Fierz identities in spaces of nonintegral dimension

Fierz transformations, which change the order of the spinor indices of direct products of $\gamma$ matrices, are studied. It is shown that the particular features of the Fierz identities in infinite-dimensional spaces (among which a space of nonintegral “dimension” $d=4-2\varepsilon$ should be included) strongly restrict the possibilities of constructing a supersymmetric dimensional regularization.