Quantum Class. Grav. 4 (1987) 1477-1486. Printed in the UK
Quantum cosmology: the supersymmetric square root
Alfred0 Maciasl-, Octavio Obreg6nt and Michael P Ryan Jri$
t Departamento de Fisica, Universidad Aut6noma Metropolitana-Iztapalapa, A Postal 55-534, Mexico D F 09340, Mexico i- Centro de Estudios Nucleares, Universidad Nacional Aut6noma de Mtxico, A Postal 70-543, Mexico D F 04510, Mexico
Received 21 April 1987
Abstract. Using N = 1 supergravity as the natural square root of gravity, we study the quantum cosmology of Bianchi type I cosmological models. This approach gives us a natural interpretation of the components of the state vector of the universe that was lacking in previous work on the square root of quantum cosmology.
1. Introduction
Recently there has been a revival of interest in quantum cosmology (see for example [ l , 21 and see [3] for a recent list of references), i.e. quantum geometrodynamics applied to metrics where homogeneity has been imposed before quantisation. This type of theory has several uses, ranging from that of a model theory for the quantisation of general relativity, where certain problems of quantum gravity can be solved in a limited context with the hope of applying these solutions to the full theory, to the idea that these models will give us roughly the overall behaviour of the actual universe in a situation where quantum mechanics is important (for a review of earlier work see [4]). One of the original difficulties in quantum cosmology in one that general relativity shares, that the Hamiltonian (in one of several ways of obtaining a time-evolution operator) of the system is quadratic in the same sense that the Hamiltonian for a relativistic particle, H 2 = p 2 + m2,is quadratic [5]. Quadratic Hamiltonians of this form lead automatically to an equation of the Klein-Gordon type (the Wheeler-DeWitt equation) for the state function that characterises the universe with all its attendant problems of interpreting these state functions as probability densities (see, however, [l]). The first idea that occurs to a physicist is to attempt a square root of this Hamiltonian of the Dirac type where H is linear in the momenta and the squared operator H 2 leads to the original Klein-Gordon-like equation. This type of square root was immediately found for quantum Bianchi-type metrics [4] and, at least for Bianchi type I universes, the resulting Dirac equation solved. The basic problem was that the solution was a vector of the form ($;) (we hesitate to use the word spinor since it would imply ideas about transformation properties under rotations in minisuperspace that we are not prepared to discuss here), and there existed no natural interpretation of these components. After the invention of supergravity, Teitelboim et a1 [6,7] showed that this theory provides a natural classical square root of gravity of the Dirac type. This square root leads immediately to the idea of interpreting the components in terms of quantum 0264-9381/87/061477
+ 10$02.50 0 1987 IOP Publishing Ltd
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A Macias, 0 Obrego'n and M P Ryan Jr
states in supergravity. However, since the Teitelboim square root is just the natural supersymmetric extension of gravity, it provides a general method for finding square root equations in quantum gravity and quantum cosmology as well as a possibility of interpreting the resulting state functions. A general formulation of canonically quantised supergravity exists [8], but in the present paper we will construct diagonal Bianchi type I quantum cosmological models in supergravity directly from the classical canonical formalism and show that a square root equation similar to the naive square root of the earlier formulation results, but with a natural interpretation of the various components as made up of eigenstates of the components of the gravitino field. The major difference will be that the state vector will have eight components instead of two, but with a natural division into four two-component vectors, each of which obeys the naive square root equations. In order to carry out this programme we will simplify supergravity as much as possible. The basic field variables in the graviton-gravitino formulation of ( N = 1) supergravity [9] are the metric, g,,(x"), and the vector-spinor gravitino field GPA, where A = 1 , . . . , 4 are spinor indices which will be suppressed when convenient, that obey the Einstein-Rarita-Schwinger system of Freedman and van Nieuwenhuizen [9]. To simplify calculations as much as possible we will assume that the can be expanded in an anticommuting basis [lo] of order two, i.e.
