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Register a new userRestoring Poincare Symmetry to the Lattice
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Added by
Alexander S. Glasser
Publication date
2019
fields
Physics
and research's language is
English
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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.
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In the following work, we pedagogically develop 5-vector theory, an evolution of scalar field theory that provides a stepping stone toward a Poincare-invariant lattice gauge theory. Defining a continuous flat background via the four-dimensional Cartesian coordinates ${x^a}$, we `lift the generators of the Poincare group so that they transform only the fields existing upon ${x^a}$, and do not transform the background ${x^a}$ itself. To facilitate this effort, we develop a non-unitary particle representation of the Poincare group, replacing the classical scalar field with a 5-vector matter field. We further augment the vierbein into a new $5times5$ funfbein, which `solders the 5-vector field to ${x^a}$. In so doing, we form a new intuition for the Poincare symmetries of scalar field theory. This effort recasts `spacetime data, stored in the derivatives of the scalar field, as `matter field data, stored in the 5-vector field itself. We discuss the physical implications of this `Poincare lift, including the readmittance of an absolute reference frame into relativistic field theory. In a companion paper, we demonstrate that this theoretical development, here construed in a continuous universe, enables the description of a discrete universe that preserves the 10 infinitesimal Poincare symmetries and their conservation laws.
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.
The relative growth of field and metric perturbations during preheating is sensitive to initial conditions set in the preceding inflationary phase. Recent work suggests this may protect super-Hubble metric perturbations from resonant amplification during preheating. We show that this possibility is fragile and sensitive to the specific form of the interactions between the inflaton and other fields. The suppression is naturally absent in two classes of preheating in which either (1) the vacua of the non-inflaton fields during inflation are deformed away from the origin, or (2) the effective masses of non-inflaton fields during inflation are small but during preheating are large. Unlike the simple toy model of a $g^2 phi^2 chi^2$ coupling, most realistic particle physics models contain these other features. Moreover, they generically lead to both adiabatic and isocurvature modes and non-Gaussian scars on super-Hubble scales. Large-scale coherent magnetic fields may also appear naturally.
In the literature it is assumed that the parton to hadron fragmentation function cannot be studied by using the lattice QCD method because of the sum over the (unobserved) outgoing hadronic states. However, in this paper we find that since the hadron formation from the partons can be studied by using the lattice QCD method, the parton to hadron fragmentation function can be studied by using the lattice QCD method by using the LSZ reduction formula for the partonic processes.
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