# What conserved quantity does Supersymmetry imply?

According to Noether's theorem, for every continuous symmetry there is a conserved quantity. What is the quantity that corresponds to Supersymmetry?

• Is it just me or is Noether's theorem a fancy way of saying $\frac{dy}{dx} = 0 \implies y(x) = y(0)\ \ \forall x$? – Mehrdad Nov 4 '17 at 6:13
• @Mehrdad, Well, yeah, in a sense. But the physics details and implications are important. – Andrei Geanta Nov 7 '17 at 10:20
• @Mehrdad No, not really, because the quantity that is invariant and the one that is conserved are not the same. Also $x$ in your first equation can be anything, while $x$ in your second equation is time. – Javier Nov 9 '17 at 20:46
• @Javier: Thank you! That's a really great point about the quantities being different. – Mehrdad Nov 9 '17 at 21:03

The conserved charged is the supercharge, as @Qmechanic told you. But then, what is supercharge? Supposing a particle of momentum $P_\mu$, the simplest SUSY implies a supercharge $Q=ip_\mu\Psi^\mu$, where $p_\mu$ is the momentum and $\Psi_\mu$ is the Grassmann variable, and you can also prove that this thing is secretly a sort of Dirac operator. I have explained some details in my blog, see the appendix of this blog post of mine: http://www.thespectrumofriemannium.com/2015/08/08/log177-scherk-susy-and-sugra/
In addition to all this, the most general SUSY algebra, as far as I know, can include beyond $P_\mu$ extra topological extensions of the SUSY algebra including central charges. So, in addition to spacetime symmetries, and mixed entities like the supercharge above, you can also get topological charges in non-trivial way. References: https://arxiv.org/abs/hep-th/9711009 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.63.2443