### Abstract

“It would be permissible to look upon the Hamiltonian form as the fundamental one, and there would then be no fundamental four-dimensional symmetry in the theory. One would have a Hamiltonian built up from four weekly [sic] vanishing functions, given by [the Hamiltonian and diffeomorphism constraints]. The usual requirement of four-dimensional symmetry in physical laws would then get replaced by the requirement that the functions have weakly vanishing P.B.'s, so that they can be provided with arbitrary coefficients in the equations of motion, corresponding to an arbitrary motion of the surface on which the state is defined.” P. A. M. Dirac, in “The theory of gravitation in Hamiltonian form,” Proc. Roy. Soc. A 246 (1958) 333-43. Introduction In its different incarnations, quantum gravity must face a diverse set of fascinating problems and difficulties, a set of issues best seen as both challenges and opportunities. One of the main problems in canonical approaches, for instance, is the issue of anomalies in the gauge algebra underlying space-time covariance. Classically, the gauge generators, given by constraints, have weakly vanishing Poisson brackets with each other: they vanish when the constraints are satisfied. After quantization, the same behavior must be realized for commutators of quantum constraints (or for Poisson brackets of effective constraints), or else the theory becomes inconsistent due to gauge anomalies. If and how canonical quantum gravity can be obtained in an anomaly-free way is an important question, not yet convincingly addressed in full generality.

Original language | English (US) |
---|---|

Title of host publication | Foundations of Space and Time |

Subtitle of host publication | Reflections on Quantum Gravity |

Publisher | Cambridge University Press |

Pages | 211-256 |

Number of pages | 46 |

Volume | 9780521114400 |

ISBN (Electronic) | 9780511920998 |

ISBN (Print) | 9780521114400 |

DOIs | |

State | Published - Jan 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

*Foundations of Space and Time: Reflections on Quantum Gravity*(Vol. 9780521114400, pp. 211-256). Cambridge University Press. https://doi.org/10.1017/CBO9780511920998.011

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*Foundations of Space and Time: Reflections on Quantum Gravity.*vol. 9780521114400, Cambridge University Press, pp. 211-256. https://doi.org/10.1017/CBO9780511920998.011

**Loop quantum gravity and cosmology.** / Bojowald, Martin.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Loop quantum gravity and cosmology

AU - Bojowald, Martin

PY - 2012/1/1

Y1 - 2012/1/1

N2 - “It would be permissible to look upon the Hamiltonian form as the fundamental one, and there would then be no fundamental four-dimensional symmetry in the theory. One would have a Hamiltonian built up from four weekly [sic] vanishing functions, given by [the Hamiltonian and diffeomorphism constraints]. The usual requirement of four-dimensional symmetry in physical laws would then get replaced by the requirement that the functions have weakly vanishing P.B.'s, so that they can be provided with arbitrary coefficients in the equations of motion, corresponding to an arbitrary motion of the surface on which the state is defined.” P. A. M. Dirac, in “The theory of gravitation in Hamiltonian form,” Proc. Roy. Soc. A 246 (1958) 333-43. Introduction In its different incarnations, quantum gravity must face a diverse set of fascinating problems and difficulties, a set of issues best seen as both challenges and opportunities. One of the main problems in canonical approaches, for instance, is the issue of anomalies in the gauge algebra underlying space-time covariance. Classically, the gauge generators, given by constraints, have weakly vanishing Poisson brackets with each other: they vanish when the constraints are satisfied. After quantization, the same behavior must be realized for commutators of quantum constraints (or for Poisson brackets of effective constraints), or else the theory becomes inconsistent due to gauge anomalies. If and how canonical quantum gravity can be obtained in an anomaly-free way is an important question, not yet convincingly addressed in full generality.

AB - “It would be permissible to look upon the Hamiltonian form as the fundamental one, and there would then be no fundamental four-dimensional symmetry in the theory. One would have a Hamiltonian built up from four weekly [sic] vanishing functions, given by [the Hamiltonian and diffeomorphism constraints]. The usual requirement of four-dimensional symmetry in physical laws would then get replaced by the requirement that the functions have weakly vanishing P.B.'s, so that they can be provided with arbitrary coefficients in the equations of motion, corresponding to an arbitrary motion of the surface on which the state is defined.” P. A. M. Dirac, in “The theory of gravitation in Hamiltonian form,” Proc. Roy. Soc. A 246 (1958) 333-43. Introduction In its different incarnations, quantum gravity must face a diverse set of fascinating problems and difficulties, a set of issues best seen as both challenges and opportunities. One of the main problems in canonical approaches, for instance, is the issue of anomalies in the gauge algebra underlying space-time covariance. Classically, the gauge generators, given by constraints, have weakly vanishing Poisson brackets with each other: they vanish when the constraints are satisfied. After quantization, the same behavior must be realized for commutators of quantum constraints (or for Poisson brackets of effective constraints), or else the theory becomes inconsistent due to gauge anomalies. If and how canonical quantum gravity can be obtained in an anomaly-free way is an important question, not yet convincingly addressed in full generality.

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U2 - 10.1017/CBO9780511920998.011

DO - 10.1017/CBO9780511920998.011

M3 - Chapter

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SN - 9780521114400

VL - 9780521114400

SP - 211

EP - 256

BT - Foundations of Space and Time

PB - Cambridge University Press

ER -