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## Quantum Geometry

Shipping: Free Within U. About this Item Ships with Tracking Number! Buy with confidence, excellent customer service! Softcover reprint of hardcover 1st ed. Bookseller Inventory n Ask Seller a Question. About this title Synopsis: This monograph presents a review and analysis of the main mathematical, physical and epistomological difficulties encountered at the foundational level by all the conventional formulations of relativistic quantum theories, ranging from relativistic quantum mechanics and quantum field theory in Minkowski space, to the various canonical and covariant approaches to quantum gravity.

Excellent customer service. This similarity led Rovelli and Smolin to a representation of the 3-geometry states.

## [gr-qc/] Quantum Geometry and Gravity

In this work, loops form a representation just as Wilson loops form a representation in lattice gauge theory. In general, the states of quantum gravity are intersecting, knotted loops or embedded graphs. One immediate consequence is that metric and geometric quantities are only defined along the loops. B eginning in , an understanding of quantum geometry emerged. Motivated by the need for geometric observables in the canonical approach to quantum gravity and placed on mathematically rigorous foundations, the new framework for the structure of space is an echo of an older, combinatorial definition of spacetime advocated by Penrose.

Space is represented as a "spin network.

### mathematics and statistics online

One tiny piece might look like this. I n the last few years a spacetime description. Q uantum gravity is the field devoted to finding the microstructure of spacetime. Is space continuous? Does spacetime geometry make sense near the initial singularity? Deep inside a black hole? These are the sort of questions a theory of quantum gravity is expected to answer. While a number of more or less exotic suggestions have been floated, no one really knows why Lambda should be so small. This "cosmological constant problem" is one of the biggest mysteries in modern physics. See Ned Wright's cosmology tutorial and Eli Michael's cosmological constant page for more about Lambda.

Popular books describe this proposal with varying degrees of accuracy. By forgetting that the "no boundary proposal" is a quantum mechanical description, though, these popularizations can sometimes be misleading. In particular, it's worth remembering that a quantum mechanical object does not have a unique, well-defined "history. But by the Heisenberg uncertainty relations, this cannot be done: we can never simultaneously exactly specify a particle's position and momentum. The Hartle-Hawking "no boundary" proposal is based on the path integral, or "sum over histories," approach to quantum mechanics, in which a probability amplitude is computed by taking a weighted sum over all possible histories that lead from an initial condition in this case, "nothing" to a final state.

In a certain approximation, this sum is dominated by a "history" in which the Universe initially has a positive-definite metric -- thus the frequent references to "imaginary time. The vacuum in quantum field theory is not really empty; it's filled with "virtual pairs" of particles and antiparticles that pop in and out of existence, with lifetimes determined by the Heisenberg uncertainty principle. When such pairs forms near the event horizon of a black hole, though, they are pulled apart by the tidal forces of gravity. Sometimes one member of a pair crosses the horizon, and can no longer recombine with its partner.

The partner can then escape to infinity, and since it carries off positive energy, the energy and thus the mass of the black hole must decrease. There is something a bit mysterious about this explanation: it requires that the particle that falls into the black hole have negative energy. Here's one way to understand what's going on. This argument is based roughly on section To start, since we're talking about quantum field theory, let's understand what "energy" means in this context. Of course, a classical configuration of a field typically does not have a single frequency, but it can be Fourier decomposed into modes with fixed frequencies.

In quantum field theory, modes with positive frequencies correspond to particles, and those with negative frequencies correspond to antiparticles. Now, here's the key observation: frequency depends on time, and in particular on the choice of a time coordinate. We know this from special relativity, of course -- two observers in relative motion will see different frequencies for the same source. In special relativity, though, while Lorentz transformations can change the magnitude of frequency, they can't change the sign, so observers moving relative to each other with constant velocities will at least agree on the difference between particles and antiparticles.

For accelerated motion this is no longer true, even in a flat spacetime. A state that looks like a vacuum to an unaccelerated observer will be seen by an accelerated observer as a thermal bath of particle-antiparticle pairs. This predicted effect, the Unruh effect, is unfortunately too small to see with presently achievable accelerations, though some physicists, most notably Schwinger, have speculated that it might have something to do with thermoluminescence. Most physicists are unconvinced. The next ingredient in the mix is the observation that, as it is sometimes put, "space and time change roles inside a black hole horizon.

The final ingredient is a description of vacuum fluctuations. One useful way to look at these is to say that when a virtual particle- antiparticle pair is created in the vacuum, the total energy remains zero, but one of the particles has positive energy while the other has negative energy. For clarity: either the particle or the antiparticle can have negative energy; there's no preference for one over the other.

