List of Symbols, Notation, and Useful Expressions

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Appendix A List of Symbols, Notation, and Useful Expressions

In this appendix the reader will ﬁnd a more detailed description of the conventions and notation used throughout this book, together with a brief description of what spinors are about, followed by a presentation of expressions that can be used to recover some of the formulas in speciﬁed chapters.

A.1 List of Symbols

κ ≡ κijk Contorsion ξ ≡ ξijk Torsion K Kähler function I ≡ I ≡ I KJ gJ GJ Kähler metric W(#) Superpotential V Vector supermultiplet #, # (Chiral) Scalar supermultiplet φ,ϕ Scalar ﬁeld V (φ) Scalar potential χ ≡ γ 0χ † For Dirac 4-spinor representation ψ† Hermitian conjugate (complex conjugate and transposition) φ∗ Complex conjugate [M]T Transpose { , } Anticommutator [ , ] Commutator θ Grassmannian variable (spinor)

Jab, JAB Lorentz generator (constraint) qX , X = 1, 2,... Minisuperspace coordinatization π μαβ Spin energy–momentum

Moniz, P.V.: Appendix. Lect. Notes Phys. 804, 207–275 (2010) DOI 10.1007/978-3-642-11570-7 c Springer-Verlag Berlin Heidelberg 2010 208 A List of Symbols, Notation, and Useful Expressions

Sμαβ Spin angular momentum D Measure in Feynman path integral

[ , ]P ≡[, ] Poisson bracket [ , ]D Dirac bracket F Superfield M P Planck mass V β+,β− Minisuperspace potential (Misner–Ryan parametrization) Z IJ Central charges ds Spacetime line element ds Minisuperspace line element F Fermion number operator eμ Coordinate basis ea Orthonormal basis (3)V Volume of 3-space (a) vμ Vector field (a) fμν Vector field strength ij π Canonical momenta to hij π φ Canonical momenta to φ π 0 Canonical momenta to N i π Canonical momenta to Ni μ a pa Canonical momenta to eμ (tetrad) ij ξ k Pk Canonical momenta to ij (torsion) J j Lorentz rotation generator Ki Lorentz boost generator Pμ Translation (Poincaré) generator εμνλσ Four-dimensional permutation

εijk Three-dimensional permutation SA SUSY generator (constraint) H⊥ Hamiltonian constraint

Hi Momentum constraints G DeWitt metric Gs DeWitt supermetric Ψ Wave function (state) of the universe Ω ADM time R Spinor-valued curvature A.1 List of Symbols 209

βij(β+,β−) Anisotropy matrices (Misner–Ryan representation) in Bianchi cosmologies ij pij Canonical momentum to β HADM ADM Hamiltonian Kij Extrinsic curvature (4)R Four-dimensional spacetime curvature (3)R Three-dimensional spatial curvature Λ Cosmological constant μ Γνλ Christoffel connection coefﬁcients , Usual derivative (vectors, tensors) (3) | and Di Four-dimensional spacetime covariant derivative with respect to the 3-metric hij, no spin connection and no torsion, with Christoffel term (vectors, tensors) ( ) ; and 4 Dμ Four-dimensional spacetime covariant derivative with respect to the 4-metric gμν, no spin connection and no torsion, with Christoffel term (vectors, tensors) ' Covariant derivative with respect to the 3-metric, including spin connection (vectors, tensors) ( Covariant derivative with respect to the 4-metric gμν, including spin connection (vectors, tensors) Dot over symbol Proper time derivative Ω2 d 3 Line element of spatial sections k Spatial curvature index a(t) ≡ eα(t) FRW scale factor N Lapse function N i Shift vector μ,ν,... World spacetime indices a, b,... Condensed superindices (either bosonic or fermionic) a, b,... Local (Lorentz) indices aˆ, bˆ,...= 1, 2, 3,... Local (Lorentz) spatial indices i, j, k Spatial indices hij Spatial 3-metric gμν Spacetime metric ηab Lorentz metric ωμ ≡{ω0,ωi } One-form basis A, A Two-spinor component indices (Weyl representation) [a] Four-spinor component indices, e.g., Dirac representation 210 A List of Symbols, Notation, and Useful Expressions

I, J,... Kähler indices J, j Representation label of SU(2) (spin state) M, N Elements of SL(2,C) La b Lorentz (transformation) matrix representation i AA pAA Momentum conjugate to ei HAA Hamiltonian and momentum constraints condensed (two-spinor components) S Lorentzian action L Lagrangian I Euclidean action S Superspace M Four-dimensional spacetime manifold

Σt Three-dimensional spatial hypersurface t Coordinate time τ Proper time n ≡{nμ} Normal to spatial hypersurface ( ) 4 Dν ≡ Dν Four-dimensional spacetime covariant derivative with respect to the 4-metric gμν, with spin connection and torsion, no Christoffel term (spinors [vectors, tensors]) ( and Dν Four-dimensional spacetime covariant derivative with respect to the 3-metric hij, with spin connection and torsion, with Christoffel term (spinors [vectors, tensors]) ' Three-dimensional analogue of Dν (spinors [vectors, tensors]) (3) (3) D j ≡ ∇ j ≡∇j Three-dimensional analogue of Dν (spinors [vectors, tensors]) (3s) (3s) D j ≡ ∇ j ≡ ∇ j Three-dimensional analogue of Dν, no torsion (spinors [vectors, tensors])

Mab, MAB Lagrange multipliers for the Lorentz constraints (Lorentz and 2-spinor indices) ωA ,ωa ,ωAA BB Bμ bμ Spin connections A ψμ Gravitino ﬁeld nμ, n AA Normal to the hypersurfaces ε AB Metric for spinor indices (Weyl representation) σ AA a Pauli matrices with two-component spinor indices (Weyl representation) σ Infeld–van der Waerden symbols A.2 Conventions and Notation 211 eaμ Tetrad aˆ eˆ i Spatial tetrad (aˆ, i = 1, 2, 3) eAA μ Tetrad with two-component spinor indices =∼ Equality provided constraints are satisﬁed (a) Gauge group index ˆ Dμ, Dˇ φ, Dμ Gauge group covariant derivatives

Q(a) Gauge group charge operator ε Infinitesimal spinor Dφ Kähler derivative D SUSY covariant derivative (SUSY superspace) %JK I Kähler connection R Kähler curvature ψ, χ, λ Spinors ψ,χ,λ Conjugate spinors, in the 2-spinor SL(2,C)Weyl representation U Minisuperspace potential P(φ, φ) SUGRA-induced potential for scalar fields T (a) Gauge group generators Q SUSY supercharge operators Gˆ Gauge group T Energy–momentum tensor D Minisuperspace dimension R Minisuperspace curvature D, dˆ, Dˆ , D˜ , Dˇ ,... Dimension of space σ Axion field 1,σ Pauli matrices A, B,... Bosonic amplitudes of SUSY Ψ AB Ei Densitized spatial triad AB A j Ashtekar connection

A.2 Conventions and Notation

= = = = −2 ≡ π Throughout this book we employ c 1 h¯ and G 1 MP , with k 8 G unless otherwise indicated. In addition, we take: • μ,ν,...as world spacetime indices with values 0, 1, 2, 3, • a, b,...as local (Lorentz) indices with values 0, 1, 2, 3, 212 A List of Symbols, Notation, and Useful Expressions

• i, j, k as spatial indices with values 1, 2, 3, • A, A as 2-spinor notation indices with values 0, 1 or 0 ,1 , respectively, •[a] as 4-spinor component indices, e.g., the Dirac representation, with values 1, 2, 3, 4.

In this book we have also chosen the signature of the 4-metric gμν (or ηab)tobe (−, +, +, +). Therefore, the 3-metric hij on the spacelike hypersurfaces has the signature (+, +, +), which gives a positive determinant. Thus the 3D totally anti- 0123 123 symmetric tensor density can be deﬁned by ε ≡ ε = ε123 =+1.

A.3 About Spinors

As the fellow explorer may already have noticed (or will notice, if he or she is venturing into this appendix prior to probing more deeply into some of the chapters) there are some idiosyncrasies in the mathematical structures of SUSY and SUGRA which are essentially related to the presence of fermions (and hence of the spinors which represent them). But why is this, and what are the main features the spinors determine? In fact, why are spinors needed to describe fermions? What are spinors and how do they come into the theory? This section is devoted (in part) to introducing this issue. Before proceeding, for completeness let us just indicate that a tetrad formalism is indeed mandatory for introduing spinors.1 The tetrad corresponds to a massless spin-2 particle, the graviton [1], i.e., corresponding to two degrees of freedom.2

Note A.1 Extending towards a simple SUSY framework, with only one generator (labeled N = 1 SUSY), the corresponding (super)multiplet will also contain a fermion with spin 3/2, the gravitino, as described in Chap. 3 of Vol. I, where the reader will ﬁnd elements of SUSY (not in this appendix).

A.3.1 Spinor Representations of the Lorentz Group

Local Poincaré invariance is the symmetry that gives rise to general relativity [2]. It contains Lorentz transformations plus translations and is in fact a semi-direct product of the Lorentz group and the group of translations in spacetime. The Lorentz

1 Tensor representations of the general linear group 4 × 4 matrices GL(4) behave as tensors under the Lorentz subgroup of transformations, but there are no such representations of GL(4) which behave as spinors under the Lorentz (sub)group (see the next section). 2 In simple terms, the tetrad has 16 components, but with four equations of motion, plus four degrees of freedom to be removed due to general coordinate invariance and six due to local Lorentz invariance, this leaves 16 − (4 + 4 + 6) = 2. A.3 About Spinors 213

J K ˆ, ˆ = , , group has six generators: three rotations aˆ and three boosts bˆ, a b 1 2 3 with commutation relations [3–5]:

[J , J ]= ε J , [K , K ]=−ε J , [J , K ]= ε K . aˆ bˆ i aˆbˆcˆ cˆ aˆ bˆ i aˆbˆcˆ cˆ aˆ bˆ i aˆbˆcˆ cˆ (A.1)

The generators of the translations are usually denoted Pμ, with3 Pμ, Pν = 0 , Ji , P j = iεijkPk , (A.2)

[Ji , P0] = 0 , Ki , P j =−iP0δij , [Ki , P0] =−iPi . (A.3)

Or alternatively, deﬁning the Lorentz generators Lμν ≡−Lνμ as L0i ≡ Ki and Lij ≡ εijkJk, the full Poincaré algebra reads Pμ, Pν = 0 , (A.4) Lμν, Lρσ =−iηνρ Lμσ + iημρ Lνσ + iηνσ Lμρ − iημσ Lνρ , (A.5) Lμν, Pρ =−iηρμPν + iηρνPμ . (A.6)

Let us focus on the mathematical structure of (A.1). The usual tensor (vector) formalism and corresponding representation is quite adequate to deal with most situations of relativistic classical physics, but there are signiﬁcant advantages in considering a more general exploration, namely from the perspective of the theory of representations of the Lorentz group. The spinor representation is very relevant indeed. For this purpose, it is usual to discuss how, under a general (inﬁnitesimal) Lorentz transformation, a general object transforms linearly, decomposing it into irreducible pieces. To do this, we take the linear combinations

± 1 J ≡ (J j ± iK j ), (A.7) j 2 and the Lorentz algebra separates into two commuting SU(2) algebras:

[J ±, J ±]= ε J ± , [J ±, J ∓]= , i j i ijk k i j 0 (A.8) i.e., each corresponding to a full angular momentum algebra. A few comments are then in order:

3 Note that we will be using either world (spacetime) indices μ, or local Lorentz indices a, b,..., and also world (space) indices i, or local spatial Lorentz indices aˆ, bˆ,..., which can be related a through the tetrad eμ (see Sect. A.2). 214 A List of Symbols, Notation, and Useful Expressions

• The representations of SU(2) are known, each labelled by an index j, with j = 0, 1/2, 1, 3/2, 2,..., giving the spin of the state (in units of h¯). The spin-j representation has dimension (2j + 1). • The representations of the Lorentz algebra (A.8) can therefore be labelled by4 (j−, j+), the dimension being (2j+ + 1)(2j− + 1), where we ﬁnd states with j taking integer steps between |j+ − j−| and j+ + j−. – In particular, (0, 0) is the scalar representation, while (0, 1/2) and (1/2, 0) both have dimension two for spin 1/2 and constitute spinorial representations.5 ( / , ) − ≡{J −} × –Inthe1 2 0 representation, J i i=1,... can be represented by 2 2 matrices and J+ = 0, viz., J− =)σ \2, which implies6 J =)σ \2, but K = iσ)\2, 7 i where we henceforth use the four 2×2 matrices σμ ≡ (1,σ ), with σ0 usually being the identity matrix and σ) ≡ σi , i = 1, 2, 3, the three Pauli matrices: 01 0 −i 10 σ 1 ≡ ,σ2 ≡ ,σ3 ≡ , (A.9) 10 i0 0 −1 satisfying σ i ,σj = 2iεijkσ k. Note the relation between Pauli matrices with 0 i upper and lower indices, namely σ =−σ0 and σ = σi , using the metric ημν. –Inthe(0, 1/2) representation we have J = σ\2butK =−iσ \2. This means that we have two types of spinors, associated respectively with (0, 1/2) and (1/2, 0), which are inequivalent representations. These will correspond in the following to the primed spinor ψ A and the unprimed spinor ψA, where A = 0, 1 and A = 0 , 1 [6–9]. The ingredients above conspire to make the group8 SL(2,C) the universal cover9 of the Lorentz group. If M is an element of SL(2,C), so is −M, and both produce the same Lorentz transformation.

4 The pairs (j−, j+) of the ﬁnite dimensional irreducible representations can also be extracted from 2 the eigenvalues j±(j± + 1) of the two Casimir operators J± (operators proportional to the identity, with the proportionality constant labelling the representation). 5 The representation (1/2, 1/2) has components with j = 1andj = 0, i.e., the spatial part and time component of a 4-vector. Moreover, taking the tensor product (1/2, 0) ⊗ (0, 1/2), we can get a 4-vector representation. 6 It should be noted from J = σ\2 that a spinor effectively rotates through half the angle that a vector rotates through (spinors are periodic only for 4π). 7 The description that follows is somewhat closer to [10, 9, 11], but different authors use other choices (see, e.g., [3, 5, 12, 13]), in particular concerning the metric signature, σ0, and some spinorial elements and expressions. See also Sect. A.4. 8 The letter S stands for ‘special’, indicating unit determinant, and the L for ‘linear’, while C denotes the complex number ﬁeld, whence SL(2,C) is the group of 2×2 complex-valued matrices. 9 So the Lorentz group is SL(2,C)/Z2,whereZ2 consists of the elements 1 and −1. Note also that SU(2) is the universal covering for the spatial rotation group SO(3). There is a two-to-one mapping of SU(2) onto SO(3). A.3 About Spinors 215

A spinor is therefore the object carrying the basic representation of SL(2,C), fun- damentally constituting a complex 2-component object (e.g., 2-component spinors, or Weyl spinors) ψA=0 ψA ≡ ψA=1 transforming under an element M according to ψ −→ ψ = B ψ , ≡ M1 M2 ∈ C , A A M A B M SL(2, ) (A.10) M3 M4 with A, B = 0, 1, labelling the components. The peculiar feature is that now the other 2-component object ψ A transforms as

∗ ψ −→ ψ = B ψ , A A M A B (A.11) which is the primed spinor introduced above, while the above ψA is the unprimed spinor.

Note A.2 The representation carried by the ψA is (1/2, 0) (matrices M). Its ∗ complex conjugate ψ A in (0, 1/2) is not equivalent: M and M constitute inequivalent representations, with the complex conjugate of (A.10) associated ∗ with (A.11), and ψ A identiﬁed with (ψA) .

Note A.3 There is no unitary matrix U such that N = UMU−1 for matrices N ≡ M∗. We have instead N = ζ M∗ζ −1 with 0 −1 ζ = ≡−iσ . 10 2

∗ This follows from the (formal) relation σ2σ σ2 =−σ , which suggests writing ζ in terms of a known matrix (appropriate for 2-component spinors), viz., σ2. It is from this feature applied to a Dirac 4-spinor χ ψ ≡ , η

that Lorentz invariants are obtained as (iσ 2χ)Tχ. (This is the Weyl representation, where χ,η are 2-component spinors transforming with M and N, respectively.) To be more concrete, with 216 A List of Symbols, Notation, and Useful Expressions

χA=0 χ ≡ χA = , χA=1

we put χ 0 χ χ A = = σ 2χ = 1 , 1 i χ −χ0

A 0 1 whence the invariant is χ χA. (Note that χ χ0 + χ χ1 = χ1χ0 − χ0χ1 = 1 0 A −χ1χ − χ0χ =−χAχ , which is quite different from the situation for μ μ spacetime vectors and tensors Aμ A = A Aμ!) This can be simpliﬁed by A AB AB 2 introducing new matrices with χ = ε χB, where (formally!) ε ≡ iσ . Likewise for

η0 η ≡ ηA = , η1

A A A B A B 2 AB 2 A B 2 η ηA , η = ε ηB , ε ≡ iσ , etc. Of course, ε ≡ iσ , ε ≡ iσ , are different objects acting on different spinors and spaces. The presence of the matrix representation is formal for computations.

