Volume 2 Issue 3 - November 16, 2007
Three-geometry and reformulation of the Wheeler–DeWitt equation
Chopin Soo

Department of Physics, National Cheng Kung University
Email: cpsoo@mail.ncku.edu.tw

Classical and Quantum Gravity 24 (2007) 1547-1555
Our current understanding of the macroscopic properties of space-time and the dynamics of the universe is based upon Einstein’s theory of General Relativity, whereas the microphysics of elementary particles is governed by the rules of quantum mechanics. A successful Quantum Theory of Gravity can be expected to yield unparalleled understanding of the nature of space-time at all scales, from the microscopic description of geometry to the large scale structure of the universe. However, endeavors to combine the demands of quantum mechanics with those of General Relativity into a unified theoretical framework in four dimensions have so far been unsuccessful; and the techniques of perturbative quantum field theories which have triumphed in the standard model of particle physics have run into mathematical inconsistencies of non-renormalizability when they are naively applied to construct quantum theories of geometry.

The program of non-perturbative quantization of gravity attempts to overcome the perturbative non-renormalizability of Einstein’s theory by constructing exact background-independent theory of quantum geometry. Physicists have also come to realize that fundamental interactions and forces are governed by the Principle of Local Gauge Invariance i.e. the Laws of Physics must remain the same under the freedom to perform certain space-time dependent transformations called gauge transformations [1]. This is achieved in theoretical physics by introducing gauge connections which act as mediators of forces.

Much excitement and insight have therefore stemmed from Ashtekar’s reformulation of General Relativity and the simplification of its constraints through the use of gauge connection and densitized triad variables [2]. General Relativity differs from other theories of fundamental forces in having the privileged role of describing the dynamics of space-time through the metric which measures space-time intervals and geometry. Conceptually, the distinction between geometrodynamics based upon the metric and gauge dynamics in Ashtekar’s reformulation is bridged by the identification of the densitized triad, Eia, from which the spatial metric gij = EiaEja/det(E) is a derived composite, as the momentum conjugate to the Ashtekar gauge connection Aia (in canonical physics, each degree of freedom is associated with a pair of variables called conjugate pair). Most intriguing too is the conjunction of the fact the Lorentz group possesses self and anti-self-dual decompositions in four and only in four dimensions with the observation that the Ashtekar gauge connection is precisely the restriction to the initial-data surface, M, of the self(or anti-self)-dual (±) projection of the connection of the Lorentz group. In a curved space-time, the tangent space at each point is isomorphic to Minkowski space which is invariant under the group of Lorentz transformations. The infusion of loop variables and subsequently spin network states by Rovelli, Smolin and others into Ashtekar’s formulation have also proved fruitful, and have yielded discrete spectra for well-defined area and volume operators. Indeed by virtue of being area and volume eigenstates, states based upon spin networks - the latter originally introduced by Penrose to explore quantum geometry [3]- are now prominent in the current manifestation of loop quantum gravity. To the extent that exact states and rigorous results are needed, simplifications and reformulations of the classical and corresponding quantum constraints of General Relativity are extremely crucial steps.

In my work a reformulation of the super-Hamiltonian constraint of General Relativity, and its associated quantum constraint, the Wheeler–DeWitt equation [4, 5], is presented. It is noted that the classical super-Hamiltonian of four-dimensional gravity as simplified by Ashtekar through the use of gauge connection and densitized triad variables can furthermore be succinctly expressed as a vanishing Poisson bracket between fundamental invariants. This observation leads naturally to a reformulation of the all-important Wheeler–DeWitt equation which determines the evolution of quantum states in canonical quantum gravity. This is discussed in the general setting when a cosmological constant λ is also present. In order to have a static a dust-filled universe which is compatible with his Field Equations, Einstein was forced to introduce a repulsive positive cosmological constant into his theory to balance the gravitational attraction of matter. It was a decision which he later regretted; but recent supernovae observations have yielded evidence for precisely such a parameter in General Relativity.

