Invariance of norm under permutations seems a reasonable assumption for state spaces. On the other hand, I now realize the answer to my question about whether permutation invariance narrows things down to p-norms is no. A simple counterexample is a linear combination of two different p-norms.
I think there might be a good reason to think in terms of norm-preserving maps. Namely, suppose the norms can be anything but the individual amplitudes don’t matter, only their ratios do. That is, states are identified not with vectors in the Hilbert space, but rays in the Hilbert space. This is the way von Neumann formulated QM, and it is equivalent to the now more common norm=1 formulation. This also seems to be the formulation Eli was implicitly using in some of his previous posts.
The usual way to formulate QM these days is, rather than ignoring the normalizations of the state vectors, one can instead just decree that the norms must always have a certain value (specifically, 1). Then we can assign meaning to the individual amplitudes rather than only their ratios. It seems likely to me that theories where only the ratios of the “amplitudes” matter, generically can be equivalently formulated as a theory with fixed norm. Thinking that only ratios matter seems a more intuitive starting point.
Psy-Kosh:
Good example with the Lorentz metric.
Invariance of norm under permutations seems a reasonable assumption for state spaces. On the other hand, I now realize the answer to my question about whether permutation invariance narrows things down to p-norms is no. A simple counterexample is a linear combination of two different p-norms.
I think there might be a good reason to think in terms of norm-preserving maps. Namely, suppose the norms can be anything but the individual amplitudes don’t matter, only their ratios do. That is, states are identified not with vectors in the Hilbert space, but rays in the Hilbert space. This is the way von Neumann formulated QM, and it is equivalent to the now more common norm=1 formulation. This also seems to be the formulation Eli was implicitly using in some of his previous posts.
The usual way to formulate QM these days is, rather than ignoring the normalizations of the state vectors, one can instead just decree that the norms must always have a certain value (specifically, 1). Then we can assign meaning to the individual amplitudes rather than only their ratios. It seems likely to me that theories where only the ratios of the “amplitudes” matter, generically can be equivalently formulated as a theory with fixed norm. Thinking that only ratios matter seems a more intuitive starting point.