## Saturday, January 22, 2011

### Loop quantization of a parameterized field theory

by Rodolfo Gambini and Jorge Pullin

• Madhavan Varadarajan, Loop quantum gravity dynamics: insights from parameterized field theory, 16 November 2010. PDF of the slides, and audio in either .wav (82MB) or .aif format (4MB). Based on joint work with Alok Laddha.

In many instances physical theories are formulated in such a way that the mathematical quantities used to describe the physics are not the bare minimum necessary. One is using redundant variables. This can be either for convenience, to make better contact with easily measurable things, or, as is the case in general relativity because we do not know how to isolate the essential variables. When one has redundant variables symmetries appear in the theory: apparently different values of the variables represent the same physics. There also are equations relating the values of the variables, called constraints. Constraints have two purposes: first they tell us that the variables are redundant and in reality there are less "free" variables than one thinks. Second, the constraints generate transformations among the variables that tell us that what appear to be two different physical configurations are in fact the same physically. Configurations that are physically equivalent through these transformations are called "gauge equivalent". When one has more than one constraint there exist consistency conditions that need to be satisfied that say that the free variables are the same no matter in which order I generate transformations with the constraints. So for instance I could start from a given configuration and transform it using constraint number 1 and then number 2 or vice-versa. Mathematically the consistency conditions state that the difference of the action of two constraints in the two orders is either zero or is proportional to a constraint. When this occurs one talks of the constraints "closing an algebra".

When one proceeds to quantize theories, the constraints have to be promoted to quantum operators. The procedure has a significant degree of ambiguity, particularly for complex field theories like general relativity. One does have some guiding tools. For instance the quantum constraints should also "close an algebra" like their classical counterparts. It is expected that achieving such requirement will cut down on the ambiguities of the quantization process and offer guidance when building the quantum theory.

The main symmetry of the general theory of relativity is called "diffeomorphism invanriance". This symmetry states that a priori all points of space-time are equivalent and can be dragged into each other. This is a very natural notion in an empty universe. Say you were lost in a ship the middle of the ocean in a very calm day so far away from the coast that you cannot see it and that the sky is overcast and at night. You could not tell where your ship is or one point of the ocean from another they will all look the same to you. The same is true in any empty universe. To start to distinguish one point from another you need to introduce objects in your universe, for instance fields. Then you can identify a point by knowing what is the value of the field at that point. Your theory will still be diffeomorphism invariant if when you drag a point into another you drag the corresponding value of the field.

Ordinary non-gravitational physics is not formulated in a diffeomorphism invariant way.