Search Results for « State-Operator Clifford Compatibility »

This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The exterior (regressive) algebra is shown to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, when we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem of Clifford algebras context, wherein the Clifford and extended Clifford algebras Cl(p,q) can be embedded in Cl(p+1,q+1), which is shown to be exactly the extended Clifford algebra. Clifford algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior products of 1-forms.

We revisit the Pauli-Clifford connection to introduce a real, grade-preserving algebraic framework for $n$-qubit quantum computation based on the tensor product $C\ell_{2,0}(\mathbb{R})^{\otimes n}$. In this setting, the bivector $J = e_{12}$ satisfies $J^{2} = -1$ and supplies the complex structure on the $J$-closure of a minimal left ideal via right multiplication, while Pauli operations arise as left actions of Clifford elements. The Peirce decomposition organizes the algebra into sector blocks determined by primitive idempotents, with nilpotent elements generating transitions between sectors. Quantum states are represented as equivalence classes modulo the left annihilator, exhibiting the quotient description underlying the minimal left ideal. Adopting a canonical stabilizer mapping, the $n$-qubit computational basis state $|0\cdots 0\rangle$ is given natively by a tensor product of these idempotents. This structural choice leads to a compatibility law that is stable under the geometric product for $n$ qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.

Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an extensive demonstration of the applications of Clifford algebras in electromagnetism using the geometric algebra G3 = Cl(3,0) as a computational model in the Maxima computer algebra system. We compare the geometric algebra-based approach with conventional symbolic tensor calculations supported by Maxima, based on the itensor package. The Clifford algebra functionality of Maxima is distributed as two new packages called clifford - for basic simplification of Clifford products, outer products, scalar products and inverses; and cliffordan - for applications of geometric calculus.

Structural Compatibility VI develops the sixth step of the Structural Compatibility series on the basis of the broader Finite-Horizon Structures I–VI framework and of the relativistic quadratic sector established in Structural Compatibility V. Starting from the Lorentz-covariant quadratic mass-shell structure obtained on irreducible external sectors, it addresses the next question left open by that result: under which additional assumptions can this quadratic relativistic layer be realized as a minimal first-order propagation law acting on a single genuine state space? The article begins by clarifying why a purely scalar second-order realization is structurally insufficient for the purposes of the present programme. Although such a realization reproduces the relativistic quadratic dispersion relation, its Cauchy problem closes only on doubled data and therefore does not provide propagation on a single complete state object in the sense required by the maintenance-compatible framework. The relativistic realization problem must therefore be reformulated as a first-order factorisation problem. The paper then introduces a translation-linear factorisation ansatz together with an explicit internal-external tensor decomposition of the propagated state space. Within this setting, the external translation generators act on the inherited relativistic sector, while the coefficient operators act on an internal sector and commute with the external translation action. Under these compatibility assumptions, and under an additional external quadratic non-degeneracy hypothesis ensuring that the external quadratic sector separates symmetric coefficients, the factorisation problem is shown to force the Lorentzian Clifford anticommutation relations. On this basis, the article derives the structural consequences of the factorisation step. A one-dimensional internal realization is excluded, so the propagated state space must carry a genuine multicomponent internal sector. In the complex irreducible case relevant to four-dimensional Lorentzian kinematics, the minimal such realization is spinorial. The resulting first-order propagation law is therefore of Dirac type, not as an independently postulated starting point, but as the minimal admissible realization selected within the stated class of factorisations. The article also shows that this first-order spinorial layer can be coupled back to the Hermitian-projective maintenance framework inherited from Structural Compatibility II–III. Under a maintenance-admissible Hamiltonian realization, and under the invariance of suitable ray-level coherence and horizon descriptors, the associated maintenance-compatible scalar is preserved along the induced projective evolution. The result is thus a conditional compatibility statement linking the quadratic relativistic sector of Structural Compatibility V to a minimal first-order Clifford-spinorial realization and to a compatible maintenance layer. The scope of the article remains deliberately limited. No gauge coupling, no interacting field structure, no second quantization, and no quantum field-theoretic completion are assumed or derived. The paper does not claim to provide a full relativistic theory of matter, but rather isolates the missing first-order structural layer that bridges quadratic relativistic kinematics and spinorial propagation within the Structural Compatibility programme. This article forms part of the Ranesis framework developed by Alexandre Ramakers.

Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. But the spin group is not the only subgroup of the Clifford algebra. An algebraist's perspective on these groups and algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible physical interpretations are indicative only, and may not work. Nevertheless, both the existence of three generations of fermions and the symmetry-breaking of the weak interaction seem to emerge naturally from an extension of the Dirac algebra from complex numbers to quaternions.

The aim of this article is to introduce the H^{\infty} -functional calculus for unbounded bisectorial operators in a Clifford module over the algebra \mathbb{R}_{n} . This work is based on the universality property of the S -functional calculus, which shows its applicability to fully Clifford operators. While recent studies have focused on bounded operators or unbounded paravector operators, we now investigate unbounded fully Clifford operators and define polynomially growing functions of them. We first generate the \omega -functional calculus for functions that exhibit an appropriate decay at zero and at infinity. We then extend to functions with a finite value at zero and at infinity. Finally, using a subsequent regularization procedure, we can define the H^{\infty} - functional calculus for the class of regularizable functions, which in particular include functions with polynomial growth at infinity and, if T is injective, also functions with polynomial growth at zero.

Several Clifford algebras that are covariant under the action of a Lie algebra $g$ can be deformed in a way consistent with the deformation of $Ug$ into a quantum group (or into a triangular Hopf algebra) $U_qg$, i.e. so as to remain covariant under the action of $U_qg$. In this report, after recalling these facts, we review our results regarding the formal realization of the elements of such ``q-deformed'' Clifford algebras as ``functions'' (polynomials) in the generators of the undeformed ones; in particular, the intruiging interplay between the original and the q-deformed symmetry. Finally, we briefly illustrate their dramatic consequences on the representation theories of the original and of the q-deformed Clifford algebra, and mention how these results could turn out to be useful in quantum physics.

Abstract The $$H^\infty $$ H ∞ -functional calculus is a two-step procedure, introduced by A. McIntosh, that allows the definition of functions of sectorial operators in Banach spaces. It plays a crucial role in the spectral theory of differential operators, as well as in their applications to evolution equations and various other fields of science. An extension of the $$H^\infty $$ H ∞ -functional calculus also exists in the hypercomplex setting, where it is based on the notion of S -spectrum. Originally this was done for sectorial quaternionic operators, but then also generalized all the way to bisectorial fully Clifford operators. In the latter setting and in Hilbert spaces, this paper now characterizes the boundedness of the $$H^\infty $$ H ∞ -functional calculus through certain quadratic estimates. Due to substantial differences in the definitions of the S -spectrum and the S -resolvent operators, the proofs of quadratic estimates in this setting face additional challenges compared to the classical theory of complex operators.

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A_1 x A_1 x A_1, A_3, B_3 and H_3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A_1 x A_1 x A_1, A_3, B_3 and H_3 yield a new simple construction of binary polyhedral groups. These in turn generate the groups A_1 x A_1 x A_1 x A_1, D_4, F_4 and H_4 of rank 4, and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D_4, F_4 and H_4 are exceptional structures, which our study sheds new light on.

The spectral theory on the $S$-spectrum originated to give quaternionic quantum mechanics a precise mathematical foundation and as a spectral theory for linear operators in vector analysis. This theory has proven to be significantly more general than initially anticipated, naturally extending to fully Clifford operators and revealing unexpected connections with the spectral theory based on the monogenic spectrum, developed over forty years ago by A. McIntosh and collaborators. In recent years, we have combined slice hyperholomorphic functions with the Fueter-Sce mapping theorem, also called Fueter-Sce extension theorem, to broaden the class of functions and operators to which the theory can be applied. This generalization has led to the definition of what we call the {\em fine structures on the $S$-spectrum}, consisting of classes of functions that admit an integral representation and their associated functional calculi. In this paper, we focus on the fine structures within the Clifford algebra setting, particularly addressing polyharmonic functions, polyanalytic functions, holomorphic Cliffordian functions and their associated functional calculi defined via integral representation formulas. Moreover, we demonstrate that the monogenic functional calculus, defined via the monogenic Cauchy formula, and the $F$-functional calculus of the fine structures, defined via the Fueter-Sce mapping theorem in integral form, yield the same operator.