Intrinsic Time Quantum Geometrodynamics

Quantum Geometrodynamics with intrinsic time development and momentric variables is presented. An underlying SU(3) group structure at each spatial point regulates the theory. The intrinsic time behavior of the theory is analyzed, together with its ground state and primordial quantum fluctuations. Cotton-York potential dominates at early times when the universe was small; the ground state naturally resolves Penrose's Weyl Curvature Hypothesis, and thermodynamic and gravitational `arrows of time' point in the same direction. Ricci scalar potential corresponding to Einstein's General Relativity emerges as a zero-point energy contribution. A new set of fundamental commutation relations without Planck's constant emerges from the unification of Gravitation and Quantum Mechanics.

A century after the birth of Einstein's General Relativity (GR), successful quantization of the gravitational field remains the preeminent challenge. Geometrodynamics bequeathed with positive-definite spatial metric is the simplest consistent framework to implement fundamental canonical commutation relations (CR) predicated on the existence of spacelike hypersurfaces. In quantum gravity, spacetime is a 'concept of limited validity' [1] and "'time" must be determined intrinsically' [2]. A full theory of Quantum Geometrodynamics dictated by first-order Schrödinger evolution in intrinsic time, i δΨ δT = H Phys. Ψ, and equipped with diffeomorphism-invariant physical Hamiltonian and time-ordering was formulated recently [3,4]. Decomposition of the fundamental geometrodynamic degrees of freedom, (q ij , π ij ), singles out the canonical pair (ln q 1 3 , π) which commutes with the remaining unimodular q ij = q − 1 3 q ij , and traceless π ij = q 1 3 π ij − 1 3 q ij π . Hodge decomposition for compact manifolds yields δ ln q 1 3 = δT + ∇ i δY i , wherein the spatially-independent δT is a 3-dimensional diffeomorphism invariant (3dDI) quantity which serves as the intrinsic time interval, whereas ∇ i δY i can be gauged away since L δ − → N ln q is a scalar density of weight one [3]. Einstein's GR (with β = 1 is a particular realization of this wider class of theories. Difficulties in implementingπ ij as self-adjoint traceless operator in the metric representation lead us to summon the momentric variable which is classicallyπ i j =q jmπ im . The fundamental CR is restriction of Klauder's affine algebra [5] to traceless momentric and unimodular part of the spatial metric, is the vielbein for the supermetricḠ ijkl =Ē m n(ij)Ē n m(kl) . Quantum mechanically, the momentric operators and CR can be explicitly realized in the metric representation bŷ which are self-adjoint on account of [ δ δqmn(x) ,Ē i j(mn) (x)] = 0. These eight momentric variables generate SL(3, R) transformations ofq ij which preserve positivity and unimodularity. Moreover, it is crucial to realize they generate, by themselves, at each spatial point, an SU (3) algebra. In fact, with Gell-Mann matrices λ A=1,..., 8 . Perturbative renormalizability of GR can be attained by adding higher derivative terms, but 4-covariance locks higher temporal and spatial derivatives to the same order, compromising the stability and unitarity of the theory [7]. Paradigm shift from 4-covariance to 3dDI not only resolves the 'problem of time', but also leads to the generic weight two (semi)positive-definite potential [3], wherein R andR i j are respectively the scalar and traceless parts of the spatial Ricci curvature, whileC i j is the Cotton-York tensor (density) which is third order in spatial derivatives and associated with dimensionless coupling constant g. In conjunction with intrinsic time evolution with H Phys , this framework presents, in quantum gravity, a new vista to surmount conceptual and technical obstacles.

