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.
Suppose your physics problem was to find your way from Florida Blvd and N. 22nd. St. to the entrance of Progress Park in Baton Rouge, Louisiana. Suppose you had "solved" your problem by buying a GPS unit. Suppose now that an earthquake takes place that does not cause much destruction but deforms the street grid as the figure shows. Your GPS is now useless as the maps it has were those prior to the earthquake. But suppose you had solved your problem by asking for directions. So you have written in a piece of paper "walk North on 22nd street, turn rigth on North St., turn left on N. 30th. St., etc. Such "solution" would still allow you to reach your destination. The reason it is still valid is that the earthquake will have moved around your trajectory and also your reference points for the turns. The end result is therefore invariant.

Similarly ordinary quantum field theories can be reformulated in such a way that they are invariant under diffeomorphisms. Basically one uses additional physical fields to label the points of space time such that when a diffeomorphism takes place the points of space and the fields one is studying move in unison, just like your directions and your route moved in unison providing an invariant final result. When one formulates them that way they are called "parameterized field theories" because the coordinates are not fixed anymore but are parameters you can vary.

This is what Madhavan Varadarajan in collaboration with Alok Laddha considered in this talk. They study a field theory in a flat 2 dimensional space-time to simplify things and formulate it in a diffeomorphism invariant fashion  (in the jargon this is called 1+1 dimensional to indicate one dimension is space and one is time). This is a subject with a long history. But what is new is that they use the techniques of loop quantum gravity to treat the problem, which in turn solves some problems that were encountered in previous attempts to treat the problem with conventional quantum field theory techniques. They find that the theory has constraints that need to be promoted to quantum operators and that they close an algebra that is very similar to the one that arises in the case of general relativity in four space-time dimensions. The advantage is that the 1+1 model is much simpler to deal with and they can complete the process of promoting the constraints to operators and check that they are consistent (they "close an algebra"). This in turn leads to valuable insights of how one would have to proceed in the case of general relativity, in particular which versions of the constraints to use and which spaces of quantum states to use.