appendixa list of symbols, notation, and useful expressions(lorentz and 2-spinor indices) ωa bμ,ω...

Appendix AList of Symbols, Notation, and UsefulExpressions

In this appendix the reader will find a more detailed description of the conventionsand notation used throughout this book, together with a brief description of whatspinors are about, followed by a presentation of expressions that can be used torecover some of the formulas in specified chapters.

A.1 List of Symbols

κ ≡ κi jk Contorsion

ξ ≡ ξi jk Torsion

K Kähler function

KIJ ≡ gI

J ≡ GIJ 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

πμαβ 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

MP Planck mass

V(β+, β−

)Minisuperspace potential (Misner–Ryan parametrization)

Z I J 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

v(a)μ Vector field

f (a)μν Vector field strength

π i j Canonical momenta to hi j

πφ Canonical momenta to φ

π0 Canonical momenta to Nπ i Canonical momenta to Ni

pμa Canonical momenta to ea

μ (tetrad)

Pi jk Canonical momenta to ξ k

i j (torsion)

J j Lorentz rotation generator

Ki Lorentz boost generator

Pμ Translation (Poincaré) generator

εμνλσ Four-dimensional permutation

εi jk 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

βi j (β+, β−) Anisotropy matrices (Misner–Ryan representation)in Bianchi cosmologies

pi j Canonical momentum to β i j

HADM ADM Hamiltonian

Ki j Extrinsic curvature(4)R Four-dimensional spacetime curvature(3)R Three-dimensional spatial curvature

Λ Cosmological constant

Γμνλ Christoffel connection coefficients

, Usual derivative (vectors, tensors)

| and (3)Di Four-dimensional spacetime covariant derivative withrespect to the 3-metric hi j , no spin connection and notorsion, with Christoffel term (vectors, tensors)

; and (4)Dμ Four-dimensional spacetime covariant derivative withrespect to the 4-metric gμν , no spin connection and notorsion, 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-metricgμν , including spin connection (vectors, tensors)

Dot over symbol Proper time derivative

dΩ23 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

a, b, . . . = 1, 2, 3, . . . Local (Lorentz) spatial indices

i, j, k Spatial indices

hi j 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


I, J, . . . Kähler indices

J, j Representation label of SU(2) (spin state)

M,N Elements of SL(2,C)

Lab Lorentz (transformation) matrix representation

piAA′ Momentum conjugate to eAA′

i

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 andtorsion, no Christoffel term (spinors [vectors, tensors])

( and Dν Four-dimensional spacetime covariant derivative withrespect to the 3-metric hi j , with spin connection andtorsion, with Christoffel term (spinors [vectors, tensors])

‖ Three-dimensional analogue of Dν (spinors [vectors,tensors])

(3)D j ≡ (3)∇ j ≡ ∇ j Three-dimensional analogue of Dν (spinors [vectors,tensors])

(3s)D j ≡ (3s)∇ 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)

ωABμ, ω

abμ, ω

AA′B B′ Spin connections

ψ Aμ Gravitino field

nμ, n AA′ Normal to the hypersurfaces

εAB 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

eai Spatial tetrad (a, i = 1, 2, 3)

eAA′μ Tetrad with two-component spinor indices

∼= Equality provided constraints are satisfied

(a) Gauge group index

Dμ, Dφ, Dμ Gauge group covariant derivatives

Q(a) Gauge group charge operator

ε Infinitesimal spinor

Dφ Kähler derivative

D SUSY covariant derivative (SUSY superspace)

%JKI Kähler connection

R Kähler curvature

ψ, χ, λ Spinors

ψ, χ, λ Conjugate spinors, in the 2-spinor SL(2,C) Weylrepresentation

U Minisuperspace potential

P(φ, φ) SUGRA-induced potential for scalar fields

T (a) Gauge group generators

Q SUSY supercharge operators

G Gauge group

T Energy–momentum tensor

D Minisuperspace dimension

R Minisuperspace curvature

D, d, D, D, D, . . . Dimension of space

σ Axion field

1, σ Pauli matrices

A,B, . . . Bosonic amplitudes of SUSY Ψ

EABi Densitized spatial triad

AABj Ashtekar connection

A.2 Conventions and Notation

Throughout this book we employ c = 1 = h and G = 1 = M−2P , with k ≡ 8πG

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,


• 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) to be(−,+,+,+). Therefore, the 3-metric hi j on the spacelike hypersurfaces has thesignature (+,+,+), which gives a positive determinant. Thus the 3D totally anti-symmetric tensor density can be defined by ε0123 ≡ ε123 = ε123 = +1.

A.3 About Spinors

As the fellow explorer may already have noticed (or will notice, if he or she isventuring into this appendix prior to probing more deeply into some of the chapters)there are some idiosyncrasies in the mathematical structures of SUSY and SUGRAwhich are essentially related to the presence of fermions (and hence of the spinorswhich represent them). But why is this, and what are the main features the spinorsdetermine? In fact, why are spinors needed to describe fermions? What are spinorsand how do they come into the theory? This section is devoted (in part) to introduc-ing this issue.

Before proceeding, for completeness let us just indicate that a tetrad formalismis indeed mandatory for introduing spinors.1 The tetrad corresponds to a masslessspin-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 gen-erator (labeled N = 1 SUSY), the corresponding (super)multiplet will alsocontain a fermion with spin 3/2, the gravitino, as described in Chap. 3 ofVol. 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-directproduct 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 underthe Lorentz subgroup of transformations, but there are no such representations of GL(4) whichbehave 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 fourdegrees of freedom to be removed due to general coordinate invariance and six due to local Lorentzinvariance, this leaves 16− (4+ 4+ 6) = 2.

A.3 About Spinors 213

group has six generators: three rotations Ja and three boosts Kb, a, b = 1, 2, 3 withcommutation relations [3–5]:

[Ja,Jb] = iεabcJc , [Ka,Kb] = −iεabcJc , [Ja,Kb] = iεabcKc . (A.1)

The generators of the translations are usually denoted Pμ, with3

[Pμ,Pν

] = 0 ,[Ji ,P j

] = iεi jkPk , (A.2)

[Ji ,P0] = 0 ,[Ki ,P j

] = −iP0δi j , [Ki ,P0] = −iPi . (A.3)

Or alternatively, defining the Lorentz generators Lμν ≡ −Lνμ as L0i ≡ Ki andLi j ≡ εi jkJk , the full Poincaré algebra reads

[Pμ,Pν

] = 0 , (A.4)

[Lμν,Lρσ

] = −iηνρLμσ + iημρLνσ + iηνσLμρ − iημσLνρ , (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 mostsituations of relativistic classical physics, but there are significant advantages inconsidering a more general exploration, namely from the perspective of the theoryof representations of the Lorentz group. The spinor representation is very relevantindeed. For this purpose, it is usual to discuss how, under a general (infinitesimal)Lorentz transformation, a general object transforms linearly, decomposing it intoirreducible pieces. To do this, we take the linear combinations

J ±j ≡ 1

2(J j ± iK j ) , (A.7)

and the Lorentz algebra separates into two commuting SU(2) algebras:

[J ±i ,J ±

j ] = iεi jkJ ±k , [J ±

i ,J ∓j ] = 0 , (A.8)

i.e., each corresponding to a full angular momentum algebra. A few comments arethen 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 a, b, . . . , which can be relatedthrough the tetrad ea

μ (see Sect. A.2).


• The representations of SU(2) are known, each labelled by an index j, withj = 0, 1/2, 1, 3/2, 2, . . . , giving the spin of the state (in units of h). The spin-jrepresentation 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 jtaking 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 representa-tions.5

– In the (1/2, 0) representation, J− ≡ {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 = i)σ\2,where we henceforth use7 the four 2×2 matrices σμ ≡ (1, σ i ), with σ0 usuallybeing the identity matrix and )σ ≡ σi , i = 1, 2, 3, the three Pauli matrices:

σ 1 ≡(

0 11 0

), σ 2 ≡

(0 −ii 0

), σ 3 ≡

(1 00 −1

), (A.9)

satisfying[σ i , σ j

] = 2iεi jkσ k . Note the relation between Pauli matrices withupper and lower indices, namely σ 0 = −σ0 and σ i = σi , using the metricημν .

– In the (0, 1/2) representation we have J = σ\2 but K = −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 cor-respond 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 ofthe Lorentz group. If M is an element of SL(2,C), so is −M, and both produce thesame Lorentz transformation.

4 The pairs (j−, j+) of the finite dimensional irreducible representations can also be extracted fromthe eigenvalues j±(j± + 1) of the two Casimir operators J2± (operators proportional to the identity,with the proportionality constant labelling the representation).5 The representation (1/2, 1/2) has components with j = 1 and j = 0, i.e., the spatial part andtime component of a 4-vector. Moreover, taking the tensor product (1/2, 0)⊗ (0, 1/2), we can geta 4-vector representation.6 It should be noted from J = σ\2 that a spinor effectively rotates through half the angle that avector rotates through (spinors are periodic only for 4π ).7 The description that follows is somewhat closer to [10, 9, 11], but different authors use otherchoices (see, e.g., [3, 5, 12, 13]), in particular concerning the metric signature, σ0, and somespinorial 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, where Z2 consists of the elements 1 and −1. Note also thatSU(2) is the universal covering for the spatial rotation group SO(3). There is a two-to-one mappingof SU(2) onto SO(3).


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 ≡[ψA=0ψA=1

]

transforming under an element M according to

ψA −→ ψ ′A = MB

AψB , M ≡[

M1 M2M3 M4

]∈ SL(2,C) , (A.10)

with A, B = 0, 1, labelling the components. The peculiar feature is that now theother 2-component object ψ A′ transforms as

ψ A′ −→ ψ ′A′ = M∗B′

A′ψ B′ , (A.11)

which is the primed spinor introduced above, while the above ψA is the unprimedspinor.

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

∗.

Note A.3 There is no unitary matrix U such that N = UMU−1 for matricesN ≡ M∗. We have instead N = ζM∗ζ−1 with

ζ =[

0 −11 0

]≡ −iσ2 .

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

ψ ≡(χ

η

),

that Lorentz invariants are obtained as (iσ 2χ)Tχ . (This is the Weyl repre-sentation, where χ, η are 2-component spinors transforming with M and N,respectively.) To be more concrete, with


χ ≡ χA =(χA=0χA=1

),

we put

χ A =(χ0

χ1

)= iσ 2χ =

(χ1−χ0

),

whence the invariant is χ AχA. (Note that χ0χ0 + χ1χ1 = χ1χ0 − χ0χ1 =−χ1χ

1 − χ0χ0 = −χAχ

A, which is quite different from the situation forspacetime vectors and tensors Aμ Aμ = Aμ Aμ!) This can be simplified byintroducing new matrices with χ A = εABχB , where (formally!) εAB ≡ iσ 2.

Likewise for

η ≡ ηA′ =(η0′

η1′

),

ηA′ηA′ , ηA′ = εA′B′ηB′ , εA′B′ ≡ iσ 2, etc. Of course, εAB ≡ iσ 2, εA′B′ ≡ iσ 2,are different objects acting on different spinors and spaces. The presence of thematrix representation is formal for computations.

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

(σμ)AA′ = {−1, σ i }AA′ , (A.12)

and

(σμ)A′A ≡ εA′B′εAB(σμ)B B′ ≡ (1,−σ i )A′A , (A.13)

where indices are raised by the antisymmetric 2-index tensors εAB and εAB givenby11

εAB = εA′B′ =[

0 1−1 0

], εAB = εA′B′ =

[0 −11 0

], (A.14)

10 Note that (A.12) and (A.13) can used to convert an O(1,3) vector into an SL(2,C) mixed bispinorbAB′ .11 Note that, in a formal matrix form, εAB = εA′B′ = iσ2, εAB = εA′B′ = −iσ 2.


with which:

ψ A = εABψB, ψA = εABψβ , ψ A′ = εA′B′ψ B′ , ψ A′ = εA′B′ψ

B′ .(A.15)

Note A.4 Note the difference between (σμ)AA′ and (σμ)A′A, i.e., the lower,upper and respective order of indices, arising from εAB = εA′B′ = iσ2 andεAB = εA′B′ = −iσ 2. To be more precise, the (‘natural’) lower (upper) A′Aindices on (σμ)AA′ , (σμ)A′A come from covariance, e.g., analysing how thematrices

γ μ ≡[

0 σμ

σμ 0

]

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

σμC D′σμA′B = −2δBC δA′

D′ , (A.16)

Tr(σμσν

) ≡ σμ

AB′σνB′A = −2gμν , (A.17)

constituting two completeness relations.In essence, it all involves the statement that σμ

AB is Hermitian, and that, fora 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 con-vention.

• Unprimed indices are always contracted from upper left to lower right (‘tento four’), while primed indices are always contracted from lower left toupper 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 thisis what we actually follow from Chap. 4 of Vol. I onwards. In particular,

εAB = εA′B′ = εAB = εA′B′ ≡[

0 1−1 0

], (A.18)

with which [note the position of terms and order of indices, in contrast with(A.14)]:


ψ A = εABψB , ψA = ψ BεB A , ψ A′ = εA′B′ψ B′ , ψ A′ = ψ B′εB′A′ .(A.19)

With (A.15) [or (A.19)], scalar products such as ψχ or ψ χ can be employed. Ofcourse, when more than one spinor is present, we have to remember that spinorsanticommute.12 Therefore, for 2-component spinors, ψ0χ1 = −χ1ψ0, and also,e.g., ψ0χ1′ = −χ1′ψ0. In more detail,

ψχ ≡ ψ AχA = εABψBχA = −εABψAχB

= −ψAχA = χ AψA = χψ , (A.20)

ψ χ ≡ ψ A′χA′ = . . . = χ A′ψ

A′ = χ ψ , (A.21)

(ψχ)† ≡ (ψ AχA)† = χ A′ψ

A′ = χ ψ = ψ χ , (A.22)

as well as

ψσμχ = ψ Aσμ

AB′χB′ , ψ σμχ = ψ A′σ

μA′BχB , (A.23)

from which, e.g.,

χσμψ = −ψ σμχ , (A.24)

χσμσνψ = ψσνσμχ , (A.25)

(χσμψ)† = ψσμχ , (A.26)

(χσμσνψ)† = ψσνσμχ , (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 literatureon SUSY and SUGRA, other representations will be met, some of which are alsoadopted 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 anotherrepresentation).

