Abstract
Supersymmetry in quantum physics is a mathematically simple phenomenon that raises deep foundational questions. To motivate these questions, I present a toy model, the supersymmetric harmonic oscillator, and its superspace representation, which adds extra anticommuting dimensions to spacetime. I then explain and comment on three foundational questions about this superspace formalism: whether superspace is a substance, whether it should count as spatiotemporal, and whether it is a necessary postulate if one wants to use the theory to unify bosons and fermions.
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Notes
Not that \((Q^{\dag })^2=0\), which means that we cannot violate the exclusion principle by replacing multiple bosons with fermions. Only one particle may be swapped. This phenomenon holds generally: SUSY doesn’t allow us to create Bose-Einstein condensates out of fermions, for example!
Roughly, a supergroup behaves like a group except that it includes some elements that multiply like Grassmann numbers.
This is closely related to North’s own view of how to understand the debate about general relativity, except that she considers substantivalism about general relativity to be a conditional view of the form: If general relativity were a fundamental theory, then spacetime would be fundamental (North 2018).
To avoid the misleading appearance of begging the question against either superspace spatiotemporality or its denial, I will do my best to refer to “the thing substantivalists are substantivalist about” not as spacetime, but as the fundamental background structure of a theory.
The two formalisms may still differ about which observable operators signify fundamental quantities, as opposed to merely physically significant (but derivative) quantities. I wish I had room in the present paper to explore the implications of this possible disagreement, but it is worth noting that it may imply further differences in explanatory power between the Minkowksi substantivalist and superspace substantivalist ontologies, beyond the ones discussed here.
Such a law will have to relate a theory’s dynamical symmetries to its absolute background geometrical structure, of course. In a theory like general relativity, for example, which admits of many solutions with different spacetime structures, we should not require that the symmetries of the force laws vary from solution to solution. But a similar law in general relativity could explain the local Lorentz covariance of the laws of general relativistic field theory, since Lorentz covariance is a local symmetry of all general relativistic spacetimes.
Knox’s account actually focuses on local inertial frames, but since the scope of this paper only includes flat spacetimes, I’ll follow Menon in leaving that complication aside and considering only the special case of global inertial frames.
Thanks to Tim Maudlin and James Wells for discussion that inspired some of the arguments presented here, to Alexander Schenkel for fielding some of my mathematical questions, and to Tushar Menon for many months of edifying discussion, correspondence and comments.
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Baker, D.J. Interpreting Supersymmetry. Erkenn 87, 2375–2396 (2022). https://doi.org/10.1007/s10670-020-00306-4
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DOI: https://doi.org/10.1007/s10670-020-00306-4