The primed and unprimed index structure10 carries to the generators, e.g., J = σ \2 and K = iσ \2, with the four σμ matrices

μ i (σ )AA ={−1,σ }AA , (A.12) and

μ A A A B AB μ i A A (σ ) ≡ ε ε (σ )BB ≡ (1, −σ ) , (A.13)

AB where indices are raised by the antisymmetric 2-index tensors ε and εAB given by11 01 0 −1 ε AB = ε A B = ,ε= ε = , (A.14) −10 AB A B 10

10 Note that (A.12) and (A.13) can used to convert an O(1,3) vector into an SL(2,C) mixed bispinor bAB . 11 AB A B 2 Note that, in a formal matrix form, ε = ε = iσ2, εAB = εA B =−iσ . A.3 About Spinors 217 with which:

A AB β A A B B ψ = ε ψB,ψA = εABψ , ψ = ε ψ B , ψ A = εA B ψ . (A.15)

μ μ A A Note A.4 Note the difference between (σ )AA and (σ ) , i.e., the lower, AB A B upper and respective order of indices, arising from ε = ε = iσ2 and 2 εAB = εA B =−iσ . To be more precise, the (‘natural’) lower (upper) A A μ μ A A indices on (σ )AA , (σ ) come from covariance, e.g., analysing how the matrices σ μ γ μ ≡ 0 σ μ 0

transform under a Lorentz transformation (the Weyl representation here). Further relations [9], in particular between σ μ and σ μ, can be written

μ σ σ A B =− δB δ A , μCD 2 C D (A.16) σ μσ ν ≡ σ μ σ ν B A =− μν , Tr AB 2g (A.17)

constituting two completeness relations. σ μ In essence, it all involves the statement that AB is Hermitian, and that, for a real vector, its corresponding spinor is also Hermitian.

Note A.5 In addition, the following points may be of interest: • Concerning (A.14) and (A.15), it must be said that this is a matter of convention. • Unprimed indices are always contracted from upper left to lower right (‘ten to four’), while primed indices are always contracted from lower left to upper right (‘eight to two’). This rule does not apply when raising or low- ering spinor indices with the ε tensor. • However, other choices can be made (see Sect. A.4) [6, 14, 7–9], and this is what we actually follow from Chap. 4 of Vol. I onwards. In particular, 01 ε AB = ε A B = ε = ε ≡ , (A.18) AB A B −10

with which [note the position of terms and order of indices, in contrast with (A.14)]: 218 A List of Symbols, Notation, and Useful Expressions

A AB B A A B B ψ = ε ψB ,ψA = ψ εBA , ψ = ε ψ B , ψ A = ψ εB A . (A.19)

With (A.15) [or (A.19)], scalar products such as ψχ or ψ χ can be employed. Of course, when more than one spinor is present, we have to remember that spinors 12 anticommute. Therefore, for 2-component spinors, ψ0χ1 =−χ1ψ0, and also, e.g., ψ0χ 1 =−χ 1 ψ0. In more detail,

A AB AB ψχ ≡ ψ χA = ε ψBχA =−ε ψAχB A A =−ψAχ = χ ψA = χψ , (A.20)

A A ψ χ ≡ ψ A χ = ...= χ A ψ = χ ψ, (A.21)

† A † A (ψχ) ≡ (ψ χA) = χ A ψ = χ ψ = ψ χ, (A.22) as well as

μ μ μ μ ψσ χ = ψ Aσ χ B , ψ σ χ = ψ σ A Bχ , AB A B (A.23) from which, e.g.,

μ μ χσ ψ =−ψ σ χ, (A.24) μ ν ν μ χσ σ ψ = ψσ σ χ, (A.25) μ μ (χσ ψ)† = ψσ χ, (A.26) μ ν ν μ (χσ σ ψ)† = ψσ σ χ, (A.27)

A.3.2 Dirac and Majorana Spinors

The above (Weyl) framework will often be used for most of this book [in particular, (A.18) and (A.19)], but if the reader goes on to study the fundamental literature on SUSY and SUGRA, other representations will be met, some of which are also adopted by a few authors in SQC. Let us therefore comment on that (see Chap. 7). In particular, we present the Dirac and Majorana spinors (associated with another representation).

12 The components of two-spinors are Grassmann numbers, anticommuting among them- selves. Therefore, complex conjugation includes the reversal of the order of the spinors, e.g., ∗ A ∗ ∗ A ∗ A (ζψ) = (ζ ψA) = (ψA) ζ = ψ A ζ = ψ ζ. A.3 About Spinors 219

A 4-component Dirac spinor is made from a 2-component unprimed spinor and a 2-component primed spinor via ψ ψ ≡ A , [a] χ A transforming as the reducible (1/2, 0) ⊕ (0, 1/2) representation of the Lorentz group, hence with ψA 0 and 0 χ A as Weyl spinors.13 But is there any need for a Dirac spinor (apart from in the Dirac equation)? The point is that, under a parity transformation [3, 5], the boost generators K change sign, whereas the generators J do not. It follows that the (j, 0) and (0, j) representations are interchanged under parity, whence 2-component spinors are not sufﬁcient to provide a full description. The Dirac 4-component spinor carries an irreducible representation of the Lorentz group extended by parity.

Note A.6 In addition, from χA † ψ ≡ ⇒ ψ = χ ,ηA , [a] ηA [a] A (A.28)

the adjoint Dirac spinor is written as (Weyl representation) ηA 0 −1 ψ ≡−ψ†γ 0 ηA, χ ⇒ ψT = ,γ0 ≡ . [a] A [a] χ A −10 (A.29) Note that ψ1ψ2 ≡ η1χ2 + χ 1η2 where the right-hand side is a Weyl 2-spinor representation, noticing that in the left-hand side we have Dirac 4-spinors, with the bar above either ψ1 or χ1 concerning different conﬁgurations and operations, and subscripts 1 and 2 merely labeling different Dirac spinors.

Note A.7 The charge conjugate spinor ψc is deﬁned by

13 There are also chiral Dirac spinors. These constitute eigenstates of γ 5 and behave differently under Lorentz transformations [see (A.33) and (A.34)]. 220 A List of Symbols, Notation, and Useful Expressions

− σ 2ηA η ψc ≡ ψT = i = A , C 2 A (A.30) iσ χ A χ

where 2 εAB =−iσ 0 C ≡ . (A.31) 0 ε A B = iσ 2 Note A.8 The Majorana condition ψ = ψc gives χ = η : χ ψ = ψc = A . χ A (A.32)

The importance of the Majorana condition is that it deﬁnes the Majorana spinor. This is invariant under charge conjugation, therefore constituting a relation (a sort of ‘reality’ condition) between ψ and the (complex) conjugate spinor ψ†. A Majorana spinor is a particular Dirac spinor, namely ψ A , ψ A

with only half as many independent components. A Majorana spinor is therefore a Dirac spinor for which the Majorana condition (A.32) is implemented: ψ = ψc χ = ψ† , i.e., A A.

Of course, whenever we have (Dirac or Majorana) 4-spinors, a related matrix framework must be used, in the form of the Dirac matrices (here in the chiral or Weyl representation): μ μ 0 σ 1 0 2 γ ≡ ,γ≡ iγ 0γ 1γ 2γ 3 = , γ 5 = 1 , σ μ 0 5 0 −1 (A.33) with γ inducing the projection operators (1 ± γ )/2 for the chiral components ψ , 5 5 A χ A of ψ, determining their helicity as right (positive) or left (negative). But there are many other options [9]: • Standard14 canonical basis: −1 0 0 σ i γ ≡ ,γ≡ . (A.34) 0 0 1 i −σ i 0

14 The standard (Dirac) representation has γ0 appropriate to describe particles (e.g., plane waves) in the rest frame. A.3 About Spinors 221

• γ ∗ =−γ A Majorana basis, where i i : −σ 2 σ 3 γ ≡ 0 ,γ≡ 0i , 0 −σ 2 0 1 iσ 3 0 (A.35) − σ 1 γ ≡ i1 0 ,γ≡ 0 i . 2 0 −i1 3 −iσ 1 0

• A basis in which the γ matrices are all real and where N = 1 SUGRA allows a real Rarita–Schwinger (gravitino) ﬁeld (see Sect. 4.1 of Vol. I) [15]: − σ 2 γ ≡ 0 1 ,γ≡ 1 0 ,γ≡ 0 i , 0 −1 0 1 0 −1 2 iσ 2 0 (A.36) 0 1 0iσ 3 γ ≡ ,γ5 ≡ iγ 0γ 1γ 2γ 3 = . 3 1 0 −iσ 3 0

Furthermore, 9 : μ ν μν γ ,γ =−2η . (A.37)

Through (A.33), the Lorentz generators then become

Lμν −→ i γ μν , 2 where μ ν ν μ μν 1 μ ν ν μ 1 σ σ − σ σ 0 γ ≡ (γ γ − γ γ ) = μ ν ν μ , (A.38) 2 2 0 σ σ − σ σ and {, } and [, ] denote the anticommutator and commutator, respectively, As expec- A ted from the representation (1/2, 0) ⊕ (0, 1/2), this determines that the ψA and χ spinors transform separately. The speciﬁc generators are also rewritten iσ μν for ψ A and iσ μν for χ A , with μν B 1 μ νC B (σ )A ≡ σ σ − (μ ↔ ν) , (A.39) 4 AC μν A 1 μA C ν (σ ) ≡ σ σ − (μ ↔ ν) . (A.40) B 4 CB 222 A List of Symbols, Notation, and Useful Expressions

A.4 Useful Expressions

In this section we present a set of formulas which will help to clarify some properties of SUSY and SUGRA, and help also to simplify some of the expressions in the book. The initial emphasis is on the 2-component spinor, but we will also indicate, whenever relevant, some speciﬁc expressions involving 4-component spinors. In a 2-component spinor notation, we can use a spinorial representation for the tetrad. This then allows us to deal with the indices of bosonic and fermionic variables in a suitable equivalent manner. To be more precise, in ﬂat space we therefore σ AA associate a spinor with any vector by the Infeld–van der Waerden symbols a , which are given by [16, 7–9]15

σ AA =−√1 ,σAA = √1 σ . 1 i i (A.41) 0 2 2

Here, 1 denotes the unit matrix and σi are the three Pauli matrices (A.9). To raise and lower the spinor indices, the different representations of the antisymmetric spinorial AB A B metric ε , εAB, ε , and εA B can be used [see Note A.5 and (A.18)]. Each of them can be written as the same matrix, given by 01 . (A.42) −10

Hence, the spinorial version of the tetrad reads

AA = a σ AA , a =−σ a AA . eμ e μ a e μ AA eμ (A.43)

Generally, for any tensor quantity T deﬁned in a (curved) spacetime, a correspond- AA AA μ ing spinorial quantity is associated through T → T = eμ T , with the inverse μ =− μ AA relation given by T eAA T . Equipped with this feature, the foliation of spacetime into spatial hypersurfaces Σt from a tetrad viewpoint (spinorial version) is straightforward: • The future-pointing unit normal vector n → nμ has a spinorial form given by

AA AA μ n = eμ n . (A.44)

• The tetrad (A.43) is thereby decomposed into timelike and spatial components AA AA e0 and ei .

15 We henceforth condense and follow the structure in Note A.5 (see Chap. 4 of Vol. I). A.4 Useful Expressions 223

• Moreover, the 3-metric is written as

=− AA . hij eAA i e j (A.45)

This metric and its inverse are used to lower and raise the spatial indices i, j, k,....

• From the deﬁnition of n AA as a future-pointing unit normal to the spatial hypersurfaces Σt , we can further retrieve

AA = AA = , n AA ei 0 and n AA n 1 (A.46)

AA AA which allow us to express n in terms of ei (see Exercise 4.3 of Vol. I). • Using the lapse function N and the shift vector N i , the timelike component of the tetrad can be decomposed according to

AA = N AA + N i AA . e0 n ei (A.47)

• Other useful formulas16 involving the spinorial tetrad eAA with the unit vector i in spinorial form n AA can be found in Sect. A.4.1. The reader and fellow explorer of SQC should note that these expressions do indeed allow one to considerably simplify, e.g., the differential equations for the bosonic functionals of the wave function of the universe (see, e.g., Chap. 5 of Vol. I).

A.4.1 Metric and Tetrad

Using the above deﬁnitions [6], we have the following relations for the timelike

AA normal vector n AA and the tetrad ei :

1 n n AB = ε B , (A.48) AA 2 A

BA 1 B n n = εA , (A.49) AA 2 √ AB 1 B AB k e e =− hijε − i hεijkn e , (A.50) AA i j 2 A AA

BA 1 B 1 BA k eAA i e =− hijεA − i√ εijkn AA e , (A.51) j 2 h

i = − ε ε . eAA i eBB n AA n BB AB A B (A.52)

16 Equations (A.44), (A.45), (A.46), and (A.47) can be contrasted with (A.60), (A.61), (A.62), (A.63), (A.64), and (A.65). 224 A List of Symbols, Notation, and Useful Expressions

From (A.50) and (A.51), by contracting with εijl, we obtain

AB l AB k i ijl AB n AA e =−n e = √ ε eAA i e , (A.53) AA 2 h j

BA l BA k i ijl BA n AA e =−n e =− √ ε eAA i e . (A.54) AA 2 h j

In addition, we have

μν AA BB AB A B g e μ e ν =−ε ε , (A.55)

AA BB gμν =−eμ e ν εABεA B , (A.56)

BA B 2eAA (μeν) = gμνεA , (A.57) AB B 2eAA (μeν) = gμνεA . (A.58)

εilm B C The normal projection of the expression DBmjD A kl, used throughout Chap. 4, is given by [17] − CAB i AA ilm B C 2i ilm B CD D AA E ≡ n ε D D = √ ε DB mje eDD kn n Bjk Bmj A kl h l A

2i ilm B CD A = √ ε DB mje e . (A.59) h l D k

Now, regarding the 4-spinor framework in Sects. 4.1.1 and 4.1.2 of Vol. I,

a Ni = e0ae i , (A.60) j a N = N j N − e0ae 0 , (A.61) na e a =− , 0 N (A.62) m e0a = N na + N ema , (A.63) γ n emγ = 0 , (A.64) ijk ⊥ ik ε γsγi = 2iγ σ , (A.65) and we have the notation ⊥ μ 0 1 i − A⊥ = A =−n Aμ = N A = A − N A , N 0 i (A.66)

ˆ N i Ai ≡ Ai − A⊥ . N (A.67) A.4 Useful Expressions 225

A.4.2 Connections and Torsion

In the second order formalism applied to N = 1 SUGRA [13], the tetrad eμCD and gravitinos ψμC , ψν D determine the explicit form of the connections.

Four-Dimensional Spacetime We will use

ab [ab] AA BB AB A B A B AB ωμ = ωμ −→ ωμ = ωμ ε + ωμ ε , (A.68) where

AB (AB) s AB AB ωμ = ωμ = ωμ + κμ , (A.69)

s AB while ωμ is the spinorial version of the torsion-free connection form

s ab aν b bν a aν bρ c ωμ = e ∂[μeν] − e ∂[μeν] − e e ecμ∂[νeρ] , (A.70)

AB (AB) ab [ab] and κμ = κμ is the spinorial version of the contorsion tensor κμ = κμ , with

AA BB AA BB νρ AB A B A B AB κμ = eμ eν κμ = κμ ε + κμ ε . (A.71)

Furthermore, explicitly, it will in this case relate to the (4D spacetime) torsion through

AA A A ξμν =−4πiGψ[μψν] , (A.72) whose tensorial version is

ξ ρ =− ρ ξ AA . μν eAA μν (A.73)

The contorsion tensor κ is deﬁned by

κμνρ = ξνμρ + ξρνμ + ξμνρ . (A.74)

Spatial Representation Within a 3D (spatial surface) representation, the novel ingredient is the expansion AA AA into n and ei . In more detail, the spin connection is written in the form

(3)ωAA BB = (3)sωAA BB + (3)κ AA BB , i i i (A.75) 226 A List of Symbols, Notation, and Useful Expressions and then decomposed into primed and unprimed parts:

(3)ωAA BB = (3)ωABε A B + (3)ωA B ε AB . i i i (A.76)

(3)ωAA BB =−(3)ωBB AA Using the antisymmetry i i , we then obtain the symmetries (3)ωAB = (3)ωBA (3)ωA B = (3)ωB A i i and i i and the explicit representations

(3) AB 1 (3) ABB (3) A B 1 (3) A BB ω = ω , ω = ω . (A.77) i 2 B i i 2 Bi

(3)sωAA BB (3)κ AA BB Analogous relations hold for i and i . Furthermore, we now have

(3)ωAB =(3s)ωAB + (3κ) AB , i i i (A.78)

(3s)ωAB where i is the spinorial version of the spatial torsion-free connection form (3s) ab bj a 1 aj bk c 1 aj b c 1 a b ω = e ∂[ j e ] − e e e ∂ j eck − e n n ∂ j eci − n ∂i n −(a ↔ b), i i 2 i 2 2 (A.79)

κ AB 17 κab and i is the spinorial version of the spatial contorsion tensor i , with

j 3κ = k 3κ = 3κ ABiε A B + 3κ A B i ε AB . AA BB i eAA eBB jki (A.81)

(3)sωAA BB The three-dimensional torsion-free spin connection i can therefore be AA AA expressed in terms of n and ei as

(3)sωAA BB = BB j ∂ AA i e [ j ei] (A.82) 1 − eAA j eBB keCC ∂ e + eAA j nBB nCC ∂ e + n AA ∂ nBB 2 i j CC k j CC i i

− AA j ∂ BB e [ j ei] 1 + eBB j eBB keCC ∂ e + eBB j n AA nCC ∂ e + nBB ∂ n AA . 2 i j CC k j CC i i

17 The 3D contorsion is simply obtained by restricting the 4D quantity: (3) κijk = κijk , (A.80) (3)κ AA BB i = AA j BB (3)κ =−(3)κ BB AA i with the spinorial contorsion e ek jki . A.4 Useful Expressions 227

A.4.3 Covariant Derivatives

Still referring to a second order formalism, the tetrad eμCD and the gravitinos

ψμC ,ψν D constitute the action variables, with which a speciﬁc covariant derivative Dμ is associated, acting only on spinor indices (not spacetime, with which Γ would be associated).

