# There's only time, no space in Quantum Mechanics

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.

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

$$S=\frac{1}{2}\int\mathrm{d}t\,g_{\mu\nu}(\phi)\,\dot{\phi}^{\mu}\dot{\phi}^{\nu}$$

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.