Erratum in Equivalent and inequivalent canonical structures of higher order theories of gravity, Published in PHYSICAL REVIEW D 96, 084025 (2017) Ranajit Mandal† , Abhik Kumar Sanyal‡ October 28, 2018 †
Dept. of Physics, University of Kalyani, West Bengal, India - 741235. Dept. of Physics, Jangipur College, Murshidabad, West Bengal, India - 742213.
‡
The correct solutions (86) to the field equations (84) and (85) are, a = a0 eHt ; φ = φ0 e−Ht ; Λ(φ) = −
φ2 1 ; V = 6αH2 + H2 φ2 . 2 48H 2
(1)
Correct expressions (92) particularly for momenta are then x = 2Hz
√ 2 2 3α x 72βH x Ha0 φ0 √ − √ √ px = Q = √ + 2 2 2 x 2H 2 2 √ √ 6Ha0 φ0 √ pz = −3αH z − 72βH3 z − 4 z Ha3 φ3 pφ = − 02 0 φ √
3 2
(2a)
√
Hence the integrals in (93) are evaluated as, √ Z 3 √ 3 3 2Ha20 φ20 √ 2α 3 48βH 2 3 2 2 x = 4αHz 2 + 96βH3 z 2 − a20 φ20 H z px dx = √ x + √ x − 2 2 2H Z 2 2√ 3 3 Ha φ 0 0 pz dz = −2αHz 2 − 48βH3 z 2 − z 2 Z √ Ha30 φ30 = Ha20 φ20 z pφ dφ = φ
(2b) (2c) (2d)
(3a) (3b) (3c)
Therefore, the correct expression for S0 in equation (94) is, 3
3
S0 = 2αHz 2 + 48βH3 z 2 −
Ha20 φ20 √ z. 2
(4)
The correct expression for the zeroth order on-shell action (95) is # Z " H2 a30 φ20 eHt 2 3 3Ht 4 3 3Ht A = Acl = 6αH a0 e + 144βH a0 e − dt. 2
(5)
On integration, the correct expression for zeroth order on-shell action reads as, A = Acl = 2αHa30 e3Ht + 48βH3 a30 e3Ht − 1 Electronic
address:
†
[email protected] ‡ sanyal
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Ha30 φ20 eHt , 2
(6)
which is the same as S0 in 4 . Since everything is fair, so one can compute the semiclassical wave function, which now reads as, 3
i
3
Ψ = ψ0 e ~ [2αHz 2 +48βH 0.0.1
3
z2−
2 Ha2 0 φ0 2
√ z]
.
(7)
Canonical Quantization
We have obtained a different phase-space Hamiltonian (??), fixing δhij = 0 = δR at the boundary. Already we have observed that the standard canonical formulation techniques (Ostrogradski’s, Dirac’s and Horowitz’) fail to produce a viable Hamiltonian of the theory. Now naturally the question - “Is the Hamiltonian and its quantum counterpart obtained following modified Horowitz’ technique viable?” must be answered. The accountability of the Hamiltonian (??) may be explored through a viable semiclassical approximation corresponding to the modified Wheeler-de-Witt equation, which reads, 2 2 n ∂ 1 ~2 ∂ ~2 ∂ 2 Ψ x i~ ∂Ψ √ =− + Ψ− − 5 + 12k Λb0 pc φ 36βx ∂x2 x ∂x 2xz 2 ∂φ2 z z ∂z 2z 2 " # (8) 3 x 2k 18kβ x 2k Λ02 x x4 12kx2 V z2 2 ˆ e Ψ. + 3α − − + + 3 + + 72k + Ψ = H 2z x z z x z 2z 2 z x b 0 pc where, n is the operator ordering index. Operator form of Λ φ appearing on the third term on the right hand side may be performed only after knowing a specific form of Λ(φ) . Specific form of Λ(φ) is also required to investigate the behaviour of the quantum theory, under certain appropriate semi-classical approximation, which may only be obtained from the solution of the classical field equations corresponding to action (??). We use the same inflationary solutions presented in (??). Correspondingly, the modified Wheeler-de-Witt equation (8) may now be expressed as 3 3αx H2 φ2 x3 i~ ∂Ψ ~2 ∂ 2 n ∂ ~2 ∂ 2 Ψ Hx2 ∂Ψ ∂ V0 z 2 √ =− + Ψ − + i~ + (φΨ) + + + Ψ, φ 9 36βx ∂x2 x ∂x 2xz 2 ∂φ2 ∂φ ∂φ 2z 2z 5 x z ∂z 2z 2 (9)
where Weyl symmetric ordering has been performed in the third term appearing on the right hand side. Now, again under a further change of variable, the above modified Wheeler-de-Witt equation, takes the look of Schr¨odinger equation, viz., ∂Ψ ~2 i~ =− ∂σ 54β
1 ∂2 n ∂ + 2 x ∂x2 x ∂x
~2 ∂ 2 Ψ Hx2 Ψ− + i~ 2 7 3xσ 3 ∂φ2 3σ 3
∂Ψ ˆ e Ψ, 2φ + Ψ + Ve Ψ = H ∂φ
(10)
3
where, σ = z 2 = a3 plays the role of internal time parameter as before. In the above, the effective potential Ve , is given by, Ve =
αx σ
2 3
+
H2 φ 2 x 3 3σ
10 3
+
2 V0 σ 3x
(11)
ˆ e allows one to write the continuity equation under the choice n = −1 , as, The hermiticity of H ∂ρ + ∇.J = 0, ∂σ
(12)
where, ρ = Ψ∗ Ψ and J = (Jx , Jφ , 0) are the probability density and the current density respectively, where i~ (ΨΨ∗,x − Ψ∗ Ψ ,x ) 54βx i~ 2Hx2 φ ∗ ∗ ∗ Jφ = Ψ Ψ 2 (ΨΨ,φ − Ψ Ψ ,φ ) − 7 3xσ 3 3σ 3 Jx =
In the process, operator ordering index here too has been fixed as n = −1 from physical argument.
(13a) (13b)
0.0.2
Semiclassical approximation
Now to check the viability of the quantum equation (10), it is required to test its behaviour under certain appropriate semi-classical approximation. For the purpose, let us express equation (9) in the form, √ n ∂ ∂Ψ ~2 ∂ 2 Ψ Hx2 ∂Ψ ~2 z ∂ 2 (14) + − i~ Ψ− + i~ 3 φ + VΨ = 0 − 3 2 2 36βx ∂x x ∂x ∂z z ∂φ 2xz 2 ∂φ where 3αx H2 φ 2 x 3 i~Hx2 V0 V= √ + + + z2. 11 5 x 2 z z2 2z 2
(15)
The above equation may be treated as time independent Schr¨odinger equation with three variables x , z and φ . Therefore, as usual, let us seek the solution of equation (14) as, i
ψ = ψ0 e ~ S(x,z,φ)
(16)
and expand S in power series of ~ as, S = S0 (x, z, φ) + ~S1 (x, z, φ) + ~2 S2 (x, z, φ) + .... .