Four-Dimensional Spacetime To be more precise, we use18

AA = ∂ AA + ωA BA + ωA AB , Dμeν μeν Bμeν B μeν (A.83)

ψ A = ∂ ψ A + ωA ψ B , Dμ ν μ ν Bμ ν (A.84)

ωA with Bμ the connection form (spinorial representation) as described in Sect. A.4.2 above.

Spatial Representation (3) Subsequently, we have the spatial covariant derivative D j acting on the spinor indices, where, for a generic tensor in spinorial form,19

(3) AA = ∂ AA + (3)ωA BA + (3)ωA AB , D j T j T B T B T (A.85)

(3)ωA (3)ωA with B and B the two parts of the spin connection [see (A.77)], using the decomposition

(3)ωAA BB = (3)sωAA BB + (3)κ AA BB . i i i (A.86)

A.4.4 (Gravitational) Canonical Momenta

Related to the presence of (covariant) derivatives in the action of N = 1 SUGRA, we CD C D retrieve the momenta conjugate to the tetrad eμ and gravitinos ψμ , ψν . The latter require the use of Dirac brackets, which are discussed elsewhere (see Appendix B and Chap. 4 of Vol. I), so we present here some elements regarding the former.

18 AA AA AA Notice that D[μeν] = ξμν ,whereξμν is the spinor version of the torsion. 19 AA ...ZZ AA ZZ μ...ν T = eμ ...eν T . 228 A List of Symbols, Notation, and Useful Expressions

For the gravitational canonical momentum, we can write δ i SN=1 ji AA j i ⊥i AA i p = −→ p =−e p , p = n p , (A.87) AA δ AA AA AA ei where we can also use (as in the pure gravitational sector, see Sect. 4.2 of Vol. I)

( ) p ij =−2πij . (A.88)

This then leads us to the issue of curvature.

A.4.5 Curvature

In fact, we can express the curvature (e.g., the intrinsic curvature or second fundamental form Kij) in spinorial terms and through the (gravitational) canonical i 20 momentum pAA , with the assistance of (A.87) and (A.88). Let us recall that

1/2 - . h ( )( ) ( ) πij ∼− K 0 ij − trK 0 hij 2k2 1/2 - . h ( ) ( ) ∼− K ij − τ ij − hij(K − τ) , (A.89) 2k2 =− a∂ + ωab , Kij ei j na na j ebi (A.90) but now emphasizing the inﬂuence of the torsion:

1 K( ) = NiIj + N j|i − hij, − 2ξ( )⊥ , (A.91) ij 2N 0 ij μ K[ij] = ξ⊥ij ≡ n ξμij . (A.92)

Four-Dimensional Spacetime The spinor-valued curvature is given by RAB = R(AB) = ∂ ωAB + ωA ωCB , μν μν 2 [μ ν] C[μ ν] (A.93)

R = μ A νRAB + A μ AB νRA B . eAA eB μν eA e μν (A.94)

20 [ij] ⊥i p and additional terms in p will appear in the Lorentz constraint Jab ↔ JABεA B + JA B εAB. A.4 Useful Expressions 229

Spatial Representation The components of the 3D curvature in terms of the spin connection read (3) AB = ∂ (3)ωAB + (3)ωA (3)ωCB , Rij 2 [i j] C[i j]

(3) A B = ∂ (3)ωA B + (3)ωA (3)ωC B . Rij 2 [i j] C [i j] (A.95)

(3)ω[AB] = (3)ω[A B ] = Because of the symmetry of i 0 and i 0, the chosen notation (3)ωA (3)ωA Bi and B i is unambiguous. The horizontal position of the indices does not need to be ﬁxed. The scalar curvature is given by

(3) = i j (3) ABε A B + (3) A B ε AB . R eAA eBB Rij R (A.96)

(3)sωAA BB The same procedure performed on i leads to the torsion-free scalar curvature: (3s) AB = ∂ (3)sωAB + (3)sωA (3)sωCB , Rij 2 [i j] C[i j]

(3s) A B = ∂ (3s)ωA B + (3s)ωA (3s)ωC B , Rij 2 [i j] C [i j] (A.97) and

(3s) = i j (3s) ABε A B + (3s) A B ε AB . R eAA eBB Rij R (A.98)

A.4.6 Decomposition with Four-Component Spinors

We now present, in a somewhat summarized manner, a few helpful formulas for the 3 + 1 decomposition of a Cartan–Sciama–Kibble (CSK) theory, and in particular, Einstein gravity with torsion (see Chap. 2 of Vol. I), using 4-component spinors [15]:

• The torsion is given by

λ 1 λ λ ξμν = Γμν − Γνμ , (A.99) 2

where eλ are basis vectors, which we will take as a coordinate basis with ei tangent to a spacelike hypersurface and the normal n with components n → nμ = (−N , 0, 0, 0), and where the components of the metric are as in (2.6), (2.7), and (2.8) of Vol. I. Then we can write 230 A List of Symbols, Notation, and Useful Expressions

λ eμ;ν = eλΓμν , (A.100)

λ (0)λ λ Γμν = Γμν − κμν , (A.101)

λ λ λ λ κμν = ξμν − ξμ ν + ξ μν , (A.102)

(0)λ λ with Γμν the Christoffel symbols and κμν the contorsion tensor. • The extrinsic curvature Kij can then be computed and retrieved from n; j by [see (2.9)ofVol.I] (4) 0 1 K ji =−N Γ = −h ji, + N j|i + Ni| j − κ ji⊥ , (A.103) ji 2N 0 1 K( ji) = −h ji, + N j|i + N j|i + τ( ) , (A.104) 2N 0 ji = ξ ,τ≡ ξ , K[ ji] ij⊥ ji 2 j⊥i (A.105)

where | denotes the derivative involving the Christoffel symbols for hij. • The curvature is then obtained from

(4) (3) ij 2 (4) α R = R − KijK + K − 2 R⊥ ⊥α , (A.106) (4) α γ β β γ ij 2 αβ R⊥ ⊥α = n n − n n − K Kij + K 1 − 2ξ ⊥nα(β . (A.107) (α (γ (β

• Finally, or almost, the Lagrangian density is √ 1/2 (3) ij 2 γ β β γ αβ hL = N h R − KijK + K − 2 n n − n n + 4ξ ⊥nα(β , (α (γ (β (A.108) from which the ﬁrst two lines of (4.17) of Vol. I are obtained. From the above, the Hamiltonian and constraints are21

H =− πik , m 2hmi |k (A.109) J ab ≡ ka b − kb a , p e k p e k (A.110) −1/2 ij 1 2 1/2 (3) 1/2 (ij) 2 H⊥ ∼ h π πij − π − h R + h τ τ(ij) − τ (A.111) 2 √ √ + ξ ξ ij − 1/2τ ξ ji + kρ − 1/2ρi + 1/2ρi ρ , h ij⊥ ⊥ 2h [ij] ⊥ hq k 2h || i 2h i

21 The reader should notice that, with the above alone, i.e., no gravitino (matter) action, we μνλ have P = 0 for the conjugate to torsion ξμνλ, whose conservation leads to ξμνλ = 0. But if the Rarita–Schwinger ﬁeld is present in an extended action, with Lagrangian terms, e.g., λμνρ ε ψλγ5γμ Dν ψρ ,then 1 ξμνλ =− ψμγλψν , 4 and torsion cannot then be ignored. A.4 Useful Expressions 231

i ab H = NH⊥ + N Hi + MabJ . (A.112)

A.4.7 Equations Used in Chap. 4

AB We employ several derivatives of functionals with respect to the tetrad e j [17]. Two of them are: δ 4 AA B C ε ilmn (D D ) (A.113) δ AB Bmj A kl e j and

δ n AA D BB . (A.114) δ AB ij e j

δ AA /δ BB First we need an explicit form for n e j . With (4.109) of Vol. I, which AA AA expresses n in terms of the tetrad, the relation n eAA i = 0 implies

AA AA AA δn eAA i δn δn 0 = eCC i = nCC n − ε C ε C + eCC j n . δ BB AA δ BB A A δ BB BB e j e j e j (A.115) In addition, we have

AA CC AA AA AA δn δn nCC n δn δn = = 2n n AA + , (A.116) δ BB δ BB CC δ BB δ BB e j e j e j e j whence

δn AA = eAA j n . (A.117) δ BB BB e j

We then use the derivative of the determinant h of the three-metric:

∂h = hijh . (A.118) ∂hij

Hence, δ h i =−2he . (A.119) δ AA AA ei

Using this and (A.48), (A.49), (A.50), (A.51), and (A.52), we can calculate the expressions 232 A List of Symbols, Notation, and Useful Expressions δ AA ilm B C n ε (D D ) δ AB Bmj A kl e j δ 1 =−4n AA εilm e D e n DB eCE e nE δ AB h Bj DD m l EE k A e j C i i 1 1 2i CB i i CB = εB δ √ 1 − 1 + − + √ 2e eBB k + e e k h 2 2 h BB k − 3i i C ij CB l = √ δ εB − 2h ε jkln e , (A.120) h k BB and δ δ 1 n AA D BB =−2inAA √ eBC e nCB δ AB ij δ AB j CC i e j e j h

2i AA BC =−√ n n eAC i . (A.121) h

Finally, we can write a transformation rule to express derivatives in terms of the AA F[ ] tetrad ei as derivatives in terms of the three-metric hij.Let e be a functional depending on the tetrad, and note that hij can be expressed in terms of the

=− AA tetrad, since we have the relation hij ei eAA j . Moreover, there is of course no inverse relation. We thus restrict the functional F by demanding that it can be written in the form F[hij]. Consequently, using the chain rule, we ﬁnd for the transformation of the functional derivatives:

δ BB CC δF δF δh jk δF e j ek = =− ε ε δ AA δh δ AA δh BC B C δ AA ei jk ei jk ei

δF δF =− ε ε CC − ε ε BB AC A C ek BA B A e j δhik δh ji δF =−2 eAA j . (A.122) δhij

AA i =−δi G[ ] Using e eAA j j , the inverse relation for an arbitrary functional hij is simply

δG 1 δG = eAA j . (A.123) δh δ AA ij 2 ei

Note that it is always possible to rewrite G[hij] in the form G[e]. References 233

References

1. Misner, C.W., Thorne, K.S., J.A. Wheeler: Gravitation, 1279pp. Freeman, San Francisco (1973) 212 2. Ortin, T.: Gravity and Strings. Cambridge University Press, Cambridge (2004) 212 3. Bailin, D., Love, A.: Supersymmetric Gauge Field Theory and String Theory. Graduate Stu- dent Series in Physics, 322pp. IOP, Bristol (1994) 213, 214, 219 4. Bilal, A.: Introduction to supersymmetry. hep-th/0101055 (2001) 213 5. Muller-Kirsten, H.J.W., Wiedemann, A.: Supersymmetry: An Introduction with Conceptual and Calculational Details, pp. 1–586. World Scientific (1987). Print-86–0955 (Kaiserslautern) (1986) 213, 214, 219 6. D’Eath, P.D.: Supersymmetric Quantum Cosmology, 252pp. Cambridge University Press, Cambridge (1996) 214, 217, 223 7. Penrose, R., Rindler, W.: Spinors and Space–Time. 1. Two-Spinor Calculus and Relativis- tic Fields. Cambridge Monographs on Mathematical Physics, 458pp. Cambridge University Press, Cambridge (1984) 214, 217, 222 8. Penrose, R., Rindler, W.: Spinors and Space–Time. 2. Spinor and Twistor Methods in Space– Time Geometry. Cambridge Monographs on Mathematical Physics, 501pp. Cambridge Uni- versity Press, Cambridge (1986) 214, 217, 222 9. Wess, J., Bagger, J.: Supersymmetry and Supergravity, 259pp. Princeton University Press, Princeton, NJ (1992) 214, 217, 220, 222 10. Martin, S.P.: A supersymmetry primer. hep-ph/9709356 (1997) 214 11. West, P.C.: Introduction to Supersymmetry and Supergravity, 425pp. World Scientific, Singa- pore (1990) 214 12. Srivastava, P.P.: Supersymmetry, Superfields, and Supergravity: An Introduction. Graduate Student Series in Physics, 162pp. Hilger, Bristol (1986) 214 13. Van Nieuwenhuizen, P.: Supergravity. Phys. Rep. 68, 189–398 (1981) 214, 225 14. D’Eath, P.D.: The canonical quantization of supergravity. Phys. Rev. D 29, 2199 (1984) 217 15. Pilati, M.: The canonical formulation of supergravity. Nucl. Phys. B 132, 138 (1978) 221, 229 16. Nilles, H.P.: Supersymmetry, supergravity and particle physics. Phys. Rep. 110, 1 (1984) 222 17. Kiefer, C., Luck, T., Moniz, P.: The semiclassical approximation to supersymmetric quantum gravity. Phys. Rev. D 72, 045006 (2005) 224, 231

Appendix B Solutions

Problems of Chap. 2

2.1 The Born–Oppenheimer Approximation and Gravitation

The Born–Oppenheimer approximation [1–4] was ﬁrst developed for molecular physics. It was then extended to quantum gravity. In brief, in the form of a recipe, we have the following:

• We use the simple Hamiltonian

P2 p2 P2 H = + + V (R, r) ≡ + h , 2M 2m 2M

with M m. • R, r denote variables for heavy (M) and light (m) particles, respectively. • The purpose is to solve (approximately) the stationary Schrödinger equation HΨ = EΨ . Then:

– Assume (and this is one of the main ingredients!) that the spectrum (in terms of eigenvalues and eigenfunctions) of the light particle is known for each con- ﬁguration R of the heavy particle,, i.e., h|n; R=ε(R)|n; R. Ψ = ( )| ; – Carry out the expansion k k R k R . • Returning to the Schrödinger equation, after multiplying by n; R|, obtain an equation for the wave functions n : 2 h¯ 2 − ∇ + εn(R) − E n(R) (B.1) 2M R

2 2 ! " h¯ h¯ 2 = n; R|∇Rk; R ∇Rk(R) + n; R|∇ k; R k(R). M 2M R k k

235 236 B Soluitions

• Now neglect the off-diagonal terms1 in (B.1): 2 2 2 ! " 1 2 h¯ A h¯ 2 [−ih¯∇−hA¯ (R)] − − n;R|∇ n;R + εn(R) n(R)= En(R). 2M 2M 2M R (B.2) The reader should note that the momentum of the slow particle has been shifted: P → P − hA¯ , where A(R) ≡−in; R|∇Rn; R. This constitutes a Berry connection.2 Concerning the application of the Born–Oppenheimer expansion to gravitation, let us add the following: 1. An expansion with respect to the large parameter M leads to satisfactory results if the relevant mass scales of non-gravitational fields are much smaller than the Planck mass. 2. Regarding the non-gravitational fields: a. Without them, an M expansion is fully equivalent to an h¯ expansion, i.e., to the usual WKB expansion for the gravitational field. b. If non-gravitational fields are present, the M expansion is analogous to a Born–Oppenheimer expansion: large (nuclear) mass → M, small (electron) mass → mass scale of the non-gravitational field.

