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Canonical Quantum Gravity as a Gauge Theory with Constraints Kuan-Jung (John) Lai Department of Physics, Theoretical Physics Group Imperial College London This dissertation is submitted in partial fulfillment of the requirements for the award of the degree of Master of Science September 2016 Acknowledgements The author would like to acknowledge the guidance provided by Professor Joao Magueijo throughout the production of this dissertation. Additional thanks are due to Jake Gordon, Jimmy Kim, and Chang Li, for their input, feedback, and company, but also for their assistance in regards to the “Tokyo Disappointment,” which occurred as the written form of this dissertation was in its initial stages. A final word of gratitude must also be expressed to Hsien-Tsang Lai, without whom none of this would have been possible. Table of contents Notation vii 1 Introduction 1.1 Formulation of a General Relativistic Theory . . . . . . . . . . . . . 1.1.1 Diffeomorphism Invariance . . . . . . . . . . . . . . . . . . . 1.1.2 Background Independence . . . . . . . . . . . . . . . . . . . 2 Gauge Theory 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Ehresmann Connection . . . . . . . . . . . . . . 2.2.1 Local Trivializations of ω . . . . . . . . . . . . 2.3 Horizontal Lifts . . . . . . . . . . . . . . . . . . . . . 2.4 Derivatives on Associated Fibre Bundles . . . . . . . 2.4.1 Example: Frame Bundles . . . . . . . . . . . . 2.5 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 General Relativity on Frame Bundles: Preliminaries 3 Gravity 3.1 Lagrangian Gravitation (I): Einstein-Hilbert 3.2 Hamiltonian Gravitation: ADM formalism . 3.2.1 ADM Formalism (I): Statics . . . . . 3.2.2 ADM Formalism (II): Dynamics . . . 3.2.3 ADM Formalism (III): Constraints . . 4 Quantum Gravity 4.1 Lagrangian Gravitation (II): Frame Fields . 4.1.1 Self-Duality, Complexification . . . 4.2 Ashtekar’s Variables . . . . . . . . . . . . . 4.2.1 Quantum Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 5 8 . . . . . . . . 9 9 11 14 17 20 23 25 28 . . . . . 33 33 35 35 38 46 . . . . 49 50 50 54 59 vi Table of contents 5 Loop Quantum Gravity 5.1 The Base Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Gauge Invariant Space . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Diffeomorphism-Invariant Space . . . . . . . . . . . . . . . . . 63 66 69 71 6 Conclusions 75 References 79 Notation Indices A, B, C, D matrix group indices a, b, c, d abstract indices i, j, k 3D spatial indices I, J, K, L Lie algebra (matrix) indices, or Minkowski indices µ, ν, σ, ρ coordinate indices s, t, u, v Lie algebra basis indices Math [·, ·] algebra commutator d exterior derivative δab Kronecker delta det determinant function δ(x) Dirac delta distribution (3) coordinate 3-form e √ qe volume form ε Levi-Civita symbol η = diag(−1, 1, 1, 1) Minkowski metric gl(n, V ) general linear algebra (real or complex) Notation viii GL(n, V ) general linear group (real or complex) L(G) Lie algebra of G £ Lie derivative ∇ covariant derivative Ω curvature 2-form ω connection 1-form ⊕ direct sum of spaces ⊗ tensor product σi Pauli matrices {·, ·} Poisson brackets Ric Ricci tensor R Ricci scalar R Riemann curvature tensor (see equation 2.31) sl(n, V ) special linear algebra (real or complex) SL(n, V ) special linear group (real or complex) so(n) special orthogonal algebra SO(n) special orthogonal group ⋆ Hodge star su(n) special unitary algebra SU (n) special unitary group TM Tangent space of M T ∗M Cotangent space of M T (p,q) M (p, q)-tensor space of M Tr trace function Notation ∧ ix exterior product Units c=1 speed of light ~=1 Planck’s constant 8πG = 1 Newton’s constant Chapter 1 Introduction There presently exist two very successful theories of nature, from which the entirely of observed physical phenomena is thought to be derived. One of these, the Standard Model of particle physics, lords over the domain of subatomic physics; its subjects are the fundamental constituents of matter: leptons, quarks and force bosons. Far down the other extremity, in the realm of stars and galaxies, the general theory of relativity reigns as chief sovereign, dictating the large-scale distribution of matter and the structure of the cosmos. Both these theories are very well-tested experimentally in the regions where they hold sway. With the additional features of a cosmological constant and neutrino mass, they have withstood all direct challenges posed by experimental and observational tests thus far. But even so, the two theories seem to be in direct conflict conceptually: general relativity produces a dynamic, malleable geometry wherein physical quantities are well-defined at any given instant, while the world-view espoused by quantum mechanics postulates a rigid background spacetime, and contains non-commuting observables that cannot be simultaneously determined to arbitrary precision, even in principle. In the regimes currently accessible to us, the two sets of laws seem to adopt a policy of “peaceful coexistence” where their borders overlap. For instance in the derivation of Hawking radiation, the black hole geometry is treated as a fixed background, and any back-reaction to the gravitational field could be ignored on grounds of being “small”. Similarly, the laws of quantum statistics may determine Introduction 2 the internal composition of dense stars, but their motions through the cosmos, the choreography of the interstellar dance, is given by orbital mechanics. We cannot, however, expect this harmony to continue deep into the very high energy scales, such as in the Planck epoch of cosmology. We will have to resolve this issue sooner or later if we wish to obtain a complete description of all physical phenomena, united under one theory. We will need a theory of gravitation that is compatible with quantum mechanics: a quantum theory of gravity. The approach to this problem that we will present, termed canonical quantum gravity, is distinguished from most other approaches to quantum gravity in that we treat general relativity on the same footing as quantum mechanics− namely, as two separate paradigms of physical theories sharing a common ancestor in Newtonian mechanics. In this context, the gravitational field is not just “one more field” to be quantized under the framework of QFT. Commonly, what the latter means is to write the spacetime metric g in the form g =η+h (1.1) for “small” perturbations h around a simple background metric η, a known solution to the Einstein equations. This expression would then be plugged into the EinsteinHilbert action, and terms polynomial in h and its derivatives would emerge under the integral sign. These are to be interpreted as the Lagrangian of a spin-2 boson, the graviton, with self-interacting terms. However, it is well-known that such methods lead to non-renormalizable theories that diverge badly in the ultraviolet regime [27]. To remedy this, a plethora of additional structure is introduced: compactified extra dimensions, supersymmetric partners, particles as strings... That such treatments are doomed to failure (or at least, plagued with mathematical complexities,) should come as no surprise in our view, like taxonomizing the banana as a tropical breed of apple. This is because such treatments disregard the truly revolutionary character of GR: that the gravitational field is in fact the same entity as spacetime itself ; it is the ocean and everything contained in its depths, and not merely the waves that ripple across its surface. 3 A necessary consequence of treating GR with full respect, then, is the fact that our theory must be non-perturbative. That is, we do not, and indeed cannot, posit any prior geometry η around which we base our theory. This will require a conceptual overhaul of our formalism, but we will be rewarded for our efforts with the following “good” features [20]: 1. four spacetime dimensions, and no need for more, 2. no ultraviolet divergence, 3. no need for supersymmetry, and 4. manifest background independence. One of the consequences of canonical quantum gravity, which we will not have time to cover, is the extraordinary result that spacetime has a granular, or discrete, structure at very short length scales (≈ 10−35 metres). In this way, we resolve the problem of ultraviolet divergences by depriving them of the space to exist, through a cutoff scale imposed by nature itself. The first and third points are also extremely attractive from a reductionist point of view, especially in light of the lack of evidence for supersymmetry in the LHC. Also of note is the fact that our program does not aim directly towards a final theory of “everything.” Unification with the other forces of nature is something left to another day and another theory. We focus solely on the limited objective of quantizing gravity. The purpose of this publication is to present a straight path to the doorstep of canonical quantum gravity, by developing the prerequisite tools along the road in a simple but thorough manner. To keep our narrative focused, we will always set the matter content of spacetime, Tab , and the cosmological constant, Λ, to zero. We will begin with a Hamiltonian formulation of GR, obtained through purely classical considerations, and continue in the tradition of Dirac and postulate that our canonical variables, given by some pair (q, p), promote to quantum operators satisfying a generalization of the identity [ q̂ , p̂ ] = i~ (1.2) Introduction 4 Our first candidate for a choice of q would perhaps most logically be the “spatial part” of the spacetime metric g. With some additional conditions on spacetime, this can always be unambiguously defined. This choice, however, leads to some very formidable non-polynomial terms which impede quantization, forcing us to go down a different path. Using instead the “square root” of the metric, a quartet of (0, 1)-tensors eI such that X ηIJ eIa eJb = gab (1.3) I,J will do the trick for the role of the conjugate momentum p. The role of q will instead be played by a generalised version of the Christoffel symbols, called the spin connection. Thusly, our theory will be formulated most naturally in terms of a gauge theory, a class of field theories that are in some ways generalizations of Maxwell’s electrodynamics. Taking the place of the matter fields in this gauge theory will be the “field of frames” eI , or, a choice of four arrows at each spacetime point representing perpendicular directions. The role of the gauge potential will be played by the spin connection, and the corresponding gauge symmetry will be the equivalence of frames eI that are related by a Lorentz transformation. The precise mathematical formulation of gauge theories already exists, and is described by the theory of principal bundles. This motivates our summary of the theory of principal bundles and Ehresmann connections in chapter 2. Since this is a mathematically terse subject that needs to be covered, we get it out of the way first, so that we may talk freely about physics afterwards without being impeded by considerations of rigour. In our third chapter, we will briefly summarize the (classical) canonical formulation of general relativity given by the ADM formalism, which we will take as the starting point of our incursion into quantum gravity. The ADM formalism presents GR as a totally constrained Hamiltonian system, and leads to the vanishing of “time” in the physical picture. This timelessness is a general feature of canonical formulations of GR, and will pop up at various points, though it will not present any problems to us, at the level of our coverage. We will also briefly discuss the general 1.1 Formulation of a General Relativistic Theory 5 theory of constrained Hamiltonian systems in that chapter, using Yang-Mills theory as an example. After that, we introduce the formalism that allows us to write the GR action in terms of the frame field, as a gauge theory. The canonical coordinates that result from this formulation are called the Ashtekar variables, and they allow us to quantize the theory with a minimal amount of effort. We will also give a brief algebraic account of the Ashtekar variables, particularly how they produce a Lorentz algebra. After that, we will see how to recover the statements of diffeomorphismand gauge-invariance from the classical constraints, as we must. In the fifth chapter, we finally set the stage for a quantum theory of gravity. Using the tools developed thus far, we will construct a manifestly diffeomorphismand gauge-invariant inner product space, which will serve as the space of physcally permissible states in Loop Quantum Gravity. In the presentation that follows, we will assume that the reader has some familiarity with the language of tangent bundles in differential geometry, on about the level of volume one of Spivak’s book, [24]. Apart from that, a level of knowledge of physics at about the level of the QFFF MSc. course at Imperial College London will also be assumed. 1.1 Formulation of a General Relativistic Theory We will begin our voyage with a minor prelude, a qualitative discussion of some features that distinguish GR from most other theories of nature, which we would like to see manifest in our quantum theory. These are diffeomorphism invariance and background independence. 