Free Hamiltonian
The free theory is characterized by SU (3) invariance generated by the momentric (whereasπ ij generate translations which do not preserve the positivity of the metric), because the Casimir invariant T A T A is related to the kinetic operator in Eq.(1) through The upshot is its spectrum can be labeled by eigenvalues of the complete commuting set at each spatial point comprising the two Casimirs For the free theory, the ground state with zero energy, |0 , corresponds to l 2 = 0 ∀x, which is an SU (3) singlet state annihilated by all the momentric operators (π i j (x)|0 = 0); and it is also 3dDI because −2∇ jπ j i generates spatial diffeomorphisms ofq ij .

Asymptotic behavior of the Hamiltonian at early and late intrinsic times
Hodge decomposition for δ ln q 1 3 and its Heisenberg equation of motion lead to d dT ln q Moreover, the Hodge decomposition also implies the change in the global intrinsic time is proportional to the logarithmic change in the volume of the universe, δT = 2 [8]. Explicitly separating out T-dependence from entities (labeled with overline) which depend only onq ij , with q-independent Cotton York tensor densityC i j which is conformally invariant. The theory is not (intrinsic)timereversal invariant; furthermore, exponential scaling behavior of q with intrinsic time implies in the limit T − T now → −∞, V /V now → 0 (i.e. early times when the universe was very small in volume), the potential V was dominated by the Cotton-York term, whereas the limit T − T now → ∞, V /V now → ∞ (i.e. late times when the universe becomes large) will be dominated by the cosmological constant term. This is compatible with current observations of our ever expanding universe, with a middle period in which curvature and cosmological terms are comparable in importance.

Early universe and Cotton-York dominance
In the era of Cotton-York dominance at the beginning of the universe,H = π †j iπ i j + g 2 2C j iC i j . A number of intriguing facts conspire to simplify and regulate the Hamiltonian: The traceless Cotton-York tensor density is expressible asC x is the 3dDI Chern-Simon functional of the affine connection ofq ij . This leads to the similarity transformation of the momentric, Moreover, [π i j ,C j i ] = 0 i.e. without zero point energy (ZPE) contribution [9]. Consequently, the Hamiltonian density is simplyH = Q †i jQ j i = Qi jQ †j i . WhileQ †i j andQ i j are related toπ i j by e ∓gW similarity transformations, they are non-Hermitian and generate two unitarily inequivalent representations of the non-compact group SL(3, R) at each spatial point; whereas the momentricπ i Initial state of the universe and Penrose's Weyl Curvature Hypothesis From the classical perspective,H = π j iπ i j + g 2 2C j iC i j attains its lowest value iff the momentric and Cotton-York tensor vanish identically, the latter being precisely the criterion for conformal flatness in three dimensions [10]. The vanishing of the momentric (hence traceless part of the classical extrinsic curvature) and spatial conformal flatness at T → −∞(q → 0) realize a Robertson-Walker Big Bang compatible with Penrose's hypothesis that the initial singularity must have vanishing 4-dimensional Weyl curvature tensor. A solar mass black hole has a Bekenstein-Hawking entropy of ∼ 10 21 per baryon. By Penrose's estimate, with 10 80 baryons in our universe, thermalization of gravitational degrees of freedom at the initial hot Big Bang would imply an entropy of 10 123 . 'Our extraordinarily special Big Bang' with low entropy [11] emerges naturally from the ground state of H P hys in the era of Cotton-York dominance; and 2nd Law thermodynamic 'arrow of time' and 'gravitational arrow of intrinsic time' (of increasing volume) point in the same direction.
Vanishing of both the traceless momentric and Cotton-York tensor implies the trace of the momentum π ∝H also vanishes, the extrinsic curvature is thus totally absent, which is the junction condition needed for Euclidean continuation of the metric (for instance, continuation to Euclidean S 4 at the conformally flat S 3 section at the throat of Lorentzian de Sitter metric). The quantum context may be in agreement with Hartle-Hawking 'no-boundary proposal' for the wavefunction of the universe [12]. But it should be noted the intrinsic time framework discussed here already allows a continuation of β in H Phys to imaginary values and Euclidean partition functions; moreover, from the formula of the emergent lapse function [3], imaginary β leads to emergent semi-classical space-times which are Euclidean in signature.
