Loop Quantum Gravity and Its Consistency

Transcription

1 Imperial College London Blackett Laboratory Department of Theoretical Physics MSc in Quantum Fields and Fundamental Forces Dissertation Loop Quantum Gravity and Its Consistency Author: Jonathan En Ze Lee CID: Supervisor: Professor Joao Magueijo September 16, 2016 Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London

2 Abstract In this dissertation, the Hamiltonian formulation of Loop Quantum Gravity is discussed and its consistency examined. Beginning with General Relativity, the Hamiltonian approach to General Relativity is reproduced and scrutinized, and then quantised. This quantisation gives us Loop Quantum Gravity. Its implementation and structure are then studied without matter content. Coupling of matter to the theory is later briefly examined. Finally, certain aspects of the consistency of the theory are examined in detail. The analysis starts off by looking at certain problems with the Hamiltonian constraint, and then proceeds to examine the familiar problem of fermion doubling. i

3 Acknowledgements I would like to thank my supervisor Professor Joao Magueijo for taking the time to mentor me despite his schedule. I am very grateful for all answers to the numerous questions I have had, as well as for the general guidance during these few months ii

4 In general we look for a new law by the following process. First we guess it. Then we compute the consequences of the guess to see what would be implied if this law that we guessed is right. Then we compare the result of the computation to nature, with experiment or experience, compare it directly with observation, to see if it works. If it disagrees with experiment it is wrong. In that simple statement is the key to science. It does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is if it disagrees with experiment it is wrong. That is all there is to it. -Richard Feynman, The Character of Physical Law, Chapter 7: Seeking New Laws iii

5 Contents 1 Introduction 1 2 Recasting of General Relativity ADM Formulation of the EH Action Hamiltonian General Relativity Connections and the Asktekar Variables The Barbero-Immirzi Variable and the New Connection 11 3 Loop Quantum Gravity Quantisation The Kinematic Hilbert Space Applying the Gauge and Diffeomorhpism Constraints SU(2) Gauge Invariance and the Spin Network Basis The Spatial Diffeomorphism Constraint The Area and Volume Operators The Area Operator The Volume Operator The Physical Significance of Spin Networks and Knot States The Hamiltonian Constraint Matter Coupling in Loop Quantum Gravity Yang-Mills Theory Fermions Coupling Scalar Coupling States in Matter Coupling The Hamiltonian and its Problems Locality and the Hamiltonian Closure of the Quantum Algebra An Alternative: The Master Constraint Fermion Doubling in LQG In Lattice Gauge Theory Scalar Field Theory on a Lattice Fermions on a Lattice Fermion Doubling In Loop Quantum Gravity Possible Resolutions Conclusion 61 iv

6 1 Introduction Many of the advances in physics since the early 20 th century have been propelled by two theories: General Relativity and Quantum Mechanics. From their inception, both these branches of physics revolutionised scientific thinking and brought about radical (at that time) new moves away from classical physics. Newtonian mechanics clearly defines space and time as two different independent and immutable entities against which measurements about matter in a system under consideration are made. As such, the universe, as defined by classical (Newtonian) mechanics, is fully deterministic. That is to say, if we knew the initial conditions for the universe we could predict what would happen at a given time. With the introduction of Special Relativity, we have had to abandon the idea that space and time are separate. Instead a single entity spacetime is considered and there is no preferred variable for time. Depending on one s frame of reference, the passage of time can be different, and there is no general notion of simultaneity. General Relativity then took this further by saying that spacetime is not some background upon which everything else lives. Rather, spacetime is actually nothing more than the gravitational field and we must treat it dynamically. Background independence enters once we realise that the matter fields no longer live on Minkowski spacetime but rather the matter and gravitational fields live on one another! It also means we can no longer consider time as being external to the dynamical fields as with special relativity, and a time t which is globally observable does not appear in General Relativity. Quantum Mechanics starts off by describing all matter as being both particles and waves. It then goes on to describe systems as wavefunctions in a superposition of physical states, which collapses into a single state under observation. Over the past few decades, Quantum mechanics has been combined with Special Relativity to give us quantum field theory, and thus the standard model which has worked extraordinarily well. Quantum field theory has given us many predictions which have been observed, making it an extremely successful theory. However, it is still background dependent since it is a pertubation theory that uses the Minkowski spacetime as the background. Each theory, taken individually, falls short in its own way since one does not incorporate the other. It is hoped that, by combining the two into a single theory of quantum gravity, we might be able to resolve many of the numerous problems which are currently outstanding. It is possible to split the approaches to quantum gravity into two main branches: pertubative and non-pertubative. The most prominent of the theories is string theory, which is pertubative. String theory proposes that particles are not points but onedimensional extended objects and with pertubation being performed on a 1

