Summer Internship Report On Introduction to Loop Quantum Gravity

Summer Internship Report On Introduction to Loop Quantum Gravity Systems

Submitted By: Abhishek Rout 14MS004 Indian Institute of Science Education and Research, Kolkata July 15 , 2017 th

Under the Guidance Of: Dr. Sudheesh Chethil Department of Physics Indian Institute of Space Science and Technology Thiruvananthapuram

Contents: 1. Introduction 2. General Relativistic approach to Gravity 2.1 Hamiltonian Formalism 2.2 Quantisation using Hamiltonian Formalism in Relativistic Systems 2.3 Introduction of Ashtekar Variables 3. Loop Quantum Gravity 3.1 Outline 3.2 Holonomies 3.3 Operators and Hilbert Space 3.4 Wilson Loops 3.5 Spin-Network States 3.6 Functional Representation in the Kinetic Hilbert Space: Cylindrical Functions 3.7 Loop Representation of the Operators 3.8 Flux 3.9 Quantisation of Area 3.10 Quantisation of Volume 4. Conclusion References

1. Introduction: General relativity was found in 1915. Quantum mechanics in 1926. A few years later, around 1930, Born, Jordan and Dirac are already capable of formalizing the quantum properties of the electromagnetic field. How long did it take to realize that the gravitational field should –most presumably– behave quantum mechanically as well? Almost no time: already in 1916 Einstein points out that quantum effects must lead to modifications in the theory of general relativity. In 1927 Oskar Klein suggests that quantum gravity should ultimately modify the concepts of space and time. In the early thirties Rosenfeld writes the first technical papers on quantum gravity, applying Pauli method for the quantization of fields with gauge groups to the linearized Einstein field equations. The relation with a linear spin-two quantum field is soon unraveled in the works of Fierz and Pauli and the spin-two quantum of the gravitational field, presumably first named “graviton” in a 1934 paper by Blokhintsev and Gal’perin , is already a familiar notion in the thirties. Bohr considers the idea of identifying the neutrino and the graviton. In 1938, Heisenberg points out that the fact that the gravitational coupling constant is dimensional is likely to cause problems with the quantum theory of the gravitational field. General relativity (GR) and quantum mechanics are two of the best verified theories of modern physics. While general relativity has been spectacularly successful in explaining the universe at astronomical and cosmological scales, quantum mechanics gives an equally coherent physical picture on small scales. However, one of the biggest unfulfilled challenges in physics remains to incorporate the two theories in the same framework. Ordinary quantum field theories, which have managed to describe the three other fundamental forces (electromagnetic, weak and strong), have failed for general relativity because it is not perturbatively renormalizable. Loop quantum gravity (LQG) is an attempt to construct a mathematically rigorous, non-perturbative, background independent formulation of quantum general relativity. GR is reformulated in terms of Ashtekar–Barbero variables, namely the densitized triad and the Ashtekar connection. The basic classical variables are taken to be the holonomies of the connection and the fluxes of the triads and these are then promoted to basic quantum operators. The quantization is not the standard Schrödinger quantization but an unitarily inequivalent choice known as loop/polymer quantization.

2. General Relativistic Approach to Gravity: Einstein’s theory of special relativity, which was later generalised to his theory of general relativity, transformed how space and time were viewed. Space and time were united into a single entity known as spacetime, and gravitational effects were incorporated into the curvature of spacetime. More importantly, spacetime itself was no longer a fixed background in which physics takes place. Instead, it was promoted to a dynamical entity influenced by the distribution of matter in it. The relationship between spacetime and matter is governed by the Einstein equations: 1

Gμν = Rμν - 2 Rgμν = Tμν where Gµν is the Einstein tensor, Rµν is the Ricci tensor that encodes some information about the curvature of spacetime given by the metric gµν , R is the Ricci scalar and Tµν is the stress-energy tensor that encodes how matter is distributed in spacetime. General relativity, however, is a purely classical theory. It does not incorporate any idea of quantum mechanics into the formulation. As early as 1916, which was only one year after Einstein finalised the theory, he pointed out that quantum effects must lead to modifications in general relativity. In general relativity, spacetime is described by a metric field g(x) on a background manifold ϻ. The dynamics of g(x) is encoded in the Einstein-Hilbert action: 1

SEH[g] = 16𝜋𝐺 ∫ d4x (√-|g|) R[Ӷ(g)] where R is the Ricci scalar and Γ(g) is the Christoffel symbol. The variation of this action with respect to the metric results in the Einstein equations in vacuum (Tµν = 0). One can add matter to the system simply by adding the relevant action to the equation.

