# Does particle spin depend on background?

As I understand it, the spin of anything is with respect to the background spacetime. So I have to wonder if the spin of elementary particles depends on the background metric, on the number of dimensions of the background, or the expansion rate, or the curvature, etc. Or is spin independent of the background spacetime? Is there any connection between the symmetries of the Standard Model and the spin of elementary particles? Any thoughts on this? Thanks.

Hi friend,

spin is just an extra degree of freedom introduced when you realise the "wavefunction"/"state-vector" of a particle requires more than one complex component to describe it, Levy-Leblond's famous 1967 paper (free download) explained this (so, in particular, Dirac's relativistic corrections are not needed to introduce spin)

You can think of this as a type of "linearisation of the Schrödinger Equation", and if you're really cool (like me) you'll extend the wave function of every particle to ~10^80 components.

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tom.stoer
Spin can be derived as an entity that is intimately related to the symmetries of spacetime. In flat Minkowski spacetime the symmetry seems to be the SO(3,1) Lorentz group. But one can enlarge this symmetry via complexification to SL(2,C) and show that this is locally isomorphic to two copies of SU(2)*SU(2). This is the reason why in 4-component Dirac spinors we have two two-component spinors (large and small component). Therefore integer and half-integer spin follow rather naturally from the symmetry group of (flat Minkowski) spacetime.

Constructing a theory of gravity coupled to spin-half fields it becomes clear that this is not possible by using the metric formalism; one must enlarge the formalism to so-called tetrads. These are something like basis vectors of a four-dim. tangent space attached to each point in spacetime. Using these tangent spaces one can "gauge" the Lorentz symmetry, i.e. one can introduce a local SL(2,C) gauge symmetry. This gauge symmetry means that one can rotate the tetrads at each spacetime differently b/c there is a connection, a gauge-field so to speak, which plays a similar role as a gauge field in ordinary gauge theories: it compensates the effect of local gauge transformations and makes the theory gauge invariant (this symmetry and the gauge field is not visible in the metric formalism).

b/c of this local SL(2,C) gauge symmetry one can say that locally a curved Riemann manifold = spacetime has the same symmetry structure as flat Minkowski space and that this allowes for the same construction of spinor fields as in Minkowski spacetime.

However there may be a global obstruction to introducing spin. A Riemann manifold that allows for a spin structure, i.e. the introduction of spinor fields, must have a certain property: it must be orientable (a Möbius strip isn't orientable) and its second Stiefel-Whitney class must vanish (unfortunately I cannot explain lucidly what this means). Therefore not all Riemann manifolds allow for the construction of spinor fields.

Assuming that the topology of spacetime remains invariant during expansion (time evolution) of the universe (which is the case in general relativity except for the formation of black holes) the spin structure does not change. It's a topological invariant.

The relation to the standard model is simply the fact that our universe allows for a spin structure and that therefore a plethora of elementary particles with spin (spin 1/2) can exist. The different particle species (electron, quarks, neutrinos, ...) in the standard model cannot be restricted or derived from the symmetry structure of spacetime. All what we can say is that spin 1/2 seems to exist and that we have a mathematical classification of manifolds which allow for spin to exist.

Spin can be derived as an entity that is intimately related to the symmetries of spacetime. In flat Minkowski spacetime the symmetry seems to be the SO(3,1) Lorentz group. But one can enlarge this symmetry via complexification to SL(2,C) and show that this is locally isomorphic to two copies of SU(2)*SU(2). This is the reason why in 4-component Dirac spinors we have two two-component spinors (large and small component). Therefore integer and half-integer spin follow rather naturally from the symmetry group of (flat Minkowski) spacetime.

Constructing a theory of gravity coupled to spin-half fields it becomes clear that this is not possible by using the metric formalism; one must enlarge the formalism to so-called tetrads. These are something like basis vectors of a four-dim. tangent space attached to each point in spacetime. Using these tangent spaces one can "gauge" the Lorentz symmetry, i.e. one can introduce a local SL(2,C) gauge symmetry. This gauge symmetry means that one can rotate the tetrads at each spacetime differently b/c there is a connection, a gauge-field so to speak, which plays a similar role as a gauge field in ordinary gauge theories: it compensates the effect of local gauge transformations and makes the theory gauge invariant (this symmetry and the gauge field is not visible in the metric formalism).

b/c of this local SL(2,C) gauge symmetry one can say that locally a curved Riemann manifold = spacetime has the same symmetry structure as flat Minkowski space and that this allowes for the same construction of spinor fields as in Minkowski spacetime.

