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## First Steps Towards a Theory of Quantum Gravity

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**First Steps Towards a Theory of Quantum Gravity**Mark Baumann Dec 6, 2006**Where to Begin?**How might we quantize gravity? Can we describe gravitational interaction by exchange of a massless particle? If so, what kind of particle? Guess #1: Scalar (spin-0) particle “Einstein-Fokker theory” Problem: No gravitational bending of light or precession of Mercury’s orbit Guess #2: Vector particle, spin-1 Problem: Gravity exhibits repulsion**Where to Begin?**Guess #3: Vector particle, spin-2 The “graviton” Perturbative approach: graviton is a perturbation to a flat background spacetime, like a gravitational wave Another approach: Start with GR and modify it to make it look more like quantum field theory.**General Relativity in a Nutshell**Einstein’s Field Equations: “Matter tells space how to curve, and curved space tells matter how to move.” --J.A. Wheeler RHS: Matter/energy LHS: Curvature of space Curvature of space is encapsulated in the “metric” g, which tells us how far apart things are. The metric is found by solving Einstein’s field equations.**General Relativity in a Nutshell**Metric in particle physics: “Minkowski” metric that describes Cartesian space and includes special relativity Metric in GR: We said we were going to modify GR. How?**Spin and Torsion**• Fundamental properties of a particle include: mass, spin • GR couples mass and the metric, a geometric property of spacetime • Right now, GR is solely a macroscopic theory. However, for a quantum theory of gravity, incorporating spin is essential! • How could we couple spin to the geometry of spacetime? • Idea: Couple spin to torsion • To understand torsion, we start with the covariant derivative and the connection**Covariant Derivatives**Covariant Derivative in quantum field theory: “Converts” global gauge invariance into local gauge invariance. Covariant Derivative in general relativity: connection The connection allows you to compare vectors from different tangent spaces. The covariant derivative is defined so that it’s a tensor. We give these the same name because they are, in fact, the same thing! The GR version is more general.**Mathematical Roadmap**Riemannian Manifold Physics done here Distance & Curvature Add a metric Manifold with Connection Covariant Derivatives Add a connection Manifold More abstraction More structure Make it locally “flat” Topological Space Add a topology Limits & Continuity Set**Torsion Defined**i.e. - it is a measure of the non-commutativity of the lower two indices of the connection. Torsion is a tensor quantity, unlike the connection. Geometrically, the torsion measures how much rotation a vector undergoes when you parallel transport it from one tangent space to another. Idea (due to Cartan, 1922): Couple spin and torsion**Connection**In classical GR, we solve the Einstein equations for the metric. The standard choice for a connection in standard GR is the Levi-Civita connection, sometimes called “Christoffel symbols.” We get this connection if we start with a metric and assume the connection is torsion-free (and also “metric-compatible”). Now we are supposing the connection might not be torsion-free. By allowing the possibility of torsion, we are “freeing” the connection to be another variable. Whereas before we solved the Einstein equations for the metric, now we wish to solve a new set of equations for both the metric and the connection. But which equations?**Einstein-Cartan Equations**Einstein’s equations are derived by varying the Einstein-Hilbert action: where V is a 4-d spacetime volume and the Lagrangian = R, which is the Ricci curvature scalar, which depends on the connection. If the connection depends on torsion, then so does R. This gives us the Einstein-Cartan action:**Einstein-Cartan Equations**Varying the Einstein-Cartan action via the usual process, we get the Einstein-Cartan equations: The first equation has reproduced Einstein’s equations. The second equation involves a spin density tensor S and a modified torsion tensor T, and therefore couples spin with torsion.**Coupling Spinors to GR**Quantum field theory uses a flat metric, but we don’t know how to do QFT on a curved spacetime. So, why not make the metric flat? In GR, we can make the metric look flat at any point by a change of basis. In other words: This new basis e is called the tetrad or vielbein basis, first introduced by Weyl (1929). Now we can proceed normally! However, what are the consequences of this change of basis?**Tetrad Formulation**At every spacetime point, we have a different basis. One consequence: the Dirac matrices are no longer constant, but depend on position as follows: Every occurrence of should be preceded by e Once we find the Dirac matrices, we can compute the metric through the usual equation: We solve for the metric by solving for the ‘s**Further Reading**Baez, J. and Muniain, J. Gauge Fields, Knots, and Gravity, volume 4 of Series on Knots and Everything. World Scientific, 1994. -- Written by a mathematician, mathematically rigorous but intuitively presented. Culminates with Ashtekar’s New Variables. Carroll, Sean M. Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley, 2003. -- A mathematical introduction to GR. Good descriptions, includes tetrad formulation. Ortín, Tomás. Gravity and Strings. Cambridge University Press, 2004.