# Resurrecting time in quantum gravity

Resurrecting time in quantum gravity

Resurrecting time in quantum gravity

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Gravity, Astrophysics, and Str**in**gs’02eds. P.P.Fiziev and M.D.TodorovSt. Kliment Ohridski University Press, Sofia, 2003An explicit **time** variable for cosmology and thematter-vacuum energy co**in**cidenceMyron BanderDepartment of Physics and Astronomy, University of California, Irv**in**e,CA 92697-4575, USAAbstract. By allow**in**g for non zero vacuum expectation values for some ofthe fields that appear **in** the Hamiltonian constra**in**t of canonical general relativitya **time** variable, with usual properties, can be identified; the constra**in**t playsthe role of the ord**in**ary Hamiltonian. The energy eigenvalues contribute to thevariation of the scale parameter similarly to the way matter density does. For auniverse described by a superposition of eigenstates or by a thermodynamic ensemblethe dom**in**ant contribution comes from energy, or equivalently effectivematter density, of the same order as the vacuum energy (cosmological constant).This may expla**in** the observed “co**in**cidence” of these two values.Keywords: **time** **in** GR, **quantum** cosmology, vacuum energy-matter densityco**in**cidenceCanonical Hamiltonian **gravity**, both classical and **quantum**, is a theory of constra**in**ts.A normal unitary evolution can be obta**in**ed only after some dynamical variable, whosecanonical conjugate will play the role of **time**, is identified and solved for as a functionof the other dynamical variables. Although this is the path to be followed **in** this work, itshould be mentioned that it is not the universal approach to the problem of **time** **in** cosmology.In the ‘no boundary’ approach of Hartle and Hawk**in**g [1] and **in** the ‘nucleation froma po**in**t’ view of Vilenk**in** [2] the existence or nonexistence of **time** is ignored and onlycorrelations between dynamical variables are looked for. In the extr**in**sic **time** approach,**time** is related to a particular foliation of a three geometry **in** four dimensional space**time**;some recent works [3] have used this method. Extensive reviews may be found **in**[4, 5] and more recently **in** [6]. As mentioned, **in** this work we shall look for a dynamicalvariable that can be identified as **time**. Some of the difficulties that this approach has hadare overcome **in** that we allow for some fields to acquire non-zero expectation values beforethe constra**in**t equations are solved. This will result **in** a Schröd**in**ger equation withthe Hamiltonian constra**in**t play**in**g the role of the Hamiltonian itself. Apply**in**g this to theevolution of the scale R(t) **in** cosmological models, we f**in**d that the energy eigenvalues E49

50 M. Bandercontribute **in** the same functional form as the matter density, which varies as 1/R 3 but ismultiplied by R 3 . A universe, whose wave function is a superposition of such eigenstates,may result **in** the effective matter density and the vacuum energy (cosmological constant)track**in**g each other. This would expla**in** the present “co**in**cidence” of these two values[7]. In obta**in****in**g these results we use the m**in**isuperspace approximation; at each step,however, we show how the terms we **in**troduce arise from a Lagrangian fully compatiblewith general relativity.Hamiltonian **gravity** and cosmology are usually treated **in** the, afore mentioned, m**in**isuperspaceapproximation, where the cont**in**uous set of gravitational and other degreesof freedom are reduced to a small number of collective, spatially constant fields. In thesimplest case, the dynamics of a Robertson-Walker-Friedman universe, with a metric[ ]drds 2 = N 2 (t)dt 2 − R 2 2(t)1 − kr 2 + r2 dΩ 2 , (1)are specified by the action for the one variable R(t)∫S = [P R dR − Ndt H G (P R , R)] (2)with P R be**in**g the momentum conjugate to R andH G = − G3πR P 2 R − 3π4G(kR − ΛR33)+ 2π 2 ρ(R)R 3 ; (3)G is Newton’s constant, Λ is the cosmological constant and ρ(R) accounts for the matterand radiation density. Vary**in**g this action with respect to the lapse function N yields theconstra**in**t H G (P R , R) = 0 and sett**in**g the variation of H G (P R , R) with respect to P Rand R to zero **in**sures the conservation of the energy-momentum stress tensor; there is noroom for both **time** and a dynamical variable that depends on it.To allow for the **in**troduction of **time** more fields have to appear. A Lagrangian for ascalar field τ,∫L τ = d 4 x √ −g gµν2 ∂ µτ∂ ν τ , (4)can be added to the one for **gravity**; **in** the m**in**isuperspace approximation this changesEq.(2) toS =∫ [( )]P R dR + π τ dτ − Ndt H G + π2 τ2R 3 , (5)where π τ is the momentum conjugate to τ. One then tries to make τ play the role of **time**for R or vice versa. A serious drawbacks of this approach is that the result**in**g Schröd**in**gerequation is hyperbolic and problems similar to those that occur **in** the first quantizedtreatment of the Kle**in**-Gordon equation also occur here. These problems would go awayif π τ were to appear l**in**early **in** the coefficient of N **in** Eq.(5). This is a goal of this work.To achieve this end we add to the follow**in**g to the Lagrangian for pure **gravity**:∫ { [ √−g gL t = d 4 µνx2 ∂ µτ∂ ν τ + bg µν ∂ µ τ∂ ν χ−c(g νν′ g λλ′ g σσ′ H νλσ H ν ′ λ ′ σ ′ + h2 ) 2] }+ dɛ µνλσ ∂ µ χH νλσ , (6)