However there may be a global obstruction to introducing spin. A Riemann manifold that allows for a spin structure, i.e. the introduction of spinor fields, must have a certain property: it must be orientable (a Möbius strip isn't orientable) and its second Stiefel-Whitney class must vanish (unfortunately I cannot explain lucidly what this means). Therefore not all Riemann manifolds allow for the construction of spinor fields.

Assuming that the topology of spacetime remains invariant during expansion (time evolution) of the universe (which is the case in general relativity except for the formation of black holes) the spin structure does not change. It's a topological invariant.

The relation to the standard model is simply the fact that our universe allows for a spin structure and that therefore a plethora of elementary particles with spin (spin 1/2) can exist. The different particle species (electron, quarks, neutrinos, ...) in the standard model cannot be restricted or derived from the symmetry structure of spacetime. All what we can say is that spin 1/2 seems to exist and that we have a mathematical classification of manifolds which allow for spin to exist.

Hi Tom,

except Spin isn't a relativistic property - look at Levy-Leblond's paper (see my post above, sorry we posted simultaneously) - you just need a multi component wave function to incorporate spin.

tom.stoer
The OP was asking for spacetime, expansion, curvature and things like that. That's why I was thinking that one should explain spin on Riemann manifolds in the context of GR. Besides that you are right, the construction of spin via SL(2,C) i.e. via SR may be too complicated to start with and a non-rel. approach should be sufficient.

However I was not aware of this paper, so thanks for the reference.

Regarding
... you just need a multi component wave function to incorporate spin.
This is not possible in general for Riemann manifolds.

There are manifolds which do not allow for half-integer spin due the above mentioned topological obstruction to the existence of spin structures. I do not know whether there are Riemann manifolds which may be relevant on GR and what exactly goes wrong when trying to construct spinors, I only know the general statement that w/o vanishing of the second Stiefel-Whitney class "there is no spin". Of course integer spin i.e. vector and tensor fields are allowed, but half-integer ius ruled out.

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The OP was asking for spacetime, expansion, curvature and things like that. That's why I was thinking that one should explain spin on Riemann manifolds in the context of GR. Besides that you are right, the construction of spin via SL(2,C) i.e. via SR may be too complicated to start with and a non-rel. approach should be sufficient.

However I was not aware of this paper, so thanks for the reference.

Yes sorry, your reply contained cool and relevant info which i didn't mean to dismiss :-)

tom.stoer
I tried to find a reference: except for the fact that differential topology and characteristic classes are difficult to understand anyway, the paper from Haefliger regarding the non-existence of spin structures is in French ;-(

A. Haefliger (1956). "Sur l’extension du groupe structural d’un espace fibré". C. R. Acad. Sci. Paris 243: 558–560.

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This is not possible in general for Riemann manifolds.

There are manifolds which do not allow for half-integer spin due the above mentioned topological obstruction to the existence of spin structures. I do not know whether there are Riemann manifolds which may be relevant on GR and what exactly goes wrong when trying to construct spinors, I only know the general statement that w/o vanishing of the second Stiefel-Whitney class "there is no spin". Of course integer spin i.e. vector and tensor fields are allowed, but half-integer ius ruled out.

Yeah, I wouldn't bother with continuous multi-dimensional geometry models, they're the epicycles of our age, maybe good for approximate modeling in simplistic situations but otherwise not reality.

Reality = linear algebra + probability

tom.stoer
Reality = linear algebra + probability
so the short answer is that spin exists b/c everything there is is a spin network; vertam ergo sum!

so the short answer is that spin exists b/c everything there is is a spin network; vertam ergo sum!

eh? (I don't know latin)

no, spin exists cos of all the complex (universe wide) linear algebraic evolution going on every 10^-43 secs or whatever

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marcus
Gold Member
Dearly Missed
so the short answer is that spin exists b/c everything there is is a spin network; vertam ergo sum!

I spin therefore I exist

Witty echo of Decartes "Cogito ergo sum". Could be deeper than Decartes.

tom.stoer
"I spin, therefore I am"

OK friend, No one understands how the quantum mechanical property of "spin" truly relates to reality, its background or whatever. My Answer: Just suppose the QM background IS the evolution equation (and everything)

tom.stoer
I think everything we written starting with post #8 can be safely deleted (by a moderator) in order not to bother the OP with 2 a.m. nonsense (my time)

Lots of stuff to consider. Thanks.

I think everything we written starting with post #8 can be safely deleted (by a moderator) in order not to bother the OP with 2 a.m. nonsense (my time)
post #8 was the best post in the thread

tom.stoer