##
**Quantum gravitation. The Feynman path integral approach.**
*(English)*
Zbl 1171.81001

Berlin: Springer (ISBN 978-3-540-85292-6/hbk; 978-3-540-85293-3/hbk). xvii, 342 p. (2009).

The monograph of Herbert W. Hamber is devoted to a very difficult direction in quantum theory, constructing a quantum version of theory of gravity. This problem is under active discussions and research during few decades. In the seventies, the general impression of quantum physicists was that, after the triumph of non-abelian gauge theories (including quantum chromodynamics and the electro-weak theory), a quantum treatment of General Relativity might be straightforward. However, the ultraviolet divergences emerging in quantization of General Relativity turned out to be a more serious obstacle than in case of the other fields. This have led to the invention of a number of special methods for quantization gravity, taking into account the qualitative difference of the gravitational field. However, these attempts including supergravity and superstrings, met with their own obstacles (in case of superstrings living in a ten-dimensional spacetime this is looking for a dynamical mechanism that would drive the compactification of spacetime from the ten dimensional string universe to our physical four-dimensional world).

In the present monograph the author concentrates on an alternative approach, that is, in his opinion, the most universal and reliable approach to construct quantum gravity. This approach is based on the usage of the manifestly covariant Feynman path integral, Wilson’s modern renormalization group and the development of lattice methods to define a regularized form of the path integral, which then allows non-perturbative calculations. In non-abelian gauge theories and in the standard model of elementary particle interactions, these instruments are working quite efficiently. Particularly, the covariant Feynman path integral approach is crucial in proving the renormalizability of non-abelian gauge theories. Modern renormalization group methods provide the derivation of the asymptotic freedom and the momentum dependence of amplitudes in terms of running coupling constants. Finally, the lattice formulation of gauge theories provide the only convincing theoretical evidence of confinement and chiral symmetry breaking in non-abelian gauge theories.

The book of Hamber covers key aspects and open issues related to a consistent regularized formulation of quantum gravity with the aid of the covariant Feynman path integral quantization. It consists of three major parts.

Part I introduces basic elements of the covariant formulation of continuum quantum gravity, with some emphasis on those issues which bear an immediate relevance for the remainder of the book. Discussion in this part includes the nature of the spin-two field, its wave equation and possible gauge choices, the Feynman propagator, the coupling of a spin-two field to matter and the implementation of a consistent local gauge invariance to all orders, ultimately leading to the Einstein gravitational action. Additional terms in the gravitational action, such as the cosmological constant and higher derivative contributions, are naturally introduced at this stage.

A number of special issues are considered in this part in connection with the perturbative weak field expansion and the background field method as applied to gravity, particularly the structure of one- and two-loop divergences in pure gravity leading to the statement of perturbative non-renormalizability for the Einstein theory in four dimensions.

Then the Feynman path integral for gravitation is introduced by closely analogizing the theory with the related Yang-Mills case, with discussing the gravitational functional measure as well as some other important aspects related to the convergence of the path integral and derived quantum averages, along with the origin of the conformal instability affecting the Euclidean case.

The methods are then summarized of Wilson’s expansion as applied to gravity, expanding the deviation of the space-time dimensions from two. As an initial motivation, but also for illustrative and pedagogical purposes, the non-linear sigma model is first introduced.

The Hamiltonian method is considered in connection with Feynman path integrals. The nature of the Hamiltonian constraint in gravity is discussed, which implies that the total energy of a quantum gravitational system is zero, resulting in the Wheeler-DeWitt equation, a SchrĂ¶dinger-like equation for the vacuum functional. The Hamiltonian method is applied as a starting point for a lattice description of quantum gravity, whose results may be regarded as complementary to those obtained via the Feynman path integral approach. The ambiguities that appear here as operator ordering problems have their correspondence in the path integral approach, associated with the choice of functional measure. The problems of Hamiltonian approach are discussed, including the lack of covariance due to the choice of time coordinate, and the difficulty of doing practical approximate non-perturbative calculations. Closely related to the Hamiltonian approach an array of semiclassical methods is presented which have been used to obtain approximate cosmological solutions to the Wheeler-DeWitt equations (they are discussed later in more detail). Some physically relevant results are exposed such as black hole radiance, and some more general issues which arise in a semiclassical treatment of quantum gravity.

Part II of the book presents the lattice theory of gravity based on Regge’s simplicial formulation, with a primary focus on the physically relevant four-dimensional case. The correspondence between lattice quantities (edges, dihedral angles, volumes, deficit angles, etc.) and continuum operators (metric, affine connection, volume element, curvature tensor etc.) is traced. It is shown how one can define discrete versions of curvature squared terms which arise in higher derivative gravity theories or as radiatively induced corrections. The lattice analogs of gravitons are introduced.

