Abstract:

The talk is a review of some results from the series of the works on the non-Lagrangian and non-Hamiltonian gauge dynamics, their BRST cohomology, symmetries, conservation laws, and quantization.

Lagrangian and/or Hamiltonian gauge dynamics have well-established machinery of solving a series of mutually related problems of principal importance for quantisation and conservation laws. In particular, given the action functional or constrained Hamiltonian equations of motion, there is a problem of finding all the gauge symmetries. This problem is solved by the Dirac-Bergmann algorithm that includes the classification of the constraints into the first and second classes, and relates the first ones to the gauge symmetries. With action functional and known gauge symmetries, applying the BV method, one can always construct the classical BRST complex for the Lagrangian dynamics. This allows one to study the classical dynamics by cohomological tools. For example, this allows to identify all the admissible classical deformations of the master action. In physical terms, this means finding all the consistent interactions in the theory with given set of fields and number of gauge symmetries.

Furthermore, the BV-BRST complex is a tool to implement the path-integral quantisation of any Lagrangian gauge field theory. Whenever the action is known for the field equations, one can relate infinitesimal symmetries and conservation laws (Noether theorem no analogue of which was known in non-variational dynamics). If classical equations are brought to the form of constrained Hamiltonian formalism, one can construct the BFV-BRST complex and carry out either deformation or path integral quantisation of the dynamics. In the commonly known form, all these methods essentially rely on the existence of the least action principle or constrained Hamiltonian formulation for the classical dynamics. In this talk, it will be explained how these methods extend to the non-variational dynamics.

In particular, I will elaborate on the procedure of bringing generic dynamics to the normal form, identifying its gauge symmetry and classifying constraints. All that is done without use of any Poisson structure. In the Hamiltonian constrained dynamics this general procedure reduces to the Dirac-Bergman algorithm. In non-Hamiltonian theory, however, no pairing exists between the constraints and gauge symmetry, unlike miltonian one. Some links are illuminated between the optimal control theory and the gauge dynamics in normal form. For example, a simple gauge-theoretical interpretation can be found for the notion of flat control. Given generic, not necessarily Lagrangian equations of motion, with known gauge symmetries and Noether identities (in non-Lagrangian theory gauge symmetries are unrelated to Noether identities, unlike the Lagrangian one), a procedure is proposed for constructing corresponding classical BRST differential for this dynamics. In Lagrangian formalism, this classical BRST complex reduces to the classical BV one. Also, if the equations of motion are brought to the normal form, the method is proposed of associating a classical BRST complex to this dynamics, which would reduce to the classical BFV one in the Hamiltonian case. All these methods are applicable to any regular gauge dynamics, and require, as an input, only the classical equations of motion.

If some auxiliary structures exist, being still less restrictive for the equations of motion, than the least action principle, or Hamiltonian form, it becomes possible to further extend the methods of Lagrangian/Hamiltonian theory to the non-variational dynamics. In particular the notion of the Lagrange anchor is introduced. Whenever the anchor is invertible, it’s inverse defines the integrating multiplier of the inverse problem of the variational calculus. So the existence of the invertible anchor is equivalent ot the least action principle. However, non-invertible anchor turns out to be sufficient to construct the path integral quantisation of the non-variational dynamics. The general method can be exemplified by path integral quantisation of well know non-Lagrangian field equations: Maxwell’s e/d in the strength tensor formalism, self-dual Y-M equations, and Donaldson-Ulehnbeck-Yau equations. In the case of Lagrangian theory, for which the anchor defines the identity map, the path integral reduces to the BV one. The Lagrange anchor, be it invertible or not, allows one to relate the infinitesimal symmetries with the characteristic cohomology classes. The latter correspond to the conservation laws, so the Lagrange anchor extends the Noether theorem to non-Lagrangian dynamics. The dynamical equations in their normal form can also admit a useful extra element, called the weak Poisson structure. This structure allows to turn the variety of the BRST cohomology classes into the Poisson algebra, even though neither original phase space, nor the constraint surface are the Poisson sub-manifolds. The existence of the weak Poisson structure turns out sufficient to provide the deformation quantisation of the dynamics. In the case of first class constrained system, this reduces to the standard BFV-BRST deformation quantisation.