2.2 Hamilton–Jacobi Equation and Emergence of ‘Classical’ Spacetime

If a solution S0 to the Hamilton–Jacobi equation is known, we can extract

δS πij = M 0 , (B.3) δhij ˙ from which hij is recovered in the form ˙ hij =−2N Kij + Ni| j + N j|i , (B.4) with Kij the components of the extrinsic curvature of 3-space, viz.,

kl Kij =−16πGGijklπ . (B.5)

Note that a choice must be made for the lapse function and the shift vector in order ˙ for hab to be speciﬁed. Once this and an ‘initial’ 3-geometry have been chosen, (B.4)

1 Neglecting the off-diagonal terms corresponds to a situation in which an interaction with environ- mental degrees of freedom allows the k (R)|k; R to decohere from each other (see Sects. 2.2.2 and 2.2.3). 2 If the fast particle eigenfunctions are complex, the connection A is nonzero. Problems of Chap. 2 237 can then be integrated (for a thorough explanation see [5]). A ‘complete’ spacetime will follow, associated with a chosen foliation and a choice of coordinates on each member of this foliation [1]: • Combine all the trajectories in superspace which describe the same spacetime into a sheaf [6]. • The lapse and shift can be chosen in such a way that these curves comprise a sheaf of geodesics. • Superposing expressions in the form of WKB solutions (bearing the gravitational field), one specific trajectory in configuration space, i.e., one specific spacetime, will be described by means of a wave packet. However, there is another point to note here. One must also discuss3 an expansion of the momentum constraints in powers of M. In fact, S0 satisfies the three equations δ S0 = . δ 0 (B.6) hab |b

The Hamilton–Jacobi equation and (B.6) are equivalent to the G00 and G0i parts of the Einstein equations, while the remaining six ﬁeld equations can be found by differentiating (B.4) with respect to t and eliminating S0 by making use of (B.4) and the Hamilton–Jacobi equation.

2.3 Factor Ordering and Semi-Classical Gravity

The terms in (2.19) are independent of the factor ordering chosen for the gravitational kinetic term. Any ambiguity in the factor ordering would have to come from the gravity–matter coupling. But let us elaborate on this [1–4]. Let us introduce a linear derivative term (parametrized by ζij) that represents an ambiguity in the factor ordering of the kinetic term. This will then appear as we ‘enter the range’ of the Schrödinger equation. In the Wheeler–DeWitt equation, we now have: 2 δ2 δ h¯ g m − Gijkl + ζij + MU + H⊥ Ψ = 0 . (B.7) 2M δhijδh jk δhij

Note that (2.11) can also be written as

2 δS0 δK 1 δ S0 Gijkl − Gijkl K = 0 , (B.8) δhij δhkl 2 δhijδhkl but with the linear term we have instead

3 Equations (B.6) will follow from the Hamilton–Jacobi equation and the Poisson bracket relations between the constraints H and Hi [7]. 238 B Soluitions 2 δS0 δK 1 δ S0 δS0 Gijkl − Gijkl + ζij K = 0 . (B.9) δhij δhkl 2 δhijδhkl δhij

0 At order M an equation also involving S1 is 2 δS0 δS1 ih¯ δ S0 Gijkl − Gijkl + Hm = 0 , (B.10) δhij δhkl 2 δhijδhkl but with the linear term we can write 2 δS0 δS1 ih¯ δ S0 δS0 Gijkl − Gijkl + ζij + Hm = 0 . (B.11) δhij δhkl 2 δhijδhkl δhij

With regard to (2.15), we now get 2 δS0 δS2 1 δS1 δS1 ih¯ δ S1 δS1 0 = Gijkl + Gijkl − Gijkl + ζij δhij δhkl 2 δhij δhkl 2 δhijδhkl δhij 1 δS δS ih¯ δ2 S +√ 1 2 − 2 . (B.12) h δφ δφ 2 δφ2

If we now further require σ˘2 to satisfy δ δσ˘ ¯ 2 δK δD ¯ 2 δ2K δK G S0 2 − h G + h G + ζ = , ijkl 2 ijkl ijkl ij 0 δhij δhkl K δhij δhkl 2K δhijδhkl δhij we obtain 2 2 δS0 δη˘ h¯ 2 δF δK δ F δF Gijkl = − Gijkl + Gijkl + ζij δhij δhkl 2F K δhij δhkl δhijδhkl δhij ih¯ δη˘ δF ih¯ δ2η˘ +√ F + √ . (B.13) h δφ δφ 2 h δφ2

Hence, the Schrödinger equation with quantum gravitational corrections, including the linear term, is now δΘ 2 δK δF δ2F δF m h 1 1 1 ih¯ = H⊥Θ + Gijkl − − ζij Θ. (B.14) δτ MF K δhij δhkl 2 δhijδhkl 2 δhij

We decompose derivatives of F into normal and tangential components to the constant S0 hypersurfaces. In more detail (see also Sect. 4.2.6), δF δF G =−i G UijH⊥F ⊕ G $op$ . ijkl ijkl m mnop kl (B.15) δhij h¯ δhmn Problems of Chap. 2 239

The ﬁrst term on the right-hand side is the normal component, with

δ δ δ −1 δ ij S0 S0 S0 1 S0 U ≡ Gijkl =− . (B.16) δhij δhij δhkl 2U δhij

The second term is the tangential component and $ij is a unit vector tangent to the ij S0 constant hypersurface, satisfying U $ij = 0. Here we use

δF op Gmnop $ $kl ≡ amn . δhmn

We then write (see Sect. 4.2.6), using (B.14),

h¯ 2 δ2F 2 δK δF δF −Gijkl + Gijkl − ζij Θ ≡ Cn ⊕ Ct , (B.17) 2MF δhijδhkl K δhij δhkl δhij wherewehave δ δH⊥ δ ≡− 1 (H⊥)2F − G S0 −F m + U H⊥F Θ Cn m ih¯ ijkl m 4MUF δhkl δhij δhij

⊥ 1 ⊥ δH δU ⊥ =− (H )2F − ih¯ −F m + H F Θ. (B.18) 4MUF m δτ Uδτ m

The reader should note that it is due to the conservation law that there are no ambi- guities here, as all terms cancel. Moreover,

2 δK δF δG δF h¯ 2 op ijkl op ij Ct ≡− Gmnop $ $kl + Gmnop $ $ 4MUF K δhkl δhmn hkl δhmn δ δF δF op op ij − Gmnop $ $kl − ζijGmnop $ $ Θ. (B.19) δhkl δhmn δhmn

For the normal components [1], δΘ π δ Hm m 4 G m 2 ⊥ ih¯ = H⊥Θ + √ (H⊥) Θ + ih¯4πG √ Θ. δτ h (3) R − 2Λ δτ h (3) R − 2Λ (B.20) 240 B Soluitions

2.4 Quantum Gravity Corrections and Unitarity vs. Non-Unitarity

The second correction term in (2.19) is pure imaginary. This may lead to a ‘complex’ situation. From the Wheeler–DeWitt equation, we get [1, 3] ⎛ ⎞ ⎛ ⎞ ↔ ↔ δ δ δ δ 1 ⎝ ∗ ⎠ 1 ⎝ ∗ ⎠ Gijkl Ψ Ψ + √ Ψ Ψ = 0 . (B.21) M δhij δhkl h δφ δφ

Applying the M expansion, at the level of the corrected Schrödinger equation we obtain ⎛ ⎞ ↔ δ ∗ h¯ δ ∗ δ 0 = (Θ Θ) + √ ⎝Θ Θ⎠ δτ 2i h δφ δφ ⎡ ⎛ ⎞ ⎤ ↔ δ δ δK δ ih¯ ⎣ ⎝ ∗ ⎠ 1 ∗ ⎦ − Gijkl Θ ψ − Θ Θ . (B.22) 2M δhij δhkl K δhij δhkl

It is the term between square brackets that is of interest, being proportional to M−1. Functionally integrating this equation over the field φ and assuming that Θ → 0for large field configurations: m d ∗ ∗ δ H⊥ DφΘ Θ = 8πG Dφ d3x ψ √ ψ. (B.23) dt δτ h (3) R − 2Λ

A violation of the Schrödinger conservation law thus comes from the above- mentioned term in (2.19).

2.5 De Sitter Space and Quantum Gravity Corrections

The following is a summary of a rather more detailed explanation presented in [3]: 1/3 ˜ • Apply a conformal transformation and write the 3-metric as hij = h hij. • The Hamilton–Jacobi equation becomes √ h δS 2 h˜ h˜ δS δS √ − 3 √0 + ik√ jl 0 0 − (3) − Λ = . ˜ ˜ 2 h R 2 0 (B.24) 16 δ h 2 h δhij δhkl √ (3) • Set R = 0 and look for a solution of the form S0 = S0( h), viz., 7 Λ √ √ S =±8 h d3x ≡±8H h d3x . (B.25) 0 3 0 Problems of Chap. 2 241

• The local time parameter4 in conﬁguration space is √ δ 3 h δS δ √ δ =− √0 √ = 3hΛ √ . (B.26) δτ 8 δ h δ h δ h

• From the conservation law ‘chosen’ for K, it follows that δK/δτ = 0, i.e., it is constant. Then the functional Schrödinger equation is (setting h¯ = 1) 1 δ2 a a3 iΘ˙ = d3x − + (∇φ)2 + m2φ2 Θ. (B.27) 2a3 δφ2 2 2

• Use the Gaussian ansatz5 ˇ 1 ˜ ˇ Θ ≡ N(t) exp − dkΩ(k, t)χˇkχˇ−k . 2

Two equations follow, but the relevant one is6

a y + 2 y + (m2a2 + k2)y = 0 , (B.28) a

where Ωˇ ≡−ia3 y˙/y, and primes denote differentiation with respect to η˜,a conformal time coordinate.7 • To the next order, there are corrections to the Schrödinger equation: δΘ π π δ Hm m 2 G m 2 2 G ⊥ ih¯ = H⊥Θ − √ (H⊥) Θ − ih¯ √ Θ. (B.29) δτ hΛ Λ δτ h

The second correction term is a source of non-unitarity. It produces an imaginary 2 3/ π 5 contribution to the energy density of the order of magnitude 29Gh¯ H0 480 Vc (reinstating c), indicating a possible instability of the system whose associated 3 −1/ 8 time scale is H0 tPl . The ﬁrst correction term in (B.29) causes a shift

√ ∂a3 δ h(x) 4 3 3 H0t This allows one to recover ≡ d y = 3H0a −→ a(t) = e . ∂t δτ(y) 3 5 d k ikx ˜ ikx The momentum representation φ(x) = ϕ(k)e ≡ dkϕk e is employed. Gaus- (2π)3 sian states are used to describe generalised vacuum states. The special form here is due to the Hamiltonian being quadratic, and the Gaussian form is preserved in time. ˙ Ωˇ 2 k2 6 Proceeding from iΩˇ = − a3 m2 + . a3 a2 1 7 We have dt = adη˜ −→ a(η)˜ =− , η˜ ∈ (0, −∞). H0η˜ 8 The prediction can be made more concrete through the following elements: 242 B Soluitions ! " ! " ! " ! " 2πG 841 H⊥ −→ H⊥ − (H⊥)2 −→ H⊥ − GhH¯ 6a3 . m m 3 2 m m π 3 0 3a H0 1382400 (B.30)

2.6 The Bunch–Davies Vacuum and Quantum Cosmology

Take the simpliﬁed Gaussian ansatz (see above)

−Ω(ˇ ) 2/ ˜ = Nˇ (t)e t fn 2 , (B.31) and insert it in (2.33) to get equations for Nˇ and Ωˇ : ˙ Nˇ Ωˇ i = , (B.32) Nˇ 2a3 ˙ Ωˇ 2 n2 iΩˇ = − a3 m2 + . (B.33) a3 a2

Using

y˙ y Ωˇ ≡−ia3 ≡−ia2 , (B.34) y y and the conformal time η, with dt = adη, we get 2 2 m y − y + + n2 y = 0 , (B.35) η H 2η2 where we take a(η) =−1/Hη. We can obtain [3]

( ) (H⊥)2 2 ( ) – Note that m is divergent, rather like Tμν x , i.e., a four-point correlation function. – But use the Bunch–Davies vacuum state (an adiabatic vacuum state) to compute the expectation value [8]. The Bunch–Davies vacuum arises as a particular solution of (B.28). It√ is de Sitter SO(3,1) invariant, reducing to the Minkowski vacuum at early times, i.e., Ωˇ → k2 + m2 as ˙ t →−∞, where the metric is essentially static and one can put Ω˙ = 0anda = 1iniΩˇ .Then ! " 29hH¯ 4a3 H⊥ 0 . m 960π 2 The ﬂuctuation of the Bunch–Davies state will be (quantum mechanically) zero, and ! " ! " 2 (H⊥)2 ≈ H⊥ . m m Problems of Chap. 3 243

i i Ωˇ − −→ − 2 3 − 2ν Zν−1 m Zν− η2 H 2 + n η2 H 2 + n 1 2η Zν 3ηH 2 Zν n2a2 m2a3 −→ (n + iaH) + i , (B.36) n2 + a2 H 2 3H where we have used the usual inﬂationary limit m2/H 2 9/4, ν 3/2−m2/3H 2, not choosing a real Bessel function for Z so that the Gaussian can be normalized. We take complex Hankel functions. If we choose a de Sitter invariant, i.e., invariant ˇ under√ SO(3,1), this reduces to the Minkowski vacuum at early times, i.e., Ω tends to n2 + m2 as t →−∞, where the metric is essentially static and one can put ˙ Ωˇ = 0 and a = 1. This is the Bunch–Davies vacuum.

Problems of Chap. 3

3.1 SUSY Breaking Conditions

Take a state | f for which (see Chap. 3 of Vol. I) 2 1 2 1 † f | P | f ∼ 'SA | f ' + S | f ≥ 0 . (B.37) 0 4 4 A

If | f is invariant under all SUSY generators, i.e., SA | f = 0, then necessarily | | = | | > S S† f P0 f 0. Conversely, if f P0 f 0, not all A and A can annihilate the state | f . This non-invariance of the vacuum state implies that a spontaneous breaking of the (super)symmetry has occurred.

3.2 No-Scale SUGRA

An interesting Kähler class of potentials are those for which we obtain no-scale SUGRA (extracted from [9–12] within the SUGRA limits of string theory), by which we mean that the effective potential for the hidden sector is flat (i.e., constant), wherein the Planck mass is the only input mass, to fix features such as the hierarchy problem, with subsequent non-gravitational radiative corrections to fix the degeneracy: • A trivial situation is described by ∗ G =−3ln φ + φ ⇒ V = 0 , ∀ φ,

where φ is the hidden sector scalar. This simple solution has a simple ﬂat potential (no scale) model, where the potential V vanishes identically, making the vacuum expectation value φ arbitrary, with an arbitrary value for the mass 244 B soluitions

of the gravitino (and the cosmological constant). These models arise naturally in (super)string compactifications, where the size of the compact dimensions is not fixed [13]. Changing them thus costs no energy and a moduli space can be defined. They correspond to Reφ. In a (SUGRA) extension of the standard model, the visible sector (of the chiral multiplets) is connected to a SUSY breaking (hidden) sector. This is all very model dependent. • An extension is ∗ r∗ 2 G =−3ln φ + φ − ϕ ϕr + ln |W| , W ∼ dpqr ϕpϕq ϕr ,

r where φ is now in the visible sector and ϕ is in the hidden sector. With da ≡ G Tarsϕs, this leads to ∂ 2 ∗ r∗ −2 W 1 V ∼ φ + φ − ϕ ϕr + Re(dadb), ∂ϕr 2

whose minimum requires

∂W = da = 0 , ∀ r, a ⇒ Vmin = 0 . ∂ϕr

3.3 Finite Energy State Degeneracy

On the one hand, from Exercise 3.1, the energy vanishes only if S| f =S†| f =0. On the other hand, let us take a state f , associated with a given energy. Since S commutes with the Hamiltonian and cannot change the energy of such a state, with S| f =$| f ,wehave0= S2| f =$S| f =$2| f leading to S| f . Hence, if a state | f is unique, it must satisfy S| f =S†| f =0, have zero energy, and necessarily be the ground state (see Note 3.6 and [14–19]).

3.4 Pair States

Notice that the energy eigenvalues of both H1 and H2 are positive semi-deﬁnite, (1,2) i.e., En ≥ 0 [see (3.40) and [20]]:

E =ψ|H|ψ∼Sψ|Sψ+S†ψ|S†ψ≥0 .

The Schrödinger equation for H1 is

H ψ(1) = A†Aψ(1) = (1)ψ(1) . 1 n n En n (B.38)

Multiplying both sides from the left by A, we get H Aψ(1) = AA†Aψ(1) = (1) Aψ(1) . 2 n n En n (B.39) Problems of Chap. 3 245

Similarly, the Schrödinger equation for A2, viz.,

A ψ(2) = AA†ψ(2) = (2)ψ(2) , 2 n n En n (B.40) implies H A†ψ(2) = A†AA†ψ(2) = (2) A†ψ(2) . 1 n n En n (B.41)

We may argue as follows: • Aψ(1) = If 0 0, the argument goes through for all the states including the ground state ,and hence all the eigenstates of the two Hamiltonians are paired, i.e., they are related by

(2) = (1) > , En En 0 (B.42)

−1/2 ψ(2) = (1) Aψ(1) , n En n (B.43) −1/2 ψ(1) = (2) A†ψ(2) , n En n (B.44) where n = 0, 1, 2,.... • Aψ(1) = (1) = If 0 0, then E0 0 and this state is unpaired, while all other states of the two Hamiltonians are paired. It is then clear that the eigenvalues and eigenfunctions of the two Hamiltonians H1 and H2 are related by

(2) = (1) , (1) = , En En+1 E0 0 (B.45)

−1/2 ψ(2) = (1) Aψ(1) , n En+1 n+1 (B.46) −1/2 ψ(1) = (2) A†ψ(2) , n+1 En n (B.47) where n = 0, 1, 2,.... Note also the following points: • Adequate state wave functions must vanish at x =±∞. From the above discus- 9 Aψ(1) = A†ψ(2) = sion and (3.42), one cannot have both 0 0 and 0 0. Only one of the two ground state energies can be zero.

