An Alternative Lattice Field Theory Formulation Inspired by Lattice Supersymmetry, -Associativity and Regularization-
- Prof. Noboru KAWAMOTO
- Prof. Noboru KAWAMOTO (Hokkaido University)
This is the second talk followed to the first one of general formulation proposing a new type of lattice field theory without lattice chiral fermion problem. It has been formulated by the momentum space by introducing a new continuum momentum on the lattice and has non-local nature in the coordinate space. The recovery of associativity of the non-local product led us to this new formulation. It turned out that this non-local lattice formulation is equivalent to the continuum theory and thus keeps all the symmetry exact, including lattice supersymmetry, that the corresponding continuum theory has. One may wonder: "How come the lattice theory be equivalent to continuum theory if it is lattice regularized ?" In fact this non-local lattice formulation is not yet regularized. We explain a new type of regularization procedure for this formulation.
Most naive regularization could be to put the system in a box.
In this case momentum becomes discrete and thus have a difficulty
to find a solution of delta function of the new momentum conservation.
We can then consider a modified lattice derivative operator with a new regularization parameter together with lattice constant. This formulation is essentially equivalent to the momentum cutoff regularization of continuum theory.
A lattice SUSY version of Ginsparg-Wilson relation requires an invertibility of the blocking transformation which requires specific form of the blocking transformation. This suggests us to introduce yet a new regularization parameter in addition to the lattice constant. We explicitly show how these regularization procedures work for 4-dimensional Wess-Zumino model and phi^4 model.
We discuss the relation between the recovery of symmetries and the introduction of the regularization parameters. The lattice SUSY can be kept exact but associativity can be broken with the introduction of the regularization parameters. The breaking of associativity generates broken gauge invariance. It is, however, expected to recover in the continuum limit without fine tuning since the formulation is equivalent to the continuum theory in the limit.
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