1.1.1 Diffeomorphism Invariance Perhaps the most distinctive feature of general relativity is the property of diffeomorphism invariance. It is also a very easily-misunderstood principle, because it is not well-expressed in purely mathematical language, and is often confused as a statement on coordinate systems. As such, it would seem a good idea to present a Introduction 6 detailed discussion of this symmetry before we begin. Suppose we have a robot, H, who has attained such a mastery of chess that it can play a perfect game on any chessboard we can give it: three-dimensional boards, toroidal boards, whatever. Given a board, H maps a configuration of pieces Ψ0 uniquely to another configuration Ψ1 = H[Ψ0 ], and another, Ψ2 = H ◦ H[Ψ0 ], and so on, ad infinitum, one move at a time. Suppose we give H a chessboard without boundary, Z2 , say, and an initial configuration Ψ0 of chess pieces. Or, we can give it the same board Z2 and the same initial configuration, but with all the pieces moved one square to the right, R[Ψ0 ]. Since H has the perfect strategy, and since the squares of the board have no intrinsic character, H would end up playing the same game. In other words, Hn ◦ R[Ψ0 ] = R ◦ Hn [Ψ0 ] (1.4) We could even warp the board so that there is a smooth kink bending it slightly to the left, or make the squares progressively smaller in one direction, shrinking as 2−n as we go upwards the board. None of these specifics matter to the robot, so long as we tell it how to modify how each of the chess pieces move, say by telling rooks to turn slightly to the right as they pass the kink, or to scale their moves in proportion to the size of the squares. The robot will still play the exact same game, evolving analogously to 1.4. However, if we change the topology of the board, say by removing a square, or by wrapping the board into a cylinder, then the robot will have to revise its strategy, because some previously legal moves will have become illegal, and vice versa. So then we no longer have any relation analogous to 1.4. This is all to say that the specific points of the board are of no intrinsic importance to H’s strategy− only the relative positioning of the pieces and the set of legal moves have any real substance. The structure of the board only enters the strategic considerations by determining, say, whether or not pawns can return to their starting squares (only possible on a toroidal board), or if there are squares inaccessable to a bishop, (there will be unless the edges of the board are “twisted” and identified,) and how far a rook can move in one direction before running into a wall, and so on. 1.1 Formulation of a General Relativistic Theory 7 Fig. 1.1 What’s in a point? That which we call chess, with any other choice of immersion, would still be the same game. In the continuum limit, we upgrade the board to a manifold M , R to a diffeomorphism φ, and the evolution given by H is replaced with the Hamiltonian, the generator of time evolutions. To say that our theory is diffeomorphism-invariant, then, is to say that 1.4 holds in the continuum limit− that the action of diffeomorphisms commutes with time evolution. Meanwhile, the transformation law on vectors v → φ∗ v ⇐⇒ v a → ∂b φa v b (1.5) corresponds to the infinitesimal, pointwise, direction-wise modification to each of the chess pieces’ movements. It must be emphasized that the character of our diffeomorphism φ is distinct from a “mere” coordinate transformation. The diffeomorphism φ : M → M maps points on the manifold M to other points, and corresponds to an active motion on the fields on M . To say that the fields φ∗ ω produce the same physics as ω is the statement that ω is diffeomorphism invariant. On the other hand, a coordinate transformation is a passive diffeomorphic transformation y ◦ x−1 : Rn → Rn , and corresponds to a change in us, changing our description of the same object. That our laws of physics are invariant under coordinate transforms means that there “really is” some field on M , ω ∈ T ∗ M , say, whom the coordinate charts are talking about in a consistent manner. In light of equation 1.5, it is quite easy to see the cause for confusion. After all, the distinction between the two “invariances” would vanish by simply setting (∂y µ/∂xν ) = (∂φa/∂xb ). However, to do so would be a confusion of the roles of the Introduction 8 abstract indices a, b, c, d and the coordinate indices µ, ν, σ, ρ, since (∂y µ/∂xν ) is not a “true” geometric object, and cannot be expressed with abstract indices. It also ignores the distinction of the active/passive roles of φ and y ◦ x−1 . For a more in-depth discussion of the abstract index notation, see section 2.4 of [28]. 1.1.2 Background Independence Background independence is a perhaps less subtle concept, but it is no less important. It is a central feature of our approach to quantum gravity, which distinguishes it from perturbative methods. Consider the Klein-Gordon Lagrangian density in Minkowski space, written in manifestly coordinate-independent notation: L(φ, dφ) = 1 η(dφ, dφ) − m2 φ2 2 (1.6) or the Maxwell equation, written similarly: d⋆ F = 4π ⋆ j (1.7) In these equations, there exists a privileged (0, 2)-tensor η (whose presence is implied in the second equation through the Hodge dual operator). η is privileged in the fact that it is a tensor field that is “exempt” from the selective process of physical evolution− that is, it is not the solution to some field equation. η possesses a place in the equations only by virtue of its “noble birth,” through the existence of a prior geometry. General relativity demonstrates its truly egalitarian character by stripping the Minkowski metric of its special status. It then institutes a field equation, the Einstein field equations, which qualifies general symmetric, Lorentz-signature (0, 2) tensors for the position previously held by η. Henceforth, the tensor field η, on M , cannot enter any physical equation except as a particular solution to the EFEs. This is the property of background independence. This property does not exclusively discriminate against the Minkowski metric, but any prior choice of metric: the anti-de Sitter metric, the Schwarzschild metric, etc. This statement must hold true even in the limit “at infinity.” That is to say that we cannot a priori postulate that the spacetime metric approaches some η at spacelike infinity, or similar. Chapter 2 Gauge Theory We begin by summarizing some fundamental results from the theory of principal bundles in differential geometry. Gauge theories are expressed most naturally in the mathematical arena of principal bundles. But this fact may be occluded by the dry and abstract mathematical definitions involved in the construction of that arena. To remedy this, we will pepper our presentation of the subject with many physically-relevant examples and constructions. The correspondence between the mathematical subjects we cover in this chapter and their physics counterparts is summarized thusly: Connection 1-forms (2.2) Horizontal lifts (2.3) Covariant derivative (2.4) Curvature 2-forms (2.5) → → → → Christoffel symbols, gauge fields Parallel transport Gauge-covariant derivatives Riemann curvature, Field strength tensor Our treatment of the subject will demonstrate a strong preference for brevity over rigour, omitting the mathematical details of uniqueness, existence, etc. Those results may be found in [12, 15, 24], among others. 2.1 Preliminaries and Definitions Definition 2.1.1 Fibre bundles: A differentiable fibre bundle is a collection (E, π, M, F, G) such that: 1. E, the total space is a differentiable manifold, 2. M , the base space is also a differentiable manifold, Gauge Theory 10 3. π, the projection is a surjection π : E → M , 4. F , the typical fibre satisfies π −1 (p) := Fp ∼ = F for each p ∈ M , 5. G, the structure group, which we will always take to be a finite-dimensional Lie group, acts on F on the left, 6. E is locally trivial, that is, there exists an open covering {Ui } of M and an associated set of diffeomorphisms {φi } with φi : Ui × F → π −1 (Ui ), satisfying π ◦ φi (p, f ) = p for each f ∈ F , 7. for Ui ∩ Uj ̸= ∅, we have φj (p, f ) = φi (p, tij (p)f ), where tij := φ−1 i,p ◦ φj,p for φi,p := φi (p, ·) : F → Fp , and {tij (p) |φi , φj } = G. In other words, “changing coordinates” is governed by G. Topologically speaking, we say that E is a trivial bundle when E ∼ =M ×F As an example, the familiar tangent bundle T Rn from multivariate calculus is a (trivial) fibre bundle with base space the manifold Rn , and has as its typical fibre the vector space (Rn , +), and structure group GL(n, R), which defines the familiar change-of-coordinates on tangents, (p, v)x ∼ (p, D(y ◦ x−1 )v))y . Our definition of the tij necessitates, straightforwardly: tii (p) = e tij (p) = t−1 ji (p) tij (p)tjk (p) = tik (p) (2.1a) (p ∈ Ui ∩ Uj ) (2.1b) (p ∈ Ui ∩ Uj ∩ Uk ) (2.1c) Definition 2.1.2 Sections: A section, on Ui ⊂ M , is a smooth σi : Ui → E, such that π ◦ σ = idUi . We shall denote the set of all sections on Ui as Γ(Ui , F ). We define σi locally instead of globally because a smooth global section may not exist. (Since F may not be a vector space, a “zero vector field” may not exist.) We will often call a section a vector field, even if the fibres F are not vector spaces, because the two terms are equivalent on tangent bundles. 2.2 The Ehresmann Connection 11 In what follows, we shall deal exclusively with two special cases of fibre bundles, endowed with some additional structure. One of these cases will have F ∼ = G, and will be what is termed a principal bundle, the natural setting for gauge theories. The other case is defined alongside a (separate) principal bundle, and will have an F endowed with vector space structure, and is known as an associated bundle, which may be used to characterize matter fields in Yang-Mills theories. Definition 2.1.3 Principal bundles: A principal fibre bundle, or just a principal bundle, P (M, G) is a fibre bundle with typical fibre equal to its structure group G, such that G acts freely (that is, without fixed point) on P on the right: (u, g) ∈ P × G is mapped to ug := Rg u ∈ P , with ug = u ⇐⇒ g = e In the context of trivializations, then, this requires that φi (p, gi )a = φi (p, gi a). Often we may call P simply a G-bundle on M . We observe that, with this additional structure, a local section σi : Ui → P very naturally defines a local trivialization on π −1 (Ui ). After all, we can define a map φi : M × G → P with φi (p, g) = Rg ◦ σi (p). In other words, in this trivialization we associate σi (p) with (p, e), and multiply on the right by the parametre gi (u) in φ−1 i (u) = (π(u), gi (u)), to “reach” the other elements on each fibre Gp . This map is smooth because σi is, and is bijective because we postulated that G act freely on P . From this we can readily see that the existence of a single smooth section defined everywhere on M implies that P ∼ = M × G is trivial. Conversely, if P is trivial, then any global trivialization defines a global section. 2.2 The Ehresmann Connection Recall that for any differentiable manifold M , we defined Tp M , the set of tangent vectors at p using an equivalence class of smooth curves γ : [−τ, τ ] → M satisfying γ(0) = p, and then we stitched the resulting fibres together with local triviality and structure group axioms, analogous to axioms 6 and 7 in definition 2.1.1. The principal bundle P (M, G) is also a differentiable manifold, and we can consider the bundle T P (M, G), a tangent bundle defined on top of a principal bundle. 12 Gauge Theory Because of the quite abstract definition, it would perhaps be instructive to picture the principal bundle P as a field of seeding dandelions: in this picture, the field (i.e., “the ground”) represents M , and we take the (spherical) head of each dandelion to be isomorphic to G, the individual spores in each head representing group elements. Consider a little man whose physical dimensions are comparable to the dandelion spores, living on top of one of the spores. The man may slide down the stalk of his individual dandelion to reach an unambiguously defined point on the ground (the projection π). The man may also walk from his spore to any other spore on the same dandelion by reading off an address (right action of G on Gp ), or he may walk from his spore to a spore on some other dandelion, in some unprescribed way, which corresponds to a general path in P . Up to an equivalence, the set of all paths that the man may traverse, then, characterizes T P . Fig. 2.1 The action of a vertical vector field is associated with movement along a single dandelion head, while a horizontal movement moves across the field. Roughly speaking, the fibres of T P will contain vectors that are in the Lie algebra of G, L(G). These are simply the tangents of a path that stays within a single fibre/dandelion. However, T P also contains vectors from T M , which result when we walk between fibres, “horizontally” with respect to the plane of “the ground.” Intuitively, then, we may well think that T P ∼ T M ⊕ L(G), a direct sum of the two 2.2 The Ehresmann Connection 13 vector spaces. In what follows, we shall elaborate on this relation. Before we may begin, we must put into precise terms the L(G) structure on T P : For each u ∈ P we can define a vector field isomorphism # : L(G) → Γ(P, T P ) with d # (2.2) A = Rexp(tA)∗ dt t=0 In other words, for differentiable f : P → R d A# f (u) = f (u exp(tA)) dt t=0 (2.3) The A# so defined is called the fundamental vector field generated by A. This isomorphism can also be taken pointwise to parametrize the set of vectors tangent to Gp at each u, which we shall term to be the vertical subspace of Tu P , Vu P . Note that A# does not trivialize T P , because the latter is not a principal bundle. Definition 2.2.1 An (Ehrenfest) connection Γ on a principal bundle is then a choice of a horizontal subspace of Tu P , labeled Hu P , that depends smoothly on u, satisfying: 1. Tu P = Vu P ⊕ Hu P 2. Every smooth vector field X may be decomposed into smooth vector fields X = XV + XH such that XV is vertical everywhere and XH is horizontal everywhere. 3. Ra∗ Hu P = Hua P Equivalently, and perhaps more commonly (in physics applications,) we see an alternative definition of a connection with an L(G)-valued 1-form, ω, called the connection 1-form. Definition 2.2.2 A linear ω : T P → L(G) is a connection 1-form if it satisfies: 1. ω(A# ) = A 2. Rg∗ ω = Adg−1 ω := g −1 ωg Because Rg∗ A# = (adg− 1 A)# , the third axiom in the first definition and the second axiom in the second are required for logical consistency. Intuitively we may think of ω as dim(L(G))-many 1-forms on T P which may be expanded in a basis on L(G): Gauge Theory 14 ω = gi ω i , so that ω may still have a “direction” after contraction with a vector. In this definition, Hu P is defined pointwise in u as the subspace of Tu P that annihilates ω. i.e., Hu P = {X ∈ Tu P | ωu (X) = 0} (2.4) so that ω is then a projection from Tu P to Vu P ∼ = L(G). Assuming X H ∈ Hu P, from the second definition we have Rg∗ ωug (X H ) = g −1 ωu (X H )g = 0 (2.5) so that ω defines horizontal subspaces that are invariant under the right action, as in the first definition. One may be misled to think that the definition of Vu P , uniquely specified by the definition of P , also uniquely specifies Hu P . After all, we could simply call Hu P the set of tangents that are orthogonal to Vu P . This would be putting the cart before the horse, because it is the connection itself that prescribes this orthogonality to begin with. For orthogonality is not defined without a metric, which, as we will see later on, is in some sense prescribed “up to a gauge” by ω. Without a connection, all we have to work with is linear independence to choose our basis vectors, which leaves some freedom of choice for the horizontal basis. Returning to our dandelion analogy: right translation (or, the finite movement generated by vertical vectors) provided our little man an unambiguous way of moving between spores on a single dandelion. Horizontal movement is not well defined a priori, but we shall see that the definition of a connection resolves this. We may then think of a choice of connection in analogy to the “bending” of the stalks of each dandelion on the field in some direction, as though in the wind. If our little man wishes to traverse the field purely horizontally with respect to the ground, then his path must take into consideration the choice of connection/the swaying of the stalks. 2.2.1 Local Trivializations of ω Here we shall illustrate how the connection 1-form relates to gauge theories. By the local triviality of P , we may consider a local trivialization on Ui ⊂ M with σi (p). 