9 Note that one would have √ x ( ) 2m ψ 2 exp W(y)dy . (B.48) 0 h¯ 246 B Soluitions

• Aψ(1) = H For 0 0, since the ground state wave function of 1 is annihilated by the operator A, this state has no SUSY partner. Knowing all the eigenfunctions of H1, we can determine the eigenfunctions of H2 using the operator A.And † conversely, using A , we can reconstruct all the eigenfunctions of H1 from those of H2 except for the ground state. • For (3.42), the normalizable zero energy ground state is associated with H1, and W is positive (negative) for large positive (negative) x, implying a zero fermion number. • If we cannot ﬁnd normalizable functions like (3.42) and (B.48), then H1 does not have a zero eigenvalue, and SUSY is broken. For W of the form wxn,ifw is positive and n odd, we get a normalizable state, but not the reverse, i.e., for w negative and n even. With W± ≡ limx→±∞ W,ifsign(W+) = sign(W−), SUSY is broken, whereas if sign(W+) =−sign(W−), SUSY remains intact, i.e., we can also write

1 Δ = sign(W+) − sign(W−) . 2

• The Witten index does not depend on the details of the superpotential being invariant under deformations that maintain their asymptotic behavior, something that is further discussed in the context of topological invariance.

If SUSY remains intact, and if the potential V (x) provides an exactly solvable situation with n bound states and ground state energy E0, one can extend to a V1(x) = V (x) − E0 setting, whose ground state energy is zero. One can obtain all the n − 1 eigenstates of H2, then obtain all the n − 2 eigenstates of H3.Inthis way, new exactly solvable settings are possible.

Problems of Chap. 4

4.1 N=1 SUGRA Action, SUSY Transformations, and Time Gauge

Let us for this discussion rewrite the N = 1 SUGRA as (see [21–23] for more details) 1 4 S2 = d xeR, (B.49) 2k2 M

= + S3/2 S3/2 S3/2 Problems of Chap. 4 247 1 4 μνρσ A A A A = d x ε ψ μeAA ν Dρψ σ − ψ μeAA ν Dρψ σ ,(B.50) 2 M B = 1 3 ˆ , S2 d x eK (B.51) k2 ∂M 1 B = 3 εijkψ A ψ A , S3/2 d x i eAA j k (B.52) 2 ∂M

a aˆ where e = det [e μ], eˆ = det [e i ], aˆ = 1, 2, 3, and the integration is over a spacetime manifold M with boundary ∂M. The boundary terms, labelled by the subscript B, are necessary for dealing with quantum amplitudes of transitions and also to obtain the correct amplitudes, with prescribed data, leading to classical solutions, i.e., with δS = 0 providing the classical ﬁeld equations [24, 25]. Recall that the curvature scalar R is given in terms of the tetrad ﬁeld e and the spin connection ωabμ by

μ ν ab R(e,ω)= ea eb R μν(ω) , (B.53) that is,

ab ab ab ac b ac b R μν(ω) = ∂μω ν − ∂νω μ + ω μωc ν − ω νωc μ . (B.54)

The variation is then δS e,ψ,ψ,ω(e,ψ,ψ) δS δS δS = δe + δψ + δψ (B.55) δ ψ,ψ,ω δψ δψ e e,ψ,ω e,ψ,ω δS δω(e,ψ,ψ) δω(e,ψ,ψ) δω(e,ψ,ψ) + δψ + δψ + δe , δω e,ψ,ψ δψ δψ δe with the help of the chain rule. It is useful to recall that ωabμ satisﬁes its own ﬁeld equations, so one can drop the last term in (B.55) after inserting ωabμ(e,ψ,ψ) into the action. Let us then vary the SUGRA action with respect to the tetrad and the gravitinos. In list form, we have the following points: • Take a a μ a δe = eδ ln det [e μ] = eδ Tr ln [e μ] = eea δe μ , (B.56)

where

ν b ν μ δea =−(δe μ)eb ea . (B.57) 248 B Soluitions

• It follows that

δ(eR) = (δe)R + e(δR) μ a μ ν ab = e ea (δe μ)R + δ(ea eb )R μν a μ μ ν ab = e (δe μ)ea R + 2ea (δeb )R μν

a μ ν μ ρ cb = eδe μ(ea R − 2ea eb ec R ρν). (B.58)

• In addition, δ(eKˆ ) =ˆe δK +ˆe−1(δeˆ)K =ˆ ω0aˆ δ i + j (δ bˆ )ω0aˆ i e i eaˆ ebˆ e j i eaˆ

=ˆωaˆ0 ( j i − i j )δ bˆ . e i eaˆ ebˆ eaˆ ebˆ e j (B.59)

• Note that we have used the time gauge here, and we are assuming it to be preserved under the SUSY transformations10 (discussed at length in Exercises 5.1–5.3).11 • Moreover12 1 4 μνρσ δS / = d x ε (δψ A μ)e ν Dρψ Aσ + ψ A μ(δe ν)Dρψ Aσ 3 2 2 AA AA M

1 4 μνρσ −1 A A B A = d x ε (2k Dμε + Λ B ψ μ)eAA ν Dρψ σ 2 M +a volume integral

1 4 μνρσ −1 A A −1 A A = d x ε 2k ∂μ(ε eAA ν Dρψ σ )−2k ε Dμ(eAA ν Dρψ σ ) 2 M +a volume integral ,

1 3 ijk A A = d x ε ε eAA i D j ψ k + a volume integral , (B.60) k ∂ M

μνρσ A 1 μνρσ A 1 μνρσ A B 10 ε Dρ Dσ ε = ε [Dρ , Dσ ]ε = ε R ρσ ε . 2 2 B 11 In brief, the usual SUGRA action is extended by the boundary terms, namely (B.52). The new ΛA ΛA feature are also the B and B terms in (B.60) and (B.61), which correspond to the Lorentz transformation terms. 12 Dμ agrees with ∂μ when it acts on objects with no free spinor indices. Problems of Chap. 4 249 1 4 μνρσ δS / =− d x ε ψ Aμ(δe ν)Dρψ A σ + ψ Aμe ν Dρ(δψ A σ ) 3 2 2 AA AA M

1 4 μνρσ A −1 A A B =− d x ε ψ μeAA ν Dρ(2k Dσ ε + Λ B ψ σ ) 2 M +a volume integral 1 4 μνρσ A A B =− d x ε ψ μeAA ν Dρ(Λ B ψ σ ) + a volume integral 2 M 1 4 μνρσ A A B A A B =− d x ε ∂ρ(ψ μeAA νΛ B ψ σ )− Dρ(ψ μeAA ν)Λ B ψ σ 2 M +a volume integral 1 3 ijk A A B =− d x ε ψ i eAA j Λ B ψ k + a volume integral . (B.61) 2 ∂ M

• Let us address two questions the reader may be wondering about: – Note that we are taking left-handed SUSY transformations [25], i.e., ε = 0, which could be the setup for the amplitude to go from prescribed data AA , ψi AA ,ψi (ei A )I on an initial surface to data (ei A)F on a final surface. AA , ψi The gravitinos filling in between (ei )I on an initial surface and data A μ AA ,ψi ψ ,ψν (ei A)F on a final surface would now be independent of ( A A). – With nμ the future-pointing unit vector normal to the boundary ∂M,ifwe assume13 that the three spacelike axes of the tetrad lie in ∂M, so that

μ na = eaμn = (1, 0, 0, 0), (B.62)

the extrinsic curvature has the simple form

ij ab 0aˆ i K = h naω j ebi = ω i eaˆ . (B.63)

The corresponding action of N = 1 SUGRA is subsequently invariant, up to a boundary term, if local supersymmetry transformations become

AA A A A A A BA A AB δe μ ≡−ik(ε ψ μ + ε ψ μ) + Λ Be μ + Λ B e μ ,(B.64)

A −1 A A B δψ μ ≡ 2k Dμε + Λ Bψ μ , (B.65)

A −1 A A B δψ μ ≡ 2k Dμε + Λ B ψ μ , (B.66)

13 This gauge choice is known as the time gauge. 250 B Soluitions

where

A AA C C i Λ B ≡−2ikn nCC ε ψ i eBA , (B.67)

and its Hermitian conjugate

A AA C C i Λ B ≡−2ikn nCC ε ψ i eAB , (B.68)

and where ε A and ε A are spacetime-dependent anticommuting ﬁelds. The inclusion of the Lorentz terms in (B.64) ensures the preservation of the gauge deﬁned in (B.62) (see Exercises 5.1–5.3).

The reader should note that the variation of the tetrad does not depend on Dμε or Dμε [see (B.64)]. Thus the volume Einsten–Hilbert sector does not involve any derivatives of ε or ε, and so cannot be integrated by parts to give a surface term. Hence, since the theory is supersymmetric, its variation must cancel with the volume terms arising from the variation of the gravitino action. From the remarks following (B.64), we are only interested in surface terms, that is, we will integrate by parts to extract all surface terms in δS3/2 which involve Dμε or Dμε. Collecting equations (B.59), (B.60), (B.61), with the volume integral terms vanishing in this SUSY context, the total variation under these local supersymmetry transformations is

1 3 ijk A A 1 3 ijk A B A δS = d x ε ε eAA i (D j ψ k) − d x ε Λ B ψ i eAA j ψ k k ∂M 2 ∂M 1 ˆ + 3 1/2ωaˆ0 ( j i − i j )δ b , d xh i eaˆ ebˆ eaˆ ebˆ e j (B.69) k2 ∂M provided that the gauge condition (B.62) is preserved. Finally, note the generic expression [25] retrieved under left-handed SUSY transformations:

2 3 ijk A (3s) A δS = d x ε ε eAA i D j ψ k , (B.70) k ∂M

(3s) where D j is the torsion-free spatial covariant derivative.

A 4.2 S0 Must Depend on the Gravitino ψ i

With the SUSY constraints and the ansatz −1 Ψ = exp i(S0G + S1 + S2G +···)/h¯ , obtain to lowest order G0, Problems of Chap. 4 251

− O(G0) ( ) δS0 [Ψ ] 1 Ψ = εijk 3s ψ A + π ψ A = . S A eAA i D j k 4 i i 0 (B.71) δeAA i

ψ A AA This must hold for arbitrary fields i and ei . Then S0 must at least depend on ( ) AA εijk 3s ψ A = ei . Otherwise we would get the condition eAA i D j k 0, which cannot hold for all fields. A Assume then that S0 does not depend on the gravitino field ψ . Integrating (B.71) i over space with an arbitrary continuous spinorial test function ε A (x) leads to

( ) δS0 I ≡ d3xε A εijke 3s D ψ A + 4πiψ A = 0 . (B.72) 0 AA i j k i δ AA ei

ψ A &→ ψ A φ( ) ε A ( ) &→ ε A ( ) −φ( ) With the replacement i i exp [ x ] and x x exp [ x ], this becomes [26] ( ) δS0 I ≡ d3xε A exp(−φ) ijke 3s D ψ A exp φ + 4πiψ A exp φ 0 AA i j k i δ AA ei

= 0 . (B.73)

" ≡ − = Now for I0 I0 I0 0, " = 3 εijk AA ( )ε A ( )ψ ( )∂ φ( ) = , I0 d x ei x x Ak x j x 0 (B.74) which cannot hold for all ﬁelds. To higher orders, the calculation for n ≥ 1 leads to

− O(Gn) δSn [Ψ ] 1 Ψ =−π nψ A = . S A 4 iG i 0 (B.75) δeAA i

AA So can we choose Sn not to depend on the bosonic ﬁeld ei ? This would be very restrictive: no proper bosonic limit would exist. We therefore dismiss this hypothe- sis. Hence, to satisfy (B.75), it is necessary to introduce a dependence on the gravitino ﬁeld at each order. This means that we must have Sn ≡ Sn[e,ψ] for all n [27].

4.3 Obtaining Equation (4.13)

Write the fermionic part of the last line as 252 B Soluitions 1 ( ) δ 1 ( ) − ih¯ 3s D εijkψ D B = εijk 3s D (ψ ψ ) i Aj A lk δψ B i Aj A k 2 l 2 - . 1 ijk (3s) (3s) = ε Di ψAj ψ + ψAj Di ψ 2 A k A k - . 1 ijk (3s) (3s) = ε D j ψAk ψ − ψAi D j ψ . 2 A i A k (B.76)

Comparing the above with the third line written in the form - . 1 ijk (3s) (3s) ε D j ψAk ψ + ψAi D j ψ , (B.77) 2 A i A k

(3s) the terms containing D j (ψ A k) will cancel out. Finally, the normal projection of (3s) ψ A the remaining term containing D j k is given by ( ) δ h¯ ( ) δ n AA ih¯εijk 3s D ψ D B = √ εijkeBC eA 3s D ψ . j Ak A li δψ B i C l j Ak δψ B l h l (B.78)

4.4 Decomposition of the Hamilton–Jacobi Equation into Bosonic and Fermionic Sectors

For the WKB wave functional, write i − i − Ψ [e,ψ]≡exp B G 1 exp (F G 1 + S +···) . (B.79) h¯ 0 h¯ 0 1

Then select the pure bosonic part B0 satisfying

δB δB 4πin AA D BB 0 0 + U = 0 , (B.80) ij δ AB δ BA e j ei i.e., corresponding to the Hamilton–Jacobi equation (4.22). This solution B0 can then be used to determine the condition for the part F0 involving fermions: δF δF δB δF 0 = 4πi ψ B 0 ECAB i 0 + ψ B 0 ECAB i 0 i δ AB Bjk δψC i δ AB Bjk δψC e j k e j k δF δF δF δB δB δF −n AA D BB 0 0 − n AA D BB 0 0 − n AA D BB 0 0 ij δ AB δ BA ij δ AB δ BA ij δ AB δ BA e j ei e j ei e j ei i ( ) δF0 + √ ijkeBC eA 3s D ψ . (B.81) i C l j Ak δψ B h l Problems of Chap. 4 253

If a solution S0 of (4.20) is to be found [omitting the term (4.26)], together with F0 = S0 − B0, then the above equation is satisﬁed.

G(s) 4.5 The DeWitt Supermetric ab Should Contain the DeWitt Metric Gijkl

Take arbitrary ‘vectors’

j ˜ i B B va ≡ AB , v˜b ≡ BA , (B.82) l ˜ k FD FC and obtain (see Chap. 4)

G(s)vav˜b =− π AA BB + BB AA j ˜ i ab 4 i n Dij n D ji BAB BBA

+ π BB ψC ε jkm A D l ˜ i 4 in j DC mi D B lk FD BBA

+ π AA ψ Bεilm B C j ˜ k . 4 in i DBmjD A kl BAB FC (B.83)

Let

δ δ j a ˜ i b B ≡ , B ≡ , AB δ AB BA δ BA e j ei

where a[e] and b[e] are two arbitrary functionals (that can also be written as a[hij] and b[hij]). Then for the ﬁrst line on the right-hand side of (B.83) (see Appendix A),

δa δb δa δb δa δb 4πi n AA D BB + n BB D AA =−32πG . ij δ AB δ BA ji δ AB δ BA iljk δh δh e j ei e j ei jk il (B.84) Therefore, for quantities that can be written in terms of the three-metric hij and the gravitino, the block B contains the usual DeWitt metric. To be more precise, it is not exactly the usual DeWitt metric due to the change of the fundamental bosonic AA ﬁeld from hij to ei . Rather it is a tetrad version of it.

4.6 Towards a Conservation Law?

Condition (4.34) can be rewritten as [27] 254 B Soluitions δ δS δS δK n AA DBB 0 K = n AA 0 DBB δ AB ij δ BA δ AB ij δ BA e j ei e j ei δ δK δ B S0 CAB i B ilm B C S0 −ψ E − ψ ε D D K i δ AB Bjk δψC i Bmj A kl δψC e j k k δ 3i C Bj CB l S0 − √ ψ + ψ ε n e . (B.85) k jkl BB δψC h k

If the right-hand side of (B.85) is equal to zero, then it can be interpreted as a conservation law (see [1] and Sect. 2.1, and also [28–31, 2, 32, 3, 4, 34]). But here, the physical context is SuperRiem(Σ). A conservation law can only be established in the highly unrealistic and special case of (i) a vanishing dependence of S0 and K on the gravitino and (ii) the assumption that S0[e] and K[e] can be expressed in the form S0[hij] and K[hij]. In fact, (B.85) then implies the conservation equation δ δS − n AA D BB 0 K 2 = 0 . (B.86) δ AB ij δ BA e j ei

However, as explained in Sect. 4.2.1 (see also Sect. 4.1), S0 ought to depend on the gravitino [27].

