89)/89.
S. Kobayashi, Transformation Groups in Differential Geometry (Springer, New York, 1972).
E.W. Mielke, Geometrodynamics of Gauge Fields On the Geometry of Yang Mills and Gravitational Gauge Theories (Akademie Verlag, Berlin, 1987).
In a fiber bundle approach, one introduces first the bundle of affine frames A(M):= P( $M^{n}$ , A(n,R), π, δ ) where π denotes the projection to the base manifold and δ the (left) action of the structure group A(n,R) on the bundle. Active affine gauge transformations are the vertical automorphisms of A(M). Similarly as the diffeomorphisms of the base manifold $M^{n}$ , they form the infinite dimensional group { cal A}(n, R):= $C^{\infty}$ ( A(M) ${times}_{Ad}$ A(n, R)). The group { cal G}{ cal L}(n, R):= $C^{\infty}$ (A(M) ${times}_{Ad}$ GL (n, R)) of linear gauge transformations and the group { cal T}(n, R):= $C^{\infty}$ (A(M) ${times}_{Ad}$ $R^{n}$ ) of local translations are subgroups of { cal A}(n, R). Taking the cross section in the associated bundle is abbreviated by $C^{\infty}$ and Ad denotes the adjoint representation with respect to GL(n,R). Because of its construction, the group of local translations { cal T}(n, R) is locally isomorphic to the group of active diffeomorphisms Diff(n,R) of the manifold, cf. V.I. Ogievetsky, Lett. Nuovo Cimento 8, 988 (1973), and S. Sternberg, Ann. Phys. (N.Y.) 162, 85 (1985). The infinite dimensional group Diff(n,R) contains the (n+ $n^{2}$ ) dimensional group A(n,R $)_{H}$ of holonomic affine transformations as a subgroup, cf. P.G. Bergmann and A.B. Komar, J. Math. Phys. 26, 2030 (1985), which is generated by the fields $P_{i}$ =- ${partial}_{i}$ and $L^{i}$ { $}_{j}$ =- $x^{i}$ ${partial}_{j}$ . Note that differentiable coordinate transformations, which leave exterior forms invariant, are regarded as passive diffeomorphisms.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1963), Vol. I.
L.K. Norris, R.O. Fulp and W.R. Davis, Phys. Lett. 79A, 278 (1980); V.N. Ponomariev, A.O. Bravinsky, and Yu.N. Obukhov, Geometrodynamical Methods and Gauge Approach to the Theory of Gravitational Interactions, in Russian (Energoatomisdat, Moscow, 1985).
Omitted end note. and references therein, is a spacetime generalization of the so called motor calculus of R. von Mises, Z. Angew. Math. Mech. 4, 155 (1924).
A. Trautman, in Differential Geometry, Symposia Matematica Vol. 12 (Academic, London, 1973), p. 139.
E. Cartan, On a Manifold with an Affine Connection and the Theory of General Relativity (Bibliopolis, Napoli, 1986).
Otherwise there also exists, for example, the nontrivial solution $T^{α}$ =a $η^{α β}$ $ξ_{β}$ and $R^{α β}$ =a $η^{α β}$ , with the dimensionful constant a.