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.,(ζψ)∗ = (ζ AψA)

∗ = (ψA)∗ (ζ A

)∗ = ψ A′ζ A′ = ψ ζ .


A 4-component Dirac spinor is made from a 2-component unprimed spinor anda 2-component primed spinor via

ψ[a] ≡(ψA

χ A′

),

transforming as the reducible (1/2, 0) ⊕ (0, 1/2) representation of the Lorentzgroup, hence with

(ψA

0

)and

(0χ A′

)

as Weyl spinors.13

But is there any need for a Dirac spinor (apart from in the Dirac equation)? Thepoint is that, under a parity transformation [3, 5], the boost generators K changesign, whereas the generators J do not. It follows that the (j, 0) and (0, j) representa-tions are interchanged under parity, whence 2-component spinors are not sufficientto provide a full description. The Dirac 4-component spinor carries an irreduciblerepresentation of the Lorentz group extended by parity.

Note A.6 In addition, from

ψ[a] ≡(χA

ηA′

)�⇒ ψ

†[a] =

(χ A′ , η

A)

, (A.28)

the adjoint Dirac spinor is written as (Weyl representation)

ψ [a] ≡ −ψ†γ 0(ηA, χ A′

)�⇒ ψT[a] =

(ηA

χ A′

), γ 0 ≡

(0 −1−1 0

).

(A.29)Note that ψ1ψ2 ≡ η1χ2 + χ1η2 where the right-hand side is a Weyl 2-spinorrepresentation, noticing that in the left-hand side we have Dirac 4-spinors,with the bar above either ψ1 or χ1 concerning different configurations andoperations, 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 differentlyunder Lorentz transformations [see (A.33) and (A.34)].


ψc ≡ CψT =(−iσ 2ηA

iσ 2χ A′

)=(ηA

χ A′

), (A.30)

where

C ≡[εAB = −iσ 2 0

0 εA′B′ = iσ 2

]. (A.31)

Note A.8 The Majorana condition ψ = ψc gives χ = η :

ψ = ψc =(χA

χ A′

). (A.32)

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

(ψA

ψ A′

),

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

†A.

Of course, whenever we have (Dirac or Majorana) 4-spinors, a related matrix frame-work must be used, in the form of the Dirac matrices (here in the chiral or Weylrepresentation):

γ μ ≡[

0 σμ

σμ 0

], γ5 ≡ iγ 0γ 1γ 2γ 3 =

[1 00 −1

],

(γ 5)2 = 1 ,

(A.33)with γ5 inducing the projection operators (1± γ5)/2 for the chiral components ψA,χ A′ of ψ , determining their helicity as right (positive) or left (negative). But thereare many other options [9]:

• Standard14 canonical basis:

γ0 ≡[−1 0

0 1

], γi ≡

[0 σ i

−σ i 0

]. (A.34)

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


• A Majorana basis, where γ ∗i = −γi :

γ0 ≡[

0 −σ 2

−σ 2 0

], γ1 ≡

[0 iσ 3

iσ 3 0

],

γ2 ≡[

i1 00 −i1

], γ3 ≡

[0 −iσ 1

−iσ 1 0

].

(A.35)

• A basis in which the γ matrices are all real and where N = 1 SUGRA allows areal Rarita–Schwinger (gravitino) field (see Sect. 4.1 of Vol. I) [15]:

γ0 ≡[

0 1−1 0

], γ1 ≡

[1 00 −1

], γ2 ≡

[0 −iσ 2

iσ 2 0

],

γ3 ≡[

0 11 0

], γ 5 ≡ iγ 0γ 1γ 2γ 3 =

[0 iσ 3

−iσ 3 0

].

(A.36)

Furthermore,

{γ μ, γ ν

} = −2ημν . (A.37)

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

Lμν −→ i

2γ μν ,

where

γ μν ≡ 1

2(γ μγ ν − γ νγ μ) = 1

2

[σμσν − σνσμ 0

0 σμσν − σνσμ

], (A.38)

and {, } and [, ] denote the anticommutator and commutator, respectively, As expec-ted from the representation (1/2, 0)⊕ (0, 1/2), this determines that the ψA and χ A′

spinors transform separately. The specific generators are also rewritten iσμν for ψA

and iσμν for χ A′ , with

(σμν)AB ≡ 1

4

[σ

μ

AC ′σνC ′B − (μ↔ ν)

], (A.39)

(σμν)B′A′ ≡ 1

4

[σμA′Cσν

C B′ − (μ↔ ν)]

. (A.40)


A.4 Useful Expressions

In this section we present a set of formulas which will help to clarify someproperties of SUSY and SUGRA, and help also to simplify some of the expres-sions in the book. The initial emphasis is on the 2-component spinor, but we willalso indicate, whenever relevant, some specific expressions involving 4-componentspinors.

In a 2-component spinor notation, we can use a spinorial representation for thetetrad. This then allows us to deal with the indices of bosonic and fermionic vari-ables in a suitable equivalent manner. To be more precise, in flat space we thereforeassociate a spinor with any vector by the Infeld–van der Waerden symbols σ AA′

a ,which are given by [16, 7–9]15

σ AA′0 = − 1√

21 , σ AA′

i = 1√2σi . (A.41)

Here, 1 denotes the unit matrix and σi are the three Pauli matrices (A.9). To raise andlower the spinor indices, the different representations of the antisymmetric spinorialmetric εAB , εAB , εA′B′ , and εA′B′ can be used [see Note A.5 and (A.18)]. Each ofthem can be written as the same matrix, given by

(0 1−1 0

). (A.42)

Hence, the spinorial version of the tetrad reads

eAA′μ = ea

μσAA′

a , eaμ = −σ a

AA′eAA′μ . (A.43)

Generally, for any tensor quantity T defined in a (curved) spacetime, a correspond-ing spinorial quantity is associated through T → T AA′ = eAA′

μ T μ, with the inverse

relation given by T μ = −eμAA′TAA′ .

Equipped with this feature, the foliation of spacetime into spatial hypersurfacesΣt from a tetrad viewpoint (spinorial version) is straightforward:

• The future-pointing unit normal vector n → nμ has a spinorial form given by

n AA′ = eAA′μ nμ . (A.44)

• The tetrad (A.43) is thereby decomposed into timelike and spatial componentseAA′

0 and eAA′i .

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

hi j = −eAA′i eAA′j . (A.45)

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

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

n AA′eAA′i = 0 and n AA′n

AA′ = 1 , (A.46)

which allow us to express n AA′ in terms of eAA′i (see Exercise 4.3 of Vol. I).

• Using the lapse function N and the shift vector N i , the timelike component ofthe tetrad can be decomposed according to

eAA′0 = Nn AA′ +N i eAA′

i . (A.47)

• Other useful formulas16 involving the spinorial tetrad eAA′i with the unit vector

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 indeedallow one to considerably simplify, e.g., the differential equations for the bosonicfunctionals 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 timelikenormal vector n AA′ and the tetrad eAA′

i :

n AA′nAB′ = 1

2εA′

B′ , (A.48)

n AA′nB A′ = 1

2εA

B , (A.49)

eAA′i eAB′j = −1

2hi jεA′

B′ − i√

hεi jkn AA′eAB′k , (A.50)

eAA′i eB A′j = −1

2hi jεA

B − i1√hεi jkn AA′e

B A′k , (A.51)

eAA′i eiB B′ = n AA′nB B′ − εABεA′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).


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

n AA′eAB′l = −n AB′ek

AA′ =i

2√

hεi jl eAA′i e

AB′j , (A.53)

n AA′eB A′l = −nB A′ek

AA′ = − i

2√

hεi jl eAA′i e

B A′j . (A.54)

In addition, we have

gμνeAA′μ eB B′

ν = −εABεA′B′ , (A.55)

gμν = −eAA′μ eB B′

ν εABεA′B′ , (A.56)

2eAA′(μeB A′ν) = gμνεA

B , (A.57)

2eAA′(μeAB′ν) = gμνεA′

B′ . (A.58)

The normal projection of the expression εilm D B′B mj DC

A′kl , used throughout Chap. 4,is given by [17]

EC AB′iB jk ≡ n AA′εilm D B′

B mj DCA′kl =

−2i√hεilm DB

B′mj e

C D′l eDD′knD

A′nAA′

= 2i√hεilm DB

B′mj e

C D′l eA

D′k . (A.59)

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

Ni = e0aeai , (A.60)

N = N jN j − e0aea0 , (A.61)

e0a = −na

N , (A.62)

e0a = Nna +Nmema , (A.63)

nγ emγ = 0 , (A.64)

εi jkγsγi = 2iγ⊥σ ik , (A.65)

and we have the notation

− A⊥ = A⊥ = −nμ Aμ = N A0 = 1

N(

A0 −N i Ai

), (A.66)

Ai ≡ Ai − N i

N A⊥ . (A.67)


A.4.2 Connections and Torsion

In the second order formalism applied to N = 1 SUGRA [13], the tetrad eμC D′and

gravitinos ψμC , ψν

D′determine the explicit form of the connections.

Four-Dimensional Spacetime

We will use

ωabμ = ω[ab]

μ −→ ωAA′B B′μ = ωAB

μ εA′B′ + ωA′B′μ εAB , (A.68)

where

ωABμ = ω(AB)

μ = sωABμ + κ AB

μ , (A.69)

while sωABμ is the spinorial version of the torsion-free connection form

sωabμ = eaν∂[μeb

ν] − ebν∂[μeaν] − eaνebρecμ∂[νec

ρ] , (A.70)

and κ ABμ = κ

(AB)μ is the spinorial version of the contorsion tensor κab

μ = κ[ab]μ , with

κ AA′B B′μ = eAA′

μ eB B′ν κνρ

μ = κ ABμ εA′B′ + κ A′B′

μ εAB . (A.71)

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

ξ AA′μν = −4π iGψ A′[μψ A

ν] , (A.72)

whose tensorial version is

ξρμν = −eρAA′ξ

AA′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 expansioninto n AA′ and eAA′

i . In more detail, the spin connection is written in the form

(3)ωAA′B B′i = (3)sωAA′B B′

i + (3)κ AA′B B′i , (A.75)


and then decomposed into primed and unprimed parts:

(3)ωAA′B B′i = (3)ωAB

i εA′B′ + (3)ωA′B′i εAB . (A.76)

Using the antisymmetry (3)ωAA′B B′i = − (3)ωB B′AA′

i , we then obtain the symmetries(3)ωAB

i = (3)ωB Ai and (3)ωA′B′

i = (3)ωB′A′i and the explicit representations

(3)ωABi = 1

2(3)ωA B B′

B′i , (3)ωA′B′i = 1

2(3)ω A′B B′

Bi . (A.77)

Analogous relations hold for (3)sωAA′B B′i and (3)κ AA′B B′

i . Furthermore, we nowhave

(3)ωABi =(3s)ωAB

i + (3)κ ABi , (A.78)

where (3s)ωABi is the spinorial version of the spatial torsion-free connection form

(3s)ωabi =

(ebj∂[ j e

ai] −

1

2eaj ebkec

i ∂ j eck − 1

2eaj nbnc∂ j eci − 1

2na∂i n

b)−(a ↔ b) ,

(A.79)

and κ ABi is the spinorial version of the spatial contorsion tensor17 κab

i , with

3κAA′B B′i = e jAA′e

kB B′

3κ jki = 3κ ABiεA′B′ + 3κ A′B′iεAB . (A.81)

The three-dimensional torsion-free spin connection (3)sωAA′B B′i can therefore be

expressed in terms of n AA′ and eAA′i as

(3)sωAA′B B′i = eB B′ j ∂ [ j eAA′

i] (A.82)

−1

2

(eAA′ j eB B′keCC ′

i ∂ j eCC ′k + eAA′ j nB B′nCC ′∂ j eCC ′i + n AA′∂i nB B′)

−eAA′ j ∂[ j eB B′i]

+1

2

(eB B′ j eB B′keCC ′

i ∂ j eCC ′k + eB B′ j n AA′nCC ′∂ j eCC ′i + nB B′∂i n AA′).

17 The 3D contorsion is simply obtained by restricting the 4D quantity:

(3)κi jk = κi jk , (A.80)

with the spinorial contorsion (3)κ AA′B B′i = eAA′ j eB B′k

(3)κ jki = − (3)κB B′AA′i .


A.4.3 Covariant Derivatives

Still referring to a second order formalism, the tetrad eμC D′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 wouldbe associated).


To be more precise, we use18

DμeAA′ν = ∂μeAA′

ν + ωABμeB A′

ν + ωA′B′μeAB′

ν , (A.83)

DμψAν = ∂μψ

Aν + ωA

BμψBν , (A.84)

with ωABμ the connection form (spinorial representation) as described in Sect. A.4.2

above.


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

(3)D j TAA′ = ∂ j T

AA′ + (3)ωAB T B A′ + (3)ωA′

B′TAB′ , (A.85)

with (3)ωAB and (3)ωA′

B′ the two parts of the spin connection [see (A.77)], using thedecomposition

(3)ωAA′B B′i = (3)sωAA′B B′

i + (3)κ AA′B B′i . (A.86)

A.4.4 (Gravitational) Canonical Momenta

Related to the presence of (covariant) derivatives in the action of N = 1 SUGRA, weretrieve the momenta conjugate to the tetrad eC D′

μ and gravitinos ψCμ , ψD′

ν . The latterrequire the use of Dirac brackets, which are discussed elsewhere (see Appendix Band Chap. 4 of Vol. I), so we present here some elements regarding the former.

18 Notice that D[μeAA′ν] = ξ AA′

μν , where ξ AA′μν is the spinor version of the torsion.

19 T AA′...Z Z ′ = eAA′μ . . . eZ Z ′

ν T μ...ν .