Problems of Chap. 5

5.1 Time Gauge and SUSY Transformations

The variation of the supergravity action can induce boundary terms. Focusing on extracting a finite quantum amplitude, we aim at exact invariance by adding boundary correction terms. This has so far been impossible for full supergravity, but for homogeneous Bianchi A models, the precise invariance of the action under the subalgebra can be restored by adding an appropriate boundary correction to the action; more precisely, under the subalgebra of left-handed SUSY generators (see [35, 21, 36, 22, 23]). Let us take a non-diagonal Bianchi class A ansatz in the time gauge. In order to preserve this gauge, the transformation laws must be modified by adding a Lorentz generator term to the supersymmetry generators. In fact, the time gauge represents a boundary fixing property (see Exercise 4.1). To preserve it, all bosonic generators must be symmetries of the boundaries, i.e., not induce translations normal to it. Any anticommutator of any fermionic generators should induce bosonic generators tangential to the boundary. Consequently, the tetrad axes 0 must remain within a 3D boundary, whence the gauge condition e i = 0 will be preserved under a supersymmetry transformation. The general SUSY transformations will not preserve this gauge. One way for- ward is through a combination of SUSY transformations added to appropriate Lorentz transformations: Problems of Chap. 5 255

AA A A A A A BA A AB δe μ =−ik ε ψ μ + ε ψ μ + Λ Be μ + Λ B e μ , (B.87)

A A 0 for some parameters Λ which are linear in ε and ε , such that δe i = 0or AA n AA δe i = 0. A solution is

A AA C C i Λ B =−2ikn nCC ε ψ i eBA , (B.88)

A AB BA where we have Λ A = 0 and Λ = Λ . Thus the extended supersymmetry transformations are

AA A A A A A BA A AB δe μ =−ik(ε ψ μ + ε ψ μ) + Λ Be μ + Λ B e μ , (B.89)

A −1 A A B δψ μ = 2k Dμε + Λ Bψ μ , (B.90)

A −1 A A B δψ μ = 2k Dμε + Λ B ψ μ . (B.91)

The addition of the Lorentz term to the SUSY transformations subsequently induces new SUSY generators in terms of the Lorentz rotation generators JBC, J B C , given by

BC SA(new) ≡ SA + φA JBC , (B.92) B C S A (new) ≡ S A + φ A J B C , (B.93) where speciﬁcally here

AB (AB) 2 AA C B i φC ≡ φC = ik n nCC ψ i e A , (B.94)

A B AB and φC is the Hermitian conjugate of φC .

5.2 Variation of N=1 SUGRA Action, Bianchi Models, and Boundary Terms

Let us now see what the action of SUGRA (or rather, its variation for left-handed SUSY transformations) becomes for Bianchi A models. The restriction to left- handed SUSY transformations means also that ε and ε are no longer Hermitian conjugates of each other: eAA μ is no longer required to be Hermitian and ψ Aμ and

ψ A μ become independent quantities [35, 21, 36, 22, 23]. For spatially homogeneous cosmologies, (B.69) takes the form

1 3 pqr A A 1 3 pqr A B A δS = d x ε ε eAA p(Dq ψ r ) − d x ε Λ B ψ peAA q ψ r k ∂ 2 ∂ M M 1 3 1/2 aˆ0 q p p q bˆ + xh ω (eˆ e ˆ − eˆ e ˆ )δe , 2 d p a b a b q (B.95) k ∂ M 256 B Soluitions

aˆ AA A A where e p, e p, ψ p, and ψ p are deﬁned by

aˆ aˆ p AA AA p e i ≡ e p(t)ω i , e i ≡ e p(t)ω i , (B.96)

A A p A A p ψ i ≡ ψ p(t)ω i , ψ i ≡ ψ p(t)ω i , (B.97)

p p i with ω˜ = ω i dx . Now, after a rather lengthy calculation using N 0 μ ≡ , ea i N eaiˆ eaiˆ we obtain the following: • For the torsion-free spin connection14

(s) aˆbˆ [bˆ aˆ]ˆcdˆ cˆ aˆbˆdˆ ω ≡ 2Q ˆ ε e ˆ − Q ε eˆ , (B.98) p c dp dˆ cp N q (s) aˆ0 1 aˆ aqˆ bˆ aˆbˆ cˆ dˆ bˆ aˆcˆdˆ ω p ≡ (e˙ p + e e pe˙ ˆ ) + Q ε ˆ ˆe pe q − Q ε e ˆ e ˆ . 2N bq N bcˆd cˆ bp dq (B.99)

• For the contorsion, aˆbˆ 2 [bqˆ aˆ] A A 1 aqˆ brˆ A A κ p ≡ ik e σ ψ [pψ q] + e e e ψ [q ψ r] , (B.100) AA 2 AA p 2 N q aˆ0 ik 1 aˆ A A aˆ aqˆ A A κ p ≡ σ ψ [pψ ] − σ + e n ψ [pψ q] 2 N AA 0 N AA AA r 1 aqˆ N aqˆ + e e ψ A [ ψ A ] − e e ψ A [ ψ A ] . (B.101) N AA p q 0 N AA p q r

• Subsequently, (s) AB 1 A BA aˆ bˆ aˆbˆ 1 aˆ aqˆ bˆ ω p = n σˆ iN e p Q − 2e ˆ Q + (e˙ p + e e pe˙ ˆ ) N A a bˆ b p 2 bq +N q aˆbˆε cˆ dˆ − bˆεaˆcˆdˆ , Q bˆcˆdˆe pe q Qcˆ ebpˆ edqˆ (B.102)

14 aˆbˆ (aˆbˆ) aˆ pq bˆ aˆ −1 Q ≡ Q ≡ e pm e q (det [e p]) . Problems of Chap. 5 257 2 N AB ik q (AC A B) A q BA r D D κ p = N e ε ψ [pψ q] + e e e ψ [q ψ r] 2N CA 2 A DD p

(A A B) r C C A BA q −n A ψ [pψ 0] − N eCC pψ [q ψ r]n A e C C A BA q q (A A B) +eCC pψ [q ψ 0]n A e + N n A ψ [pψ q] . (B.103)

• And last but not least,

aˆ aˆ AA δe p =−σ AA δe p ˆ ˆ ˆ ˆ = σ a ε A ψ A + εabc q ε A ψ A . ik AA p k ebˆ ecpˆ n AA q (B.104)

• Substituting (B), (B.103), and (B.104) into (B.95), we obtain

aˆ A A 1 s aˆbˆ δS = ςdet [e ]ε ψ σˆ e ˆ Q p s k aAA b i r aˆ s ts s aˆ q + e e˙ r eˆ +˙e t h − 2e e˙ q eˆ 4kN AA a AA AA a i p cˆ s dˆ aˆbˆ + N σ e ˆ e p Q ε ˆ 2kN AA b aˆcˆd ˆ ˆ ˆ ˆ +N pσ bs d acε cAAˆ e e p Q aˆbˆdˆ k pr qs 1 q ps + n χpqr h h − e χpq h 2 AA 2 AA 0 k s pq s(r q)p + e χp q h − e χq r h h 2N AA 0 AA p 0 N p k s qr s(t q)r + e χpqr h + e χtpqh h , 2N AA AA r (B.105)

where 3 p ς ≡ d x det [ω i ]

and

A A χpqr ≡ ψ [pψ q]eAA r , A A χp0q ≡ ψ [pψ 0]eAA q , A A χpq0 ≡ ψ [pψ q]n AA . (B.106) 258 B Soluitions

5.3 Invariance of N=1 SUGRA Action, Bianchi Models, and Boundary Terms

It is possible to restore the invariance of the action of N = 1 SUGRA restricted to the Bianchi class A setting, under a subalgebra of (left!) SUSY generators [35, 21, 36, 22, 23]. If we add B d 1 S ≡ i W + Wψ (B.107) dt g 2 to the supergravity Lagrangian, where

1 pq pqr W ≡ ςm h , Wψ ≡ iςε ψ A ψ A e , (B.108) g k2 pq p q AA r then invariance of the N = 1 SUGRA action is restored under left-handed SUSY transformations. AA aˆ To check this, besides the transformation for e p and e p, we also need [23]

A 2i B AA aˆ bˆ aˆbˆ δψ = ε n σˆ e Q − 2e ˆ Q p k aAB p bˆ b p

1 B AA aˆ aqˆ bˆ − ε n σˆ e˙ + e e e˙ ˆ kN aAB p p bq

2 q B AA bˆ cˆ aˆdˆ cˆ aˆbˆdˆ − N ε n σˆ e e Q ε ˆ ˆ + eˆ e ˆ Q ε kN aAB p q dbcˆ cp bq dˆ

r (A C B ) A 1 B AA r t D D −ikε e ε ψ [pψ r] − ikε e e e ψ [r ψ t] B AC 2 AB DD p

ik ( ) ik ( ) + ε A ψ B ψ A − N q ε A ψ B ψ A N B n A [p 0] N B n A [p q]

ik ik − εB AA q ψC ψC + N q εB AA r ψC ψC N n eAB eCC p [q 0] N n eAB eCC p [r q]

C AA C q B −2ikε n nCC ψ q eAB ψ p . (B.109)

The signiﬁcance of the above is that the existence of a non-trivial, boundary- preserving (sub)algebra allows the theory to be SUSY invariant and admits a Nicolai map.15 But the addition of the boundary term also presents a signiﬁcant bonus which underlies the (perhaps crucial) developments in Sect. 5.2.3 of Vol. I and Sect. 4.1.1

15 Each bosonic component of the wave function will evolve under supersymmetry according to a type of Fokker–Planck equation. In Bianchi class A cosmologies, for the quantum states that permit Nicolai maps, just two states appear, corresponding to the empty and filled fermion states (see Sects. 3.3 and 3.4). Problems of Chap. 5 259 of this volume: it simplifies the canonical formulation of the theory compared with the formulation presented in Chap. 4 of Vol.I. In particular, it leads to simplifications in the new Dirac brackets, whereupon the quantization provides a wider range of potential tools [37–40].

5.4 Quantization of Bianchi Models and Boundary Terms

The exact invariance of the N = 1 SUGRA Bianchi A theory under the action of the left-handed generators is due to (B.107). For right-handed SUSY transformations (ε = 0), the correction term would have been the Hermitian conjugate. Note that Wψ is just the fermion number F, and recall that only the Euclidean version of a supersymmetric theory is known to admit a Nicolai map. Hence we take

B d 1 S ≡ W + Wψ , (B.110) E dτ g 2 B d 1 S ≡− W + Wψ . (B.111) E dτ g 2

B =− B With SE SE , this can be written [23] d 1 pq 1 pqr A A L −→ L± ≡ L ± ςm h pq + iςε ψ pψ q e . (B.112) dτ k2 2 AA r

Now with I is the Euclidean action introduced in Sect. 2.7 of Vol. I, deﬁne

1 I± ≡ I ± W + Wψ , (B.113) g 2

± p from which there follows an expression for the new momentum p AA :

± ∂ I± p p ≡ AA AA ∂e p

p 1 pqr 2 pq =−ip ± iςε ψ ψAr ∓ ςm e . (B.114) AA 2 A q k2 AA q

AA Then the following Dirac brackets become possible among the variables e p, ± p A A p AA , ψ p, and ψ p: 260 B Soluitions - . AA BB e p, e q = 0 , (B.115) D - .

AA ± q A A q e p, p BB = ε Bε B δp , (B.116) D 9 : + + p, q = , p AA p BB D 0 (B.117) 9 : − − p, q = , p AA p BB D 0 (B.118) 9 : ψ A ,ψB = , p q D 0 (B.119) - . A B ψ p, ψ q = 0 , (B.120) D - .

A A 1 AA ψ p, ψ q = D pq , (B.121) D σ - . AA B e p,ψ q = 0 , (B.122) D - . AA B e p, ψ q = 0 , (B.123) D 9 : + p,ψB = , p AA q D 0 (B.124) - . − p B p AA , ψ q = 0 , (B.125) D - .

+ p B prs B p AA , ψ q =−iε ψ A s DA rq , (B.126) D 9 : − p,ψB = ε prsψ B , p AA q D i As D A qr (B.127) where (B.117), (B.118), (B.124), and (B.125) are indeed much simpler then the ones without (B.113). The Hamiltonian becomes (see Exercise 5.1)

AA A A H ≡−e 0HAA + ψ 0SA(new) + S A (new)ψ 0 C AB C A B − ωAB0 + ψ 0φCAB J − ωA B 0 + ψ 0φC A B J , (B.128) Problems of Chap. 5 261

where we now have

A p − JAB = ie(A p B)A p , (B.129) A p + J A B = ie (A p AB )p , (B.130) together with

1 2 A + p B C S ( ) =− k ψ p p + φ J , (B.131) A new 2 AA A B C

1 2 A − p BC S ( ) = k ψ p p + φA JBC . (B.132) A new 2 AA The reader can indeed conﬁrm that the resulting SUSY constraints become simpler. p Note also that, in HAA ≡−n AA H⊥ + eAA Hp, the form of HAA is compli- cated, but we can write - . 2 (new) (new) HAA S , S + terms proportional to J and J k2 A A D 4 k 1 − p + q BB pqr + s B B = p p D qp + iε p D ψAr ψ p 4 ς AB BA BA B sq

pqr − s B B 1 1/2 p B Bq +iε p AB DB qsψ A r ψ p + ςh n BB ψ A ψ [q ψAp]ψ 2 1 pqr s B B + iςε e ψ ψ q ψA[pψ s] . (B.133) 2 BB A r

Quantum mechanically [23], we can obtain an explicit representation of the differ- 16 ential operator S A (new) as ∂ ∂ + 1 2 1 2 1/2 AC q S ( ) = h¯k ψ A − h¯k h n ψC D , (B.134) A new p AA CA q p AC 2 ∂e p 2 ∂e p

− while the differential operator SA(new) is represented by ∂ ∂ − 1 2 1 2 1/2 CA q S ( ) =− h¯k ψ A + h¯k h n ψC D , (B.135) A new p AA AC q p CA 2 ∂e p 2 ∂e p

± ± corresponding to the ± in the action in (B.113). So S A(new) and S A (new) are related to the original operators S A(new) and S A (new) by the following transformations (see Sect. 3.3):

16 In the following, τ should be considered as a mere parameter, although in a path integral presentation, it would be the Euclidean time. 262 B Soluitions

+ −Wg/h¯ Wg/h¯ S A(new) ≡ e S A(new)e , (B.136)

+ −Wg/h¯ Wg/h¯ S A (new) ≡ e S A (new)e , (B.137)

− Wg/h¯ −Wg/h¯ S A(new) ≡ e S A(new)e , (B.138)

− Wg/h¯ −Wg/h¯ S A (new) ≡ e S A (new)e . (B.139)

The transformed wave functions Ψ +, Ψ − are therefore associated with wave functions Ψ(e,ψ,τ), Ψ(e, ψ,τ) by

+ − / Ψ (e,ψ,τ) = e Wg h¯ Ψ(e,ψ,τ), (B.140) − / Ψ (e, ψ,τ) = eWg h¯ Ψ(e, ψ,τ) . (B.141)

+ p In summary, we may take the (Euclidean) representations of the operators p AA , A AA A ψ p acting on a wave function Ψ(e p,ψ p) by means of

+ ∂ p p =−h¯ , (B.142) AA AA ∂e p

¯ ∂ ψ A = h AA , p D qp A (B.143) ς ∂ψ q where

+ + B C S A (new) = S A + φ A J B C , (B.144)

− − BC S A(new) = S A + φA J BC , (B.145) and

+ 1 ∂ S = h¯k2ψ A , (B.146) A p AA 2 ∂e p

− 1 ∂ S =− h¯k2ψ A . (B.147) A p AA 2 ∂e p

Problems of Chap. 6

6.1 Spinor Symmetry

/ AA = ( )1 2 A BA Since eAA i n 0, it follows that 2h ie A i n is symmetric. Problems of Chap. 6 263

6.2 FRW with Cosmological Constant

The (dimensionally) reduced Lagrangian and Hamiltonian can be expressed, after integrating over the spatial coordinates, as =˙ωζ + θ˙ ηA − H ω,ζ; θ ,ηA , L A A (B.148) √ γ 2 1/2 2 1 A H = 4 12Vζ iω − ω + ζ − $ γ θAθ 12 4 √ γ −1/2 A A + 4 12Vζ ηA 2 (i − ω) θ + η , (B.149) 3$

i where the gauge AAB0 = ψA0 = N = 0, N = 1 was used. The classical evolution is given by [41, 24, 42, 43]

2 √ dω γ / = [ω, H] = 12Vζ 1 2 , (B.150) dt 3 √ dθ − / A =−8 12Vζ 1 2 (i − ω) θ A , (B.151) dt σ √ d 1/2 A −1/2 A = 4 12V −σ (i − 2ω) θ + 2ζ ηAθ , (B.152) dt √ dη − / 1 = 4 12Vζ 1 2 $ γ θ A − 2 (i − ω) ηA , (B.153) dt 2 with the relations (from the Hamilton–Jacobi equation) ∂ Y 12 2 1 A ζ = = −iω + ω + $ γ θAθ , (B.154) ∂ω γ 2 4