$+A
= $pAIEI
+ $pAZE2
(1.1)
where the $ , A , ( ~ a ) are ordinary functions and the E, are constants that obey E ~ E ,= -&]E:. It is obvious that the field degrees of freedom that we will quantise are the but we will write the quantum operators as j P A The Bianchi-type metric we will use is . d s 2 = (N'-NJN,) dt2-2N, dtw1-e-2n(') e2P(') w ' w l (1.2)
where Cl( t ) is a scalar and p,,( t ) is a 3 x 3 matrix, and the lapse and shift functions are N ( t ) and NI(t ) respectively. The 1-forms w ' are the invariant 1-forms appropriate to a particular Bianchi universe, and here we will take w ' = dx' to obtain a type I model. We will take quantum cosmology to mean the quantisation of the homogeneous type I model with $ = $ t ) and the full set of dynamical variables is a( ) , PI,( t ) and , ( , t t ) . We will also take p, diagonal and use the Misner parametrisation [5], p, = diag (p++&p-, p+-&p-, -2p+). For comparison we will write the naive square root equation of motion. A representation can be chosen in which the equation for the vector
can be written as [4] (1.3) where the aiare the ordinary Pauli matrices. The major problem we will encounter in this quantum cosmology is the well known one of trying to construct an action that is a function only of the dynamical variables mentioned above and that gives the full Einstein and Rarita-Schwinger equations [4]. An example where this fails is the diagonal Ellis-MacCallum class B [ 111 Bianchi-type models where the Einstein equations are usually non-diagonal and an integration by parts in the non-coordinated basis that gives non-zero contribution is automatically dropped [ 121. The homogenised
Quantum cosmology: the supersymmetric square root
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supergravity Lagrangian for diagonal type I models suffers from the first of these diseases, since the Einstein equations are non-diagonal. We will show that these non-diagonal equations reduce to a set of algebraic constraints on the $w which we will leave as a set of constraints on the final solution. A possible problem that is not encountered is that classical solutions for diagonal Bianchi type I models might not exist. This is not necessarily serious, since quantum solutions could still be found, but could cause problems in homogeneous models where quantum solutions have no inhomogeneous fluctuations to drive them. This problem does not arise here since classical solutions to our problem are known [ 131. The paper is organised as follows. In 2 we construct the homogeneous Lagrangian 2 for the model, and in 8 3 compare the full Einstein and Rarita-Schwinger equations for the metric and gravitino field with the equations obtained from the variation of 2. From this comparison we derive the algebraic constraints due to the non-diagonal Einstein equations, and in 4 we write the quantum equations constructed from the square root formalism of Teitelboim et aI [ 6 , 7 ] . This last section also includes suggestions for further research.
2. The Lagrangian
As mentioned in 1 we restrict ourselves to the case
*w
= &I*PI
+ E2*ll2
with
c2sl= 0. The lagrangian of supergravity then reduces to
2 = iJ-g R - ;E
where we take
wLypu
(c47 s y y D p ~ u
where
;&ab
=
4[ e a (dPebu a ehP + e 2 e )
- spwY
recp e ,,I - ( a
-
b)
(2.4)
K w Y p = - s w Y p + S,,
s,,
= ii4wYp*v
are the (zero-torsion) Ricci rotation coefficients, the contorsion and torsion tensors respectively. Note that in our Lagrangian all terms of third and higher order have disappeared because of the assumption (2.1). For the y matrices we use the real
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A Maci'as, 0 Obrego'n and M P Ryan Jr
Majorana representation: 0 0 0 - i 0 - i o 0 0
O O O
0
0
0
0 0
i
0 0 0 - i
-i
0 0
0
0
-i
where 4 = GTC with C = -i yo. We will write the metric of a general Bianchi-type model as ds2 = ( N 2- NJN,)d t 2- 7 3 dt
f
- e p 2 c 1 ( ' ) e 2'P ( r ) I
W'WJ
(2.6)
where R ( t ) is a scalar, P,( t ) is a 3 x 3 matrix, N = N ( t ) and N , = N E t() . The differential forms W ' are invariant 1-forms appropriate to the model under consideration and obey dw' = -LC'Jk g JA w k 2 (2.7)
where CJk are the group structure constants of the group of motions associated with the particular Bianchi-type model under consideration. We will need an orthonormal basis in order to write the gravitino equations of motion, and we choose
er'
=N
e(') = 0
e t ) = e"e,p
N,
e(J)=
,
e Ret.