2.2 The Ehresmann Connection 15 In the context of gauge theories, the gauge field Ai is then defined as Ai = σi∗ ω ∈ L(G) ⊗ T ∗ Ui (2.6) i Let us make the connection with Yang-Mills theory. In SU (2) theory, Aµ = Aiµ σ2i is an su(2)-valued 1-form on R4 . In entry-level QFT, we (implicitly) assumed that the principal bundle was trivial, so that a single trivialization sufficed to define the gauge field on all of spacetime. We also postulated that Aµ transform as: Aµ (p) → U (p)(Aµ (p) + ∂µ )U (p)−1 (2.7) where U : R4 → SU (2) is called a gauge transformation. This transformation is a passive one, and is the result of changing between trivializations on P . In our generalized theory, we seek to reproduce 2.7. Indeed, a series of straightforward calculations show that on Ui ∩ Uj ̸= ∅, we must have, Aj (p) = t−1 ij (p) (Ai (p) + d) tij (p) (2.8) where, as before, the tij (p) ∈ G are transition functions. As a matter of practicality, though, ω is almost always defined in the direction opposite to our presentation: by stitching together 1-forms {Ai } on open neighbourhoods {Ui }, and then defining their relation to each other via equation 2.8 on the overlap. We can thusly construct non-trivial bundle structure in a straightforward way, by “twisting” the gauge forms “by hand” on the overlap of the charts. An example of this can be seen in 3-dimensional electrostatics, in the case of the Dirac monopole: there we could define, in polar coordinates AN (x) = ig(1 − cos θ)dφ θ ∈ UN := [0, π/2 + ε] (2.9a) AS (x) = −ig(1 + cos θ)dφ θ ∈ US := [π/2 − ε, π] (2.9b) where g ∈ R, and ε > 0 so that the charts overlap, and on the overlap: AN = AS + d(2gφ) (2.10) Gauge Theory 16 so that tN S = exp(2igφ) in equation 2.8. Furthermore, periodicity of the φ coordinate requires 2g ∈ Z. The punchline of this construction is revealed by defining B = dA to be the magnetic field, and integrating B over a sphere of radius r0 , Z S2 B= Z θ>π/2 = = Z dAN + AN − S1 Z 2π Z S1 Z θ<π/2 dAS As (2.11) d(2gφ) 0 = 4πg The first line is true because for small ε, the overlap UN ∩ US is of arbitrarily small size. The second line uses Stokes’ theorem, and the minus sign occurs because of the choice of orientation. But this implies that1 Z r<r0 dB = Z S2 B (2.12) = 4πg which is independent of r0 , so that clearly divB = 4πgδ (3) (x), and the configuration of fields corresponds to a magnetic charge. This rather straightforward example illustrates the wealth of physical scenarios that may be described elegantly in the language of principal bundles. Inverting the definition, we may obtain an ω ∈ L(G) ⊗ T ∗ P , working “from the bottom up," by stitiching together (as we did just now) an appropriate collection of Ai on open patches Ui , and using the associated trivializations {Ui , σi , gi } to define: ω|Ui = gi−1 π ∗ Ai gi + g −1 dgi (2.13) this satisfies equation 2.6, because by definition gi ◦ σi = e identically on Ui , and σ ∗ π ∗ = (π ◦ σ)∗ is the identity map on T ∗ P . 1 B is a 2-form, so divB = dB 2.3 Horizontal Lifts 2.3 17 Horizontal Lifts Our current situation is quite similar to that in introductory general relativity, where we did not have any prior means of comparing the vector spaces Tp M and Tq M for p ̸= q. In the presence of a metric, though, we defined a transportation of a vector v ∈ Tp M → Tq M that depended on a choice of path γ : [−τ, τ ] → M and the metric gab , through the Levi-Civita connection. In other words, v ∈ Tp M would be transported to Xq , the value at q of a vector field X satisfying ∇γ̇ X = 0, and Xp = v. In a coordinate system: d (X µ ◦ γ) + γ̇ σ (Γµ νσ X ν ) ◦ γ = 0 dt (2.14) This is the parallel transport equation. By the fundamental theorem of ODEs, this equation always has a unique, suitably differentiable solution. We would like to import this technique into the theory of principal bundles, to compare fibres Gp and Gq . This shall be done in the following: Definition 2.3.1 Horizontal lift: Let P (M, G) be a principal fibre bundle over M with group G. Consider a continuously differentiable curve γ : [0, 1] → M with γ(0) = p. Then a curve γ̃ : [0, 1] → P with π ◦ γ̃ = γ is called a horizontal lift of γ if its tangent γ̃∗ dtd is horizontal everywhere. In other words, for a given curve γ, our dandelion-man γ̃ walks along the “shortest possible path” in P (that is, without any extraneous vertical motion). His “shadow,” projected on M traces the curve γ, with the same velocity as γ. The horizontal lift defines a unique transport between the fibres of P , Gp and Gq : Theorem 2.3.2 Given a γ : [0, 1] → M with γ(0) = π(u) there exists a unique horizontal lift of γ such that γ̃(0) = u. To prove this, let us work on a coordinate patch (Ui , σi , gi ). It would be no loss to write γ̃(t) = vit ait , where vit := σi (γ(t)) is a curve with vi0 = u and ait is a curve in the structure group G with ai0 = e. Then we write d γ̃(t) = v̇it ait + vit ȧit dt by the Leibniz rule on T P , and q̇(t) denotes the tangent of q at t. (2.15) Gauge Theory 18 (For convenience, and since we are working exclusively with matrix groups, we may as well take G to be its fundamental representation on GL(n, C) and define all the relevant products there. In general, however, there is no need for such heavy-handed methods, and the proof works for finite-dimensional Lie groups that are not matrix groups, too. [12]) We then impose the horizontal condition: 0 = ω(v̇it ) = ω(Rait ∗ v̇it ) + ω(vit ȧit ) d ait = ada−1 ω(v̇it ) + a−1 it it dt ! d d ∗ + a−1 = ada−1 σi ω γ∗ ait it it dt t=0 dt d = Ai (X)ait + ait dt (2.16) The first line follows by the definition of the connection, the second by linearity, the third by the definition of the connection again, and the fact that vit ȧit is actually # (a−1 it ȧit ) at vit . The fourth line is true because vit = σi ◦ γ(t), and in the fifth line we defined X = γ∗ dtd , the tangent field to γ, and right-multiplied the whole expression by ait . With our embedding into GL(n, C), the result is a linear system of n2 coupled first-order ODEs in ait , which has a unique and suitably differentiable solution. In fact, we can write an expression for the solution right away: ( ait = ai0 I − ∞ Yn X ˜ n=1 Z ti−1 i=1 dti 0 ) Yn ˜ i=1 (Ai (X(ti ))) (2.17) QN where t0 := t, and ˜ i=k (Ti ) is a glyph that represents an instruction to write down Tk Tk+1 · · · TN , or just 1, if k > N . This is not necessarily a multiplicative product, but is purely formal. This is reminiscent of Dyson’s formula for scattering in QFT [18]: ( U (t, T ) = I − ∞ Yn X ˜ n=1 i=1 Z ti−1 T dti Yn ˜ i=1 ) (iHI (ti )) (2.18a) 2.3 Horizontal Lifts 19 which is written more compactly as U (t, T ) = T exp −i Z t T dτ HI (τ ) (2.18b) The exact same combinatorial argument allows us to write ai1 = ai0 P exp − = P exp − Z γ Z Ai [0,1] Ai d γ∗ dt !! (2.19) where P is the path-ordering symbol, a glyph that indicates that products in the power-series expression are to be path-ordered, in analogy to the time-order with the path parametre replacing the time. So, finally, the horizontal lift of γ is2 γ̃(t) = σi (γ(t)) P exp − Z [0,t] Ai d γ∗ dt !! (2.20) This expression extends straightforwardly to cases where γ extends outside of Ui : simply transform according to equation 2.8 on the overlap region, and apply equation 2.20 again in the new coordinate patch. For any u′ = ug in the same fibre as u, γ̃(t)g is a horizontal lift of γ that starts at u′ . And thus the theorem is proved. Equation 2.20 should be independent of a choice of trivialization, because γ̃ ⊂ T P is− but it depends quite clearly on a choice of trivialization σi ; we will see how this works out later, when we have developed a suitable vocabulary. One might think that we could have written the parallel transport on (psuedo-) Riemannian manifolds as a curve γ̃ in T M such that π ◦ γ̃ = γ. Indeed, as we shall see later, the parallel transport may indeed be formulated this way, but γ̃ will be a curve on a GL(n, R)-bundle on M , rather than in T M itself. When γ is a loop, so that γ(1) = γ(0) := p, we generally do not have γ̃(1) = γ̃(0), so that γ̃ maps Gp to Gp via right multiplication by an element in G. This is commonly called the holonomy group at p, a subgroup of G, and its elements are 2 This is well defined because the path-ordered exponential converges: see page 235 of [3] for a simple proof. Gauge Theory 20 often written U (ω, γ) = P exp − I γ Ai (2.21) We may then define a “group” of loops, then, with multiplication γ2 ◦ γ1 defined by traversing γ1 first, then γ2 and γ −1 to be the loop γ traversed in the opposite direction. The direction of γ matters, but not its parametrization. In that case, U (ω, γ2 ◦ γ1 ) = U (ω, γ2 )U (ω, γ1 ) (2.22a) U (ω, γ −1 ◦ γ) = e (2.22b) the first identity can be seen by first transporting along γ1 , then γ2 , which is the same as transporting along γ2 ◦ γ1 . The second identity is true because reversing the direction of γ reverses the sign on the integral, and eA e−A = id. But this is not truly a group, though, because γ −1 ◦ γ is not the identity curve. Generally, the word holonomy is used even for cases where γ is not a loop. We shall be liberal with our terminology, and adopt this usage when it is convenient. 2.4 Derivatives on Associated Fibre Bundles In classical Yang-Mills theory, the gauge fields A began as book-keeping devices to preserve the form of the physical equations, as an answer to the postulate of invariance under local gauge transformations. The matter fields were vectors fields whose vectors belonged to a vector space of a certain sort, which had a left action of local rotations by the so-called gauge group. In SU (2) Yang-Mills theory, this vector space was C2 , on which SU (2) acted through its fundamental representation. We defined a covariant derivative on these matter fields ψ in a rather ad-hoc way, by insisting that the action of the covariant derivative operator D transform as Dψ → U Dψ under local gauge transformations U . The gauge field A was then tacked on to the definition of D, and postulated to transform according to equation 2.7 in order to cancel the inhomogeneous term in the local transform. Thus we obtained D = d + A. We now present an alternative but ultimately equivalent definition of the gauge derivative− one that in fact generalizes to the covariant derivative from classical GR. 2.4 Derivatives on Associated Fibre Bundles 21 Definition 2.4.1 Associated fibre bundle: Let P (M, G) be a principal bundle and F a manifold (in particular a vector space, for us) on which G acts on the left. Consider the product manifold P × F , with an equivalence relation (u, ξ) ∼ (ua−1 , aξ). The quotient space of P × F by this equivalency , E(M, F, G, P ), is called the fibre bundle associated with P . Sometimes we will also write E = P ×G F . As a fibre bundle, E has base space M , typical fibre F , which we will always take to be endowed with a vector space structure, and structure group G. It inherits a projection πE from P . To see this, consider a trivialization on P, Ui , so that the product manifold has the local equivalence(p, g, ξ) ∼ (p, ga−1 , aξ), so that the intermediary G on the trivialization becomes extraneous, and πE ((p, g, ξ)) has πE−1 (Ui ) ∼ = Ui × F . Immediately from this definition, we can define a map from P × F to E, with the right product given by uξ = [(u, ξ)], the right-hand side being the equivalence class on E. This product is well defined because u(aξ) = (ua)ξ for all a ∈ G. The elaborate and rather opaque construction above was to formulate the idea of a parallel transport unambiguously. Definition 2.4.2 Parallel transport: Consider a curve γ on M . We say that a section of E, s, is parallel transported along γ if s(t) has a representative (γ̃(t), ξ(t)) for which ξ is constant, and γ̃ is any horizontal lift of γ. We showed earlier that horizontal lifts of the same curve differ by right multiplication by some constant a ∈ G, so that the specific choice of lift does not matter. i.e., for γ̃ ′ = γ̃(t)a, we have (γ̃(t)a, ξ(t)) ∼ (γ̃(t), aξ(t)), and aξ(t) is constant if ξ(γ(t)) is. In other words, ξ(t) is parallel transported if its “relative angle” to a γ̃ remains constant, i.e., ξ rotates with the holonomy along γ. Definition 2.4.3 Covariant derivative: Consider any curve γ : [−τ, τ ] → M with tangent X and p = γ(0). Let s be a section of E defined on some neighbourhood of p. We can write s|γ with representative s(γ(t)) = (γ̃(t), η(γ(t))), where γ̃ is any horizontal lift of γ. Then s0 (t) = [(γ̃(t), η(p))] is a parallel transported section along γ. Then the (gauge) covariant derivative is defined as DX s = lim [s(γ(h)) − s0 (h)] /h h→0 " = d γ̃(0), η(t) dt t=0 !# (2.23) Gauge Theory 22 where addition is inherited from F , acting on the second argument. In other words, we take nearby values of s and parallel transport it back to p with γ to compare the two vector spaces, and then take their first-order difference. This definition does not depend on the specific choice of γ nor its horizontal lift. The first claim is fairly obvious, and the second is evident from the equivalence class. The covariant derivative D : T M × Γ(M, E) → Γ(M, E) is “actually” a derivative operator. That is, it is linear in its first and second arguments, and obeys a Leibniz rule when the second argument is multiplied by a non-constant function. Thus we could instead choose to interpret the derivative operator as a D : Γ(M, E) → T ∗ M × Γ(M, E). As far as practical calculations are concerned, the covariant derivative is far more conveniently expressed in local coordinates. To this end, Let us choose a basis {ea } of F , the typical fibre of E and a corresponding basis {ea } of F ∗ so that ea (eb ) = δ a b , a local trivialization σi , and coordinate chart {xµ } on Ui . With respect to this trivialization we can unambiguously write an explicit representative s = (σi , ξi ) = (σ, ξia ea ). Without proof, " DX s|p = σi (p), d a ξ ◦ γ + X µ Aiµ a b ξib dt i ! !# ea (2.24a) t=0 The L(G) coefficient of the connection is interpreted as a linear Aiµ : V → V , defined by the representation of L(G) on V , so this map is defined unambiguously as an endomorphism on V . Oftentimes we will forgo the cumbersome notational structure on E, and write simply: DX ξi = X µ ∂µ ξia + Aiµ a b ξib ea (2.24b) so that DX ea = X µ Aiµ b a eb (2.24c) where the local trivialization σi is implicitly understood. And so finally, we can write in abstract index notation, (Dµ ξi )a = ∂µ ξia + Aiµ a b ξib (2.24d) 2.4 Derivatives on Associated Fibre Bundles 23 The same expression is true in a different trivialization, given by σj = σi tij : DX (tij ξj ) = X µ tij ∂µ ξja + Aiµ a b ξjb ea (2.25) = tij DX ξj and (σi , DX ξi ) = (σi , tij DX ξj ) ∼ (σj , DX ξj ). We may now make explicit the coordinate-independence of equation 2.20. Writing γ̃(t) = σi (γ(t))ait with the same specifications as in equation 2.20, and a section σj := σi tij : (γ̃(t), ξ) ∼ (σi (t)ait , ξi ) = (σj (t)t−1 ij (t)ait , tij (0)ξj ) (2.26) ∼ (σj (t)t−1 ij (t)ait tij (0), ξj ) ! = (σj (t)ajt , ξj ) So aj0 P exp − Z [0,t] Aj d γ∗ dt !! = t−1 ij (t)ai0 P exp − Z [0,t] Ai d γ∗ dt !! tij (0) (2.27) As a demonstration of this mathematical machinery in action, we immediately apply it to an example, defining along the way some crucial concepts we will use later on in our quest for a quantum theory of gravity. 2.4.1 Example: Frame Bundles Let M be some n-dimensional differentiable manifold, and T M its tangent bundle. At every point p ∈ M , we can choose a basis of n linearly independent vectors {Xi }ni=1 ⊂ Tp M . Such a u = {Xi }ni=1 is called a linear frame at p. Denote the set of all [choices of linear frames at all points in M ] as F (M ). We will construct a principal fibre bundle out of this manifold: Definition 2.4.4 Frame bundles: Consider a coordinate system {xµ } on an open patch Ui ⊂ M , which naturally gives a local section of F (M ), σi = {∂/∂xµ }. We P may expand the vectors in any linear frame on Ui as Xi |p = µ Xiµ |p (∂/∂xµ ). Clearly then Xiµ ∈ GL(n, R) parametrizes F (M )p for each p, so that the fibres of F (M ) are Gauge Theory 24 diffeomorphic to GL(n, R). Changing coordinates to {y µ } shows that the structure group is GL(n, R). Finally, we endow F (M ) with a GL(n, R) action from the right, ua = { i ai j Xi }nj=1 . π The result is that F (M ) → M is a principal GL(n, R)-bundle, which we call the frame bundle over M . P A linear frame u at p ∈ M has a natural left action on vectors i v i ei ∈ Rn , given by P P u( i v i ei ) = i v i Xi |p ∈ Tp M , where ei ∈ Rn are basis vectors. We may then consider the tangent bundle T M as an Rn -bundle associated with F (M )(M, GL(n, R)). In other words, T M ∼ = F (M )×GL(n,R) Rn with the isomorphism [(u, ξ)] → uξ defined earlier. P By extension, we can also use F (M ) to contextualize the tensor bundles T (p,q) M of (p, q)-tensor fields, with linear combinations of the action on basis elements: a ...a ueb11...bqp → Xa∗1 · · · Xa∗r Xb1 · · · Xbs ⊂ Tp(r,s) M .3 With these isomorphisms, then, a local section u|Ui of F (M ) may instead be defined as a linear u : Rn → Γ(Ui , T Ui ). We will often make use of this more versatile notation over the formal definition. These elaborate definitions allow us to recover some already-familiar results from elementary differential geometry: consider a gl(n, R)-valued 1-form on M = Rn , Γ, which defines the connection on F (M )(M, GL(n, R)). The covariant derivative of a vector field v ∈ Γ(M, T M ) can be written in a coordinate system with equation 2.24d: (Dµ v)ν = ∂µ v ν + Γµ ν σ v σ (2.28) And from this, the representation of L(G) on tensor products of T M gives the correct expression for the covariant derivative of tensor fields of arbitrary valence. Furthermore, by equation 2.8 we straightforwardly find the transformation of the ′ ′ ′ connection under coordinate change {xµ } → {y µ }, so that txy µ µ = ∂y µ /∂xµ : Γµ ν σ 3 ∂xν → ν′ ∂y ′ Γµ′ ν′ ′ ∂y σ ∂ ∂y σ + σ′ ∂xσ ∂xµ ∂xσ ! ∂xµ ∂y µ′ (2.29) Of course, this is just the action of the representation of the gl(n, R) algebra on the tensor-ified r−2 s−2 space T M ⊗ · · · ⊗ T M ⊗ T ∗ M ⊗ · · · ⊗ T ∗ M . 