The difference between Einstein's general relativity and its Cartan extension is analyzed within the scenario of asymptotic safety of quantum gravity. In particular, we focus on the four-fermion interaction which distinguishes the Einstein-Cartan theory from its Riemannian limit. 1. Introduction In the coupling of gravity to Dirac type spinor fields [1], it is at times surmised that the Einstein-Cartan (EC) theory [2] is superior to standard General Relativity (GR), inasmuch as the involved torsion tensor of Cartan [3, 4] can accommodate the spin of fundamental Fermions of electrons and quarks in gravity. However, classically,the effects of spin and torsion cannot be detected by Lageos or Gravity Probe B [5] and would be significant only at densities of matter that are very high but nevertheless smaller than the Planck density at which quantum gravitational effects are believed to dominate. It was even claimed [6] that EC theory may avert the problem of singularities in cosmology, but for a coupling to Dirac fields, the opposite happens [7–9]. Recently, it has been stressed by Weinberg [10–12] that the Riemann-Cartan (RC) connection , a one-form, is just a deformation of the Christoffel connection by the (con-)tortion tensor-valued one-form , at least from the field theoretical point of view. Although algebraically complying with [13], this argument has been refuted [14] on the basis of the special geometrical interpretation [15, 16] of Cartan’s torsion. It is well-known [17, 18] that EC theory coupled to the Dirac field is effectively GR with an additional four-fermion (FF) interaction. However, such contact interactions are perturbatively nonrenormalizable in without Chern-Simons (CS) terms [19], which was one of the reasons for giving up Fermi’s theory of the beta decay. Since GR with a cosmological constant appears to be asymptotically safe, in the scenario [20] first devised by Weinberg [21], one may ask [22] what the situation in EC theory is, where Cartan’s algebraic equation relates torsion to spin, that is, to the axial current in the case of Dirac fields, on dimensional grounds coupled with gravitational strength. 2. Dirac Fields in Riemann-Cartan Spacetime In our notation [13, 23–25], a Dirac field is a bispinor-valued zero-form for which denotes the Dirac adjoint and is the exterior covariant derivative with respect to the RC connection one-form , providing a minimal gravitational coupling. In the manifestly Hermitian formulation, the Dirac Lagrangian is given by the four-form where is the Clifford algebra-valued coframe, obeying , and is
References
[1]
H. Weyl, “Gravitation and the electron,” Proceedings of the National Academy of Sciences of the United States of America, vol. 15, no. 4, pp. 323–334, 1929.
[2]
A. Trautman, “On the structure of the Einstein-Cartan equations,” in Differential Geometry, vol. 12 of Symposia Mathematica, pp. 139–1162, Academic Press, London, UK, 1973.
[3]
é. Cartan, On Manifolds with an Affine Connection and the Theory of General Relativity, 1924, English translation of the French original, Bibliopolis, Napoli, Italy, 1986.
[4]
F. W. Hehl, “Book review,” General Relativity and Gravitation, vol. 21, no. 3, pp. 315–317, 1989.
[5]
C. M. Will, “Finally, results from Gravity Probe B,” Physics, vol. 4, p. 43, 2011.
[6]
A. Trautman, “Spin and torsion may avert gravitational singularities,” Nature Physical Science, vol. 242, pp. 7–8, 1973.
[7]
R. F. O'Connell, “Contact interactions in the Einstein and Einstein-Cartan-Sciama-Kibble (ECSK) theories of gravitation,” Physical Review Letters, vol. 37, no. 25, pp. 1653–1655, 1976.
[8]
R. F. O'Connell, “Contact interactions in the Einstein and Einstein-Cartan-Sciama-Kibble (ECSK) theories of gravitation,” Physical Review Letters, vol. 38, no. 6, p. 298, 1977.
[9]
R. F. O'Connell, “Attractive spin-spin contact interactions in the Einstein-Cartan-Sciama-Kibble torsion theory of gravitation,” Physical Review D, vol. 16, no. 4, p. 1247, 1977.
[10]
S. Weinberg, “Einstein's mistakes,” Physics Today, vol. 58, no. 11, pp. 31–35, 2005.
[11]
S. Weinberg, “Weinberg replies,” Physics Today, vol. 59, p. 1516, 2006.
[12]
F. W. Hehl and S. Weinberg, “Note on the torsion tensor,” Physics Today, vol. 60, no. 3, 2007.
[13]
F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne'eman, “Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance,” Physics Report, vol. 258, no. 1-2, pp. 1–171, 1995.