2.1 Hamiltonian Formalism: 𝑑𝑞 𝑖 (𝑡) ) 𝑑𝑡

The dynamics of a non-relativistic system is specified by a Lagrangian L(qi ,vi ) = L(q i(t),

where i

i

runs over the number of degrees of freedom m of the system and q are values in the configuration space. The allowed motion is then given by an extremum of the action: 𝑡

𝑑𝑞 𝑖 (𝑡)

1

𝑑𝑡

S[q] = ∫𝑡 2 dt L(q i(t),

)

The extrema are given by the Euler-Lagrange equations or equivalently Hamilton’s equations. The Hamilton-Jacobi equation, which can be derived by taking the classical limit of the Schrödinger equation, is 𝜕𝑆 (𝑞 𝑖 ,𝑡) 𝜕𝑡

+ 𝐻(

𝜕𝑆 (𝑞 𝑖 ,𝑡) 𝜕𝑞 𝑖

,qi)=0

where H is the non-relativistic Hamiltonian and S(q i ,t) is the action evaluated along the classical trajectory with end-point (q i ,t). Solutions of the Hamilton-Jacobi equation can be found in the form S(q i,Q i ,t) = E t −W(q i ,Q i ), where E is a constant and W, the characteristic Hamilton-Jacobi function, satisfies

H(

𝜕𝑊 (𝑞 𝑖 ,𝑄 𝑖 ) 𝜕𝑞 𝑖

,qi)=E

S(q i ,Q i ,t) is called the principal Hamilton-Jacobi function. Once those have been found, the solutions of the equation of motion q i can be found by first calculating

P i(q i, Q i, t) = −

𝜕𝑆 (𝑞 𝑖 ,𝑄 𝑖 ,𝑡) 𝜕𝑄 𝑖

and then inverting the equation to q i (t) = q i (Q i,P i ,t). P i are integration constants.

2.2 Quantisation using Hamiltonian Formalism in relativistic systems: The Hamilton-Jacobi formulation can be extended to include relativistic systems.The relativistic Hamilton-Jacobi equation is given by:

𝐻(

𝜕𝑆 (𝑞 𝑎 ) 𝜕𝑞 𝑎

,qa)=0

where q a are the observables. Notice that this is simpler, since an external time variable cannot be specified. The evolution equation is:

f(qa, Pi , Qi) =

𝜕𝑆 (𝑞 𝑎 ,𝑄 𝑖 ) 𝜕𝑄 𝑖

+ Pi = 0

In order to quantise a theory using canonical methods, one has to rewrite it in the Hamiltonian form first. To put a theory into its Hamiltonian form, we have to identify the appropriate configuration variables and define the corresponding conjugate momenta, which are related to the temporal derivative of the configuration variables. Thus, we have to identify a “time” parameter in the theory. In the case of general relativity, this procedure departs from manifest covariance of the theory. This requirement is automatically satisfied if we set spacetime (M,g µν ) to be globally hyperbolic. In such spacetime, we can define a global time function t and foliates it into a family of Cauchy hypersurfaces Σ labelled by t . We can then describe general relativity in terms of spatial metric hab on Σ evolving through time t ∈ R. To make the split into space and time assume that the four-dimensional spacetime manifold M with a metric gμν can be split as M≈ Σ × R, where Σ is a three-dimensional spatial slice. Let t label the constant spatial slices Σ and nµ be a normal to Σ. Then define the spatial three-metric hab as

hab ≡ gab + nanb

It is easiest to think that in this expression a,b = 1,2,3. Then one can decompose a vector t a into components normal and tangential to Σ as

t a = Nna + Na N is called the lapse and is related to moving between spatial slices, while N a is called the shift vector which transports along the spatial slice Σ. Here t has indices 0 to 3 but only N not Na contributes to the zeroth component.

Fig 2.1: The foliation introduced for the canonical formulation of General Relativity. With the 3+1 decomposition above, a general metric can be written as:

ds2 = −N2dt2 + hab(dxa + Nadt)(dxb + Nbdt) The information about the spacetime metric g µν is now completely encoded in h ab , N and Na . An important quantity in the canonical description is the extrinsic curvature. The extrinsic curvature measures the rate of change of the spatial metric along the congruence defined by na and therefore gives an idea of the “bending” of spatial surfaces in spacetime. In other words, the extrinsic curvature measures the change in the three-dimensional metric moving from one slice to another. Given a Cauchy hyper surface and the associated Riemannian metric, its relation to the neighbouring hyper surface is given by the extrinsic curvature 1

Kab = - 2𝑁 (ḧ ab − ∇aNb − ∇bNa ) where the dot indicates derivative with respect to t and ∇c is the covariant derivative on the hypersurface, compatible with h ab . Written in terms of the structures on Σ, the Einstein-Hilbert action takes the form: 1

S =16𝜋𝐺 ∫ d4x (√-|h|) (R – KabKab –( Kaa)2) By analysing the action, we can derive the canonical momentum conjugate to hab ,

p ab(x) = 𝛿ℎ

𝛿𝐿 𝑎𝑏 (𝑥)

= 2𝑁𝛿𝛿𝐿𝐾 = 𝑎𝑏

√|ℎ| 16𝜋𝐺

(Kab – Kc c hab)

These variables satisfy the Poisson Bracket:

{ hab(x), pcd(y) } = 𝛿 a c 𝛿 b d 𝛿 (x,y) The canonical momenta conjugate to N and Na , on the other hand, are zero since no time derivative of these variables appear in the action. These types of variables are known as Lagrange multipliers and are not actually configuration variables. So, we identify that in the Hamiltonian formulation of general relativity, the only configuration variable is the spatial metric, hab . The phase space variables are then hab and pcd satisfying Poisson bracket as given above. Although the Lagrange multipliers do not serve as configuration variables, they do have an important role. Setting the action to be invariant under arbitrary variation of the Lagrange multipliers yields a set of equations of the form Ci= 0, where Ci are known as the constraints of the system. These constraint equations are meant to be applied after the Poisson bracket structure of the canonical variables is constructed. For any system with constraints, not all points on the phase space are physically relevant. Instead, only those that satisfy all the constraint equations are physically relevant. For general relativity, the variation with respect to N gives the Hamiltonian or scalar constraint:

C grav =

16𝜋𝐺

1

𝑁√|ℎ|

(pabpab - 2(pc c)2 ) - 16𝜋𝐺 R √|ℎ|

And with respect to Na the variation gives us the diffeomorphism constraint or vector constraint:

Ca grav = -2Dbpab Given an action of a system, one will now proceed to calculate the Hamiltonian that would generate the time evolution for the system. In our case now, the Hamiltonian obtained from the action is:

Hgrav = ∫d3x( ḣabpab – Lgrav) 16𝜋𝐺

1

𝑁√|ℎ|

= ∫d3x ( √|ℎ| (pabpab - 2(pcc)2 ) +2pabDaNb - 16𝜋𝐺 R) = ∫d3x ( NCgrav + NaCagrav) One should immediately recognise that for physically relevant situations, the Hamiltonian vanishes since it consists entirely of constraints. This gives rise to the problem mentioned at the beginning of this section, namely the problem of time. With the vanishing Hamiltonian, one is led to a theory with apparently no “time evolution”.

2.3 Introduction of Ashtekar Variables: In 1986, Ashtekar introduced a new complex variable that puts general relativity in the language of gauge theory. To introduce the Ashtekar-Barbero variable, we first consider a set of three vectors ei= eai∂a , where i = 1,2,3, at each point in space. These vectors are taken to be orthonormal to each other; that is:

h ab e ai ebj= δ ij We can then invert this to obtain:

h ab = δij e ia e jb

As a result of this, general relativity can be expressed in terms of a three dimensional SU(2) connection A ia and a real three dimensional momentum conjugate, the densitised triad E ai=√|q| E ai with: 𝜕𝑆 [𝐴] 𝜕𝐴 𝑖𝑎

= Eai

Both sets of indices run from 1 to 3. Indices a denote vector indices in curved space and i are internal indices raised and lowered with the flat metric δ ij. The variables in his formulation have the Poisson bracket:

{ Aia(x), Ebj(y) } = 8 π γ G δ ij δ baδ(x,y) where γ is a complex constant called Barbero-Immirzi parameter. Theories with different γ’s are related by canonical transformation variables.

The connection A ia is related to the spin connection Γ ia = Γ ajk εjki and the extrinsic curvature by:

A ia = Γ ia + γK ia with K the extrinsic curvature K ia = K ab E ai / √|q|

In terms of these variables the Einstein-Hilbert action, with R the Ricci scalar, becomes: 1

L =8𝜋𝐺𝛾 ∫ d3x (Eai Ӓia + Nεijk Eai Ebj F kab + NaEbi F iab + λi(DaEa)i) where γ has been set to 1 and D a is the covariant derivative with A ik and F iab is the curvature.