7 Minkowski spacetime background. For non-pertubative theories, the key is that background independence is an immutable fact, and so it is the starting point. Such theories include Causal Dynamical Triangualation, Causal Set Theory and Non-Commutative Geometry, among others. The focus of this dissertation, however, will be on Loop Quantum Gravity. The main idea behind Loop Quantum Gravity is to quantise General Relativity. We begin by separating general spacetime manifolds in a 3 1 manner so that d 4 x dtd 3 x. This requires the Einstein-Hilbert action to be rewritten so that it is expressed as a product manifold. Once this is done, the next step is to move on to a Hamiltonian formulation for General Relativity from the Langrangian formalism. Upon doing so, we end up with constraints, and find that Hamiltonian General Relativity is a fully constrained system. The configuration variable of this system is the spatial part of the metric. This system gives us two constraints which generate spatial diffeomorphism and time translations when we take their poisson bracket with the phase space variables. The variables of the hypersurface metric and its conjugate momenta, however, are not ideal for quantisation, and the connection of the theory was found to be better for such a task. By using the connection instead as the configuration variable of the system, General Relativity can now be re-expressed as a gauge theory much like in the fashion of field theories. This re-expression also gives us a third constraint, the Gauss constraint since the connection has more degrees of freedom than the spatial metric. In order now to quantise this theory, the variables are promoted to operators that act on wavefunctions and the poisson brackets become commutation relations between these operators. Unfortunately, if you simply promote the connection and its conjugate momentum to operators in that fashion, they take a wavefunction out of the kinematic state space. Therefore, the holonomy (parallel propagator) of the connection along curves is preferred as it keeps the wavefunction it acts on within the Hilbert space. Its conjugate momentum is then defined as the Electric flux integrated across a surface, so that when taking the commutator of the holonomy and its conjugate, you get a non-distributional result. The usage of holonomy as the basic variable of the theory is where the name Loop Quantum Gravity was derived. The Gaussian constraint and spatial diffeomorphism constraint are applied in order to find the space of states that are SU(2) and spatially diffeomorphism invariant. It is then possible to construct area and volume operators for the theory from the classical expressions but they must be regulated appropriately. Upon examination of these operators, they are found to have the spin networks (states of LQG) as eigenvectors and give a discrete spectrum of eigenvalues! Area and volume of space have been derived to be quantised in, Loop Quantum Gravity, instead of assumed. The remaining constraint, the generator of time translation, is often 2

8 called the Hamiltonian Constraint as it determines the dynamics of the system. In order to quantise it, this constraint must be regularised much like the area and volume operators. There is much ambiguity in the quantum form of the Hamiltonian Constraint due to the way it is defined. As such, its action is not always straightforward to state, although a naive version of its action will be presented later. Having outlined the theory, it must be consistent with already established physics. Any theory of quantum gravity cannot violate the theories that have preceded it when the previous theories have already been backed up by volumes of experimental evidence. One of the problems encountered involves the commutators of the constraints in the quantum theory. In the classical theory, spacetime covariance is recovered via the algebra of the constraints when scrutinising their poisson brackets. When moving to the quantum theory, examining the constraint algebra is not as simple due to a few obstacles. It is then necessary to say that the algebra closes in a weaker fashion than usual. This weaker notion of closure then brings into question the recovery of full spacetime diffeomorphism group. Another problem that is encountered is fermion doubling. It was a problem that was first encountered in Lattice Gauge Theory and extends to Loop Quantum Gravity. The mathematical origin of the problem is that, when looking at the propagator of the free Dirac theory, the introduction of lattice momentum doubles the number of fermions in the theory (for each dimension that is put on the lattice). This is noted by the fact that the lattice propagator has an inappropriate number of singularities (and thus particles). Since this is not seen in reality, it can be considered to be inconsistent. Potential solutions can be pulled from those that have been in use by Lattice Gauge Theory although there may be alternative formulations of Loop Quantum Gravity that avoid this problem altogether. There exist other concerns about the consistency of Loop Quantum Gravity with known physics besides these two main areas that will be covered here. Problems with consistency are an important area to pursue. Hopefully research in this area will either eventually allow Loop Quantum Gravity to be considered a viable theory for quantum gravity with observations to be proven by experiment, or to be shown as inconsistent and therefore allowing researchers to look into more fruitful theories. 3

9 2 Recasting of General Relativity 2.1 ADM Formulation of the EH Action In desiring to quantise Einstein s general relativity in a Hamiltonian manner, the action must first be cast in a form that allows us to reach a canonical formulation. The first step in this direction has beeen done by Richard Arnowitt, Stanley Deser and Charles W. Misner, and the resulting action was called the ADM action [1]. In moving to this formalism, we need to assume the manifold M as being globally hyperbolic, 1 which allows for a foliation of the manifold into a one-parameter family of hypersurfaces. This means that the manifold can be thought of as the product manifold (R Σ), where Σ is a generic 3-dimensional manifold, and we take the real line to be our time direction. By doing this, one might worry that general covariance is broken by the separation time and space in this manner, and that some of the content of general relativity is lost. However, we later see that general covariance is in fact still present in the classical theory within this new Hamiltonian formulation. 2 The starting point is the Einstein-Hilbert action (1), which needs to be altered to reflect this decomposition so that we can move to the Hamiltonian formulation(κ = 16πG and set c = 1). This action is equivalent (by variation of the metric) to the vacuum Einstein equations and, if you desire to add matter content, you could add relevant terms to the action. A temporary distinction will be made between the Ricci scalar for the full manifold R (M) and the Ricci scalar of the surface Σ which is R (Σ). [2] S = 1 d 4 x det(g) R (M) (1) κ M The decomposition of the full manifold means that it can be foliated into a family of hypersurfaces Σ t labelled by the value t along the real line. We can then define the embedding of these surfaces as 3 x µ t = xµ t (ya ) := x µ (t, y a ), where x µ are the coordinates of M, y a are the coordinates of Σ t, and the t labels of x µ t differentiate the embedding for different surfaces. The foliation is defined to be arbitary as we see that this freedom in foliation is equivalent to the diffeomorphism group Diff(M), since the action is not changed by the foliation.[2] We can represent an arbitrary foliation of the manifold by defining the deformation vector field as shown below. 1 The assumption that we have a globally hyperbolic manifold works while we are still considering the classical theory. However the assumption is not true for the quantum theory as topology changes are possible, but this is not a hindrance to the construction of the theory. For more details on this specific topic, see T.Thiemann s book [2] and the relevant references it gives. 2 In general you would need to add boundary terms to (1) but for simplicity this is not done here, [2] has more information. 3 a, b, c,... = 1, 2, 3 are the spatial indices and µ, ν... = 0, 1, 2, 3 are the spacetime indices. 4

