Covariant Loop Quantum Gravity

Transcription

1 Covariant Loop Quantum Gravity Josh Kirklin 10th March 2016 Quantum physics and general relativity are well known to be at odds. The most popular option for solving this problem is String Theory. We will consider an alternative: Covariant Loop Quantum Gravity. While String Theory takes a perturbative approach, with strings as quantum objects embedded in a background spacetime, Covariant LQG treats spacetime itself as an exact quantum entity, and is thus really a theory of quantum geometry. We will see that for quantum physics and general relativity to work together, we must treat spacetime as granular. To this end, we will investigate Regge calculus and the 2-complex discretisation of spacetime. Quantisation leads to a spectrum of lengths between any two points, and a natural expansion of spacetime states in a spin network basis. The introduction of a positive cosmological constant helpfully fixes any divergences in the action. Although we will mainly look at the 3D case, I will give a descriptive account of the difficulties that arise when moving to 4D. If there is time we will briefly look at some applications. Much of this talk was prepared using [Rovelli and Vidotto, 2015]. Contents 1 Introduction 2 2 Tetrad-connection formulation 3 3 Spacetime discretisation Regge calculus complex discretisation Boundaries Quantisation Length quantisation Gauge Spin network basis Transition amplitude Infrared divergence 13 6 Four dimensions 14 7 Black hole entropy 15 1

2 1 Introduction General Relativity is non-renormalisable. If we try to quantise the Einstein-Hilbert action, we get ultraviolet divergences resulting from arbitrarily small spacetime regions, and it is impossible to make these go away with any standard mechanisms. In this way, GR and quantum physics are at odds. The most popular choice for solving this problem is String Theory (although in this term s String Theory course we didn t go into very much detail on quantum gravity, slightly disappointingly). It takes a perturbative approach; strings are objects embedded in a background spacetime, and gravitons are certain oscillation modes of these strings. I m going to talk about the second most popular option: Loop Quantum Gravity. People who study LQG aren t satisfied with the fact that String Theory is perturbative, and that it is based on a background. LQG instead treats spacetime itself as an exact quantum entity. Suppose we want to know the distance between two particles to a high accuracy. The uncertainty principle implies that we will need to put in a lot of energy to do this. At some point, we put in so much energy that a black hole will form. But then the black hole has non-zero extent, and we cannot reliably resolve distances inside of it. Energy Schwarzschild radius r = 2M accessible uncertainty l P x Therefore together general relativity and quantum physics imply that there is some lower limit on distances. By examining the various relevant equations we can deduce that this minimum length is on the order of a Plank length l P = G c m. (1) So spacetime must be granular at a very fine lengthscale. One main idea in LQG is to embrace this granularity, and to discretise GR before quantising. 2

3 2 Tetrad-connection formulation Covariant LQG is formulated in terms of the tetrad formalism of GR, so I ll briefly review it. Instead of a metric g µν, we use a tetrad i.e. a set of four covector fields e I, I = 0, 1, 2, 3, related to the metric by g µν = e I µe J ν η IJ. We can invert this relation to find that the covector fields are orthonormal. They also satisfy a local Lorentz gauge invariance under e I Λ I J ej. e 1 e 3 e 0 e 2 Thus we have a set of frame-independent axes at each point in space. Recall that in Yang-Mills theory we use a member of the gauge group s Lie algebra as a connection; we do the same here, defining ω IJ = ω JI, a member of the Lie algebra of the Lorentz group, to be the connection of the gravitaitonal field. Together the tetrad and the connection define a Cartan geometry (more general than Riemann in that it includes torsion). Using form notation we define the torsion 2-form by the first Cartan equation and the curvature 2-form by the second Cartan equation T I = de I + ω I J e J, (2) F I J = dω I J + ω I K ω K J. (3) It can be shown that a vanishing torsion implies a unique solution ω[e] for the connection, called the spin connection (Levi-Civita). The curvature of the spin connection is directly related to the Riemann tensor by R µ νρσ = e µ I ej ν F I Jρσ. A little work shows we can write the Einstein-Hilbert action S[g] = 1 d 4 x gr (4) 2 as S[e] = e I e J ( F ) IJ (5) Note that we can let ω be a free variable. This gives the action (omitting contracted indices) S[e, ω] = e e F [ω]. (6) Varying ω in this action, we actually recover the condition that the torsion vanishes, i.e. that ω is the spin connection. 3

