Is there a method to renormalize a theory without using perturbative expansions for the divergences? For example, is there a method to get masses and other renormalized quantities without using expansions and counterterms?
O have heard about Lattice Gauge theory
but what other solutions of examples of non perturbative renormalization (numerical or analytical ) are there?
Answer
The standard nonperturbative way (that provided rigorous constructions in 1+1D and 1+2D QFTs) is constructing the Euclidean (imaginary time) field theory as a limit of lattice theories, and then using analytic continuation to real time via the Osterwalder--Schrader theorem.
In 1+3D, there is so far no rigorous construction of an interacting QFT, but neither is there a corresponding no-go theorem.
In 1+1D, there are also lots of exactly solvable QFTs, where the nonperturbative solution is obtained by the quantum inverse scattering method.
http://en.wikipedia.org/wiki/Quantum_inverse_scattering_method
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