Noether’s theorems in a general setting. Reducible graded Lagrangians

Volume 110 / 2016

Gennadi Sardanashvily Banach Center Publications 110 (2016), 239-255 MSC: Primary 70S05; Secondary 58C50. DOI: 10.4064/bc110-0-16


Noether’s first and second theorems are formulated in a general setting of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. Such Lagrangian theory is characterized by a hierarchy of higher-stage Noether identities described by a Koszul–Tate chain complex. Noether’s second theorems associate to this complex a cochain sequence whose ascent operator defines higher-stage gauge symmetries of the Grassmann-graded Lagrangian system. This operator is extended to a nilpotent BRST operator that provides a BRST extension of the original Lagrangian theory. Noether’s first theorem is formulated as a straightforward corollary of the global variational formula. It associates to any gauge symmetry a conserved current which is proved to be a total differential on-shell.


  • Gennadi SardanashvilyDepartment of Theoretical Physics
    Moscow State University
    119991 Moscow, Russia

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