Boosting equal time bound states

TL;DR

This paper develops an explicit method to boost relativistic bound states at equal time in QED and QCD in 1+1 dimensions, ensuring the wave function transforms correctly under Lorentz boosts in the Born approximation.

Contribution

It constructs the Poincaré generators and derives the transformation law of the wave function for bound states in 1+1 dimensional gauge theories at lowest order in ar, maintaining gauge conditions.

Findings

Wave function shape is independent of CM momentum in a specific variable.

Lorentz contraction depends on the potential and position, proportional to 1/(E-V(x)).

Boosted states remain eigenstates of P^0 and P^1 with transformed eigenvalues.

Abstract

We present an explicit and exact boost of a relativistic bound state defined at equal time of the constituents in the Born approximation (lowest order in hbar). To this end, we construct the Poincar\'e generators of QED and QCD in D=1+1 dimensions, using Gauss' law to express A^0 in terms of the fermion fields in A^1=0 gauge. We determine the fermion-antifermion bound states in the Born approximation as eigenstates of the time and space translation generators P^0 and P^1. The boost operator is combined with a gauge transformation so as to maintain the gauge condition A^1=0 in the new frame. We verify that the boosted state remains an eigenstate of P^0 and P^1 with appropriately transformed eigenvalues and determine the transformation law of the equal-time, relativistic wave function. The shape of the wave function is independent of the CM momentum when expressed in terms of a variable, which is quadratically related to the distance x between the fermions. As a consequence, the Lorentz contraction of the wave function is proportional to 1/(E-V(x)) and thus depends on x via the linear potential V(x).