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The nontrivial fixed point discovered for $��^4$-marginal couplings in tensorial group field theories have been showed to be incompatible with Ward-Takahashi identities. In this previous analysis we have stated that the case of models with interactions of order greater than four could probably lead to a fixed point compatible with local Ward's identities. In this paper we focus on a rank-4 Abelian $��^6$-just renormalizable tensorial group field theory and describe the renormalization group flow over the sub-theory space where Ward constraint is satisfied along the flow, by using an improved version of the effective vertex expansion. We show that this model exhibit a nontrivial fixed points in this constrained subspace. Finally, the well-known asymptotically freedom of this model is highlighted., 35 pages

I describe here the attempt to introduce spin-charge separation in Schrodinger equation. The construction we present here gives a decomposed Schrodinger spinor that has one problem: Its absolute value can only have value between 0 and ${1}{2}$. The problem we solve is to expand and generalize this construction so that one can have a Schrodinger spinor with absolute value that are arbitrary non-negative numbers. It may be that one has to introduce a set of different decompositions to cover all nonnegative values, that is to introduce patches over $\mathbb{R}_{+}^{3}$ so that in each patch one has a different representation. It seems that the decomposition has a direct relation to so called entangled states that have been discussed very much in connection of e.g. quantum computing, and we would like to find this relation and discuss it in detail., 24 pages. Improved version, some statements corrected, more comments

This work prolongs, using an operator method, the investigations started in our recent paper J. Math. Phys. 51., 102108 on the spectrum and states of the harmonic oscillator on twisted Moyal plane, where rather a Moyal-star-algebraic approach was used. The physical spectrum and states of the harmonic oscillator on twisted Moyal space, obtained here by solving the corresponding differential equation, are similar to those of the ordinary Moyal space, with different parameters. This fortunately contrasts with the previous study which produced unexpected results, i.e. infinitely degenerate states with energies depending on the coordinate functions., Comment: 19 pages, 2 figures

We prove that the rank 3 analogue of the tensor model defined in [arXiv:1111.4997 [hep-th]] is renormalizable at all orders of perturbation. The proof is given in the momentum space. The one-loop $��$- and $��$-functions of the model are also determined. We find that the model with a unique coupling constant for all interactions and a unique wave function renormalization is asymptotically free in the UV., 42 pages, 14 figures; improved version, some statements corrected, more comments

Harmonic oscillator, in 2-dimensional noncommutative phase space with non-vanishing momentum-momentum commutators, is studied using an algebraic approach. The corresponding eigenvalue problem is solved and discussed.

A Hamiltonian formulation for the Grosse-Wulkenhaar $��^{4}_{\star D}$ model is performed. The study is based on $D+1$ dimensional space-time formulation of $D$ dimensional non-local theories. The analysis of constraints shows that the secondary constraints describe the Euler-Lagrange equations of motion. Relevant tensors are computed and analyzed.