3 edited tags

You and I are having a conversation:

"Okay," I say,"I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles."

"Something like that"

"...And then we can add these as local symmetries to space by restricting a connection on some priciple bundle with a nice little lagrangian..."

"Again, something like that."

"But this all seems pretty 'top-down'- I mean, why aren't we trying to see what matter looks like?" I reel off a somewhat overblown soliloquy on that old John Wheeler quote about empty space (free wine from the afternoon's colloquium, evident in its effect), but you have stopped listening.

"String theory?"

"Don't get me started on string theory! A bunch of guys who never learned to apply Occam's razor is what that is- 'ooh it's not working, must be because we need more assumptions'- or something to keep the differential geometers busy 'til they unfreeze Einstein!" You look offended. "What?!! I'm joking!!"

"Sure."

"Still, though, they're wrong- I mean; Donaldson's Theorem is a dead give away isn't it?! Here we are hurtling through topological $\mathbb{R}^4$, having this conversation, surrounded by artefacts of differential structure: someone's proved that this is the only space of this sort in which these artefacts can occur in some sensibly invariant way- and with a proper continuum of possibilities, too- and we're talking about string theory! Why isn't everyone in the mathematical physics world trying to crack the puzzle of the different structures in topological $\mathbb{R}^4$? I'm not saying it's going to be electron= Casson handle, but it must be worth at least looking. For crying out loud, why aren't we all looking at exotic $\mathbb{R}^4$?!

You resist the temptation to give me a withering look, clear your throat, and say:

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# Theoretical physics: Why not just R^4?

You and I are having a conversation:

"Okay," I say,"I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles."

"Something like that"

"...And then we can add these as local symmetries to space by restricting a connection on some priciple bundle with a nice little lagrangian..."

"Again, something like that."

"But this all seems pretty 'top-down'- I mean, why aren't we trying to see what matter looks like?" I reel off a somewhat overblown soliloquy on that old John Wheeler quote about empty space (free wine from the afternoon's colloquium, evident in its effect), but you have stopped listening.

"String theory?"

"Don't get me started on string theory! A bunch of guys who never learned to apply Occam's razor is what that is- 'ooh it's not working, must be because we need more assumptions'- or something to keep the differential geometers busy 'til they unfreeze Einstein!" You look offended. "What?!! I'm joking!!"

"Sure."

"Still, though, they're wrong- I mean; Donaldson's Theorem is a dead give away isn't it?! Here we are hurtling through topological $\mathbb{R}^4$, having this conversation, surrounded by artefacts of differential structure: someone's proved that this is the only space of this sort in which these artefacts can occur in some sensibly invariant way- and with a proper continuum of possibilities, too- and we're talking about string theory! Why isn't everyone in the mathematical physics world trying to crack the puzzle of the different structures in topological $\mathbb{R}^4$? I'm not saying it's going to be electron= Casson handle, but it must be worth at least looking. For crying out loud, why aren't we all looking at exotic $\mathbb{R}^4$?!

You resist the temptation to give me a withering look, clear your throat, and say: