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World Journal of Advanced Research and Reviews

4-dimensional lattice models for the quantum range

Gudrun Kalmbach H.E.

Published Oct 13, 2021• 0 citations
Publisher

Abstract

Some finite subspace models L are presented for quantum structures which replace the use of countable infinite Hilbert space H dimensions. A maximal Boolean sublattice, called block, is 24, where its four atoms directly above 0εL, base vectors of H in 24 are drawn as four points on an interval. Blocks can overlap in one or two atoms. Different kinds of operators can map one block onto another and interpretations are given such that subspaces can carry on their base vector tuple real, complex or quaternionic numbers, energies, symmetries and generate coordinate lines. Describing states of physical systems is done using L and its applications for dynamical modelling. They don‘t need the infinte dimensional vectors of H. L has in the first model 11 blocks and 24 atoms (figure 1). They correspond to the 24 elements of the tetrahedral S4 symmetry. S4 arises from a spin-line rgb-graviton whirl operator with center at the tip of a tetrahedron and a nucleon triangle base with three quarks as vertices. The triangles factor group D3 of S4 is due to the CPT Klein normal subgroup Z2 x Z2 of S4 . It has a strong interaction SI rotor for the nucleons inner dynamics which is used for integrating functions, exchanging energies of nucleon with its environment and setting barycentrical coordinates in the triangle. At their intersection B as barycenter sets a Higgs boson or field the rescaled quark mass of a nucleon. Each factor class of one element from D3 assigns to it a color charge, a coordinate, an energy vector and a symmetry. Symmetries attached can be different according to interactions involved. Every atom of L has then a specific character with different properties.Three characters are added to octonian base vectors, listed by their indices as n = 0,1,…,7, and named for the atoms of L as na, nb, nc. The structure and element attributes of the finite subspace lattices L are desribed in many examples and models which technical constructed run macroscopically. Several models are described below. Example, the color charge whirl as rgb-graviton projection operator maps the block 2c3b5a6a to 0a1a2a3a. The symmetries change dimension from 3x3- to 2x2-matrices. From SU(3) are λ1 on 3b mapped to the SU(2) x-coordinate Pauli matrix σ1, from λ2 on 5a to σ2 y-coordinate and from λ3 on 6a to σ3 z-coordinate of real Euclidean space R³. The SU(3) matrices have complex w3 = z +ict, w2 = (iy,f), w1 = (x,m) coordinates. In figure 3 is shown how a rotation of two proton tetrahedrons for fusion changes the two linearly independent wj vectors to the 1-dimensional x,y,z base vectors. In deuteron then on one coordinate line sit with Cooper paire u-d-quarks at the ends the Heisenberg coupled energy or space vector rays 15 (x,m), m mass measured in kg, x in meter, 23 (iy,E(rot)), E(rot) rotational energy measured in Joule J, y in meter, 46 (ict,f), t time measured in seconds, f = 1/∆t frequency s inverse time interval measured in Hz. The six color charges are red r on +x as octonian coordinate 1, green g on +y as 2 , blue b on -z as 6, turquoise on -x as 5, magenta on -y as 3, yellow on +z as 4..

DOI
10.30574/wjarr.2021.12.1.0003

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4-dimensional lattice models for the quantum range

Gudrun Kalmbach H.E.

