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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-662-49170-6_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-49168-3
Online ISBN: 978-3-662-49170-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)