A Condensed Matter Interpretation of SM Fermions and Gauge Fields (original) (raw)

Space-geometric interpretation of standard model fermions

Based on the geometric interpretation of the Dirac equation as an evolution equation on the three-dimensional exterior bundle /(R^3), we propose the bundle (T x / x /)(R^3) as a geometric interpretation of all standard model fermions. The generalization to curved background requires an ADM decomposition M^4=M^3 x R and gives the bundle (T x / x /)(M^3). As a consequence of the geometric character of the bundle there is no necessity to introduce a tetrad or triad formalism. Our space-geometric interpretation associates colors as well as fermion generations with directions in space, electromagnetic charge with the degree of a differential form, and weak interactions with the Hodge star operator. The space-geometric interpretation leads to different physical predictions about the connection of the SM with gravity, but gives no such differences on Minkowski background.

Yang–Mills gauge theories from simple fermionic lattice models

Physics Letters A, 2009

A doublet of three-dimensional Dirac fermions can effectively describe the low energy spectrum of a fermionic cubic lattice. We employ this fermion doubling to encode a non-Abelian SU (2) charge in the fundamental representation. We explicitly demonstrate that suitable distortion of the tunnelling couplings can introduce a scalar and a Yang-Mills field in the effective low energy description, both coupled to the Dirac fermions. The simplicity of the model suggests its physical implementation with ultra-cold atoms or molecules.

A doubler-free lattice theory for QCD based on geometric fermions

We present doubler-free gauge-invariant lattice vector gauge action for some real representations of Wilson gauge fields on an octet of fermions. It is based on a geometric representation of the Dirac equation as an evolution equation on the three-dimensional exterior bundle /(R^3) for a single bispinor and of the bundle (/\x/)(R^3) for an octet. We find doubler-free lattice Dirac operators for above bundles. A gauge-invariant connection with Wilson lattice gauge fields is possible for some real representations of the gauge group. The QCD action of SU(3) is of this type. Application in lattice QCD seems useful: We don't have to waste time and memory for doublers as well as for correction terms to suppress them.

The Dirac Equation in Six-dimensional SO(3, 3) Symmetry Group and a Non-chiral "Electroweak" Theory

We propose a model of electroweak interactions without chirality in a sixdimensional spacetime with 3 time-like and 3 space-like coordinates, which allows a geometrical meaning for gauge symmetries. The spacetime interval ds 2 = dx μ dx μ is left invariant under the symmetry group SO(3, 3). We obtain the six-dimensional version of the Dirac gamma matrices, μ , and write down a Dirac-like Lagrangian density, L = iψ μ ∇ μ ψ. The spinor ψ can be decomposed into two Dirac spinors, ψ 1 and ψ 2 , interpreted as the electron and neutrino fields, respectively. In six-dimensional spacetime the electron and neutrino fields appear as parts of the same entity in a natural manner. The SO(3, 3) Lorentz symmetry group is locally broken to the observable SO(1, 3) Lorentz group, with only one observable time component, t z. The t z-axis may not be the same at all points of the spacetime, and the effect of breaking the SO(3, 3) spacetime symmetry group locally to an SO(1, 3) Lorentz group, is perceived by the observers as the existence of the gauge fields. We interpret the origin of mass and gauge interactions as a consequence of extra time dimensions, without the need of introducing the so-called Higgs mechanism for the generation of mass. Further, in our 'toy' model, we are able to give a geometric meaning to the electromagnetic and non-Abelian gauge symmetries.

A new fermion Hamiltonian for lattice gauge theory

Nuclear Physics B - Proceedings Supplements, 2002

We formulate Hamiltonian vector-like lattice gauge theory using the overlap formula for the spatial fermionic part, H f. We define a chiral charge, Q5 which commutes with H f , but not with the electric field term. There is an interesting relation between the chiral charge and the fermion energy with consequences for chiral anomalies.

3-dimensionalSU(2) lattice gauge theory in terms of gauge invariant variables

Zeitschrift für Physik C Particles and Fields, 1982

As a first step towards a duality transformation for the SU(2) lattice gauge theory in 3 dimensions, the integration over all gauge variant variables is performed explicitly after introducing gauge invariant auxiliary variables. The resulting new Hamiltonian is complex and involves a sum over closed loops. Each of these loops is confined to an elementary cube of a dual lattice. Like in a previous investigation for the 0(4) symmetric Heisenberg ferromagnet Riihl's boson representation is used to derive the result.