Explicitly,

Davi = ∂av i + εijk A jav k F iab = ∂a Aib - ∂b Aia + εijk A ja A kb The lapse N and shift Na and the gauge parameter λi are Lagrange multipliers. This means that there are seven constraints given by:

G i = DaE ai = 0, Vb = E ai F iab = 0, H = εijk E ai E bj F kab = 0 G is called the Gauss’ law, V is the momentum or vector constraint and H is the Hamiltonian constraint. The total Hamiltonian of GR is a linear combination of these constraints as can be seen from the Lagrangian. On a further note, GR is a totally constrained system with the Hamiltonian written explicitly in Ashtekar variables i.e.

H = ∫ d3x (Nεijk E ai E bj F kab + NaE bi F iab + λi(DaEa)i) This is for Barbero-Immirzi parameter γ = 1. If one restricts γ to a real number then one obtains Lorentzian GR where everything else stays the same. We can fully describe general relativity using these new variables together with the constraints in the appropriate forms.

3. Loop Quantum Gravity: 3.1 Outline: General relativity has two main properties: diffeomorphism invariance and background independence. General relativity could be viewed as a field theory from the form of the EinsteinHilbert action but this form does not influence the notions of space and time. Only diffeomorphism invariance and background independence are important. The task is to find a background independent quantum field theory or a general relativistic quantum field theory. Loop quantum gravity is based on the assumption that quantum mechanics and general relativity are correct. It assumes back- ground independence and does not attempt to unify forces, only to quantise gravity. Spacetime is assumed to be four-dimensional and no supersymmetry is required. Loop quantum gravity although looking for a general relativistic quantum field theory does not use conventional quantum field theory because quantum field theory is defined in a background dependent way. Instead loop quantum gravity uses the Hilbert space of states, operators and transition amplitudes of traditional quantum mechanics. The canonical algebra of fields with positive and negative frequency components is replaced by an algebra of matrices of parallel transport along closed curves. These matrices are called holonomies or Wilson loops. The holonomies are the essence of loop quantum gravity as they, on quantisation, become operators that create loop states. A loop state transforms under an infinitesimal transformation into an equivalent representation of the same state. Finite transformations change the state into a different one. This is because only the relative position of the loop with respect to other loops is significant. The states in loop quantum gravity are solutions of the generalised Schrödinger equation called the Wheeler-DeWitt equation

HΨ = 0 where H is the relativistic Hamiltonian or Hamiltonian constraint and Ψ is the space of solutions to the equation. The right-hand side is zero since space and time are on an equal footing in loop quantum gravity, so there is no external time variable to differentiate by. The space of solutions can be expressed in terms of an orthogonal basis of spin network states. Spin network states are finite linear combinations of loop states. In loop quantum gravity, a basis state |S> describing space is represented by a collection of connected curves known as a knotted spin network (s-knot) state. H acts only on the nodes of a spin network, hence a loop or a set of nonintersecting loops solves the Wheeler-DeWitt equation. The curves are initially defined to be embedded in a Riemannian manifold Σ. However, due to the diffeomorphism invariance inherited from general relativity, the positions of the curves on the manifold lose their meanings. Important information is encoded in the combinatorics between the structures associated with the curves and their meeting points.

3.2 Holonomies: Holonomies are important because all observables that are functions of the connection only, can be expressed in a basis of holonomies. A holonomy H(γ) is the parallel transport along a closed curve γ with a basepoint. The holonomy has the same information in it as the curvature: knowing all holonomies of the one-form connection A ia defines the connection uniquely.. holonomy and flux, smear A ia and E bj fields, respectively, in a background independent way. Parallel transport around a closed curve l is given by the path ordered exponential

H A(l) = P exp ∫𝑙 𝐴(𝑦)𝑑𝑦a where A a is the connection. H is the holonomy . Note that the holonomy is coordinate-independent, but is not gauge-invariant. Under a gauge transformation, the holonomy transforms as

h e [A] → g s(e) h e [A]g −1t(e) where s(e) denotes the source or starting point of e and t(e) denotes the target or ending point. One can introduce an equivalence relation where one identifies all closed curves that lead to the same holonomy for all smooth connections. Two closed curves are equivalent if one can be continuously deformed to the other. All loops which are equivalent form an equivalence class. The equivalence classes of closed curves form a group structure, and holonomies can be thought of as a map from this group onto a Lie group G. Technically the equivalence classes of closed curves are called loops and the group a group of loops. Functions of the elements in the group of loops are called wavefunctions and these form the loop representationThere are several ways of defining these equivalence classes - an example of one is the following. Let p1 , p2 and q be open curves and m1 and m2 closed curves. Then if m1 = p1 ◦p2 and m2 = p1 ◦ q ◦ q −1 ◦ p2 then m1 ∼ m2 . These equivalence classes are called loops and they are identified with Greek letters .The inverse is defined as the curve travelled in the opposite direction: the inverse γ −1 of the loop γ satisfies γ ◦γ −1 = i with i being the set of closed curves equivalent to the null curve. With this definition of a loop, the holonomy has the properties