The Chern–Simons functional [6] is a mathematical geometric invariant first studied by Shiing-Shen Chern and J. Simons. It has served as a fertile link between quantum field theories of three and four dimensions. In General Relativity with Ashtekar variables, it has the additional significance of being the carrier of information of both intrinsic and extrinsic curvatures. Although it is not as extensively explored in spin networks in present formulations of loop quantum gravity as in quantum field theories, the expectation that the Chern–Simons invariant of the Ashtekar potential, , has a very significant, and even direct, role in four-dimensional quantum gravity is in fact borne out by the discussions of our work. While the volume operator has become a preeminent object in the current manifestation of loop quantum gravity, the Chern–Simons functional can be of equal significance. Both these fundamental objects appear explicitly in my reformulation of the Wheeler–DeWitt equation. The reformulation not only highlights the role of gauge-invariant 3-geometry elements in the Wheeler–DeWitt equation, but also spells out which specific functionals of 3-geometry are involved. With the prescription of Dirac quantization [7], the reformulated Wheeler–DeWitt equation for self(or anti-self)-dual Ashtekar variables is recast as , where Lp ~ 10-35 m is the Planck length and ν2 = det(E) (the determinant of densitized triad). It has a number of remarkable properties: (1)This equation for the full theory (not just a particular sector of General Relativity) is not merely symbolic but is in fact expressed explicitly in terms of the gauge-invariant 3-geometry element which is none other than the Chern–Simons functional of the Ashtekar connection. This is to be contrasted with the traditional formulation with metric variables, wherein the equation in terms of abstract 3-geometry G was symbolically[4]. (2)There are now no factor ordering ambiguities for the composite operators and C[A] which have clear geometric meanings; and each of which is made up of commuting variables. Ordering `ambiguities’ that were present in super-Hamiltonian constraint in the transition from classical to quantum theory have been decided by requiring that the Wheeler–DeWitt equation is expressible in terms of these 3-geometry elements. (3)It has long been known that three-dimensional diffeomorphism invariance would require the quantum states to be functionals of 3-geometries [4, 5]. Therefore, one may surmise that, albeit a nontrivial endeavor, it ought to be possible to express the Wheeler–DeWitt equation of the full theory in terms of explicit 3-geometry elements. However, the constraint is also required to be satisfied at each spatial point on the initial-data surface. Both these requirements are remarkably realized in the reformulation here in that the Wheeler–DeWitt constraint is cast in terms of the commutator between at each spatial point and the Chern–Simons functional of the Ashtekar connection.

Unlike the Gauss Law and super-momentum constraints, which are kinematic constraints enforcing invariance under local internal SO(3) gauge transformations and three-dimensional general coordinate transformations respectively, and which have straightforward group-theoretic interpretations, the ordering ambiguity of the non-commuting variables in the Wheeler-DeWitt equation is a more intricate matter. There is no unique prescription for defining a quantum theory from its classical correspondence. Thus the ordering problem has to be decided through other means, for instance, through mathematical consistency (sometimes expediency) and the closure of the quantum algebra. Even so, these may or may not yield a unique ordering. With complex Ashtekar variables, the Hermiticity of the super-Hamiltonian too cannot be adopted as a criterion. Often when dealing with the factor ordering of a complicated constraint, an initial motivation is needed; and a specific ordering is assumed first before checking the consistency of the composite operator. The proposed quantum constraint considered here is motivated by 3-geometry considerations, is polynomial in the basic conjugate variables, and is also demonstrated to correspond to a precise simple ordering of the quantum operators. It may thus help to resolve the factor ordering ambiguity in the extrapolation from classical to quantum gravity.

References
  • Yang C N and Mills R L 1954 Conservation of Isotopic Spin and Isotopic Gauge Invariance Phys. Rev. 96 191
  • Ashtekar A 1986 New Variables for Classical and Quantum Gravity Phys. Rev. Lett. 57 2244
  • Penrose R 1971 Angular momentum: an approach to combinational space-time, in Quantum Theory and Beyond ed T Bastin (Cambridge University Press, Cambridge)
  • Wheeler J A 1968 Superspace and the Nature of Quantum Geometrodynamics, Battelle Rencontres: Lectures in Mathematics and Physics ed C M DeWitt and J A Wheeler (Benjamin, New York)
  • DeWitt B S 1967 Quantum Theory of Gravity. I. The Canonical Theory Phys. Rev. 160 1113
  • Chern S S and Simons J 1974 Characteristic forms and geometric invariants, Annals Math. 99 48
  • Dirac P A M 1964 Lectures on Quantum Mechanics (Yeshiva University Press, New York)
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