A conformal group approach to the Dirac–Kähler system on the lattice

Abstract

Abstract. Starting from the representation of the (n − 1) + n−dimensional Lorentz pseudo-sphere on the projective space PR n,n , we propose a method to derive a class of solutions underlying to a Dirac-Kähler type equation on the lattice. We make use of the Cayley transform ϕ(w) = 1 + w 1 − w to show that the resulting group representation arise from the same mathematical framework as the conformal group representation in terms of the general linear group GL (2, Γ(n − 1, n − 1) ∪ {0}). That allows us to describe such class of solutions as a commutative n−ary product, involving the quasi-monomials ϕ (z j ) − x j h (x j ∈ hZ) with membership in the paravector space R ⊕ Re j e n+j .

On fractional semidiscrete Dirac operators of Lévy–Leblond type

“…Let us now recall the basic setup and results from the series of papers [19][20][21]41] to discuss further aspects of our construction. In the following, we will use the nilpotents 𝔣 and 𝔣 † of 𝐶𝓁 1,1 to introduce firstly on ℂ ⊗ 𝐶𝓁 𝑛+1,𝑛+1 , with 𝐶𝓁 𝑛+1,𝑛+1 = 𝐶𝓁…”

“…Let us now recall the basic setup and results from the series of papers [19–21, 41] to discuss further aspects of our construction. In the following, we will use the nilpotents $${\mathfrak {f}}$$ and $${\mathfrak {f}}^\dagger$$ of $$C \hspace{-1.00006pt}\ell _{1,1}$$ to introduce firstly on $${\mathbb {C}}\otimes C \hspace{-1.00006pt}\ell _{n+1,n+1}$$, with $$C \hspace{-1.00006pt}\ell _{n+1,n+1}=C \hspace{-1.00006pt}\ell _{1,1}\otimes C \hspace{-1.00006pt}\ell _{n,n}$$, the semidiscrete Dirac‐type operator $$$$\begin{align} { }_\theta \mathcal {D}_{h,t}:=D_h+{\mathfrak {f}}\partial _t +{\mathfrak {f}}^\dagger \text{e}^{-i\theta }, \end{align}$$$$carrying the discrete Dirac operator …”

“…Originally highlighted in [24] and on the series of papers [9,18], such fact was only duly clarified in author's recent paper [19], through the canonical isomorphism 𝐶𝓁 𝑛,𝑛 ≅ End(𝐶𝓁 0,𝑛 ) between the Clifford algebra of signature (𝑛, 𝑛), 𝐶𝓁 𝑛,𝑛 , and the algebra of endomorphisms acting on the Clifford algebra of signature (0, 𝑛), 𝐶𝓁 0,𝑛 . This in turn allows us to establish a canonical correspondence between the discrete counterpart of the Dirac-Kähler operator, 𝑑 − 𝑑 * , and the multivector approximation of Dirac operator, 𝐷 ℎ , over ℎℤ 𝑛 (see also [41]).…”

“…Originally highlighted in [24] and on the series of papers [9, 18], such fact was only duly clarified in author's recent paper [19], through the canonical isomorphism $$C \hspace{-1.00006pt}\ell _{n,n}\cong \mbox{End}(C \hspace{-1.00006pt}\ell _{0,n})$$ between the Clifford algebra of signature $$(n,n)$$, $$C \hspace{-1.00006pt}\ell _{n,n}$$, and the algebra of endomorphisms acting on the Clifford algebra of signature (0, n ), $$C \hspace{-1.00006pt}\ell _{0,n}$$. This in turn allows us to establish a canonical correspondence between the discrete counterpart of the Dirac–Kähler operator, $$d-d^*$$, and the multivector approximation of Dirac operator, $$D_h$$, over $$h{\mathbb {Z}}^n$$ (see also [41]).…”

“…In this paper, we will focus our attention on the time‐fractional and space‐fractional Dirac operators, carrying the parameters $$0<\alpha \le 1$$, $$\beta \ge 1$$ and $$|\theta |\le \frac{\alpha \pi }{2}$$. On the construction neatly summarized in Figure 1, the notation $$\Delta _h$$ stands for the discrete Laplacian on the lattice $$h{\mathbb {Z}}^n$$ considered in several author's contributions such as [19, 41].…”