Quantum Fields in Curved Space-times

[MManuel Abad]

Hello, people:

I've been wondering about the definition of Quantum Fields in Curved Space-times (CS). I know that, in flat space-time (Minkowski), the fields are defined as irreducible representations of the universal covering group SU(2)xSU(2) of SO(4) (which is basically the Lorentz group SO(1,3) under a Wick rotation of time).

Nevertheless, in general (that is in CS) your field theory is not invariant under Lorentz transformations. How then are fields defined in CS? Could you explain your answer and give some references, please?

Thanks

 

[Ben Niehoff]

The simplest way is to use orthonormal frames. In this language, the tangent bundle can be thought of as a principal SO(1,3) bundle (or SO(4) after Wick rotation) over spacetime. Then fields take values in this SO(1,3) bundle or in some derived bundle (i.e., other representations of SO(1,3)).

There is some subtlety with spinors, as not all curved manifolds admit spinors (those that do have what is called a "spin structure").

 

[MManuel Abad]

Oh! These orthonormal frames are simply the tetrads, aren't they?

So no, instead of asking the fields to be representations of SU(2)xSU(2), we ask their contractions with the Vierbein to be irreps of this group. Is this correct?

Thankyou!

 

[Ben Niehoff]

Yes, although for half-integer spin you should think "Clifford action" instead of "contraction".