The Cotton-York interaction is introduced through the extensionπ i j → e gWπ i j e −gW =Q i j ; thusQ i j andQ †i j respectively annihilate the state e ±gW |0 . Moreover both these states are annihilated byH becauseQ †i jQ j i =Q i jQ †j i , due to the absence of Cotton-York ZPE. So any linear combination is a zero energy state. From the quantum perspective, the classical conformally flat configuration with vanishing Cotton-York tensor is the extremum of W , and thus precisely a saddle point for the ground state wavefunction wherein H ijkl (x, y) =

Emergence of Einstein-Hilbert Gravity
Ricci curvature terms become increasingly important in the potential after the initial era of Cotton-York dominance. They can be introduced in a manner which preserves the underlying structure which regulate the Hamiltonian by extending the Chern-Simons action with 3dDI invariants of the spatial metric. This not only guarantees 3dDI invariance; but also makes the Hamiltonian density the square-root of a (semi)positive-definite and self-adjoint object Q †i jQ j i ; and ensures the preservation of all these properties even under renormalization of the coupling constants. In increasing order of spatial derivatives, these invariants are Λ √ qd 3 x, EH = b √ qRd 3 x, and the Chern-Simons functional of the affine connection with dimensionless coupling constant. Even higher derivative curvature invariants will come along with super-renormalizable dimensional coupling constants, while the cosmological constant volume term commutes withπ i j due to the traceless projectorĒ i j(mn) . To wit, only the Einstein-Hilbert action in 3 dimensions and the Chern-Simons functional are relevant i.e. total W T = g wherein (again due to theĒ i j(mn) projector) only the traceless part of the Ricci tensor remains. The Hamiltonian density is thenH wherein the ZPE from incorporating the Einstein-Hilbert action in Remarkably, the potential for Einstein's theory, which is the Ricci scalar, and a (positive) c-number term emerge. This means the simple Hamiltonian density Q †i jQ j i (with all its aforementioned advantages) already contains Einstein's GR with cosmological constant. Furthermore,R i j and the Cotton-York tensor only appear in the highercurvature higher-derivative combination (gC j i +a √ qR j i )(gC i j +a √ qR i j ) (these 'non-GR' terms are automatically absent in homogeneous FLRW cosmology (that the Weyl Curvature Hypothesis holds in the Cotton-York era has been addressed), and also in constant curvature slicings of Painlevé-Gullstrand solutions of black holes[16]). Consequently, except for Cotton-York preponderance at very early times, Einstein's GR dominates at low curvatures and long wavelengths in a theory in which 'four-dimensional symmetry is not a fundamental property of the physical world'[17].
). TT conditions impose 4 restrictions on the symmetric α (ij) P hys (x), leaving exactly 2 free parameters. The action of U P hys (α T T ) = e − i (αT T ) i jπ j i d 3 x (which is thus local SL(3, R) modulo spatial diffeomorphism) on any 3dDI wavefunction would result in an inequivalent state. The caveat is TT conditions require a particular background metric to be defined. However, in Ref.
[19] a basis of infinitely squeezed states was explicitly realized by Gaussian wavefunctionals Ψ . 3dDI is recovered in the limit of zero Gaussian width with divergent lim ǫ→0fǫ → δ(0). These localized Newton-Wigner states are infinitely peaked at q B ij which can be deployed to actualize the TT conditions. The action of U P hys (α T T ) on these states would thus generate 2 infinitesimal local physical excitations at each spatial point.