The Dirac equation and a non-chiral electroweak theory in six dimensional spacetime from a locally gauged SO (3, 3) symmetry group

Arxiv preprint arXiv:0901.4230, 2009

A toy model for the electroweak interactions(without chirality) is proposed in a six dimensional spacetime with 3 timelike and 3 spacelike coordinates. The spacetime interval ds 2 = dxµdx µ is left invariant under the symmetry group SO(3, 3). We obtain the six-dimensional version of the Dirac gamma matrices, Γµ, and write down a Dirac-like lagrangian density, L = iψΓ µ ∇µψ. The spinor ψ is decomposed into two Dirac spinors, ψ1 and ψ2, which we interpret as the electron and neutrino fields, respectively. In six-dimensional spacetime the electron and neutrino fields are then merged in a natural manner. The SO(3, 3) Lorentz symmetry group must be locally broken to the observable SO(1, 3) Lorentz group, with only one observable time component, tz. The tz-axis may not be the same at all points of the spacetime and the effect of breaking the SO(3, 3) spacetime symmetry group locally to an SO(1, 3) Lorentz group is perceived by the observers as the existence of the gauge fields. The origin of mass may be attributed to the remaining two hidden timelike dimensions. We interpret the origin of mass and gauge interactions as a consequence of extra time dimensions, without the need of the so-called Higgs mechanism for the generation of mass. Further, we are able to give a geometric meaning to the electromagnetic and non-abelian gauge symmetries.

Fermionized spin systems and the boson-fermion mapping in (2+1)-dimensional gauge theory

Physics Letters B, 1989

A generalization of the Jordan-Wigner transformation is used to present 2-dimensional quantum spin systems with s= ½ as particular lattice gauge theories with spinless fermionic variables. It is argued that for magnetically ordered phases the large wavelength limit produces a concrete physical realization of continuum topologically massive electrodynamics in 2 + 1 dimensions with fermionic matter. We also give an explicit example of transmutation from Fermi to Bose statistics in (2+ 1)-dimensional gauge theory with a Chern-Simons term. The Jordan-Wigner transformation [ 1 ] is an important tool for the analysis of 1-dimensional spin systems. It maps an s= ~ quantum spin chain onto a lattice fermion system. This gives an exact solution of the 1-dimensional quantum x-y model [ 2 ] and is useful for perturbative or renormalization group analyses of other more complicated models. ~. Recently a generalization of this transformation to 2 dimensions has been used to study representations of (2 + 1)-dimensional continuum gauge theories which exhibit exotic spin and exchange statistics of elementary particles [4 ]. It has also been used to study the fermionic excitations of the resonating valence bond ansatz for the ground state of the 2-dimensional Hubbard model [ 5 ]. In this letter we use it to fermionize a spin system [6] ~2. We show that the critical behavior of a fermionized Heisenberg antiferromagnet can be represented by continuum (2 + 1)-dimensional electrodynamics with fermionic matter and where the gauge field action has a Chern-Simons term. We use this correspondence to show that the bare fermionic matter of the continuum field theory, when dressed by its interaction with topologically massive gauge fields, are relativistic bosons which coincide with the bosonic spin waves of the antiferromagnet. This gives an explicit nonperturbative example of the transmutation of spin and statistics in (2+ 1)-dimensional electrodynamics with a Chern-Simons term ~3. Consider a 2-dimensional lattice x and SO (3) Lie algebra [S~, S~] = i~at'cS~(x, y) with the irreducible s = ½ representation. This algebra can be represented by the fermionic variables ~' and ~u t with {~'x, ~) = ~(x, y) and all other anticommutators vanishing through the transformation

Fermions in quantum gravity

We study the quantum fermions+gravity system, that is, the gravitational counterpart of QED. We start from the standard Einstein-Weyl theory, reformulated in terms of Ashtekar variables; and we construct its non-perturbative quantum theory by extending the loop representation of general relativity.

Fermions, strings, and gauge fields in lattice spin models

Physical Review B, 2003

We investigate the general properties of lattice spin models with emerging fermionic excitations. We argue that fermions always come in pairs and their creation operator always has a string-like structure with the newly created particles appearing at the endpoints of the string. The physical implication of this structure is that the fermions always couple to a nontrivial gauge field. We present exactly soluble examples of this phenomenon in 2 and 3 dimensions. Our analysis is based on an algebraic formula that relates the statistics of a lattice particle to the properties of its hopping operators. This approach has the advantage that it works in any number of dimensions -unlike the flux-binding picture developed in FQH theory.