An explicit **time** variable for cosmology and the matter-vacuum energy co**in**cidence 51τ is the field whose spatially constant part will ultimately play the role of **time**. χ isa cyclic field, 0 ≤ χ < 2π and H νλσ is an antisymmetric three **in**dexed tensor field.With h 2 positive the third term **in** eq. (6) ensures that the spatial parts of H acquire anexpectation√value. It should be noted that due to the ɛ µνλσ **in** the coefficient of d, no−g is necessary for this term and, by design, no (∂χ) 2 appears **in** the Lagrangian; thecyclic nature of the field χ can result from χ be**in**g the phase of a regular complex fieldwhose magnitude is fixed to some vacuum expectation value. A k**in**etic energy for H νλσcan occur as can a potential **in**volv**in**g τ. We will show that this produces a π τ appear**in**gl**in**early **in** the constra**in**t.The m**in**isuperspace approximation to (6) is∫L tMS = dt[R 32N( ) 2 dτ+ b R3 dτ dχdt N dt dtReplac**in**g H ijk by its vacuum expectation valuethe Hamiltonian correspond**in**g to L tMS isH tMS =−cNR 3 ( H ijk H ijk /R 6 − h 2) 2+ ddχdt ɛijk H ijk]. (7)< H ijk >= R3 hɛ ijk6N [b 2 R 3 bπ τ (π χ − dhR 3 ) − 1 ]2 (π χ − dhR 3 ) 2 . (9)In the fixed π χ = 0 (a small value for the parameter b would favor π χ = 0) sectorwe recover a term l**in**ear **in** π τ . Rescal**in**g τ we obta**in**, **in**stead of the Wheeler-de-Wittequation, a Schröd**in**ger equation with H G act**in**g as a normal Hamiltonian [8],(8)−i ∂∂τ + H G = 0 . (10)The contribution of an eigenvalue E of Eq.(10) for H G correspond**in**g to Eq.(3) cannotbe dist**in**guished from the contributions of matter, ρ M (R) ∼ 1/R 3 , to the evolution of thescale parameter R; thus we may consider (E − 2π 2 ρ M R 3 ) to be the effective energyeigenvalue, or equivalently (ρ M − E/2πR 3 ) the effective matter density. Interest**in**gresults obta**in** for a universe that, rather then be**in**g a state with a def**in**ite energy, is asuperposition of such eigenstates∫Ψ(R, τ) = dEf(E)ψ E (R)e −iEτ ; (11)ψ E (R) is an eigenfunction of H G and f(E) describes the energy wave packet. Thepossibility now arises that at different R’s different E’s dom**in**ate the **in**tegral **in** Eq.(11).Should the E and R dependence ln ψ E (R) appear **in** the comb**in**ation( ) ΛRln ψ E (R) = R 3 3g , (12)E

52 M. Banderwith g some function and with ln[f(E)] vary**in**g no faster than c|E|, the dom**in**ant contributionto Ψ(R, τ) **in** Eq.(11) comes from E ∼ ΛR 3 . As was remarked earlier, Econtributes to the evolution of R the same way as ρ M R 3 and thus for any given R theeffective ρ M ∼ Λ. Thus the observation that today ρ M ∼ Λ [7] may not be a co**in**cidencebut may hold true at all scale factors R [9].An example of ψ E (R) scal**in**g accord**in**g to Eq.(12) is the WKB approximation for theobservationally favored case of a flat, k = 0, universe [7] **in** an epoch where the radiationdensity may be neglected; ψ E is a comb**in**ation of(ψ (±)E (R) = exp ± 3π ∫ R2G i0√ )Λr4dr3 − 4GEr . (13)3πThe **in**tegrals **in** the exponents of the above have the desired scal**in**g property. For a universedescribed by a thermodynamic density matrix at a temperature 1/β the dom**in**antcontributions to the diagonal density matrix elements, those that account for the distributionof the scale factor R,∫ρ(R, R) = dEψ E (R)e −βE ψE(R) ∗ , (14)likewise come from E ∼ ΛR 3 .References[1] Hartle, J. B. and Hawk**in**g, S. W. (1983) Phys.Rev. D28 2960.[2] L**in**de, A. D. (1984) JETP 60, 211; Vilenk**in**, A. (1987) Phys.Rev. D37 888.[3] Giribet, G. and Simeone, C., gr-qc/0106027; Simeone, C., gr-qc/0108081.[4] Kuchař, K. V. (1992) **in** Proc. the 4th Canadian Conference on General Relativity and RelativisticAstrophysics, G. Kunstatter, D. V**in**cent and J. Williams (Eds.), World Scientific, S**in**gapore.[5] Isham, C. J. (1993) **in** Integrable Systems, Quantum Groups and Quantum Field Theories,L.I. Ibort and M.A. Rodriguez (Eds.), Kluwer Academic Publishers, London, (see also grqc/9210011)[6] Barv**in**sky, A. O., **in** Proc.the 9th Marcel Grossmann Meet**in**g On Recent Developments InTheoretical And Experimental General Relativity, Gravitation And Relativistic Field Theories(MG 9) 2-9 July 2000, Rome, Italy, (see also gr-qc/0101046).[7] Perlmutter, S., et al. (1998) Nature 391 51; (1998) Astrophys. J. 517 565; Riess, A., et al.(1998) Astron. J. 116 1009; Garnovich, P. M., et al. (1998) Astophys. J. 509 74.[8] The last term **in** Eq.(9) contributes to the cosmological constant.[9] An alternate suggestion that it is the effective cosmological constant that varies has recentlybeen made by K. Griest (see astro-ph/0202052).