10 Figure 1: The foliation of spacetime and the deformation field. Source: [2] ( x T µ µ ) (t, y) (x) := t x=x(t,y) = N(x)n µ (x) + N µ (x) (2) n µ is the normal to the hypersurface, the shift N µ lies tangent to the surface and the lapse N is the constant of proportionality of the normal. Figure (1) displays the situation described by these vectors. n µ therefore is determined by the requirements that g µν n µ n ν = 0 and g µν n µ x ν,a. We start with the first and second fundamental form of Σ t, which are q µν and K µν respectively. The first fundamental form helps to define the metric on the surface while the second fundamental form is the extrinsic curvature of the surface. q µν (x) := g µν + n µ n ν (3) K µν (x) := q ρ µq σ ν ρ n σ (4) R (M) = R (Σ) + [K µν K µν K 2 ] 2 µ (n ν ν n µ n µ ν n ν ) (5) These are used to express R (M) in terms of R (Σ) in what is called the Codazzi equation (5).[2] We then pull back these quantities onto the hypersurface. The pull back of the volume form Ω(x) gives us the equation (8). We define the extrinsic curvature K ab of Σ in equation (7) and the pullback of the metric on Σ in equation (6). To do the full calculation of then taking 5

11 the Einstein Hilbert action and putting it in the terms of Σ is relatively long and can be found in [2], where the preceding equations are derived. After the calculation, one arrives at the ADM action shown below (9). q ab (t, y) := ( x µ,ax ν,b q µν) (x(t, y)) = gµν (x(t, y))x µ,a(t, y)x ν,b(t, y) (6) K ab (t, y) := ( x µ,ax ν,b K ( µν) (x(t, y)) = x µ,a x ν,b ) µ n ν (t, y) (7) S = 1 κ Ω(x) := det(g) d 4 x N 2 det(q) dtd 3 y (8) R dt d 3 y N 2 det(q)(r Σ + [K ab K ab (Ka) a 2 ]) (9) Σ 2.2 Hamiltonian General Relativity We now want to find a canonical pair for general relativity, so that we can cast it in a Hamiltonian manner. The configuration variables are given as q ab (t, y), N(t, y) and N a (t, y), and we now seek their conjugate momenta. In doing so we see that, since the action lacks the time derivation of the latter two, we have only one conjugate momentum and two constraints. Let Π and Π a be the momenta of N and N a while C and C a are the treatment of these momenta as constraints. P ab (t, y) := δs δq ab (t, y) = sign(n) 1 det(q)[k ab q ab (K c κ c)] (10) C(t, y) := Π = C a (t, y) := Π a = δs = 0 (11) δṅ(t, y) δs δ N a (t, y) = 0 (12) In accordance with Dirac s treatment of constrained systems [3], these constraints are known as primary constraints and they are first class. The Lagrange multipliers λ and λ a and introduced for these primary constraints. We then cast the action in to the form in (13) via a Legendre transform. { [ S = dt d 3 y q ab (P, q, N, N)P ab + ṄΠ + N a Π a q ab P ab + λc + λ a C a R Σ κ 1 ]} N 2 det(q)(r σ + [K ab K ab (Ka) a 2 ])(P, q, N, N) (13) 6

12 S = := R R dt dt Σ Σ { d 3 y q ab P ab + ṄΠ + { d 3 y q ab P ab + ṄΠ + [ ]} N a Π a λc + λ a C a + N a H a + N H } N a Π a κh (14) The term H is what would normally be called the Hamiltonian, and is composed by the primary constraints plus two extra terms, H a and H. The smearing of these terms can be defined for arbitrary functions f and f as C(f) := Σ d3 y fc and C( f) := Σ d3 y f a C a. We then want to take the poisson brackets { C( f), H} and {C(f), H} to ensure that they are equal to zero. This is not a given (as per Dirac) and the constraints are applied only after the brackets have been taken. It is indeed the case that they are not equal to zero, but give the equations (15). { C( f), H} = H( f) ( ) N {C(f), H} = H N f (15) The terms H a and H are then called the secondary constraints, since for any f and f the equations (15) must be equal to zero. They are still considered first class ( constraints. ) We call H( f) the spatial diffeomorphism constraint while H N N f is called the Hamiltonian constraint. Once again, it is necessary to check the poisson brackets of the secondary constraints. They give the poisson bracket (16) and can be cast as the constraint algebra in the form of a Dirac algebra.(17) {H, H( f)} = H(L N f) H(L f N ) {H, H(f)} = H(L N f) + H( N( N, f, q) (16) { H( f)), H( g)} = κ H(L f g) { H( f), H(f)} = κh(l f f) {H(f), H(g)} = κ H( N(f, g, q)) (17) Noting that the constraints 4 C and C do not contribute to the equations of motion, we arrive at what is called the canonical ADM action (18).[2] 4 Examining the equations of motion for the shift and Lapse from (14), we find Ṅ a = λ a and Ṅ = λ. It follows that, Since λa, λ are arbitrary, unspecified functions we see that also the trajectory of lapse and shift is completely arbitrary. [2] 7

