Abstract
A Gaussian integral over Grassmann variables yields a Pfaffian. Its connection with the determinant is derived. The Jacobian for transformations of Grassmann variables under the integral is presented.
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- 1.
The sequence of substitutions can be singular, even if \(\det (\partial \zeta /\partial \eta )\not =0\). Then one may use an infinitesimally close non-singular transformation and finally perform the limit. For the transformation ζ 1 = η 2, ζ 2 = η 1 one may use \(\zeta _{1} =\eta _{2} + c\eta _{1}\), ζ 2 = η 1 with \(c \rightarrow 0\). We leave it to the reader to calculate D 1 and D 2 and consider the product.
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© 2016 Springer-Verlag Berlin Heidelberg
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Wegner, F. (2016). Substitution of Variables, Gauss Integrals II. In: Supermathematics and its Applications in Statistical Physics. Lecture Notes in Physics, vol 920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49170-6_5
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DOI: https://doi.org/10.1007/978-3-662-49170-6_5
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-49168-3
Online ISBN: 978-3-662-49170-6
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