I was studying this book about the conformal group, and it explains the generators of the group in this way: 1 2
I don’t get why you can’t obtain the generators in d=1 following this reasoning. Everything there is also valid in d=1 except the L generator is zero. How to generalize this reasoning in d=1?
For example in this paper https://inspirehep.net/literature/108211 there are the generators in d=1.
The book is Conformal Field Theory by Philippe di Francesco.
SOLVED: The solution was explained later in the book for the case d=2 and the reasoning is the same as in the case d=1: 3
Notice that if d=1 the conformal map equation does not put any constraint on the scale factor f(x), any smooth and invertible map is conformal. When d=1, every generator vanishes except translation, CST, and dilation. One example of such a conformal theory is conformal quantum mechanics, symmetric under SL(2,R), see:
https://arxiv.org/abs/1506.05596
You don't have to generalize the reasoning in your book, as the discussion is perfectly valid when d=1. (You get three generators, one of which is the Hamiltonian of the system, the other two are just "t" times the preceding generator, you can check the commutators and confirm that this is indeed a SL(2,R) algebra)