Abstract
An infinite dimensional Grassmann algebra on a compact Riemannian manifold is constructed by means of rigged Hilbert spaces of differential forms. We give a notion of \(p\)-form of order \(\alpha \) on a product manifold and define a wedge product of these forms. The set of involutive generators of infinite dimensional Grassmann algebra which can be used for geometric approach to ghost fields appearing in quantized gauge theory is introduced. We extend our approach to vector bundles and construct an infinite dimensional Grassmann algebra with generators by means of the rigged Hilbert spaces of sections of a vector bundle.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Thierry-Mieg, J.: Geometrical reinterpretation of Faddeev-Popov ghost particles and BRS transformations. J. Math. Phys. 21(12), 2834–2838 (1980)
Thierry-Mieg, J.: Explicit classical construction of the Faddeev-Popov ghost field. Nuovo Cimento A(11), 56(4), 396–404 (1980)
Abramov, V., Lumiste, Ü.: Superspace with underlying Banach bundle of connections and supersymmetry of effective action (Russian). Izv. Vyssh. Uchebn. Zaved., Mat. 1, 3–12 (1986)
Berezin, F.A.: The Method of Second Quantization. Academic Press, New York (1966)
Faddeev, D.D., Slavnov, A.A.: Gauge Fields: Introduction to Quantum Theory. Benjamin/Cummings Publishing Company, New York (1980)
De Rham, G.: Differentable Manifolds. Springer, Berlin (1984)
Acknowledgments
The authors gratefully acknowledge the financial support of the Estonian Science Foundation under the research grant ETF9328, target finance grant SF0180039s08 and Estonian Doctoral School in Mathematics and Statistics.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Abramov, V., Vajakas, J. (2014). Geometric Approach to Ghost Fields. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds) Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics & Statistics, vol 85. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55361-5_27
Download citation
DOI: https://doi.org/10.1007/978-3-642-55361-5_27
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-55360-8
Online ISBN: 978-3-642-55361-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)