Lagrangian versus Quantization

We discuss examples of systems which can be quantized consistently, although they do not admit a Lagrangian description. Whether a given set of equations of motion admits or not a Lagrangian formulation has been an interesting issue for a long time. As early as 1887, Helmholtz formulated necessary and sufficient conditions for this to happen, and the problem has a rich history [1]. More recently, motivated by some unpublished work of Feynman [2], a connection was made between the existence of a Lagrangian and the commutation relations satisfied by a given system [3, 4]. Ref. [3] concluded that under quite general conditions, including commutativity of the coordinates, [qi, qj ] = 0, the equations of motion of a point particle admit a Lagrangian formulation. The purpose of this note is to demonstrate the reverse, namely that noncommutativity of the coordinates forbids a Lagrangian formulation (therefore a Lagrangian implies commutativity). This happens in all but a few cases, which we all identify. On the other hand, an extended Hamiltonian formulation always remains available. It permits quantization of the system in any of the three usual formalisms: operatorial, wave-function, or path integral. Several examples will be used to illustrate the properties of such unusual systems. We work in a (2+1)-dimensional space, although our considerations easily extend to higher dimensions, and assume that [q1, q2] = iθ 6= 0. (1) ∗On leave from: National Institute of Nuclear Physics and Engineering P.O. Box MG-6, 76900 Bucharest, Romania; e-mail: [email protected].