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Usually, a supersymmetry transformation is carried out on bosonic and fermionic fields which are functions of the coordinates (or on a superfield which is a function of real and fermionic coordinates). But, is it possible to interpret supersymmetry transformations as coordinate transformations on the set of coordinates $(x^0,\ldots,x^N,\theta_1,\ldots,\theta_M)$?

The problem I see is that the coordinates would transform something like $x^\mu\rightarrow x^\mu+\theta\sigma\bar{\theta}$ which is no longer a real (or complex) number, but a commuting Grassmann number. Can one make sens of a coordinate position no longer being a real number?

Edit: To clarify, this is NOT about confusion in what happens when adding real numbers with commuting grassmann numbers in general. That the lagrangian in QFT for example is not a real number, but a commuting grassmann number, is fine. What I am confused about is really how to make sense of coordinates that are grassmannian. Coordinates are supposed to describe a position in spacetime/on a manifold, and it seems to me that it is essential that a position is a standard real number.

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  • $\begingroup$ what kind of mathematical object is the sum of a real/complex number and a grassmannian? It is obviously closed under sum, and apparently under multiplication, but probably not a full fledged field, since no inverse $\endgroup$ – lurscher Jan 5 '15 at 15:11
  • $\begingroup$ Essentially a duplicate of this Phys.SE post. $\endgroup$ – Qmechanic Jan 5 '15 at 15:15
  • $\begingroup$ @Qmechanic: No it is not a duplicate of this (but it is essentially the same as the unanswered comment in the first reply). I have edited my post to clarify! $\endgroup$ – Jonathan Lindgren Jan 5 '15 at 15:23
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Comments to the question (v3):

  1. Recall that a supernumber $z=z_B+z_S$ consists of a body $z_B$ (which always belongs to $\mathbb{C}$) and a soul $z_S$ (which only belongs to $\mathbb{C}$ if it is zero), cf. e.g. this Phys.SE post.

  2. An observable/measurable quantity can only consist of ordinary numbers (belonging to $\mathbb{C}$). It does not make sense to measure a soul-valued output in an actual experiment.

  3. Souls are indeterminates that appear in intermediate formulars, but are integrated (or differentiated) out in the final result.

  4. In a superspace formulation of a field theory, a Grasmann-even spacetime coordinates $x^{\mu}$ in superspace is promoted to a supernumber, and is not necessarily an ordinary number.

  5. A supersymmetry-translation of a Grasmann-even spacetime coordinate $x^{\mu}$ only changes the soul (but not the body) of $x^{\mu}$.

  6. Note that in the mathematical definition of a supermanifold, the focus of the theory is not on spacetime coordinates per se, but (very loosely speaking) rather on certain algebras of functions of spacetime. See also e.g. Refs. 1-3 for details.

References:

  1. Pierre Deligne and John W. Morgan, Notes on Supersymmetry (following Joseph Bernstein). In Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, American Mathematical Society (1999) 41–97.

  2. V.S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes 11, 2004.

  3. C. Sachse, A Categorical Formulation of Superalgebra and Supergeometry, arXiv:0802.4067.

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