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Appendix A List of Symbols, Notation, and Useful Expressions In this appendix the reader will find a more detailed description of the conventions and notation used throughout this book, together with a brief description of what spinors are about, followed by a presentation of expressions that can be used to recover some of the formulas in specified chapters. A.1 List of Symbols κ ≡ κijk Contorsion ξ ≡ ξijk Torsion K Kähler function I ≡ I ≡ I KJ gJ GJ Kähler metric W(#) Superpotential V Vector supermultiplet #, # (Chiral) Scalar supermultiplet φ,ϕ Scalar field V (φ) Scalar potential χ ≡ γ 0χ † For Dirac 4-spinor representation ψ† Hermitian conjugate (complex conjugate and transposition) φ∗ Complex conjugate [M]T Transpose { , } Anticommutator [ , ] Commutator θ Grassmannian variable (spinor) Jab, JAB Lorentz generator (constraint) qX , X = 1, 2,... Minisuperspace coordinatization π μαβ Spin energy–momentum Moniz, P.V.: Appendix. Lect. Notes Phys. 804, 207–275 (2010) DOI 10.1007/978-3-642-11570-7 c Springer-Verlag Berlin Heidelberg 2010 208 A List of Symbols, Notation, and Useful Expressions Sμαβ Spin angular momentum D Measure in Feynman path integral [ , ]P ≡[, ] Poisson bracket [ , ]D Dirac bracket F Superfield M P Planck mass V β+,β− Minisuperspace potential (Misner–Ryan parametrization) Z IJ Central charges ds Spacetime line element ds Minisuperspace line element F Fermion number operator eμ Coordinate basis ea Orthonormal basis (3)V Volume of 3-space (a) vμ Vector field (a) fμν Vector field strength ij π Canonical momenta to hij π φ Canonical momenta to φ π 0 Canonical momenta to N i π Canonical momenta to Ni μ a pa Canonical momenta to eμ (tetrad) ij ξ k Pk Canonical momenta to ij (torsion) J j Lorentz rotation generator Ki Lorentz boost generator Pμ Translation (Poincaré) generator εμνλσ Four-dimensional permutation εijk Three-dimensional permutation SA SUSY generator (constraint) H⊥ Hamiltonian constraint Hi Momentum constraints G DeWitt metric Gs DeWitt supermetric Ψ Wave function (state) of the universe Ω ADM time R Spinor-valued curvature A.1 List of Symbols 209 βij(β+,β−) Anisotropy matrices (Misner–Ryan representation) in Bianchi cosmologies ij pij Canonical momentum to β HADM ADM Hamiltonian Kij Extrinsic curvature (4)R Four-dimensional spacetime curvature (3)R Three-dimensional spatial curvature Λ Cosmological constant μ Γνλ Christoffel connection coefficients , Usual derivative (vectors, tensors) (3) | and Di Four-dimensional spacetime covariant derivative with respect to the 3-metric hij, no spin connection and no torsion, with Christoffel term (vectors, tensors) ( ) ; and 4 Dμ Four-dimensional spacetime covariant derivative with respect to the 4-metric gμν, no spin connection and no torsion, with Christoffel term (vectors, tensors) ' Covariant derivative with respect to the 3-metric, including spin connection (vectors, tensors) ( Covariant derivative with respect to the 4-metric gμν, including spin connection (vectors, tensors) Dot over symbol Proper time derivative Ω2 d 3 Line element of spatial sections k Spatial curvature index a(t) ≡ eα(t) FRW scale factor N Lapse function N i Shift vector μ,ν,... World spacetime indices a, b,... Condensed superindices (either bosonic or fermionic) a, b,... Local (Lorentz) indices aˆ, bˆ,...= 1, 2, 3,... Local (Lorentz) spatial indices i, j, k Spatial indices hij Spatial 3-metric gμν Spacetime metric ηab Lorentz metric ωμ ≡{ω0,ωi } One-form basis A, A Two-spinor component indices (Weyl representation) [a] Four-spinor component indices, e.g., Dirac representation 210 A List of Symbols, Notation, and Useful Expressions I, J,... Kähler indices J, j Representation label of SU(2) (spin state) M, N Elements of SL(2,C) La b Lorentz (transformation) matrix representation i AA pAA Momentum conjugate to ei HAA Hamiltonian and momentum constraints condensed (two-spinor components) S Lorentzian action L Lagrangian I Euclidean action S Superspace M Four-dimensional spacetime manifold Σt Three-dimensional spatial hypersurface t Coordinate time τ Proper time n ≡{nμ} Normal to spatial hypersurface ( ) 4 Dν ≡ Dν Four-dimensional spacetime covariant derivative with respect to the 4-metric gμν, with spin connection and torsion, no Christoffel term (spinors [vectors, tensors]) ( and Dν Four-dimensional spacetime covariant derivative with respect to the 3-metric hij, with spin connection and torsion, with Christoffel term (spinors [vectors, tensors]) ' Three-dimensional analogue of Dν (spinors [vectors, tensors]) (3) (3) D j ≡ ∇ j ≡∇j Three-dimensional analogue of Dν (spinors [vectors, tensors]) (3s) (3s) D j ≡ ∇ j ≡ ∇ j Three-dimensional analogue of Dν, no torsion (spinors [vectors, tensors]) Mab, MAB Lagrange multipliers for the Lorentz constraints (Lorentz and 2-spinor indices) ωA ,ωa ,ωAA BB Bμ bμ Spin connections A ψμ Gravitino field nμ, n AA Normal to the hypersurfaces ε AB