13 The naive Hamiltonian H is once again the term in the square brackets and consists of only the linear combination of secondary constraints. S = 1 dt d 3 y{ q ab P ab [N a H a + N H]} (18) κ R Σ { H( N), q ab } = κ(l N q) ab {H( N ), q ab } = κ(l Nn q) ab (19) { H( N), P ab } = κ(l N P ) ab {H( N ), P µν } = qµν NH 2 N det(q)[q µρ q νσ q µν q ρσ ]R (M) ρσ + (L Nn P ) µν (20) To show that these generate spatial diffeomorphism and time translations, first take their Poisson bracket with the pulled-back metric q ab. Following source [2], we get the results (19) which indeed show that H leads to spatial diffeomorphisms, while H gives diffeomorphisms orthogonal to the spatial surface. (L a [ ] is the Lie derivative of [ ] with respect to a) If we attempt this for the conjugate momentum, we arrive at the same conclusion for H. However for H, we see that it can only be interpreted as the generator in the direction orthogonal to the hypersurfaces once we apply the constraints where the vacuum equations motion are true. 5 Thus for any tensor t ab built from these canonical variables q ab and P ab, if the equations of motion hold, H a generates a flow along the spatial direction that preserves the foliation of M while H generates a flow in the direction orthogonal to the hypersurfaces. The question that is still open is whether or not we recovered the group Diff(M). It is apparent that we still at least have diffeomorphism invariance for the hypersurface Σ, thanks to the spatial diffeomorphism constraint. What we need is for the constraint algebra to be a Lie algebra that returns Diff(M) to us. Unfortunately this would not seem to be the case. While the first two poisson brackets of (17) between the constraints are closed without involving the phase space, the poisson bracket of the Hamiltonian constraint instead returns with what is called a structure function instead of a structure constant since it involves the metric q. Therefore, the constraint algebra is not a Lie algebra! Instead it is what is known as a Dirac algebra. How then is covariance recovered for this situation? Papers by Bergmann and Komar have analysed such groups. 6 Let such groups be D(M). We see that if we restrict ourselves to the solutions (equations of motion), D(M) = Diff(M). Only when we go off-shell do these two groups differ, but in the classical general relativity this does not come into play. The inherent 5 The vacuum equations of motion: R M µν 1 2 RM g µν = 0 6 See [2] for more details. 8

14 presence of this difference is because the Diff(M) group is related to the kinematical symmetry and does not care about the Lagrangian, while D(M) is related to the dynamical symmetry and therefore would be sensitive to the Lagrangian s form. This is therefore the reason that the two match under the equations of motion, and thus we confirm that general covariance is still present within the canonical formulation of classical general relativity. ([4],[2]) The answer however, is not so easy once we get into the quantisation of Hamiltonian general relativity, as the question of closure of the constraint algebra is called into question. It is represents one of the outstanding open problems within Loop Quantum Gravity and will be revisited in Chapter Connections and the Asktekar Variables At this point, while we have general relativity in a canonical form, the variables that appear in the previous section did not allow for much progress in quantisation to be made. The main problem in using such variables lie with quantising the Hamiltonian constraint as it eludes a simple interpretation, unlike the diffeomorphism constraint. 7 Quantisation of the Hamiltonian constraint needs to be done directly and such a task was not completed in a general sense. The inability to define an inner product and observables for the quantum theory meant that this line of research had reached a dead end.[5] 8 In 1986 A. Ashtekar introduced new canonical variables that allowed for quantisation of the theory, and it is these variables that will be covered in this section. While the new variables were introduced in a spinorial formulation, this section will introduce the new variables using the triads instead in order to simplify things. ([6],[2]) q ab := e j ae k b δ jk (21) G(Λ) := σ G ab := K j [a ej b] = 0 (22) d 3 xλ jk K aj E a k ; G j = ɛ jkl K k a E a l (23) We first move to the local frame fields in 3-dimensions, so that the noncoordinate basis is defined using co-dreibeins e i a as in (21). (Let the indices i,j...=1,2,3) These new indices carry the representation of so(3) and the 7 Application of the diffeomorphism constraint can be simplified as just a requirement that wavefunctions are invariant under spatial diffeomorphisms. 8 A slightly clearer explanation of the difficulties encountered can be found in [2] and [5] although it is a focus of neither book. 9

15 metric is by definition invariant under such transformations. As the Cartan- Killing metric for this group is just the identity matrix, up/down placing of the (i,j,k) indices does not matter. Using the dreibeins, we can define the extrinsic curvature using the new indices as K ab = K(a i ei b), and can define the new constraint (22) as simply the fact that K ab, by definition, is symmetric. We can also rewrite it using the densitised Triad Ei a = det(q)e a i and replacing e a i with Ea i. In fact using these, we can define an extended phase space (Ka, i Ei a ). By using the constraint (22), we get back ADM phase space. The proof involves the rotational constraint (23) where Λ jk is an antisymmetric matrix and it is a scalar that takes values in so(3). ((22) can equally be expressed as (23)) Writing q ab and P ab as functions of Ei a and Ka, i we then can check that their poission brackets are reduced to the correct results after applying the constraints. [2] D a v j = a v j + Γ ajk v k (24) D a e j b = ae j b Γc ab ej c + Γ ajk e k b = 0 (25) Γ ajk = e b k [ ae j b Γc ab ej c] (26) Examining the action of the (metric compatible) covariant derivative on the tensors, it can be found that it s action on a tensor with so(3) indices can be defined as (24), where Γ ajk is the spin connection and can be defined from the Levi-Civita connection as (26). Also note that the covariant derivative acting on the triad gives zero (25). This also applies for the densitised triad. G jk = ( a E a + [A a, E a ]) jk (27) G j = ɛ jkl Ka k El a (28) ( E a ) j = D a + ɛ jkl Ka k El a γ ( E a ) [ ] j E = a + ɛ jkl Γ k a + γka k a l := D(γ) a E a(γ) j (29) γ γ A j(γ) a := Γ j a + γk j a (30) D a (γ) v j := a v l + ɛ jkl Aa k(γ) v l ; D a (γ) u b := D a u b (31) In order to express the rotation constraint as the Gauss constraint of a gauge theory, we want to have (27) for some potential field A. Before this is 10

