the components of the normal as a contravariant vector n. Notice now that n fi n ff = 1 and hence ( 0= (4) r i (n fi n ff ) = (4) r i n fi n ff + n fi

Transcription

1 1 Introduction On the following pages we show how to quantize the Bianci V cosmology using three different methods: The Square Root Hamiltonian approach, The geometrodynamic approach and the approach which leads to Wheeler-Dewitt equations (we could say Standard Quantization). In Section we relate the formulas developed on the +1 split of spacetime by [MTW pg ]. The main formulas will be the equations of Gauss and Codazzi. We then show how to recover the Lagrangian in [MTW, eq. 1.84] from the Gauss and Codazzi equations. Section will be the details on how to recover the super-hamiltonian" and super-momentum" in [MTW, eq. 1.9, eq. 1.9] using the standard Hamiltonian approach. Section 4 will use the Bianchi V metric tensor to illustrate the general approach of Section. Section concludes with the three different quantization schemes. +1 Split of Spacetime The basic idea of the +1 split of spacetime is to write the four metric, (4) g fffi [ff; fi =1::4] in terms of the three metric, () g ij = g ij, N i (shift) and N (lapse) [i,j=1..] where the shift and lapse are to be specified from the outside" and basically describe how to recover the 4 geometry from a family of space-like slices [see MTW ]. In term of covariant components we can write (4) (4) g g fffi = (4) 00 g 0k Nm N = m N N k (4) : (1) g i0 g ij N i g ik In terms of contraviant components (i.e. the inverse of (4) g fffi )we can write Let (4) g fffi = (4) g 00 (4) g 0k 1 = (4) g i0 g ij N N i N N k N g ik N i N k N : () n ff =( N;0; 0; 0) () be the components of a time-like, unit normal covariant vector n. Raising () with () gives 1 n fi i = N ; N (4) N 1

2 the components of the normal as a contravariant vector n. Notice now that n fi n ff = 1 and hence ( 0= (4) r i (n fi n ff ) = (4) r i n fi n ff + n fi (4) r i n ff = (4) r i n fi n ff + n fi (4) r i g fffi g flfi n fl (insert ffi fl ff) = (4) r i n fi n ff + n fi g fffi (4) r i n fi (raise and use g mk is covariant constant) = (4) r i n fi n ff + n ff (4) r i n fi (lowering) = (4) r i n fi n ff : That is, (4) r i n is perpindicular to n and hence must point" in the spatial direction. This indicates that we can write It then follows that (4) r i n = K j i e j: (5) K im = K j i g jm = K j i e j e m = e m (4) r i n = r i ( e m n)+n (4) r i e m =0+n (4) r i e m where we used e m and n perpindicular. It can be shown that (4) r i e m = K ij n n n +() a ij e a (6) It can be shown that the covariant derivative of n using () reduces to n i;m = N (4) 0 im and hence K im = n i;m where if we use the definition of it follows easily [see MTW 1.67] that we get the extrinsic curvature tensor K im = 1 Nijm + N kji g im;t N Now the idea is to write the Riemann tensor, (4) Rijk m in terms of extrinsic curvature and () R m ijk. These are exactly the Gauss and Codazzi equations. In a straightforward calculation which uses (6) [MTW, 1.75, 1.76] arrive at the Gauss and Codazzi equations (4) R m ijk = () R m ijk + 1 Kij K m k K ik K m j n n (4) R 0 ijk = 1 Kijjk K ikjj n n (7) In previous notes of mine (ein.tex), Einsteins equations are derived from the Lagrangian S = 1 Z p (4) g(l geometry + L matter)d 4 x 16ß (8) (9)