2.5 Curvature 25 The 1-form index µ plays a very distinct role from that of the gl(n, R) indices ν, σ so we now see see that Γ only resembles a tensor superficially. Note, though, that we have not imposed either the metric-compatibility nor the torsion-free condition, and so that the above conditions hold for general connections Γ. 2.5 Curvature Consider Maxwell theory on Rn . Here, the gauge group is abelian U (1): the group elements all commute, and Ai is a general imaginary-valued 1-form. We can then rewrite equation 2.21 with Stokes’ equation: I Z U (ω, γ) = P exp − γ = P exp − Γ Ai Fi (2.30) where Γ is any 2-surface with boundary γ, and Fi is the Maxwell tensor dAi . If Ai is a gradient (pure gauge) along γ, then Fi vanishes identically, and the holonomy group becomes trivial. In the section above, we briefly described parallel transport on (psuedo-) Riemannian manifolds in terms of a connection on F (M )(M, GL(n, R)). In that context, it should be obvious that parallel transport on a (psuedo-) Riemannian manifold is trivial if and only if the curvature is uniformly zero, i.e., the metric is flat. Indeed, the definition of the Riemann curvature tensor [15, 28] (∇X ∇Y − ∇Y ∇X ) Z − ∇[X,Y ] Z = R(X, Y )Z (2.31a) or, in abstract index notation, Rabc d ωd = (∇a ∇b − ∇b ∇a ) ωc (2.31b) is exactly the measure of the change in fields when parallel transported along a small loop. Clearly, then, the Maxwell tensor plays a role similar to the curvature tensor in U (1) gauge theory. In what follows, we will strive to make this similarity more precise. Gauge Theory 26 Definition 2.5.1 Covariant exterior derivative: Consider an L(G)-valued r-form η on a principal bundle P (M, G), η ∈ L(G) ⊗ Ωr (P ). We define the covariant exterior derivative D as follows: Dη(X1 , · · · , Xr+1 ) = dη(hX1 , · · · , hXr+1 ) (2.32) where d is the exterior derivative on P , hX is the horizontal component of the vector field X, and dη := gi dη i However, the covariant exterior derivative is not “actually” a covariant derivative, because it generally does not satisfy D2 η = 0 in general. Definition 2.5.2 Curvature 2-form: Let ω be a connection 1-form on a principal bundle P (M, G). The curvature 2-form Ω ∈ L(G) ⊗ Ω2 (P ) is defined by Ω = Dω (2.33) From the definition of the covariant exterior derivative, and the definition of ω (specifically, the right-invariance of Hu P and the right action on ω, Rg∗ ω = g −1 ωg) we have: Rg∗ Ω = g −1 Ωg (2.34) Borrowing the Lie algebra structure, we can define a commutator between an L(G)-valued p-form η = η s τs and L(G)-valued q-forms ζ = ζ s τs : [ η , ζ ] = [ τs , τ t ] η s ∧ ζ t (2.35) where [·, ·] is the Lie bracket on L(G), ∧ is the exterior product on P , and τ s is a basis on L(G). When a product can be defined between elements of L(G), (e.g., via the fundamental matrix representation,) we may just as well define an exterior product: η ∧ ζ := η s ∧ ζ t τs τt (2.36) or, (η ∧ ζ)I K = η I J ∧ ζ J K (2.37) where I, J are L(G) indices, is the exterior product of a “matrix of forms”. Note that [ η , η ](X, Y ) = 2[ η(X), η(Y ) ] 2.5 Curvature 27 The curvature 2-form satisfies one of the two Cartan structure equations, Ω = dω + ω ∧ ω (2.38) which can be proven by decomposing its input vectors X1 , X2 into horizontal and vertical components, then using linearity and skew-symmetry to reduce into the proof into three distinct cases. [15] Now consider a local trivialization σi . The curvature in this coordinate system is Fi : = σi∗ Ω = dAi + Ai ∧ Ai (2.39a) Then by introducing a coordinate system on M , we can write Aiµν = ∂µ Aiν − ∂ν Aiµ + [ Aiµ , Aiν ] (2.39b) which is just the familiar Yang-Mills field strength tensor when G is a gauge group. Switching to the trivialization σj = σi tij , it is quite straightforward to show that Fj = t−1 ij Fi tij (2.40) from which we may recover the action of the field strength tensor under gauge transformations. In fact, it can be shown that any L(G)-valued p-form φ satisfies Dφ = dφ + [φ, ω] (2.41) For a proof of this, see page 79 of [12], and proceed by induction. When P (M, G) is the frame bundle F (M ), then writing the connection 1-form as Γµ I J with the gl(n, R)-indices I, J written out explicitly as before, we obtain: (Rµν )I J = ∂µ Γν I J − ∂ν Γµ I J + Γµ I K Γν K J − Γν I K Γν K J (2.42) which is extremely similar to the familiar Riemann curvature tensor, but for the fact that the two objects live in different spaces: Rabc d is a (1, 3)-tensor field over M , and RI J is a gl(n)-valued 2-form− but see the next section. This expression does in fact transform like a tensor under coordinate change, because equation 2.40 gives the correct GL(n) transformation. Gauge Theory 28 We conclude this section by stating the Bianchi identity: 0 = dΩ − d2 ω − dω ∧ ω + ω ∧ dω = dΩ + (ω ∧ Ω) − (Ω ∧ ω) (2.43) = DΩ The first equation is from taking the (normal) exterior derivative of Ω, and the third follows from the second because ω annihilates horizontal vectors. It can be shown that on a frame bundle F (M ), this expression translates to the familiar Bianchi identity on (psuedo-) Riemannian manifolds, ∇[e Rab]c d = 0 2.6 GR on Frame Bundles: Preliminaries Let (M, g) be an n-dimensional Lorentzian manifold. We present here the formalism of general relativity in a “non-coordinate basis”: on some open cover {Ui } of M , we may diagonalize the metric locally on each Ui with vector fields {eI }Ui so that g(eI , eJ )|Ui = ηIJ (2.44a) Where η = diag(−1, 1, 1, · · · ). Alternatively, we could have begun by defining {θI }Ui the basis dual to {eI }Ui so that g|Ui = ηIJ θI ⊗ θJ (2.44b) which is equivalent to equation 2.44a. In a coordinate basis on M , we can expand eI = eµI ∂µ , and θI = eIµ dxµ , where the eIµ , eµI are matrix inverses of each other, eIµ eµJ = δ I J , because the bases are defined as duals of each other, θI (eJ ) = δJI . Equations 2.44a, 2.44b are equivalent to a condition on {eIµ }: eIµ eJν ηIJ = gµν (2.45) which determines {eIµ } up to an SO(n−1, 1)-rotation ΛI J satisfying ΛI K ηIJ ΛJ L = ηKL (2.46) 2.6 General Relativity on Frame Bundles: Preliminaries 29 such a frame is called a vierbien, or tetrad (resp. German and Greek) in four dimensions, or vielbein in n-dimensions. In the language of principal bundles formulated in this chapter, {eI }Ui is a local section of the frame bundle F (M )(M, GL(n, R)) that in particular solves equation 2.44a. When g is a solution to the Einstein equations, we may very well confer upon the vielbein {eI } and its dual {θI } the distinguished title of gravitational field, instead of g. Then instead of formulating general relativity as a theory about the evolution of the metric g, we could cast {θI } in the starring role of a gravitational gauge theory of frames. (In what follows, we will drop the explicit reference to coordinate charts {Ui }.) Co-starring in this re-formalism is the connection (gl(n)-valued) 1-form ω on F (M ), which features in the other Cartan’s structure equation: dθI + ω I J ∧ θJ = ΘI (2.47a) (which can be taken to be the definition of the torsion 2-form ΘI .) In particular, ω is said to be torsion free if it solves ΘI = 0. In a coordinate basis, a torsion-free connection solves the equation ∂[µ eIν] + eJ[ν ωµ] I J = 0 (2.47b) Taking the exterior derivative of equation 2.47a and using 2.38 gives the other Bianchi identity: dΘI + ω I K ∧ ΘK = ΩI K ∧ θK (2.48) The operator d is the exterior derivative on the frame bundle, so the above equations are statements on the linear frame defined by {θI }, and not the dual vector fields θI individually. In the absence of torsion, it can be shown that the Bianchi identity reduces to the familiar symmetry property of the Riemann tensor: R[abc] d = 0. ([24], p.288) The metric-compatibility condition manifests in the frame bundle formalism as ηIK ω K J = −ηJK ω K I ωIJ = −ωJI (2.49) which is to say that the values of ω are in particular restricted to the so(n − 1, 1) subspace of gl(n), reflecting the symmetry imposed by equation 2.45. It should be Gauge Theory 30 clear that the metric-compatibility and torsion free conditions really do correspond to the familiar conditions ∇X (ηIJ θI θJ ) = 0 and [∇X , ∇Y ]f = 0 respectively, where ∇ is the covariant derivative operator associated with the connection ω on the associated bundle T (r,s) M . The unique ([24], p. 237) metric-compatible, torsionfree connection is called the Levi-Civita connection. The connection 1-form ω associated with it is often called the spin connection. In a coordinate basis, the spin connection is related to the Christoffel symbols, in keeping with 2.13, ωµ I J = eIν Γµ ν σ eσJ + eIν ∂µ eJσ (2.50) or we can solve for its components using metric compatibility and vanishing of torsion: 1 ν σ ν ωµIJ = eI ∂[µ eν]J + eσJ ∂[σ eµ]I − eK e e ∂ e (2.51) [σ ν]K µ I J 2 We can also explicitly write out the natural isomorphisms on the F (M )-associated bundles that we defined in section 2.4.1, in terms of the frame components {eµI } and its inverse {eIµ }. e.g., for vector fields, v I → v I eI (2.52a) v µ → eIµ v µ (2.52b) ωI → ωI eI (2.52c) ωµ → eµI ωµ (2.52d) and for dual fields, too: so we may write the frame field and its dual with abstract indices from now on, eaI , eIa , as “transformations between indices.” They change abstract indices on T M and T ∗ M , {a, b, c · · · }, to so(n−1, 1) (Minkowski/internal) indicies on Rn , {I, J, K · · · }, and vice versa. This generalizes to tensors of arbitrary valence. e.g., Ωab I J edI eJc = Rabc d (2.53) so the Riemann tensor is “actually” the curvature, after all.Finally, we end this section with the Ricci tensor and scalar written in terms of {eµI }: RicIJ = Ωµν K I eµK eνJ (2.54a) 2.6 General Relativity on Frame Bundles: Preliminaries R = Ωµν IJ eµI eνJ 31 (2.54b) which will come in handy later, when we resolve to express GR fully in terms of frame bundles, in chapter 4. Chapter 3 Gravity Now that we have properly settled the matter of mathematical prerequisites, it is now time to start talking about the main focus of this publication: a canonical formulation of quantum gravity. By the adjective “canonical”, we mean “pertaining to Hamiltonian mechanics,” the machinery underlying time evolution. We can obtain a Hamiltonian function describing gravitational fields by performing a Legendre transform on a suitable Lagrangian function, provided to us by the Einstein-Hilbert action. We will briefly review this in section 3.1. The Hamiltonian function thus obtained is called the ADM Hamiltonian, which we will discuss in section 3.2. It describes the evolution of a field qab , the spatial 3-metric, with respect to a fiducial time parametre τ . We will set the stage for this Hamiltonian formalism by defining a time function, etc. Then we will examine the evolution of qab in time. The Hamiltonian function thus obtained turns out to be describe a totally constrained system, motivating a summary of the general treatment of such systems in the First Interlude. Finally, we conclude the chapter with a discussion of the constraints in the ADM Hamiltonian. 3.1 Lagrangian Gravitation (I): Einstein-Hilbert We begin this chapter by presenting the already-familiar Einstein-Hilbert action. Let (M, g) be an n-dimensional manifold M with a Lorentzian metric g. The Gravity 34 Einstein-Hilbert action is: SEH [g] = = Z ⋆ R √ R −ge ZM (3.1) M √ where ⋆ is the Hodge dual operator, and −ge is the volume form on M , whose value in a coordinate basis is given by the square root of the determinant of g. This action reproduces the vacuum Einstein field equations (without a cosmological constant) upon extremization in variations of g, δSEH |g = √ √ (δR) −ge + Rδ( −ge) Z (3.2) M ! =0 Note the following identities, [28, 3] √ √ 1 δ( −ge) = − gab δg ab −ge 2 (3.3a) δR = Ricab δg ab + ∇a ωa (3.3b) where ∇a ωa is a “total divergence” of some quantity which we will eliminate by using Stokes’ theorem and assuming that ω falls to zero “at infinity.” 1 Then, δ Z √ √ 1 Ricab + Rgab δg ab −ge R −ge = 2 M M Z (3.4) which is zero if and only if Ricab + 12 Rgab = 0, and we’ve recovered the vacuum Einstein equations. For the sake of generality, we could add to the gravitational 1 This is the divergence theorem on general manifolds: given a 1-form ω, by Stokes’ theorem we have Z Z ⋆ d⋆ ω = ω S ∂S Which is metric-dependent: there is geometric information from g encoded into ⋆ . To recover the abstract-index expression, note that we can use any torsion-free derivative operator ∇ in the definition of the exterior derivative, because the total-antisymmetry property annihilates any dependence on the connection coefficients. So we might as well use the metric-compatible ∇, √ √ which gives the neat expression d⋆ ω = ∇[an −gω a e|a|a1 ···an−1 ] = −g∇[an ω a e|a|a1 ···an−1 ] , so that the abstract index notation expresses straightforwardly: Z Z a ∇ ωa = ωa n a S ∂S 3.2 Hamiltonian Gravitation: ADM formalism 35 Lagrangian LG a Lagrangian LM encapsulating the totality of gravitating matter, and impose Z √ δ (LM + LG ) −ge = 0 (3.5) M This gives the general Einstein field equations (with factors of 8πG restored,) 1 Gab := Ricab + Rgab = 8πGTab 2 for 8πGTab := − ∂LM 1 Lgab + ∂g ab 2 (3.6) (3.7) In what follows, however, we will always assume Tab = 0. 3.2 Hamiltonian Gravitation: ADM formalism A notion of “time evolution” does not arise naturally in the context of general relativity. When coupled to general matter fields, the Einstein field equations solve everything that happens in spacetime “all at once,” as though the matter and gravitational fields were conspiring together to create the illusion of causality: space bending and matter moving, intertangled but predetermined, dancing to a well-scripted choreography. There are no dynamics in this picture: the river does not flow, but rather water is rained into it from above. This is quite distinct to the usual notion of time evolution in physics, wherein an instantaneous shapshot of a physical system is mapped to the next instant, in accordance with the evolution law as decreed by the Hamiltonian of the system, who is the sole sovereign. But this is to be expected of a general spacetime, wherein a global definition of time coordinate does not necessarily exist− without this, the very concept of a “time evolution” becomes ill-defined. However, there exists a condition on the causal structure of spacetime that is equivalent to the existence of such a coordinate, through which we may hope to restore a notion of time evolution. This condition is global hyperbolicity. 