16 done however, observe that a canonical rescaling of the pair (Ka, i Ei a ) can be done as (γka, i Ea i γ ) := (Ki(γ) a, E a(γ) i ). Using the (28) expression of the rotational constraint combined with (25), and the recognizing Levi-Civita tensor as the generator of SO(3), we get (29). Therefore it is possible to identify the potential field A as the combination of the Levi-Civita connection and the extrinsic curvature (30). The free parameter γ is called the Immirzi- Barbero variable. The full name of this new connection A can be given as the Sen-Ashtekar-Immirzi-Barbero connection. This connection defines a new covariant derivative which acts as (31) on tensors with so(3) indices, and tensors with spatial indices respectively. F j(γ) ab = R j ab + 2γD [ak j b] + γ2 ɛ jkl K k a K l b (32) Introducing the field strength tensor F abjl v l = [D a, D b ]v j = ɛ jkl F k ab vl, which is the the curvature given the connection A, and using R abjl v l = [D a, D b ]v j = ɛ jkl R k ab vl we can express the field strength as (32). H a = H a + f j ag j H = H + f j G j (33) G j = D a (γ) E a(γ) j H a = F j(γ) ab E b(γ) j H = [F j(γ) ab (γ 2 + 1)ɛ jmn Ka m Kb n ] ɛ jklek a Eb l (34) det(q) S = 1 dt d 3 y{ A κ i(γ) a E a(γ) i [Λ j G j + N a H a + N H]} (35) R Σ Turning now to the Gauss, Diffeomorphism and Hamiltonian constraints, it can be found that the expression of the Hamiltonian and Diffeomorphism constraints contain pieces which are proportional to the Gauss constraint in the manner shown in (33). Considering that moving from the old variables (Ka, i Ei a ) to the new ones (Ai(γ) a, E a(γ) i ) is a canonical transformation, and following the reasoning in [2] we see that it is equivalent to work with the constraints redefined as (34). H and H a no longer have pieces proportional to G j. Finally, using these constraints and the new canonical variables we can finally write the Einstein Hilbert action as (35) The Barbero-Immirzi Variable and the New Connection Ashtekar originally chose to set γ = ±i keeping in mind that this would greatly simplify the Hamiltonian Constraint by eliminating the second term and thus making it polynomial after a factor of det(q) has been multiplied 11

17 out. Unfortunately, what then occurs is that it becomes necessary to enforce reality conditions (36), as the theory only should have SU(2) gauge transformations, but allowing the connection to be complex without restriction in turn allows for SL(2, C) transformations. Since these conditions are nonpolynomial it becomes difficult to implement them upon quantisation.[2] E (β) β = E(β) β ; [ A (β) Γ ] β = [ A (β) Γ ] β (36) It was found however, that T.Thiemann s regularisation of the full Hamiltonian makes it unnecessary to simplify the Hamiltonian. In addition to this J. Fernando Barbero G had shown that you can have Lorentzian General Relativity with just the real connection.[7] There exist criticisms of Lorentz Covariance in this connection [8], but it seems a connection to Lorentzian General relativity can still be made (with certain caveats).[9] As a result, much of the work that has been done in this field pertains to the real connection and dispenses with the need for the tricky reality conditions. The real connection is often referred to as the Barbero connection, while setting γ = ±i gives the Ashtekar connection. 12

18 3 Loop Quantum Gravity 3.1 Quantisation In this section, the Sen-Ashtekar-Immirzi-Barbero connection is taken to be real so that we will be dealing with the gauge group SU(2) instead of the group SL(2, C). (γ = 1 will be used for convenience here unless explicitly stated.) The reason underlying this is that the group SL(2, C) is non compact, which does not allow us to directly apply numerous techniques from Yang-Mills theory. Furthermore, most of the progress that has been made in Loop Quantum Gravity has been with the real connection. Specifically, the Euclidean theory will be covered in this section as it is by far more approachable, and the action of its Hamiltonian is easier to describe. 9 While the full Lorentz theory will not be fully described by the results of the Euclidean theory, the quantisation is largely the same and very often we can extend the results to the Lorentz theory. This section will follow the book by Carlo Rovelli [10] with deviations to other sources. For an extremely detailed account one could look at Thomas Thiemann s book as well [2]. To quantise Hamiltonian general relativity, we first consider wave functions of the connection Φ[A] which are functionals on the configuration space G (space of the 3-dimensional connections defined on Σ). The canonical variables are promoted to operators that act on these wave functions (37) and poisson brackets to commutation relations of the operators [10]. The Hamiltonian constraint acting on a state HΦ = 0 gives us the Wheeler- Dewitt equation and governs the dynamics of our system. The remaining two constraints are the conditions that we must have gauge and diffeomorphism invariant states. What we want to find is the suitable Gelfund triple S H S where S is a suitable space of the functionals Φ[A] so that we have a kinematic state space. Â i a(τ)φ[a] = A i a(τ)φ[a] 8πGÊa 1 δ i (τ)φ[a] = i δa i Φ[A] (37) a(τ) A few changes are still required to the canonical variables before we embark to find the rigged Hilbert space for quantisation. Since the Poisson bracket between A j a and Ej a turns out to be a distribution, we might first think of smearing these variables with test functions fa, j Fj a as our first attempt (38). However, if we use the (smeared) Gauss constraint (which is the generator of gauge transformation), and take its poisson bracket with the 9 This is because in the Hamiltonian constraint the term (β 2 + 1) becomes (β 2 1) for the euclidean theory, which then means that for γ = 1 the term will cancel, simplifying the constraint in an identical manner that Ashtekar originally did for the Lorentizan theory. 13

