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Initially the label $a$ represent the site on a lattice with sites separated by distance $l$. i.e., the discrete position for each of a collection of oscillators, $q_a$, with an oscillator at each lattice site. When we take the continuum limit, $l\to0$, the label $a$ becomes continuous, i.e. it becomes the position variable $x$. We now have an infinite ...

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In general, bound states in lattice QCD are found by analyzing correlation functions of operators with the appropriate quantum numbers. This works both for baryons and mesons, even for pure-glue states like glueballs. See for example these introductory lecture notes: http://arxiv.org/abs/hep-lat/9807028. Chapter 17 should answer your questions in more ...

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The first thing to realise here, is that in your Hamiltonian formulation, the lattice constant is implicitly stated as being $b=1$ (I'm calling it b, not to confuse it with the lattice operators). Next you can identify, if you're working in two dimensions, $x=am$ and $y=bn$ yielding \hat{H}_0=-J(\sum_{(x/b),(y/b)} ...

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