4 3 Spacetime discretisation 3.1 Regge calculus This is a discretisation of general relativity introduced by Tullio Regge. We need a few definitions. A d-simplex is the set obtained by taking the convex hull of d + 1 vertices in R d, i.e. the smallest convex subset of R d containing the vertices. In d = 0 this is a point, in d = 1 this is a line segment, in d = 2 this is a triangle, in d = 3 this is a tetrahedron. Note that a d-simplex s boundary is given by the union of d + 1 (d 1)-simplices (called faces). A d-simplex forms a metric space by inheriting the Euclidean metric of R d. d = 0 point d = 1 segment d = 2 triangle d = 3 tetrahedron d = 4 4-simplex etc.... A Regge space (M, L s ) in d dimensions is a d-dimensional metric space obtained by gluing d-simplices along matching boundary (d 1)-simplices. These are often referred to as triangulations. L s Note that the metric of a Regge space is uniquely determined by the lengths L s of its line segments s; these are taken to be the free variables of the theory (subject to some constraints). In a Regge space of d dimensions, a kind of curvature is generated on the (d 2)-simplices, also called hinges. Consider a point P in a d = 2 Regge space. P is contained in a certain number of triangles; let the angle of triangle t at P be θ t (this angle is determined by the lengths L s by elementary geometry). Clearly, if space is flat at P, then these angles will sum to 2π. If the space has curvature at P, the sum will deviate from 2π. 4

5 flat curved Thus we define the Regge curvature at P by δ P (L s ) = 2π t θ t (L s ) (7) Similar logic extends the definition to higher dimensions; the Regge curvature is the deficit angle around a (d 2)-simplex. It is harder to draw. Geometrical interpretation is simple: if we parallel transport a vector around a hinge, the vector changes by the deficit angle. δ δ The Regge action of a Regge space (M, L s ) is given by S M (L s ) = h A h (L s )δ h (L s ). (8) The sum is over the hinges, and A h is the area (or length, volume etc.) of the hinge h. We can approximate any Riemann manifold by triangulating it; i.e. for any ɛ we can choose a triangulation such that the difference between the Riemann distance and the Regge distance is less than ɛ. Riemann manifold Regge space approximate 5

6 The continuum limit is ɛ 0, and remarkably Regge showed that the Regge action converges to the Einstein-Hilbert action in this limit. Regge calculus is a good discretisation of general relativity. Unfortunately, the Regge discretisation is not very good for the quantum theory, for two reasons. It is based on metric variables (the lengths L s ), meaning fermion coupling is impossible. More importantly, the free variables L s are subject to triangle inequalities. Thus the configuration space is very complicated. We will look at another method based on the tetrad-connection formulation of general relativity complex discretisation Given any triangulation, we can construct its dual by replacing each n-simplex with a (d n)-dimensional convex hull in the following way. Consider d = 3. Then we replace each tetrahedron with a vertex, each triangle between two tetrahedra with an edge connecting their respective vertices, and each segment with a face. The resulting structure is known as a 2-complex, and it is on this that we will discretise. From now on we will focus on d = 3 because it is easier than d = 4. We will treat each edge e as oriented, and associate to it a Lorentz group element U e (Ue 1 for the opposite orientation). In the same way that the connection provides a description of change over infinitesimal distances, these group elements describe how the gauge field changes as we move between the vertices of the lattice. The relation between the group elements and the connection is given by U e = P e ω e SU(2), (9) where P denotes that we take a path ordered exponential; this is commonly called the holonomy. This expression for U e is invariant under all local gauge transformations, except for those that act a the boundary points of the edge; thus the local gauge symmetry is reduced to one that acts at vertices only. The elements transform as U e λ se U e λ 1 t e, (10) where s and t denote the start and end point of an edge respectively. 6