19 smeared connection we see that it transforms inhomogeneously as a gauge potential rather than in the adjoint representation. E(f) := {G(Λ), F (A)} = β κ 2 σ d 3 xfae j j a ; F (A) := σ d 3 xf a j A j a (38) d 3 xf a j [ a (Λ j ) + ɛ jkl A k aλ l ] (39) How then do we construct something that does transform in the adjoint? This problem has not only been considered by Loop Quantum Gravity, but in fact it has been well studied in Lattice Gauge Theories. The solution that has been found is the method of Wilson Loops.[11] Let s start by defining parallel transport from connections and then holonomies. Say that we have a curve γ and parameter t defined below, as well as a covariant derivative a defined by a connection A. The requirement that a co-vector E b is parallel transported along γ is written as γ a a E b = 0. Expanding this equation, we get to the partial differential equation (41), and integrate it so that we get the following line.[12] γ : [0, 1] Σ; s x µ (s); γ(0) = I (40) γ a (s) a E b (s) = ig γ d (s)a d (s)e b (s) E b (s) = E b (0) ig s 0 dt γ a (t)a a (t)e b (t) (41) We want to eliminate of the term E b (t) from the right-hand side of the equation (41). To do this we iterate the equation by inserting the equation into itself to arrive at the expression (42) which is now a sum to infinity with E b (0). We can also define the path ordering operator P which pushes operators with larger t to the left. Using the formula (43), we can then express this as the path ordered exponential( 44) and gives the parallel propagator h γ (A). E b (s) = ) (( ig) n ds 1...ds n γ a 1 (s 1 )A a1 (s 1 )... γ an (s n )A an (s n ) E b (0) s 1... s n 0 (42) n=0 t 1... t n 0 dt 1...dt n γ a1 (t 1 )A a1 (t 1 )... γ an (t n )A an (t n ) = 1 n! P ( s 0 ) n γ a (t)a a (t)dt (43) 14

20 [ ( s )] E b (t) = P exp ig γ a (t)a a (t)dt E b (0) = h γ (A)E b (0) (44) 0 When the starting and ending point of the curve are the same point, the parallel propagator is instead known as the holonomy 10 U(A, γ). The holonomy can be defined as a linear transformation at a point p and can be interpreted as the failure of the parallel transport around a loop to preserve the tensor. The holonomy can be said to be a group element determined by the connection A and path γ. Going back to the Asktekar connection A a, let us write it as the oneform A = A i aτ i dx a where τ i = i 2 σ i is the su(2) Lie algebra basis, and σ i are the Pauli matrices. We can do this as we know that A transforms as a gauge potential in the adjoint representation, and the Lie algebra of SO(3) and SU(2) are isomorphic to one another. If we then examine the gauge transformation of the holonomy, we find that it transforms nicely in the adjoint representation of SU(2) and the connection gets smeared along one dimension. Just like the wavefunctions, the holonomy (given a curve γ) is a functional on the configuration space G. For completeness, how then do we smear the conjugate operator Ej a? For reasons to be discussed later, the form of E that we are using will be E i which is the functional derivative smeared across a two dimensional surface S. Moving along, Φ[A] can now be expressed using the holonomies as a basis. Say that there is a collection Γ of smooth oriented paths {γ i : i = 1,..., l} which are embedded in the hypersurface Σ. Let this collection be ordered. We also have a smooth function of group elements f(u 1,..., U l ) known as a cylindrical function. A brief definition of the cylindrical function is that it is a function of classical configuration space (see source [5] for a better description and [2] for a more rigorous one). We are then able write a functional of the connection Φ[A] as (45), where (Γ, f) defines the functional. We can then define S as the space of these functionals for all Γ and f. A scalar product for S is defined in (46) for two wavefunctions defined by the same ordered oriented graph Γ, but different functions f and g. du is the Haar measure on SU(2). Φ Γ,f [A] = f(u(a, γ 1 ),..., U(A, γ l )) (45) Φ Γ,f Φ Γ,g du 1 du l f (U 1,..., U l )g (U 1,..., U l ) (46) This product can be simply extended if the two wavefunctions differ in their graphs merely by ordering or orientation. 10 Unfortunately, the terms holonomy are and parallel propagator are often used interchangably in the literature, so little distinction is made here. 15

21 However, what happens if the functionals Φ Γ,f and Φ Γ,g are defined for different Γ? In this case, we start off by considering the union of the l and l curves of the two graphs (Γ = Γ Γ ), and define the new functions in the manner g (U 1,..., U l ) = g(u 1,..., U l, U l +1,..., U l +l ). We can now define the scalar product of these two different Φ as (47). Φ Γ,f Φ Γ,g Φ Γ,f Φ Γ,g (47) Aside: Initally, Loop Quantum Gravity was constructed as a theory of loop states provided by the case of Γ being a single closed curve α and the function f was the trace of the holonomy (tr). This state is written as α = Φ α. To express this in terms of the connection A, we have (48). Φ α,tr [A] = A α = tru(a, α) (48) Φ α 2 = du tru 2 = 1 (49) Φ {α} [A] = Φ α1 [A]...Φ αn [A] (50) Φ[α] = Φ α Φ = ( ) dµ 0 [A]tr P e αa Φ[A] (51) The norm is then given by the scalar product we defined earlier and gives us (49). We can have a multiloop state which is a finite collection {α} of loops defined in (50). A functional in loop space is given by (51), where we can see that it looks like a sort of Fourier transform from the space of connections to the space of loops. The reason that this representation fell out of favour is that the loops here form an over complete basis and result in complicated non-linear relations between the different elements of the basis. 3.2 The Kinematic Hilbert Space Now that a general idea of how to construct basic wavefunctions Φ[A] has been covered, let us examine the kinematic Hilbert space in detail. The space S is the space of linear finite combinations of the states Φ[A]. Then the kinematic Hilbert space H kin is defined as the space of all linear superpositions of these wavefunctions with a finite norm.[13] Mathematically, we can say that it is the space of the Cauchy sequences Φ n, where Φ m Φ n converges to zero.[10] The dual of S is S, and is defined as the space of sequences Φ n such that Φ n Φ converges for all Φ in S.[10] This gives us the full definition of the rigged Hilbert space that we mentioned earlier. We use this definition, as the scalar product defined earlier is diffeomorphism 16

