Self-adjoint Wheeler-DeWitt Operators, the Problem of Time and the Wave Function of the Universe Joshua Feinberg University of Haifa at Oranim & Technion Analytic and algebraic methods in physics 6-9 June 2016, Villa Lanna, Prague (baed on an old work with Yoav Peleg: Phys. Rev. D52(1995)1988) Thursday, June 9, 16
• self-adjoint Schrodinger operator with inverted quartic potential − x 4 • quantum-cosmological application • (motion along real axis, position x is bona- fide observable) Thursday, June 9, 16
Outline: • Introduction and set-up of problem: Robertson-Walker minisuperspace • Non-empty Robertson-Walker minisuperspace • inner products, boundary conditions, and the problem of time • Domain of the self-adjoint hamiltonian Thursday, June 9, 16
Roberston-Walker Minisuperspace * a theory for the Universe in the Planch epoch (where quantum gravity kicks-in), before commencement of inflation. * according to cosmomolgical principles, the Universe is described by an isotropic and homogeneous geometry. The standard model for this is the Robertson-Walker metric: ds 2 = − N 2 ⊥ d η 2 + a 2 ( η ) d Ω 2 3 standard line element on the unit three- sphere a ( η ) - the scale factor: the only dynamical degree of freedom N ⊥ - the lapse-function: a pure gauge d.o.f., non-dynamical 1 = 3 π conventions: ~ = c = 1 , G N = M 2 4 p Thursday, June 9, 16
• superspace - space of all possible quantum states of the metric • minisuperspace - restriction to simple geometry like RW (with only one dynamical d.o.f., reminiscent of the collective coordinate method for quantizing solitons) Thursday, June 9, 16
pure gravitational action for RW metric: a 2 − Λ a 3 ✓ ◆ � Z 1 − ˙ S g = d η N ⊥ a N 2 3 ⊥ cosmological constant; Λ ≥ 0 Λ a = da ˙ arbitrary evolution parameter; η d η gauge symmetry: reparametrization invariance under η → η 0 ( η ) together with N ? d η = N 0 ? d η 0 einbein, square root of g ηη a ( η ) = a 0 ( η 0 ) scalar under reprametrization Thursday, June 9, 16
a 2 − Λ a 3 ✓ ◆ � Z 1 − ˙ S g = d η N ⊥ a N 2 3 ⊥ is a pure gauge non-dynamical d.o.f., it has no conjugate momentum N ⊥ P a = ∂ L a = − 2 a ˙ a ∂ ˙ N ⊥ ✓ 1 ◆ g 2 = Λ 4 aP 2 a + a − g 2 a 3 H = − N ⊥ 3 gauge invariance yields the constraint (in Dirac’s parlance - a secondary first class constraint) − ∂ H = 1 a + a − g 2 a 3 = 0 4 aP 2 ∂ N ⊥ (it is the analog of the Gauss’ Law first class constraint in QED) Thursday, June 9, 16
− ∂ H = 1 a + a − g 2 a 3 = 0 the constraint 4 aP 2 ∂ N ⊥ requires a gauge fixing condition, e.g.: * N ⊥ = const . 6 = 0 in this gauge, becomes essentially the proper time η τ ** a more interesting gauge condition is the conformal time gague : η = t − a 4 potential N ⊥ = a ( t ) in which the hamiltonian has the standard kinetic term ✓ 1 ✓ P 2 ◆ ◆ 4 aP 2 a + a − g 2 a 3 4 + a 2 − g 2 a 4 a H = − N ⊥ = − Thursday, June 9, 16
Quantization (in coordinate representation) a → ˆ a P a = − i ∂ P a → ˆ ∂ a physical states should be annihilated by the constraint: ✓ 1 ◆ 4 aP 2 a + a − g 2 a 3 Ψ ( a ) = 0 − This is the Wheeler-DeWitt equation (ignored operator ordering issues here, they are not very important for our purposes) wave function of the Universe Ψ ( a ) a wave function for the physical state of the Universe, describing it in a gauge-invariant manner the analog of the WDW equation in QED is that the Gauss-Law constraint annihilates all physical states: ( r · E � 4 πρ ) | phys i = 0 Thursday, June 9, 16
the WDW equation describes essentially a particle moving in the potential V ( a ) = a 2 − g 2 a 4 with zero total energy. finite classical time-of-flight to infinity! Z ∞ da T = < ∞ p − V ( a ) a 0 > 1 /g Based on semiclassical considerations, a wave packet released from some finite ,will reach a infinity in finite (conformal) time. Thus, we have to impose appropriate boundary conditions at infinity, in order to preserve quantum probability, i.e., ensure unitary time evolution Since , appropriate b.c. have to be imposed at as well a ≥ 0 a = 0 Defining a self-adjoint extension of the hamiltonian is inevitable Thursday, June 9, 16