7 We can also define a group element for each (oriented) face in the lattice by the ordered product of its edges: U f = U e1 U e2... U ek (11) U e2 U e1 U e3 U f U e4 U e5 This is a discrete version of the curvature. One way to rationalise this is to think of U f as the holonomy of the connection around a closed loop f. By Stokes theorem U f = P exp ω = P exp F. (12) f In particular, if spacetime is flat in f, then U f is the identity. Deviation of U f from the identity is then in direct analogy with curvature. Using (9), we can see that the trace of U f is gauge-invariant: tr U f = tr U e1 U e2... U ek (13) tr λ se1 U e1 λ 1 t e1 λ se2 } {{ } =e U e2 λ 1 t e2 λ se3 } {{ } =e f... λ 1 t ek 1 λ sek } {{ } =e U ek λ 1 t ek (14) = tr λ 1 t ek λ se1 U e1 U e2... U ek (15) }{{} =e Recall that the trace of the curvature is the Ricci scalar, which plays a key role in the Einstein-Hilbert action. We might then expect tr U f to play a similar role in a discretised action. In fact we will need to consider something else to obtain the full action; this is where the tetrad (triad) comes into play. For each segment s of the original triangulation, we associate a vector L i s, given by L i s = e i R 3, (16) s where this definition is understood to be in a gauge where the connection is constant along the segment. Since each segment in the triangulation corresponds to a face in the 2-complex, these are really defined on each face, and it is convenient to view them as elements of the su(2) algebra (it can be shown that they transform in the appropriate way): So the variables of the discretised theory are L f = L i fτ i (17) 7

8 U e SU(2) for each edge e, and L f su(2) for each face f. The action we form from these variables is S = 1 8πG tr(l f U f ), (18) f and it can be shown that in the continuum limit this is equivalent to the Einstein-Hilbert action. 3.3 Boundaries In Hamiltonian mechanics, we have the Hamilton function I(x, t; x, t ), which gives the total action of the path of a system between (x, t) and (x, t ). The quantum analog is the transition amplitude W (x, t; x, t ), determining the probability of a certain set of boundary values. In both cases, although we are interested in the process that occurs between (x, t) and (x, t ), we probe this process by specifying properties of the boundary of the worldline. (x, t ) (x, t) The same will be true in gravity. Consider some subset U of a manifold M obeying Einstein s equations. The properties of spacetime inside U is the process here, and the boundary U is what we use to probe it. U U M 8

9 So let s look at the boundary in a 2-complex. Suppose the original triangulation is only defined in a certain region. Then it will have a boundary consisting of a set of triangles. The triangles become edges in the 2-complex. We call the outer endpoints of these edges nodes. The nodes are connected by links, and together the nodes and links form the boundary graph. links nodes The boundary variables are defined on each link l: U l is the group element defined on each link. L l is the algebra element defined on the face corresponding to each link. Suppose there are L links. su(2) is the cotangent space of SU(2), so we have that the space of boundary variables is T SU(2) L. The cotangent space is well known to have a phase space structure (see e.g. applications of differential geometry to physics), so this is in fact a boundary phase space, with U l the positions, and L l the momenta. The Poisson brackets are: 4 Quantisation To quantise, we will need two things: {U l, U l } = 0, (19) {U l, L i l } = 8πGδ ll U lτ i, (20) {L i l, L j l } = 8πGδ ll ɛ ij k Lk l. (21) A Hilbert space describing the quantum states of the boundary geometry. A transition amplitude for these boundary states. U l are the coordinates, so we define the Hilbert space of the boundary graph Γ as H Γ = L 2 [SU(2) L ], (22) so states are wavefunctions ψ(u l ). The scalar product is given by ψ φ = du l ψ(u l )φ(u l ), (23) SU(2) L 9