K. Hayashi and T. Nakano, Prog. Theor. Phys. 38, 491 (1967); K. Hayashi and T. Shirafuji, ibid. 64, 866 (1980); ibid. 80, 711 (1988).
G. Grignani and G. Nardelli, Phys. Rev. D 45, 2719 (1992).
R.U. Sexl and H.K. Urbantke, Gravitation und Kosmologie, 2nd ed. (Bibliographisches Institut, Mannheim, 1983).
G. Sardanashvily and M. Gogbershvily, Mod. Phys. Lett. A 2, 609 (1987).
J. Hennig and J. Nitsch, Gen. Relativ. Gravit. 13, 947 (1981).
A. Trautman, Czech. J. Phys. B 29, 107 (1979).
K.A. Pilch, Lett. Math. Phys. 4, 49 (1980).
P.K. Smrz, J. Math. Phys. 28, 2824 (1987).
L. O'Raifeartaigh, in Differential Geometry, Group Representations, and Quantization, edited by J.D. Hennig, W. Lücke, and J. Tolar, Lecture Notes in Physics, Vol. 379 (Springer, Berlin, 1991), p. 99.
F.W. Hehl, J.D. McCrea, E.W. Mielke and Y. Ne'eman, Found. Phys. 19, 1075 (1989); Phys. Rep. (to be published); J.D. McCrea, Class. Quantum Grav. 9, 553 (1992).
K.S. Stelle and P.C. West, Phys. Rev. D 21, 1466 (1980).
S. Gotzes and A.C. Hirshfeld, Ann. Phys. (N.Y.) 203, 410 (1990).
J.D. Hennig, in Conformal Geometry and Spacetime Gauge Theories, Proceedings of the 2nd International Wigner Conference, Goslar, 1991, edited by H.D. Doebner et al. (World Scientific, Singapore, 1992).
The plus sign is in accordance with our earlier conventions for active trans for mations.
R.D. Hecht, F.W. Hehl, J.D. McCrea, E.W. Mielke and Y. Ne'eman, Phys. Lett. A 172, 13 (1992).
Y. Ne'eman, in Differential Geometrical Methods in Mathematical Physics, edited by K. Bleuler, H.R. Petry, and A. Reetz, Lecture Notes in Mathematics, Vol. 676 (Springer Verlag, Berlin, 1978), p. 189.
J. M. Nester, in An Introduction to Kaluza Klein Theories, Proceedings of the Workshop, Chalk River, Ontario, 1983, edited by H.C. Lee (World Scientific, Singapore, 1984), p. 83.
T.W.B. Kibble and K.S. Stelle, in Progress in Quantum Field Theory: Festschrift for Umezawa, edited by H. Ezawa and S. Kamefuchi (Elsevier, New York, 1986), p. 57.
G. Stephenson, Nuovo Cimento 9, 263 (1958).
C.W. Kilmister and D.J. Newman, Proc. Cambridge Philos. Soc. 57, 851 (1961).
C.N. Yang, Phys. Rev. Lett. 33, 445 (1974).
K.S. Stelle, Phys. Rev. D 16, 953 (1977).
C. Y. Lee and Y. Ne'eman, Phys. Lett. B 242, 59 (1990); C. Y. Lee, Class. Quantum Grav. 9, 2001 (1992).
T.T. Burwick, A.H. Chamseddine and K.A. Meissner, Phys. Lett. B 284, 11 (1992).
G. Grignani and G. Nardelli, Phys. Lett. B 300, 38 (1993).
S.B. Giddings, Phys. Lett. B 268, 17 (1991).
J. Greensite, Phys. Lett. B 300, 34 (1993).
A.D. Sakharov, Zh. Eksp. Teor. Fiz. 87, 375 (1984) [Sov. Phys. JETP 60, 214 (1984)].
E.W. Mielke, Gen. Relativ. Gravit. 8, 175 (1977).
M.Yu. Konstantinov and V.N. Melnikov, Class. Quantum Grav. 3, 401 (1986).

