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Full Four-Dimensional Diffeomorphism Invariants and Their Role in Quantum

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Full Four-Dimensional Diffeomorphism Invariants and Their Role in Quantum Intrinsic GR Full four-dimensional diffeomorphism invariants and their role in quantum theories of gravity Donald Salisbury Austin College, USA Balfest80 DIAS, Dublin, January 25, 2018 Intrinsic GR Collaborators Work performed in collaboration in part with Josep Pons, J¨urgen Renn, Larry Shepley, Kurt Sundermeyer See arXiv:1508.01277v5 Intrinsic GR Overview 1 Introduction 2 Brief history of Hamiltonian general covariance 3 Gauge conditions and associated invariants 4 Intrinsic gravitational Hamilton-Jacobi approach 5 Implications for loop quantum gravity? Intrinsic GR Introduction 1. INTRODUCTION Intrinsic GR Introduction Introduction Focus of this talk: What are some implications for an eventual quantum theory of gravity of the classical evolution of the metric with respect to temporal and spatial landmarks constructed with the use of Weyl curvature scalars. All variables are diffeomorphism invariants. Questions to be addressed: What is the status of the multiplicity of observer-based evolution in classical general relativity. How can the fully relational approach be applied to loop quantum gravity? Is there a meaningful quantum generalization to non-commuting intrinsic coordinate algebras in loop quantum gravity? Intrinsic GR Brief history of Hamiltonian general covariance 2. BRIEF HISTORY OF HAMILTONIAN GENERAL COVARIANCE Intrinsic GR Brief history of Hamiltonian general covariance Rosenfeld's 1930 tetrad gravitational Lagrangian \Zur Quantelung der Wellenfelder", Annalen der Physik 397, 113 (1930) Translation by Salisbury and Sundermeyer [Rosenfeld, 2017] 1 1 µ ν I LJ I LJ L = (−g) 2 E E ! ! − ! ! 2κ I J µ L ν ν L µ 1 −! +< (−g)1=2 i ¯γµ @ + Ω − m ¯ + L 2 µ µ em µ I Tetrads EI , Rotation coefficients !µ L, Fermion field , 1 I J spinor connection Ωµ = 4 γ γ !µIJ . Intrinsic GR Brief history of Hamiltonian general covariance Rosenfeld's 1930 tetrad Hamiltonian density Rosenfeld invented a systematic procedure for solving for the _ µ velocities EI in terms of the conjugate momenta given that the @2L Jacobian matrix _ µ _ ν is singular. Although he did not do this @EI EJ explicitly for this model, the result (see [Salisbury & Sundermeyer, 2017] ) is h ab a yi I [IJ] H = H0 gab; p ; Aa; p ; ; + λI F + λIJ F + λF I [IJ] where F , F and F are primary constraints and λI , λIJ and λ are arbitrary spacetime functions. Preceeding Bergmann and Dirac by twenty years! See [Salisbury, 2009]. Intrinsic GR Brief history of Hamiltonian general covariance Rosenfeld's infinitesimal phase space symmetry generator Rosenfeld proved that the vanishing Noether charge generated the correct variations of all of the phase space variables under all of the local symmetries. Most importantly for us is that the active variations under the infinitesimal coordinate transformations x0µ = xµ + ξµ(x) are correct. His conserved and vanishing generating density is I _0 I _a _0 aI ν a ν 0 a −F e0I ξ − F eaI ξ − FA0ξ − p eνI ξ;a − p Aνξ;a − HA0ξ − Gaξ e e _ a y IJ −Fξ + p ξ;a + i p ξ − i p y ξ + F[IJ]ξ = 0 ~c ~c Intrinsic GR Brief history of Hamiltonian general covariance Peter Bergmann and Paul Dirac We leap two decades forward to the contributions to constrained Hamiltonian dynamics of Peter Bergmann and Paul Dirac - beginning in 1949. Dirac never concerned himself with the phase space realization of the full general covariance group. See his Vancouver lectures, [Dirac, 1950] [Dirac, 1951] [Bergmann, 1949], later with Jim Anderson [Anderson & Bergmann, 1951] and numerous collaborators including [Goldberg, 1953] did concern themselves with this symmetry. In particular a joint publication with Ralph Schiller [Bergmann & Schiller, 1953] explicitly employed the vanishing Noether charge. Intrinsic GR Brief history of Hamiltonian general covariance Non-realizability of the diffeomorphism Lie algebra But there is an obstacle to the realization of finite diffeomorphisms, explicitly recognized by Bergmann. One can see this in the Lie algebra µ µ ν µ ν µ 0 µ 0 ξ3 = ξ1,νξ2 − ξ2,νξ1 = ξ1;0ξ2 − ξ2;0ξ1 + ::: Repeated commutators lead to higher and higher order time derivatives. Intrinsic GR Brief history of Hamiltonian general covariance Dirac's resolution Dirac's solution was probably inspired by his student Paul Weiss: write the infinitesimal variations as a sum of perpendicular and tangent increments, µ µ 0 µ a ξ = n + δa This results in the familiar metric dependent Dirac algebra. Intrinsic GR Brief history of Hamiltonian general covariance The Bergmann - Komar group [Bergmann & Komar, 1972] interpreted this algebra as representing a compulsory metric-dependent transformation group Intrinsic GR Gauge conditions and associated invariants 3. GAUGE CONDITIONS AND ASSOCIATED INVARIANTS Intrinsic GR