[14]
M. Blagojevic and F. W. Hehl, Eds., Gauge Theories of Gravitation: A Reader with Commentaries, Imperial College Press, London, UK, 2013.
[15]
D. K. Wise, “MacDowell-Mansouri gravity and Cartan geometry,” Classical and Quantum Gravity, vol. 27, no. 15, Article ID 155010, pp. 1–26, 2010.
[16]
H. F. Westman and T. G. Zlosnik, “Cartan gravity, matter fields, and the gauge principle,” Annals of Physics, vol. 334, pp. 157–197, 2013.
[17]
F. W. Hehl and B. K. Datta, “Nonlinear spinor equation and asymmetric connection in general relativity,” Journal of Mathematical Physics, vol. 12, no. 7, pp. 1334–1339, 1971.
[18]
E. W. Mielke and E. Sanchez Romero, “Cosmological evolution of a torsion-induced quintaxion,” Physical Review D, vol. 73, no. 4, Article ID 043521, 2006.
[19]
V. S. Alves, M. Gomes, S. V. L. Pinheiro, and A. J. da Silva, “Four-fermion field theories and the Chern-Simons field: a renormalization group study,” Physical Review D, vol. 60, no. 2, pp. 1–4, 1999.
[20]
M. Reuter and F. Saueressig, “Quantum Einstein gravity,” New Journal of Physics, vol. 14, Article ID 055022, 2012.
[21]
S. Weinberg, “Ultraviolet divergences in quantum theories of gravitation,” in General Relativity: An Einstein Centenary Survey, S. W. Hawking and W. Israel, Eds., chapter 16, p. 790, Cambridge University Press, 1979.
[22]
E. W. Mielke, “Asymptotic safety of the Cartan induced four-fermion interaction?” in Proceedings of the Thirteenth Marcel Grossman Meeting on General Relativity, K. Rosquist, R. T. Jantzen, and R. Ruffini, Eds., pp. 1–3, World Scientific, Singapore, 2013.
[23]
F. W. Hehl, J. Lemke, and E. W. Mielke, “Two lectures on fermions and gravity,” in Geometry and Theoretical Physics, J. Debrus and A. C. Hirshfeld, Eds., pp. 56–140, Springer, Berlin, Germany, 1991.
[24]
J. Lemke, E. W. Mielke, and F. W. Hehl, “？quivalenzprinzip für Materiewellen? Experimente mit Neutronen, Atomen, Neutrinos…,” Physik in unserer Zeit, vol. 25, no. 1, pp. 36–43, 1994.
[25]
E. W. Mielke, “Consistent coupling to Dirac fields in teleparallelism: comment on ‘metric-afflne approach to teleparallel gravity’,” Physical Review D, vol. 69, no. 12, Article ID 128501, 2004.
[26]
D. Kreimer and E. W. Mielke, “Comment on ‘topological invariants, instantons, and the chiral anomaly on spaces with torsion’,” Physical Review D, vol. 63, no. 4, Article ID 048501, 2001.
[27]
S. Yajima and T. Kimura, “Anomalies in four-dimensional curved space with torsion,” Progress of Theoretical Physics, vol. 74, no. 4, pp. 866–880, 1985.
[28]
E. W. Mielke, “Anomalies and Gravity,” in Particles and Fields, Commemorative Volume of the Division of Particles and Fields of the Mexican Physical Society, Morelia, Michoacán, 6–12 November 2005, Part B, M. A. Pérez, L. F. Urrutia, and L. Villase？or, Eds., vol. 857 of AIP Conference Proceedings, pp. 246–257, American Institute of Physics, Melville, NY, USA, 2006.
[29]
E. W. Mielke, “Chern-Simons solutions of the chiral teleparallelism constraints of gravity,” Nuclear Physics B, vol. 622, no. 1-2, pp. 457–471, 2002.
[30]
E. W. Mielke, “Einstein-Weyl gravity from a topological SL(5, R) gauge invariant action,” Advances in Applied Clifford Algebras, vol. 22, pp. 803–817, 2012.