H(γ1 ◦ γ2 ) = H(γ1 )H(γ2 ) H(γ −1 ) = (H(γ)) −1

3.3 Operators and Hilbert Space:

When we consider our first assumption that the spatial metric h ab and its conjugate momentum p ab as the basic variables and using Dirac’s procedure, these variables were promoted to operators on a kinematical Hilbert space Hkin .

hab → ĥab , pab → p̂

ab

such that the Poisson bracket between them is promoted to a commutation relation:

[ĥab (x), p̂ cd(y) ] =iћ 𝛿 a c 𝛿 b d 𝛿 (x,y) Then, one chooses a representation space to study the action of the operators on a general quantum state |Ψ> ∈ Hkin . For instance, in metric representation, we would have:

ĥab(x)Ψ[h ab (x)] = h ab (x)Ψ[h ab (x)] 𝛿

p̂ cd(x)Ψ[hab (x)] = −𝑖ћ 𝛿ℎ Ψ[h ab (x)] 𝑐𝑑

Among the elements of H kin , only those that are annihilated by both quantum versions of constraints,

ĉ grav Ψ[h ab (x)] = 0 ĉ grav aΨ[h ab (x)] = 0 are physically relevant. They are elements of the physical Hilbert space H phys . One can also implement the diffeomorphism constraint alone first to identify a diffeomorphism-invariant Hilbert space H diff . In this way, one will have a chain of Hilbert space construction. Loop quantum gravity, more or less, follows a similar path. Instead of working the metric variables (hab ,p cd ), we will work with the connection variables (Aia ,E bj ). As we have mentioned above, using these variables introduces another constraint, the Hamiltonian constraint as mentioned earlier. So, we introduce the concept of Wilson Loops which are a great contender for being the basis states for our loop representation.

3.4 Wilson Loops: Any gauge invariant quantity involving the connection A can be written in terms of traces of holonomies or Wilson loops:

WA(γ) = Tr [P exp( 𝑖

𝛾

𝑑𝑦aAa )]

They have vanishing Poisson brackets which means that they are observables too. Wilson loops have two useful properties which together imply that Wilson loops are an overcomplete basis of the Gauss’ law constraint. There properties are called Mandelstam identities and the reconstruction property. They follow from trace identities of N ×N matrices and reflect the structure of the gauge group in consideration. The first type of Mandelstam identity follows directly from the cyclicity of traces. For any gauge group of any dimension

W(γ1 ◦ γ2 ) = W(γ2 ◦ γ1 ) The second type of identity is a restriction which guarantees that the Wilson loops are traces of N × N matrices. These identities can be derived by considering the vanishing of an N + 1 dimensional antisymmetric matrix in N dimensions with matrix representation indices A and B, For SU(2)

W(γ1 ◦ γ2 ) = W(γ2 ◦ γ1 ), W(γ1 )W(γ2 ) = W(γ1 ◦ γ2 −1) + W(γ1 ◦ γ2 ), W(γ) = W(γ −1 ) In general W(i) = N and |W(γ)| ≤ |W(ı)| = N. There is a recurrence relation which allows the calculation of the Mandelstam identities of this second kind but it is avoided for confusions. The reconstruction property describes whether one can reconstruct the holonomy given a function of loops that satisfy the Mandelstam identities. It was proved in that given a function W(γ) satisfying the Mandelstam constraints one can reconstruct the holonomy. Wilson loops satisfy the Mandelstam identities so they uniquely define the holonomy. Having defined the Wilson loops, wavefunctions ψ can be expressed in the basis of the loops by the following expansion,

ψ(γ) = ∫ dA W ∗A(γ)ψ[A] The holonomy constructed from Wilson loops is a representation of the group of loops and the traces of the representation satisfy the Mandelstam identities. Any gauge invariant function can be expressed as a combination of products of Wilson loops. Loops are a solution of the Hamiltonian constraint in Ashtekar variables. This was an important discovery in the history of loop quantum gravity. However, the loop basis is an overcomplete basis because of the Mandelstam identities. In order to remove the overcompleteness one can use spin network states, which are a basis of states for the quantum theory.

