Chia-Hsiung Tze

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 latters 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.

Background independent quantum mechanics and gravity

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.

TOWARDS A BACKGROUND INDEPENDENT QUANTUM THEORY OF GRAVITY

Any canonical quantum theory can be understood to arise from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This geometric perspective offers a novel, background independent non-perturbative formulation of quantum gravity. We invoke a quantum version of the equivalence principle, which requires both the statistical and symplectic geometries of canonical quantum theory to be fully dynamical quantities. Our approach sheds new light on such basic issues of quantum gravity as the nature of observables, the problem of time, and the physics of the vacuum. In particular, the observed numerical smallness of the cosmological constant can be rationalized in this approach.

Generalized self-dual bosanic membranes, vector cross-products and analyticity in higher dimensions

Abstract By generalizing the notion of self-duality for chiral currents and gauge fields to Nambu-Goto membranes, we exhibit a remarkable class of self-dual classical bosonic p -branes in n spacetime dimensions. The latter exist whenever a p -fold vector cross-product can be defined in R n . Their connections to generalized cauchy-Riemann analyticity, to exceptional geometries and supermembranes in critical dimensions 7 and 8 in particular are discussed.

Modeling Time’s Arrow

Quantum gravity, the initial low entropy state of the Universe, and the problem of time are interlocking puzzles. In this article, we address the origin of the arrow of time from a cosmological perspective motivated by a novel approach to quantum gravitation. Our proposal is based on a quantum counterpart of the equivalence principle, a general covariance of the dynamical phase space. We discuss how the nonlinear dynamics of such a system provides a natural description for cosmological evolution in the early Universe. We also underscore connections between the proposed non-perturbative quantum gravity model and fundamental questions in non-equilibrium statistical physics.

What is quantum theory of gravity

We present a line by line derivation of canonical quantum mechanics stemming from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This viewpoint can be naturally extended to provide a conceptually novel, non-perturbative formulation of quantum gravity. Possible observational implications of this new approach are briefly mentioned.

Global dynamics of electric and magnetic membranes on the complex, quaternionic and octonionic hopf bundles

Abstract As a prelude to chiral field soliton realizations, we present a unified path space formulation of global dynamics for systems of relativistic point particles, 2-sphere, 6-sphere membranes and their rank-1, -3 and -7 tensor U(1) gauge fields in D=3, 7 and 15 spacetime dimensions. The path spaces are realized by the complex ( C ), quaternionic ( H ) and octonionic (Ω) Hopf bundles. These systems have an electric-magnetic duality in D=4, 8 and 16 dimensions. Their hypercomplex symplectic Kahler structures generalize the geometric prequantization condition for the point charge-monopole complex. In a subsequent analysis of Aharonov-Bohm effects induced by the above tensor U(1) gauge fields in three D-dimensional KP1 σ-models with a Hopf-Chern-Simons term where (D, K )=(3, C ) , (7, H ) and (15, Ω), these global Wess-Zumino-Novikov-Witten actions play a key role in the London-Nielsen-Olesen limit of very thin membrane solitons.

On the Octonionic Nahm Equations and Self-Dual Membranes in 9 Dimensions

While circumstances have prevented us from being in festive and beautiful Bregenz on this happy occasion, along with our best wishes we are delighted to contribute this article as a small token of our friendship and admiration for Professor L. C. Biedenharn on his 70th birthday. Our topic relates to one of the projects on which we have been working during the past year - we hope that it may appeal to Larry as it makes use of some exceptional objects dear to him, namely octonions and, appropriately enough, may hint at a bridge between rotations of tops and quantum algebras. These pages contain some of our preliminary results.

Classical gauge theories on the coadjoint orbits of infinite dimensional groups

Abstract We reformulate several classical gauge theories on the coadjoint orbits of the semidirect product of the gauge group and the Weyl group. The construction is given for the Yang-Mills theories in arbitrary spacetime dimension d , Chern-Simons topological theory ( d =3) and higher dimensional topological models of Horowitz ( d ⩾4).