Momenta fields and the derivative expansion

The Polchinski exact renormalization group equation for a scalar field theory in arbitrary dimensions is translated, by means of a covariant Hamiltonian formalism, into a partial differential equation for an effective Hamiltonian density that depends on an infinite tower of momenta fields with higher spin. A natural approximation scheme is then expanding the Hamiltonian in momenta with increasing rank. The first order of this expansion, one next to the local potential approximation, is regulator-independent and already includes infinitely many derivative interactions. Further truncating this down to a quadratic dependence on the momenta leads to an alternative to the first order of the derivative expansion, which is used to compute $η=0.03616(1)$ for the critical exponent of the three dimensional Ising model.

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