3 where L g = (4) R, the Ricci scalar. The Lagrangian we use in these notes will again be the Ricci scalar but this time written in terms of () R and extrinsic curvature rather than just the metric. This is done by contracting the Gauss and Codazzi equations: (4) R ik = (4) R m imk = () R ik + 1 n n (K imk m k K ik Km) m (first contraction) raise with g!! ia g ia R ik = (4) R a k = () R a k + 1 n n (Ka m Km k K a k Km m) (4) R = (4) R a a = () R + 1 n n (Ka m Km a K a a Km m) (second contraction) Now we introduce the notation TrK = K m m and TrK = K i j Kj i so the two contractions of the first Gauss Codazzi equation (8) yields (4) R = () R + 1 n n (TrK (TrK) ) (10) As for the second Gauss Codazzi equation (9) it is shown in [MTW, pg. 519] that 1 (4) R 0 i0i = 1 n n n n ((TrK) TrK ) (11) Adding (10) and (11) gives (4) R = () R + 1 and finally using that n n = 1, p (4) g = N we write the final action S = 1 Z 16ß n n ((TrK) TrK ) (1) p () g and assuming no matter p [R + TrK (TrK) ]N gd 4 x (1) z } L modified where R and g are the three dimensional Ricci scalar and metric tensor. H modif ied From L modif ied In this section, using the standard relationship between the Hamiltonian and Lagrangian (i.e H = p _q L), we derive the super-hamiltonian and the super- Momentum. It must be said that this argument will not pay attention to the divergence terms. This is not so bad as the variational principle does not deal with the boundaries (using the divergence theorem we can kill the

4 boundary terms, see ein.tex for the details on where a divergence occurs). From section 1 we recall that the modified Lagrangian was given by L mod = [R + TrK (TrK) ]Ng 1= = [R + K i j Kj i Ka a Kb b]ng 1= = [R + g mi g nj K mj K ni g ra g sb K ra K sb ]Ng 1= = RNg 1= + Ng 1= 6 g mi g nj 4 4N (N mjj + N jjm _g mj )(N nji + N ijn _g ni ) 7 5 z } g Ng 1= 6 ra g sb 4 4N (N rja + N ajr _g ra )(N sjb + N bjs _g sb ) 7 5 (using (7)) z } (b) (a) Now recall that _q First we compute so by analogy we need to compute ßcd _g _g cd = Ng 1=» g mi g nj Φ mj ( ffi 4N cd )(N nji + N ijn _g ni )+( fficd)(n ni mjj + N jjm _g mj ) Ψ Λ g ci g nd K ni g md g cj K mj = Ng1= N = g 1= K cdλ : (14) _g cd = Ng 1= g ra g sb = Ng1= N g cd K b b + g cd KaΛ a = g1=» ( ffira) cd K sb N +( fficd sb) K ra N Λ g ra g sb ffi cd ra K sb + g ra g sb ffi cd sb K ra = Λ g 1= g cd TrK : (15) Adding (14) and (15) yields the congugate momenta [MTW, eq. 1.91] ß cd = g 1= g cd TrK K cdλ (16) 4

5 Thinking now thatwe want to parallel H = p _q L we need to compute _g ab. From (16) we have ß ij = g 1= g ij TrK K ijλ =) g 1= ß ij g ij TrK = g ni g mj K nm =) N[g 1= ß ij g ij TrK]=g ni g mj (N njm + N mjn _g nm ) Now notice that =) Ng ia g jb [g 1= ß ij g ij TrK]=ffi n a ffim b (N njm + N mjn _g nm ) =) N[g 1= ß ab g ab TrK]+(N ajb + N bja )= _g ab (17) g ij ß ij g 1= + g ij K ij = g ij g ij TrK =) g 1= TrΠ+TrK =TrK =) 1 g 1= TrΠ=TrK (18) Substituting (18) into (17) gives [MTW, eq ] _g ab =Ng ß 1= ab 1 g abtrπ +(N ajb + N bja ) (19) Again paralleling H = p _q L, we compute H = ß ij _g ij L. This will require the use of two formulas. The first follows easily from (18): 1 g 1= TrΠ=TrK =) (TrK) = 1 4 g 1 (TrΠ) (0) The second follows from (16): and hence ß ij = g 1= g ij TrK K ijλ =) ß j a = g 1= (ffi j a TrK Kj a) (TrΠ) = ß j a ßa j = g[(ffi j a TrK Kj a)(ffi a j TrK Ka j )] = g[ffi j j (TrK) (TrK) TrK ]=g[(trk) (TrK) + TrK ] = g[(trk) + TrK ] =) TrK = g 1 TrΠ (TrK) (1) 5