3.2.1 ADM Formalism (I): Statics Let (M, g) be a 4-dimensional Lorentzian spacetime that satisfies global hyperbolicity. That is, there exists a diffeomorphism φ : R × Σ → M such that for every Gravity 36 t ∈ R, φ−1 (M )|t ∼ = Σ, and the 1-form dφ is dual to a timelike vector. In other words, we can introduce a time function that slices spacetime into “time zones” t → Σt := φ−1 (M )|t such that t is strictly increasing for future-bound observers. Let n be the timelike vector field normal to each Σt on M , defined by φp∗ dtd for p ∈ Σ. It is no loss to assume g(n, n) = −1, so that we may define a Riemannian metric on Σt : qab := gab + na nb (3.8) This gives a “spatial projection” of Tp M to the subspace Tp Σt of spacelike vectors:2 , given by v a → qba v b . With a coordinate system {xµ } on Σ, φ gives us spatial coordinates (with a little abuse of notation,) φt∗ (∂/∂xµ ). This completes a basis on T M with n. We can define the extrinsic curvature of Σt as: 1 Kab = £n qab 2 (3.9) £n being the Lie derivative with respect to n. This tensor field quantifies the nonintrinsic curvature of the embeddings of Σ into 4-dimensional M . It complements the 3-dimensional intrinsic curvature on Σ given by qab , which we will denote (3) Rabc d . Let us now introduce a suitably differentiable congruence γ(τ, x) of general (that is, they need not be normal to Σt , nor to be geodesics.) timelike curves on M parametrized by τ , with timelike tangents ∂τ . ∂τ |t need not be normal to Σt , but by a suitable reparametrization we can make the congruence satisfy γ(T, Σ) = ΣT . That is, γ replicates the slicing of the spacetime into time zones generated by φ, but the flow of the curves produces some “spatial translation” across each slice Σt . From these definitions, it is evident that ∂τ = N n + N 2 (3.10a) The index on qab is raised with gab . Unless explicitly stated, we will always raise and lower indices with gab . 3.2 Hamiltonian Gravitation: ADM formalism 37 for some spacelike N , and a normalization constant N , defined by N = −g(∂τ , n) (3.10b) N = ∂τ + g(∂τ , n)n (3.10c) so then we can write 1 (∂τ − N ) (3.10d) N N is called the lapse, and N the shift, because they quantify the infinitesimal motions through time and space, respectively, generated by ∂τ . n= Fig. 3.1 The decomposition of ∂τ . Figure taken from [3]. Using the Riemannian metric qab , we may define on each Σt the covariant differential operator (3) ∇a , the curvature tensor (3)Rabc d , the Ricci tensor, (3)Ricab , and the Ricci scalar (3)R, all of which are defined in the usual way on Σ by the metric (φ∗t q)ab , then pushed forward. Together with the extrinsic curvature, the 4-dimensional curvature tensor can be expressed as in terms of these objects: (see [3], p. 422) nd Rabc d = (3) ∇a Kbc − (3) ∇b Kac (3.11a) qed Rabc e = (3)Rabc d + Kbc Ka d − Kac K d b (3.11b) These are collectively known as the Gauss-Codazzi equations. These lead to the following expressions for the Einstein tensor on M : Gab na nb = − 1 (3) R + (Kaa )2 + Kab K ab 2 (3.12a) Gravity 38 Gac na qbc = (3) ∇a Kba − (3) ∇b Kaa (3.12b) These cannot describe any dynamics in the EFEs, since the right-hand side of these equations only depend on qab in Σt . Setting the left-hand side to zero, as in a vacuum, they can instead be interpreted as “constraints” imposed on the Cauchy data which q must satisfy to be a legal initial state, much like the constraint on the electric field divE = 0 in a vacuum Maxwell theory. 3.2.2 ADM Formalism (II): Dynamics Let us now interpret the congruence γ(τ, x) as the world lines of a set of timelike observers. The crux of the ADM formalism, then, is to formulate general relativity as a description of the evolution in τ of the 3-metric qab as a field theory on Σ. To do this (in a vacuum, and without a cosmological constant, to keep things from getting out of hand,) we must first produce a Hamiltonian out of the Einstein-Hilbert action. And to do that, we start by taking as our canonical coordinate qab , and defining its time derivative q̇ab : = £τ qab = 2NKab + £N qab (3.13) The Einstein-Hilbert action gives the Lagrangian, in terms of qab : L(q, q̇, t) = Z (3) Σt √ R + (Kaa )2 + Kab K ab N q (3)e (3.14a) √ √ √ where it may be verified that N q = g, and so q (3)e is the volume form on Σt . Then we have a Lagrangian density: L(q, q̇) = (3) √ R + (Kaa )2 + Kab K ab N q (3.14b) and a conjugate momentum density to qab : δL δ q̇ab √ ab = q K − Kcc q ab pab = (3.15) It is crucial to note that these are not scalars or tensors, but rather the duals thereof: scalar densities and tensor densities, whose natural habitats are under integral signs. Then we perform a Legendre transform on L to obtain the Hamiltonian 3.2 Hamiltonian Gravitation: ADM formalism 39 density: H(q, p) = pab q̇ab − L √ = q (N C + N a Ca ) where −1 (3) C := − R + |q| 1 p pab − (pcc )2 2 ab (3.16) (3.17a) Ca := −2(3) ∇b |q|−1/2 pab (3.17b) R √ Now that we have a Hamiltonian H = Σ H q (3)e, we can define Poisson brackets on functionals of q, p. Z {f, g} := Σt ! δf δg δf δg √ (3) − ab q e(z) ab δqab (z) δp (z) δp (z) δqab (z) (3.18) which gives the familiar relations: n o qab (x), pcd (y) = δac δbd + δad δbc δ (3) (x − y) (3.19a) {qab , qcd } = 0 (3.19b) n (3.19c) o pab , pcd = 0 and, finally, we obtain the time evolution of the state coordinates: q̇ab = {qab , H} (3.20a) ṗab = {pab , H} (3.20b) By the equivalence of Hamiltonian and Lagrangian mechanics, these must be equivalent to the remaining Einstein field equations, Gab qca qdb = 0, complementary to equation 3.12. These expressions can be written out explicitly, but they are rather long without being particularly enlightening, and we have no real use for them in what follows, so we will omit them here (they are in Appendix C of [28]). Before we proceed any further, though, the Hamiltonian, equation 3.16, has a rather disturbing property. That is, for on-shell states, it is uniformly zero. For consider gab , a solution to the vacuum Einstein equations: Gab [g] = 0. Writing out equation 3.12 in terms of the phase space parametres (qab , pcd )|g , we can see that Gravity 40 this translates into na nb Gab [g] = 0 ⇐⇒ C|g = 0 (3.21a) nb qac Gbc [g] = 0 ⇐⇒ Ca |g = 0 (3.21b) H|g = 0 (3.22) so then, Equation 3.22 is not to be interpreted, however, as a statement that our theory is trivial. It is not any more trivial than, say, finding points on the surface P P 2 1/2 2 3 {(zi ) ∈ CP82 |( 33 − ( 82 i=1 (zi + zi+1 ) ) i=34 zi−1 zi ) = 0}, which is an analogous probP lem where the equation H = 0 plays the role of the defining equation ( 33 i=1 (zi + P82 2 1/2 2 zi+1 ) ) − ( i=34 zi−1 zi ) = 0. This requires some elaboration, provided below. First Interlude: Constrained Hamiltonian Systems The idea of constraint equations on phase space should not be too alien to us; the form in which a physical problem is presented to us is generally not expressed without extraneous degrees of freedom. Consider for example a particle that is constrained to move on the surface of a 2-sphere in 3-space: {⃗q ∈ R3 |q12 + q22 + q32 = 1}. The apparent number of degrees of freedom is 3, but really there are only 2: ( qθ , qφ ), the longitudinal and latitudinal ones. But even without resorting to this rather artificial case, there is already a sufficiently general case of a constrained system with which we are familiar: Yang-Mills theory. In this section we will converse about general constrained systems, and then we will invoke the specific case of Yang-Mills theory as an analogy to shed some light on GR. A general theory of constrained Hamiltonian systems was pioneered by Dirac in [5], in the context of (sub)atomic physics on flat or curved spaces. However, this formalism has sufficient generality to find an application even in a description of the gravitational field. We begin our discussion with a system with finitely many degrees of freedom, described by a Lagrangian L with phase coordinates (qi , q̇i ), and a canonical choice of time parametre t. Though manifest Lorentz invariance is violated by this choice, it can be restored in the equations of motion, as we will see later. 3 zi are homogeneous coordinates on CP. As far as I know this is just an arbitrary, convoluted surface. 3.2 Hamiltonian Gravitation: ADM formalism 41 We introduce the canonical momenta, pi = ∂L ∂ q̇i (3.23) and a set of constraints on the parametres, (3.24) φm (qi , pi ) = 0 which could be φ(qi , pi ) = q12 + q22 + q32 − 1, for example. Performing a Legendre transform, we obtain the Hamiltonian from the Lagrangian: H= X pi q̇i − L (3.25) i and define Poisson brackets: {f, g} := X i ∂f ∂g ∂f ∂g − ∂qi ∂pi ∂pi ∂qi ! (3.26) which gives {qi , pj } = δij (3.27a) {qi , qj } = 0 (3.27b) {pi , pj } = 0 (3.27c) {f, H} =: d f dt (3.27d) In particular, the last equation gives d ∂H qi = dt ∂pi (3.28a) d ∂H pi = − dt ∂qi (3.28b) The Poisson brackets thus defined are anti-symmetric, linear in both arguments, and satisfy a Jacobi equation, etc.. All of this is already familiar to us. To impose the constraints {φm }, we crudely tack on to the system a set of new “artificial coordinates,” the Lagrange multipliers λm , and their conjugate momenta Gravity 42 µm , and write a new Hamiltonian: HT (qi , pi , λm , µm ) := H(qi , pi ) + X λm φm (3.29) m This new Hamiltonian does not depend on µm , the conjugate momenta; they manifest themselves in the dynamics through their absence: Re-defining the Poisson brackets, etc., with respect to HT now, the equation of motion in λm gives, through equation 3.28: φm = 0 (3.30a) (3.30b) λ̇ = 0 In the picture described by the “total Hamiltonian” HT , the constraints φm = 0 are slipped into the physics as the equations of motion of the (artificial) state coordinates λm . A crucial subtlety is that in this picture, the constraint φm = 0 is “emergent,” and falls out of the equations of motion, instead of being imposed “from above”. In other words, we cannot assume that φm = 0 before we “press the play button” and evolve the system in time by imposing the equations of motion. Applying the constraint prematurely will result in incorrect dynamics. Dirac calls these such equations “weak equations” in [5], and the symbol ≈ denotes such a weak equality. e.g., φm (q, p) ≈ 0. In particular, a set of constraints {φm } are said to be of first-order if it defines a canonical algebra: {φi , φj }T = X Cijk φk (3.31) k where the Poisson bracket { , }T is defined with respect to the total Hamiltoi nian. The “structure constants”Cjk are antisymmetric in j, k, because of the antisymmetry of the Poisson brackets. Of course, the additional terms in HT , λm φm alter the bracket structure, and therefore the time evolution, of the original physical parametres (qi , pi ) in a non-trivial way. The first-order correction to the time evolution thusly produced is quantified by X f (1) = {f, λm φm }T (3.32) m 3.2 Hamiltonian Gravitation: ADM formalism 43 which is just {f, HT }T − {f, H}, to first order.4 As an example, say we have a particle on the real plane, parametrized by ⃗q, p⃗, that is constrained to move on a circular rail of radius 1, φ(⃗q, p⃗) = qx2 +qy2 −1. We take the classical Hamiltonian for free particles H = p⃗ 2 /2m and tack on the Lagrange multiplier λ, so p2x + p2y HT (⃗q, p⃗, λ, µ) = + λ qx2 + qy2 − 1 (3.33) 2m and obtain pi q̇i = (3.34a) m ṗi = −2λqi (3.34b) meaning λ qi m The equations in λ become constraints on the boundary conditions, q̈i = −2 (3.34c) qx2 + qy2 = 1 (3.35a) λ̇ = 0 (3.35b) giving the correct solutions of the type (qx (t), qy (t)) = (cos(ωt + φ0 ), sin(ωt + φ0 )) for q a constant angular frequency ω := 2λ/m. We could invert this argument and identify as Lagrange multipliers the canonical coordinates B that have vanishing or absent conjugate momenta. Then the terms in the Hamiltonian that are multiples of B can be collected together to form a constraint equation. The simplest way to identify such cases is when the corresponding Lagrangian L does not depend on Ḃ. That is, pB := ∂L/∂ Ḃ = 0 If we make the jump right away to infinite degrees of freedom (i.e., field theory) without looking down at the gaping crevasse below (covered in Chapter 13 of [9]), then we see that Yang-Mills theory is a perfect example of such a system. The SU (N ) Yang-Mills action on 4-dimensional Minkowski space may be written 4 In the cases that we review, the first order correction is sufficient to determine the entire corrected dynamics. In general cases, however, constraints of the second order may occur, producing non-linear corrections to the dynamics, of which equation 3.32 is just the linear correction. See [5]. Gravity 44 in the language of chapter 2: 1Z ⋆ J SY M [ω] = Ω I ∧ ΩI J 2 R4 (3.36a) or in more familiar notation, (assuming the bundle structure is trivial,) Z 1Z s SY M [A] = − dt d3xFsab Fab 3 4 R R (3.36b) where Fab = Fsab λs is the expansion with respect to a basis 5 {λs } on su(n) satisfying tr(λs λt ) = −δ st /2. In particular, we have broken manifest relativistic invariance with a choice of time t, which will define time in what follows. Taking the state coordinates to be (As0 (⃗x), Asi (⃗x)), we have a set of conjugate momenta (πs (⃗x), Esi (⃗x)):6 ∂L =0 ∂ Ȧ0 ∂L Ei = = η ij Ȧj − ∂j A0 + [A0 , Aj ] ∂ Ȧi π= (3.37) and we immediately see that we have n-many continua of Lagrange multipliers {As0 (x)}. From the Legendre transform, we obtain a Hamiltonian density: H = Esi Ȧsi − L 1 i s = Es Ei + Bsi Bis − As0 (Di E i )s 2 (3.38) where Bi = εijk Fjk is a function of spatial derivatives of the state parametres {Ai }, and Di Ei := −(D⋆ F)0 = ∂i Ei + [Ai , Ei ]. Manifest gauge invariance is lost, because of the second term in H, but we will recover it eventually. Defining δF (x) = δ (3) (x − y) (3.39) δF (y) 5 We can make this manifestly representation-independent by defining Killing forms on su(n), if we want. The result is identical. 6 To avoid any possible confusion, we reiterate that {abc} are abstract indices, and {ijk} are spatial indices on Minkowski space with an explicit choice of time. The argument of the fields is also purely spatial, because time is now something we evolve in, and not merely a (continuous) index. We have denoted the argument as ⃗x here to make that fact explicit, but it will be just x in the following. 3.2 Hamiltonian Gravitation: ADM formalism and 45 δF δG δG δF {F, G} = dz − a s a δAs (z) δEa (z) δAs (z) δEas (z) R3 Z ! 3 (3.40) with the understanding that E0 = π in the sum. It can then be seen that the equation of motion for π is equivalent to the time component of the Yang-Mills equation of motion (D⋆ F)0 = 0: π̇s (x) = {πs (x), H} δπs (x) δH δH δπs (x) = dz − r a 3 δAa (z) δEr (z) δAra (z) δEra (z) R δH =− s δA0 (x) Z ! 