(17)
Now inserting the expressions (16) and (17) in equation (14) and equating the coefficients of different powers of ~ to zero, one obtains the following set of equations (upto second order) √ 2 S0,φ z 2 Hx2 S0,x + φS0,φ + V(x, z, φ) = 0 (18a) 3 + S0,z − 36βx z3 2xz 2 √ √ √ i z in z iS0,φφ zS0,x S1,x S0,φ S1,φ Hx2 − S0,xx − S − + S + + − φS1,φ = 0. (18b) 0,x 1,z 3 3 36βx 36βx2 18βx z3 2xz 2 xz 2 √ √ √ zS0,x S2,x zS1,xx S1,φφ Hx2 n zS1,x S0,φ S2,φ − i + S − −i −i + φS2,φ = 0, (18c) 2,z 3 3 18βx 36βx 36βx2 z3 xz 2 xz 2 which are to be solved successively to find S0 (x, z, φ), S1 (x, z, φ) and S2 (x, z, φ) and so on. Now identifying S0,x with px ; S0,z with pz and S0,φ with pφ one can recover the classical Hamiltonian constraint equation H = 0 , given in equation (??) from equation (18a). Thus, S0 (x, z) can now be expressed as, Z S0 =
Z pz dz +
Z px dx +
pφ dφ
(19)
apart from a constant of integration which may be absorbed in ψ0 . The integrals in the above expression may be evaluated using the classical solution for k = 0 , presented in equation (??), the definition of pz , pφ in (??) and px = N Q. Further, recalling the expression for Q given in (??), the relation, x = z˙ , where, z = a2 , the choice n = −1 , for which probability interpretation holds, and using the solution (??), x , px , pz , and pφ are found as, The corrected expressions for momenta of equation (119) are, x = 2Hz √ 3√ px = 36 2βH 2 x √ √ Ha20 φ20 pz = −6α zH − 72βH3 z − √ 2 z 3 3 2Ha0 φ0 pφ = − 3φ2 Correct form of the integrals in expressions (120) are Z √ 3 3 3 px dx = 24 2βH 2 x 2 = 96βH3 z 2 ; Z √ 3 3 pz dz = −4αHz 2 − 48βH3 z 2 − Ha20 φ20 z; Z √ 2Ha30 φ30 2 = Ha20 φ20 z. pφ dφ = 3φ 3
(20a) (20b) (20c) (20d)
(21a) (21b) (21c)
Therefore the corrected expression for S0 appearing in equation (121) is, 3
3
S0 = −4αHz 2 + 48βH3 z 2 −
H 2 2√ a φ z. 3 0 0
(22)
The corrected on-shell action (122) reads as, Z H2 3 2 Ht 2 3 3Ht 4 3 3Ht A = Acl = a φ e dt. −12αH a0 e + 144βH a0 e − 3 0 0
(23)
Integration then gives the corrected form of the expression (123) as
A = Acl = −4αHa30 e3Ht + 48βH3 a30 e3Ht −
H 3 2 Ht a φ e , 3 0 0
(24)
which is the same as we obtained in (22). At this end, the corrected form of the wave function (124) takes the form, i
h
ψ = ψ0 e ~ 0.0.3
3
3
2 2 −4αHz 2 +48βH3 z 2 − H 3 a0 φ0
√ i z
.
(25)
First order approximation
Now for n = −1 , equation (18b) can be expressed as, √ z 1 Hx2 i φS1,φ = 0. iS0,xx − 2S0,x S1,x − S0,x − − (iS − 2S S ) + S − 0,φφ 0,φ 1,φ 1,z 3 36βx x z3 2xz 2
(26)
Using the expression for S0 in (22), we can write S1,z from the above equation as i A1 + A22 − A3 S1,z = 4 z 2A2z A3 Hz , A0 + z + 2 (12βH2 −α) + H(1 − 32H3 ) + (1 + 12βH2 2 24(12βH −α) . On integration the form of a20 φ20
(12βH2 −α) 12βH2
where, A0 =
8H3 ) and A1 =
+ 2(1 − 32H3 ) , A2 =
A3 =
S1 in principle may be found as,
S1 = if (z).
(27) a20 φ20 (1−32H3 ) 288βH2
and
(28)
Hence the wavefunction (128) up to first-order approximation is expressed as, i
ψ = ψ01 e ~
h i 3 3 2 2√ −4αHz 2 +48βH3 z 2 − H 3 a0 φ0 z
,
(29)
where, ψ01 = Ψ0 ef (z) .
(30)
Thus, first-order approximation only modifies the prefactor, keeping the oscillatory behavior of the wave function intact. The oscillatory behaviour of the wavefunction indicates that the region is classically allowed and the wavefunction is strongly peaked about a set of exponential solutions to the classical field equations. This establishes the correspondence between the quantum equation and the classical equations, which was not possible to establish with other techniques which fix, as already mentioned, hij and Kij at the boundary. This clearly dictates that modified Horowitz’ formalism which on the contrary fixes hij and R at the boundary, only ends up with a viable quantum theory.
