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Scalar fields have dimension 1, spinor fields dimension 3/2, and vector bosons like the photon dimension 1. According to the principles of renormalizability (along with others), this restricts the possible interactions a field can participate in, by restricting the possible combinations of fields in a Lagrangian. However, if we had a dimensionless field, we could have a kinetic term with four derivatives, and every combination of that field could exist with a coupling constant of dimension 4.

I can see one problem in how this field might couple to other fields. I just don't have the depth of understanding to see where the problems lead.

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From some point of view the answer is the Poincare symmetry. Really, the dimension of the field is determined by the lagrangian (its kinetic term), which, in its turn, is constructed in a way such that it generates equations of motion for the given field. The (free) equations of motion for the fields realizing irreducible representations of the Poincare group are first and second order in derivatives. Therefore the lagrangian also has to contain only terms with first and second order of derivatives, which means that the fields can't be dimensionless (since there are no fourth derivatives kinetic term).

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