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Register a new userThe Z3-graded extension of the Poincare algebra
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Authors
Richard Kerner
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A Z3 symmetric generalization of the Dirac equation was proposed in recent series of papers, where its properties and solutions discussed. The generalized Dirac operator acts on coloured spinors composed out of six Pauli spinors, describing three colours and particle-antiparticle degrees of freedom characterizing a single quark state, thus combining Z2 x Z_2 x Z_3 symmetries of 12-component generalized wave functions. Spinorial representation of the Z3-graded generalized Lorentz algebra was introduced, leading to the appearance of extra Z2 x Z2 x Z3 symmetries, probably englobing the symmetries of isospin, flavors and families. The present article proposes a construction of Z3-graded extension of the Poincare algebra. It turns out that such a generalization requires introduction of extended 12-dimensional Minkowskian space-time containing the usual 4-dimensional space-time as a subspace, and two other mutually conjugate replicas with complex-valued vectors and metric tensors. Representation in terms of differential operators and generalized Casimir operators are introduced and their symmetry properties are briefly discussed.
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The wave equation generalizing the Dirac operator to the Z3-graded case is introduced, whose diagonalization leads to a sixth-order equation. It intertwines not only quark and anti-quark state as well as the u and d quarks, but also the three colors, and is therefore invariant under the product group Z2 x Z2 x Z3. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry and of the SU(2) x U(1) that arise automatically in this model, leading to the full bosonic gauge sector of the Standard Model.
We investigate a non-trivial extension of the $D-$dimensional Poincare algebra. Matrix representations are obtained. The bosonic multiplets contain antisymmetric tensor fields. It turns out that this symmetry acts in a natural geometric way on these $p-$forms. Some field theoretical aspects of this symmetry are studied and invariant Lagrangians are explicitly given.
The following work demonstrates the viability of Poincare symmetry in a discrete universe. We develop the technology of the discrete principal Poincare bundle to describe the pairing of (1) a hypercubic lattice `base manifold labeled by integer vertices-denoted ${mathbf{n}}={(n_t,n_x,n_y,n_z)}$-with (2) a Poincare structure group. We develop lattice 5-vector theory, which describes a non-unitary representation of the Poincare group whose dynamics and gauge transformations on the lattice closely resemble those of a scalar field in spacetime. We demonstrate that such a theory generates discrete dynamics with the complete infinitesimal symmetry-and associated invariants-of the Poincare group. Following our companion paper, we `lift the Poincare gauge symmetries to act only on vertical matter and solder fields, and recast `spacetime data--stored in the $partial_muphi(x)$ kinetic terms of a free scalar field theory--as `matter field data-stored in the $phi^mu[mathbf{n}]$ components of the 5-vector field itself. We gauge 5-vector theory to describe a lattice gauge theory of gravity, and discuss the physical implications of a discrete, Poincare-invariant theory.
We study the symmetric subquotient decomposition of the associated graded algebras $A^*$ of a non-homogeneous commutative Artinian Gorenstein (AG) algebra $A$. This decomposition arises from the stratification of $A^*$ by a sequence of ideals $A^*=C_A(0)supset C_A(1)supsetcdots$ whose successive quotients $Q(a)=C(a)/C(a+1)$ are reflexive $A^*$ modules. These were introduced by the first author, and have been used more recently by several groups, especially those interested in short Gorenstein algebras, and in the scheme length (cactus rank) of forms. For us a Gorenstein sequence is an integer sequence $H$ occurring as the Hilbert function for an AG algebra $A$, that is not necessarily homogeneous. Such a Hilbert function $H(A)$ is the sum of symmetric non-negative sequences $H_A(a)=H(Q_A(a))$, each having center of symmetry $(j-a)/2$ where $j$ is the socle degree of $A$: we call these the symmetry conditions, and the decomposition $mathcal{D}(A)=(H_A(0),H_A(1),ldots)$ the symmetric decomposition of $H(A)$. We here study which sequences may occur as the summands $H_A(a)$: in particular we construct in a systematic way examples of AG algebras $A$ for which $H_A(a)$ can have interior zeroes, as $H_A(a)=(0,s,0,ldots,0,s,0)$. We also study the symmetric decomposition sets $mathcal{D}(A)$, and in particular determine which sequences $H_A(a)$ can be non-zero when the dual generator is linear in a subset of the variables. Several groups have studied exotic summands of the Macaulay dual generator $F$. Studying these, we recall a normal form for the Macaulay dual generator of an AG algebra that has no exotic summands. We apply this to Gorenstein algebras that are connected sums. We give throughout many examples and counterexamples, and conclude with some open questions about symmetric decomposition.
We show how the discrete symmetries $Z_2$ and $Z_3$ combined with the superposition principle result in the $SL(2, {bf C})$-symmetry of quantum states. The role of Paulis exclusion principle in the derivation of the SL(2, C) symmetry is put forward as the source of the macroscopically observed Lorentz symmetry, then it is generalized for the case of the Z3 grading replacing the usual Z2 grading, leading to ternary commutation relations. We discuss the cubic and ternary generalizations of Grassmann algebra. Invariant cubic forms are introduced, and their symmetry group is shown to be the $SL(2,C)$ group The wave equation generalizing the Dirac operator to the Z3-graded case is constructed. Its diagonalization leads to a sixth-order equation. The solutions cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry.
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