For the gravitational canonical momentum, we can write

piAA′ =

δSN=1

δeAA′i

−→ p ji = −eAA′ j piAA′ , p⊥i = n AA′ pi

AA′ , (A.87)

where we can also use (as in the pure gravitational sector, see Sect. 4.2 of Vol. I)

p(i j) = −2π i j . (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 fun-damental form Ki j ) in spinorial terms and through the (gravitational) canonicalmomentum pi

AA′ , with the assistance of (A.87) and (A.88). Let us recall that20

π i j ∼ −h1/2

2k2

{K (0)(i j) −

[trK (0)

]hi j}

∼ −h1/2

2k2

{[K (i j) − τ (i j)

]− hi j (K − τ)

}, (A.89)

Ki j = −eai ∂ j na + naω

abj ebi , (A.90)

but now emphasizing the influence of the torsion:

K(i j) = 1

2N(Ni I j +N j |i − hi j,0

)− 2ξ(i j)⊥ , (A.91)

K[i j] = ξ⊥i j ≡ nμξμi j . (A.92)


The spinor-valued curvature is given by

RABμν = R(AB)

μν = 2(∂[μωAB

ν] + ωAC[μωC B

ν])

, (A.93)

R = eμAA′eA′νB RAB

μν + eA′μA eAB′νRA′B′

μν . (A.94)

20 p[i j] and additional terms in p⊥i will appear in the Lorentz constraint Jab ↔ JABεA′B′ +JA′B′εAB .



The components of the 3D curvature in terms of the spin connection read

(3)R ABi j = 2

[∂[i (3)ωAB

j] + (3)ωAC[i (3)ωC B

j]]

,

(3)RA′B′i j = 2

[∂[i (3)ωA′B′

j] + (3)ωA′C ′[i

(3)ωC ′B′j]

]. (A.95)

Because of the symmetry of (3)ω[AB]i = 0 and (3)ω

[A′B′]i = 0, the chosen notation

(3)ωABi and (3)ωA′

B′i is unambiguous. The horizontal position of the indices does notneed to be fixed. The scalar curvature is given by

(3)R = eiAA′e

jB B′

[(3)R AB

i j εA′B′ + (3)RA′B′

εAB]

. (A.96)

The same procedure performed on (3)sωAA′B B′i leads to the torsion-free scalar

curvature:

(3s)R ABi j = 2

[∂[i (3)sωAB

j] + (3)sωAC[i (3)sωC B

j]]

,

(3s)RA′B′i j = 2

[∂[i (3s)ωA′B′

j] + (3s)ωA′C ′[i

(3s)ωC ′B′j]

], (A.97)

and

(3s)R = eiAA′e

jB B′

[(3s)R AB

i j εA′B′ + (3s)RA′B′

εAB]

. (A.98)

A.4.6 Decomposition with Four-Component Spinors

We now present, in a somewhat summarized manner, a few helpful formulas for the3 + 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

ξμνλ = 1

2

(Γμν

λ − Γνμλ), (A.99)

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


eμ;ν = eλΓμνλ , (A.100)

Γμνλ = Γ (0)λ

μν − κμνλ , (A.101)

κμνλ = ξμν

λ − ξμλν + ξλ

μν , (A.102)

with Γ(0)λμν the Christoffel symbols and κμν

λ the contorsion tensor.• The extrinsic curvature Ki j can then be computed and retrieved from n; j by [see

(2.9) of Vol. I]

K ji = −N (4)Γ 0j i =

1

2N(−h ji,0 +N j |i +Ni | j

)− κ j i⊥ , (A.103)

K( j i) = 1

2N(−h ji,0 +N j |i +N j |i

)+ τ( j i) , (A.104)

K[ j i] = ξi j⊥ , τ j i ≡ 2ξ j⊥i , (A.105)

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

(4)R = (3)R − Ki j K i j + K 2 − 2(4)R⊥α⊥α , (A.106)

(4)R⊥α⊥α =(

nγ nβ(α − nβnγ

(γ)(β − K i j Ki j + K 21− 2ξαβ⊥nα(β . (A.107)

• Finally, or almost, the Lagrangian density is

√hL = N h1/2

[(3)R − Ki j K i j + K 2 − 2

(nγ nβ

(α − nβnγ(γ)(β + 4ξαβ⊥nα(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

Hm = −2hmiπik|k , (A.109)

J ab ≡ pkaebk − pkbea

k , (A.110)

H⊥ ∼ h−1/2(π i jπi j − 1

2π2)− h1/2 (3)R + h1/2

[τ (i j)τ(i j) − τ 2

](A.111)

+√hξi j⊥ξ i j⊥ − 2h1/2τ[i j]ξ j i⊥ +√

hqkρk − 2h1/2ρi|| i + 2h1/2ρiρi ,

21 The reader should notice that, with the above alone, i.e., no gravitino (matter) action, wehave Pμνλ = 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.,ελμνρψλγ5γμDνψρ , then

ξμνλ = −1

4ψμγλψν ,

and torsion cannot then be ignored.


H = NH⊥ +N iHi +MabJ ab . (A.112)

A.4.7 Equations Used in Chap. 4

We employ several derivatives of functionals with respect to the tetrad eAB′j [17].

Two of them are:

ε4ilmn AA′ δ

δeAB′j

(D B′B mj DC

A′kl) (A.113)

and

n AA′ δ

δeAB′j

DB B′i j . (A.114)

First we need an explicit form for δn AA′/δeB B′j . With (4.109) of Vol. I, which

expresses n AA′ in terms of the tetrad, the relation n AA′eAA′i = 0 implies

0 = eCC ′i δn AA′eAA′iδeB B′

j

= nCC ′n AA′δn AA′

δeB B′j

− εACεA′

C ′ δn AA′

δeB B′j

+ eCC ′ j nB B′ .

(A.115)In addition, we have

δn AA′

δeB B′j

= δnCC ′nCC ′n AA′

δeB B′j

= 2nCC ′nAA′ δn AA′

δeB B′j

+ δn AA′

δeB B′j

, (A.116)

whence

δn AA′

δeB B′j

= eAA′ j nB B′ . (A.117)

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

∂h

∂hi j= hi j h . (A.118)

Hence,

δh

δeAA′i

= −2heiAA′ . (A.119)

Using this and (A.48), (A.49), (A.50), (A.51), and (A.52), we can calculate theexpressions


n AA′εilm δ

δeAB′j

(D B′B mj DC

A′kl)

= −4n AA′εilm δ

δeAB′j

(1

he D′

B j eDD′mnDB′eC E ′l eE E ′knE

A′)

= εBCδi

ki√h

(1− 1+ 1

2− 1

2

)+ 2i√

h

(2eC B′i eB B′k + ei

B B′eC B′k

)

= −3i√hδi

kεBC − 2hi jε jkln

C B′elB B′ , (A.120)

and

n AA′ δ

δeAB′j

DB B′i j = −2in AA′ δ

δeAB′j

(1√h

eBC ′j eCC ′i n

C B′)

= − 2i√h

n AA′nBC ′eAC ′i . (A.121)

Finally, we can write a transformation rule to express derivatives in terms of thetetrad eAA′

i as derivatives in terms of the three-metric hi j . Let F[e] be a func-tional depending on the tetrad, and note that hi j can be expressed in terms of thetetrad, since we have the relation hi j = −eAA′

i eAA′ j . Moreover, there is of courseno inverse relation. We thus restrict the functional F by demanding that it can bewritten in the form F[hi j ]. Consequently, using the chain rule, we find for the trans-formation of the functional derivatives:

δFδeAA′

i

= δFδh jk

δh jk

δeAA′i

= − δFδh jk

εBCεB′C ′δeB B′

j eCC ′k

δeAA′i

= − δFδhik

εACεA′C ′eCC ′k − δF

δh jiεB AεB′A′e

B B′j

= −2δFδhi j

eAA′ j . (A.122)

Using eAA′i eAA′ j = −δij , the inverse relation for an arbitrary functional G[hi j ] is

simply

δGδhi j

= 1

2eAA′ j δG

δeAA′i

. (A.123)

Note that it is always possible to rewrite G[hi j ] in the form G[e].

References 233

References

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2. Ortin, T.: Gravity and Strings. Cambridge University Press, Cambridge (2004) 2123. Bailin, D., Love, A.: Supersymmetric Gauge Field Theory and String Theory. Graduate Stu-

dent Series in Physics, 322pp. IOP, Bristol (1994) 213, 214, 2194. Bilal, A.: Introduction to supersymmetry. hep-th/0101055 (2001) 2135. 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 UniversityPress, 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

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224, 231

Appendix BSolutions

Problems of Chap. 2

2.1 The Born–Oppenheimer Approximation and Gravitation

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

• We use the simple Hamiltonian

H = P2

2M+ p2

2m+ V (R, r) ≡ P2

2M+ h ,

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 termsof 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〉.

– Carry out the expansion Ψ =∑k �k(R)|k; R〉.• Returning to the Schrödinger equation, after multiplying by 〈n; R|, obtain an

equation for the wave functions �n :

[− h2

2M∇2

R + εn(R)− E

]�n(R) (B.1)

= h2

M

∑k

〈n; R|∇Rk; R〉 ∇R�k(R)+ h2

2M

∑k

⟨n; R|∇2

Rk; R⟩�k(R) .

235

236 B Soluitions

• Now neglect the off-diagonal terms1 in (B.1):

{1

2M[−ih∇ − h A(R)]2− h2 A2

2M− h2

2M

⟨n;R|∇2

Rn;R⟩+ εn(R)

}�n(R)=E�n(R).

(B.2)

The reader should note that the momentum of the slow particle has been shifted:P → P − h A, where A(R) ≡ −i〈n; R|∇Rn; R〉. This constitutes a Berry connec-tion.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 resultsif the relevant mass scales of non-gravitational fields are much smaller than thePlanck mass.

2. Regarding the non-gravitational fields:

a. Without them, an M expansion is fully equivalent to an h expansion, i.e., tothe usual WKB expansion for the gravitational field.

b. If non-gravitational fields are present, the M expansion is analogous to aBorn–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

π i j = MδS0

δhi j, (B.3)

from which hi j is recovered in the form

hi j = −2N Ki j +Ni | j +N j |i , (B.4)

with Ki j the components of the extrinsic curvature of 3-space, viz.,

Ki j = −16πGGi jklπkl . (B.5)

Note that a choice must be made for the lapse function and the shift vector in orderfor 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. 2.2.2and 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’ spacetimewill follow, associated with a chosen foliation and a choice of coordinates on eachmember of this foliation [1]:

• Combine all the trajectories in superspace which describe the same spacetimeinto a sheaf [6].

• The lapse and shift can be chosen in such a way that these curves comprise asheaf of geodesics.

• Superposing expressions in the form of WKB solutions (bearing the gravitationalfield), 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 ofthe momentum constraints in powers of M. In fact, S0 satisfies the three equations

(δS0

δhab

)|b= 0 . (B.6)

The Hamilton–Jacobi equation and (B.6) are equivalent to the G00 and G0i partsof the Einstein equations, while the remaining six field equations can be found bydifferentiating (B.4) with respect to t and eliminating S0 by making use of (B.4) andthe 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 gravita-tional kinetic term. Any ambiguity in the factor ordering would have to come fromthe gravity–matter coupling. But let us elaborate on this [1–4].

Let us introduce a linear derivative term (parametrized by ζi j ) that represents anambiguity 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, wenow have:

(− h2

2MGi jkl

δ2

δhi jδh jk+ ζi j

δ

δhi j+MU g +Hm⊥

)Ψ = 0 . (B.7)

Note that (2.11) can also be written as

Gi jklδS0

δhi j

δKδhkl

− 1

2Gi jkl

δ2S0

δhi jδhklK = 0 , (B.8)

but with the linear term we have instead

3 Equations (B.6) will follow from the Hamilton–Jacobi equation and the Poisson bracket relationsbetween the constraints H and Hi [7].

238 B Soluitions

Gi jklδS0

δhi j

δKδhkl

− 1

2

(Gi jkl

δ2S0

δhi jδhkl+ ζi j

δS0

δhi j

)K = 0 . (B.9)

At order M0 an equation also involving S1 is

Gi jklδS0

δhi j

δS1

δhkl− ih

2

(Gi jkl

δ2S0

δhi jδhkl

)+Hm = 0 , (B.10)

but with the linear term we can write

Gi jklδS0

δhi j

δS1

δhkl− ih

2

(Gi jkl

δ2S0

δhi jδhkl+ ζi j

δS0

δhi j

)+Hm = 0 . (B.11)

With regard to (2.15), we now get

0 = Gi jklδS0

δhi j

δS2

δhkl+ 1

2Gi jkl

δS1

δhi j

δS1

δhkl− ih

2

(Gi jkl

δ2S1

δhi jδhkl+ ζi j

δS1

δhi j

)

+ 1√h

(δS1

δφ

δS2

δφ− ih

2

δ2S2

δφ2

). (B.12)

If we now further require σ2 to satisfy

Gi jklδS0

δhi j

δσ2

δhkl− h2

K2Gi jkl

δKδhi j

δDδhkl

+ h2

2K

(Gi jkl

δ2Kδhi jδhkl

+ ζi jδKδhi j

)= 0 ,

we obtain

Gi jklδS0

δhi j

δη

δhkl= h2

2F

(− 2

KGi jklδFδhi j

δKδhkl

+ Gi jklδ2F

δhi jδhkl+ ζi j

δFδhi j

)

+ ih√hF δη

δφ

δFδφ

+ ih

2√

h

δ2η

δφ2. (B.13)

Hence, the Schrödinger equation with quantum gravitational corrections, includingthe linear term, is now

ihδΘ

δτ= Hm⊥Θ + h2

MF Gi jkl

(1

KδKδhi j

δFδhkl

− 1

2

δ2Fδhi jδhkl

− 1

2ζi j

δFδhi j

)Θ . (B.14)

We decompose derivatives of F into normal and tangential components to the con-stant S0 hypersurfaces. In more detail (see also Sect. 4.2.6),

Gi jklδFδhi j

= − i

hGi jklU i jH⊥

mF ⊕ GmnopδF

δhmn$op$kl . (B.15)


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

U i j ≡ δS0

δhi j

(Gi jkl

δS0

δhi j

δS0

δhkl

)−1

= − 1

2U

δS0

δhi j. (B.16)

The second term is the tangential component and $i j is a unit vector tangent to theS0 constant hypersurface, satisfying U i j$i j = 0. Here we use

GmnopδF

δhmn$op$kl ≡ amn .