6 Now we compute» ß ij _g ij L mod = ß ij Ng ß 1= ij 1 g ijtrπ +(N ijj + N jji ) Ng 1= [R + TrK (TrK) ]» = Ng 1= TrΠ 1 (TrΠ) + ß ij (N ijj + N jji ) Ng 1= [R + TrK (TrK) ]» = Ng 1= TrΠ 1 (TrΠ) + ß ij (N ijj + N jji ) (using (0) and (1)) Ng 1= R Ng 1= g 1 TrΠ g 1 (TrΠ) + Ng 1= 4 g 1 (TrΠ) = Ng 1=» TrΠ 1 (TrΠ) Ng 1= R Ng 1= TrΠ + 1 g 1= (TrΠ) +ß ij (N ijj + N jji ) = Ng 1=» TrΠ 1 (TrΠ) Ng 1= ß ik jk N k +(ß ij N i ) jj +(ß ij N j ) ji z } divergence terms=0 Finally we have [MTW, 1.9, 1.9] H mod = Ng 1=» TrΠ 1 (TrΠ) g 1= R z } H +N k ( ß ik jk z } H k ) () where H is called the super-hamiltonian and H k is called the super-momentum. The Hamiltonian is simply written as H mod = NH + N k H k The next section takes the general computations from this section and redoes them using a specific example, the Bianchi V cosmology. 4 Quantization of the Bianchi V Cosmology This section has a MAPLE program associated with it (bianchi5.mws). This section will relay the main points of the program but for the details see Appendix 1. The Maple program is broken into three sections: Standard (Dirac), Square Root Hamiltonian and Geometrodynamic Quantization. The following section will be split similarily: 6

7 4.1 Standard (Dirac) Quantization The three dimensional part of Bianchi V metric tensor is given by 0 1 e Ω 4fi g ij 0 e Ω x+fi p ++ fi 0 A 0 0 e Ω x+fi p + fi where the g 00 component of the full four dimensional metric is N. It is easily computed that p g = e Ω x As the metric is diagonal we let shift=n i =0 and thus we can compute the extrinsic curvature (i.e. the components of the diagonal tensor) as K 11 = e Ω 4fi + ( _Ω 4 fi _ + ) N p K = e Ω x+fi ++ fi ( _Ω+ fi _ + + p fi _ ) N p K = e Ω x+fi + fi ( Ω+ _ fi _ + p fi _ ) N Using the inverse of the metric it can be shown that TrK = _ Ω N It can also be shown that the Ricci scalar is given by TrK = ( _ Ω + _ fi + + _ fi ) N : () R = e Ω 4fi + : As can be seen from (1) wenowhave the ingredients to build the Lagrangian L mod = e Ω x (e Ω+4fi + 4 _Ω +4 fi _ + +4 fi _ ) : Nß Recalling that S = R Ng 1= [L mod ]dtd x we can write Z " e Ω S = e x (e Ω+4fi + 4 _Ω +4 fi _ + +4 fi _ # ) dt Nß Z e Ω (e Ω+4fi + = 4 _Ω +4 fi _ + +4 fi _ ) V dt Nß and hence think of the Lagrangian as d x: L mod = V Nß e Ω (e Ω+4fi + 4 _ Ω +4 _ fi + +4 _ fi ) 7

8 Now shifting to the Hamiltonian picture where, e.g., we find that P Ω = V _Ω solving 4Nß e Ω =) _Ω = 4NV ß e Ω P Ω H mod = P Ω _Ω+P fi+ fi+ _ + P fi fi _ L Vß V = N e Ω (P Ω P fi + Pfi )+ ß e Ω+4fi + (4) where the term in parentheses is playing the role of the super-hamiltonian, H. Varying with respect to N yields Z Z Z ffis = ffi(h)dt = ffi(nh)dt = H(ffiN)dt =0 =) H =0: So the constraint equation is given by P Ω P fi + P fi = 9V From H mod we can write Hamiltons = + 8ß e = = fi+ V eω P = fi+ V eω Ω = f(p = g(p Ω;P fi ± ;fi +;N) 64ß e 4Ω+4fi + : (5) Given then a curve in the configuration space which satisfies Hamiltons equations we can then derive the Hamilton-Jacobi equations by varying one of endpoints in both space and time [see MTW, pg. 487]. The Hamilton-Jacobi equations in general follow from ffis = pffiq Hffit and are given by ffis ffiq ffis ffit = p = H q; ffis ffiq ;t : 8