3 (3.41) = Di Esi where H = R3 d3 xH. For reference, the evolution of Esi gives the spatial Yang-Mills equations, and the evolution of Asi gives the definition of Esi we started with. So then we have a “generalized Gauss’ law” constraint, (Di E i )s = 0 at every point in R3 . R We are already done, but generally working with a continuum of constraints “asis” is somewhat dangerous, because divergent expressions involving Dirac deltas may occur. So the constraints are typically written in scalar form, “smeared out” with an su(n)-valued distribution: G(λ) = Z R3 P(µ) = d3zλs (Di E i )s Z R3 d3zµs πs (3.42a) (3.42b) for any su(n)-valued distributions λ, µ. The constraints are indeed “of first order”: {G(λ1 ), G(λ2 )} = 0 (3.43a) {P(λ1 ), G(λ2 )} = 0 (3.43b) {G(λ1 ), G(λ2 )} = G([λ1 , λ2 ]) (3.43c) and they “alter” the dynamics, in the sense of equation 3.32, in an interesting way. {G(λ), Ai (x)} = −Di λ(x) (3.44a) Gravity 46 n o G(λ), Ei (x) = [Ei (x), λ(x)] (3.44b) Recall that when we perform infinitesimal gauge transformations U ≈ 1 + λ on Aa generated by small λ, equation 2.7 yields precisely δAa (x) = Di λ(x) (3.45a) δFab (x) = [λ(x), Fab (x)] (3.45b) Thus, G(λ) teaches Hamiltonian Yang-Mills theory, which is a priori “ignorant” of gauge symmetry, how to perform infinitesimal gauge transformations in λ. When all possible λ are thus considered, the concept of gauge symmetry re-emerges at last. But the gauge invariance is only on the spatial components of the fields, because the zeroth component has “privileged character” due to our choice of time. The other constraint has only one non-trivial bracket, and addresses this: {P(µ), A0 (x)} = −µ(x) (3.46) which is just the statement that A0 is a Lagrange multiplier, and can be taken to be any constant. (End of First Interlude.) 3.2.3 ADM Formalism (III): Constraints Returning to our discussion of the ADM Hamiltonian, we may now interpret ( C, Ca ) as constraints, and (N , N a ) as Lagrange multipliers. Following the typical convention on constrained Hamiltonian field theories, we define the “smeared out” constraints: C(N ) := Z −1 (3) N − R + |q| Σt C(N ) := −2 Z Σt 1 p pab − (pcc )2 2 ab Na (3) ∇b |q|−1/2 pab √ q (3)e √ q (3)e (3.47a) (3.47b) That N and N a are arbitrary (fields of) Lagrange multipliers should come as no surprise to us, because they arose through our selection of ∂τ , a fiducial choice of “time” which is not an intrinsic description of the actual physical system. The constraints are of first order: {C(N ), C(M )} = C(£M N ) (3.48a) 3.2 Hamiltonian Gravitation: ADM formalism 47 {C(N ), C(M)} = C(£N M) (3.48b) {C(N ), C(M)} = C((N ∂ i M − M∂ i N )∂i ) (3.48c) And analogously to the Yang-Mills case, the constraints “generate” a symmetry of the system: {f (q, p), C(M )} = £M f (q, p) (3.49) This is exactly the diffeomorphism invariance that distinguishes general relativity from other theories of nature. Specifically, it is the invariance of the system with respect to “Lie dragging” the fields with the one-parametre family of spatial diffeomorphisms φt : Σ → Σ, along the integral curves of N ∈ Γ(Σ, T Σ). The scalar constraint complements C as an expression of the invariance of the physics under reparametrization of the observers γ in τ , thus giving diffeomorphisminvariance in the temporal direction as well. [3] Indeed, the argument on the right-hand side of equation 3.48c is of the form of a Lie derivative,£(N n) (Mn), projected onto Σ− which suggests that the constraints ( C, Ca ) might be interpreted as one set of constraints that expresses invariance under Lie dragging in four dimensions. It is very satisfying to see this symmetry emerge out of our formalism, without any additional effort on our part. So then, of the ten “independent” values of gab , four ( N , N a ) are Lagrange multipliers, and there are four algebraic constraints ( C, Ca ) that must be set to zero, leaving us with just two “true” degrees of freedom. A field theorist would identify these with the spin-up and spin-down polarizations of the spin-2 graviton field. The task of obtaining a quantum theory out of the formalism thus presented seems daunting, to put things mildly: the Hamiltonian contains terms that are seri√ ously non-polynomial in the fields (e.g., q). There are also nigh-insurmountable problems concerning the definition of an inner product on the Hilbert space, and √ so on, but the problem with factors of q will suffice to dissuade us from going down this particular path to quantizing gravity. But suppose, for now, that we have obtained a quantum theory out of the ADM formalism, and we have a Hamiltonian Ĥ. Then the constraints C(N ), C(N ) ≈ 0 ˆ )ψ = Ĉ(N )ψ = 0. In other translates to a statement on the physical states: C(N Gravity 48 words, Ĥψphys = 0 (3.50) This is an example of a Wheeler-DeWitt equation. In this context, the entirely of the physics is specified by the kinematics- there is no room for the dynamics (time) to have any say in the physics! The entirely of the universe, and all of the events that ever occur within it, are determined in one fell swoop. This is the same situation as in the interpretation we offered at the beginning of this section, of solving the Einstein equations in their original form. Just when it seemed that we obtained a fully dynamical formalism of general relativity, it slips away like sand between our fingers. Chapter 4 Quantum Gravity In this chapter, we begin our incursion into quantum gravity. We hinted in section 2.6 that we could formulate general relativity in the context of frame bundles as a gravitational gauge theory on the frame bundle F (M ), and we will do that now. We start again with the Einstein-Hilbert action, but written in terms of frame fields and the connection this time. This is done by rewriting the Ricci scalar with equation 2.54b. This action will not suffice, but it turns out that a modified version of it will yield a Hamiltonian that does the trick for quantization. We will summarize the algebraic details that make this possible in section 4.1.1. The Hamiltonian function thus obtained will be expressed in terms of canonical coordinates (A, E), named in analogy to the gauge and electric fields in Maxwell theory. We will discuss the form of the theory in terms of these new variables in section 4.2. In the Second Interlude, we will outline the program of obtaining a quantum theory from a Hamiltonian theory, through a procedure called canonical quantization. The role of the constraints in the resultant quantum theory will be covered in 4.2.1. Quantum Gravity 50 4.1 Lagrangian Gravitation (II): Frame Fields We may rewrite the vacuum Einstein-Hilbert action in terms of the frame fields {eIµ } and the connection ω: SP [g] = → S[e, ω] = Z ZM M ⋆ R ⋆ µ ν ΩIJ µν [ω]eI eJ (4.1) This action is called the Palatini action. A priori, ω can be any gl(n, R)-valued 1-form, but a simple derivation shows that, for variations in ω, this action is extremized by the Levi-Civita connection. On the other hand, variations in eµI are extremized by solutions to the Einstein field equations (see Appendix C of [28]). Thus this action produces equivalent descriptions of the same physics, and so we might want to perform a Legendre transform on this new action as a second attempt at quantizing gravity. But as it turns out, this new formalism obtains the same equations as does the ADM formalism, expressed in terms of the frame fields, ([1] p. 62) and gives rise to second-order constraints. We mainly present it here to motivate the “self-dual action,” described in the following. 4.1.1 Self-Duality, Complexification ⃗ fields and the In classical Maxwell theory, we were able to combine the electric E ⃗ fields together to form the Riemann-Silberstein vector [13] magnetic B ⃗ := E ⃗ + iB ⃗ S (4.2) which satisfies the complexified Maxwell equations (in a vacuum), i ∂ ⃗ ⃗ + iB ⃗ S = curl E ∂t (4.3a) ⃗=0 divS (4.3b) This vector transforms neatly under general Lorentz transformations, and its vector ⃗ ·S ⃗ = (E 2 − B 2 ) + 2i(E ⃗ · B) ⃗ is a sum of invariant scalars. This may seem like norm S a miraculous coincidence, but all that we’ve really done here is rewrite the Maxwell 4.1 Lagrangian Gravitation (II): Frame Fields 51 field strength tensor F by grouping it into three “anti-self-dual” components, ⃗ = (F0i + iFjk ) dx0 ∧ dxi S (4.4a) with the understanding that ε0ijk = 1. In Lorentz-covariant notation, S = FIJ + iεIJ KL FKL dxI ∧ dxJ (4.4b) As we will see in what follows, there is an analogous way of rewriting the fields in GR that will prove to be extremely useful. To see what we mean by “(anti-) self-duality”, consider a 2n-dimensional Riemannian manifold. Here the Hodge dual operator satisfies (⋆ )2 = 1, so we can decompose arbitrary n-forms as T = +T + −T (4.5) for self-dual + T satisfying ⋆ (+ T ) = + T , and anti-self-dual − T satisfying ⋆ (− T ) = −− T 1 ± T := (T ± ⋆ T ) (4.6) 2 On our 4-dimensional Lorentzian manifolds we are not so lucky; (⋆ )2 = −1, so the best we can do for a definition of self-duality is ⋆ (+ F ) = i+ F , and anti-self-duality, ⋆ − ( F ) = −i− F : 1 ± (4.7) F := (F ∓ i⋆ F ) 2 and verily, the Riemann-Silberstein “vector” satisfies ⋆ S i = −iS i . What has happened here is that we complexified the Lorentz Lie algebra, and decomposed it into two mutually commuting complex-valued su(2) modules: so(1, 3) → so(1, 3)C ∼ = + su(2)C ⊕ − su(2)C (4.8) ∼ = + sl(2, C) ⊕ − sl(2, C) because sl(2, C) ∼ = su(2)C , where gC is shorthand for g ⊗ C. Let us make this decomposition explicit: recall the familar basis vectors of so(1, 3), Mst , defined with components Mst I J = −(η sI δJt − η tI δJs ); recall also that we Quantum Gravity 52 could define (see, e.g., Section 4.2 of [16], ) 1 Ji := εijk Mjk 2 (4.9) Ki := M0i satisfying [Ji , Jj ] = εijk Jk [Ki , Kj ] = −εijk Jk [Ji , Kj ] = εijk Kk (4.10) Then complexifying, ± Ai := 1 (Ji ± iKi ) 2 (4.11) we obtain two commutating copies of su(2)C : [+Ai , +Aj ] = +Ak εijk [−Ai , −Aj ] = −Ak εijk [+Ai , −Aj ] = 0 (4.12) or, writing 4.11 more suggestively, in terms of Mst , i i Mst ∓ εst uv Muv Ast := 2 2 ± (4.13) But we want (anti)-self-duality on the I, J, K algebra indices, not on their labels s, t, u. So we define instead: ± MstIJ i i MstIJ ∓ εIJ KL MstKL := 2 2 (4.14) But the two definitions are in fact equivalent, because MstIJ = −(δIs δJt −δJs δIt ), which IJ KL = εst uv Muv . implies εIJ KL Muv We can take complex linear combinations of +Mi := +M0i as a basis for + su(2)C , or, we can take real linear combinations of +Mst as a basis for + sl(2, C) that is labeled by two antisymmetric indices. Both choices give 6 independent parametres, as we would expect. Crucially, the self-dual basis transforms “correctly” under Lorentz transforms, so that the basis indices s, t, u can be treated as Lorentz indices. ′ ′ This can be seen by noting that Λaa Λbb εa′ b′ c′ d′ = Λcc′ Λdd′ εabcd in equation 4.14, so we have ′ ′ Λss Λtt ±Ms′ t′ = ±Mst (4.15) In the same vein, we complexify the tangent bundle T M and denote the complexification T M C , and similarly for the frame bundle: F (M ) → F (M )C . So, local 4.1 Lagrangian Gravitation (II): Frame Fields 53 sections of the complexified frame bundle map the “internal” Minkowski C4 vectors at a point p ∈ M to tangents on the complexified tangent fibre Tp M C . Then consider a gl(n, C)-valued connection 1-form, Aa I J , on F (M )C . We can antisymmetrize in the internal indices it by defining 1 A′a IJ := (AaIJ − AaJI ) 2 (4.16) with the I, J indices lowered with the internal metric ηIJ . This puts A′a in the complexified Lorentz algebra. We can then define its self-dual (in the internal indices) part: 1 i KL ′ ′ + ′ (Aa )IJ := Aa IJ − εIJ AaKL (4.17) 2 2 We can expand an antisymmetric, self-dual connection 1-form Aa I J with respect to the basis +Mst I J on sl(2, C): + I Aa I J = Ast a ( Mst ) J (4.18) This will come in handy later on. IJ A self-dual curvature +Fab can be built out of a general connection by taking its self-dual anti-symmetrized part +A′a , then building it as usual. Writing out the components explicitly, + IJ Fab [A] IJ + ′ = Fab [ A] IJ = ∂a +A′ b − ∂b +A′ IJ + [+A′a , +A′b ]IJ (4.19) Linear combinations and derivatives of self-dual A are easily verified to be self-dual. Commutators are self-dual too, because the construction 4.14 satisfies 4.12. Thus IJ we can see that the curvature Fab [A] of self-dual, antisymmetric connections A is itself self-dual. Then we define a self-dual action out of the complexifed Palatini action: Z ⋆ a b + IJ S[e, A] = eI eJ Fab [A] (4.20) M Or, we could just restrict the space of connections to those that are already self-dual and antisymmetric and write instead, SSD [e, +A] = Z M ⋆ IJ + eaI ebJ Fab [ A] (4.21) Quantum Gravity 54 We shall favour the latter expression, because it eliminates equivalent solutions. This action is, quite unusually, complex-valued. Note that, a priori, we have no constraints on the tetrad that extremizes this action, and so the metric gab = eIa eJb ηIJ may very well be complex. We will not concern ourselves with this peculiarity. Instead, we will satisfy ourselves with the knowledge that the self-dual action is extremized by solutions (eS , AS ) that solve the self-dual half of the complexified vacuum EFEs ([3], p. 441-443): 0 = + Ricab [eS , AS ] + 1+ Rgab [eS , AS ] 2 (4.22) where + Ric, and + R are defined with +F [AS ] contracted in the usual way with the frame field eS . This in turn implies that + Ric = 0. Furthermore, one also obtains that AS is the self-dual part of the complexified spin connection: ([1], p. 45-47) (deS )I = −(ωS )I J ∧ (eS )J i 1 (ωS )aIJ − εIJ KL (ωS )aKL = 2 2 (AS )aIJ (4.23a) (4.23b) with this in mind, it would be temping to demand all physical quantities be built out of self-dual objects. However, we do not currently have a good way of associating the frame field eIa with self-duality. 4.2 Ashtekar’s Variables Let us reinstate the formalism of section 3.2.1. That is, let us bring back the foliation of M with Σ and its normals n, and the timelike observers γ(t, x) with their corresponding tangent vectors ∂τ , etc. But now instead of working with EinsteinHilbert action and the phase space coordinates (qab , pab ) defined by the 3-metric, we will use the self-dual action, and “new” phase space coordinates (A, E), which are analogous to the vector potential and electric fields. In this section, and what follows it, we will often without impunity inter-mix our indices, especially the index types {i, j, k}, {I, J, K, L} and {s, t, u, v}, because they all correspond to Minkowski indices in the present context, raised and lowered by η. 4.2 Ashtekar’s Variables 55 Furthermore, consider the expansion of a 2-form Fab in the form of 4.18 with st i 0i components Fab . We will often write Fab for spatial index i when we mean Fab . We can do this unambiguously because we now have a privileged choice of time provided by n. Then, an index of 0 will always correspond to contraction with nI := eaI na . e.g., ε0JKL := nI εIJKL . A is simply the spatial part of the projection of the self-dual connection onto each Σt : Aia := iqab A0i (4.24) b On the other hand, E will be the “self-dual-compatible” part of a spatial triad tangent to Σ built from the frame field: EIa := qba ebI and a ELI := √ q n[L EI]a i a − εLI JK nJ EK 2 (4.25a) (4.25b) nI := eaI na has by construction nI = δ0I , so this simplifies into a Eia : = E0i √ = qEia (4.25c) The (A, E) are called the Ashtekar (new) Variables. They relate to the ADM variables thusly: Aia = (3) Γia − iKai (4.26a) E ia Eib = |q|q ab (4.26b) I bI where (3) Γia := (3) Γ0i a and Ka = Kab e . We can perform a Legendre transform on the self-dual Lagrangian and obtain a Hamiltonian ([8], p. 176), H(Aia , Eia ) = − Z Σ i e |q|−1/2 N εijk Eia Ejb Fabk + τ aAia (Db E b )i + N a Eib Fab (3) (4.27) where Da = (3)∇a is the covariant derivative on Σ. τ a := ∂τ is an independent parametre, so we might as well write τ a Aia = λi . The Hamiltonian is once again a sum of constraints, with Lagrange multipliers N , N a , and λi . These new constraints Quantum Gravity 56 are just the usual suspects in disguise. Smeared out, One is related to the scalar ADM constraint C(N ), the Hamiltonian or scalar constraint, H(N ) := Z e|q|−1/2 N εijk Eia Ejb Fabk (3) Σ (4.28a) and the vector constraint, V(N ) := Z i eN a Eib Fab (4.28b) eλi (Da E a )i (4.28c) (3) Σ and the last one is a Gauss law, G(λ) := Z Σ (3) The last constraint excludes the additional redundancy