22 and locally gauge invariant. It gives real classical observables as self-adjoint operators. The strict conditions that the scalar product satisfies are necessary so that we have a consistent theory that gives correct classical limit. Furthermore we have that the loop states Φ α can be normalized. One of the criticisms that might be raised is that the H kin is non-separable and stems from the spatial hypersurface Σ which is a continuum.[13] However, we see that as we go to the physical Hilbert space H, the non-separability of the original space was just gauge freedom and the physical Hilbert space itself is separable. [10] For a given collection of paths Γ, the cylindrical functions with support on Γ make up the space H kin Γ H kin which is finite dimensional. This is the space of square integrable functions of SU(2) L where L is the number of paths in Γ. Consequently if we have a another graph Γ Γ then the space H kin Γ is proper subspace of H kin Γ. This structure gives a projective family of kinetic Hilbert spaces, where H kin is known as the projective limit of this. Using the Peter-Weyl theorem, we can find a basis for H kin. The theorem states that: A basis on the Hilbert space of L 2 functions on SU(2) is given by the matrix elements of the irreducible representations of the group. [10] Following [10], such representations are labelled by their spin j and the Hilbert spaces on which they are defined are labelled H j and their modules are v α. Therefore we have matrices labelled by the representation they are in and which group element they correspond to in (52). In this case we are using the holonomies as the group elements of SU(2), and the indices α and β label the matrix elements. R (j)α β(u) = U j, α, β (52) Consider again a graph that is a collection of paths Γ = {γ i ; i = 1,..., L}. Putting the previous information together we can obtain a basis for the subspace H kin Γ. By picking an ordering and orientation for Γ, we can then define a basis as (53). We can also represent this basis for H kin Γ as the tensor product of the matrix elements defined earlier (54). In order that the vectors are an orthonormal basis in H kin we only take the states where j only takes the values ( 1 2, 1, 3 2,...), and not the singlet representation (j = 0). To see the reason for this, consider two graphs Γ Γ. The same vector appears in both the Hilbert spaces for Γ and Γ. However any vector of Hkin Γ belongs to the singlet representation of the loops that are in Γ but not Γ. By eliminating the vectors which have (j = 0) for any j we eliminate this redundancy. Γ, j l, α l, β l Γ, j 1,..., j L, α 1,..., α L, β 1,..., β 1,..., β l (53) A Γ, j l, α l, β l = R (j 1)α 1 β1 (U(A, c 1 ))...R (j L)α L βl (U(A, c L )) (54) 17

23 We can then define the proper graph subspace Hkin Γ, where HΓ kin is spanned by the basis states of HΓ kin with the extra condition j l > 0. All the proper subspaces are orthogonal, and span the full kinematic Hilbert space which we can now define in terms of the proper subspaces as in (55), for all possible graphs Γ (including the Γ = graph). Without detail, H kin is the space of square integrable functions on the extended configuation space discussed earlier, with the Ashtekar-Lewandoski measure.[10] H kin Γ H Γ kin (55) One of the last questions that we can pose, before leaving the topic of kinematic state space behind, is the invariance of the scalar product. The transformations of the connection A i a mean that the kinematical state space S kin carries a natural representation for local SU(2) and spatial diffeomorphisms Dif f(σ). Furthermore, because of the way that the scalar product was defined (46), we find that it is invariant under transforms of these groups, and thus H kin carries a unitary representation of these groups.[10] As discussed earlier, A i a transforms as a gauge potential while the parallel propagator U[A, γ] transforms homogenously as (56). We then define the cylindrical functions under gauge transformations as f λ in (57), and then the transformation of the quantum state therefore as (58). Given these definitions, we can see that the scalar product is indeed invariant under local gauge transformations. The basis states Γ, j l, α j, β l transform as (59), where i l and f l represent the points a path l begin and end respectively. U[A, γ] U[A λ, γ] = λ(x γ f )U[A, γ]λ 1 (x γ i ) (56) f λ (U 1,..., U L ) = f(λ(x γ 1 f )U[A, γ 1]λ 1 (x γ 1 i ),..., λ(x γ L f )U[A, c L ]λ 1 (x γ L i ) (57) Φ Γ,f (A) [U λ Φ Γ,f ](A) = Φ Γ,f (A λ 1) = Φ Γ,fλ 1 (A) (58) U λ Γ, j l, α l, β l = R (j 1)α 1 α 1 (λ 1 (x f1 ))R (j 1)β 1 β 1 (λ(x i1 ))...R (j L)α L α L (λ 1 (x fl ))R (j L)β L β L (λ(x il ) Γ, j l, α l, β l (59) What about under diffeomorphisms? Let s consider a slightly larger group called extended diffeomorphisms Diff, the reasons behind this will be mentioned in section (3.3.2) Extended diffeomorphisms are invertible maps φ : Σ Σ so that the map and its inverse are continuous and are infinitely differentiable everywhere except at a finite number of points.[14] 18

24 The connection A i a transforms as a one-form as expected, and S kin has the representation U φ of Diff defined by (60). The holonomy transforms as (61), which is the statement that shifting the connection by φ is the same as dragging the curve γ. The cylindrical function defined by the pair (Γ, f) is shifted to a new function defined by (φγ, f). Turning back to the inner product, we see that it depends only on the functions f and g and not the graph. It is therefore invariant under the extended diffeomorphism. U φ Φ(A) = Φ((φ ) 1 A) (60) U[A, γ] U[φ A, γ] = U[A, φ 1 γ] (61) 3.3 Applying the Gauge and Diffeomorhpism Constraints In the previous section, the kinematic Hilbert space was described as the space of arbitrary functionals of the connection. To reach the physical Hilbert space H phys, we must apply the constraints in (34) one by one to the quantum theory. This Hilbert space will therefore be the space of functionals that are solutions to the Wheeler-DeWitt equations (also called the Hamiltonian constraint) and are invariant under diffeomorphisms and SU(2) gauge transformations. The constraints will be applied in the order as shown by (62), where H 0 is the space of states invariant under local gauge transformations and H diff is the space of states invariant under extended diffeomorphisms and local gauge transformations. H kin H 0 H diff H phys (62) These two constraints are approached separately from the Hamiltonian constraint as the application to the theory is not difficult. We are able to construct their corresponding quantum operators and find the Hilbert spaces H 0 and H diff without difficulty. The quantisation and application of the Hamiltonian constraint is a bit more tricky and will be approached in a separate subsection. Furthermore, there are problems that still exist with the Hamiltonian constraint and such problems will be discussed in a separate section SU(2) Gauge Invariance and the Spin Network Basis On the one hand, we could formally apply the diffeomorphism constraint via quantisation to find the space H 0. However, we can also observe that with the multiloop states that were discussed earlier, we already had a basis that spans H The multiloop states were the basis of choice in the earlier 11 The full mathematical derivation and treatment of H 0 can be found in source [2]. 19