Gauge conditions and associated invariants The Legendre projectability requirement We have an mathematical justification for the Dirac decomposition. It is required in order that configuration-velocity variations be projectable under the Legendre transformation to phase space, [Pons et al. , 1997] Intrinsic GR Gauge conditions and associated invariants Ashtekar-Barbero-Immirzi Lagrangian [Pons et al. , 2000], [Pons & Salisbury, 2002] The connection is α i i −1 i α i i −1 i a a Aa := !a − α Ka; A0 := Ω0 − α KaN + αTi N;a The Lagrangian is ∼_ ∼ ≈ ∼ L = α−1αAi T a + NaH + NαH + αAi αH ; ABI a i c ∼ 0 0 i where the secondary constraints are α ∼ α ∼a α ∼ ∼bα i Hi := −α DaT i = 0; Ha := αT i Fba = 0 and ≈ 1 ∼ ∼ αH := − T aT b(−α2αF ij + (1 + α2)3Rij ) = 0: 0 2 i j ab ab Intrinsic GR Gauge conditions and associated invariants Legendre-projectable infinitesimal symmetries It is noteworthy that when gauge symmetries are present in addition to diffeomorphism symmetry, the transformations of some of the additional gauge variables under the change of coordinates 0µ µ µ 0 µ a x = x − n ξ − δa ξ ; are no longer projectable. See [Pons et al. , 2000]. In this case the α i a variation of A0 acquires unprojectable time derivatives of N, N , α i and A0. But these can be removed by adding to the variation an µα i −1 bi SO(3) rotation with descriptor −n Aµ + αN T N;b ≈ α This is in fact the variation generated by H0. Intrinsic GR Gauge conditions and associated invariants The complete generator of infinitesimal symmetry transformations The primary constraints are the variables canonically conjugate to the lapse, shift, and time component of the connection. Combination of primary and secondary constraints, all first class, allows one to construct the full set of gauge generators 0 00 _A B C A Gξ = PAξ + (HA + PC 00 N CAB0 )ξ ; C 00 where CAB0 are the structure functions associated with the Poisson brackets of the secondary constraints and the descriptors ξ are infinitesimal arbitrary functions of spacetime coordinates appearing in the projectable infinitesimal coordinate transformations 0µ µ µ 0 µ a x = x − n ξ + δa ξ ; ∼ i α and local SO(3) gauge rotations - generated by ξ Hi . Intrinsic GR Gauge conditions and associated invariants Time evolution versus diffeomorphisms The evolution in time is generated by Z Z ∼ ∼ ≈ d3xH = d3x −αAi αH + NaαH + NαH + λ PA : ABI 0 i a ∼ 0 A R 3 But the finite diffeomorphism generator exp s d x Gξ(t) transforms solutions into new solutions. Intrinsic GR Gauge conditions and associated invariants Enlargement of phase space Note that the lapse function N, the shift Na and the nought α i component of the connection A0 must be retained as canonical variables. Note also that contrary to popular belief, the Hamiltonian formulation does not fix a time foliation. New foliations result in new multipliers λA and new Hamiltonians as a consequence of the time dependence of the Hamiltonian. Intrinsic GR Intrinsic gravitational Hamilton-Jacobi approach 4. INTRINSIC GRAVITATIONAL HAMILTON-JACOBI APPROACH Intrinsic GR Intrinsic gravitational Hamilton-Jacobi approach An implementation of Rovelli's partial variable program Now that we have the full diffeomorphism group at our disposal, we can employ it to establish correlations between partial variables. One possible implementation, in principle, is to locate temporal and spatial landmarks by referring to curvature even in the vacuum case. There are of course many more possibilities when matter is present. We will employ these landmarks as \intrinsic" coordinates. Such coordinates must be formed from spacetime ∼ µ α i b scalars. Thus we choose X [ Aa; T j ]. In the vacuum case we propose the use of the four Weyl curvature scalars, as originally suggested by [Komar, 1958]. They are quadratic and cubic in the Weyl tensor. [Bergmann & Komar, 1960] showed that they are expressible solely in terms of the three metric and its conjugate momenta. The same ∼ α i b logic demonstrates dependence only on Aa and T j Intrinsic GR Intrinsic gravitational Hamilton-Jacobi approach Weyl scalars in the Ashtekar program In fact, we have for the complex connections that the Newman-Penrose scalars are 1 (i abc j) ij = t a F : 4 ∼ bc (See [Salisbury et al. , 1994], summarizing results originally due to [Capovilla et al. , 1991]). The Weyl scalars are in turn expressible j i j k i as I = i j and J = i j k . (See [Penrose & Rindler, 1988]) Intrinsic GR Intrinsic gravitational Hamilton-Jacobi approach Proof of principle Weyl scalars can always serve as coordinates in a local Riemann normal coordinate system. Overlapping patches can then cover the entire manifold. 1 Expand the metric in the neighborhood of any spacetime event to third order. See, for example, [Brewin, 2009] 1 1 g (x) = g − xρxσR − xρxσxκ@ R + O(4) µν µν 3 µνρσ 6 κ µνρσ 2 Keep only the linear in xµ contributions to I and J and solve for the xµ in terms of the I and J.
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