Metric for spinor indices (Weyl representation) σ AA a Pauli matrices with two-component spinor indices (Weyl representation) σ Infeld–van der Waerden symbols A.2 Conventions and Notation 211 eaμ Tetrad aˆ eˆ i Spatial tetrad (aˆ, i = 1, 2, 3) eAA μ Tetrad with two-component spinor indices =∼ Equality provided constraints are satisfied (a) Gauge group index ˆ Dμ, Dˇ φ, Dμ Gauge group covariant derivatives Q(a) Gauge group charge operator ε Infinitesimal spinor Dφ Kähler derivative D SUSY covariant derivative (SUSY superspace) %JK I Kähler connection R Kähler curvature ψ, χ, λ Spinors ψ,χ,λ Conjugate spinors, in the 2-spinor SL(2,C)Weyl representation U Minisuperspace potential P(φ, φ) SUGRA-induced potential for scalar fields T (a) Gauge group generators Q SUSY supercharge operators Gˆ Gauge group T Energy–momentum tensor D Minisuperspace dimension R Minisuperspace curvature D, dˆ, Dˆ , D˜ , Dˇ ,... Dimension of space σ Axion field 1,σ Pauli matrices A, B,... Bosonic amplitudes of SUSY Ψ AB Ei Densitized spatial triad AB A j Ashtekar connection A.2 Conventions and Notation = = = = −2 ≡ π Throughout this book we employ c 1 h¯ and G 1 MP , with k 8 G unless otherwise indicated. In addition, we take: • μ,ν,...as world spacetime indices with values 0, 1, 2, 3, • a, b,...as local (Lorentz) indices with values 0, 1, 2, 3, 212 A List of Symbols, Notation, and Useful Expressions • i, j, k as spatial indices with values 1, 2, 3, • A, A as 2-spinor notation indices with values 0, 1 or 0 ,1 , respectively, •[a] as 4-spinor component indices, e.g., the Dirac representation, with values 1, 2, 3, 4. In this book we have also chosen the signature of the 4-metric gμν (or ηab)tobe (−, +, +, +). Therefore, the 3-metric hij on the spacelike hypersurfaces has the signature (+, +, +), which gives a positive determinant. Thus the 3D totally anti- 0123 123 symmetric tensor density can be defined by ε ≡ ε = ε123 =+1. A.3 About Spinors As the fellow explorer may already have noticed (or will notice, if he or she is venturing into this appendix prior to probing more deeply into some of the chapters) there are some idiosyncrasies in the mathematical structures of SUSY and SUGRA which are essentially related to the presence of fermions (and hence of the spinors which represent them). But why is this, and what are the main features the spinors determine? In fact, why are spinors needed to describe fermions? What are spinors and how do they come into the theory? This section is devoted (in part) to introduc- ing this issue. Before proceeding, for completeness let us just indicate that a tetrad formalism is indeed mandatory for introduing spinors.1 The tetrad corresponds to a massless spin-2 particle, the graviton [1], i.e., corresponding to two degrees of freedom.2 Note A.1 Extending towards a simple SUSY framework, with only one gen- erator (labeled N = 1 SUSY), the corresponding (super)multiplet will also contain a fermion with spin 3/2, the gravitino, as described in Chap. 3 of Vol. I, where the reader will find elements of SUSY (not in this appendix). A.3.1 Spinor Representations of the Lorentz Group Local Poincaré invariance is the symmetry that gives rise to general relativity [2]. It contains Lorentz transformations plus translations and is in fact a semi-direct product of the Lorentz group and the group of translations in spacetime. The Lorentz 1 Tensor representations of the general linear group 4 × 4 matrices GL(4) behave as tensors under the Lorentz subgroup of transformations, but there are no such representations of GL(4) which behave as spinors under the Lorentz (sub)group (see the next section). 2 In simple terms, the tetrad has 16 components, but with four equations of motion, plus four degrees of freedom to be removed due to general coordinate invariance and six due to local Lorentz invariance, this leaves 16 − (4 + 4 + 6) = 2. A.3 About Spinors 213 J K ˆ, ˆ = , , group has six generators: three rotations aˆ and three boosts bˆ, a b 1 2 3 with commutation relations [3–5]: [J , J ]= ε J , [K , K ]=−ε J , [J , K ]= ε K . aˆ bˆ i aˆbˆcˆ cˆ aˆ bˆ i aˆbˆcˆ cˆ aˆ bˆ i aˆbˆcˆ cˆ (A.1) The generators of the translations are usually denoted Pμ, with3 Pμ, Pν = 0 , Ji , P j = iεijkPk , (A.2) [Ji , P0] = 0 , Ki , P j =−iP0δij , [Ki , P0] =−iPi . (A.3) Or alternatively, defining the Lorentz generators Lμν ≡−Lνμ as L0i ≡ Ki and Lij ≡ εijkJk, the full Poincaré algebra reads Pμ, Pν = 0 , (A.4) Lμν, Lρσ =−iηνρ Lμσ + iημρ Lνσ + iηνσ Lμρ − iημσ Lνρ , (A.5) Lμν, Pρ =−iηρμPν + iηρνPμ . (A.6) Let us focus on the mathematical structure of (A.1). The usual tensor (vector) formalism and corresponding representation is quite adequate to deal with most situations of relativistic classical physics, but there are significant advantages in considering a more general exploration, namely from the perspective of the theory of representations of the Lorentz group.
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