The coupling of the matter fields to lattice gravity is described in terms of the new fields localized on vertices as well as the corresponding dual volumes which enter the definition of the kinetic terms for those fields. The fermion case is considered on the basis of vierbein fields introduced within each simplex, and an appropriate spin rotation matrix relating spinors between neighboring simplices. The formulation of fractional spin fields on a simplicial lattice is shown to be useful in formulating a lattice discretization of supergravity. The simplicial lattice formulation is compared with other discrete approaches to quantum.

Later in his book the author deals with the interesting problem of what gravitational observables should look like, that is which expectation values of operators have physical interpretation in the context of a manifestly covariant formulation. Such averages include expectation values of the integrated scalar curvature and other related quantities (for example curvature squared terms), as well as correlations of operators at fixed geodesic distance referred to as bi-local operators. Another set of physical averages refer to the geometric nature of space-time itself, such as the fractal dimension. One more set of physical observables correspond to the gravitational analog of the Wilson loop, which provides information about the parallel transport of vectors, and therefore on the effective curvature, around large near-planar loops, and the correlation between particle world-lines, which gives the static gravitational potential.

Part III of the book discusses applications of the lattice theory to non-perturbative gravity. A discrete formulation combined with numerical tools are reviewed as an essential step towards a quantitative and controlled investigation of the physical content of the theory. Specifically, lattice gravity in four dimensions is characterized by two phases: a weak coupling degenerate polymer-like phase, and a strong coupling smooth phase with small average curvature. The somewhat technical aspect of the determination of universal critical exponents is outlined, together with a brief discussion of how the lattice continuum limit has to be approached in the vicinity of a non-trivial ultraviolet fixed point.

The determination of non-trivial scaling dimensions in the vicinity of the fixed point leads to a discussion of the renormalization group properties of fundamental couplings: their scale dependence and the emergence of physical renormalization group invariant quantities (the fundamental gravitational correlation length and the gravitational condensate). These are discussed next, with an eye towards physical applications: the physical nature of the expected quantum corrections to the gravitational coupling, and the relation of the two phases of lattice gravity to the two opposite scenarios of gravitational screening (for weak coupling, and therefore unphysical due to the branched polymer nature of this phase) versus anti-screening (for strong coupling, and therefore physical).

The book is oriented on the physicists interested in quantum gravity. It will be useful both for the first acquaintance with the specific features of the Feynman path integrals in gravity and for those who are already working with such integrals but need more wide knowledge of the corresponding methods.

In the present monograph the author concentrates on an alternative approach, that is, in his opinion, the most universal and reliable approach to construct quantum gravity. This approach is based on the usage of the manifestly covariant Feynman path integral, Wilson’s modern renormalization group and the development of lattice methods to define a regularized form of the path integral, which then allows non-perturbative calculations. In non-abelian gauge theories and in the standard model of elementary particle interactions, these instruments are working quite efficiently. Particularly, the covariant Feynman path integral approach is crucial in proving the renormalizability of non-abelian gauge theories. Modern renormalization group methods provide the derivation of the asymptotic freedom and the momentum dependence of amplitudes in terms of running coupling constants. Finally, the lattice formulation of gauge theories provide the only convincing theoretical evidence of confinement and chiral symmetry breaking in non-abelian gauge theories.

The book of Hamber covers key aspects and open issues related to a consistent regularized formulation of quantum gravity with the aid of the covariant Feynman path integral quantization. It consists of three major parts.

Part I introduces basic elements of the covariant formulation of continuum quantum gravity, with some emphasis on those issues which bear an immediate relevance for the remainder of the book. Discussion in this part includes the nature of the spin-two field, its wave equation and possible gauge choices, the Feynman propagator, the coupling of a spin-two field to matter and the implementation of a consistent local gauge invariance to all orders, ultimately leading to the Einstein gravitational action. Additional terms in the gravitational action, such as the cosmological constant and higher derivative contributions, are naturally introduced at this stage.

A number of special issues are considered in this part in connection with the perturbative weak field expansion and the background field method as applied to gravity, particularly the structure of one- and two-loop divergences in pure gravity leading to the statement of perturbative non-renormalizability for the Einstein theory in four dimensions.

Then the Feynman path integral for gravitation is introduced by closely analogizing the theory with the related Yang-Mills case, with discussing the gravitational functional measure as well as some other important aspects related to the convergence of the path integral and derived quantum averages, along with the origin of the conformal instability affecting the Euclidean case.

The methods are then summarized of Wilson’s expansion as applied to gravity, expanding the deviation of the space-time dimensions from two. As an initial motivation, but also for illustrative and pedagogical purposes, the non-linear sigma model is first introduced.

