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We argue that the demand of background independence in a quantum theory of gravity calls for an extension of standard geometric quantum mechanics. We discuss a possible kinematical and dynamical generalization of the latter by way of a quantum covariance of the state space. Specifically, we apply our scheme to the problem of a background independent formulation of matrix theory. DOI: 10.1103/PhysRevD.68.061501 PACS number͑s͒: 04.60.Ϫm, 11.25.Yb The quest for a consistent, unified quantum theory of matter and gravity remains very much an open issue, despite great progress in string theory ͓1͔ and other approaches to quantum gravity ͓2͔. At the outset either quantum mechanics ͑QM͒ or general relativity ͑GR͒ or both should give way to a new substructure. From the predominant viewpoint in string theory it is GR that needs replacing, while QM is complete by itself. This stand is well motivated when GR is taken as valid only at low energy scales ͓1͔. In fact, the most general diffeomorphism invariant effective action is derivable from the consistency requirements ͑i.e., conformal invariance͒ of a perturbative string theory. Alas, a nonperturbative, background independent understanding of this remarkable fact is still lacking. Most of the other attempts at quantizing gravity ͓2͔, while stressing the relational, background independent nature of GR ͑and thus arguing against the effective field theoretic point of view and for a nonstandard approach to quantization of gravity͒ still keep to the canonical structure of QM. 1What is the physical rationale for changing the canonical quantum mechanical structure? And if such a rationale is found, is it theoretically or empirically compelling? In the negative, is not the canonical structure somehow unique, and if so, what does this imply for the foundational issues of quantum theory, of a theory of quantum gravity, in particular?This paper puts forth a radical but, in our view, a justified approach to extending QM. Motivated by the physical requirement of background independence and the need to make room for gravity at the quantum level, we are led to a rather drastic extension of standard QM: to wit, we modify both its dynamics and kinematics, and thereby the very symplectic and Riemannian structures that underlie its geometric foundations.2 The upshot of our proposal is that the space of quantum states (events) becomes dynamical and that the dynamical geometric information is described in terms of a nonlinear diffeomorphism invariant theory, in such a way that the space of quantum events is nonlinearly inter-related with the generator of quantum dynamics-the Hamiltonian.We briefly review the key features of geometric standard QM ͓5͔ ͑for reviews of this approach consult ͓6-10͔͒. Pure states are points of an infinite dimensional Kahler manifold P(H), the complex projective space of the Hilbert space H. Equivalently, P(H) is a real manifold with an integrable almost complex structure J. As such it has a Kahler metric given by ͉͗͘, the Hermitian inner product of two states ͉͗...

In an explicit, unified, and covariant formulation, we study and generalize the exceptional vector products in R8 of Zvengrowski, Gray, and Kleinfeld. We derive the associated general quadratic, cubic, and quartic G2 invariant algebraic identities and uncover an octonionic counterpart to the d=4 quaternionic duality. When restricted to seven dimensions the latter is an algebraic statement of absolute parallelism on S7. We further link up with the Ogievetski–Tzeitlin vector product and obtain explicit tensor forms of the SO(7) and G2 structure constants. SO(8) transformations related by Triality are characterized by means of invariant tensors. Possible physical applications are discussed.

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.

We offer an interpretation of super-quantum correlations in terms of a "doubly" quantum theory. We argue that string theory, viewed as a quantum theory with two deformation parameters, the string tension α ′ and the string coupling constant gs, is such a super-quantum theory, one that transgresses the usual quantum violations of Bell's inequalities. We also discuss the → ∞ limit of quantum mechanics in this context. As a super-quantum theory, string theory should display distinct experimentally observable super-correlations of entangled stringy states.