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Canonical quantization of so-called non-Lagrangian systems. (English) Zbl 1191.81143

Summary: We present an approach to the canonical quantization of systems with equations of motion that are historically called non-Lagrangian equations. Our viewpoint of this problem is the following: despite the fact that a set of differential equations cannot be directly identified with a set of Euler-Lagrange equations, one can reformulate such a set in an equivalent first-order form that can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. It turns out that in the general case the hamiltonization and canonical quantization of such an action are non-trivial problems, since the theory involves time-dependent constraints. We adopt the general approach of hamiltonization and canonical quantization for such theories as described by D.M. Gitman and I.V. Tyutin [Quantization of fields with constraints. Springer Series in Nuclear and Particle Physics. (1990; Zbl 1113.81300)] to the case under consideration. There exists an ambiguity (that cannot be reduced to the addition of a total time derivative) in associating a Lagrange function with a given set of equations. We present a complete description of this ambiguity. The proposed scheme is applied to the quantization of a general quadratic theory. In addition, we consider the quantization of a damped oscillator and of a radiating point-like charge.

MSC:

81S10 Geometry and quantization, symplectic methods

Citations:

Zbl 1113.81300
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References:

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