## The Spin-Foam Approach to Quantum Gravity

Now, finally, here's a way to understand Hawking radiation. Picture a virtual pair created outside a black hole event horizon. One of the particles will have a positive energy E, the other a negative energy -E, with energy defined in terms of a time coordinate outside the horizon.

Suppose, though, that in this time the negative-energy particle crosses the horizon. The criterion for it to continue to exist as a real particle is now that it must have positive energy relative to the timelike coordinate inside the horizon , i. This can occur regardless of its energy relative to an external time coordinate. So the black hole can absorb the negative-energy particle from a vacuum fluctuation without violating the uncertainty principle, leaving its positive-energy partner free to escape to infinity.

The effect on the energy of the black hole, as seen from the outside that is, relative to an external timelike coordinate is that it decreases by an amount equal to the energy carried off to infinity by the positive-energy particle. Total energy is conserved, because it always was, throughout the process -- the net energy of the particle-antiparticle pair was zero.

So the black hole can lose energy to vacuum fluctuations, but it can't gain energy.

See the relativity FAQs, here , for a related but slightly different description of black hole thermodynamics and Hawking radiation. One way to understand the simplification is the following.

- Quantum Gravity?
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It is a standard result of general relativity, however, that n of the Einstein field equations are constraints on initial conditions rather than dynamical equations. These constraints eliminate n degrees of freedom per point. Another n degrees of freedom per point can be removed by using the freedom to choose n coordinates. In four spacetime dimensions, this gives the four phase space degrees of freedom of ordinary general relativity, two gravitational wave polarizations and their time derivatives.

The Einstein tensor, in turn, is fixed uniquely, through the Einstein field equations, by the distribution of matter. As a result, there are no propagating gravitational degrees of freedom -- the geometry of spacetime at a point is almost entirely determined by the amount and type of matter at that point. At first sight, this is too strong a restriction. It's not much of a test to be able to quantize a theory with no degrees of freedom, and general relativity is supposed to be about curved spacetimes, not flat ones.

But this sort of counting argument can miss a finite number of "global" degrees of freedom. The simplest example of such degrees of freedom is the following:. Consider a flat, square piece of paper, with the following "gluing" rule: a point at any edge is to be considered the same as the corresponding point at the opposite edge.

Such a space is topologically a torus , and is sometimes called the "video game model" of the torus. When you reach one edge, you automatically pop back in at the opposite edge, as in many video games. Geometrically, this space is flat -- all of the rules of Euclidean geometry hold in any small finite region -- because, after all, any small region looks just like a region of the piece of paper you started with.

It's a fun exercise to convince yourself that this is true even for regions that contain an "edge. Now change this model a little bit by starting with a parallelogram rather than a square. This gives you a different manifold for each different choice of parallelogram, up to some subtle symmetries the "mapping class group". Each of these manifolds is flat, but they are geometrically distinguishable. In fact, this construction gives you a three-parameter family of flat spaces with torus topology: there's one parameter for the length of each side of the parallelogram, plus one for the angle between two adjacent sides.

The constraints fix the overall scale in terms of two parameters say, one side length and one angle , but these two parameters have an interesting and nontrivial evolution. More complicated topologies give more parameters. So do point particles, which can be represented as conical "defects" in space. Note that this does not contradict the earlier counting argument. There are still only finitely many total degrees of freedom, rather than one or more degrees of freedom per point.

But this finite number of degrees of freedom still has a very interesting dynamics, and the theory is rich enough to test many standard approaches to quantum gravity. Two-dimensional spacetimes also provide an interesting testing ground for quantum gravity.

As you might guess from the earlier counting argument, ordinary general relativity does not make sense in two spacetime dimensions -- the count gives "-2 degrees of freedom per point. For a beautiful nontechnical introduction to topologies like the "video game torus," see Jeff Weeks' book, The Shape of Space. This task is rather difficult, since we don't yet have a quantum theory of gravity. But there may be reasonable approximations that can be used to obtain partial information.

Among the popular approaches are various saddle point approximations to the path integral including approximations of the no boundary proposal and "minisuperspace models," models in which all but a finite number of degrees of freedom of the gravitational field are "frozen out" and held fixed. The quantum geometry program has recently made some interesting progress in such minisuperspace cosmology -- see, for example this review by Bojowald. The states of loop quantum gravity are described by "spin networks," graphs whose edges are labeled by spins and whose vertices are labeled by "intertwiners" think Clebsch-Gordan coefficients that tell how to combine spins.

Geometric operators like area, constructed from the spacetime metric, act on these states, changing the network. The dynamics of general relativity comes in through the Hamiltonian constraint, a not-fully-understood operator condition that determines the admissible spin networks. Loop quantum gravity has had important successes in black hole physics and in quantum cosmology.