In the preceding discussions, the entity δ(0) which denotes the 3-dimensional coincidence limit, lim x→y δ(x − y), was left untouched, with the understanding that it can be regularized, for instance, by normalized Gaussians of infinitesimal but non-zero width. However, the underlying SU (3) structure already provides unambiguous guidance on how to regularize the theory. The Hamiltonian assumes the elegant form, wherein δ(0) β d 3 x is a dimensionless volume element, its divergence to be absorbed by renormalization of β [20]. With the cancelation of on both sides of the Schrödinger equation, our universe is described by a fundamental equation with dimensionless Hamiltonian and intrinsic time. What is paramount to causality is not the actual dimension of time (an exemplar is intrinsic time interval measured with dimensionless redshift in FLRW cosmology), but the sequence and ordering in time. Even as will continue to leave its imprints in physics in the conversion factor between SU (3) generators T A and the momentric (hence momentum of the gravitational field), unification of gravitation and quantum mechanics comes with the demotion of its elementary significance. With dimensionless fundamental variables, the CR are[21] Quantum essence is still embodied in the non-commutativity, but Planck's constant is absent.
C. Soo, 'Wavefunction of the universe and Chern-Simons perturbation theory', Class. Quantum Grav. 19 (2002) 1051). Two configurations, Γ and Γg n , related by transformation gn with winding number n will then produce the same value ; and the wave function will be periodic under gn.
[14] The partition function of the total WT can be defined, in the sense that the combined 3-dimensional Einstein-Hilbert and Chern-Simons actions is renormalizable even though the dimension of b is 1 L . In fact the Cotton-York Chern-Simons theory is the UV completion of the Einstein-Hilbert action. Perturbing about the flat metric (which is an extremum of the combined action) yields the Hessian for transverse traceless δqij modes as bδ ik δ jl − gδ jk ǫ iml ∂m ∂ 2 δ(x − y). At large momenta, Chern-Simons theory dominates the propagator with 1/k 3 behavior, which signifies renormalizability, as loop integrals are over d 3 k and the vertices are at most cubic in k ; while at low momenta the Einstein-Hilbert 1/k 2 behavior dominates. Saddle point steepest descent computation about the extremum of W is a good approximation in the event of large g (i.e. the limit of small coupling, 1 √ g , in the vertices in Chern-Simons perturbation theory), and the quadratic term of the Hessian will be the main contribution, so we expect 1/k 3 two-point correlation functions in primordial quantum fluctuations, tempered by 1/k 2 behavior of Einstein's theory for small enough values of k to make b gk significant. 0)). The heat kernel, K(ǫ; x, y) with limǫ→0K(ǫ; x, y) = δ(x − y), presents, in the coincidence limit and with infinitesimal ǫ, the means to regularize ∇ 2 δ(0) for generic metrics. In terms of Seeley-DeWitt coefficients bn, 2σ(x, y) the square of the geodesic length, ∆V the Van Vleck determinant, K(ǫ; x, y) = (4πǫ) −3/2 ∆ 1/2 V (x, y) q(y)e −σ(x,y)/2ǫ ∞ n=0 bn(x, y; ∇ 2 )ǫ n , wherein ǫ is of dimension L 2 . Thus the heat kernel equation implies ∇ 2 K(ǫ; x, y) = ∂K(ǫ;x,y) ∂ǫ → −( 3 2ǫ − b1)δ(0) in the x = y and infinitesimal ǫ limit. Seeley-DeWitt coefficient b1 = R 6 yields [π i j , ib √ q R j i ] = − 5 12 b 2 δ(0) √ q(5R − 9 ǫ ). [18] In the full diffeomorphism generator, Hi = −2q ik ∇jπ kj = Di − 2 3 ∇iπ, the last term separately generates diffeomorphisms of (ln q 1 3 ,π) and commutes with the momentric and spatial unimodular d.o.f.
[20] It is conventional for the spatial label x to carry the dimension of length, even though it is non-dynamical and a mere dummy variable to be integrated over. But it makes more physical sense to take partial derivatives, and calculate curvatures, with respect to a dimensionless variable X = x/L. To wit, WT = gW [Γ(X)] + (bL) q(X)R(X)d 3 X, with dimensionless coupling constant (bL); furthermore, δ(0)