2nd LeCosPA International Symposium, Nat.Taiwan U. (December 17th 2015) Quantum geometrodynamics with intrinsic time development 許 Chopin Soo 祖 斌 Department of Physics, Nat. Cheng Kung U, Taiwan Everything? What about (let’s not forget) the `problem of time’ ? A very wise man once said “What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know”, pondering the mystery of what time really is, Saint Augustine (of Hippo) (354-430 A.D.) wrote in his Confessions《懺悔錄》, “Si nemo ex me quaerat scio; Si quaerente explicare velim, nescio ’’ Time: We all have some intuitive understanding of time. But what is time? Where does it comes from? Physical theory of space and time Einstein’s GR: Classical space-time <-> (pseudo-Riemannian)Geometry Quantum Gravity: “`Spacetime’ - a concept of limited applicability” ---J. A. Wheeler What, if anything at all, plays the role of “time” in Quantum Gravity? Importance of time: 1)Quantum probabilities are normalized at fixed instances of time 2)Time <-> Hamiltonian as generator of time translation/evolution Hamiltonian <-> Energy Time: Is it 1) fundamental (present even in Quantum Gravity)? 2) emergent (present in the (semi)-classical context only)? 3) an illusion (!)? … Nemeti, Istvan et al. arXiv:0811.2910 [gr-qc] Getting to know Einstein’s theory Spacetime tells matter how to move; matter tells spacetime how to curve John Wheeler, in Geons, Black Holes, and Quantum Foam p.235 This saying has seeped into popular culture and public discourse; but with all due respect to Wheeler, it is the Hamiltonian which tells everything (including the metric) how to move! from The Early Universe (E. Kolb & M. Turner) AMD(1959) ADM-50 (2009) => “frozen in time” t Matrix elements: Suggestion: DeWitt: Superspace metric has (- + + + + +) signature. 1/4 –ve mode: Intrinsic time ζ =[det(qij)] But Wheeler-DeWitt-Equation and Hamilton-Jacobi Eq. still 2nd order in “intrinsic time” (which is multi-fingered) Wheeler: Three Dimensions, NOT Four! A curved 4D space-time is an emergent, (semi-)classical, concept of limited applicability in Quantum Gravity. ?=>? Then `4-covariance’ cannot be fundamental to Quantum Gravity! DeWitt Metric on Superspace: Deformation parameter Supermetric: (=1 for DeWitt supermetric) in GR Only 3D diffeomorphism symmetry => Generalized supermetric Symplectic potential of Geometrodynamics: 1st order in (intrinsic) time Schrodinger and Hamilton-Jacobi Eqs with NON-VANISHING Hamiltonian generating “time” evolution. But still in Schwinger-Tomonaga form with many-fingered x-dependent ”intrinsic time interval” δln q(x). Is 4d diffeomorphism invariance present a)in classical GR? b)in quantum gravity? How is intrinsic time interval correlated to space-time intervals of classical solutions of Einstein’s theory? Dirac Algebra *Structure FUNCTIONS (depending on dynamical variables) *Not the algebra of 4D diffeomorphisms! Role of constraints in Einstein’s theory of GR: On-shell (modulo constraints +EOM), The constraints do generate 4d diffeomorphisms Eventhough Dirac algebra is NOT algebra of 4d diffeomorphisms [ ] Completely arbitrary N and Ni ? Wouldn’t that make everything associated to dt arbitrary? a priori The a posteriori lapse function, N, of Einstein’s theory: General Relativity a posteriori relation for N and dispense(!) with the Hamiltonian constraint Similarly for effectively reproduces the a posteriori form of N in GR, The advantage is that this will allow us to tweak/modify the potential without getting into inconsistencies with closure of the constraint algebra, and yet yield the same dynamics in the limit when the potential is that of GR Proposal: (P. Horava) give up 4d Diff. Invariance in favor of 3d Diff. Invariance for power-counting renormalizability Disadvantage of 4d Diff. Inv.: Tweaking potential terms (+ higher spatial derivatives) changing the kinetic term (+ higher time derivatives) Strategy: improve convergence by adding higher SPATIAL derivatives, without affecting the time derivatives (kinetic term) Skip ?! Question: If “time” (albeit intrinsic time variable) were a quantum field (part of the quantized metric in QG), would this not run into problems with Pauli’s No-Go theorem (time should be a parameter (c-number) and not a quantum variable) ? Problem(s) of Time: 1)Problem of Time in Gravity ---> intrinsic time variable to “deparametrize the theory” (correlate the changes of TT d.o.f. with intrinsic time development) 2)Problem of Time in Quantum Mechanics (Pauli’s no-go theorem) 3)Problem of Many-Fingered (Schwinger-Tomonaga) Time 2)Problem of time in Quantum Mechanics Circumventing Pauli’s no-go theorem (time should not be an operator) “Covariant” extension of [x,p]=iЋ pg.63 dT/dt = 1 => [T,H] = iℏ Observable T would function as a perfect "clock" M D Srinivas and R Vijayalakshmi Pramana, Vol. 16, No. 3, March 1981, pp. 173-199 http://tompashby.info/ Proc. R. Soc. Lond. A. 1926 111 758 Extended phase space P.B. Circumventing Pauli’s Theorem: Following and extending Dirac’s logic and paradigm : 1)time in all of QM can be traced to an extra conjugate pair 2)This extra pair must be universal to all systems since we apply the universal Schrodinger equation (with the same t) albeit with different H’s to different systems of interest in QM 3)Naturally this extra pair should come from variables which determine the structure of spacetime i.e. Gravitation Lo and behold ! (t, -W) <-> (ln q1/3,Π) H - W = 0 <-> = 0 Caveat: In QG, (ln q1/3(x), Π(x)) are FIELD variables => Schwinger-Tomonaga many-fingered time. Meaning of δT δT ~ change in logarithm of volume of the universe 1)This is exact in the FULL theory and NOT minisuperspace or FRW model 2)Observationally, our Universe has ALWAYS increased in size and lnV~Time 3) In FRW models, time interval between then (a) and now (a0) is δlnV = 3ln(a0/a) ; a0/a = (1+ z); z = redshift … What QG took away, QG gives back in another way What replaces the “time-ordering of events “ which underpins causality? Time-ordered development (evolution) of quantum states (the physical quantum state is the fundamental gauge-invariant entity that can be ordered in time) Dyson series Modifications to the theory Original metric and conjugate momentum variable (Klauder’s) Momentric , instead of momentum variable John Klauder (U. Florida) Momentric is better than momentum ! => Quantizate in metric, but not in momentric representation may(?) help to explain why our universe seems inherently metric, and “not conjugately realized” “Improvements” to quantum theory of GR General Relativity without paradigm of space-time covariance, and resolution of the Problem of Time, C. S. and Hoi-Lai Yu, Prog. Theor. Exp. Phys. (2014) 013E01..