The Hamiltonian method is considered in connection with Feynman path integrals. The nature of the Hamiltonian constraint in gravity is discussed, which implies that the total energy of a quantum gravitational system is zero, resulting in the Wheeler-DeWitt equation, a SchrĂ¶dinger-like equation for the vacuum functional. The Hamiltonian method is applied as a starting point for a lattice description of quantum gravity, whose results may be regarded as complementary to those obtained via the Feynman path integral approach. The ambiguities that appear here as operator ordering problems have their correspondence in the path integral approach, associated with the choice of functional measure. The problems of Hamiltonian approach are discussed, including the lack of covariance due to the choice of time coordinate, and the difficulty of doing practical approximate non-perturbative calculations. Closely related to the Hamiltonian approach an array of semiclassical methods is presented which have been used to obtain approximate cosmological solutions to the Wheeler-DeWitt equations (they are discussed later in more detail). Some physically relevant results are exposed such as black hole radiance, and some more general issues which arise in a semiclassical treatment of quantum gravity.

Part II of the book presents the lattice theory of gravity based on Regge’s simplicial formulation, with a primary focus on the physically relevant four-dimensional case. The correspondence between lattice quantities (edges, dihedral angles, volumes, deficit angles, etc.) and continuum operators (metric, affine connection, volume element, curvature tensor etc.) is traced. It is shown how one can define discrete versions of curvature squared terms which arise in higher derivative gravity theories or as radiatively induced corrections. The lattice analogs of gravitons are introduced.

The coupling of the matter fields to lattice gravity is described in terms of the new fields localized on vertices as well as the corresponding dual volumes which enter the definition of the kinetic terms for those fields. The fermion case is considered on the basis of vierbein fields introduced within each simplex, and an appropriate spin rotation matrix relating spinors between neighboring simplices. The formulation of fractional spin fields on a simplicial lattice is shown to be useful in formulating a lattice discretization of supergravity. The simplicial lattice formulation is compared with other discrete approaches to quantum.

Later in his book the author deals with the interesting problem of what gravitational observables should look like, that is which expectation values of operators have physical interpretation in the context of a manifestly covariant formulation. Such averages include expectation values of the integrated scalar curvature and other related quantities (for example curvature squared terms), as well as correlations of operators at fixed geodesic distance referred to as bi-local operators. Another set of physical averages refer to the geometric nature of space-time itself, such as the fractal dimension. One more set of physical observables correspond to the gravitational analog of the Wilson loop, which provides information about the parallel transport of vectors, and therefore on the effective curvature, around large near-planar loops, and the correlation between particle world-lines, which gives the static gravitational potential.

Part III of the book discusses applications of the lattice theory to non-perturbative gravity. A discrete formulation combined with numerical tools are reviewed as an essential step towards a quantitative and controlled investigation of the physical content of the theory. Specifically, lattice gravity in four dimensions is characterized by two phases: a weak coupling degenerate polymer-like phase, and a strong coupling smooth phase with small average curvature. The somewhat technical aspect of the determination of universal critical exponents is outlined, together with a brief discussion of how the lattice continuum limit has to be approached in the vicinity of a non-trivial ultraviolet fixed point.

The determination of non-trivial scaling dimensions in the vicinity of the fixed point leads to a discussion of the renormalization group properties of fundamental couplings: their scale dependence and the emergence of physical renormalization group invariant quantities (the fundamental gravitational correlation length and the gravitational condensate). These are discussed next, with an eye towards physical applications: the physical nature of the expected quantum corrections to the gravitational coupling, and the relation of the two phases of lattice gravity to the two opposite scenarios of gravitational screening (for weak coupling, and therefore unphysical due to the branched polymer nature of this phase) versus anti-screening (for strong coupling, and therefore physical).

The book is oriented on the physicists interested in quantum gravity. It will be useful both for the first acquaintance with the specific features of the Feynman path integrals in gravity and for those who are already working with such integrals but need more wide knowledge of the corresponding methods.

Reviewer: Michael B. Mensky (Moskva)

### MSC:

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

81V17 | Gravitational interaction in quantum theory |

81S40 | Path integrals in quantum mechanics |

83C45 | Quantization of the gravitational field |

81T16 | Nonperturbative methods of renormalization applied to problems in quantum field theory |

81T17 | Renormalization group methods applied to problems in quantum field theory |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

81V22 | Unified quantum theories |

81T25 | Quantum field theory on lattices |

81T18 | Feynman diagrams |

81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |

70H45 | Constrained dynamics, Dirac’s theory of constraints |

83-08 | Computational methods for problems pertaining to relativity and gravitational theory |