In this lecture (44:23) Nathan Seiberg: Topics in 2+1 Dimensional Quantum Field Theories 2.

Nathan Seiberg says there's no space in QM and therefore fermions have spin 0. This sounds pretty revolting and the audience is confused, too. I think we all had many spin systems in QM where electrons carry 1/2 spin.

So what does he mean?

Is it because there's only a time derivative in the Schrödinger equation

$$i\hbar \partial_{t} \psi=H\psi$$

and the Hamiltonian is an abstract object independent of any space representations?

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    $\begingroup$ Apparently these days, when a quantum field theorist (emphasis on theorist) says "quantum mechanics", they mean "quantum field theory in 0+1 dimensions". $\endgroup$ – Mitchell Porter May 21 '19 at 2:45
  • $\begingroup$ It's less sensational if we notice that it's not "usual" dimensionality, and that in lower dimensions all sorts of interesting effects appear in both classical and quantum physics. Graphene properties make a good example. $\endgroup$ – luk32 May 21 '19 at 8:55

Apparently Seiberg follows the definition that for a field $\phi: M\to N$ from a spacetime $M\cong\mathbb{R}^{1,n-1}$ to a target space $N$ the spin of the field $\phi$ refers to the representation of the Lorentz group $O(1,n-1)$ of spacetime $M$.

Now in QM/point mechanics, the spacetime $M\cong\mathbb{R}$ is just time with no space, so the Lorentz group $O(1)\cong\mathbb{Z}_2$ is 2 points, and therefore the spin does not make sense, or is zero by definition.

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    $\begingroup$ NB: $\mathbb Z_2$ does not admit projective representations, cf. $H^2(\mathbb Z_2,U(1))=0$. $\endgroup$ – AccidentalFourierTransform May 20 '19 at 20:48

When Seiberg defines quantum mechanics, what he means is a one-dimensional quantum field theory, typically a sigma model, which describes the world-line action of a point particle. For example, a one-dimensional quantum field theory with action


naturally describes the motion of a massless particle with world-line coordinates $\phi^{\mu}(t)$.

In these models, there are two points of view which can be taken: an internal (world-line) point of view, or an external (target-space) point of view. The first considers the manifold of interest to be the world-line itself, while the second treats the manifold as the target space.

Now, when Seiberg is discussing Fermions in quantum mechanics, what he really means is Fermionic living on the world-line. Such fields have spin zero since the Lorentz group of a $0+1$ dimensional manifold has no spin representation.


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