One of the key problemsin our era is the problem of quantum gravity- whether we can realize quantummechanics and general relativity (GR) as a single consistent theory. One logical way ofgetting a quantum theory of gravity would be to obtain a perturbation expansionof canonical GR, and quantizing it using the rules in quantum field theory. Butwe see that the terms of this perturbation expansion are divergent; in factthere are infinitely many divergences 1, making it necessary to have infinitelymany counter-terms to be of any physical significance. This is the infamous non-renormalizability of gravityand there are many intuitive arguments for the same 2.

For example, Newton’sgravitational constant multiplied by energy is used as a coupling constant inGR. At high energies, coupling constant is also high which results in aninfinite number of divergent Feynman diagrams.To proceed from here, wecould assume GR is not a fundamental theory but a low energy limit of a moregeneral underlying theory, which leads to the approach of string theory. Or wecould assume perturbation expansion in Newton’s constant is not well defined,but that GR can still be quantized non- perturbatively. This leads to theapproach of loop quantum gravity. LQG begins with GR and tries to integratequantum features, while string theory, begins with quantum field theory and triesto add GR to its framework.Non- perturbativeapproach leads to a background independent theory.

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There are arguments that thecorrect quantum theory of gravity must be background independent 3. Ofcourse, if a theory of quantum gravity pre-supposes GR to be valid, it (also)must be background independent. FormalismQuantizing GRT withoutusing a perturbation expansion yields a plethora of difficulties. Fornon-perturbative quantization of GR we assume the Einstein Hilbert action fromwhich GR is derived is exact and not a low-energy limit of a more general,underlying theory. There are three constraint equations- the Hamiltonian, Gaussand diffeomorphism that come from this approach.

 To describe GR in a canonical way; we separatespace and time with a method called 3+1 decomposition from which we getLagrangian density and the Hamiltonian density. The constraint equations followfrom the Hamiltonian density. This was done for the first time by Arnowitt,Deser and Misner in 1962 4.

This is the ADM formulation of GeneralRelativity. From here one can quantize gravity by calculating the poissonbracket  And replacing the variablesby operators The constraint equationshows that Hamiltonian density  at all times. The Schrödinger equation reducesto .

From this we get the Wheeler DeWittequation   Where {displaystyle {hat {H}}(x)}  is the Hamiltonian constraint inquantized general relativity and  stands for the wave function of the universe.The equations turn out to highly singular and no known solutions to theseconstraint equations exist so far. To overcome this, Abbhay Ashtekar,introduced a new set of variables (Ashtekar variables) which turn theconstraint equations into simple polynomials. The initial promisethat the Ashtekar variables would simplify the constraint equations waslessened due to the necessity of introducing the Barbero-Immirzi parameter inthe new variables. When we choose the Barbero-Immirzi parameter to be complex,it gives polynomial constraint equations. But this choice leads to a complexphase space of GR.

To obtain physical solutions reality conditions must beimposed. In the classical case this is not a problem, but after quantizing thetheory it becomes a major problem to find such reality conditions. Thus thiscomplex form is abandoned.Also, the numericalvalue of the Barbero Immirzi parameter is currently fixed by matching the semi-classical black hole entropy, as calculatedby Stephen Hawking, and the counting ofmicrostates in loop quantum gravity. There are, however, no physical reasonsfor this value.In the complex formwhen the theory is quantized, the metric is no longer a simple operator. Thisis an indication that theories like LQG, which uses the Ashtekar variables toquantize GR will have difficulties finding semi-classical states.

Also thechange of the metric to ashtekar variables still does not yield any results.Thus we need another change of variables- the loop representation.It was found thatcertain functionals, loops, annihilate the Hamiltonian constraint. They dependonly on the Ashtekar variables through the trace of the holonomy, a measure ofthe change of the direction of a vector when its parallel transported over aclosed circle (a loop) i.e. they are SO(3)-invariant or rotational invariant. Space can be viewed as networksof loops called spin networks5.

A spin network, as formulated byPenrose 6 is a kind of diagram in which each line segment representsthe world line of a “unit”. Three linesegments join at each vertex. A vertex may be interpreted as an event in whicheither a single unit splits into two or two units collide and join into asingle unit. More technically, aspin network is a directed graph whose edges areassociated with irreducible representations ofa compact Lie group andwhose vertices are associatedwith intertwiners of the edge representationsadjacent to it. A spin network, immersed into a manifold, can be used to definea functional on the spaceof connections on thismanifold. In fact a loop is a closed spin network (For example, certainlinear combinations of Wilson loops are called spin network states).

  The evolution of a spin network overtime is called a spin foam which is about the size of the Plancklength. Spin foam is a topological structure made out of two-dimensional facesthat represents one of the configurations that must be summed to obtain aFeynman’s path integral description of quantum gravity. A spin networkrepresents a “quantum state” of the gravitational field on a3-dimensional hypersurface. The set of all possible spin networks is countable;it constitutes a basis of LQG Hilbert space.

In LQG space and timeare quantized, it gives aphysical picture of spacetime where space and time are “granular”,analogous to photons in quantum electrodynamics or discrete values of angularmomentum and energy in quantum mechanics. For example, quantization of areas: the operatorof the area A of a two-dimensional surface ? should have adiscrete spectrum. Every spin network isan eigenstate ofeach such operator, and the area eigenvalue equals Where the sum goes over all intersections i of ? withthe spin network and  is the Planck length   isthe Immirzi parameter and  = 0, 1/2, 1, 3/2,.

.. is the spin associatedwith the link i of the spin network. The two-dimensional area istherefore “concentrated” in the intersections with the spin network.According to this formula, the lowest possible non-zero eigenvalueof the area operator corresponds to a link that carries spin 1/2representation.

Assuming an Immirziparameter on the order of 1, this gives the smallest possible measurablearea of ~10?66 cm2.Criticismfor LQG·       LQGmight not get general relativity in the semi classical limit·       PossibleLorentz violation·       Testability-though this remains a problem for all existing theories of quantum gravity Advantages of LQG·       Itis a background independent approach·       Itdoesn’t invoke extra mathematical structures other than that dealt by quantumfield theory and general relativity·       LQGnaturally gives the discreteness of space time 


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