To disambiguate terminology: By “dual space” I mean the standard meaning in QM; we go from a complex vector space V to a space V∨ of linear maps V→C, with the bra/ket duality as a special case. Perhaps I should avoid using “dual” but this explains my earlier usage.
By “complex conjugate space” (sometimes abbreviated as “conjugate space”) I mean specifically the ¯¯¯¯V construction, a completely formal mathematical operation. (To avoid confusion I could say “complex conjugate space”)
Since “complex conjugate space” is an entirely formal construction, it doesn’t necessarily have a physical meaning. Or it could have multiple physical meanings as different isos with V.
OP doesn’t try to go lower-level than Hilbert space; what follows is my attempt to engage with the level you’re talking about:
A “duality” like thing is “polarity of representation of U(1)±” as implied by the category [U(1)±,VectR]. Where to simplify in OP, I’m always using the polarity between V and ¯¯¯¯V in the representation, but this is not strictly necessary.
The correspondence with the SO(n) model might be: When you are representing an element of [BSO(n),VectR] as a complex vector space (B means “delooping groupoid”), you are picking out a sub-group of SO(n) iso to U(1)≅SO(2). To get to a Hilbert space you clearly have to at least pick out a U(1) subgroup.
Another idea is to find a functor [U(1)±,BO(n)]. This does not necessarily get you an O(2) subgroup, because J and J−1 in U(1)± need not map to the same group element of O(n). (Hence spinors and so on)
If you have a representation [BO(n),VectR] and a groupoid functor [U(1)±,BO(n)] you can trivially compose to get a polar representation [U(1)±,VectR]. That gives you, as the positive component, something like a Hilbert space, and as the negative component, something formally iso to its complex conjugate space (though with a better physical interpretation).
At this point you can use the formal isos: ¯¯¯¯V≅V∨ (Riesz representation), and the iso from ¯¯¯¯V to the polar negative of your Hilbert-like space (where polar negation is from the [U(1)±,VectR] representation). This gives you nice physical interpretations of bra/kets, density matrices, observables, and so on, through formal isos.
(I am not sure how much I’m understanding or how much is connecting; feel free to ignore irrelevant detail)
To disambiguate terminology: By “dual space” I mean the standard meaning in QM; we go from a complex vector space V to a space V∨ of linear maps V→C, with the bra/ket duality as a special case. Perhaps I should avoid using “dual” but this explains my earlier usage.
By “complex conjugate space” (sometimes abbreviated as “conjugate space”) I mean specifically the ¯¯¯¯V construction, a completely formal mathematical operation. (To avoid confusion I could say “complex conjugate space”)
Since “complex conjugate space” is an entirely formal construction, it doesn’t necessarily have a physical meaning. Or it could have multiple physical meanings as different isos with V.
OP doesn’t try to go lower-level than Hilbert space; what follows is my attempt to engage with the level you’re talking about:
A “duality” like thing is “polarity of representation of U(1)±” as implied by the category [U(1)±,VectR]. Where to simplify in OP, I’m always using the polarity between V and ¯¯¯¯V in the representation, but this is not strictly necessary.
The correspondence with the SO(n) model might be: When you are representing an element of [BSO(n),VectR] as a complex vector space (B means “delooping groupoid”), you are picking out a sub-group of SO(n) iso to U(1)≅SO(2). To get to a Hilbert space you clearly have to at least pick out a U(1) subgroup.
Another idea is to find a functor [U(1)±,BO(n)]. This does not necessarily get you an O(2) subgroup, because J and J−1 in U(1)± need not map to the same group element of O(n). (Hence spinors and so on)
If you have a representation [BO(n),VectR] and a groupoid functor [U(1)±,BO(n)] you can trivially compose to get a polar representation [U(1)±,VectR]. That gives you, as the positive component, something like a Hilbert space, and as the negative component, something formally iso to its complex conjugate space (though with a better physical interpretation).
At this point you can use the formal isos: ¯¯¯¯V≅V∨ (Riesz representation), and the iso from ¯¯¯¯V to the polar negative of your Hilbert-like space (where polar negation is from the [U(1)±,VectR] representation). This gives you nice physical interpretations of bra/kets, density matrices, observables, and so on, through formal isos.
(I am not sure how much I’m understanding or how much is connecting; feel free to ignore irrelevant detail)