3.5 Spin-network States:

The definition of a spin network state is as follows: Consider a set of curves or “links” that only overlap at the ends of the curves or “nodes”. The curves are oriented, and each node has a multiplicity labelled by m, which denotes the number of curves coming into and going out of the node. The set of curves forms a graph Γ. In addition, the links carry a non-trivial irreducible representation j i , and the nodes carry an intertwiner n k . An intertwiner is an invariant tensor in the product space of a set of representations: in this case the intertwiner n k is in the product space that the representations of the adjacent links form. Properties:    

Spin networks are an orthonormal basis for gauge invariant functions, or the kinematical state space of loop quantum gravity. They have an inner product defined 1 if two spin network states are related by a diffeomorphism and 0 otherwise. Spin networks are a linear combination of loop states. Spin network states are not diffeomorphism invariant but they are used to define a diffeomorphism invariant state called a spin-knot or s-knot . A diffeomorphism changes a spin network state to an orthogonal one or changes the orientation of the links. There is a finite number of states related by changes in link orientation. Let them all be equivalent. If two spin network states are knotted in a different way then they are orthogonal. In order to make them orthonormal, a diagonalisation has to be done. So then, the type of knot and the labelling of the links and nodes form the spin-knot.

Fig: A spin network with nodes ni and links ji.

3.6 Functional Representation in the Kinetic Hilbert Space Cylindrical Functions:

Instead of just a single curve e as in the definition of holonomy above, consider the set of curves in spin states Γ defined as a collection of oriented paths e ∈ Σ meeting at most at their endpoints . The paths are usually referred to as links or edges in loop quantum gravity literature. Given a graph Γ ∈ Σ with L links, one can associate a smooth function f : SU(2) → C with it. A cylindrical function is a couple (Γ,f), which in connection representation is defined as a functional of A given by:

< A|Γ,f > = ψ (Γ,f) [A] = f(h e1[A],...,h eL[A]) where ei with i = 1,...,L are the links of the corresponding graph Γ. The space of all functions f associated with a particular graph Γ is denoted as Cyl Γ . The kinematical Hilbert space H kin is then defined as H kin ≡ ⊕ Γ ε Σ H Γ with the scalar product:

< Γ 1 ,f 1 |Γ 2 ,f 2 > ≡ An example of cylindrical function is the case where each e is a single closed curve (that is, a loop) α and f is the trace function, Tr . For a graph with a collection of loops, that is, Γ ≡ α = (α 1 ,...,α n ), we have

< A|α > = ψα[A] = ψ α1 [A]...ψ αn [A] = Tr h α1 [A]...Tr h αn [A] where Tr hα [A] is the Wilson Loop representation for the system. Historically, they were found to be exact solutions to the quantum version of the Gauss constraint and Hamiltonian constraint. This suggested that one can expand a general quantum state Ψ*A+ in terms of these multi-loop states

< A|Ψ > ≡ Ψ[A] =

𝛼

𝛹(𝛼)ψα[A]

where Ψ(α) is the loop space representation of the state |Ψ >.

3.7 Loop Representation of Operators: Following the quantum theory in the loop basis is constructed along these lines. All quantities can be constructed from the connection Aia and the conjugate E ai . Their quantum equivalents are defined on the functionals or states Ψ*A+ as :

 ia Ψ[A] = A ia Ψ[A] 1

𝛿

8𝜋𝐺

Ê ai Ψ[A] = −𝑖ћ 𝛿𝐴𝑖 (𝑥) Ψ[A] 𝑎

In other words the action of A on a state is simply multiplication and the conjugate is a functional 1

derivative. The factor 8𝜋𝐺 can be set to one. Notice that the right hand sides do not live in the same space as Ψ[A] so use holonomies as the operators instead of A and E. A holonomy operator ĥe [A] simply acts as a multiplicative factor in connection representation:

ĥe [A]Ψ[A] = he [A]Ψ[A] The connection can be replaced by the holonomy U(A,γ). The momentum E requires a bit more work and in fact the action of E is to pick out intersections of the holonomy with a two-dimensional surface:

Ê i(S)U(A,γ) = ± iћ U(A,γ1 ) τ i U(A,γ2 ) i.e. Ê on a holonomy is equivalent to inserting a matrix ±iћ τi where E and the holonomy curve intersect. The holonomy γ is then split into γ1 and γ2 . This comes from the fact that E can be rewritten in smeared over a two-dimensional plane as

Ei(s) = -iћ ∫𝑆 dσ1dσ2εabc

𝜕𝑥 𝑏 (𝜎 ) 𝜕𝑥 𝑐 (𝜎) 𝜕𝜎 1

𝜕𝜎 2

where σ are coordinates on the two-dimensional surface. The derivative of U by A is given by: 𝛿 𝛿𝐴𝑖𝑎 (𝑥)

𝑈(𝐴, 𝛾) = ∫ds γ a(s) 𝞭 3(γ(s), x) [U(A,γ1 )τi U(A,γ2 )]

with γ 1 and γ 2 being the two parts into which the point s separates the curve γ and τ the Pauli matrices. The operators Tr(U) (a closed loop) and E give the unique representation, the loop representation of a quantised diffeomorphism invariant theory. This is the LOST theorem (for Lewandowski, Okolow, Sahlmann and Thiemann).