Some finite subspace models L are presented for quantum structures which replace the use of countable infinite Hilbert space H dimensions. A maximal Boolean sublattice, called block, is 24, where its four atoms directly above 0εL, base vectors of H in 24 are drawn as four points on an interval. Blocks can overlap in one or two atoms. Different kinds of operators can map one block onto another and interpretations are given such that subspaces can carry on their base vector tuple real, complex or quaternionic numbers, energies, symmetries and generate coordinate lines. Describing states of physical systems is done using L and its applications for dynamical modelling. They don‘t need the infinte dimensional vectors of H. L has in the first model 11 blocks and 24 atoms (figure 1). They correspond to the 24 elements of the tetrahedral S4 symmetry. S4 arises from a spin-line rgb-graviton whirl operator with center at the tip of a tetrahedron and a nucleon triangle base with three quarks as vertices. The triangles factor group D3 of S4 is due to the CPT Klein normal subgroup Z2 x Z2 of S4 . It has a strong interaction SI rotor for the nucleons inner dynamics which is used for integrating functions, exchanging energies of nucleon with its environment and setting barycentrical coordinates in the triangle. At their intersection B as barycenter sets a Higgs boson or field the rescaled quark mass of a nucleon. Each factor class of one element from D3 assigns to it a color charge, a coordinate, an energy vector and a symmetry. Symmetries attached can be different according to interactions involved. Every atom of L has then a specific character with different properties.Three characters are added to octonian base vectors, listed by their indices as n = 0,1,…,7, and named for the atoms of L as na, nb, nc. The structure and element attributes of the finite subspace lattices L are desribed in many examples and models which technical constructed run macroscopically. Several models are described below. Example, the color charge whirl as rgb-graviton projection operator maps the block 2c3b5a6a to 0a1a2a3a. The symmetries change dimension from 3x3- to 2x2-matrices. From SU(3) are λ1 on 3b mapped to the SU(2) x-coordinate Pauli matrix σ1, from λ2 on 5a to σ2 y-coordinate and from λ3 on 6a to σ3 z-coordinate of real Euclidean space R³. The SU(3) matrices have complex w3 = z +ict, w2 = (iy,f), w1 = (x,m) coordinates. In figure 3 is shown how a rotation of two proton tetrahedrons for fusion changes the two linearly independent wj vectors to the 1-dimensional x,y,z base vectors. In deuteron then on one coordinate line sit with Cooper paire u-d-quarks at the ends the Heisenberg coupled energy or space vector rays 15 (x,m), m mass measured in kg, x in meter, 23 (iy,E(rot)), E(rot) rotational energy measured in Joule J, y in meter, 46 (ict,f), t time measured in seconds, f = 1/∆t frequency s inverse time interval measured in Hz. The six color charges are red r on +x as octonian coordinate 1, green g on +y as 2 , blue b on -z as 6, turquoise on -x as 5, magenta on -y as 3, yellow on +z as 4.

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Topological Color Codes and Two-Body Quantum Lattice Hamiltonians

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Topological color codes are among the stabilizer codes with remarkable properties from the quantum information perspective. In this paper, we construct a lattice, the so-called ruby lattice, with coordination number 4 governed by a two-body Hamiltonian. In a particular regime of coupling constants, in a strong coupling limit, degenerate perturbation theory implies that the low-energy spectrum of the model can be described by a many-body effective Hamiltonian, which encodes the color code as its ground state subspace. Ground state subspace corresponds to a vortex-free sector. The gauge symmetry Z2×Z2 of the color code could already be realized by identifying three distinct plaquette operators on the ruby lattice. All plaquette operators commute with each other and with the Hamiltonian being integrals of motion. Plaquettes are extended to closed strings or string-net structures. Non-contractible closed strings winding the space commute with Hamiltonian but not always with each other. This gives rise to exact topological degeneracy of the model. A connection to 2-colexes can be established via the coloring of the strings. We discuss it at the non-perturbative level. The particular structure of the two-body Hamiltonian provides a fruitful interpretation in terms of mapping onto bosons coupled to effective spins. We show that high-energy excitations of the model have fermionic statistics. They form three families of high-energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. The emergence of invisible charges is related to the string-net structure of the model. The emerging fermions are coupled to nontrivial gauge fields. We show that for particular 2-colexes, the fermions can see the background fluxes in the ground state. Also, we use the Jordan–Wigner transformation in order to test the integrability of the model via introducing Majorana fermions. The four-valent structure of the lattice prevents the fermionized Hamiltonian from being reduced to a quadratic form owing to interacting gauge fields. We also propose another construction for the two-body Hamiltonian based on the connection between color codes and cluster states. The corresponding two-body Hamiltonian encodes a cluster state defined on a bipartite lattice as its low-energy spectrum, and subsequent selective measurements give rise to the color code model. We discuss this latter approach along with the construction based on the ruby lattice.

