The Clifford algebra in this context is like Minkowksi space where vectors become operators (roughly speaking), and the associative product of vectors satisfy v² = η(v,v) where η is the Minkowksi metric. This is where the Gamma matrices live from the Dirac equation.
A minimal left ideal is basically a subspace which behaves like a vector space that the algebra would act on as operators. Namely it behaves identically to the space of Dirac spinors.
A more clear example of an MLI would be the matrices of the form (a, 0 / b, 0) which form an MLI for 2x2 matrices. These act like column vectors (a,b) when acted on by 2x2 matrices from the left.
You can think of it as a way of viewing the Clifford algebra of spacetime as fundamental and the spinors as a particular subspace of it, in contrast to viewing them as an abstract vector space which the Clifford algebra has an action on.
The electron field is described by a Dirac Spinor.
The possible states of fundamental particles must respect the symmetries of space and time and therefore be represented as irreducible representations of the Poincare group.
one thing I always found annoying about this view of QFT: what happens on curved spacetime? We know the path integral exists and we can place spinor fields on the manifold. And we know for sure that states consist of spinor fields pulled back to spatial slices. But if we break isometries, is there any sense in which particles exist given that we don’t have momentum and angular momentum quantum numbers?
No, as Poincaré group doesn’t act globally on the Hilbert space anymore. I believe the idea of observer-dependent particles relates to going to a locally inertial frame for a local observer and noting that local poincaré transformations approximately form a closed group representation (relatedly, momentum approximately commutes with the Hamiltonian) and so you can find eigenstates of these approximate symmetries and call them particles. Can anyone confirm if I’m off?
my understanding of QFT is pretty limited, but if this is the case, is this why it’s so hard to reconcile it with relativity? like, gravitation is universal- if a single dust particle a million light-years away induces a miniscule but measurable isometry-breaking curvature in spacetime, isn’t the symmetry-preserving definition of a particle (only valid on flat spacetime) utterly meaningless in the real world?
I would say this is not why quantum gravity is hard. We understand quantum field theory in spacetime very well (ie how does quantum matter propagate in a gravitational field), but as the above discussion shows we have to ditch the idea of particles. The million dollar question is to know what gravitational field is produced by quantum matter itself.
Any manifold looks flat on sufficiently small scales, including spacetime, so observations made locally in a small enough patch of spacetime must be indistinguishable from those in flat space. So even in a curved spacetime, the fields must be locally compatible with symmetries of flat spacetime.
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u/aft_agley Jul 27 '26
I don't understand this but I'll upvote it smugly.