10 where we do the integrals by treating SU(2) as a round sphere. There is a natural covariant derivative operator on the functions ψ(u l ) given by (J i ψ)(u) = i d dt ψ(uetτ i ) (24) t=0 where τ i are the generators of SU(2). We then define L i l = 8π GJ l i. It is simple to show then that the operators U l and L i l satisfy the appropriate quantised algebra [U l, U l ] = 0, (25) [U l, L i l ] = i8π Gδ ll U lτ i, (26) [L i l, L j l ] = i8π Gδ ll ɛ ij k Lk l. (27) By the normal arguments, this approaches the classical structure in the limit Length quantisation The length L s of a segment s in a triangulation is given by L f. If s is a boundary segment, then this length is L l, where l is the link that crosses that segment. Note that J is a generator of SU(2)), and its square J 2 is the SU(2) Casimir. This is the same mathematically as the angular momentum operator, so it has eigenvalues j(j + 1) for half integer js. L l is just a constant multiplied by J l, so we obtain a discrete spectrum of segment lengths L jl = 8π G j l (j l + 1). (28) This is the first hint that we might be on the right track: lengths have become granular. 4.2 Gauge Note that in our boundary Hilbert space, there are many non-physical states that are not gauge-invariant and thus non-physical. The states that are gauge-invariant must satisfy or equivalently for each node n ψ(u l ) = ψ(λ sl U l Λ 1 t l ) where Λ n SU(2), (29) (L l1 + L l2 + L l3 )ψ = 0 (30) where the three links are those at n, since this operator generates SU(2) transformations at n. This has a nice geometrical interpretation. Each L gives the vector describing the side of the triangle corresponding to n, so this is exactly the condition that this triangle closes. The proper subspace of H Γ where this condition is satisfied is K Γ = L 2 [SU(2) L /SU(2) N ] Γ. (31) 10

11 4.3 Spin network basis It turns out that the quantum numbers j l are sufficient to label a basis of K Γ. To see this, we will explicitly construct K Γ. Firstly, consider representations of SU(2) with spin j. These have bases labelled by the magnetic quantum number m, and the action of U SU(2) is given by the Wigner matrices D j (U). We have (by the Peter-Weyl theorem) du D j m n D j mn = 1 2j + 1 δjj δ mm δ nn, (32) The Wigner matrices form an orthogonal basis of the full Hilbert space H = L 2 (SU(2)). In other words, H can be decomposed into a sum of finite dimensional subspaces H j of spin j. The basis elements are maps from H j to itself, and can thus be viewed as elements of H j H j. Therefore we have L 2 [SU(2)] = j H j H j (33) and so L 2 [SU(2) L ] = H j H j = H jl H jl, (34) l j j l l where the sum in j l is over all possible link spin configurations. The two Hilbert spaces associated with each link belong to each end of the link, since each transforms according to the gauge transformations at their respective ends. In this way each node n has three Hilbert spaces associated to it, so we can write L 2 [SU(2) L ] = H j1 H j2 H j3 (35) j l n We want the space of gauge invariant states, so restrict to the invariant part of any set of spaces transforming at the same node: L 2 [SU(2) L /SU(2) N ] = Inv SU(2) (H j1 H j2 H j3 ) (36) j l n Note that Inv SU(2) (H j1 H j2 H j3 ) is exactly the spin 0 part of H j1 H j2 H j3. By the normal expansion of spin products, this only exists if the three spins satisfy j 1 j 2 < j 3 < j 1 + j 2. (37) This is the triangle inequality! If it is satisfied, the spin 0 part is one dimensional and Inv SU(2) (H j1 H j2 H j3 ) = C, and therefore L 2 [SU(2) L /SU(2) N ] = j l C, (38) where the sum is over those j l that satisfy the triangular inequalities. But this statement is identical to saying that there is a basis j l labeled by j l satisfying triangle inequalities. The basis of such states is known as the spin network basis, and any generic quantum state in loop quantum gravity can be written as a superposition of spin-network states ψ = j l C jl j l (39) 11