∂Y $ ηA = =−6 ( − ω) θ A , i (B.155) ∂θA γ ω2 ω3 12 i 1 A Y =− − + $ γ (i − ω) θAθ , (B.156) γ 2 2 3 4 √ where $ , γ are rescalings of Λ =−6Υ 2 and $ = 2/3. A comment is now in order. In (B.150) and (B.152), the presence of the Grass- mann variables suggests that the solution process begins with the bosonic sector, i.e., obtaining an anti-de Sitter background, namely, an H 4 hyperboloid, for a negative cosmological constant (see Sect. 5.1.3 of Vol. I). The following step (to the 264 B Soluitions next order) is to solve (B.151), (B.153) which are linear in fermions, providing the evolution of the gravitino in the background (recall the analysis and discussion in Sect. 4.2). As expected, we consistently insert these results into (B.150), (B.152), thereby obtaining fermionic corrections to ω and ζ . For more details, the reader should consult [24, 44, 42, 43]. 6.3 N=2 Hidden SUSY and Ashtekar’s Variables

Another element that emerges is the (hidden) N = 2 supersymmetrization of Bianchi cosmologies straight from bosonic general relativity, using Ashtekar’s new variables (see Chap. 6). In the following, we summarize the main features of [45]. Restricting to a minisuperspace at a pure gravity level, the constraint equations (not with a 2-spinor form) are:

i ai J = DaE , (B.157) C = ai i , b E Fab (B.158) H = ε ai bj ˜ k , ijkE E Rab (B.159)

i i D i where Rab is the curvature of Aa, and a the covariant derivative formed with Aa. a Restricting to Bianchi cosmologies, we introduce a basis of vectors Xi [see (6.54)– (6.57)], and then expand into new variables:

˜ a ≡ ˇ i a , E j E j Xi (B.160) ˜ j = ˇ j χi . Aa Ai a (B.161)

i j Dropping the diacritic to simplify the notation, the quantities E j and Ai only depend on time. Inserting these substitutions into the constraint equations [41], we obtain17

J i ≡ k ij + εijk m , C jkE AmjEk (B.162) H =− j i m + εimn i , k Ei AmC jk E j A jmAkn (B.163) H = εijk p m n + m n( i j − i j ). C mnEi E j Apk Ei E j AmAn AnAm (B.164)

Moreover, the constraint equations can be written in a more compact and rather interesting fashion. Introducing the variables

i ≡ i k , Q j EkA j (B.165) the Hamiltonian constraint for all Bianchi class A models can be written as

17 i = ( 1, 2, 3) i = ( , , ) Further restricting to diagonal models, E j diag E E E and A j diag A1 A2 A3 . Problems of Chap. 6 265

∗i k ∗i j H ≡ Q k Q i − Q i Q j . (B.166)

All the dynamics of all class A diagonal models can be further summarized by

H = Q1 Q2 + Q1 Q3 + Q2 Q1 + Q2 Q3 + Q3 Q1 + Q3 Q2 ij = G Qi Q j , (B.167) where i, j = 1,...,3, and ⎡ ⎤ 011 1 ⎢ ⎥ Gij = ⎣ 101⎦ . (B.168) 2 110

Using the Misner–Ryan parametrization with βij given by √ √ βij = diag(β+ + 3β−,β+ − 3β−, −2β+), it is possible to ﬁnd solutions of the form [46] Ψ [α, β±]=C[α, β±] exp −I (α, β±) .

This involves requiring I to be the solution of the Euclidean Hamilton–Jacobi equation for this particular model. The relation with the Ashtekar variables is that, if we have a wave function ΨA given in terms of the Ashtekar variables [47], we can reconstruct the wave function in terms of the Misner–Ryan/ variables by choosing Ψ = (± )Ψ = aΓ i 3 MR exp iIA A, where IA is given by IA 2i Ei ad x, with IA equal to i ∓iI for the Bianchi IX case. Γ a is the corresponding (compatible) spin connection. But where is the (‘hidden’ N = 2) SUSY? The above Hamiltonian suggests (see Chap. 8 of Vol. I) introducing the expressions

i S ≡ ψ Qi , (B.169) S ≡ ψi ∗ , Qi (B.170) for the SUSY constraints, with

ψi ψ j + ψi ψ j = Gij , (B.171) and G as given above. The Hamiltonian constraint becomes

1 H = SS + SS . (B.172) 2

In quantum mechanical terms, one realization is Ψ [A,η], with 266 B Soluitions ∂ Ei Ψ [A]= Ψ [A] , (B.173) ∂Ai

Ai Ψ [A]=Ai Ψ [A] , (B.174)

i together with ψi = ηi , ψ = Gij∂/∂ηj . The equations become

i SΨ [A,η]= Qi η Ψ [A,η] , (B.175)

∗ ∂ SΨ [A,η]=Qˆ Ψ [A,η] , (B.176) i ∂ηi and using the reality conditions, we get

∂ i η Ai Ψ [A,η]=0 , (B.177) ∂Ai ∂ ∂ ki (Ak − 2 Γ k) G Ψ [A,η]=0 . (B.178) ∂Ak ∂ηi

The only solution admitted by this system of equations is Ψ [A]=constant, but it has a nontrivial form in terms of the Misner–Ryan variables.

Problems of Chap. 7

7.1 An Avant Garde (Non-SUSY) Square Root Formulation

Take the corresponding Hamiltonian constraint to be

2 2 2 − pΩ + p+ + p− = 0 , (B.179) with the convention in [48] (see Sect. 7.2 for more details on the expressions used here). With the quantum correspondence

∂ ∂ pΩ −→ − ih¯ , p± −→ − ih¯ , (B.180) ∂Ω ∂β± we may write

∂2Ψ ∂2Ψ ∂2Ψ = + . 2 2 2 (B.181) ∂Ω ∂β+ ∂β− Problems of Chap. 7 267

Now ‘linearize’ the square operator

∂2Ψ ∂2Ψ + , 2 2 ∂β+ ∂β− to obtain 1/2 ∂Ψ ∂2Ψ ∂2Ψ ∂Ψ ∂Ψ ∂Ψ =± + −→ = α+ − α− , i 2 2 i i i (B.182) ∂Ω ∂β+ ∂β− ∂Ω ∂β+ ∂β−

2 where α± are matrices satisfying α± = 1 and which anticommute! The minimal rank for such matrices is two. Then taking α± to be the Pauli matrices, namely α+ ≡ σ1, α− ≡ σ2, Ψ must be a two-component ‘vector’ (spinor). Writing Ψ+ Ψ = , (B.183) Ψ− we obtain plane wave solutions Ψ± ∼ exp i (p+β+ + p−β− − EΩ) , (B.184)

/ where ± (p+β+ + p−β−)1 2 = E and p+, p−,Ω are constants. The two spinor degrees of freedom were described in [48] as ‘disturbing’. In fact, the two solutions correspond to expanding (E > 0) and contracting (E < 0) universes, but how should one then interpret the ‘spin’ states Ψ± in terms of physical attributes of the universe? The reader should remember that this was in 1972! Researchers had to wait for SUGRA in order to obtain a clearer view (see Sect. 7.2). The above is a possible representation within SQC, i.e., SUGRA as applied to cosmology, using the matrix representation for fermionic momenta! It is altogether astonishing how this emerged back in 1972, before the full development of SUSY and SUGRA.

7.2 From the Lorentz Constraints to the Lorentz Conditions

−1 We ﬁnd that ΨIII =−(J12) J13ΨIV. Then show that J12ΨII = J23ΨIV, whence

−1 −1 J23 = J12 (J13) J23 (J12) J13 , −1 −1 J12 = J23 (J13) J12 (J23) J13 , −1 −1 J13 = J23 (J12) J13 (J23) J12 . (B.185)

BC Using JA = ε0ABCJ /2, the result follows. 268 B Soluitions

7.3 From the Lorentz Constraints to the Non-diagonal ‘Missing’ Equations

First obtain the equations of motion and Lorentz constraints for the diagonal Bianchi type IX cosmological model [49]. Then, e.g., substitute the J 01 and the J 23 component expressions in the (0, 1) component of the Einstein–Rarita–Schwinger equation. This reduces to √ ˙ ˙ ˇ 2 ˇ −3β+ + 3β_ ψ2γ ψ1 = 0 . (B.186)

At the quantum level (in this matrix representation approach!), (B.186) yields the same result as the Lorentz constraints. Now with this result, we can analyze the (2, 3) component of the Einstein–Rarita–Schwinger equation. The last term is once ˇ 2 ˇ more ψ2γ ψ1 and the first and second terms cancel out. The first of the three remaining terms is quadratic in the gravitino field components. The other two terms ˇ can also be reduced to quadratic terms in ψi , multiplied by linear combinations of ˙ ˙ ˙ ˇ Ω, β+, and β_, with the help of the equations for ψi . The same procedure is applied to, e.g., the (0, 2) and (0, 3) components of the Einstein–Rarita–Schwinger equa- ˇ tion. Substituting into all the i = j equations, and the ψi equations are used once more. The quadratic expressions resulting from this procedure for all the equations i = j are:

ˇ 0 1 2 ˇ 0 2 3 ˇ ψ1γ γ γ φ1 , ψ2γ γ γ ψ1 ,

0 1 3 0 1 3 ˇ φ1γ γ γ φ1 , φ1γ γ γ ψ2 ,

ˇ 0 2 3 0 2 3 ˇ ψ1γ γ γ φ1 , φ3γ γ γ ψ1 ,

ˇ 0 1 2 1 2 3 ˇ ψ2γ γ γ φ2 , φ3γ γ γ ψ1 ,

ˇ 0 1 3 0 1 2 ˇ ψ2γ γ γ φ2 , φ1γ γ γ ψ3 ,

ˇ 0 2 3 0 1 2 ˇ ψ2γ γ γ φ2 , φ3γ γ γ ψ2 ,

ˇ 0 1 2 0 1 3 ˇ ψ3γ γ γ φ3 , φ2γ γ γ ψ3 ,

ˇ 0 1 3 1 2 3 ˇ ψ3γ γ γ φ3 , φ3γ γ γ ψ2 ,

ˇ 0 2 3 1 2 3 ˇ ψ3γ γ γ φ3 , φ1γ γ γ ψ2 . (B.187)

ˇ These expressions at the quantum level (i.e., proceeding from the variables ψi ab to χi ) lead to the same wave function that has been obtained by applying the J constraints. Therefore the equations i = j yield the same result as the Lorentz constraints J ab. Problems of Chap. 8 269

Problems of Chap. 8

8.1 Invariance of the Action Under N=2 Conformal SUSY

For the the action (8.39), we get (see also [50])

· 1 f Z X Y Z δS = GXY(q )q˙ q˙ + f N U(q ) dt , (B.188) 2 N under the reparametrization t → t + f (t) with

· δq X (t) = f (t)q˙ X , δN (t) = ( f N ) . (B.189)

Hence, we get a total derivative. We are interested in the action (8.41), whose variation is G i XY X Y Z δS = Dη LDη DηQ DηQ + W(Q ) (B.190) grav 2 2N G XY X Y Z +Dη LDη DηQ DηQ + W(Q ) dηdηdt , 2N under (8.7), (8.8), (8.12), and

i i δQ = LQ˙ + DηLDηQ + DηLDηQ . (B.191) 2 2

This is a total derivative, so the action is indeed invariant, as claimed.

8.2 Dirac Brackets in N=2 Conformal SUSY

From ψ(t), χ(t), and λ(t), we have the following second class constraints:

i pψ ≡ πψ + ψ = 0 , (B.192) 3 i p ≡ π + λ = 0 , (B.193) ψ ψ 3

i J pI (χ) ≡ πI (χ) − G χ = 0 , (B.194) 2κ2 JI i p (χ) ≡ π (χ) − G χ J = 0 , (B.195) I I 2κ2 IJ 270 B Soluitions

i J ΠI (λ) ≡ πI (λ) − G λ = 0 , (B.196) 2κ2 JI i p (λ) ≡ π (λ) − G χ J = 0 , (B.197) I I 2κ2 IJ

˙ I ˙ I where πψ = dL/dψ, πI (χ) = dL/dχ˙ , and πI (λ) = dL/dλ are the momenta conjugate to the anticommuting variables ψ(t), χ(t), and λ(t), respectively. Then write the matrices

2 i i C = i , C (χ) =− G , C (λ) =− G , (B.198) ψψ 3 IJ k2 IJ IJ k2 IJ and their inverses

ψψ − 3 C = (C ) 1 =− i , C IJ(χ) = ik2GIJ , C IJ(λ) = ik2GIJ . ψψ 2 (B.199) The Dirac brackets [ , ]D are (recall Chap. 4 of Vol. I) −1 ik [ f, g]D =[f, g]− f, pi (C ) pk, g , (B.200) and allow elimination of the momenta conjugate to the fermionic variables, leaving us with the following non-zero Dirac bracket relations:

[a,πa] = [a,πa] =−1 , (B.201) D D φ ,πJ = φ ,πJ =−δ J , I φ I φ I (B.202) D φ ,πJ = φ ,πJ =−δ J , I φ I φ (B.203) D I χ I , χ J =−ik2GI J , (B.204) D λI , λJ =−ik2GI J , (B.205) D

3 ψ, ψ = i . (B.206) D 2

8.3 FRW Wave Function in N=2 Conformal SUSY

With k = 0, 1, take the action to be (see [51, 52]) Problems of Chap. 8 271

S = SFRW + Smat , √ A 1 k 2 S = 6 − DηADηA + A dηdηdt , FRW 2k2 N 2k2 A3 1 3 S = DηΦDηΦ − 2A W(#) dηdηdt , (B.207) mat 2 N whereweuse

# = φ(t) + iηχ (t) + iηχ (t) + F (t)ηη (B.208) for the component of the (scalar) matter superfields #(t,η,η), with #+ = #. As often indicated, after the integration over the Grassmann complex coordinates η and η, and making the following redefinition of the ‘fermion’ fields (Grassmann variables)

1 ψ(t) −→ a−1/2(t)ψ(t), χ(t) −→ a−3/2(t)χ(t), 3 we ﬁnd the Lagrangian in which the ﬁeld F(t) is auxiliary, and they can be elim- inated with the help of their equations of motion. In addition, we use the Taylor expansion for the superpotential:

∂W 1 ∂2W W(#) = W(φ) + (# − φ) + (# − φ)2 +··· , ∂φ 2 ∂φ2 with # as given above.18 In terms of components of the superﬁelds A, N, and #,the Lagrangian then reads (somewhat more simply than in Chap. 8) √ 2 3 a(Da) 2 k / L =− + iψDψ + a1 2(ψ ψ − ψ ψ) k2 N 3 k 0 0 √ 3 2 1 − 3k a (Dφ) + Na 1 kψψ + Na + − iχDχ 3 k2 2 N √ √ 3 − − kNa 1χχ − k2NW(φ)ψψ − 6 kNW(φ)a2 2 3 ik −Na3V (φ) + k2NW(φ)χχ + Dφ(ψχ + ψχ) 2 2 2 2 ∂ W(φ) ∂W(φ) k − / −2N χχ − kN (ψχ − ψχ)+ a 3 2(ψ ψ − ψ ψ)χχ ∂φ2 ∂φ 4 0 0 ∂W(φ) −ka3/2(ψ ψ − ψ ψ)W(φ) + a3/2 (ψ χ − ψ χ) , 0 0 ∂φ 0 0 (B.209)

18 This series expansion ends at this point, in the second term, since (# − φ) is nilpotent. 272 B Soluitions where

ik i Da =˙a − a−1/2(ψ ψ + ψ ψ) , Dφ = φ˙ − a−3/2(ψ χ + ψ χ) , 6 0 0 2 0 0 are the supercovariant derivatives, and

i i Dψ = ψ˙ − V ψ, Dχ =˙χ − V χ, 2 2 are the U(1) covariant derivatives. The potential for the homogeneous scalar fields is ∂W(φ) 2 V (φ) = − k2W2(φ) , 2 ∂φ 3 (B.210) and consists of two terms. One is the potential for the scalar field in the case of global supersymmetry. The potential (B.210) is not positive semi-definite, in contrast with what happens in standard supersymmetric quantum mechanics. Indeed, the present model describing the minisuperspace approach to supergravity coupled to matter, allows supersymmetry breaking when the vacuum energy is equal to zero V (φ) = 0. Quantization requires for the Dirac brackets

3 ψ, ψ = i , [χ,χ] =−i , (B.211) D 2 D whence

9 : 3 ψ, ψ =− , {χ,χ} = 1 , (B.212) 2

[a,πa] =−i , φ,πφ =−i . (B.213)

Antisymmetrizing, i.e., writing a bilinear combination in the form of the commutators, e.g.,

1 χχ −→ [χ,χ] , 2 this leads eventually to eigenstates of the Hamiltonian with four components: ⎡ ⎤ Ψ1(a,φ) ⎢ ⎥ ⎢ Ψ2(a,φ)⎥ Ψ(a,φ)= ⎢ ⎥ , (B.214) ⎣ Ψ3(a,φ)⎦ Ψ4(a,φ) References 273 using instead ψS − ψS Ψ and χS − χS Ψ , with a matrix representation for ψ, ψ, χ, and χ. Then only Ψ1 or Ψ4 have the right behaviour when a →∞, because the other components are inﬁnite as a →∞. We thus get the partial differential equations √ − / ∂ / / 3 − / −a 1 2 − 6W(φ)a3 2 + 6 kM2 a1 2 + a 3 2 Ψ = 0 , (B.215) ∂a Pl 4 4 ∂ ∂W(φ) + a3 Ψ = , ∂φ 2 ∂φ 4 0 (B.216)

√ − / ∂ / / 3 − / −a 1 2 + 6W(φ)a3 2 − 6 kM2 a1 2 + a 3 2 Ψ = 0 , (B.217) ∂a Pl 4 1 ∂ ∂W(φ) − a3 Ψ = , ∂φ 2 ∂φ 1 0 (B.218) with solutions √ Ψ −→ Ψ ( ,φ) 3/4 − w(φ) 3 + 2 2 , 4 a a exp 2 a 3 kMPla (B.219)

√ Ψ −→ Ψ ( ,φ) 3/4 w(φ) 3 − 2 2 . 1 a a exp 2 a 3 kMPla (B.220)

These are eigenstates of the Hamiltonian with zero energy and also with zero fermionic number.

