Abstract
We present an exact computation of effective Hamiltonians for an elementary model obtained from the Yukawa theory by going to the limit of bare fermions being infinitely heavy and bare bosons being at rest with respect to the fermions that emit or absorb them. The coupling constant can be arbitrarily large. The Hamiltonians are computed by solving the differential equation of the renormalization group procedure for effective particles (RGPEP). Physical fermions, defined in the model as eigenstates of the effective Hamiltonians, are obtained in the form of an effective fermion dressed with a coherent state of effective bosons. The model computation illustrates the method that can be used in perturbative computations of effective Hamiltonians for realistic theories. It shows the mechanism by which the perturbative expansion and Tamm-Dancoff approximation increase in accuracy along the RGPEP evolution.
Highlights
Complexity of relativistic quantum field theory (QFT) implies a need for approximate computational methods
This paper presents a novel, exact renormalization group computation of effective Hamiltonians in a model that results from drastic but precisely specified simplifications of QFT, so that one can see the steps that would have to be reconstructed in an analogous computation of the effective Hamiltonians and their spectra in QFT
If the model were divergent, as it is the case for bare Hamiltonians in local QFT, H0 would be supplied with the counterterms that would be computed from the condition that the coefficients of effective particle operators in Ht for any finite, fixed value of t are not sensitive to the adopted regularization of the divergences
Summary
Complexity of relativistic quantum field theory (QFT) implies a need for approximate computational methods. This variety of models that can be solved suggests to the author it should be clearly stated that the main purpose here is different It is to apply a recently formulated renormalization group equation for Hamiltonians of QFT, to a simplified, onelevel model for fermions coupled to bosons in a way that is analogous to the Yukawa theory coupling. In application to local QFT, the RGPEP has been recently illustrated in [15], which includes references to the previous works In all these examples, one is forced to use the approximations that are not under precise control, such as the mentioned earlier weak-coupling expansion [2] or a limitation on a number of virtual particles, called the Tamm-Dancoff (TD) approximation [16,17].
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