Scalar Fields on a Random Lattice

Lee next generalizes discrete quantum mechanics to the case of a massless scalar field j) interacting with an arbitrary external current J ([tdlee85a], [tdlee85b]). 

In continuum field theory, the field f) x, t) is a dynamical variable and the coordinates X and t are only parameters. With the conjugate momentum f (T, t), and Hamiltonian operator 

To discretize this therrry, Lee assumes that in any n-dimensional volume O — L V, there can be at most N measurements that determine the space-time position Xi of the, observation and the value of the field d i at The ratio p = N/Q, = is therefore a fumlamental constant of the theory, 

A discrete action, 5a, may be defined by as.sinuing that only neighboring pairs of sites (say, i and j) are coupled. Each pair contributes to the action a term 4 i - 4 N, weighted by a factor Aij, Since different links lij can have different lengths and orientations, it is reasonable to assume that they should also contribute 

Using this simplicial decomposition, we may now define a dual lattice constructively as follows [tdleeSSb]. Let i be an arbitrary site in , We say that a point p G D belongs to i if i is the nearest site to p in . The dual to site i is the volume iJi = p p belongs to f. We call Wj an n-dimensional cell.f The dual lattice of the random lattice consists of the cells into which s volume D is divided using the above construction. 

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