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Index

A D Ashtekar connection, 112 Decoherence Ashtekar–Jacobson formulation, 116 quantum cosmology, 21 N = 1 SUGRA, 116 SQC, 74 DeWitt metric, 71 DeWitt supermetric, 71, 74, 75, 78 B N = 1 SUGRA, 71 Bianchi I matrix representation, 133 F Bianchi IX Fayet–Iliopoulos potential, 35 matrix representation, 138 Fermionic differential operator representation, Born–Oppenheimer 87, 96, 101 semiclassical (quantum) gravity, 14 supersymmetry breaking, 87 Boundary conditions, 5, 19, 21 Fermionic momentum inﬁnite wall proposal, 5, 28 differential operator representation, 6, 7 no boundary, 5, 19, 21 matrix operator representation, 87 tunneling, 5, 19, 21 matrix representation, 7 FRW model C matrix representation, 147 Cartan–Sciama–Kibble N = 1 SUGRA, 229 G torsion, 229 General relativity Chiral superﬁelds, 35 constraints Classical SUSY spin 1/2 particle DeWitt metric, 71 Dirac equation, 130 Goldstino, 38 quantization, 130 Grassmann numbers, 218 Constraints, 4 Gravitational canonical momenta, 227 Contorsion Gravitino mass, 41 decomposing 3D connection, 227 Conventions and notation, 211 H Cosmological constant Hamiltonian SUSY breaking, 41 minisuperspace, 20 Covariant derivative, 227 Hamiltonian constraint minisuperspace, 20 N = 1 SUGRA, 67 Curvature, 228 SQC extrinsic, 228 FRW, 92 intrinsic, 228 generic gauge supermatter FRW, 92 spinor valued, 228 scalar supermultiplet FRW, 89

277 278 Index

Hamiltonian formulation Lorentz constraint, 128 N = 1 SUGRA, 6, 7 Minisuperspace, 4, 20 general relativity, 5 covariant derivative, 20 supergravity, 6, 7 curvature, 20 Hamilton–Jacobi equation, 15 D’Alembert equation, 20 N = 1 SUGRA, 68, 70 dimension, 20 body and soul, 70 factor ordering problem, 20 bosonic and fermionic parts, 69–71, 75 Hamiltonian, 20 spacetime emergence, 70 Laplace–Bertrami, 20 supermanifolds, 71 minisupermetric, 20 superRiemaniann geometries, 71 validity, 20 SUSY spacetime, 70 Wheeler–DeWitt equation, 20 quantum cosmology, 15 perturbations, 20 quantum gravity, 15 WKB approximation, 21 Hidden supersymmetries, 7 Momentum constraints SQC, 7 N = 1 SUGRA, 67 M-theory, 4 I Infeld–van der Waerden symbols, 222 N Inflationary cosmology, 28 Nicolai maps, 55 Nilpotency, 70 K Kähler P geometry, 90, 91 Pauli matrices, 222 metric, 90, 91 spinors, 222 potential, 90, 91 Q L Quantum constraints List of symbols, 207 N = 1 SUGRA, 67 Loop quantum cosmology, 111 Quantum cosmology, 4, 6, 7 Loop quantum gravity, 111 decoherence, 21 Ashtekar connection, 112 Hamilton–Jacobi equation, 15 Barbero–Immirzi parameter, 112 historical, 4 connection quantization, 114 inflation, 13, 19, 21, 28 Hamiltonian formulation, 114 vacuum, 21 reality conditions, 112 minisuperspace, 6, 19, 21 Lorentz conditions beyond, 19 matrix representation, 152 Hamiltonian, 20 rest frame solution, 153 perturbations, 19, 21 Lorentz constraint vacuum state, 21 matrix representation, 147, 155 validity, 20 SQC WKB, 21 Bianchi, 96 observational, 13, 19, 66 FRW, 92 predictions, 13, 19 generic gauge supermatter FRW, 92 vacuum state, 21 Lorentz constraint trivialization Schrödinger equation, 15 matrix representation, 153 semiclassical, 13, 19, 66 Lorentz group, 214 semiclassical limit, 7 representations, 214 structure formation, 19 spinors, 214 vacuum state, 21 summary and review, 31 M superstring, 5, 28 Matrix representation, 127 infinite wall proposal, 5, 28 fermionic momenta, 127 landscape problem, 5, 28 Index 279

Tomonaga–Schwinger equation, 15, 74 Spinors, 212 vacuum state Dirac, 218 structure formation, 21 Infeld–van der Waerden symbols, 222 Quantum FRW solutions Lorentz (group algebra), 213 fermionic differential operator Lorentz (group representations), 212 representation Lorentz transformations, 212 generic gauge supermatter, 94 Majorana, 218 Quantum gravity, 4 Pauli matrices, 222 Hamilton–Jacobi equation, 15 primed, 214, 216 observational context, 4 tetrad, 212 Schrödinger equation, 15 unprimed, 214, 216 semiclassical, 13, 66 Weyl, 218 superspace, 4, 19, 21, 61, 70 SQC, 5, 87 Tomonaga–Schwinger equation, 15 achievements, 7 Quantum gravity corrections, 17 Ashtekar–Jacobson formulation, 111, 118, Schrödinger equation, 17 120 R Bianchi class A models, 120 Reality conditions, 112 FRW with cosmological constant, 118 N = 1 SUGRA, 116 Bianchi, 96, 101 Ansatze, 96, 101 S Lorentz constraints, 96 Schrödinger equation, 15, 74 Bianchi I quantum cosmology, 15 Matrix representation, 133 quantum gravity, 15 Bianchi IX quantum gravity corrections, 17 Matrix representation, 138 N = 1 SUGRA, 75, 78, 79 Bianchi minisuperspace reduction, 96, 101 normal component, 79 Lagrangian, 96 SQC and observation, 81 N = 2 SUGRA, 101 tangential component, 79 Bianchi wave function of the universe, 96, SQC, 74 101 Semiclassical expansion connection/loop variables, 7 = N 1 SUGRA, 66 connection quantization, 111 Semiclassical (quantum) gravity, 13 decoherence, 74 Born–Oppenheimer, 13 fermionic differential operator Hamilton–Jacobi equation, 15 representation, 6, 7, 87, 96, 101 observational, 13 fermionic momenta Schrödinger equation, 15, 16 matrix representation, 127 corrections, 17 FRW, 88 time functional, 15 constraints, 92 time functional, 15 gauge ﬁelds ansatze, 91 Tomonaga–Schwinger equation, 15, 16 time functional, 15 generic gauge supermatter, 91 Semiclassical N = 1 SUGRA, 61, 68 Hamiltonian constraint, 92 Born–Oppenheimer, 68 Lorentz constraints, 92 observational, 68 SUSY constraint, 92 Schrödinger equation, 74 FRW generic gauge supermatter sector time functional, 74 quantum states, 96 time functional, 74 FRW model Tomonaga–Schwinger equation, 74 matrix representation, 147 time functional, 74 FRW scalar supermultiplet sector SL(2,C), 214 quantum states, 89 Spacetime foliation FRW wave function of the universe SUGRA, 222 generic gauge supermatter, 94 280 Index

further exploration, 87 Witten index, 45 generic gauge supermatter FRW SUGRA constraints, 92 spacetime foliation, 222 Hamiltonian constraint, 92 summary and review, 56 Lorentz constraints, 92 N = 1 SUGRA, 72, 74, 75 SUSY constraint, 92 bosonic and fermionic parts, 74, 75 wave function of the universe, 94 conservation law, 253 hidden N = 2SUSY,7 simplified form, 73 hidden supersymmetries, 7 N = 1 SUGRA, 3, 4 loop quantization, 111 affinely connected spaces, 229 Lorentz conditions Ashtekar–Jacobson formulation, 116 matrix representation, 152 Ashtekar–Jacobson Hamiltonian Lorentz constraint formulation, 116 matrix representation, 128, 147 Ashtekar–Jacobson quantization, 117 Lorentz constraint trivialization Bianchi class A models, 120 matrix representation, 153 FRW with cosmological constant, 118 matrix fermionic operator representation, bosonic states, 62 7, 87 Cartan–Sciama–Kibble, 229 matrix representation, 127 CFOP argument, 61 Bianchi I, 133 DeWitt supermetric, 71, 75, 78 Bianchi IX, 138 Schrödinger equation, 74, 75 FRW, 147 finite fermion number states, 63 historical background, 129 gravitino mass Lorentz boost components, 155 super Higgs effect, 41 Lorentz constraint, 128 Hamilton–Jacobi equation, 68, 70 Taub, 143 body and soul, 70 non-supersymmetric square root, 129 bosonic and fermionic parts, 69–71, 75 observational, 61, 66, 75 DeWitt supermetric, 71, 75 observational insights, 81 spacetime emergence, 70 open issues, 6, 7, 19, 21 supermanifolds, 71 observational context, 6, 21 superRiemaniann geometries, 71 SUSY breaking, 6 SUSY spacetime, 70 rest frame solution Hamiltonian constraint matrix representation, 153 normal projection, 67 scalar supermultiplet Bianchi Hamiltonian formulation, 6, 7 quantum states, 96, 101 infinite fermion number states, 63 scalar supermultiplet Bianchi class A matrix representation, 130 quantum states, 96 momentum, 67 scalar supermultiplet FRW physical states, 61 constraints, 89 bosonic states, 62 Hamiltonian constraint, 89 CFOP argument, 61 quantum states, 91 D’Eath’s claims, 61 wave function of the universe, 89 finite fermion number, 63 Schrödinger equation, 74 infinite fermion number, 63 semiclassical, 61, 66, 75 quantum constraints, 67 N = 1 SUGRA, 87 Hamiltonian, 67 N = 2 SUGRA, 87, 101 quantum states, 61 Taub model Riemann–Cartan, 229 matrix representation, 143 torsion, 229 wave function of the universe, 6, 89, 94, Schrödinger equation 96, 101, 106 bosonic and fermionic parts, 74, 75 SQM using DeWitt supermetric, 74, 75 SUSY breaking, 45 Schrödinger equation, 72, 74, 75 Index 281

conservation law, 253 Hamiltonian, 177 DeWitt supermetric, 78 Bianchi class A, 169 normal component, 79 FRW universe, 164 quantum gravity corrections, 75, 78 Hamiltonian formulation, 168 simplified form, 73 quantum cosmology, 181 SQC and observation, 81 supersymmetry breaking, 183 tangential component, 79 time reparametrization, 164 using DeWitt supermetric, 74 N = 2 Supersymmetric quantum mechanics, second-order formalism, 225 43 semiclassical expansion, 66 N = 2 supersymmetric quantum mechanics semiclassical formulation, 61 Fock space, 51 semiclassical limit, 7 Nicolai maps, 55 SQC, 87 quantum states, 51 Bianchi ansatze, 96 Fock space, 51 FRW, 88 sigma model, 48 FRW gauge fields ansatze, 91 quantum states, 51 generic gauge supermatter, 91 N = 8 SUGRA, 3 super DeWitt metric, 74 Superfield description, 163 super Higgs effect, 39 N = 2 conformal supersymmetry, 163 gravitino mass, 41 Superfields, 35, 47 supermatter chiral, 35 gaugino condensate, 40 vector, 35 SUSY breaking, 39 Supergravity superRiem(Σ), 61, 70, 78 Hamiltonian formulation, 6, 7 normal part, 78 square root, 5, 6 tangential part, 78 Supergravity (SUGRA), 6, 7 SUSY breaking, 39 SuperRiem(Σ), 61, 70, 78 cosmological constant, 41 normal part, 78 gaugino condensate, 40 tangential part, 78 gravitino mass, 41 Superspace super Higgs effect, 39 quantum gravity, 19, 21 Tomonaga–Schwinger equation, 74 Superspace (N = 1 SUGRA) Wheeler–DeWitt equation, 67 canonical quantization, 67 N = 2 (local) conformal supersymmetry, 163 Dirac quantization, 67 N = 2SQM,43 Superspace (gravity), 61, 70 Nicolai Maps, 55 Superspace (SUSY), 47 quantum states, 51 interpretation, 47 Fock space, 51 supertranslations, 47 Topology and vacuum states, 52 Superstring, 4 N = 2 SUGRA, 3, 7 cosmological models, 7 SQC, 87, 101 dualities, 5 Bianchi ansatze, 101 quantum cosmology, 5, 28 N = 2 SUGRA (SQC) infinite wall proposal, 5, 28 Bianchi quantum states landscape problem, 5, 28 vector field sector, 106 Supersymmetric quantum cosmology (SQC), Bianchi wave function of the universe, 106 5, 87 vector field sector Bianchi Bianchi class A global O(2), 103 Ansatze, 96, 101 Bianchi class A local O(2), 105 FRW, 88 Bianchi class A quantum states, 101 gauge fields ansatze, 91 Bianchi quantum states, 106 generic gauge supermatter, 91 N = 2 conformal supersymmetry generic gauge supermatter sector (complex) scalar fields, 172 FRW quantum states, 96 282 Index

scalar supermultiplet Bianchi vector field sector, 101 quantum states, 96, 101 SUSY breaking scalar supermultiplet Bianchi class A SQM, 45 quantum states, 96 N = 1 SUGRA scalar supermultiplet FRW super Higgs effect, 42 quantum states, 91 SUSY constraint scalar supermultiplet sector SQC FRW quantum states, 89 FRW, 92 Supersymmetric quantum mechanics (SQM), generic gauge supermatter FRW, 92 7, 35, 43, 88 SUSY FRW quantum states, 51 fermionic differential operator Fock space, 51 representation SUSY,6,7,35 gauge fields sector, 91 breaking, 35, 36, 183 generic gauge supermatter, 91 conditions, 36 supersymmetry breaking, 87 cosmological constant, 41 generic gauge supermatter sector fermionic masses, 39 quantum solutions, 96 gaugino condensate, 40 scalar supermultiplet sector goldstino, 38 quantum solutions, 89 gravitino mass, 41 supersymmetry breaking, 87 mechanisms, 37 N = 1 SUGRA, 39 T super Higgs effect, 39, 41 Taub model chiral superfields, 35 matrix representation, 143 Tetrad, 223 Fayet–Iliopoulos potential, 35 metric, 223 goldstino, 38 tetrad decomposition Grassmannian variables, 47 spinorial version, 222 Nicolai maps, 55 Time functional N = 2SQM,43 semiclassical N = 1 SUGRA, 74 Fock space, 51 semiclassical (quantum) gravity, 15 quantum states, 51 Tomonaga–Schwinger equation, 15, 74 summary and review, 56 quantum cosmology, 15, 74 superfields, 35, 47 quantum gravity, 15 supermultiplet, 35, 90 N = 1 SUGRA, 74 chiral, 35, 90 Torsion, 225 vector, 35, 90 Cartan–Sciama–Kibble, 229 superspace, 43 connections, 225 supersymmetric quantum mechanics, 35, 43, 88 U quantum states, 51 Unit normal vector supertranslations, 47 spinorial form, 222 vector superfields, 35 Useful expressions, 222 SUSY Bianchi fermionic differential operator V representation Vector superfields, 35 quantum states, 96 scalar supermultiplet W quantum solutions, 96 Wave function of the universe, 4, 6, 19–21, 89, SUSY Bianchi class A 94, 96, 101, 106 fermionic differential operator perturbations, 20 representation, 96, 101 SQC, 6, 89, 94, 96, 101, 106 scalar supermultiplet sector, 96 Bianchi, 96, 101 N = 2 SUGRA, 101 generic gauge supermatter FRW, 94 Index 283

scalar supermultiplet FRW, 89 N = 1 SUGRA, 67 N = 2 SUGRA (SQC) semiclassical limit, 67 Bianchi, 106 Witten index, 45 WKB approximation, 21 SQM, 45 Wheeler–DeWitt equation, 6, 20 WKB approximation minisuperspace, 20 perturbations, 20 minisuperspace, 21 perturbations, 20 quantum cosmology, 21 polynomial structure, 111 wave function of the universe, 21