Hilbert series and operator bases with derivatives in effective field theories

We introduce a systematic framework for counting and finding independent operators in effective field theories, taking into account the redundancies associated with use of the classical equations of motion and integration by parts. By working in momentum space, we show that the enumeration problem can be mapped onto that of understanding a polynomial ring in the field momenta. All-order information about the number of independent operators in an effective field theory is encoded in a geometrical object of the ring known as the Hilbert series. We obtain the Hilbert series for the theory of N real scalar fields in (0+1) dimensions--an example, free of space-time and internal symmetries, where aspects of our framework are most transparent. Although this is as simple a theory involving derivatives as one could imagine, it provides fruitful lessons to be carried into studies of more complicated theories: we find surprising and rich structure from an interplay between integration by parts and equations of motion and a connection with SL(2,C) representation theory which controls the structure of the operator basis.