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

Qmechanic
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Nemo
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    Is it just me or is Noether's theorem a fancy way of saying $\frac{dy}{dx} = 0 \implies y(x) = y(0)\ \ \forall x$? – user541686 Nov 04 '17 at 06:13
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    @Mehrdad, Well, yeah, in a sense. But the physics details and implications are important. – Nemo Nov 07 '17 at 10:20
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    @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 09 '17 at 20:46
  • @Javier: Thank you! That's a really great point about the quantities being different. – user541686 Nov 09 '17 at 21:03

2 Answers2

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A supercharge is the conserved quantity that corresponds to a supersymmetry. It generates a supersymmetry & commutes with the Hamiltonian.

A supercharge is a Grassmann-odd quantity. Noether's theorem works fine for supermanifolds, cf. e.g. this Phys.SE post.

However, note that one cannot measure (the expectation value of) a Grassmann-odd quantity directly in an experiment, cf. e.g. my Phys.SE answer here. In other words, the experimental consequences of a conserved supercharge are extracted by other indirect means.

Qmechanic
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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/

Moreover, two tips, FOR THE SIMPLEST SUSY: 1) the SUSY transformation of the supercharge is, up to a multiplicative constant, THE SUSY LAGRANGIAN, and 2) the SQUARE of the supercharge is THE SUSY HAMILTONIAN.

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

riemannium
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