# Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity?

## Abstract

**:**

## 1. Introduction

## 2. Reduction and Quantization

#### 2.1. Symplectic Reduction and Geometric Quantization

#### 2.2. Extended Phase Space Quantization and Quantum Constraint Reduction

#### 2.3. Does Quantisation Commute with Reduction?

## 3. Canonical General Relativity; Reduction and Quantization

#### 3.1. The ADM Formalism and the Constraint Algebra

#### 3.2. Reduction and Quantisation of the Momentum Constraints

#### 3.3. Classical Hamiltonian Constraints, Reduction and the Problem of Time

#### 3.4. How Should We Interpret the Quantum Hamiltonian Constraints?

## 4. Conclusions

## Acknowledgements

## References

- Dirac, P.A.M. Lecture on Quantum Mechanics; Dover: Mineola, NY, USA, 1964. [Google Scholar]
- Henneaux, M.; Teitelboim, C. Quantization of Gauge Systems; Princeton University Press: Princeton, NJ, USA, 1992. [Google Scholar]
- Butterfield, J. On symplectic reduction in classical mechanics. In Handbook of Philosophy of Physics; Butterfield, J., Earman, J., Eds.; Elsevier: Oxford, UK, 2007; pp. 1–132. [Google Scholar]
- Gotay, M.J.; Nester, J.M.; Hinds, G. Presymplectic manifolds and the Dirac-Bergman theory of constraints. J. Math. Phys.
**1978**, 19, 2388–2399. [Google Scholar] [CrossRef] - Souriau, J.M. Structure of Dynamical Systems: A Symplectic View of Physics; Cushman-de Vries, C.H., Trans., Cushman, R.H., Tuynman, G.M., Eds.; Birkhauser: Cambridge, MA, USA, 2007. [Google Scholar]
- Echeverria-Enriquez, A.; Munoz-Lecanda, M. Mathematical foundations of geometric quantization, 1999. Available at: http://arxiv.org/abs/math-ph/9904008 (accessed on 31 March 2011).
- Landsman, N.P. Between classical and quantum. In Handbook of Philosophy of Physics; Butterfield, J., Earman, J., Eds.; Elsevier: Oxford, UK, 2007; pp. 417–554. [Google Scholar]
- Guillemin, V.; Sternberg, S. Geometric quantization and multiplicities of group representations. Invent. Math.
**1982**, 67, 515–538. [Google Scholar] [CrossRef] - Gotay, M. Constraints, reduction, and quantization. J. Math. Phys.
**1986**, 27, 2051–2066. [Google Scholar] [CrossRef] - Duistermaat, H.; Guillemin, V.; Meinrenken, E.; Wu, S. Symplectic reduction and Riemann-Roch for circle actions. Math. Res. Lett.
**1995**, 2, 259–266. [Google Scholar] [CrossRef] - Conrady, F.; Freidel, L. Quantum geometry from phase space reduction. J. Math. Phys.
**2009**, 50, 123510–123539. [Google Scholar] [CrossRef] - Kuchař, K. The problem of time in canonical quantization of relativistic systems. In Conceptual Problems of Quantum Gravity; Ashtekar, A., Stachel, J., Eds.; Springer: New York, NY, USA, 1991; pp. 141–171. [Google Scholar]
- Isham, C. Canonical quantum gravity and the problem of time, 1992. Available at: http://arxiv.org/abs/ grqc/9210011 (accessed on 31 March 2011).
- Anderson, E. The Problem of Time in Quantum Gravity, 2010. Available at: http://arxiv.org/pdf/ 1009.2157 (accessed on 31 March 2011).
- Pons, J.M.; Salisbury, D.C.; Sundermeyer, K.A. Observables in classical canonical gravity: folklore demystified. J. Phy. Conf. Ser.