3.8 Flux: Given a surface S ∈ Σ with local coordinates ya a flux FSf is defined as:

F Sf [E] = ∫𝑆 d2y na E ai f i 1

𝜕𝑥 𝑏 𝜕𝑥 𝑐

2

𝜕𝑦 𝜇 𝜕𝑦 𝜈

where fi is an SU(2)-valued function, na = εabc ε μν

is the co-normal to the surface S and xa

is the local coordinate of Σ. From the expression, it is obvious that the flux is both coordinate independent and gauge-invariant. The action of a flux operator is as follows:

F̂

Si

ψ[A] = −8𝜋𝛾𝑖ћ ∫𝑆 d2y na

𝛿 𝛿𝐴 𝑖𝑎 (𝑥(𝑦 ))

𝜓[𝐴]

Now, when we replace 𝜓[𝐴] with the holonomy U(A,γ) as mentioned above we have a relation as:

F̂

Si

U(A,γ) = ± 8𝜋𝛾𝑖ћ G U(A,γ1 ) τ i U(A,γ2 )

The sign + or - depends on the relative orientation of the curve and the surface. An important composite operator to consider is the scalar product of two fluxes,

F̂

2S

= 𝛿 ij F̂

Si

F̂

Sj

Instead of inserting just a single generator τ (j)i, this composite operator will insert a product of them, that is, 𝛿 ij τ (i)j τ (j)i which is equal to –j(j+1)I (Casmir operator) which can be commuted through U(A, γ) and we obtain a very simple result:

F̂

2S

U(A,γ) =( 8𝜋𝛾ћ G)2 j(j+1) U(A,γ)

3.9 Quantisation of Area: It turns out that the area of any physical surface is quantised. This is the main result of loop quantum gravity: spacetime is quantised. The argument for the discreteness of the area operator goes as follows: The area of a 2d surface S in 3d space is given by

A = ∫𝑆 dx1dx2 √|h| which using Ashtekar variables can be written as

A = ∫𝑆 dx1dx2 √ (δ ij naE ainbE bj ) ≈ lim𝑁→∞

𝑁 𝑛=1 𝑆 n

= lim𝑁→∞

√ (δi j naE ainbE bj )

𝑁 𝑛=1

√ (δij F Si F Sj )

where the integral has been written as the limit of a Riemann sum in the second line. The third line follows from the fact that for a small enough surface δS, the flux can be approximated as:

F Sf [E] ≈ δS na E ai To get the result of the area operator on the whole spin network we split the area into small sections so that each section only intersects with one line of the spin network. Then the action of the area operator is:

ÂS|S> = ћ

𝑖

√ (ji(ji +1) |S>

for the intersection points i and the representation j i of the spin network line at the point i.

Fig: The spin network intersects with the surface which is divided into section. Each intersection gives a contribution to the area eigenvalue.

Putting physical units back in and using the Barbero connection instead of the real connection we get:

ÂS |S> = 8𝜋𝛾ћ Gc -3 = 8𝜋γ(lP )2

𝑖

𝑖

√ (ji(ji +1) |S>

√ (ji(ji +1) |S>

which is clearly discrete and finite. Note that the smallest possible area is of order Plank area l2P if the Immirzi parameter γ is taken to be of order unity. Hence, the smallest quantum of area from j = 1/2 for γ = 1 is (4√3)πћGc −3 ≈ 10 -66 cm2 . This discreteness is a result of quantisation in LQG rather than postulated. The area expression contains the Barbero-Immirzi parameter which classically did not affect the theory but in the quantum theory it affects the physics.

