Quantum world in 2+1 dimensions

Don Ross, James Ladyman

Classical probability models of incentive response are inadequate in "large worlds," where the dimensions of relative risk and the dimensions of similarity in outcome comparisons typically differ. Quantum probability models for choice in large worlds may be motivated pragmatically - there is no third theory - or metaphysically: statistical processing in the brain adapts to the true scale-relative structure of the universe.

Gianluca Stefanucci

Gianluca Stefanucci

Hannes Herrmann, Michael J. W. Hall, Howard M. Wiseman +1 more

Recently the Many-Interacting-Worlds (MIW) approach to a quantum theory without wave functions was proposed. This approach leads quite naturally to numerical integrators of the Schr\odinger equation. It has been suggested that such integrators may feature advantages over fixed-grid methods for higher numbers of degrees of freedom. However, as yet, little is known about concrete MIW models for more than one spatial dimension and/or more than one particle. In this work we develop the MIW approach further to treat arbitrary degrees of freedom, and provide a systematic study of a corresponding numerical implementation for computing one-particle ground and excited states in one dimension, and ground states in two spatial dimensions. With this step towards the treatment of higher degrees of freedom we hope to stimulate their further study.

Victor V. Albert, David Aasen, Wenqing Xu +3 more

E. Rundblad, Valery Labunets, Péter Novák

We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW 2N+1 (Z N m n,Z N m n, Z m n) over the ring Z m n. This group is the discrete version of the real Heisenberg group HW 2N+1 (R N , R N , R), where R is the real field. These functions have one dimensional and m 1 , m 2 ,..., m n -dimensions matrix valued spectral components (for irreducible representations of HW 2N+1 . The family of all ID representations gives classical world world (CW). Various m i -dimension representations (i = 1,2,...,n) map classical world (CW) into quantum worlds QW(m i ) of ith resolution (i = 1,2,....n). Worlds QW(m 1 ) and QW(m n ) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.