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

h2

2MF

(−Gi jkl

δ2Fδhi jδhkl

+ 2

KGi jklδKδhi j

δFδhkl

− ζi jδFδhi j

)Θ ≡ Cn ⊕ Ct , (B.17)

where we have

Cn ≡ − 1

4MUF

[(H⊥

m)2F − ihGi jklδS0

δhkl

(−F δH⊥

m

δhi j+ δU

δhi jH⊥

mF)]

Θ

= − 1

4MUF

[(H⊥

m)2F − ih

(−F δH⊥

m

δτ+ δU

UδτH⊥

mF)]

Θ . (B.18)

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

Ct ≡ − h2

4MUF

[2

KδKδhkl

(Gmnop

δFδhmn

$op)$kl + δGi jkl

hkl

(Gmnop

δFδhmn

$op)$i j

− δ

δhkl

(Gmnop

δFδhmn

$op)$kl − ζi jGmnop

δFδhmn

$op$i j]Θ . (B.19)

For the normal components [1],

ihδΘ

δτ= Hm⊥Θ + 4πG√

h[(3)R − 2Λ

] (Hm⊥)2Θ + ih4πGδ

δτ

[Hm⊥√

h[(3)R − 2Λ

]]Θ .

(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]

1

MGi jkl

δ

δhi j

⎛⎝Ψ ∗

↔δ

δhklΨ

⎞⎠+ 1√

h

δ

δφ

⎛⎝Ψ ∗

↔δ

δφΨ

⎞⎠ = 0 . (B.21)

Applying the M expansion, at the level of the corrected Schrödinger equation weobtain

0 = δ

δτ(Θ∗Θ)+ h

2i√

h

δ

δφ

⎛⎝Θ∗

↔δ

δφΘ

⎞⎠

− ih

2MGi jkl

⎡⎣ δ

δhi j

⎛⎝Θ∗

↔δ

δhklψ

⎞⎠− 1

KδKδhi j

Θ∗ δ

δhklΘ

⎤⎦ . (B.22)

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 → 0 forlarge field configurations:

d

dt

∫DφΘ∗Θ = 8πG

∫Dφ

∫d3x ψ∗ δ

δτ

{Hm⊥√

h[(3)R − 2Λ

]}ψ . (B.23)

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]:

• Apply a conformal transformation and write the 3-metric as hi j = h1/3hi j .• The Hamilton–Jacobi equation becomes

− 3√

h

16

(δS0

δ√

h

)2

+ hik h jl

2√

h

δS0

δhi j

δS0

δhkl− 2

√h[(3)R − 2Λ

]= 0 . (B.24)

• Set (3)R = 0 and look for a solution of the form S0 = S0(√

h), viz.,

S0 = ±8

√Λ

3

∫ √h d3x ≡ ±8H0

∫ √h d3x . (B.25)


• The local time parameter4 in configuration space is

δ

δτ= −3

√h

8

δS0

δ√

h

δ

δ√

h= √

3hΛδ

δ√

h. (B.26)

• From the conservation law ‘chosen’ for K, it follows that δK/δτ = 0, i.e., it isconstant. Then the functional Schrödinger equation is (setting h = 1)

iΘ =∫

d3x

[− 1

2a3

δ2

δφ2+ a

2(∇φ)2 + a3

2m2φ2

]Θ . (B.27)

• Use the Gaussian ansatz5

Θ ≡ N (t) exp

[−1

2

∫dkΩ(k, t)χk χ−k

].

Two equations follow, but the relevant one is6

y′′ + 2a′

ay′ + (m2a2 + k2)y = 0 , (B.28)

where Ω ≡ −ia3 y/y, and primes denote differentiation with respect to η, aconformal time coordinate.7

• To the next order, there are corrections to the Schrödinger equation:

ihδΘ

δτ= Hm⊥Θ − 2πG√

hΛ(Hm⊥)

2Θ − ih

2πG

Λ

δ

δτ

(Hm⊥√h

)Θ . (B.29)

The second correction term is a source of non-unitarity. It produces an imaginarycontribution to the energy density of the order of magnitude 29Gh2 H3

0 /480πV c5

(reinstating c), indicating a possible instability of the system whose associated

time scale is(

H−10 /tPl

)3. The first correction term in (B.29) causes a shift8

4 This allows one to recover∂a3

∂t≡∫

d3 yδ√

h(x)δτ(y)

= 3H0a3 −→ a(t) = eH0t .

5 The momentum representation φ(x) =∫

d3k

(2π)3ϕ(k)eikx ≡

∫dkϕkeikx is employed. Gaus-

sian states are used to describe generalised vacuum states. The special form here is due to theHamiltonian being quadratic, and the Gaussian form is preserved in time.

6 Proceeding from i ˙Ω = Ω2

a3− a3

(m2 + k2

a2

).

7 We have dt = adη −→ a(η) = − 1

H0η, η ∈ (0,−∞) .

8 The prediction can be made more concrete through the following elements:

242 B Soluitions

⟨H⊥

m

⟩−→

⟨H⊥

m

⟩− 2πG

3a3 H20

⟨(H⊥

m)2⟩−→

⟨H⊥

m

⟩− 841

1382400π3Gh H6

0 a3 .

(B.30)

2.6 The Bunch–Davies Vacuum and Quantum Cosmology

Take the simplified Gaussian ansatz (see above)

� = N (t)e−Ω(t) f 2n /2 , (B.31)

and insert it in (2.33) to get equations for N and Ω :

i˙N

N= Ω

2a3, (B.32)

i ˙Ω = Ω2

a3− a3

(m2 + n2

a2

). (B.33)

Using

Ω ≡ −ia3 y

y≡ −ia2 y′

y, (B.34)

and the conformal time η, with dt = adη, we get

y′′ − 2

ηy′ +

(m2

H2η2+ n2

)y = 0 , (B.35)

where we take a(η) = −1/Hη. We can obtain [3]

– Note that⟨(H⊥

m)2⟩

is divergent, rather like 〈T 2μν(x)〉, i.e., a four-point correlation function.

– But use the Bunch–Davies vacuum state (an adiabatic vacuum state) to compute the expectationvalue [8]. The Bunch–Davies vacuum arises as a particular solution of (B.28). It is de SitterSO(3,1) invariant, reducing to the Minkowski vacuum at early times, i.e., Ω → √

k2 + m2 as

t →−∞, where the metric is essentially static and one can put Ω = 0 and a = 1 in i ˙Ω . Then

⟨H⊥

m

⟩� 29h H4

0 a3

960π2.

The fluctuation of the Bunch–Davies state will be (quantum mechanically) zero, and⟨(H⊥

m)2⟩≈⟨H⊥

m

⟩2.


Ω � − i

η2 H2(

3− 2ν

2η+ n

Zν−1

Zν

) −→ − i

η2 H2(

m2

3ηH2+ n

Zν−1

Zν

)

−→ n2a2

n2 + a2 H2(n + iaH)+ i

m2a3

3H, (B.36)

where we have used the usual inflationary limit m2/H2 � 9/4, ν � 3/2−m2/3H2,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., invariantunder SO(3,1), this reduces to the Minkowski vacuum at early times, i.e., Ω tendsto√

n2 + m2 as t → −∞, where the metric is essentially static and one can put˙Ω = 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)

〈 f | P0 | f 〉 ∼ 1

4‖SA | f 〉‖2 + 1

4

∥∥∥S†A | f 〉

∥∥∥2 ≥ 0 . (B.37)

If | f 〉 is invariant under all SUSY generators, i.e., SA | f 〉 = 0, then necessarily〈 f | P0 | f 〉 = 0. Conversely, if 〈 f | P0 | f 〉 > 0, not all SA and S†

A can annihilatethe state | f 〉. This non-invariance of the vacuum state implies that a spontaneousbreaking of the (super)symmetry has occurred.

3.2 No-Scale SUGRA

An interesting Kähler class of potentials are those for which we obtain no-scaleSUGRA (extracted from [9–12] within the SUGRA limits of string theory), bywhich we mean that the effective potential for the hidden sector is flat (i.e., con-stant), wherein the Planck mass is the only input mass, to fix features such as thehierarchy problem, with subsequent non-gravitational radiative corrections to fix thedegeneracy:

• A trivial situation is described by

G = −3 ln(φ + φ∗

) �⇒ V = 0 , ∀φ ,

where φ is the hidden sector scalar. This simple solution has a simple flat poten-tial (no scale) model, where the potential V vanishes identically, making thevacuum expectation value 〈φ〉 arbitrary, with an arbitrary value for the mass

244 B soluitions

of the gravitino (and the cosmological constant). These models arise naturallyin (super)string compactifications, where the size of the compact dimensions isnot fixed [13]. Changing them thus costs no energy and a moduli space can bedefined. 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 (hid-den) sector. This is all very model dependent.

• An extension is

G = −3 ln(φ + φ∗ − ϕr∗ϕr

)+ ln |W|2 , W ∼ dpqrϕpϕqϕr ,

where φ is now in the visible sector and ϕ is in the hidden sector. With da ≡ Gr

Tarsϕs , this leads to

V ∼ (φ + φ∗ − ϕr∗ϕr)−2

∣∣∣∣∂W∂ϕr

∣∣∣∣2

+ 1

2Re(dadb) ,

whose minimum requires

∂W∂ϕr

= da = 0 , ∀ r, a �⇒ Vmin = 0 .

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 Scommutes with the Hamiltonian and cannot change the energy of such a state, withS| f ′〉 = $| f ′〉, we have 0 = 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, andnecessarily 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,i.e., E (1,2)

n ≥ 0 [see (3.40) and [20]]:

E = 〈ψ |H|ψ〉 ∼ 〈Sψ |Sψ〉 + 〈S†ψ |S†ψ〉 ≥ 0 .

The Schrödinger equation for H1 is

H1ψ(1)n = A†Aψ(1)

n = E (1)n ψ(1)

n . (B.38)

Multiplying both sides from the left by A, we get

H2

(Aψ(1)

n

)= AA†Aψ(1)

n = E (1)n

(Aψ(1)

n

). (B.39)


Similarly, the Schrödinger equation for A2, viz.,

A2ψ(2)n = AA†ψ(2)

n = E (2)n ψ(2)

n , (B.40)

implies

H1

(A†ψ(2)

n

)= A†AA†ψ(2)

n = E (2)n

(A†ψ(2)

n

). (B.41)

We may argue as follows:

• If Aψ(1)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., theyare related by

E (2)n = E (1)

n > 0 , (B.42)

ψ(2)n =

[E (1)

n

]−1/2Aψ(1)

n , (B.43)

ψ(1)n =

[E (2)

n

]−1/2A†ψ(2)

n , (B.44)

where n = 0, 1, 2, . . ..• If Aψ

(1)0 = 0, then E (1)

0 = 0 and this state is unpaired, while all other states ofthe two Hamiltonians are paired. It is then clear that the eigenvalues and eigen-functions of the two Hamiltonians H1 and H2 are related by

E (2)n = E (1)

n+1 , E (1)0 = 0 , (B.45)

ψ(2)n =

[E (1)

n+1

]−1/2Aψ

(1)n+1 , (B.46)

ψ(1)n+1 =

[E (2)

n

]−1/2A†ψ(2)

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-sion and (3.42), one cannot have both9 Aψ

(1)0 = 0 and A†ψ

(2)0 = 0. Only one of

the two ground state energies can be zero.

9 Note that one would have

ψ(2)0 � exp

[√2m

h

∫ x

W(y)dy

]. (B.48)

246 B Soluitions

• For Aψ(1)0 = 0, since the ground state wave function of H1 is annihilated by

the operator A, this state has no SUSY partner. Knowing all the eigenfunctionsof H1, we can determine the eigenfunctions of H2 using the operator A. Andconversely, using A†, we can reconstruct all the eigenfunctions of H1 from thoseof H2 except for the ground state.

• For (3.42), the normalizable zero energy ground state is associated with H1, andW is positive (negative) for large positive (negative) x , implying a zero fermionnumber.

• If we cannot find normalizable functions like (3.42) and (B.48), then H1 doesnot have a zero eigenvalue, and SUSY is broken. For W of the form wxn , if w

is positive and n odd, we get a normalizable state, but not the reverse, i.e., forw negative and n even. With W± ≡ limx→±∞ W, if sign(W+) = sign(W−),SUSY is broken, whereas if sign(W+) = −sign(W−), SUSY remains intact, i.e.,we can also write

Δ = 1

2

[sign(W+)− sign(W−)

].