produced by our transfer from a geometrodynamic theory to an SU (2) gauge theory: qab has 6 independent components at each point in Σ, with C providing 1 constraint and Ca , 3, for a total of 2 degrees of freedom. On the other hand, Aia has 9 components; Gi provides 3 constraints, Va , 3, and H, 1. This leaves 2 degrees of freedom again. The constraints are, once again, of the first order. We can show this explicitly, but a slight re-arrangement will make things considerably neater: linear combinations of constraints are again constraints, so we can define C(N ) := V(N ) − G(N a Aa ) (4.29) This is the diffeomorphism constraint. It is related to the ADM constraint C(N ) in a way that way that shall be elucidated later. We define the Poisson brackets, {f, g} = Z Σ δG δG δF δF − e(z) a δAia (z) δEi (z) δAia (z) δEia (z) ! (3) (4.30) and this yields {Aia (x), Ejb (y)} = iδab δji δ (3) (x − y) {Aia , Ajb } = 0 {Eia , Ejb } = 0 {C(N ), C(M )} = C(£N M ) (4.31) (4.32a) 4.2 Ashtekar’s Variables 57 {C(N ), H(M)} = H(£N M) (4.32b) {G(λ), G(µ)} = G([λ, µ]) (4.32c) {C(N ), G(µ)} = G(£N µ) (4.32d) {G(λ), H(M)} = 0 (4.32e) {H(N ), H(M)} = C(K) (4.32f) K a = |q|1/2 Eia E bi (N ∂b M − M∂b N ) (4.33) where Observe that the ADM constraints C, C generate the same algebra as the new constraints H, C So, similarly to the ADM case, the bracket of C(N ) with a scalar function f in the canonical coordinates is proportional to its Lie derivative in N , so the C(N ) constraint generates infinitesimal “Lie dragging” 3-diffeomorphisms across Σ. e.g., {C(N ), f (A)} = £N f (A) (4.34) and the scalar constraint generates diffeomorphism invariance in the temporal direction parallel to n, Lie dragging the fields into the future, or the past. This can be interpreted as a form of time evolution, hence the association of H with Hamiltonian functions. As in the Yang-Mills case, G generates an SU (2) gauge symmetry, which corresponds to the invariance of the theory under local spatial rotations. Second Interlude: Canonical Quantization Consider a free non-relativistic particle on the real line R. The canonical formalism offers a relatively straightforward path from the classical system to a quantum theory: we “promote” the phase space coordinates q, p into elements q̂, p̂ of an algebra A, and functions f (q, p) are promoted to algebra elements as well, denoted fˆ. The commutation relations on the promoted algebra is subject to conditions given by the Poisson bracket: 1 \ {f, g} = [ fˆ, ĝ ] i~ (4.35) Quantum Gravity 58 Then we let the elements of the algebra act on some vector space of functions on R, which will make up a Hilbert space H. In the “position basis” for the particle on the line, we find that the condition 1 \ {q, p} = [ q̂, p̂ ] = 1 i~ (4.36) is satisfied by operators q̂, p̂ defined by q̂Ψ(x) = xΨ(x) (4.37a) ∂Ψ (x) (4.37b) ∂x A natural choice of inner product on H that satisfies the necessary axioms of bilinearlity, symmetry and positive-definiteness is p̂Ψ(x) = −i~ ⟨Ψ, Φ⟩ = Z dx Ψ̄(x)Φ(x) (4.38) R so we might identify H with the space of square-integrable functions L2 (R), and define its inner product with the L2 product. In some situations, however, the Hilbert space axioms may be too strict. Say we want to make use of the basis of momentum eigenfunctions ψk (x) = √12π eikx . But this basis is not square integrable on the line with the usual integration measure. However, in the context of distributions/generalized functions, we can write Z dx ψ̄k1 (x)ψk2 (x) = δ (3) (k1 − k2 ) (4.39) R which is, in a sense, a limit on the L2 norm of ⟨ψ1 , ψ2 ⟩ where ψ1 → ψk1 and ψ2 → ψk2 . This motivates the idea of a “rigged Hilbert space,” or a Gefland triple, which defines the Hilbert space in terms of a dense subspace Φ ⊂ H, and its dual, Φ∗ ⊃ H. Then we “rig up” H by defining its inner products with the duality of Φ, Φ∗ . In other words, ⟨Ψ, Φ⟩H = Ψ(Φ). This is well defined because Ψ ∈ H ⊂ Φ, and Φ ∈ H = H∗ ⊂ Φ∗ , by the Riesz representation theorem. For a more detailed discussion, see chapter 5, or [4]. 4.2 Ashtekar’s Variables 59 Let us make this whole procedure more precise. In [1], Ashtekar outlines a program for the quantization of a general constrained Hamiltonian system, based on the method of Dirac in [5]. The key steps are as follows: 1. Choose a subspace of the functions on the classical phase space associated with classical variables, denoted S. Each element in S is to be promoted to a quantum operator (on a yet-to-be defined Hilbert space H). In particular, S has to be general enough to parametrize the phase space, and restrictive enough to make promotion unambiguous, without factoring problems. 2. Promote each F ∈ S to an operator F̂ , and define a commutation relation [·, ·] such that 1 \ {F, G} = [ F̂ , Ĝ ] (4.40) i~ 3. Find a representation of the resulting algebra A as the set of endomorphisms on a complex vector space V. 4. Restrict V to the set of vectors annihilated by all of the promoted constraints φ̂m . This linear subspace will represent the space of physical quantum states Vphys 5. Define an inner product ⟨· , ·⟩ on Vphys , associated with an involution operator ⋆ on A, which defines self-adjointedness on the resulting Hilbert space. 6. Patch up the set of operators A so that they, and their adjoints, A⋆ , commute with the constraints under the inner product. Then Vphys is closed under the action of elements in A. Each of these must be addressed in order to obtain a canonically quantized theory. (End of Second Interlude.) 4.2.1 Quantum Constraints Before we set sail for quantum seas, though, let us meditate a bit on what a hypothetical quantum theory of gravity must look like. The bracket relations 4.31 afford a straightforward representation of the promoted phase coordinates. In the Quantum Gravity 60 “connection representation,” where states are functionals of the connection A, Âia (x)Ψ[A] = Aia (x)Ψ[A] (4.41a) δ Ψ[A] δAia (x) (4.41b) Êia (x)Ψ[A] = so then [ Âia (x), Êjb (y) ] = δab δji δ (3) (x − y) (4.42) Let us consider the Gauss constraint G(λ). By construction, a physical state Ψ must be annihilated by Ĝ(λ) for every su(2)-valued distribution λ. Without even explicitly choosing a promotion, the Poisson bracket (similar to equation 3.44a) and equation 5.8 already show that h i 1 + εĜ(λ) Ψ[A] = Ψ[A − εDλ] ! (4.43) = Ψ[A] for small ε. This result is very good; it means that the space of physical states does not distinguish between expressions that differ by a gauge transformation. Ignoring for now the constraint algebra 4.32, we can promote this to a quantum constraint, if only formally: Ĝ(λ)Ψ[A] = Z eλi Da (3) Σ δΨ[A] δAia (z) (4.44) Here we’ve made the choice of ordering the factor of A in D to the left of E. The diffeomorphism constraint receives a similar treatment: from the Poisson bracket and equation 5.8 again, h i 1 + εĈ(N ) Ψ[A] = Ψ[A] + ε£N Ψ[A] ≈ Ψ[A + ε£N A] (4.45) ! = Ψ[A] This means that the physics is undisturbed if we slide the fields on Σ along the flows of N . 4.2 Ashtekar’s Variables 61 To promote C(N ), we may formally promote the vector constraint V(N ) first, V̂(N )Ψ[A] = Z i eN a F̂ab (3) Σ δΨ[A] δAib (z) (4.46) which gives Z i eN a F̂ab + Âia Db (3) Σ δΨ[A] δAib (z) (4.47) i might be ∂a Âib − ∂b Âia − εijk Aˆja Âkb . where a good operator ordering on F̂ab Up to operator-ordering, the promotion of the constraints C, G yielded terms that were merely polynomial in the fields and their derivatives. The Hamiltonian constraint is a lot more troublesome. The integrals that smear the constraints G and C are integrated with a coordinate-dependent 3-form (3)e(z) 1 rather than √ the coordinate-independent volume form q (3)e This is because we absorbed the √ factors of q into E, saving us from having to deal with the non-polynomial terms that plagued the ADM formalism. The Hamiltonian constraint, however, has a factor of |q|−1/2 which cannot be dealt with thusly. So it seems that we cannot even formally write down a promotion of H, never mind ordering the operators to fit the constraint algebra. There exists, however, a clever trick with which we may rewrite H(N ), introduced by Thiemann, which may be found in section 10 of [26]. First we define a classical-valued function V purely in terms of the triad 2 : V (E) : = Z s (3) e Σ = Z 1 εijk εabc Eia Ejb Ekc 3! √ (3) q e (4.49) Σ 1 2 e.g., in Cartesian coordinates, (3)e = dx ∧ dy ∧ dz; in spherical coordinates, (3)e = dr ∧ dθ ∧ dφ. √ We can work out factors of q quick and dirty, as follows: ea eb = qab =⇒ det(ea eb ) = det qab √ =⇒ det(ea ) = sgn(e) q a 3/2 =⇒ |det(E )| = |q| (4.48) a |det(e )| a =⇒ |det(E )| = |q| √ because E a = qea , and the identities det(ea ) = det(ea )−1 and det(c E a ) = c3 det(E a ), the latter holding for 3 × 3 matrices. Quantum Gravity 62 which outputs the 3-volume of the spatial manifold Σt . When promoted into a quantum operator V̂ , it is called the volume operator. The trick, then, is in noticing that {Aia , V } = |q|−1/2 εijk εabc Ebj Eck (4.50) so that H(N ) can be rewritten H(N ) = Z eN {Akc , V }Fkab εabc (3) (4.51) After quantization, we can work with eigenfunctions of V̂ so that {Akc , V } promotes to a polynomial or Taylor series in A. We will not concern ourselves too much with the specifics of the Hamiltonian constraint in what follows, however. Chapter 5 Loop Quantum Gravity Our treatment of the constraints in the First Interlude is quite contrasted to that of the Second. In the former, we slipped in to the Hamiltonian a set of artificial coordinates λm , and the constraints φm = 0 emerged as an extra set of equations of motion in the associated momenta µm . In the quantum theory, however, the constraints were strong-armed in to the theory by banishing from the space of physical states Vphys all that which is not annihilated by the quantized constraints φ̂m , “by hand”. In the case of canonical quantum gravity, a hybrid approach may be used. We construct our space of states by selecting those that are annihilated by the diffeomorphism and Gauss constraints, and the Hamiltonian constraint is imposed through a “dynamical” equation on the resulting space: 0 = ĤΨ[A] = εijk F̂abk δ δ Ψ[A] δAia δAjb (5.1) The right-hand side of this is reminiscent of the non-relativistic Hamiltonian for a free particle, 1 ∂ ∂ Ĥ1p ψ(x) = − δ ab a b ψ(x) (5.2) 2 ∂x ∂x which perhaps illustrates the purely kinematic nature of the symmetry generated by the Hamilton constraint1 . 1 States that solve ĤΨ = 0 are annihilated by the Hamiltonian constraint, but the converse might not be true. Loop Quantum Gravity 64 In this context, equation 5.1 is the Wheeler-DeWitt equation, because all the other constraints that make up the full Hamiltonian are satisfied by definition. It can be thought of as an equation describing time evolution, since it is related to the Hamiltonian constraint, the generator of temporal diffeomorphisms. However, it is not a “true” dynamical equation because there is no explicit reference to time, and thus no parametre relative to which evolution occurs. We note that, up to now, we have been using the self-dual part of a complexified Lorentz algebra to define our field A: AaIJ i i = ωaIJ − εIJ KL ωaKL 2 2 (5.3) where ωa is the connection 1-form that solves equation 2.47. This simplified matters because A was then a connection 1-form, with a straightforward projection on to Σ by taking its 0i components. On Σ, A has a very simple su(2) ∼ = so(3) representation, which is just the angular momentum subalgebra of the (self-dual) Lorentz algebra. However, we also had to deal with the unhappier consequences of pushing our real physics out onto the complex plane, leading to complex Einstein equations. One might find it more convenient to work with the real-valued field A′aIJ = ωaIJ 1 + εIJ KL ωaKL 2 (5.4) This field would be self-dual if we were working with a Riemannian metric instead of a Lorentzian one. It is manifestly real-valued because complexification does not occur in the first place. However, A′ does not transform as a connection under general Lorentz transformations, because we lose the transformation property from 4.15. i.e., A′ does not belong to a representation of the Lorentz algebra. Nevertheless, this A′ is still a field on R × Σ, and it transforms properly under spatial SU (2) rotations because the field has an su(2) representation when projected on Σ: the two terms in the definition 5.4 transform the same way under pure rotations, but not under boosts. It must be observed however that using this field will lead to a severe change in our physics, because the theory obtained from this field would turn out to describe 65 a Riemannian 4-dimensional spacetime metric. Because the field A no longer represents a Lorentz algebra, we lose the hyperbolic structure of time in our theory, and obtain instead a Euclidean “time” [20]. On the other hand, problems with the non-compactness of SL(2, C) give rise to unsolved problematic complications which prevent us from using the self-dual connection in the theory [7]. We seem to have found ourselves in a bit of a quandary, where neither treatment is acceptable. However, we will be more modest with our ambitions in our coverage of quantum gravity: we shall set the stage with the connection representation of the kinematical space of states that are annihilated by the diffeomorphism and Gauss constraints, and then simply bow out. Our construction of the constrained space of physical states H will be simplified because we already know that our constraints C(N ) and G(λ) are generators of diffeomorphisms and gauge rotations, e.g., from equations 4.43 and 4.45. Thus we need only to select functionals Ψ[A] that are manifestly gauge- and diffeomorphism-invariant to fill up the space H. In what follows, we will formulate H as a Gefland triple S ⊂ H ⊂ S ∗ , rather than deal with the stricter laws of Hilbert spaces. Following Chapter 6 of [20], we will start in section 5.1 with a space of select functionals Ψ[A], denoted K, which will be the base space upon which we build our space of states. Then in section 5.2, we build gauge invariant linear combinations of the basis states of K to obtain a basis that spans a manifestly gauge-invariant space K0 . Finally, we identify states that are “Lie-diffeomorphic” to each other on Σt to obtain KD = H, the kinematic, constrained space of states in section 5.3 . This is the final destination in our coverage of Canonical Quantum Gravity: a rigged Hilbert space of functionals Ψ[A] on Σ, with argument A, a su(2)-valued 1-form that transforms as an SU (2) connection under local gauge rotations. As an aside: if had we started from the Palatini action and defined our canonical variable A as A = (3) Γia + βKai (5.5) Loop Quantum Gravity 66 the Hamiltonian constraint has the more convoluted general form2 H(N ) = Z " (3) −1/2 e|q| Σ N Eia Ejb ! β2 + 1 i Kb]j ε Fabk + 2 K[a 2 β ijk # (5.6) β is called the Barbero-Immirzi parametre. For the choice β = −i, we get the Ashtekar variable A, and the second term in the sum vanishes; for β = 1 we get the real valued field A′ . Had we chosen the real field, the quantum commutator in equation 4.42 would have instead been [ Âia (x), Êjb (y) ] = iδab δji δ (3) (x − y) (5.7) which would have yielded the operators Âia (x)Ψ[A] = Aia (x)Ψ[A] Êia (x)Ψ[A] = −i δ Ψ[A] δAia (x) (5.8a) (5.8b) analogous to the one-particle non-relativistic theory. Since we will be working with the real valued field A′ (henceforth, simply A,), we will need to deal with a much more complicated expression of the Wheeler-DeWitt equation. We will not need to concern ourselves with this additional complexity, or with commutators in general, in what follows; we only state these for the sake of completeness. 