**2010**, 222, 012018. [Google Scholar] [CrossRef] - Kuchař, K. Time and interpretations of quantum gravity. In Proceedings of the Fourth Canadian Conference on General Relativity and Relativistic Astrophysics, University of Winnipeg, 16-18 May, 1991; Kunstatter, G., Vincent, D., Williams, J., Eds.; World Scientific: Singapore, 1992; pp. 211–314. [Google Scholar]
- Barbour, J. The timelessness of quantum gravity: I. The evidence from the classical theory. Class. Quant. Grav.
**1994**, 1, 2853–2874. [Google Scholar] [CrossRef] - Thiemann, T. Modern Canonical Quantum General Relativity; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Giulini, D.; Marolf, D. On the generality of refined algebraic quantization. Class. Quant. Grav.
**1999**, 16, 2479–2488. [Google Scholar] [CrossRef] - Giulini, D.; Marolf, D. A uniqueness theorem for constraint quantization. Class. Quant. Grav.
**1999**, 16, 2489–2506. [Google Scholar] [CrossRef] - Thiemann, T. The Phoneix Project: Master Constraint Programme, 2003. Available at: http://arxiv.org/abs/ gr-qc/0305080 (accessed on 31 March 2011).
- Dittrich, B.; Thiemann, T. Testing the master constraint programme for loop quantum gravity: I. General framework. Class. Quant. Grav.
**2006**, 23, 1025–1065. [Google Scholar] [CrossRef] - Walters, P. An Introduction to Ergodic Theory; Springer-Verlag: New York, NY, USA, 1981. [Google Scholar]
- Corichi, A. On the geometry of quantum constrained systems, 2008. Available at: http://arxiv.org/ abs/0801.1119 (accessed on 31 March 2011).
- Arnowitt, R.; Deser, S.; Misner, C.W. The Dynamics of General Relativity. In Gravitation: An Introduction to Current Research; Witten, L., Ed.; Wiley: New York, NY, USA, 1962; pp. 227–239. [Google Scholar]
- Isham, C.; Kuchař, K. Representations of spacetime diffeomorphisms. I. canonical parametrized field theories. Ann. Phys.
**1985**, 164, 288–315. [Google Scholar] [CrossRef] - Wheeler, J.A. Superspace and the nature of quantum geometrodynamics. In Lectures in Mathematics and Physics; DeWitt, C.M., Wheeler, J.A., Eds.; Benjamin: New York, NY, USA, 1968; pp. 242–307. [Google Scholar]
- Giulini, D. The Superspace of geometrodynamics. Gen. Relativ. Gravit.
**2009**, 41, 785–815. [Google Scholar] [CrossRef] - Barbour, J.; Foster, B. Constraints and gauge transformations: Dirac’s theorem is not always valid, 2008. Available at: http://arxiv.org/pdf/0808.1223 (accessed on 31 March 2011).
- Thébault, K.P.Y. Symplectic reduction and the problem of time in nonrelativistic mechanics, 2010. Available at: http://philsci-archive.pitt.edu/8433/1/NRPoT.pdf (accessed on 31 March 2011).
- Rovelli, C. Quantum Gravity; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Belot, G. The representation of time and change in mechanics. In Handbook of Philosophy of Physics; Butterfield, J., Earman, J., Eds.; Elsevier: Oxford, UK, 2007; pp. 133–228. [Google Scholar]
- Dittrich, B. Partial and complete observables for canonical general relativity. Class. Quant. Grav.
**2006**, 23, 6155–6185. [Google Scholar] [CrossRef] - Dittrich, B. Partial and complete observables for Hamiltonian constrained systems. Gen. Rel. Grav.
**2007**, 39, 1891–1927. [Google Scholar] [CrossRef]

