Nambu Quantum Mechanics: A Nonlinear Generalization of Geometric Quantum Mechanics

We propose a generalization of the standard geometric formulation of quantum mechanics, based on the classical Nambu dynamics of free Euler tops. This extended quantum mechanics has in lieu of the standard exponential time evolution, a nonlinear temporal evolution given by Jacobi elliptic functions. In the limit where latter’s moduli parameters are set to zero, the usual geometric formulation of quantum mechanics, based on the Kahler structure of a complex projective Hilbert space, is recovered. We point out various novel features of this extended quantum mechanics, including its geometric aspects. Our approach sheds a new light on the problem of quantization of Nambu dynamics. Finally, we argue that the structure of this nonlinear quantum mechanics is natural from the point of view of string theory. e-mail: [email protected] e-mail: [email protected] Generalizations of quantum mechanics are difficult. One idea which has often been exploited is to make the Schrödinger equation non-linear (see for example [1]). Another approach is to extend the quantum phase space from the usual complex projective space to an arbitrary Kahler manifold. Yet another avenue is to enlarge complex quantum mechanics by changing the coefficients on the Hilbert space to quaternions [2] or octonions [3, 4]. In this article a more radical approach to this question is investigated. What is proposed is a modification of the kinematics and dynamics, the very symplectic and Riemannian structure of geometric complex quantum mechanics. Any search for generalizations of quantum mechanics has to have a well defined motivation. One possible general starting point is provided by the observation that the evolution of fundamental physical theories, characterized by appearance of new dimensionful parameters (new constants of nature), can be mathematically understood from the point of view of deformation theory [5]. In particular, relativity theory, quantum mechanics and quantum field theory can be understood mathematically as deformations of unstable structures [6]. An example of an unstable algebraic structure is non-relativistic classical mechanics. By deforming an unstable structure, such as classical non-relativistic mechanics, via dimensionful deformation parameters, the speed of light c and the Planck constant h̄, one obtains new stable structures special relativity and quantum mechanics. Likewise, relativistic quantum mechanics (quantum field theory) can be obtained through a double (c and h̄) deformation. It is natural to expect that there is a further deformation via one more dimensionful constant, the Planck length lP . The resulting structure could be expected to form a stable structural basis for a quantum theory of gravity. A closely related idea has appeared in open string field theory, as originally formulated by Witten [7]. There, the deformation parameters are α and h̄. The classical open string field theory lagrangian is based on the use of the string field (which involves an expansion to all orders in α) and a star product which is defined in terms of the world-sheet path integral, also involving α. The full quantum string field theory is thus, in principle, an example of a one-parameter (α) deformation of quantum mechanics. In this letter we lay the basis for a generalized quantum mechanics based on the classical Nambu dynamics [9] of Euler’s asymmetric top. This Nambu quantum mechanics, naturally possesses besides Planck constant, new deformation parameters. One of its defining experimental signatures is a nonlinear time evolution generated by Jacobian elliptic functions, as compared to the standard exponential time evolution of standard quantum mechanics. The new deformation parameters are given by the moduli of the elliptic functions. In the limit An algebraic structure is termed stable (or rigid) for a class of deformations if any deformation in this class leads to an equivalent (isomorphic) structure. Similarly, one can also intuit that string theory calls for a generalization of quantum mechanics from the existence of the minimal length uncertainty relations in the framework of perturbative string theory [8].