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Main Question or Discussion Point
Abstract:
I propose a simple cellular lattice model with configuration space [itex]Aff(3)(Z^3)[/itex].
The model allows to obtain, in the continuous limit, all fermions of the SM, the gauge group of the SM, it's action on the fermionic sector, and an effective Lorentz metric.
1 a) Importance of a unification of all observed particles and fields does not need further comment. b) The novelty of the approach is obvious. c) The SM itself is a phenomenological theory, which does not explain its particle and fields content at all. d) This gap is filled by the new model, which allows to explain the particle content of the fermionic and gauge sector of the SM completely. Unification of the SM with gravity and the quantization of gravity are other important problems which may be solved in the new approach: The gravitational field is included too. Moreover, the theory is, essentially, a condensed matter theory on a Newtonian background, and, therefore, a theory which can be canonically quantized.
2.) For details and references see the attached article. Further information can be found at iljaschmelzer.de/clm and iljaschmelzer.de/glet.
Basic Ideas: We consider a lattice of simple cells in R^3. The state of each cell is described by an affine deformation from a standard reference cell. Therefore, the configuration space of the lattice model is [itex]Aff(3)(Z^3)[/itex], resp. the phase space [itex]Aff(3)\otimes C(Z^3)[/itex].
We have a lattice Dirac equation on this lattice. Because of an analogon of the fermion doubling effect, this equation gives, in the continuous limit, a Dirac equation on [itex]Aff(3)\otimes C\otimes \Lambda(R^3)[/itex]. We identify the eight complex fields [itex]C\otimes \Lambda(R^3)[/itex] with electroweak doublets. The twelf components of the 3x(3+1) affine matrix can be easily identified with twelf electroweak doublets (3 generations x (3 quark colors + leptons)) of the SM.
The lattice model gives two natural things to be preserved by gauge fields: Euclidean symmetry and the symplectic structure of the phase space. We show that these requirements explain the following properties of the SM gauge group:
1.) They preserve generations and act on the three generations in the same way
2.) There exists some direction in each generation preserved by all gauge fields (righthanded neutrinos)
3.) The gauge groups are unitary
We consider the way gauge actions can be defined on the lattice. We find that Wilson gauge fields cannot have different charges inside electroweak doublets. As a consequence, the maximal possible Wilson gauge group is [itex]U(3) \cong SU(3)_c \times U(1)_B[/itex].
We show that deformations of the underlying lattice Z^3 require correction terms which, in the continuous limit, lead to another set of effective gauge fields. These gauge fields have to preserve electroweak doublets and are generated by the operators I_i and gamma^5. Combined with the previously found requirements, this gives electroweak [itex]U(2)_L \times U(1)_R[/itex], so that righthanded neutrinos have charge 0. In combination with U(1)_B we can obtain hypercharge Y and, therefore, all SM gauge fields.
As a last requirement, we use anomaly freedom. All additional fields (the diagonals of [itex]SU(3)_c[/itex] and [itex]SU(2)_L[/itex]) are anomalous, thus, the SM gauge group appears to be a maximal anomalyfree subgroup of the group [itex]U(3) \times U(2)_L \times U(1)_R[/itex] we have obtained.
The gravitational field can be incorporated too. We construct an effective Lorentz metric using density, threevelocity and stress tensor. This decomposition is, essentially, the ADM decomposition in the preferred frame of the Newtonian background. We have continuity and Euler equations, which correspond to the harmonic condition for the effective metric. We require that they appear as Noether conservation laws. This allows to derive the general Lagrangian of the theory: It is the sum of a term which enforces the harmonic gauge and the most general Lagrangian of general relativity. As a consequence, in the resulting theory we has an exact Einstein equivalence principle, and obtain, in a natural limit, the Einstein equations of GR.
As far, the considerations have been classical. But an important step of the quantization is also done: We have found a way to obtain anticommuting fermion fields starting from canonical quatization of the lattice theory. This method requires the introduction of some additional heavy bosons.
Some questions have been left to future research: The mass terms, which require symmetry breaking, dynamics of the gauge fields, quantization of gauge fields and gravity, and renormalization.
