We formulate a renormalization group (RG) transformation through a unitary transformation which diagonalize the Hamiltonian in the high frequency modes, allowing the definition of an effective low-energy theory. We apply this to the λΦ 4 -scalar theory.
We develop the idea that renormalization, decoupling of heavy particle effects from low energy physics and the construction of effective field theories are intimately linked to the momentum space entanglement of disparate modes of an interacting quantum field theory. Using unitary transformations to decouple these modes at the perturbative level, we show in a scalar field theoretical model with light and heavy fields, how renormalization may be consistently implemented and how the low energy effective field theory can be constructed. We also obtain a renormalization group equation in this framework and apply it to the scalar field theoretical model.
After a brief review of the definition and properties of the quantum effective Hamiltonian action, we describe its renormalization flow by a functional renormalization group equation. This equation can be used for a nonperturbative quantization and study of theories with bare Hamiltonians that are not quadratic in the momenta. As an example, the vacuum energy and gap of quantum mechanical models are computed. Extensions of this framework to quantum field theories are discussed. In particular, one possible Lorentz-covariant approach for simple scalar field theories is developed. Fermionic degrees of freedom, being naturally described by a first order formulation, can easily be accommodated in this approach.
We propose a non-perturbative renormalization procedure for light-front Hamiltonian. Wilson's renormalization group prescription is applied to λΦ4-model within a light-front Fock space truncated up to three-body state. Renormalized Hamiltonian which preserves low-energy physics for arbitrary renormalization transformations is presented. Renormalization group equations for mass parameter μ2 and coupling constant λ are derived by integrating high invariant mass space and rescaling transverse momenta and fields.
Gradient flow has proved useful in the definition and measurement of renormalized quantities on the lattice. Recently, the fact that it suppresses high-modes of the field has been used to construct new, continuous RG transformations both analytically and on the lattice, distinct from the usual blocking techniques in spin models and gauge theories. In this work, we discuss such a formulation for scalar field theory, and we present preliminary numerical results for its application to the determination of critical exponents at the Wilson-Fisher fixed point of three-dimensional ϕ4 theory.
Gradient flow has proved useful in the definition and measurement of renormalized quantities on the lattice. Recently, the fact that it suppresses high-modes of the field has been used to construct new, continuous RG transformations both analytically and on the lattice, distinct from the usual blocking techniques in spin models and gauge theories. In this work, we discuss such a formulation for scalar field theory, and we present preliminary numerical results for its application to the determination of critical exponents at the Wilson-Fisher fixed point of three-dimensional ϕ4 theory.
We propose inverse renormalization group transformations within the context of quantum field theory that produce the appropriate critical fixed point structure, give rise to inverse flows in parameter space, and evade the critical slowing down effect in calculations pertinent to criticality. Given configurations of the two-dimensional ϕ^{4} scalar field theory on sizes as small as V=8^{2}, we apply the inverse transformations to produce rescaled systems of size up to V^{'}=512^{2} which we utilize to extract two critical exponents. We conclude by discussing how the approach is generally applicable to any method that successfully produces configurations from a statistical ensemble and how it can give novel insights into the structure of the renormalization group.