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I have some ideas about the reason:

1 - it's not part of the theory to have a conserved interval

2 - there's no way to have a metric in a complex Hilbert space

On the other hand, in QM / QFT conservation of probability seems to be as important as a metric interval is in Special/ General Relativity. So that confuses me.

Probably the answer is that as QM/QFT are worked out in Hilbert Spaces, there's an inner product, which plays the role of a "metric interval" as we know in Special/General Relativity?