5.1 The Base Space On Σ, the field Aa has an su(2)-valued representation: Aa = Aia 2 σi 2i (5.9) The form of the diffeomorphism and Gauss constraints would also be changed, of course. However, it will not change the fact that they would still be generators of diffeomorphism and gauge symmetries, so their exact form will not matter. 5.1 The Base Space 67 Observe that a family of manifestly gauge-invariant functionals in A may be constructed out of the set of holonomies, by simply taking their trace: Hγ [A] = Tr P exp − I γ Ai (5.10) where γ is any closed loop with an orientation. That this is gauge invariant can be seen from equation 2.27 for a loop. It turns out that these functionals span the space of of gauge invariant functionals on Σ; indeed, this property was the historical impetus for the study of loops in canonical quantum gravity, hence the umbrella term of Loop Quantum Gravity. With this basis, the (smooth 1-parametre) diffeomorphism invariance on A can be transferred over as an equivalence relation on the set of loops γ: P exp − I ∗ φ A = P exp − γ ! I A (5.11) φ(γ) So we need only γ which are homotopically distinct to parametrize our basis on the space of states. i.e., classes of knots and links that cannot be smoothly deformed into one another. This leads to the surprising cameo appearance of knot theory in quantum gravity. [8, 3] However, even with this equivalence relation, this basis is over-complete; this fact would lead to undesirable redundancies in describing physical states. Instead, we will end up considering the better-tempered basis of spin network states, which will be a continuation of the spirit underlying this line of inquiry. We start by defining a set of “cylindrical” functions, S, which will be a dense subspace of the base space K in the Gefland triple. Consider: 1. A function f : SU (2)L → C for L > 0 ∈ N, 2. An ordered3 collection Γ of L oriented non-trivial paths in Σ, (γ1 , · · · , γL ), 3. A map (A, γ) → U (A, γ) ∈ SU (2) given by P exp − 3 That is, (γ1 , γ2 ) ̸= (γ2 , γ1 ) unless γ1 = γ2 , and so on. H γ A Loop Quantum Gravity 68 then we can define a set of functionals Ψ[A]: ΨΓ,f [A] := f (U (A, γ1 ), · · · , U (A, γL )) (5.12) Choosing f that are not trivial (i.e., constant) in any of their variables, S is defined as the space of ΨΓ,f for all sufficiently differentiable f and Γ. We may be well-concerned about the enormous size of this space, since there are very many distinct choices of oriented paths γ in Σ, and the fact that we are taking ordered collections of these loops can only make things worse. We will see later when we build KD that diffeomorphism invariance drastically reduces the size of this space, in a way similar to the reduction afforded by 5.11. We can define an inner product right away: ⟨ΦΓ′ ,g , ΨΓ,f ⟩ = Z dU1 · · · dUL g(U1 , · · · UL )f (U1 , · · · UL ) if Γ = Γ′ 0 if Γ ̸= Γ′ (5.13) where dU is the Haar measure on SU (2), which is just a normalized integration measure on the 3-sphere. We define K as the limit of sequences {Ψn }n∈N ⊂ S for which ||Ψn || = ⟨Ψn , Ψn ⟩ converges. S ∗ , being the dual space to S, has the natural definition as the limits of {Ψn }n∈N for which ⟨Ψn , Ψ⟩ converges, for every Ψ ∈ S.4 This makes it manifestly clear that S is a dense subset of K with the metric topology defined by the L2 norm. The definition of K as the completion of S with respect to the L2 norm means that we can take not just finite linear combinations of the “basis” states ΨΓ,f , but in fact arbitrary sums, so long as the sum has a finite norm. That we should be allowed to do this in our space of states is completely reasonable. For example, in the 1D infinite square well in NRQM, we often had to express wavefunctions in P n2 π 2 terms of series of the energy eigenfunctions: ψ(x) = ∞ n=1 an sin( 2L2 x), because merely finite sums would generally not suffice. 4 In L2 , we identify functions that differ by a function of norm zero. i.e., ||f − g|| = 0 ⇐⇒ f ∼ g. In other words, the limits are well-defined and unique up to an equivalency. 5.2 The Gauge Invariant Space 69 Similarly, S ∗ accommodates “states” that do not have a well-defined finite norm, but do have a well-defined inner product with physical states. Examples of these are the momentum and position eigenfunctions of a free particle on the line, eipx and δ(x − y), respectively. By our definition of the inner product in 5.13, the base space K is already neatly divided into a set of mutually orthogonal subspaces parametrized by the ordered collections Γ, KΓ ∼ = L2 [SU (2)L ]. In what follows, it will be convenient to find an orthonormal basis for each KΓ . There is a ready-made such basis given by the Peter-Weyl theorem, which implies that L2 [SU (2)] decomposes into an orthogonal direct sum of finite-dimensional irreducible representations (irreps) of SU (2). In other words, the space is spanned by the (2j +1)×(2j +1) matrix representations of SU (2), R(j)a b [U ], for non-zero halfintegers j. The j labels the particular representation to which the matrix belongs. For computational simplicity, we will label these basis states directly with their matrix components. So then a basis |j, a, b⟩ of L2 [SU (2)] may be defined, satisfying ⟨U |j, a, b⟩ = R(j)a b [U ] (5.14) for a fixed, faithful spin-j representation R(j) , for each positive half-integer j. The extension to L2 [SU (2)L ] is just the tensor product of L-many such basis vectors, |Γ, J, A, B⟩ := |j1 , a1 , b1 ⟩ · · · |jL , aL , bL ⟩, for multi-indices J, A, B. And thus, in the connection representation, ⟨A|Γ, J, A, B⟩ = R(j1 )a1 b1 [U (A, γ1 )] · · · R(jL )aL bL [U (A, γL )] When all distinct Γ are considered, this becomes a basis on K = 5.2 L Γ (5.15) KΓ . The Gauge Invariant Space To construct K0 , we need to find linear combinations of the basis vectors |Γ, J, A, B⟩ that are invariant under local SU (2) transformations. The fact that these are associated vector spaces to SU (2) irreps makes this rather straightforward: consider the action of a local SU (2) rotation λ : Σ → SU (2) on |Γ, J, A, B⟩. From equations Loop Quantum Gravity 70 2.27 and 5.15, ⟨A|Uλ |Γ, J, A, B⟩ =⟨λ(A + d)λ−1 |Γ, J, A, B⟩ ′ =R(j1 )a1 a′1 [λ(γ1,1 )] R(j1 )b1 b1 [λ−1 (γ1,0 )] · · · ′ R(jL )aL a′L [λ(γL,1 )] R(jL )bL bL [λ−1 (γL,0 )] (5.16a) ⟨A|Γ, J, A′ , B ′ ⟩ where γn,0 is the initial point of γn , and γn,1 is its endpoint, and by unitarity, ⟨A|Uλ = ⟨Uλ−1 A| = ⟨λ(A + d)λ−1 |. Since this holds for arbitrary fields A, and since neither the basis nor the rotation matrices depend on A, we can write ′ Uλ |Γ, J, A, B⟩ = R(j1 )a1 a′1 [λ(γ1,1 )] R(j1 )b1 b1 [λ−1 (γ1,0 )] · · · ′ R(jL )aL a′L [λ(γL,1 )] R(jL )bL bL [λ−1 (γL,0 )] ′ (5.16b) ′ |Γ, J, A , B ⟩ Now, if we could find a collection of SU (2) tensors I that trivialize rotations, e.g., I[jm , · · · jn ]am ···an R(jm )a1 a′m [U ] · · · R(jn )an a′n [U ] = Ia′m ···a′n [jm , · · · jn ] (5.17) then by a suitable identification of initial and end points of the curves γn , we can write down gauge-invariant states as, for example, ′ ′ |S⟩ :=I[j1 , · · · jL ]a′1 ···a′L I[j1 , · · · jL ]b1 ···bL |Γ, J, A, B⟩ (5.18) This state corresponds to the particular case where all the curves in Γ start at the same point, p, and all end at another point q. If we wanted to join end points then I would have raised b indices mixed in. In that case, the b indices would correspond to the conjugate representations, R−1 . The tensors I[J] “shrug off” the effects of a local SU (2) rotation, leaving the overall state |S⟩ unchanged. Thus the collection of tensors I[J], the intertwiners of the representations of SU (2), find for us states |S⟩ that are gauge invariant. These intertwiners are in fact no strangers to us: when only joining initial points, they are simply the Clebsch-Gordan coefficients cm1 ···mn that produce trivial spin-0 representations out of tensor products of spin-(jm , · · · jn ) irreps. For example, for 5.3 The Diffeomorphism-Invariant Space 71 j1 = 1/2, j2 = 1/2, this is the tensor cm1 m2 that gives the singlet state combination5 |0, 0⟩ = X cm1 m2 |m1 , 1/2⟩|m2 , 1/2⟩ 1 = √ (| ↑↓⟩ − | ↓↑⟩) 2 (5.19) The connection is apparent: it is only the spin-0 representation that is invariant under general spatial rotations [10]. This motivates the definition of spin network states |Γ, J, In ⟩, each labeled by a triplet (Γ, J, IC ) and associated pictorially with a web with L links and N nodes. This state assigns a spin-jn irrep (called a colouring,) and an orientation to each “link” γn , and an intertwiner IC to each node, labeled by an index C, which trivializes the (tensor product of the) irreps carried by each of the γi that meet on the node. These states comprise an orthonormal set of states, as states with different choices of colouring J or intertwiners I are orthogonal [20]. We can use the method of Young tableaux to take the product j1 ⊗ · · · ⊗ jn ; a set of links can only form a node if their irreps form a trivial representation. Then the span of spin network basis states |Γ, J, In ⟩ gives us gauge-invariant S0 , and K0 and S0∗ can be defined with the same completions with respect to the L2 inner product as last time. 5.3 The Diffeomorphism-Invariant Space The last piece of the puzzle is brought to us by means of an equivalence relation. Consider an arbitrary cylindrical function ΨΓ,f [A] ∈ S, and a family of diffeomorphisms φt , smooth in t ∈ [0, 1], for which φ0 is the identity map. Then from the definition, we have ΨΓ,f [φt∗ A] = Ψφ−1 [A] (5.20) t (Γ),f 5 When end points are involved, we will need to invert the signs on the corresponding index m, because they are in the conjugate representation. If we are exclusively joining endpoints, the result will be unchanged, but mixing endpoints and initial points requires extra care to the signs on the matrix indices m. Loop Quantum Gravity 72 where φ(Γ) is the ordered collection of oriented curves (φ(γ1 ), · · · φ(γn )) for Γ = (γ1 , · · · γn ). This can be seen by pulling back the integral, as in equation 5.11. So then, the condition of diffeomorphism-invariance suggests that we identify basis states in S0 |Γ, J, In ⟩ ∼ |Γ′ , J, In ⟩ ⇐ ∃ smooth φt : φ0 (Γ) = Γ; φ1 (Γ) = Γ′ (5.21) This relation identifies a large number of mutually orthogonal subspaces S0Γ′ ∼ S0Γ , drastically reducing the size of the space of distinct physical states; it saves our physical space of states from being spanned by a continuum of basis vectors to just countably many of them. To make this more clear, we define an equivalence on the oriented graphs Γ instead: ∃φt : φt (Γ) = Γ′ ⇐⇒ Γ ∼ Γ′ (5.22) Since φt is smooth in t, it must preserve the ordering and orientation of the links. These equivalence classes are called (oriented) s-knot classes, in analogy to knot classes. We will denote s-knots classes as [K] = Γ/ ∼, and a representative of [K] as K. Indeed, the class of s-knots classes reduces to the class of knot classes when we restrict it to graphs consisting of only one (necessarily closed) curve, and Σ ∼ = R3 . So now we can label states on KD by identifying spin network states with equivalent oriented graphs K, as |[K], J, IC ⟩. With this identification, the inner product on cylindrical functions must then undergo the modification ⟨ΦΓ′ ,g , ΨΓ,f ⟩ = Z 0 dU1 · · · dUL g(U1 , · · · UL )f (U1 , · · · UL ) if Γ ∼ Γ′ if Γ Γ′ (5.23) so that |[K], J, In ⟩ forms an orthonormal basis on SD . Taking the same L2 comple∗ tions as before for KD and SD , then, we are done. So for states in the physical space of states, only the inter-weaving of the oriented, ordered links γ, their colourings J and the choice of intertwiners I determine the state− not the points on Σ to which they are mapped. In fact, the only role that Σ plays in defining KD is as the topological background through which the graphs 5.3 The Diffeomorphism-Invariant Space 73 Γ are deformed and identified. Then we no longer need to think of spin networks as being embedded in Σ and occupying space, they are space− we are at last free of the crutch of the “chessboard” that is the manifold Σ, and we can now play the game directly, without extraneous descriptions of the state pertaining to absolute positions and gauge. Fig. 5.1 Two inequivalent knots, and a pair of equivalent s-knots. Chapter 6 Conclusions ∗ Now that we have constructed the kinematic space of states SD ⊂ KD ⊂ SD , we are finished, so far as this dissertation is concerned. But in a much broader sense, we have only just started. The board is set, and the pieces assembled− now it’s finally time to play. But alas, we can only cover so much material in the amount of space/time allocated to us, and we will have to sign off here. Proceeding further down the path set out by our narrative, the next step would logically be a coverage of the geometric operators  and V̂ , respectively the quantum area and volume operators. The (unquantized) area operator has the expression Z q A(σ) = (2)e na Eia nb E bi (6.1) σ q where σ is a 2-dimensional manifold embedded in Σ, and (2)e na Eia nb E bi is its volume form. The spectrum of the quantum operator can be determined, and is of the form (restoring physical constants,) −3 q Â(σ)|S⟩ = 8πG~c γ j(j + 1)|S⟩ (6.2) where γ is again the Barbero-Immirzi parametre. The spectrum of the area operator thus gives a quantitative and testable prediction of loop quantum gravity. Its smallest permissible eigenvalue, for γ = 1, is equal to √ A0 = 4 3πG~c−3 (6.3) Conclusions 76 and corresponds to the quantum of area. Thus spacetime is discretized on the Planck scale, cutting the Gordian Knot on the problem of renormalization. Perhaps also surprising is the fact that the loop quantum gravity formalism already has a known exact solution when the cosmological constant Λ is non-zero. It is given by 6 (6.4) ΨCS [A] = exp − SCS (A) Λ where 2 SCS (A) = Tr A ∧ dA + A ∧ A ∧ A 3 Σ Z (6.5) This is the Chern-Simons state [3]. It is gauge- and 4-diffeomorphism-invariant, and therefore it solves ĤΨCS [A] = 0 (6.6) The physical interpretation of this state, however, is still not entirely clear. Though still an immature science, loop quantum gravity has already found applications, in cosmology and black hole thermodynamics. Very surprisingly, in LQG cosmology, the inflationary phase can be derived analytically through the quantum properties of the gravitational field itself, without the use of, say, a scalar inflation field. Furthermore, the initial cosmological (big bang) “singularity” is no longer singular, due to a lower bound on the permissible values of volume. The latter property is exactly the what we would like to see in a consistent quantum theory of gravity that describes the early universe. [20] In black hole thermodynamics, an expression for the entropy of a black hole in terms of its area is usually given as SBH = kA 4G~ (6.7) This result is commonly known as the Bekenstein-Hawking entropy. A calculation of the same quantity through LQG seems to set the Immirzi-Barbero parametre √ at γ = ln(2)/π 3, but the interpretation of this result is not presently very well understood, either. [20] 77 There is also a formulation of the theory in terms of path integrals and transition amplitudes, which gives rise to spin foams, a generalization of spin networks... All of this is to say that we have only here covered one edge of the tip of the iceberg. Research in LQG remains extremely active today, with multiple teams worldwide working in collaboration to develop it further. References [1] Abhay Ashtekar. Lectures on non-perturbative canonical gravity. World Scientific, 1991. [2] John C Baez. 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