12 4.4 Transition amplitude Recall that the action for a 2-complex dual to a triangulation is S = 1 8πG tr(u f L f ). (40) f To compute the transition amplitude of the theory, we use a path integral: W (U l ) = N du e dl f e i 8π G f tr(uflf) (41) With a bit of work it is possible to write this in the spin network basis, and the result is very nice (no integrals!): W (j l ) = N ( 1) jf (2j f + 1) ( 1) Jv {6j} (42) j f f v The sum is over the association of a spin to each face, respecting the triangular inequalities. J v = 6 a=1 j a where the j a are the spins of the faces adjacent to the vertex v. {6j} is the Wigner 6j symbol, a symbol with certain symmetries involving the j a. To define it, we first define the Wigner 3j symbol ( ) ι m 1m 2 m 3 j1 j = 2 j 3. (43) m 1 m 2 m 3 (This is not a matrix.) This is the unique(!) object with three indices in SU(2) representations that is invariant when the three indices are acted upon by a group transformation; it is closely related to the Clebsh-Gordon coefficients. The 6j symbol is then a certain contraction over the indices of four 3j symbols. This seems a little absurd. Why should this expression involving basic combinatorial objects in SU(2) representation theory have anything to do with general relativity? Consider a flat geometrical tetrahedron; its geometry is determined by its sides L 1,, L 6. Associate six spins to these with j a = L a 1/2. Let its volume be V. Now consider the asymptotic behaviour of the 6j symbol involving these spins for large j. The following was proven in 1998: {6j} j ( 1 cos S + π ) 12πV 4 (44) S is the Regge action of the tetrahedron. Now this cosine is like e is + e is, and with large j we can approximate the sum in the transition amplitude as an integral. The 12

13 transition amplitude thus begins to look a lot like the path integral in the Regge calculus. As we have already stated, as we refine the triangulation, the Regge action approaches the Einstein-Hilbert action. So we can believe that in the classical and continuum limits, this theory approaches general relativity. This 3D Quantum Gravity is known as the Ponzano-Regge model. 5 Infrared divergence One consequence of the presence of a discrete length spectrum is that it is not possible to have an arbitrarily small spacetime region, and so the ultraviolet divergences I mentioned at the beginning of this talk simply do not occur. However, with the theory in its current form, we do have infrared divergences; i.e. divergences arising from arbitrarily large spacetime regions. In general it is possible to have 2-complices on which edge lengths can become arbitrarily large, and in these cases the transition amplitude diverges. This is bad, but the theory is not doomed. A modification was found in 1992 that removes the divergences. It involves a new parameter q = e iπ/r for some integer r, and the amplitudes are given by W q (j l ) = N j f ( 1) jf d q (j f ) ( 1) Jv {6j} q. (45) f v The quantities d q (j) and {6j} q are the quantum dimension and quantum 6j symbol appearing in the representation theory of the quantum group SU(2) q. (Quantum groups are certain deformations of Lie groups, not really quantum, and not really groups. I don t know much about them so you may have to take what follows as given.) In the limit r, this converges on the previous model. This amplitude is independent of the triangulation. The quantum dimension is given by d q (j) = ( 1) 2j q2j+1 q 2j 1 q q 1 (46) 13

14 We can plot this on a graph. d q (j) j Note that there is a maximum value of j, and this increases as q 1. Spins only up to a maximum value j max r 2 (47) 2 enter this theory, and the finiteness of the ranges of the js makes the amplitude manifestly finite. As well as a minimum length for each segment, we now have a maximum length. There is a serendipitous physical interpretation of this modification. Before we went from the 2-complex transition amplitude to the Regge amplitude by arguing about the large j behaviour of {6j}; we saw that we recover general relativity. Now we have {6j} q, and it was shown in 1992 that for large j, by similar arguments we once again recover GR, but this time, we have a cosmological constant, given by Λ = π2 (r G) 2. (48) Using the current estimates for the cosmological constant, this leads to a value for r of around The corresponding length eigenvalue is about four times the diameter of the observable universe, and perhaps this is more than just a coincidence. This is excellent. Two open issues in fundamental physics - the presence of a small (and positive) cosmological constant, and the existence of quantum field theoretical divergences - resolve one another. This modification of the Ponzano-Regge model is known as the Turaev-Viro model. 6 Four dimensions So far, everything I have talked about has been in d = 3. This has been relatively simple because GR in 3D has no internal degrees of freedom. In 4D however, this is not true - we have two internal degrees of freedom. This makes things much more difficult, and in fact several decades passed between the full formulation of the 3D theory and that of the 4D theory. In 3D the gauge group is SU(2). In 4D we must use SL(2, C). 14

