Von Neumann’s letters discuss specifically what would today be called quantum dynamics on state-spaces having a noncommutative geometry. More broadly, it’s von Neumann’s view that:

[Quantum field theory] is not at all rigorous the way it is done, but it must either have a rigrous equivalent or something is totally wrong with the way that quantum mechanics is now proceeding. (I think that “public opinion” among physicists has accepted too easily that a rigorous equivalent must exist.) Either alternatives would be interesting and significant and the sooner a broad mathematical audience familiarizes itself with this challenge, the better are the chances that the matter will be resolved.

Today, seventy years later, these questions are still regarded as subtle, difficult, and open.

]]>I guess I just don’t see the big deal. QFT in 3+1 does not exist. Physicists can still calculate what goes on in collider experiments, it’s just that these techniques are way further from a coherent mathematical theory than anbody thought possible.

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