Spacetime linear canonical transform and the uncertainty principles (Q6052754)

The spacetime algebra (Clifford algebra) \(\mathcal{C}\ell_{(1,3)}\) is defined as an associative noncommutative algebra generated by the orthonormal basis \(\left\{ e_t, e_1, e_2, e_3 \right\}\), of the real four-dimensional non-Euclidean vector space \(\mathbb{R}^{(1,3)}\), governed by the multiplication rule \[ e_i e_j+e_j e_i=2\epsilon_j \delta_{i,j},\; i,j\in \left\{ t,1,2,3 \right\}, \] and \[ \epsilon_j=1, \mbox{ for } j=1,2,3, \mbox{ and } \epsilon_j=-1,\mbox{ for } j=t. \] The spacetime algebra \(\mathcal{C}\ell_{(1,3)}\) is a 16-dimensional space. The paper is concerned with spacetime algebra-valued function \[ f: \mathbb{R}^{(1,3)}\rightarrow \mathcal{C}\ell_{(1,3)}, \] in particular, with those functions in the space \( L^1\left( \mathbb{R}^{(1,3)} , \mathcal{C}\ell_{(1,3)} \right)\). The linear canonical transform (LCT) is an integral transform whose kernel depends on 4 parameters \(a,b,c,d\) such that \(ad-bc=1.\) It generalizes other integral transforms, including the Fourier and fractional Fourier transforms. As in the classical case, the author introduces the notion of the spacetime geometric algebra-valued linear canonical transform as a generalization of spacetime Fourier transform by replacing the right-side (spatial) Fourier kernel with a more general LCT kernel. The spacetime LCT is a non-commutative multi-vector linear canonical transform that acts on functions from the four-dimensional spacetime algebra \(\mathbb{R}^{(1,3)}\) to the 16-dimensional Clifford geometric algebra \(\mathcal{C}\ell_{(1,3)}.\) The author then investigates fundamental properties of the proposed spacetime LCT, such as its translation, scaling, and modulation covariances, and generalizes Heisenberg's uncertainty principle to spacetime algebra-valued signals subject to the spacetime linear canonical transformation.