On Hamiltonian formulations of the Schr¨ odinger system L´aszl´o ´ A. Gergely Astronomical Observatory and Department of Experimental Physics, University of Szeged, D´om t´ er 9, Szeged, H-6720 Hungary We review and compare different variational formulations for the Schr¨ odinger field. Some of them rely on the addition of a conveniently chosen total time derivative to the hermitic Lagrangian. Alternatively, the Dirac-Bergmann algorithm yields the Schr¨ odinger equation first as a consistency condition in the full phase space, second as canonical equation in the reduced phase space. The two methods lead to the same (reduced) Hamiltonian. As a third possibility, the Faddeev-Jackiw method is shown to be a shortcut of the Dirac method. By implementing the quantization scheme for systems with second class constraints, inconsistencies of previous treatments are eliminated. I. INTRODUCTION Outstanding equation of modern physics, the Schr¨ odinger equation has multiple and deep connections with integral principles. Historically, Schr¨ odinger obtained his equation guided by the beautiful analogy between the Fermat principle and the principle of least action [1]. Moreover, motivated by a remark of Dirac [2], Feynman has derived the Schr¨ odinger equation from the Huygens principle, realizing the first step towards his path integral approach [3], [4]. (For recent recent reviews see [5], [6].) The Schr¨odinger field is equally a popular choice to illustrate how second quantization proceeds. At a closer inspection, however, the power and beauty of the variational approach is obstructed by an aesthetical bug. The reason: the abundance of dynamical variables, some of them being redundant. The Lagrangian yielding the nonrelativistic Schr¨ odinger equation is linear in the time derivatives of the fields Ψ and its complex conjugate Ψ ∗ . Thus the Legendre transformation does not lead to an unambiguous Hamiltonian. Various Hamiltonians, all having Schr¨ odinger’s equation as canonical equation can be found in textbooks. They depend either on two pairs of canonical variables [7], or - as a result of adding a total time derivative to the hermitic Lagrangian - just on a single (complex or real) canonical pair [8]- [10]. In the next section we briefly rewiev these approaches, pointing out both the weakness and ingenuity in sweeping away the problem. There are two equivalent methods to circumvent this difficulty. In section 3 we treat the Schr¨odinger field as a constrained system, applying the Dirac-Bergmann algorithm [11]- [13]. As the system has second class constraints, the dynamics involves Dirac brackets. We present two alternatives to the existing derivations of the Schr¨odinger equation. First, the consistency requirement yields the Schr¨ odinger equation in the form of a weak equation on the full phase space. Second, by a suitable canonical transformation we introduce new canonical coordinates, containing the constraints. The Dirac bracket of the full phase space becomes the Poisson bracket of the reduced phase space and the Schr¨ odinger equation is a canonical equation. As a bonus we recover the Hamiltonian obtained by addition of a properly chosen time-derivative. At the end of Section 3 we present an alternative discussion in terms of real variables: the real and imaginary parts of the complex field Ψ. In Section 4 we apply the Faddeev-Jackiw scheme developed for Lagrangians, which are first order in ”velocities” [14], [15]. We verify that the fundamental brackets of the Faddeev-Jackiw approach coincide with the Dirac brackets. Finally in the fifth section we follow the canonical quantization scheme [11]- [13], [16], giving an operator repre- sentation of the Dirac bracket algebra of the canonical variables. This is equivalent with the quantization of the Faddeev-Jackiw fundamental bracket. Our approach avoids the interpretational inconsistencies of some standard treatments [7], [8], already pointed out by Tassie [17] and is a viable alternative to the existing reduced phase space quantization schemes [9], [10]. By imposing the second class constraints as operator identities, second quantization proceeds smoothly. II. STANDARD VARIATIONAL PROCEDURES. A REVIEW A. Two pairs of complex variables The action for the Schr¨ odinger field cf. Henley and Thirring [7] is: S[ψ,ψ ∗ ]=  dt  dr L (2.1) 1