I propose a simple cellular lattice model with configuration space [itex]Aff(3)(Z^3)[/itex].
The model allows to obtain, in the continuous limit, all fermions of the SM, the gauge group of the SM, it's action on the fermionic sector, and an effective Lorentz metric.
1 a) Importance of a unification of all observed particles and fields does not need further comment. b) The novelty of the approach is obvious. c) The SM itself is a phenomenological theory, which does not explain its particle and fields content at all. d) This gap is filled by the new model, which allows to explain the particle content of the fermionic and gauge sector of the SM completely. Unification of the SM with gravity and the quantization of gravity are other important problems which may be solved in the new approach: The gravitational field is included too. Moreover, the theory is, essentially, a condensed matter theory on a Newtonian background, and, therefore, a theory which can be canonically quantized.
2.) For details and references see the attached article. Further information can be found at iljaschmelzer.de/clm and iljaschmelzer.de/glet.
Basic Ideas: We consider a lattice of simple cells in R^3. The state of each cell is described by an affine deformation from a standard reference cell. Therefore, the configuration space of the lattice model is [itex]Aff(3)(Z^3)[/itex], resp. the phase space [itex]Aff(3)\otimes C(Z^3)[/itex].
We have a lattice Dirac equation on this lattice. Because of an analogon of the fermion doubling effect, this equation gives, in the continuous limit, a Dirac equation on [itex]Aff(3)\otimes C\otimes \Lambda(R^3)[/itex]. We identify the eight complex fields [itex]C\otimes \Lambda(R^3)[/itex] with electroweak doublets. The twelf components of the 3x(3+1) affine matrix can be easily identified with twelf electroweak doublets (3 generations x (3 quark colors + leptons)) of the SM.
The lattice model gives two natural things to be preserved by gauge fields: Euclidean symmetry and the symplectic structure of the phase space. We show that these requirements explain the following properties of the SM gauge group:
1.) They preserve generations and act on the three generations in the same way
2.) There exists some direction in each generation preserved by all gauge fields (righthanded neutrinos)
3.) The gauge groups are unitary
We consider the way gauge actions can be defined on the lattice. We find that Wilson gauge fields cannot have different charges inside electroweak doublets. As a consequence, the maximal possible Wilson gauge group is [itex]U(3) \cong SU(3)_c \times U(1)_B[/itex].
We show that deformations of the underlying lattice Z^3 require correction terms which, in the continuous limit, lead to another set of effective gauge fields. These gauge fields have to preserve electroweak doublets and are generated by the operators I_i and gamma^5. Combined with the previously found requirements, this gives electroweak [itex]U(2)_L \times U(1)_R[/itex], so that righthanded neutrinos have charge 0. In combination with U(1)_B we can obtain hypercharge Y and, therefore, all SM gauge fields.
As a last requirement, we use anomaly freedom. All additional fields (the diagonals of [itex]SU(3)_c[/itex] and [itex]SU(2)_L[/itex]) are anomalous, thus, the SM gauge group appears to be a maximal anomalyfree subgroup of the group [itex]U(3) \times U(2)_L \times U(1)_R[/itex] we have obtained.
The gravitational field can be incorporated too. We construct an effective Lorentz metric using density, threevelocity and stress tensor. This decomposition is, essentially, the ADM decomposition in the preferred frame of the Newtonian background. We have continuity and Euler equations, which correspond to the harmonic condition for the effective metric. We require that they appear as Noether conservation laws. This allows to derive the general Lagrangian of the theory: It is the sum of a term which enforces the harmonic gauge and the most general Lagrangian of general relativity. As a consequence, in the resulting theory we has an exact Einstein equivalence principle, and obtain, in a natural limit, the Einstein equations of GR.
As far, the considerations have been classical. But an important step of the quantization is also done: We have found a way to obtain anticommuting fermion fields starting from canonical quatization of the lattice theory. This method requires the introduction of some additional heavy bosons.
Some questions have been left to future research: The mass terms, which require symmetry breaking, dynamics of the gauge fields, quantization of gauge fields and gravity, and renormalization.
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