Wednesday, February 17, 2010

Loop Quantum Gravity: Fasciantion With Field Lines.

(The second post of the LQG series. )

The founders of electromagnetic theory made a remarkable discovery: forces seem to be related to field lines that begin and end on charges.  (See figure to the right.)  In fact, Faraday thought of these field lines as 'lines for force' that were really propagating through space.

Furthermore, it has been discovered that all "gauge-theories" such as electricity and magnetism or Yang-Mills theories can be described by field lines that represent "the holonomy" of the gauge potential of the theory.  Holonomy is a fancy way of classifying the structure of the gauge potential that is related to the exponential of the line integral along some loop alpha of the gauge potential.

For example, for electricity and magnetism that has gauge potential A, the holonomy along some loop alpha is this:
Where U is the holonomy.

Non-canonical quantization.

Now, in a canonical quantization of a theory we create what are called raising and lowering operators that give rise to the different states in the theory.  Objects in your theory become excitations of your field. These excitations are related to how many creation or annihilation operators are needed to create the state.

In the "holonomy" approach to quantization, instead of quantizing the theory using canonical raising and lowering operators, you quantize by promoting the holonomy to an operator.  These operators act on each state creating a new state in the same way creation and annihilation do the job in the canonical approach to quantum mechanics.

This means, excitations of the field quantized in this way give rise to "loop states". (Hence the name loop quantum gravity.)  Hence, imagining filed lines looping around emerging from charges may not be such a bad way to think about things after all.  Loops of field lines are the excitations of the field.

Take away message if you fell asleep:
Quantum theories can be thought of as excitations of some underlying quantum field.  The excitations are related to operators that quantize the theory.  It turns out, we can quantize theories by using the holonomy or field lines of the gauge potential. (The things in the figure above.)  By quantizing a theory in this way, as is done in loop quantum gravity, what emerges is idea that excitations of the underlying quantum field are actually loops.  (Loops much like the field lines imagined by the founders of the electromagnetic theory.)

So, loop quantum gravity is in fact a loopy theory.

1 comment:

  1. This is an interesting way of thinking about the Wilson loops / spin networks. Nice.


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