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

If SUSY remains intact, and if the potential V (x) provides an exactly solvablesituation with n bound states and ground state energy E0, one can extend to aV1(x) = V (x) − E0 setting, whose ground state energy is zero. One can obtainall the n − 1 eigenstates of H2, then obtain all the n − 2 eigenstates of H3. In thisway, 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 moredetails)

S2 = 1

2k2

∫M

d4x eR , (B.49)

S3/2 = S′3/2 + S′3/2


= 1

2

∫M

d4x εμνρσ(ψ A′

μeAA′ν DρψAσ − ψ A

μeAA′ν DρψA′

σ

),(B.50)

SB2 =

1

k2

∫∂M

d3x eK , (B.51)

SB3/2 =

1

2

∫∂M

d3x εi jkψ Ai eAA′ jψ A′

k , (B.52)

where e = det [eaμ], e = det [ea

i ], a = 1, 2, 3, and the integration is over a space-time manifold M with boundary ∂M. The boundary terms, labelled by the subscriptB, are necessary for dealing with quantum amplitudes of transitions and also toobtain 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 thespin connection ωab

μ by

R(e, ω) = eaμeb

ν Rabμν(ω) , (B.53)

that is,

Rabμν(ω) = ∂μω

abν − ∂νω

abμ + ωac

μωcbν − ωac

νωcbμ . (B.54)

The variation is then

δS(e, ψ,ψ,ω(e, ψ,ψ)

)

= δeδS

δe ψ,ψ,ω+ δψ

δS

δψ e,ψ,ω

+ δψδS

δψ e,ψ,ω

(B.55)

+ δS

δω e,ψ,ψ

[δω(e, ψ,ψ)

δψδψ + δω(e, ψ,ψ)

δψδψ + δω(e, ψ,ψ)

δeδe

],

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 thegravitinos. In list form, we have the following points:

• Take

δe = eδ(ln det [ea

μ]) = eδ

(Tr ln [ea

μ]) = eea

μδeaμ , (B.56)

where

δeaν = −(δeb

μ)ebνea

μ . (B.57)

248 B Soluitions

• It follows that

δ(eR) = (δe)R + e(δR)

= e[ea

μ(δeaμ)R + δ(ea

μebν)Rab

μν

]

= e[(δea

μ)eaμR + 2ea

μ(δebν)Rab

μν

]

= eδeaμ(ea

μR − 2eaνeb

μecρ Rcb

ρν) . (B.58)

• In addition,

δ(eK ) = e[δK + e−1(δe)K

]

= e[ω0a

iδeai + eb

j (δebj )ω

0ai ea

i]

= eωa0i (ea

j ebi − ea

i ebj )δeb

j . (B.59)

• Note that we have used the time gauge here, and we are assuming it to bepreserved under the SUSY transformations10 (discussed at length in Exercises5.1–5.3).11

• Moreover12

δS′3/2 = 1

2

∫M

d4x εμνρσ[(δψ A′

μ)eAA′ν DρψAσ + ψ A′

μ(δeAA′ν)DρψAσ

]

= 1

2

∫M

d4x εμνρσ (2k−1 DμεA′ +ΛA′

B′ψ B′μ)eAA′ν Dρψ

Aσ

+a volume integral

= 1

2

∫M

d4x εμνρσ[2k−1∂μ(ε

A′eAA′ν DρψAσ )−2k−1εA′ Dμ(eAA′ν Dρψ

Aσ )]

+a volume integral ,

= 1

k

∫∂M

d3x εi jkεA′eAA′i D jψA

k + a volume integral , (B.60)

10 εμνρσ Dρ Dσ εA′ = 1

2εμνρσ [Dρ, Dσ ]εA′ = 1

2εμνρσ R A′

B′ρσ εB′ .

11 In brief, the usual SUGRA action is extended by the boundary terms, namely (B.52). The newfeature are also the ΛA

B and ΛA′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.


δS′3/2 =−1

2

∫M

d4x εμνρσ[ψ A

μ(δeAA′ν)DρψA′

σ + ψ AμeAA′ν Dρ(δψ

A′σ )]

=−1

2

∫M

d4x εμνρσψ AμeAA′ν Dρ(2k−1 Dσ ε

A′ +ΛA′B′ψ B′

σ )

+a volume integral

=−1

2

∫M

d4x εμνρσψ AμeAA′ν Dρ(Λ

A′B′ψ B′

σ )+ a volume integral

=−1

2

∫M

d4x εμνρσ[∂ρ(ψ

AμeAA′νΛA′

B′ψ B′σ )−Dρ(ψ

AμeAA′ν)ΛA′

B′ψ B′σ

]

+a volume integral

=−1

2

∫∂M

d3x εi jkψ Ai eAA′ jΛA′

B′ψ B′k + a volume integral . (B.61)

• 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(eAA′

i , ψ iA′ )I on an initial surface to data (eAA′

i , ψ iA)F on a final surface.

The gravitinos filling in between (eAA′i , ψ i

A′ )I on an initial surface and data

(eAA′i , ψ i

A)F on a final surface would now be independent of (ψμ

A′ , ψνA).

– With nμ the future-pointing unit vector normal to the boundary ∂M, if weassume13 that the three spacelike axes of the tetrad lie in ∂M, so that

na = eaμnμ = (1, 0, 0, 0) , (B.62)

the extrinsic curvature has the simple form

K = hi j naωab

j ebi = ω0ai ea

i . (B.63)

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

δeAA′μ ≡ −ik(εAψ A′

μ + εA′ψ Aμ)+ΛA

BeB A′μ +ΛA′

B′eAB′μ ,(B.64)

δψ Aμ ≡ 2k−1 Dμε

A +ΛABψ

Bμ , (B.65)

δψ A′μ ≡ 2k−1 Dμε


μ , (B.66)

13 This gauge choice is known as the time gauge.

250 B Soluitions

where

ΛAB ≡ −2ikn AA′nCC ′ε

CψC ′i eB A′ i , (B.67)

and its Hermitian conjugate

ΛA′B′ ≡ −2ikn AA′nCC ′ε

C ′ψCi eAB′ i , (B.68)

and where εA and εA′ are spacetime-dependent anticommuting fields. Theinclusion of the Lorentz terms in (B.64) ensures the preservation of the gaugedefined 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 anyderivatives 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 volumeterms 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 toextract all surface terms in δS3/2 which involve Dμε or Dμε.

Collecting equations (B.59), (B.60), (B.61), with the volume integral terms van-ishing in this SUSY context, the total variation under these local supersymmetrytransformations is

δS = 1

k

∫∂M

d3x εi jkεA′eAA′i (D jψA

k)− 1

2

∫∂M

d3x εi jkΛA′B′ψ B′

i eAA′ jψ Ak

+ 1

k2

∫∂M

d3x h1/2ωa0i (ea

j ebi − ea

i ebj )δeb

j , (B.69)

provided that the gauge condition (B.62) is preserved.Finally, note the generic expression [25] retrieved under left-handed SUSY trans-

formations:

δS = 2

k

∫∂M

d3x εi jkεA′eAA′i(3s)D jψ

Ak , (B.70)

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

4.2 S0 Must Depend on the Gravitino ψ Ai

With the SUSY constraints and the ansatz

Ψ = exp[i(S0G−1 + S1 + S2G + · · · )/h

],

obtain to lowest order G0,


[Ψ ]−1S A′ΨO(G0)= εi jkeAA′i

(3s)D jψAk + 4π iψ A

iδS0

δeAA′ i= 0 . (B.71)

This must hold for arbitrary fields ψ Ai and eAA′

i . Then S0 must at least depend on

eAA′i . Otherwise we would get the condition εi jkeAA′i (3s)D jψ

Ak = 0, which cannot

hold for all fields.Assume then that S0 does not depend on the gravitino field ψ A

i . Integrating (B.71)

over space with an arbitrary continuous spinorial test function εA′(x) leads to

I0 ≡∫

d3xεA′[εi jkeAA′i

(3s)D jψAk + 4π iψ A

iδS0

δeAA′i

]= 0 . (B.72)

With the replacement ψ Ai &→ ψ A

i exp [φ(x)] and εA′(x) &→ εA′(x) exp [−φ(x)],this becomes [26]

I ′0 ≡∫

d3xεA′ exp(−φ)

[εi jkeAA′i

(3s)D j

(ψ A

k expφ)+ 4π iψ A

iδS0

δeAA′i

expφ

]

= 0 . (B.73)

Now for "I0 ≡ I0 − I ′0 = 0,

"I0 =∫

d3x εi jkeAA′i (x)εA′(x)ψAk(x)∂ jφ(x) = 0 , (B.74)

which cannot hold for all fields.To higher orders, the calculation for n ≥ 1 leads to

[Ψ ]−1S A′ΨO(Gn)= −4π iGnψ A

iδSn

δeAA′ i= 0 . (B.75)

So can we choose Sn not to depend on the bosonic field eAA′i ? 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 grav-itino 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

2ih (3s)Di

(εi jkψAj DB

A′lkδ

δψ Bl

)= 1

2εi jk (3s)Di (ψAjψ A′k)

= 1

2εi jk

{[(3s)DiψAj

]ψ A′k + ψAj

[(3s)Diψ A′k

]}

= 1

2εi jk

{[(3s)D jψAk

]ψ A′i − ψAi

[(3s)D jψ A′k

]}.

(B.76)

Comparing the above with the third line written in the form

1

2εi jk

{[(3s)D jψAk

]ψ A′i + ψAi

[(3s)D jψ A′k

]}, (B.77)

the terms containing (3s)D j (ψ A′k) will cancel out. Finally, the normal projection ofthe remaining term containing (3s)D jψ

Ak is given by

n AA′ ihεi jk[(3s)D jψAk

]DB

A′liδ

δψ Bl

= h√hεi jkeBC ′

i eAC ′l[(3s)D jψAk

] δ

δψ Bl

.

(B.78)

4.4 Decomposition of the Hamilton–Jacobi Equationinto Bosonic and Fermionic Sectors

For the WKB wave functional, write

Ψ [e, ψ] ≡ exp

(i

hB0G−1

)exp

[i

h(F0G−1 + S1 + · · · )

]. (B.79)

Then select the pure bosonic part B0 satisfying

4π in AA′ DB B′i j

δB0

δeAB′j

δB0

δeB A′i

+U = 0 , (B.80)

i.e., corresponding to the Hamilton–Jacobi equation (4.22). This solution B0 canthen be used to determine the condition for the part F0 involving fermions:

0 = 4π i

(ψ B

iδF0

δeAB′j

EC AB′iB jk

δF0

δψCk

+ ψ Bi

δB0

δeAB′j

EC AB′iB jk

δF0

δψCk

−n AA′ DB B′i j

δF0

δeAB′j

δF0

δeB A′i

− n AA′ DB B′i j

δF0

δeAB′j

δB0

δeB A′i

− n AA′ DB B′i j

δB0

δeAB′j

δF0

δeB A′i

)

+ i√hεi jkeBC ′

i eAC ′l[(3s)D jψAk

] δF0

δψ Bl

. (B.81)


If a solution S0 of (4.20) is to be found [omitting the term (4.26)], together withF0 = S0 − B0, then the above equation is satisfied.

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

Take arbitrary ‘vectors’

va ≡[

B jAB′

FlD

], vb ≡

[Bi

B A′

FkC

], (B.82)

and obtain (see Chap. 4)

G(s)abv

avb = −4π i(

n AA′ DB B′i j + nB B′ D AA′

j i

)B j

AB′ BiB A′

+4π inB B′ψCj ε jkm DC

A′mi DD

B′lk FlD Bi

B A′

+4π in AA′ψ Bi εilm D B′

B mj DCA′kl B j

AB′ FkC . (B.83)

Let

B jAB′ ≡

δa

δeAB′j

, BiB A′ ≡

δb

δeB A′i

,

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

4π i

(n AA′ DB B′

i jδa

δeAB′j

δb

δeB A′i

+ nB B′ D AA′j i

δa

δeAB′j

δb

δeB A′i

)= −32πGil jk

δa

δh jk

δb

δhil.

(B.84)Therefore, for quantities that can be written in terms of the three-metric hi j and thegravitino, the block B contains the usual DeWitt metric. To be more precise, it isnot exactly the usual DeWitt metric due to the change of the fundamental bosonicfield from hi j to eAA′

i . 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

n AA′ δ

δeAB′j

(DB B′

i jδS0

δeB A′i

K)= n AA′ δS0

δeAB′j

DB B′i j

δKδeB A′

i

−ψBi

δS0

δeAB′j

EC AB′iB jk

δKδψC

k

− ψBi εilm D B′

B mj DCA′kl

δS0

δψCk

K

−(

3i√hψC

k + ψB j ε jklnC B′el

B B′)

δS0

δψCk

. (B.85)

If the right-hand side of (B.85) is equal to zero, then it can be interpreted as aconservation 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 establishedin the highly unrealistic and special case of (i) a vanishing dependence of S0 and Kon the gravitino and (ii) the assumption that S0[e] and K[e] can be expressed in theform S0[hi j ] and K[hi j ]. In fact, (B.85) then implies the conservation equation

n AA′ δ

δeAB′j

(DB B′

i jδS0

δeB A′i

K−2

)= 0 . (B.86)

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

Problems of Chap. 5

5.1 Time Gauge and SUSY Transformations

The variation of the supergravity action can induce boundary terms. Focusing onextracting a finite quantum amplitude, we aim at exact invariance by adding bound-ary correction terms. This has so far been impossible for full supergravity, but forhomogeneous Bianchi A models, the precise invariance of the action under thesubalgebra can be restored by adding an appropriate boundary correction to theaction; 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 timegauge. In order to preserve this gauge, the transformation laws must be modifiedby adding a Lorentz generator term to the supersymmetry generators. In fact, thetime gauge represents a boundary fixing property (see Exercise 4.1). To preserveit, all bosonic generators must be symmetries of the boundaries, i.e., not inducetranslations normal to it. Any anticommutator of any fermionic generators shouldinduce bosonic generators tangential to the boundary. Consequently, the tetrad axesmust remain within a 3D boundary, whence the gauge condition e0

i = 0 will bepreserved 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 appropriateLorentz transformations:


δeAA′μ = −ik

(εAψ A′

μ + εA′ψ Aμ

)+ΛA

BeB A′μ +ΛA′

B′eAB′μ , (B.87)

for some parameters Λ which are linear in εA and εA′ , such that δe0i = 0 or

n AA′δeAA′i = 0.

A solution is

ΛAB = −2ikn AA′nCC ′ε

CψC ′i eB A′ i , (B.88)

where we have ΛAA = 0 and ΛAB = ΛB A. Thus the extended supersymmetry

transformations are

δeAA′μ = −ik(εAψ A′

μ + εA′ψ Aμ)+ΛA

BeB A′μ +ΛA′

B′eAB′μ , (B.89)

δψ Aμ = 2k−1 Dμε

A +ΛABψ

Bμ , (B.90)

δψ A′μ = 2k−1 Dμε


μ . (B.91)

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

SA(new) ≡ SA + φABCJBC , (B.92)

S A′(new) ≡ S A′ + φA′B′C ′J B′C ′ , (B.93)

where specifically here

φCAB ≡ φC

(AB) = ik2n AA′nCC ′ψC ′i eB

A′ i , (B.94)

and φC ′ A′B′ is the Hermitian conjugate of φC

AB .

