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Supersymmetric Hilbert space.

G C Rota1, J A Stein

  • 1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

Proceedings of the National Academy of Sciences of the United States of America
|January 1, 1990
PubMed
Summary
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This study generalizes symmetric bilinear forms to include positive and negative signatures, revealing a connection to exterior algebra. This advances invariant theory and uncovers dualities in supersymmetric spaces.

Area of Science:

  • Algebraic Geometry
  • Mathematical Physics

Background:

  • Symmetric bilinear forms are fundamental in linear algebra.
  • Understanding structures with mixed signatures is an ongoing challenge.

Purpose of the Study:

  • To generalize symmetric bilinear forms to include "supersymmetric variables" with positive and negative signatures.
  • To explore the relationship between this generalized structure and exterior algebra.
  • To derive a supersymmetric extension of the second fundamental theorem of invariant theory.

Main Methods:

  • A supersymmetric extension of the standard basis theorem.
  • Generalization of symmetric bilinear forms to incorporate mixed signatures.

Main Results:

  • The generalized structure is shown to be isomorphic to the exterior algebra of a vector space.

Related Experiment Videos

  • A supersymmetric extension of the second fundamental theorem of invariant theory is established as a corollary.
  • A natural duality is demonstrated between supersymmetric Hilbert space and supersymplectic space.

    • Conclusions:

    • The generalization provides a unified framework for bilinear forms with mixed signatures.
    • The isomorphism to exterior algebra simplifies the study of these structures.
    • The established duality deepens the understanding of supersymmetric mathematical physics concepts.