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edited Sep 11, 2020 at 17:07

I have the following two impressions about fermions in physics. I'm confused about their accuracy, and their compatibility:

To consider the behavior of a fermion, whose intrinsic spin is described by a representation $V$ of the group $Spin(p,q)$, on a pseudo-Riemannian manifold $M$ of signature $(p,q)$, you first introduce a spin structure on $M$. Then the fermion field is a section of the bundle associated to $V$.
To consider the behavior of a fermion on a pseudo-Riemannian manifold $M$ of signature $(p,q)$, you first turn $M$ into a supermanifold. Then the fermion field is a superfunction on $M$ with some constraints coming from its intrinsic spin.

Question: Is either of (1) or (2) close to accurate? What major points or subtleties have I missed? If both are close to accurate, then how does one "translate" between the formalism of (1) and the formalism of (2)?

I have the following two impressions about fermions in physics. I'm confused about their accuracy, and their compatibility:

To consider the behavior of a fermion, whose intrinsic spin is described by a representation $V$ of the group $Spin(p,q)$, on a pseudo-Riemannian manifold $M$ of signature $(p,q)$, you first introduce a spin structure on $M$. Then the fermion field is a section of the bundle associated to $V$.
To consider the behavior of a fermion on a pseudo-Riemannian manifold $M$ of signature $(p,q)$, you first turn $M$ into a supermanifold. Then the fermion field is a superfunction on $M$ with some constraints coming from its intrinsic spin.

Question: Is either of (1) or (2) accurate? If both are accurate, then how does one "translate" between the formalism of (1) and the formalism of (2)?

I have the following two impressions about fermions in physics. I'm confused about their accuracy, and their compatibility:

To consider the behavior of a fermion, whose intrinsic spin is described by a representation $V$ of the group $Spin(p,q)$, on a pseudo-Riemannian manifold $M$ of signature $(p,q)$, you first introduce a spin structure on $M$. Then the fermion field is a section of the bundle associated to $V$.
To consider the behavior of a fermion on a pseudo-Riemannian manifold $M$ of signature $(p,q)$, you first turn $M$ into a supermanifold. Then the fermion field is a superfunction on $M$ with some constraints coming from its intrinsic spin.

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asked Sep 11, 2020 at 16:54

Tim Campion

What is the relationship between spinors and supermanifolds and fermions?

I have the following two impressions about fermions in physics. I'm confused about their accuracy, and their compatibility:

To consider the behavior of a fermion, whose intrinsic spin is described by a representation $V$ of the group $Spin(p,q)$, on a pseudo-Riemannian manifold $M$ of signature $(p,q)$, you first introduce a spin structure on $M$. Then the fermion field is a section of the bundle associated to $V$.
To consider the behavior of a fermion on a pseudo-Riemannian manifold $M$ of signature $(p,q)$, you first turn $M$ into a supermanifold. Then the fermion field is a superfunction on $M$ with some constraints coming from its intrinsic spin.

Question: Is either of (1) or (2) accurate? If both are accurate, then how does one "translate" between the formalism of (1) and the formalism of (2)?