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-handedSUSY transformations) becomes for Bianchi A models. The restriction to left-handed SUSY transformations means also that ε and ε are no longer Hermitianconjugates 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

δS = 1

k

∫∂M

d3x ε pqrεA′eAA′ p(DqψA

r )− 1

2

∫∂M

d3x ε pqrΛA′B′ψ B′

peAA′qψ Ar

+ 1

k2

∫∂M

d3x h1/2ωa0p(ea

qebp − ea

pebq)δeb

q , (B.95)

256 B Soluitions

where eap, eAA′

p, ψ Ap, and ψ A′

p are defined by

eai ≡ ea

p(t)ωp

i , eAA′i ≡ eAA′

p(t)ωp

i , (B.96)

ψ Ai ≡ ψ A

p(t)ωp

i , ψ A′i ≡ ψ A′

p(t)ωp

i , (B.97)

with ωp = ωpi dxi . Now, after a rather lengthy calculation using

eaμ ≡(

N 0

N i eai eai

),

we obtain the following:

• For the torsion-free spin connection14

(s)ωabp ≡ 2Qc

[bεa]cd ed p − Qcdεabd ec p , (B.98)

(s)ωa0p ≡ 1

2N (eap + eaqeb

pebq)+N q

N(

Qabεbcd ecped

q − Qbcε

acd eb pedq

).

(B.99)

• For the contorsion,

κ abp ≡ ik2

(e[bqσ a]

AA′ψ A′ [pψ Aq] + 1

2eaqebr eAA′ pψ A′ [qψ A

r ])

, (B.100)

κ a0p ≡ ik2

2

[1

N σ aAA′ψ A′ [pψ A

0] −(N q

N σ aAA′ + eaqn AA′

)ψ A′ [pψ A

q]

+ 1

N eaqeAA′ pψ A′ [qψ A0] − Nr

NeaqeAA′ pψ A′ [qψ A

r ]]

. (B.101)

• Subsequently,

(s)ωABp = 1

N n AA′σa

B A′[

iN(

eap Qb

b− 2eb p Qab

)+ 1

2(ea

p + eaqebpebq)

+N q(

Qabεbcd ecped

q − Qbcε

acd eb pedq

)], (B.102)

14 Qab ≡ Q(ab) ≡ eapm pq eb

q (det [eap])−1.


κ ABp = ik2

2N

[N eC A′qε

(ACψ A′ [pψ B)q] + N

2eA

A′qeB A′r eDD′ pψD′[qψD

r ]

−n(AA′ψ A′ [pψ B)

0] −N r eCC ′ pψC ′ [qψCr ]n A

A′eB A′q

+eCC ′ pψC ′ [qψC0]n A

A′eB A′q +N qn(AA′ψ A′ [pψ B)

q]]

. (B.103)

• And last but not least,

δeap = −σ a

AA′δeAA′p

= ikσ aAA′ε

A′ψ Ap + kεabceb

qec pn AA′εA′ψ A

q . (B.104)

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

δS = ςdet [eap]εA′ψ A

s

{1

kσa AA′eb

s Qab

+ i

4kN

(eAA′r ea

r eas + eAA′ t h

ts − 2eAA′ s eaqea

q)

+ i

2kN(N pσ c

AA′ebsed

p Qabεacd

+N pσcAA′ebsed

p Qacεabd

)

+k2

(n AA′χpqr h pr hqs − 1

2eAA′qχpq0h ps

)

+ k2N

[eAA′ sχp0q h pq − eAA′ pχq0r hs(r hq)p

]

+kN p

2N[eAA′ sχpqr hqr + eAA′rχtpq hs(t hq)r

]},

(B.105)

where

ς ≡∫

d3x det [ωpi ]

and

χpqr ≡ ψ A′ [pψ Aq]eAA′r ,

χp0q ≡ ψ A′ [pψ A0]eAA′q ,

χpq0 ≡ ψ A′ [pψ Aq]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 tothe Bianchi class A setting, under a subalgebra of (left!) SUSY generators [35, 21,36, 22, 23]. If we add

SB′ ≡ id

dt

(Wg + 1

2Wψ

)(B.107)

to the supergravity Lagrangian, where

Wg ≡ 1

k2ςm pq h pq , Wψ ≡ iςε pqrψ A′

pψA

qeAA′r , (B.108)

then invariance of the N = 1 SUGRA action is restored under left-handed SUSYtransformations.

To check this, besides the transformation for eAA′p and ea

p, we also need [23]

δψ A′p = 2i

kεB′n AA′σa AB′

(ea

p Qbb− 2eb p Qab

)

− 1

kN εB′n AA′σa AB′(

eap + eaqeb

pebq

)

− 2

kN N qεB′n AA′σa AB′(

ebpec

q Qadεd bc + ec pebq Qcdεabd

)

−ikεB′eAC ′rε(A′C ′ψ B′)[pψ A

r ] − 1

2ikεB′eAA′r eAB′ t eDD′ pψD′

[rψDt]

+ ikN εB′n A

(A′ψ B′)[pψ A0] − ik

N N qεB′n A(A′ψ B′)[pψ A

q]

− ikN εB′n AA′eAB′qeCC ′ pψC ′

[qψC0] + ik

N N qεB′n AA′eAB′r eCC ′ pψC ′[rψC

q]

−2ikεC ′n AA′nCC ′ψC

qeAB′qψ B′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 Nicolaimap.15

But the addition of the boundary term also presents a significant bonus whichunderlies 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 toa type of Fokker–Planck equation. In Bianchi class A cosmologies, for the quantum states thatpermit Nicolai maps, just two states appear, corresponding to the empty and filled fermion states(see Sects. 3.3 and 3.4).


of this volume: it simplifies the canonical formulation of the theory compared withthe formulation presented in Chap. 4 of Vol. I. In particular, it leads to simplificationsin the new Dirac brackets, whereupon the quantization provides a wider range ofpotential 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 theleft-handed generators is due to (B.107). For right-handed SUSY transformations(ε = 0), the correction term would have been the Hermitian conjugate. Note thatWψ is just the fermion number F , and recall that only the Euclidean version of asupersymmetric theory is known to admit a Nicolai map. Hence we take

SB′E ≡ d

dτ

(Wg + 1

2Wψ

), (B.110)

SB′E ≡ − d

dτ

(Wg + 1

2Wψ

). (B.111)

With SB′E = −SB′

E , this can be written [23]

L −→ L± ≡ L ± d

dτ

(1

k2ςm pq h pq + 1

2iςε pqrψ A′

pψA

qeAA′r

). (B.112)

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

I± ≡ I ±(

Wg + 1

2Wψ

), (B.113)

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

p±AA′p ≡ ∂ I±

∂eAA′p

= −ipAA′ p ± 1

2iςε pqrψ A′qψAr ∓ 2

k2ςm pqeAA′q . (B.114)

Then the following Dirac brackets become possible among the variables eAA′p,

p±AA′ p, ψ Ap, and ψ A′

p:

260 B Soluitions

{eAA′

p, eB B′q

}D= 0 , (B.115)

{eAA′

p, p±B B′q}

D= εA

BεA′

B′δqp , (B.116)

{p+AA′

p, p+B B′q}

D = 0 , (B.117)

{p−AA′

p, p−B B′q}

D = 0 , (B.118)

{ψ A

p, ψB

q}

D = 0 , (B.119)

{ψ A′

p, ψB′

q

}D= 0 , (B.120)

{ψ A

p, ψA′

q

}D= 1

σD AA′

pq , (B.121)

{eAA′

p, ψB

q

}D= 0 , (B.122)

{eAA′

p, ψB′

q

}D= 0 , (B.123)

{p+AA′

p, ψ Bq}

D = 0 , (B.124)

{p−AA′

p, ψ B′q

}D= 0 , (B.125)

{p+AA′

p, ψ B′q

}D= −iε prsψ A′s DA

B′rq , (B.126)

{p−AA′

p, ψ Bq}

D = iε prsψAs DBA′qr , (B.127)

where (B.117), (B.118), (B.124), and (B.125) are indeed much simpler then the oneswithout (B.113).

The Hamiltonian becomes (see Exercise 5.1)

H ≡ −eAA′0HAA′ + ψ A

0SA(new) + S A′(new)ψA′

0

− (ωAB0 + ψC0φC AB

)J AB −

(ωA′B′0 + ψC ′

0φC ′A′B′)J A′B′ , (B.128)


where we now have

JAB = ie(AA′ p p−B)A′ p , (B.129)

J A′B′ = ieA(A′

p p+AB′)p , (B.130)

together with

S A′(new) = −1

2k2ψ A

p p+AA′p + φA′

B′C ′J B′C ′ , (B.131)

SA(new) = 1

2k2ψ A′

p p−AA′p + φA

BCJBC . (B.132)

The reader can indeed confirm that the resulting SUSY constraints become simpler.Note also that, in HAA′ ≡ −n AA′H⊥ + eAA′ pHp, the form of HAA′ is compli-

cated, but we can write

HAA′ � 2

k2

{S(new)

A ,S(new)

A′}

D+ terms proportional to J and J

= k4

4

(1

ςp−AB′

p p+B A′q DB B′

qp + iε pqr p+B A′s DB

B′sqψArψB′

p

+iε pqr p−AB′s DB

B′qsψ A′rψ B

p + 1

2ςh1/2nB B′ψ A′ pψ B′ [qψAp]ψ Bq

+1

2iςε pqr eB B′ sψ A′rψ B

qψA[pψ B′s])

. (B.133)

Quantum mechanically [23], we can obtain an explicit representation of the differ-

ential operator16 S A′(new) as

S A′(new)+ = 1

2hk2ψ A

p∂

∂eAA′p− 1

2hk2h1/2nC A′ψC

q D AC ′qp

∂

∂eAC ′p, (B.134)

while the differential operator SA(new)− is represented by

S A(new)− = −1

2hk2ψ A′

p∂

∂eAA′p+1

2hk2h1/2n AC ′ψC ′

q DC A′p

q ∂

∂eC A′p, (B.135)

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 transforma-tions (see Sect. 3.3):

16 In the following, τ should be considered as a mere parameter, although in a path integral pre-sentation, it would be the Euclidean time.

262 B Soluitions

S A(new)+ ≡ e−Wg/hS A(new)e

Wg/h , (B.136)

S A′(new)+ ≡ e−Wg/hS A′(new)e

Wg/h , (B.137)

S A(new)− ≡ eWg/hS A(new)e

−Wg/h , (B.138)

S A′(new)− ≡ eWg/hS A′(new)e

−Wg/h . (B.139)

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

Ψ+(e, ψ, τ) = e−Wg/hΨ (e, ψ, τ) , (B.140)

Ψ−(e, ψ, τ) = eWg/hΨ (e, ψ, τ) . (B.141)

In summary, we may take the (Euclidean) representations of the operators p+AA′ p,ψ A′

p acting on a wave function Ψ (eAA′p, ψ

Ap) by means of

p+AA′p = −h

∂

∂eAA′p, (B.142)

ψ A′p = h

ςD AA′

qp∂

∂ψ Aq

, (B.143)

where

S A′(new)+ = S A′

+ + φA′B′C ′J B′C ′ , (B.144)

S A(new)− = S A

− + φABC J BC , (B.145)

and

S A′+ = 1

2hk2ψ A

p∂

∂eAA′p, (B.146)

S A− = −1

2hk2ψ A′

p∂

∂eAA′p. (B.147)

Problems of Chap. 6

6.1 Spinor Symmetry

Since eAA′i n AA′ = 0, it follows that (2h)1/2ieAA′i n

B A′ is symmetric.


6.2 FRW with Cosmological Constant

The (dimensionally) reduced Lagrangian and Hamiltonian can be expressed, afterintegrating over the spatial coordinates, as

L = ωζ + θAηA −H

(ω, ζ ; θ A, η

A)

, (B.148)

H = 4√

12Vζ 1/2(

iω − ω2 + γ ′2

12ζ − 1

4$′γ ′θAθ

A)

+ 4√

12Vζ−1/2ηA

[2 (i− ω) θ A + γ ′

3$′ηA]

, (B.149)

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

dω

dt= [ω, H ] = γ ′2

3

√12Vζ 1/2 , (B.150)

dθA

dt= −8

√12Vζ−1/2 (i− ω) θ A , (B.151)

dσ

dt= 4

√12V

[−σ 1/2 (i− 2ω) θ A + 2ζ−1/2ηAθ

A]

, (B.152)

dη

dt= 4

√12Vζ−1/2

[1

2$′γ ′θ A − 2 (i− ω) ηA

], (B.153)

with the relations (from the Hamilton–Jacobi equation)

ζ = ∂Y

∂ω= 12

γ ′2

(−iω + ω2 + 1

4$′γ ′θAθ

A)

, (B.154)

ηA = ∂Y

∂θA= −6$′

γ ′ (i− ω) θ A , (B.155)

Y = − 12

γ ′2

[iω2

2− ω3

3+ 1

4$′γ ′ (i− ω) θAθ

A]

, (B.156)

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 H4 hyperboloid, for a neg-ative 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 theevolution of the gravitino in the background (recall the analysis and discussion inSect. 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 readershould 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 ofBianchi cosmologies straight from bosonic general relativity, using Ashtekar’s newvariables (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:

J i = DaEai , (B.157)

Cb = Eai Fiab , (B.158)

H = εi jkEai Ebj Rkab , (B.159)

where Riab is the curvature of Ai

a , and Da the covariant derivative formed with Aia .

Restricting to Bianchi cosmologies, we introduce a basis of vectors Xai [see (6.54)–

(6.57)], and then expand into new variables:

Eaj ≡ E i

j Xai , (B.160)

A ja = A j

i χia . (B.161)

Dropping the diacritic to simplify the notation, the quantities Eij and A j

i only depend

on time. Inserting these substitutions into the constraint equations [41], we obtain17

J i ≡ CkjkEi j + εi jkAmj Em

k , (B.162)

Hk = −E ji Ai

mCmjk + εimnEi

j A jmAkn , (B.163)

H = εi jkC pmnEm

i Enj Apk + Em

i Enj (A

imA j

n − AinA j

m) . (B.164)

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

Qij ≡ Ei

kAkj , (B.165)

the Hamiltonian constraint for all Bianchi class A models can be written as

17 Further restricting to diagonal models, Eij = diag(E1,E2,E3) and Ai

j = diag(A1,A2,A3).


H ≡ Q∗ik Qk

i − Q∗ii Q j

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

= G i j Qi Q j , (B.167)

where i, j = 1, . . . , 3, and

Gi j = 1

2

⎡⎢⎣

0 1 1

1 0 1

1 1 0

⎤⎥⎦ . (B.168)

Using the Misner–Ryan parametrization with βi j given by

βi j = 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 equa-tion for this particular model. The relation with the Ashtekar variables is that, ifwe have a wave function ΨA given in terms of the Ashtekar variables [47], we canreconstruct the wave function in terms of the Misner–Ryan variables by choosingΨMR = exp(±iIA)ΨA, where IA is given by IA = 2i

∫Ea

i Γiad3x , with IA equal to

∓iI for the Bianchi IX case. Γ ia is the corresponding (compatible) spin connection.

But where is the (‘hidden’ N = 2) SUSY? The above Hamiltonian suggests (seeChap. 8 of Vol. I) introducing the expressions

S ≡ ψ i Qi , (B.169)

S ≡ ψ i Q∗i , (B.170)

for the SUSY constraints, with

ψ iψ j + ψ iψ j = Gi j , (B.171)

and G as given above. The Hamiltonian constraint becomes

H = 1

2

(SS + SS

). (B.172)

In quantum mechanical terms, one realization is Ψ [A, η], with

266 B Soluitions

EiΨ [A] = ∂

∂AiΨ [A] , (B.173)

AiΨ [A] = AiΨ [A] , (B.174)

together with ψ i = ηi , ψi = Gi j∂/∂η j . The equations become

SΨ [A, η] = QiηiΨ [A, η] , (B.175)

SΨ [A, η] = Q∗i

∂

∂ηiΨ [A, η] , (B.176)

and using the reality conditions, we get

ηi Ai∂

∂AiΨ [A, η] = 0 , (B.177)

(Ak − 2 Γ k)∂

∂AkGki ∂

∂ηiΨ [A, η] = 0 . (B.178)

The only solution admitted by this system of equations is Ψ [A] = constant, but ithas 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

− p2Ω + p2+ + p2− = 0 , (B.179)

with the convention in [48] (see Sect. 7.2 for more details on the expressions usedhere). With the quantum correspondence

pΩ −→ −ih∂

∂Ω, p± −→ −ih

∂

∂β±, (B.180)

we may write

∂2Ψ

∂Ω2= ∂2Ψ

∂β2++ ∂2Ψ

∂β2−. (B.181)


Now ‘linearize’ the square operator

∂2Ψ

∂β2++ ∂2Ψ

∂β2−,

to obtain

i∂Ψ

∂Ω= ±

(∂2Ψ

∂β2++ ∂2Ψ

∂β2−

)1/2

−→ i∂Ψ

∂Ω= iα+

∂Ψ

∂β+− iα−

∂Ψ

∂β−, (B.182)

where α± are matrices satisfying α2± = 1 and which anticommute! The minimalrank for such matrices is two. Then taking α± to be the Pauli matrices, namelyα+ ≡ σ1, α− ≡ σ2, Ψ must be a two-component ‘vector’ (spinor). Writing

Ψ =(Ψ+Ψ−

), (B.183)

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 spinordegrees of freedom were described in [48] as ‘disturbing’. In fact, the two solutionscorrespond to expanding (E > 0) and contracting (E < 0) universes, but howshould one then interpret the ‘spin’ states Ψ± in terms of physical attributes of theuniverse? The reader should remember that this was in 1972! Researchers had towait for SUGRA in order to obtain a clearer view (see Sect. 7.2). The above is apossible representation within SQC, i.e., SUGRA as applied to cosmology, usingthe matrix representation for fermionic momenta! It is altogether astonishing howthis emerged back in 1972, before the full development of SUSY and SUGRA.

7.2 From the Lorentz Constraints to the Lorentz Conditions

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

J23 = J12 (J13)−1 J23 (J12)

−1 J13 ,

J12 = J23 (J13)−1 J12 (J23)

−1 J13 ,

J13 = J23 (J12)−1 J13 (J23)

−1 J12 . (B.185)

Using JA = ε0ABCJ BC/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 Bianchitype IX cosmological model [49]. Then, e.g., substitute the J 01 and the J 23 com-ponent expressions in the (0, 1) component of the Einstein–Rarita–Schwinger equa-tion. This reduces to

(−3β+ +

√3β_

)ψ2γ

2ψ1 = 0 . (B.186)

At the quantum level (in this matrix representation approach!), (B.186) yields thesame 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

more ψ2γ2ψ1 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 termscan also be reduced to quadratic terms in ψ i , multiplied by linear combinations ofΩ , β+, and β_, with the help of the equations for ψi . The same procedure is appliedto, e.g., the (0, 2) and (0, 3) components of the Einstein–Rarita–Schwinger equa-tion. Substituting into all the i �= j equations, and the ψ i equations are used oncemore. The quadratic expressions resulting from this procedure for all the equationsi �= j are:

ψ1γ0γ 1γ 2φ1 , ψ2γ

0γ 2γ 3ψ1 ,

φ1γ0γ 1γ 3φ1 , φ1γ

0γ 1γ 3ψ2 ,

ψ1γ0γ 2γ 3φ1 , φ3γ

0γ 2γ 3ψ1 ,

ψ2γ0γ 1γ 2φ2 , φ3γ

1γ 2γ 3ψ1 ,

ψ2γ0γ 1γ 3φ2 , φ1γ

0γ 1γ 2ψ3 ,

ψ2γ0γ 2γ 3φ2 , φ3γ

0γ 1γ 2ψ2 ,

ψ3γ0γ 1γ 2φ3 , φ2γ

0γ 1γ 3ψ3 ,

ψ3γ0γ 1γ 3φ3 , φ3γ

1γ 2γ 3ψ2 ,

ψ3γ0γ 2γ 3φ3 , φ1γ

1γ 2γ 3ψ2 . (B.187)

These expressions at the quantum level (i.e., proceeding from the variables ψ i

to χi ) lead to the same wave function that has been obtained by applying the J ab

constraints. Therefore the equations i �= j yield the same result as the Lorentzconstraints J ab.


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])

δS = 1

2

∫ [f

N GXY (qZ )q X qY + f NU (q Z )

]·dt , (B.188)

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

δSgrav = i

2

∫ {Dη

[LDη

(GXY

2NDηQXDηQY + W(QZ )

)](B.190)

+Dη[LDη

(GXY

2NDηQXDηQY + W(QZ )

)]}dηdηdt ,

under (8.7), (8.8), (8.12), and

δQ = LQ+ i

2DηLDηQ+ i

2DηLDηQ . (B.191)

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:

pψ ≡ πψ + i

3ψ = 0 , (B.192)

pψ ≡ πψ +i

3λ = 0 , (B.193)

pI (χ) ≡ πI (χ)− i

2κ2GJ Iχ

J = 0 , (B.194)

pI (χ) ≡ πI (χ)− i

2κ2GI Jχ

J = 0 , (B.195)

270 B Soluitions

ΠI (λ) ≡ πI (λ)− i

2κ2GJ Iλ

J = 0 , (B.196)

pI (λ) ≡ πI (λ)−i

2κ2GI Jχ

J = 0 , (B.197)

where πψ = dL/dψ , πI (χ) = dL/dχ I , and πI (λ) = dL/dλI are the momentaconjugate to the anticommuting variables ψ(t), χ(t), and λ(t), respectively.

Then write the matrices

Cψψ =2

3i , CI J (χ) = − i

k2GI J , CI J (λ) = − i

k2GI J , (B.198)

and their inverses

Cψψ = (Cψψ)−1 = −3

2i , C I J (χ) = ik2GI J , C I J (λ) = ik2GI J .

(B.199)The Dirac brackets [ , ]D are (recall Chap. 4 of Vol. I)

[ f, g]D = [ f, g] − [ f,pi](C−1)ik [pk, g

], (B.200)

and allow elimination of the momenta conjugate to the fermionic variables, leavingus with the following non-zero Dirac bracket relations:

[a, πa]D = [a, πa]D = −1 , (B.201)[φI , π

Jφ

]D=[φI , π

Jφ

]= −δ J

I , (B.202)

[φ I , π

Jφ

]D=[φ I , π

Jφ

]= −δ J

I, (B.203)

[χ I , χ J

]D= −ik2GI J , (B.204)

[λI , λJ

]D= −ik2GI J , (B.205)

[ψ,ψ

]D =

3

2i . (B.206)

8.3 FRW Wave Function in N=2 Conformal SUSY

With k = 0, 1, take the action to be (see [51, 52])


S = SFRW + Smat ,

SFRW =∫

6

(− 1

2k2

ANDηADηA+

√k

2k2A2

)dηdηdt ,

Smat =∫ [

1

2

A3

NDηΦDηΦ − 2A3W(#)

]dηdηdt , (B.207)

where we use

# = 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 (Grassmannvariables)

ψ(t) −→ 1

3a−1/2(t)ψ(t) , χ(t) −→ a−3/2(t)χ(t) ,

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 Taylorexpansion for the superpotential:

W(#) = W(φ)+ ∂W∂φ

(#− φ)+ 1

2

∂2W∂φ2

(#− φ)2 + · · · ,

with # as given above.18 In terms of components of the superfields A, N, and #, theLagrangian then reads (somewhat more simply than in Chap. 8)

L = − 3

k2

a(Da)2

N+ 2

3iψDψ +

√k

ka1/2(ψ0ψ − ψ0ψ)

+1

3Na−1

√kψψ + 3k

k2Na + a3

2

(Dφ)2

N− iχDχ

−3

2

√kNa−1χχ − k2NW(φ)ψψ − 6

√kNW(φ)a2

−Na3V (φ)+ 3

2k2NW(φ)χχ + ik

2Dφ(ψχ + ψχ)

−2N∂2W(φ)

∂φ2χχ − kN

∂W(φ)

∂φ(ψχ − ψχ)+ k2

4a−3/2(ψ0ψ − ψ0ψ)χχ

−ka3/2(ψ0ψ − ψ0ψ)W(φ)+ a3/2 ∂W(φ)

∂φ(ψ0χ − ψ0χ) , (B.209)

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

272 B Soluitions

where

Da = a − ik6

a−1/2(ψ0ψ + ψ0ψ) , Dφ = φ − i

2a−3/2(ψ0χ + ψ0χ) ,

are the supercovariant derivatives, and

Dψ = ψ − i

2Vψ , Dχ = χ − i

2Vχ ,

are the U(1) covariant derivatives. The potential for the homogeneous scalar fields is

V (φ) = 2

[∂W(φ)

∂φ

]2

− 3k2W2(φ) , (B.210)

and consists of two terms. One is the potential for the scalar field in the case of globalsupersymmetry. The potential (B.210) is not positive semi-definite, in contrast withwhat happens in standard supersymmetric quantum mechanics. Indeed, the presentmodel 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

[ψ,ψ

]D =

3

2i , [χ, χ ]D = −i , (B.211)

whence

{ψ,ψ

} = −3

2, {χ, χ} = 1 , (B.212)

[a, πa] = −i ,[φ, πφ

] = −i . (B.213)

Antisymmetrizing, i.e., writing a bilinear combination in the form of the commuta-tors, e.g.,

χχ −→ 1

2[χ, χ ] ,

this leads eventually to eigenstates of the Hamiltonian with four components:

Ψ (a, φ) =

⎡⎢⎢⎢⎣

Ψ1(a, φ)

Ψ2(a, φ)

Ψ3(a, φ)

Ψ4(a, φ)

⎤⎥⎥⎥⎦ , (B.214)

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 → ∞, becausethe other components are infinite as a → ∞. We thus get the partial differentialequations

[−a−1/2 ∂

∂a− 6W(φ)a3/2 + 6

√k M2

Pla1/2 + 3

4a−3/2

]Ψ4 = 0 , (B.215)

[∂

∂φ+ 2a3 ∂W(φ)

∂φ

]Ψ4 = 0 , (B.216)

[−a−1/2 ∂

∂a+ 6W(φ)a3/2 − 6

√k M2

Pla1/2 + 3

4a−3/2

]Ψ1 = 0 , (B.217)

[∂

∂φ− 2a3 ∂W(φ)

∂φ

]Ψ1 = 0 , (B.218)

with solutions

Ψ −→ Ψ4(a, φ) � a3/4 exp[−2w(φ) a3 + 3

√k M2

Pla2]

, (B.219)

Ψ −→ Ψ1(a, φ) � a3/4 exp[2w(φ) a3 − 3

√k M2

Pla2]

. (B.220)

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

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Index

AAshtekar connection, 112Ashtekar–Jacobson formulation, 116

N = 1 SUGRA, 116

BBianchi I

matrix representation, 133Bianchi IX

matrix representation, 138Born–Oppenheimer

semiclassical (quantum) gravity, 14Boundary conditions, 5, 19, 21

infinite wall proposal, 5, 28no boundary, 5, 19, 21tunneling, 5, 19, 21

CCartan–Sciama–Kibble

N = 1 SUGRA, 229torsion, 229

Chiral superfields, 35Classical SUSY spin 1/2 particle

Dirac equation, 130quantization, 130

Constraints, 4Contorsion

decomposing 3D connection, 227Conventions and notation, 211Cosmological constant

SUSY breaking, 41Covariant derivative, 227

minisuperspace, 20Curvature, 228

extrinsic, 228intrinsic, 228spinor valued, 228

DDecoherence

quantum cosmology, 21SQC, 74

DeWitt metric, 71DeWitt supermetric, 71, 74, 75, 78

N = 1 SUGRA, 71

FFayet–Iliopoulos potential, 35Fermionic differential operator representation,

87, 96, 101supersymmetry breaking, 87

Fermionic momentumdifferential operator representation, 6, 7matrix operator representation, 87matrix representation, 7

FRW modelmatrix representation, 147

GGeneral relativity

constraintsDeWitt metric, 71

Goldstino, 38Grassmann numbers, 218Gravitational canonical momenta, 227Gravitino mass, 41

