相关论文: Quantum Geometry and the Planck Scale

The ordinary quantum theory points out that general relativity is negligible for spatial distances up to the Planck scale. Consistency in the foundations of the quantum theory requires a``soft'' spacetime structure of the general relativity…

Nonrelativistic quantum mechanics is commonly formulated in terms of wavefunctions (probability amplitudes) obeying the static and the time-dependent Schroedinger equations (SE). Despite the success of this representation of the quantum…

A new framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully flat quantum homogeneous…

We examine the notion of symmetry in quantum field theory from a fundamental representation theoretic point of view. This leads us to a generalization expressed in terms of quantum groups and braided categories. It also unifies the…

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial…

Quantum field planes furnish a noncommutative differential algebra $\Omega$ which substitutes for the commutative algebra of functions and forms on a contractible manifold. The data required in their construction come from a quantum field…

Usually the generators of a quantum group are assumed to be commutative with the noncommuting coordinates of a quantum plane. We have relaxed the assumption and investigated its consequences. Not only does a two-parameter quantum group…

General relativity successfully describes space-times at scales that we can observe and probe today, but it cannot be complete as a consequence of singularity theorems. For a long time there have been indications that quantum gravity will…

Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of…

Semigroup algebras admit certain `coherent' deformations which, in the special case of a path algebra, may associate a periodic function to an evolving path; for a particle moving freely on a straight line after an initial impulse, the wave…

The nonadiabatic geometric phase in a time dependent quantum evolution is shown to provide an intrinsic concept of time having dual properties relative to the external time. A nontrivial extension of the ordinary quantum mechanics is thus…

Quantum Poincar\'e-Weyl group in two dimensional quantum Minkowski space-time is considered and an appriopriate relativistic kinematics is investigated. It is claimed that a consistent approach to the above questions demands a kind of a…

The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor…

The traditional formalism of non-relativistic quantum theory allows the state of a quantum system to extend across space, but only restricts it to a single instant in time, leading to distinction between theoretical treatments of spatial…

We develop a generalised gauge theory in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory,…

It is shown that the recently geometric formulation of quantum mechanics implies the use of Weyl geometry. It is discussed that the natural framework for both gravity and quantum is Weyl geometry. At the end a Weyl invariant theory is…

We find a quantum group structure in two-dimensional motions of a nonrelativistic electron in a uniform magnetic field and in a periodic potential. The representation basis of the quantum algebra is composed of wavefunctions of the system.…

We advocate that the dual picture of spacetime noncommutativity , i.e. the existence of a curved momentum space, could be a way out to solve some of the open conceptual problems in the field, such as the basis dependence of observables. In…

Quantum geometry, characterized by the quantum geometric tensor, is pivotal in diverse physical phenomena in quantum materials. In condensed matter systems, quantum geometry refers to the geoemtric properties of Bloch states in the…

In this talk I review a series of recent conceptual developments at the interface of the quantum and gravitational realms. Wherever possible, I comment on the possibility to probe the interface experimentally. It is concluded that the…