## Notes

1. | If such a “commutation” proof is available then the observables will be connected by a far stronger relationship of being unitarily equivalent—however for our purposes a weaker representational notion will prove more useful. |

2. | In particular their phase space action cannot be unambiguously connected to the cotangent bundle projection of a symmetry group acting on the tangent bundle see [15] and references therein. |

3. | |

4. | These particular constraints are primary constraints since they arise directly from the definition of the canonical momenta. It is also possible that there exist secondary constraints but for the purposes of this paper this distinction will be unimportant. |

5. | Symplectic geometries on the other hand equip us with a Poisson bracket structure such that $\Omega ({X}_{f},\xb7)=\{f,\xb7\}=\mathbf{d}f$ gives a unique ${X}_{f}$ for every $f\in {C}^{\infty}(\Gamma )$. |

6. | The first is assured since the quotient we are taking is of a manifold by a sectional foliation see [5] p. 42 and pp. 82–83. It is a sectional foliation because the orbits which partition $\Sigma $ constitute manifolds which are suitably transverse. The second is assured because the quotient is of a presymplectic manifold with respect to the kernel of its own presymplectic form and it can be shown that this implies that the resulting quotient manifold will be endowed with a closed two form with a kernel of zero dimension—i.e., it will have a symplectic geometry see [5] theorem 9.10. |

7. | Explicitly we require that: (1) ${\mathcal{H}}_{Q}$ is a separable complex Hilbert space. The elements $\mid \psi \rangle \in {\mathcal{H}}_{Q}$ are the quantum wave functions and the elements ${\mid \psi \rangle}_{\mathbb{C}}\in \mathbf{P}{\mathcal{H}}_{Q}$ are the quantum states where $\mathbf{P}{\mathcal{H}}_{Q}$ is the projective Hilbert space; (2) A is a one to one map taking the classical observables $f\in {\Omega}^{0}(\mathcal{M})$ to the self adjoint operators ${A}_{f}$ on ${\mathcal{H}}_{Q}$ such that: (i) ${A}_{f+g}={A}_{f}+{A}_{g}$; (ii) ${A}_{\lambda f}=\lambda {A}_{f}$$\forall \lambda \in \mathbb{C}$; (iii) ${A}_{1}=I{d}_{{\mathcal{H}}_{Q}}$; (3) $[{A}_{f},{A}_{g}]=i\hslash {A}_{\{f,g\}}$ (i.e., A is a Lie algebra morphism up to a factor); (4) For a complete set of classical observables $\{{f}_{j}\}$, ${\mathcal{H}}_{Q}$ is irreducible under the action of the set $\{{A}_{{f}_{j}}\}$. |

8. | There is also the additional problem that if the constraints depend non-polynomially on the field variables then it may prove impossible to find a rigorously defined representation of them on the ${\mathcal{H}}_{aux}$. This issue is particularly pressing for the constraints of canonical general relativity and leads, in that case, to the introduction of Ashtekar variables. However, neither this formal issue, nor the structure of the new variables, have any particular bearing on the our more conceptual concerns regarding the nature of quantum Hamiltonian constraints. Their discussion can, therefore, be reasonably neglected for the purposes of this non-explicit treatment. |

9. | i.e., we have that $U(g)f[\varphi ]=f(U{g}^{-1}\varphi )$, $\forall \varphi \in \Phi $. |

10. | $\widehat{\mathcal{O}}U(g)\mid \varphi \rangle =U(g)\widehat{\mathcal{O}}\mid \varphi \rangle $, $\forall g\in G$, $\varphi \in \Phi $. |

11. | A right (left) Haar measure is a positive measure on a group invariant under right (left) translations. For the uni-modular case which we are restricting ourselves to, the left and right Haar measures agree. |

12. | |

13. | See [28] for detailed discussion of the metric and topological structure of $\mathcal{S}$. |

14. | See [18] §9 for extensive details of such a methodology for the Dirac type quantisation of the momentum constraints in the context of Ashtekar variables. |

15. | |

16. | We should here, more properly, be speaking of the Hamiltonian constraints as reformulated in Ashtekar variables rather than those expressed in normal ADM variables. However, since the reformulated Hamiltonian constraints close with the same Poisson bracket structure (as they must), this difference is immaterial to our current purpose—although it will become important within a more explicit treatment. |

17. | Here the $\widehat{{\phi}_{i}}$ are a countable and closeable set of operators which need not be self-adjoint nor form a Lie algebra but are such that $\{0\}$ lies only in their common point spectrum. |

18. | Our ability to apply these classical geometrical terms in the quantum context derives from the symplectic structure encoded in the space of rays associated with any Hilbert space. See [24] and references therein for more details. |

19. |

© 2011 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

## Share and Cite

**MDPI and ACS Style**

Thébault, K.P.Y. Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity? *Symmetry* **2011**, *3*, 134-154.
https://doi.org/10.3390/sym3020134

**AMA Style**

Thébault KPY. Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity? *Symmetry*. 2011; 3(2):134-154.
https://doi.org/10.3390/sym3020134

**Chicago/Turabian Style**

Thébault, Karim P. Y. 2011. "Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity?" *Symmetry* 3, no. 2: 134-154.
https://doi.org/10.3390/sym3020134