[31]
E. W. Mielke, “Symmetry breaking in topological quantum gravity,” International Journal of Modern Physics D, vol. 22, no. 5, Article ID 1330009, 2013.
[32]
J.-E. Daum and M. Reuter, “Renormalization group flow of the Holst action,” Physics Letters B, vol. 710, no. 1, pp. 215–218, 2012.
[33]
J.-E. Daum and M. Reuter, “Einstein-Cartan gravity, asymptotic safety, and the running Immirzi parameter,” Annals of Physics, vol. 334, pp. 351–419, 2013.
[34]
E. W. Mielke, “Topologically modified teleparallelism, passing through the Nieh-Yan functional,” Physical Review D, vol. 80, Article ID 067502, 4 pages, 2009.
[35]
E. W. Mielke, “Ashtekar's complex variables in general relativity and its teleparallelism equivalent,” Annals of Physics, vol. 219, no. 1, pp. 78–108, 1992.
[36]
R. D. Hecht, J. M. Nester, and V. V. Zhytnikov, “Some Poincaré gauge theory Lagrangians with well-posed initial value problems,” Physics Letters A, vol. 222, no. 1-2, pp. 37–42, 1996.
[37]
F.-H. Ho and J. M. Nester, “Poincaré gauge theory with coupled even and odd parity spin-0 modes: cosmological normal modes,” Annalen der Physik, vol. 524, no. 2, pp. 97–106, 2012.
[38]
J. Bjorken, “Darkness: what comprises empty space?” Annalen der Physik, vol. 525, no. 5, pp. A67–A79, 2013.
[39]
K. Falls, D. F. Litim, K. Nikolakopoulos, and C. Rahmede, “A bootstrap towards asymptotic safety,” http://arxiv.org/abs/1301.4191.
[40]
F. E. Schunck, F. V. Kusmartsev, and E. W. Mielke, “Dark matter problem and effective curvature Lagrangians,” General Relativity and Gravitation, vol. 37, no. 8, pp. 1427–1433, 2005.
[41]
R. Kuhfuss and J. Nitsch, “Propagating modes in gauge field theories of gravity,” General Relativity and Gravitation, vol. 18, no. 12, pp. 1207–1227, 1986.
[42]
C.-Y. Lee and Y. Ne'eman, “Renormalization of gauge-affine gravity,” Physics Letters B, vol. 242, no. 1, pp. 59–63, 1990.
[43]
A. Eichhorn and H. Gies, “Light fermions in quantum gravity,” New Journal of Physics, vol. 13, Article ID 125012, 2011.
[44]
M.-J. Schwindt and C. Wetterich, “Asymptotically free four-fermion interactions and electroweak symmetry breaking,” Physical Review D, vol. 81, no. 5, Article ID 055005, 2010.
[45]
F. Bazzocchi, M. Fabbrichesi, R. Percacci, A. Tonero, and L. Vecchi, “Fermions and Goldstone bosons in an asymptotically safe model,” Physics Letters B, vol. 705, no. 4, pp. 388–392, 2011.
[46]
D. Benedetti and S. Speziale, “Perturbative quantum gravity with the Immirzi parameter,” Journal of High Energy Physics, vol. 2011, no. 6, article 107, 2011.
[47]
R. Hojman, C. Mukku, and W. A. Sayed, “Parity violation in metric-torsion theories of gravitation,” Physical Review D, vol. 22, no. 8, pp. 1915–1921, 1980.
[48]
Y. N. Obukhov and F. W. Hehl, “Extended Einstein-Cartan theory a la Diakonov: the field equations,” Physics Letters B, vol. 713, p. 321, 2012.
[49]
N. J. Pop？awski, “Four-fermion interaction from torsion as dark energy,” General Relativity and Gravitation, vol. 44, no. 2, pp. 491–499, 2012.
[50]
I. B. Khriplovich, “Gravitational four-fermion interaction on the Planck scale,” Physics Letters B, vol. 709, no. 3, pp. 111–113, 2012.
[51]
I. B. Khriplovich and A. S. Rudenko, “Cosmology constrains gravitational four-fermion interaction,” Journal of Cosmology and Astroparticle Physics, vol. 2012, article 040, 2012.
[52]
H. Nicolai, “Quantum gravity: the view from particle physics,” http://arxiv.org/abs/1301.5481.