Nonempty RW Minisuperspaces for concreteness, consider filling the Universe with scalar field conformally coupled to the RW metric (ignore, for simplicity, quantum perturbations to the metric itself) φ ( η , x ) scalar field φ ( η ) homogeneous and isotropic case χ = π a φ rescaled scalar field total action for gravity and matter fields: S tot = S g + S matter = a 2 χ 2 + a − g 2 a 3 − χ 2 � Z + a ˙ − a ˙ d η N ⊥ N 2 N 2 a ⊥ ⊥ Thursday, June 9, 16
total hamiltonian H W DW = H tot = 1 � − N ⊥ a − 1 4 P 2 4 P 2 χ + a 2 − g 2 a 4 − χ 2 a quantization - in the coordinate representation: P χ = − i ∂ P a = − i ∂ ∂χ , ∂ a Wheeler-DeWitt equation: ∂ 2 ∂ 2 ✓ ◆ − 1 ∂ a 2 + a 2 − g 2 a 4 + 1 ∂χ 2 − χ 2 Ψ ( a, χ ) = 0 4 4 Thursday, June 9, 16
separation of variables: Ψ ( a, χ ) = ψ a ( a ) ψ χ ( χ ) ∂ 2 ✓ − 1 ◆ ∂ a 2 + a 2 − g 2 a 4 ψ a ( a ) = E ψ a ( a ) 4 ∂ 2 ✓ ◆ − 1 ∂χ 2 + χ 2 ψ χ ( χ ) = E ψ χ ( χ ) 4 we obtained two Schrodinger equations is some non-negative eigenvalue, either of the first equation or the second one E E ≥ 0 due to energy condition on matter fields the space of physical states is spanned by all these solutions need to equip them with time-independent inner product, in order to form the Hilbert space of physical states a well-defined inner product has to be time independent so that there be no conflict between time evolution of physical states and the definition of Hilbert space at each time slice Thursday, June 9, 16
Inner Products, Boundary Conditions, and the “Problem of Time” Quantization requires gauge-fixing, namely, a definition of “time”. The freedom left in making such a gauge choice leads to the “problem of time” in quantum cosmology. In our simple model, we can either choose matter fields as clock (i.e., as time coordinate), or the scale factor. These choices lead to utterly different Hilbert spaces of physical states, and consequently, to different physical realities. Self-adjointness of the (spatial part) of the WDW operator plays an important role in all these considerations Thursday, June 9, 16
Construction of inner products: ← → currents J 1 , 2 a, χ = i Ψ ∗ ∂ a, χ Ψ 2 1 Ψ 1 , 2 any pair of solutions of the WDW equation ∂ 2 ∂ 2 ✓ − 1 ∂ a 2 + a 2 − g 2 a 4 + 1 ◆ ∂χ 2 − χ 2 Ψ ( a, χ ) = 0 4 4 current conservation: can easily check the identity ∂ a J 1 , 2 − ∂ χ J 1 , 2 = 0 a χ Thursday, June 9, 16
choice number 1: scale factor as time a = t inner product: ∞ 1 ( a, χ ) ! Z h Ψ 1 | Ψ 2 i ( a ) = � i d χ [ Ψ ∗ ∂ a Ψ 2 ( a, χ )] | a = t =const . −∞ ∞ Z d χ J 1 , 2 = � a −∞ impose time independence: ∂ t h Ψ 1 | Ψ 2 i ( a ) = ∂ a h Ψ 1 | Ψ 2 i ( a ) = J χ ( χ = �1 ) � J χ ( χ = + 1 ) time independence of inner product implies J χ ( χ = + ∞ ) − J χ ( χ = −∞ ) = 0 Thursday, June 9, 16
This condition holds automatically due to finiteness of the inner product, which means spatial motion ψ χ ,n ( χ ) along follows that of the harmonic oscillator, with being the usual HO wave functions χ and the corresponding eigenvalues, and of course E n = n + 1 / 2 J χ ( χ = + ∞ ) = J χ ( χ = −∞ ) = 0 The spatial part of the WDW operator is automatically self-adjoint in this domain and there are no further constraints Thursday, June 9, 16
choice number 2: matter field as time χ = t inner product: ∞ 1 ( a, χ ) ! Z h Ψ 1 | Ψ 2 i ( χ ) = � i da [ Ψ ∗ ∂ χ Ψ 2 ( a, χ )] | χ = t =const . 0 ∞ Z daJ 1 , 2 = � χ 0 impose time independence: ∂ t h Ψ 1 | Ψ 2 i ( χ ) = ∂ χ h Ψ 1 | Ψ 2 i ( χ ) = J a ( a = 0) � J a ( a = + 1 ) time independence of inner product implies J 1 , 2 a ( a = + ∞ ) − J 1 , 2 a ( a = 0) = 0 Thursday, June 9, 16
Unlike the previous case, this condition sets non-trivial constraints on the spectrum of the spatial part of the WDW equation ∂ 2 ✓ − 1 ◆ ∂ a 2 + a 2 − g 2 a 4 ψ a ( a ) = E ψ a ( a ) 4 and requires the Schrodinger operator be self-adjoint. This is a non-trivial demand, since the potential energy is not bounded from below. Thursday, June 9, 16
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