9 For our specific problem we find that our constraint equation becomes the Einstein-Hamilton-Jacobi equation ffis ffis ffis = 9V ffiω ffifi + ffifi 64ß e 4Ω+4fi + : and the evolution is given by (since H=0) ffis ffit =0 So here is a problem: By doing Standard (Dirac) quantization we have an evolution equation which says that the action is constant in time. We now quantize the E-H-J equations using the canonical scheme ffis ffiω ; 7! ~i ; ffifi + 7! ~i ; ffis 7! ~i to obtain the Wheeler-Dewitt ψ = 64ß e 4 ^Ω+4 ^ fi + ψ: (6) and the trivial state function evolution equation (Schröedinger =0 4. Square Root Hamiltonian Quantization The Square Root Hamiltonian approach starts from the standard Lagrangian and imposes the condition from Dirac quantization that H=0 and hence L mod = P Ω _Ω+P fi+ _ fi+ + P fi _ fi H mod = P Ω _Ω+P fi+ _ fi+ + P fi _ fi : Now the idea is to salvage the Hamiltonian picture. This is done by solving the constraint equation (5) for P Ω and to let this be the new Hamiltonian, H sqrt. This gives H sqrt = 1 r 64Pfi Pfi + 9V e 4Ω+4fi + : ß For this to be the Hamiltonian we must have that _Ω=1. We can do this if we let t =Ω. This gives the new action + S sqrt = P + H sqrt 9

10 from which we can derives Hamiltons equations then Hamilton-Jacobi equations and finally the quantized version (using the same canonical quantization from section 4.1). The interesting thing about this method is that to salvage the Hamiltonian structure we were forced to choose Ω=t. Now according to Hamiltons equations (and using _Ω = 1)we obtain _Ω Ω = 4NV ß e Ω P Ω =1 which forces us to fix our control over lapse by N sqrt = e Ω 4V ß P Ω 4. Geometrodynamic Quantization The idea behind Geometrodynamic Quantization is similar to the Square Root formalism in that Ω is not treated as a dynamical variable right. But rather than let a constraint fix this for us(because we saw previously this fixed N) take Ω out of the Hamiltonian right from the outset. That is, we use the same Lagrangians as the previous two methods but the Hamiltonian becomes H dyn = P fi+ _ fi+ + P fi _ fi L mod = Nß V (P fi + + Pfi )eω VN ß e Ω+4fi + + V _ 8Nß e Ω Ω (7) Hamiltons equations become (the same as Dirac Quantization except for a minus sign dyn =@fi + and missing the Ω + = VN 8ß e Ω+4fi = dyn = fi+ V eω P dyn = fi+ V eω P fi Varying with repect to N gives the new constraint equation ß ffih dyn = V (P fi + + Pfi )eω VN ß e Ω+4fi + V 8N ß e Ω ffin =0 =) ß V (P fi + + P fi )eω VN ß e Ω+4fi + =) ß V (P fi + + P fi )eω = VN ß e Ω+4fi V 8N ß e Ω =0 V 8N ß e Ω (8)

11 Now recall that (4) in Dirac Quantization the Hamiltonian was written as H = NH where H was independent of N and hence gave rise to the constraint H = 0 which in turn gave a trivial Schrödinger equation. Geometrodynamic is different in this sense because H = NH where H is dependent on N and hence gives rise to a non-trivial Schrödinger = ^H dyn ( ^Pfi ± ; ^Ω; ^ fi + ; ^_Ω;N)ψ where (making everything in (7) an operator) ^H dyn = Nß V ( ^P fi + + ^P fi ^Ω )e VN ß e ^Ω+4 fi ^ + + V ^Ω 8Nß e _^Ω such that 7! ^Pfi ± = ~i ffifi ± The following will be a what is a function of what" argument describing how the constraint (8) and different choices of time come in play. Assume that we have solved Schrödinger's equation. Call this solution Ψ s ( ^Pfi ± ; ^Ω; ^ fi + ; ^_Ω;N): Now applying the constraint equation (7) gives Ψ s (^Ω; ^ fi + ; ^_ Ω;N): If we use N=1 for our time (i.e. proper time as N dt + g ii dx i dx i = dt + g ii dx i dx i ) then Ψ s (^Ω; fi ^ + ; ^_Ω): But if we use Yorke's time then t = TrK by () and hence we have which gives _Ω = _Ω(N; t) = Nt Ψ s ( ^ fi + ;N;t): =) Ω=Ω(N; t) 5 Things Still To Be Done ffl Modify the MAPLE code to be more robust in the sense of being able to input any cosmology and having it run straight through to give the Wheeler-Dewitt equations and the Schrödinger equation for both Square Root Quantization and Geometrodynamic Quantization. (Note: The places in the code where there where cut and pastes are clearly marked) 11

12 ffl Go through the solving of each of the Schrödinger equations to see the actual form of the wave functions. ffl And of course think more about time and the subtleties of the previous arguments. 1