[c1] R.P. Feynman, Lectures on Gravitation, compiled by F.B. Morinigo and W.G. Wagner (California Institute of Technology, Pasadena, California, 1962/63).

[c2] F.W. Hehl, P. von der Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48, 393 (1976); see also F.W. Hehl, in Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity, Proceedings of the International School, Erice, Italy, 1979, edited by P.G. Bergmann and V. de Sabbata,NATO Advanced Study Institute Series B: Physics, Vol. 58 (Plenum, New York, 1980), p. 5.

[c3] G.T. Horowitz, Class. Quantum Grav. 8, 587 (1991).

[c4] C.J. Isham, in Recent Aspects of Quantum Fields, Proceedings of the XXXth International Univer sitäts wo chen für Kernphysik, Schladming, Austria, 1991, edited by H. Mitter and H. Gausterer,Lecture Notes in Physics, Vol. 396 (Springer, Berlin, 1991), p. 123.

[c5] E.W. Mielke, Ann. Phys. (N.Y.) 219, 78 (1992).

[c6] P. Baekler, E.W. Mielke and F.W. Hehl, Nuovo Cimento B 107, 91 (1992).

[c7] This is analogous to the connection representation in which the ``triad density" is represented by the functional derivative ${}^{underline {*}}{underline}$ { vartheta $}^{B}$ := δ / δ { buildrel pm over { underline{A $}_{B}$ }} with respect to the canonically conjugate Ashtekar variable buildrel pm over{ underline{A $}_{B}$ }. This provides a mapping from the Hamiltonian constraint of gravity with a cosmological term to the Chern Simons three form [7] of the Ashtekar Sen connection, see B. Brügmann, R. Gambini and J. Pullin, Nucl. Phys. B385, 587 (1992).

[c8] E. Witten, Nucl. Phys. B311, 46 (1988/89)/89.

[c9] S. Kobayashi, Transformation Groups in Differential Geometry (Springer, New York, 1972).

[c10] E.W. Mielke, Geometrodynamics of Gauge Fields On the Geometry of Yang Mills and Gravitational Gauge Theories (Akademie Verlag, Berlin, 1987).

[c11] In a fiber bundle approach, one introduces first the bundle of affine frames A(M):= P( $M^{n}$ , A(n,R), π, δ ) where π denotes the projection to the base manifold and δ the (left) action of the structure group A(n,R) on the bundle. Active affine gauge transformations are the vertical automorphisms of A(M). Similarly as the diffeomorphisms of the base manifold $M^{n}$ , they form the infinite dimensional group { cal A}(n, R):= $C^{\infty}$ ( A(M) ${times}_{Ad}$ A(n, R)). The group { cal G}{ cal L}(n, R):= $C^{\infty}$ (A(M) ${times}_{Ad}$ GL (n, R)) of linear gauge transformations and the group { cal T}(n, R):= $C^{\infty}$ (A(M) ${times}_{Ad}$ $R^{n}$ ) of local translations are subgroups of { cal A}(n, R). Taking the cross section in the associated bundle is abbreviated by $C^{\infty}$ and Ad denotes the adjoint representation with respect to GL(n,R). Because of its construction, the group of local translations { cal T}(n, R) is locally isomorphic to the group of active diffeomorphisms Diff(n,R) of the manifold, cf. V.I. Ogievetsky, Lett. Nuovo Cimento 8, 988 (1973), and S. Sternberg, Ann. Phys. (N.Y.) 162, 85 (1985). The infinite dimensional group Diff(n,R) contains the (n+ $n^{2}$ ) dimensional group A(n,R $)_{H}$ of holonomic affine transformations as a subgroup, cf. P.G. Bergmann and A.B. Komar, J. Math. Phys. 26, 2030 (1985), which is generated by the fields $P_{i}$ =- ${partial}_{i}$ and $L^{i}$ { $}_{j}$ =- $x^{i}$ ${partial}_{j}$ . Note that differentiable coordinate transformations, which leave exterior forms invariant, are regarded as passive diffeomorphisms.

[c12] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1963), Vol. I.

[c13] L.K. Norris, R.O. Fulp and W.R. Davis, Phys. Lett. 79A, 278 (1980); V.N. Ponomariev, A.O. Bravinsky, and Yu.N. Obukhov, Geometrodynamical Methods and Gauge Approach to the Theory of Gravitational Interactions, in Russian (Energoatomisdat, Moscow, 1985).

[c14] Omitted end note. and references therein, is a spacetime generalization of the so called motor calculus of R. von Mises, Z. Angew. Math. Mech. 4, 155 (1924).

[c15] A. Trautman, in Differential Geometry, Symposia Matematica Vol. 12 (Academic, London, 1973), p. 139.

[c16] E. Cartan, On a Manifold with an Affine Connection and the Theory of General Relativity (Bibliopolis, Napoli, 1986).

[c17] Otherwise there also exists, for example, the nontrivial solution $T^{α}$ =a $η^{α β}$ $ξ_{β}$ and $R^{α β}$ =a $η^{α β}$ , with the dimensionful constant a.

[c18] K. Hayashi and T. Nakano, Prog. Theor. Phys. 38, 491 (1967); K. Hayashi and T. Shirafuji, ibid. 64, 866 (1980); ibid. 80, 711 (1988).