15 One key difference corresponding to the addition of a new dimension is that it is no longer distance which has a discrete angular momentum-like spectrum. Instead, it is area: A j = 8πγ G j(j + 1) (49) Here γ is a parameter of the theory known as the Barbero-Immirzi constant. We can observe another difficulty. In the 3D theory, lengths were sufficient to fully describe the triangulation; a triangle is uniquely defined by the three lengths of its sides. However, a tetrahedron is not uniquely defined by the areas of its faces, so the area quantum numbers are no longer sufficient to fully describe boundary states. I admit I haven t had the time to understand the 4D theory in much detail, so I ll say no more. 7 Black hole entropy It is a famous result of semiclassical physics that black holes have entropy. This entropy is given by the Bekenstein-Hawking formula S BH = kc3 A. (50) 4 G The fact that this entropy is finite and proportional to the area already seems to suggest area granularity, consistent with Loop Quantum Gravity. In fact it is possible to use LQG to compute this entropy directly. To start, consider a Schwarzschild black hole of mass M. This mass is equivalent to the ADM energy measured at. The same energy measured at a radius R is scaled by a redshift factor: M E(R) = (51) 1 2GM R Consider an observer at close to the horizon at R = 2GM + r where r 2GM. The energy can then be approximated as 2GM E M. (52) r The physical distance of the observer from the horizon is d = g rr r 2GMr, (53) so we can express the energy measured at a physical distance d as E = 2GM 2. (54) d Recall that for the observer to stay near the horizon, it must maintain an acceleration a = 1/d. Also the area of the black hole horizon is A = 4π(2GM) 2. Thus, for a static observer near the horizon, with acceleration a, the black-hole energy is E = aa 8πG. (55) 15

16 We will work in the microcanonical framework. The problem of computing the entropy of the black hole then reduces to counting the number of possible states with a given area A. In classical theory, this diverges, but in loop quantum gravity, it is finite (since the area spectrum is discrete). The surface of the black hole will be punctured by the links of a 2-complex. The areas associated with these links sum to give the total area of the black hole. links black hole Consider one such link. If the area has spin j, its area is given by (49). Since it carries a spin j representation, there are 2j + 1 independent states in which the system can be. Thus the probability for having spin j at inverse temperature β is given by p j (β) (2j + 1)e βe (56) where e βe is the Boltzmann factor. Substituting in (55) (since the link is near the horizon), together with (49), gives p j (β) (2j + 1)e βaγ j(j+1), (57) where we have absorbed the contributions of the areas of the other links into the proportionality factor. This proportionality factor (Z 1 (β)) 1 is fixed by unitarity j p j = 1, and is the partition function Z 1 (β) = j (2j + 1)e βaγ j(j+1). (58) If N links puncture the horizon, then the total partition function is Z = Z1 N. Recall the standard thermodynamical relation E = T S + F where F = T log Z is the Helmholtz free energy. Since the observer has acceleration a, we set T to the Unruh temperature a 2π. Then we have S = E/T F/T (59) = 2π aa a 8πG + N log Z 1 (60) = A 4G + N log (2j + 1)e 2πγ j(j+1). (61) j 16

17 If we can set the logarithm to zero, then this replicates exactly the Bekenstein-Hawking entropy (up to constants that have been set equal to 1). In fact we can, if we choose γ appropriately; numerically we obtain γ = γ This argument was a bit non-rigorous, but more refined calculations have been made, including fully dynamical derivations, and treatments with the grand canonical ensemble. I won t go into them here as they require the full 4D theory, but the main result is that the entropy is finite and proportional to area, and that if γ = γ 0 then the Bekenstein-Hawking result is replicated exactly. References [Rovelli and Vidotto, 2015] Rovelli, C. and Vidotto, F. (2015). Covariant loop quantum gravity : an elementary introduction to quantum gravity and spinfoam theory. Cambridge University Press, Cambridge. 17