(2.8)
We will be interested in a type I model with diagonal P, where CJk 0 and P can be = parametrised by
PIJ= diag(P+ + a -, P+ - aP-,2 P + ) . P The Ricci rotation coefficients now become
(2.9)
(2.10)
W i ( j k )= K i ( j k )
where a dot denotes the derivative with respect to R-time and indices in parentheses refer to the tangent flat space. We would now like to calculate the Lagrangian (2.2) for this case. We will make the following change of variables for the gravitino field [14]: (2.11)
Quantum cosmology: the supersymmetric square root In terms of these variables (2.2) becomes
044= ( 3 e - 3 n / ~ ) ( f i 2 - f i : -65)
1481
-(1/4N)&[741(-2fi
-B+ -a&)Y242(-2fi -b++&fi-) +
&~~(-2fi+2fi+)]40
0 3
+ Y34+2fi + 2 j + ) 1 +(1/4N)[6,y1(-2h-8+-&8-)
+ 42Y2(-2fi-b++&B-)+
-$[(-3b+-ab-)$iY
+(2&B-)62Y
2 0 1 1
Y Y 43+(-2&fi-)&iY1Y0Y242
Y Y 4i+(-38++&8-)6zY2YoY343
3 0 2
+ (3B + + a -14 Y 3YOY l 4 1 8 3
+(38+-m-)&3Y Y Y 421-1[61Y1Y0(Y242+ Y363) (2.12) + 6 2 ~ 2 ~ o (+~y 3 & ) + 4 3 ~ 3 ~ 0 ( ~ & ) ] ) . 1& + Y~ 1 & At this stage we have made no attempt to combine the terms in 4o and 6, into one term (this could have been done by means of the relation f j y 4 = -677) or to use the identity 6 y p * y y p 7= - f j y P y Y y + to simplify the terms of the form $e-yyI*yyyp4p. We do this to facilitate comparison of the dynamical equations obtained from the Lagrangian (2.12) and the equations of motion calculated directly from the Einstein and Rarita-Schwinger equations of N = 1 supergravity.
3. Equations of motion
The Einstein and Rarita-Schwinger equations for N
R,(,) -$e,,,$
= =1
supergravity are
TdU)
eArup s yyI*DyIc$0 y = where (3.3) For the basis (2.8) with pij given by (2.9) and for the assumption (2.1), we find that for the gauge choices N = 1, Ni = 0, (3.1)-(3.3) reduce to a set of equations for p * , R and c$~ that are given in the appendix. is There no guarantee that the equations of motion obtained by varying the Lagrangian (2.12) will be equivalent to (3.1)-(3.3), since we have imposed the condition of homogeneity on 2 before variation, something that is not necessarily consistent with variation followed by homogenisation. This means that we must check the variational equations against the true ones given above. If we vary 2 with respect to and c$i we obtain the Rarita-Schwinger equations (A17) and (A18). Equation ( A l ) also follows if we vary with respect to N.Variation with respect to R, p+ and p- gives
sn+2(~-3h2)+3ie3n(c$0,y4j)= 0
w++ tB+ -3.i.lB+) - d i e 3 n { [ 6 0 ( ~ 4+ ~ 1
+3(61Y
I O
- 24~ ~~ ~ 1 1 4
3
? 43+c$ZY
3
2
0
43))=O
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A Maci'as, 0 Obrego'n and M P Ryan J r
1
sp_-,&(@--3h&) -$ie"{(&y
2 0 3
y 0 y 3 +3)'+2(&y1yoy242)'
-($2Y
Y Y 43)'+[~0(Y141-Y242)1'}=o.
These equations are equivalent to the equations (Al)-(A4). If we now vary with respect to Ni, arrive at the ( i o ) Einstein-Rarita-Schwinger equations. Given all of these we equations, the ( O i ) Einstein equations are automatically fulfilled. While the variational equations are not exactly the full Einstein and Rarita-Schwinger equations, they form an equivalent set, at least as far as we have gone. Since we have inserted the diagonal metric given by (2.10) directly into the Lagrangian, we have automatically lost all (ij) Einstein equations where i # j . In many cases in quantum cosmology, these equations are identities, so no information is lost. Here, however, it can be seen from the appendix that this does not hold. In our problem these equations are a set of extra constraints on the problem that must be taken care of in some way. It is easy to show that they can be reduced a set of algebraic constraints on the vector-spinor components of the gravitino field. In the next section we will discuss this problem in more detail.