[c19] G. Grignani and G. Nardelli, Phys. Rev. D 45, 2719 (1992).

[c20] R.U. Sexl and H.K. Urbantke, Gravitation und Kosmologie, 2nd ed. (Bibliographisches Institut, Mannheim, 1983).

[c21] G. Sardanashvily and M. Gogbershvily, Mod. Phys. Lett. A 2, 609 (1987).

[c22] J. Hennig and J. Nitsch, Gen. Relativ. Gravit. 13, 947 (1981).

[c23] A. Trautman, Czech. J. Phys. B 29, 107 (1979).

[c24] K.A. Pilch, Lett. Math. Phys. 4, 49 (1980).

[c25] P.K. Smrz, J. Math. Phys. 28, 2824 (1987).

[c26] L. O'Raifeartaigh, in Differential Geometry, Group Representations, and Quantization, edited by J.D. Hennig, W. Lücke, and J. Tolar, Lecture Notes in Physics, Vol. 379 (Springer, Berlin, 1991), p. 99.

[c27] F.W. Hehl, J.D. McCrea, E.W. Mielke and Y. Ne'eman, Found. Phys. 19, 1075 (1989); Phys. Rep. (to be published); J.D. McCrea, Class. Quantum Grav. 9, 553 (1992).

[c28] K.S. Stelle and P.C. West, Phys. Rev. D 21, 1466 (1980).

[c29] S. Gotzes and A.C. Hirshfeld, Ann. Phys. (N.Y.) 203, 410 (1990).

[c30] J.D. Hennig, in Conformal Geometry and Spacetime Gauge Theories, Proceedings of the 2nd International Wigner Conference, Goslar, 1991, edited by H.D. Doebner et al. (World Scientific, Singapore, 1992).

[c31] The plus sign is in accordance with our earlier conventions for active trans for mations.

[c32] R.D. Hecht, F.W. Hehl, J.D. McCrea, E.W. Mielke and Y. Ne'eman, Phys. Lett. A 172, 13 (1992).

[c33] Y. Ne'eman, in Differential Geometrical Methods in Mathematical Physics, edited by K. Bleuler, H.R. Petry, and A. Reetz, Lecture Notes in Mathematics, Vol. 676 (Springer Verlag, Berlin, 1978), p. 189.

[c34] J. M. Nester, in An Introduction to Kaluza Klein Theories, Proceedings of the Workshop, Chalk River, Ontario, 1983, edited by H.C. Lee (World Scientific, Singapore, 1984), p. 83.

[c35] T.W.B. Kibble and K.S. Stelle, in Progress in Quantum Field Theory: Festschrift for Umezawa, edited by H. Ezawa and S. Kamefuchi (Elsevier, New York, 1986), p. 57.

[c36] G. Stephenson, Nuovo Cimento 9, 263 (1958).

[c37] C.W. Kilmister and D.J. Newman, Proc. Cambridge Philos. Soc. 57, 851 (1961).

[c38] C.N. Yang, Phys. Rev. Lett. 33, 445 (1974).

[c39] K.S. Stelle, Phys. Rev. D 16, 953 (1977).

[c40] C. Y. Lee and Y. Ne'eman, Phys. Lett. B 242, 59 (1990); C. Y. Lee, Class. Quantum Grav. 9, 2001 (1992).

[c41] T.T. Burwick, A.H. Chamseddine and K.A. Meissner, Phys. Lett. B 284, 11 (1992).

[c42] G. Grignani and G. Nardelli, Phys. Lett. B 300, 38 (1993).

[c43] S.B. Giddings, Phys. Lett. B 268, 17 (1991).

[c44] J. Greensite, Phys. Lett. B 300, 34 (1993).

[c45] A.D. Sakharov, Zh. Eksp. Teor. Fiz. 87, 375 (1984) [Sov. Phys. JETP 60, 214 (1984)].

[c46] E.W. Mielke, Gen. Relativ. Gravit. 8, 175 (1977).

[c47] M.Yu. Konstantinov and V.N. Melnikov, Class. Quantum Grav. 3, 401 (1986).

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Avoiding degenerate coframes in an affine gauge approach to quantum gravity

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