HHamiltonian

minisuperspace, 20Hamiltonian constraint

N = 1 SUGRA, 67SQC

FRW, 92generic gauge supermatter FRW, 92scalar supermultiplet FRW, 89

277

278 Index

Hamiltonian formulationN = 1 SUGRA, 6, 7general relativity, 5supergravity, 6, 7

Hamilton–Jacobi equation, 15N = 1 SUGRA, 68, 70

body and soul, 70bosonic and fermionic parts, 69–71, 75spacetime emergence, 70supermanifolds, 71superRiemaniann geometries, 71SUSY spacetime, 70

quantum cosmology, 15quantum gravity, 15

Hidden supersymmetries, 7SQC, 7

IInfeld–van der Waerden symbols, 222Inflationary cosmology, 28

KKähler

geometry, 90, 91metric, 90, 91potential, 90, 91

LList of symbols, 207Loop quantum cosmology, 111Loop quantum gravity, 111

Ashtekar connection, 112Barbero–Immirzi parameter, 112connection quantization, 114Hamiltonian formulation, 114reality conditions, 112

Lorentz conditionsmatrix representation, 152rest frame solution, 153

Lorentz constraintmatrix representation, 147, 155SQC

Bianchi, 96FRW, 92generic gauge supermatter FRW, 92

Lorentz constraint trivializationmatrix representation, 153

Lorentz group, 214representations, 214spinors, 214

MMatrix representation, 127

fermionic momenta, 127

Lorentz constraint, 128Minisuperspace, 4, 20

covariant derivative, 20curvature, 20D’Alembert equation, 20dimension, 20factor ordering problem, 20Hamiltonian, 20Laplace–Bertrami, 20minisupermetric, 20validity, 20Wheeler–DeWitt equation, 20

perturbations, 20WKB approximation, 21

Momentum constraintsN = 1 SUGRA, 67

M-theory, 4

NNicolai maps, 55Nilpotency, 70

PPauli matrices, 222

spinors, 222

QQuantum constraints

N = 1 SUGRA, 67Quantum cosmology, 4, 6, 7

decoherence, 21Hamilton–Jacobi equation, 15historical, 4inflation, 13, 19, 21, 28

vacuum, 21minisuperspace, 6, 19, 21

beyond, 19Hamiltonian, 20perturbations, 19, 21vacuum state, 21validity, 20WKB, 21

observational, 13, 19, 66predictions, 13, 19

vacuum state, 21Schrödinger equation, 15semiclassical, 13, 19, 66semiclassical limit, 7structure formation, 19

vacuum state, 21summary and review, 31superstring, 5, 28

infinite wall proposal, 5, 28landscape problem, 5, 28

Index 279

Tomonaga–Schwinger equation, 15, 74vacuum state

structure formation, 21Quantum FRW solutions

fermionic differential operatorrepresentation

generic gauge supermatter, 94Quantum gravity, 4

Hamilton–Jacobi equation, 15observational context, 4Schrödinger equation, 15semiclassical, 13, 66superspace, 4, 19, 21, 61, 70Tomonaga–Schwinger equation, 15

Quantum gravity corrections, 17Schrödinger equation, 17

RReality conditions, 112

N = 1 SUGRA, 116

SSchrödinger equation, 15, 74

quantum cosmology, 15quantum gravity, 15quantum gravity corrections, 17

N = 1 SUGRA, 75, 78, 79normal component, 79SQC and observation, 81tangential component, 79

SQC, 74Semiclassical expansion

N = 1 SUGRA, 66Semiclassical (quantum) gravity, 13

Born–Oppenheimer, 13Hamilton–Jacobi equation, 15observational, 13Schrödinger equation, 15, 16

corrections, 17time functional, 15

time functional, 15Tomonaga–Schwinger equation, 15, 16

time functional, 15Semiclassical N = 1 SUGRA, 61, 68

Born–Oppenheimer, 68observational, 68Schrödinger equation, 74

time functional, 74time functional, 74Tomonaga–Schwinger equation, 74

time functional, 74SL(2,C), 214Spacetime foliation

SUGRA, 222

Spinors, 212Dirac, 218Infeld–van der Waerden symbols, 222Lorentz (group algebra), 213Lorentz (group representations), 212Lorentz transformations, 212Majorana, 218Pauli matrices, 222primed, 214, 216tetrad, 212unprimed, 214, 216Weyl, 218

SQC, 5, 87achievements, 7Ashtekar–Jacobson formulation, 111, 118,120

Bianchi class A models, 120FRW with cosmological constant, 118

Bianchi, 96, 101Ansatze, 96, 101Lorentz constraints, 96

Bianchi IMatrix representation, 133

Bianchi IXMatrix representation, 138

Bianchi minisuperspace reduction, 96, 101Lagrangian, 96N = 2 SUGRA, 101

Bianchi wave function of the universe, 96,101connection/loop variables, 7connection quantization, 111decoherence, 74fermionic differential operatorrepresentation, 6, 7, 87, 96, 101fermionic momenta

matrix representation, 127FRW, 88

constraints, 92gauge fields ansatze, 91generic gauge supermatter, 91Hamiltonian constraint, 92Lorentz constraints, 92SUSY constraint, 92

FRW generic gauge supermatter sectorquantum states, 96

FRW modelmatrix representation, 147

FRW scalar supermultiplet sectorquantum states, 89

FRW wave function of the universegeneric gauge supermatter, 94

280 Index

further exploration, 87generic gauge supermatter FRW

constraints, 92Hamiltonian constraint, 92Lorentz constraints, 92SUSY constraint, 92wave function of the universe, 94

hidden N = 2 SUSY, 7hidden supersymmetries, 7loop quantization, 111Lorentz conditions

matrix representation, 152Lorentz constraint

matrix representation, 128, 147Lorentz constraint trivialization

matrix representation, 153matrix fermionic operator representation,7, 87matrix representation, 127

Bianchi I, 133Bianchi IX, 138FRW, 147historical background, 129Lorentz boost components, 155Lorentz constraint, 128Taub, 143

non-supersymmetric square root, 129observational, 61, 66, 75observational insights, 81open issues, 6, 7, 19, 21

observational context, 6, 21SUSY breaking, 6

rest frame solutionmatrix representation, 153

scalar supermultiplet Bianchiquantum states, 96, 101

scalar supermultiplet Bianchi class Aquantum states, 96

scalar supermultiplet FRWconstraints, 89Hamiltonian constraint, 89quantum states, 91wave function of the universe, 89

Schrödinger equation, 74semiclassical, 61, 66, 75N = 1 SUGRA, 87N = 2 SUGRA, 87, 101Taub model

matrix representation, 143wave function of the universe, 6, 89, 94,96, 101, 106

SQMSUSY breaking, 45

Witten index, 45SUGRA

spacetime foliation, 222summary and review, 56N = 1 SUGRA, 72, 74, 75

bosonic and fermionic parts, 74, 75conservation law, 253simplified form, 73

N = 1 SUGRA, 3, 4affinely connected spaces, 229Ashtekar–Jacobson formulation, 116Ashtekar–Jacobson Hamiltonianformulation, 116Ashtekar–Jacobson quantization, 117

Bianchi class A models, 120FRW with cosmological constant, 118

bosonic states, 62Cartan–Sciama–Kibble, 229CFOP argument, 61DeWitt supermetric, 71, 75, 78

Schrödinger equation, 74, 75finite fermion number states, 63gravitino mass

super Higgs effect, 41Hamilton–Jacobi equation, 68, 70

body and soul, 70bosonic and fermionic parts, 69–71, 75DeWitt supermetric, 71, 75spacetime emergence, 70supermanifolds, 71superRiemaniann geometries, 71SUSY spacetime, 70

Hamiltonian constraintnormal projection, 67

Hamiltonian formulation, 6, 7infinite fermion number states, 63matrix representation, 130momentum, 67physical states, 61

bosonic states, 62CFOP argument, 61D’Eath’s claims, 61finite fermion number, 63infinite fermion number, 63

quantum constraints, 67Hamiltonian, 67

quantum states, 61Riemann–Cartan, 229

torsion, 229Schrödinger equation

bosonic and fermionic parts, 74, 75using DeWitt supermetric, 74, 75

Schrödinger equation, 72, 74, 75

Index 281

conservation law, 253DeWitt supermetric, 78normal component, 79quantum gravity corrections, 75, 78simplified form, 73SQC and observation, 81tangential component, 79using DeWitt supermetric, 74

second-order formalism, 225semiclassical expansion, 66semiclassical formulation, 61semiclassical limit, 7SQC, 87

Bianchi ansatze, 96FRW, 88FRW gauge fields ansatze, 91generic gauge supermatter, 91

super DeWitt metric, 74super Higgs effect, 39

gravitino mass, 41supermatter

gaugino condensate, 40SUSY breaking, 39

superRiem(Σ), 61, 70, 78normal part, 78tangential part, 78

SUSY breaking, 39cosmological constant, 41gaugino condensate, 40gravitino mass, 41super Higgs effect, 39

Tomonaga–Schwinger equation, 74Wheeler–DeWitt equation, 67

N = 2 (local) conformal supersymmetry, 163N = 2 SQM, 43

Nicolai Maps, 55quantum states, 51

Fock space, 51Topology and vacuum states, 52

N = 2 SUGRA, 3, 7SQC, 87, 101

Bianchi ansatze, 101N = 2 SUGRA (SQC)

Bianchi quantum statesvector field sector, 106

Bianchi wave function of the universe, 106vector field sector

Bianchi class A global O(2), 103Bianchi class A local O(2), 105Bianchi class A quantum states, 101Bianchi quantum states, 106

N = 2 conformal supersymmetry(complex) scalar fields, 172

Hamiltonian, 177Bianchi class A, 169FRW universe, 164Hamiltonian formulation, 168quantum cosmology, 181supersymmetry breaking, 183time reparametrization, 164

N = 2 Supersymmetric quantum mechanics,43

N = 2 supersymmetric quantum mechanicsFock space, 51Nicolai maps, 55quantum states, 51

Fock space, 51sigma model, 48

quantum states, 51N = 8 SUGRA, 3Superfield description, 163

N = 2 conformal supersymmetry, 163Superfields, 35, 47

chiral, 35vector, 35

SupergravityHamiltonian formulation, 6, 7square root, 5, 6

Supergravity (SUGRA), 6, 7SuperRiem(Σ), 61, 70, 78

normal part, 78tangential part, 78

Superspacequantum gravity, 19, 21

Superspace (N = 1 SUGRA)canonical quantization, 67Dirac quantization, 67

Superspace (gravity), 61, 70Superspace (SUSY), 47

interpretation, 47supertranslations, 47

Superstring, 4cosmological models, 7dualities, 5quantum cosmology, 5, 28

infinite wall proposal, 5, 28landscape problem, 5, 28

Supersymmetric quantum cosmology (SQC),5, 87Bianchi

Ansatze, 96, 101FRW, 88

gauge fields ansatze, 91generic gauge supermatter, 91

generic gauge supermatter sectorFRW quantum states, 96

282 Index

scalar supermultiplet Bianchiquantum states, 96, 101

scalar supermultiplet Bianchi class Aquantum states, 96

scalar supermultiplet FRWquantum states, 91

scalar supermultiplet sectorFRW quantum states, 89

Supersymmetric quantum mechanics (SQM),7, 35, 43, 88quantum states, 51

Fock space, 51SUSY, 6, 7, 35

breaking, 35, 36, 183conditions, 36cosmological constant, 41fermionic masses, 39gaugino condensate, 40goldstino, 38gravitino mass, 41mechanisms, 37N = 1 SUGRA, 39super Higgs effect, 39, 41

chiral superfields, 35Fayet–Iliopoulos potential, 35goldstino, 38Grassmannian variables, 47Nicolai maps, 55N = 2 SQM, 43

Fock space, 51quantum states, 51

summary and review, 56superfields, 35, 47supermultiplet, 35, 90

chiral, 35, 90vector, 35, 90

superspace, 43supersymmetric quantum mechanics, 35,43, 88

quantum states, 51supertranslations, 47vector superfields, 35

SUSY Bianchifermionic differential operatorrepresentation

quantum states, 96scalar supermultiplet

quantum solutions, 96SUSY Bianchi class A

fermionic differential operatorrepresentation, 96, 101

scalar supermultiplet sector, 96N = 2 SUGRA, 101

vector field sector, 101SUSY breaking

SQM, 45N = 1 SUGRA

super Higgs effect, 42SUSY constraint

SQCFRW, 92generic gauge supermatter FRW, 92

SUSY FRWfermionic differential operatorrepresentation

gauge fields sector, 91generic gauge supermatter, 91supersymmetry breaking, 87

generic gauge supermatter sectorquantum solutions, 96

scalar supermultiplet sectorquantum solutions, 89supersymmetry breaking, 87

TTaub model

matrix representation, 143Tetrad, 223

metric, 223tetrad decomposition

spinorial version, 222Time functional

semiclassical N = 1 SUGRA, 74semiclassical (quantum) gravity, 15

Tomonaga–Schwinger equation, 15, 74quantum cosmology, 15, 74quantum gravity, 15N = 1 SUGRA, 74

Torsion, 225Cartan–Sciama–Kibble, 229connections, 225

UUnit normal vector

spinorial form, 222Useful expressions, 222

VVector superfields, 35

WWave function of the universe, 4, 6, 19–21, 89,

94, 96, 101, 106perturbations, 20SQC, 6, 89, 94, 96, 101, 106

Bianchi, 96, 101generic gauge supermatter FRW, 94

Index 283

scalar supermultiplet FRW, 89N = 2 SUGRA (SQC)

Bianchi, 106WKB approximation, 21

Wheeler–DeWitt equation, 6, 20minisuperspace, 20

perturbations, 20perturbations, 20polynomial structure, 111

N = 1 SUGRA, 67semiclassical limit, 67

Witten index, 45SQM, 45

WKB approximationminisuperspace, 21quantum cosmology, 21wave function of the universe, 21

appendixa list of symbols, notation, and useful expressions(lorentz and 2-spinor indices) ωa bμ,ω...

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