3.10 Quantisation of Volume: The definition of the volume operator V̂ R of a region R follows a similar strategy. One starts by dividing the region into small cells c of coordinate size ε3 and writing the classical expression for volume as a Riemann sum:

V = ∫𝑅 d3x √|h| = ∫𝑅 d3x √|E| 1

𝑁 𝑛=1

≈ 𝑙𝑖𝑚𝑁→∞

ε3n √ (3! εabc εijk E ai(x)E bj(x)E ck(x) )

where x is an arbitrary point inside the small cell c. In the construction of the Ashtekar-Lewandoski volume operator, for example, one proceeds by choosing three non-coincident surfaces (S1 ,S2 ,S3 ) within each small cell . This allows the volume to be written in terms of fluxes, which are then quantised by turning them into operators:

V̂ R = 𝑙𝑖𝑚𝑁→∞

𝑁 𝑛 =1

1

ε3n √ (3! εabc ε ijk F̂ i (S an) F̂ j (S bn ) F̂ k (S cn) )

The volume operator only acts on nodes. More precisely the volume operator vanishes if the region R does not contain a node of four or more spin network lines. Since this method relies on the choice of three surfaces, one has to carry an averaging procedure over all possible choices of surfaces. This results in a volume operator with the following discrete spectrum:

V̂ R |S> = ( 8𝜋γlP2) 3/2 κ0

𝑛𝜀𝑆

√(

1 48

εijk

(𝑒,𝑒 ′ ,𝑒 ′′ )

ε(e, e’, e’’) τ ei τ e’j τ e’’k ) |S>

where n are nodes in the spin network state |S> and (e, e’,e’’ ) runs over all possible sets of three links at the node n. The factor κ0 is a constant that came from the averaging procedure, and ε(e,e’,e’’) is a function associated with the relative orientation of the three links. The end result is that the action of the volume operator on a spin network state is of the elegant form:

V̂ R | Γ, jl, i1, ......iN > = (16πћ G γ) 3/2

𝑛

V i’i | Γ, jl, i1, ......iN >

Where | Γ, jl, i1, ......iN > = |S> is a spin network state (Γ graph, jl irreducible representations labelling links and in intertwiners of nodes) and V i’i are matrices computed such that they depend on the nodes of the spin network. The conclusion is that the volume operator has a discrete spectrum like the area operator. It must be noted that the area operator and the volume operator below are not gauge invariant operators. Hence, they cannot be directly taken to represent physical quantities. However, for certain reasons they are thought to imply that spacetime is discrete. Firstly, it is true that the area and volume operators have discrete spectra in any gauge. As the discreteness depends on commutators of the operators and not the precise description of the operators or observables, for any area the discreteness prevails. Also, discreteness is independent of the dynamics of the system. This gives the basis for saying that LQG predicts a discrete area and volume.

4. Conclusion:         

Loop quantum gravity is a nonperturbative background independent formulation of quantised general relativity. It builds on the Hamiltonian formalism of general relativity where four-dimensional spacetime is split into three-dimensional spatial slices and a time direction. The action of general relativity has constraints. These constraints can be expressed neatly in terms of the Ashtekar variables which use an SU(2) connection and its momentum conjugate. This introduces a variable called the Barbero-Immirzi parameter which has an important role in the quantum theory. A basis of solutions for the constraints are the Wilson loops. These can be developed further to define the spin network basis which is a complete basis for the quantum theory. Using the spin network basis, the system of constraints can be quantised. It is found that the area and volume operators of loop quantum gravity have discrete eigenvalues, which implies that spacetime is quantised. The size of the quanta are of Planck scale and proportional to the Immirzi parameter. This dicreteness of space is the reason why loop quantum gravity is UV divergence free.

The most appealing feature of the theory is its lack of additional a priori assumptions about the nature of quantum spacetime. The inputs are only quantum mechanics and general relativity, which have been very successful within their regime of applicability. The quantisation of spacetime has interesting consequences in the context of cosmology. Loop quantum cosmology is a truncated version of the full loop quantum gravity theory and the exact relationship between loop cosmology and the full theory is still under investigation. However, the result that spacetime is quantised implies at least for the FLRW model and homogeneous cosmology models that singularities are resolved. The Big Bang is replaced by the Big Bounce where the classical singularity is avoided due to the loop quantum gravity effects. Early universe inflation is automatically predicted and later stopped by loop quantum cosmology. Loop cosmology is a rapidly developing area where current research aims to consider models of increasing complexity to see whether results from simpler models carry over. In particular, interacting matter and inhomogeneities cannot be analysed using current methods. Another application of loop quantum gravity is the black hole entropy calculation which fixes the value of the Immirzi parameter. More recent developments of loop quantum gravity have been created in the spinfoam formalism which is a path integral formulation of loop quantum gravity. Most importantly, this formalism allows the calculation of n-point functions and the graviton propagator. There is much work to do in the topic of n-point functions because it is a very recently founded topic in loop quantum gravity. While loop quantum gravity does not attempt to unify matter and gravity it can incorporate matter easily. The theory makes predictions and stands on a rigorous foundation, which cannot be said of string theory. However, it has not been proved that general relativity is the low-energy limit of loop quantum gravity, nor has a choice been made of the best way to define a Lorentzian version of the theory.

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