4. Quantum cosmology
The Lagrangian (2.12) can be put into Hamiltonian form in the usual way (defining p + = a=Y/d@+, etc). As usual with spinor Lagrangians, the momenta conjugate to the are essentially the 4Ithemselves, so we do not bother to rename them. We now have
3 = p + j + + p - f i - +pn.i2+-~e3"(-p:,+p:+p2) + iNk e 3Rpa& ( Y1481 + i W 1 e 3 Y + + a p - 1 $I( Yi411 p +&iN2e3n(p+ a p - ) $ z ( y ' 4 , ) + ~ i N 3 e 3 R ( 2 p + ) $ 3 ( y 1 4 , )
+ & i ~ e ~ " [ - ( 3 p + &p-)$, y1yoy343 (3p+- &p_)42y2yoy343 + -
+ (3p++J5p-)&r Y Y 41 + (3P+ -ap-)$3Y3YoY242 2 0 1 - 2 a p - $1 Y YOY 2 4 2 + 2 4 - 4 2 Y Y Y 4 1 1 ++$oie3"[Pn(Y '41)-b+( + Y 2 4 2 -2Y343) Y14i
3
0
1
-$&-(Y141
- Y 2 4 2 ) 1 -Aie3R[pR(&Y')
(4.1) -+p+($1Y1+62Y2-263Y3) -MP-($1Y1 -&Y2)140. The non-dynamical variables N , N , and 4omultiply the constraints Ro,XIand S. We hope here that these three constraints satisfy the closed algebra discovered by Teitelboim [6,7] (this is not guaranteed for the homogenised constraints), or a similar one that will allow us to maintain the 'square root' property of the full algebra. If we use the homogeneous form of the Dirac brackets of [6] and [15], (4.2) converted to anticommutation relations, it is possible to show that in our case the constraints S and Eo have the 'square root property'
{SA(x), SB(x'))=
Y$BXps(%
x')
{SI, SI, S i ) = s n e ( Y O ) ~ ~ [ % O + ~ ~ , Y ' Y ~ ) I S I I ~~( {S29 3 3 ) = { S 2 , S2)=872e
{SI, 3 2 1
3 1
311
3 1
311
(~0)23[Xo+
18(i&J~'~o),Szl 18(i6j~J~o)3~31 (4.3)
= -{S3,
S3) = i i + ' " ( ~ O ) 3 2 [ % +
{SA,
1 {S4, 3 ) = {S4, S4)=ih$3n(Yo)dxo+ ~ ~ ( ~ & Y ' Y O ) ~ S ~ I SB}=o A # B.
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The homogenisation of the Lagrangian has only led to two extra terms in the algebra that are proportional to S and depend only on the canonical variables of the theory. Since they are proportional to S they are automatically weakly zero and do not add new constraints to the algebra. In order to quantise the problem we have outlined above, we will convert p a , p + , p - and to operators that act on a state function W of the universe that in principle will give us the probability of finding the universe in a state with given values of R, pi and + i . The constraints S, X o and Xiare now operators, and the equation SW = 0 is the equation that determines q . It is easy to see that the usual constraint in Bianchi-type cosmologies, XoY = &e3n(p & - p : -p_)W = 0 that produces a KleinGordon-like equation will be a consequence of SW = 0 through the algebra (4.3). The operator S has four spinor components similar to SI = n e
1
3 0
[(411
+ 4 2 4 - 4 3 2 ) P R - &(411+ 424+2432)p+ -+&(411-
424)P-I
(4.4)
where we have arbitrarily chosen a factor ordering. We will now try to realise the operators that make up the S. The momenta pn, p + and p - will take the usual form ia/aR, -ia/ap+ [ 6 ] . We will then find a matrix realisation of the 4iAwhich gives {4iA, 4jjs} t i ( ~ ~ 7Since { ) A ,~ ~ 0,.A # B, we can take each of the SA to operate on = ~ S S,} = orthogonal subspaces, and we can write S A 9 = 0 in the form
where each of the SA will be a matrix operator of the smallest rank possible that produces the appropriate algebra for S. For each SA we see that it can be written as
ne
1
3R
(MAIpR+ MA2p++MA3P-)
(4.6) (4.7)
or :,n operator form n e 3n (iM4,a/aR-iMA,a/ap+-iMA3a/ap-). 1 The only non-zero anticommutator containing each SA is
isA,S A } = T3g 1n e3R [ - & i e 3 ( - p : + p : + p 2 ) ]
which implies the following algebra for the MA,: MAI}=ii {MA],
(MA2, MA2}={MA3, M ~ 3 } = - i i { M A , ,M A , ) = o
(i#j).
For all A the minimum possible rank for a matrix representation of the MAI is two. A convenient representation is
MA1 = (3i/8)a3
MA^ = i( 3i/ 8)I2U2
MA, = i(3i/8)/*a1
(4.8)
and the operator equation S A W , = 0, where the T Aare the two-dimensional vectors that make up W, reduces to (4.9) Note that for each W A this is just equation (1.3) that was derived from the simplest possible square root of the Wheeler-DeWitt equation.
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The supergravity square root approach gives us exactly what was missing in the ad hoc square root approach, an interpretation of the two components of the V Avector in terms of states of differing probability of finding possible values of c$i. In the next section we will discuss this interpretation.
5. Discussion of solutions and conclusions
For each V Athe solution to (4.9) is
where p+ are constants and CAI= c A 2 / f i = -(p+-ip-)/fiE. The tw...