25 days of LQG, but overcompleteness of the basis together with the non-linear relation between the basis states made it difficult and complex to use. Instead, the idea of spin networks as a basis was introducted by Rovelli and Smolin [15] in 1995, and form an orthonormal basis. 12 As will be demonstrated later, they are finite linear combination of the multiloop states. Let an ordered collection of oriented curves be denoted as the graph Γ, and let the end points of the curves be called nodes. Assume that if the curves γ Γ overlap, it is only at the nodes. Lets call the curves links, and associate an irreducible representation j of the gauge group with a link l so we have j l (non-trivial representations). The number of links beginning(ending) at a node is called its outgoing(ingoing) multiplicity m out (m in ). The total multiplicity is defined as m = m out + m in. We associate what is called an intertwiner i n to each node n, which map from one representation to another effectively connecting the representations of different links. Consider a graph with L links and N nodes. We can then define a spin network state with the triplet S = (Γ, j n, i n ), where Γ is the graph, j l is the choice of spin representation for each of the L links, and i n is the choice of intertwiner for each of the N nodes. The choice of j l and i n is called a colouring of the links and nodes. Given a general spin network state S = (Γ, j n, i n ), we want to relate it to the basis states Γ, j l, α l, β l that appeared in the earlier section. This basis has L number of α l (up) and L number of β l (down) indices. We find that the set of intertwiners i n is in precisely the dual of the representation. Consequently, we can write the spin network state as the contraction of the two as per (63). The contraction between the indicies occur between representation and intertwiners when a link (representation) ends (β l ) or starts (α l ) at a node (intertwiner). The choice of intertwiner is limited by the representations that end and start from it but this limitation does not mean that the choice is necessarily unique. The local SU(2) invariance of S is evident from the observation of the transformation of the basis Γ, j l, α l, β l and the invariance of the intertwiners. We can write the spin network state as a functional of the connection A i a as in (64). Each representation R (j l) lives in the tensor product space of Hj l H jl and so the first bracket of the ( ) RHS (64) lives in the space l Hj l H jl while the second bracket lives in the dual of this space. S v β 1...β n1 i 1 α1 v β n β n2...α n1 i 2 αn1...v β (n N 1 )+1...β L +1...α n2 i N α(nn 1 Γ, j l, α l, β l )+1...α L (63) 12 The application of spin networks to quantum gravity is from [15] but the original idea of spin networks can actually be though as starting with Penrose.[15] 20

26 ( ) ( ) Φ S [A] = A S R (jl) (U[A, γ l ]). i n l n (64) The basis of the spin networks is therefore labelled by the choice of Γ and the colouring of the links and nodes in that graph. As a reminder, the states for which j = 0 are not included to avoid redundacy. We know from before, that the states Γ, j l, α l, β l form a basis in H kin, and we then used the intertwiners to form a set of basis states that were locally gauge invariant. Technically the Γ labels in the spin network basis representents any unordered and unorientated graph, but the colouring of the links and nodes chooses the ordering and orientation for Γ.[10] The space S 0 (of the Gelfand triple for H 0 ) is the space of any finite linear combination of the spin network states S. Aside: To look at how multiloops and spin network states are related, let s first look at the map ɛ, which is the map between a representation j and its dual j. The object ɛ ab is actually the totally antisymmetric tensor, and allows us to raise and lower indices. The second fact we approach is that a representation with spin j can be written a tensor product of the defining (j defining = 1 2 ) representation 2j times which is fully symmetrised on its up and down indices (separately). The re-casting of R (j) (U) is shown in (65). R (j)α 1,...,α 2j β1,...,b 2j (U) = U (α 1 (β1...u α 2j) β2j ) (65) ζ1,...,ζ v 2j α1,..,α 2j,β 1,...,β 2j = ɛ α1 β 1...ɛ αaβa δ ζ) 1 β a+1...δ ζ b β δζ b+1 2j α a+1...δ ζ 2j α 2j (66) The intertwiners then become simple combintations of δa b and ɛ ab and a general decomposition is demonstrated in (66).(with j = a+c, j = a+b and j = b + c) If the parallel propagator of two curves (γ 1, γ 2 ) are intertwined by the delta δa b as in (67), then the resultant object is the propagator of the curve made by joining the two curves and is denoted as γ 1 #γ 2 = γ 3. On the other hand, if we have the two curves joined by ɛ ab, then we use the fact that ɛ ab U b cɛ cd = (U 1 ) d a to give the combination (γ 1 1 #γ 2). This decomposition of the representations and the intertwiners gives us the decomposition of a spin network state into a linear combination of multiloop states, and a better mathematical treatment can be found in [10]. A graphical idea of this is to replace a spin j link with 2j parallel links and then connect it each end to one other (non-parallel)link that is at the same node as itself. The nonredundant permutations of this give us the decomposition into multiloop states. 21