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Topological color codes and two-body quantum lattice Hamiltonians

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Topological color codes are among the stabilizer codes with remarkable properties from the quantum information perspective. In this paper, we construct a lattice, the so-called ruby lattice, with coordination number 4 governed by a two-body Hamiltonian. In a particular regime of coupling constants, in a strong coupling limit, degenerate perturbation theory implies that the low-energy spectrum of the model can be described by a many-body effective Hamiltonian, which encodes the color code as its ground state subspace. Ground state subspace corresponds to a vortex-free sector. The gauge symmetry Z 2 × Z 2 of the color code could already be realized by identifying three distinct plaquette operators on the ruby lattice. All plaquette operators commute with each other and with the Hamiltonian being integrals of motion. Plaquettes are extended to closed strings or string-net structures. Non-contractible closed strings winding the space commute with Hamiltonian but not always with each other. This gives rise to exact topological degeneracy of the model. A connection to 2-colexes can be established via the coloring of the strings. We discuss it at the non-perturbative level. The particular structure of the two-body Hamiltonian provides a fruitful interpretation in terms of mapping onto bosons coupled to effective spins. We show that high-energy excitations of the model have fermionic statistics. They form three families of high-energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. The emergence of invisible charges is related to the string-net structure of the model. The emerging fermions are coupled to nontrivial gauge fields. We show that for particular 2-colexes, the fermions can see the background fluxes in the ground state. Also, we use the Jordan–Wigner transformation in order to test the integrability of the model via introducing Majorana fermions. The four-valent structure of the lattice prevents the fermionized Hamiltonian from being reduced to a quadratic form owing to interacting gauge fields. We also propose another construction for the two-body Hamiltonian based on the connection between color codes and cluster states. The corresponding two-body Hamiltonian encodes a cluster state defined on a bipartite lattice as its low-energy spectrum, and subsequent selective measurements give rise to the color code model. We discuss this latter approach along with the construction based on the ruby lattice.

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Some finite subspace models L are presented for quantum structures which replace the use of countable infinite Hilbert space H dimensions. A maximal Boolean sublattice, called block, is 24, where its four atoms directly above 0εL, base vectors of H in 24 are drawn as four points on an interval. Blocks can overlap in one or two atoms. Different kinds of operators can map one block onto another and interpretations are given such that subspaces can carry on their base vector tuple real, complex or quaternionic numbers, energies, symmetries and generate coordinate lines. Describing states of physical systems is done using L and its applications for dynamical modelling. They don‘t need the infinte dimensional vectors of H. L has in the first model 11 blocks and 24 atoms (figure 1). They correspond to the 24 elements of the tetrahedral S4 symmetry. S4 arises from a spin-line rgb-graviton whirl operator with center at the tip of a tetrahedron and a nucleon triangle base with three quarks as vertices. The triangles factor group D3 of S4 is due to the CPT Klein normal subgroup Z2 x Z2 of S4 . It has a strong interaction SI rotor for the nucleons inner dynamics which is used for integrating functions, exchanging energies of nucleon with its environment and setting barycentrical coordinates in the triangle. At their intersection B as barycenter sets a Higgs boson or field the rescaled quark mass of a nucleon. Each factor class of one element from D3 assigns to it a color charge, a coordinate, an energy vector and a symmetry. Symmetries attached can be different according to interactions involved. Every atom of L has then a specific character with different properties.Three characters are added to octonian base vectors, listed by their indices as n = 0,1,…,7, and named for the atoms of L as na, nb, nc. The structure and element attributes of the finite subspace lattices L are desribed in many examples and models which technical constructed run macroscopically. Several models are described below. Example, the color charge whirl as rgb-graviton projection operator maps the block 2c3b5a6a to 0a1a2a3a. The symmetries change dimension from 3x3- to 2x2-matrices. From SU(3) are λ1 on 3b mapped to the SU(2) x-coordinate Pauli matrix σ1, from λ2 on 5a to σ2 y-coordinate and from λ3 on 6a to σ3 z-coordinate of real Euclidean space R³. The SU(3) matrices have complex w3 = z +ict, w2 = (iy,f), w1 = (x,m) coordinates. In figure 3 is shown how a rotation of two proton tetrahedrons for fusion changes the two linearly independent wj vectors to the 1-dimensional x,y,z base vectors. In deuteron then on one coordinate line sit with Cooper paire u-d-quarks at the ends the Heisenberg coupled energy or space vector rays 15 (x,m), m mass measured in kg, x in meter, 23 (iy,E(rot)), E(rot) rotational energy measured in Joule J, y in meter, 46 (ict,f), t time measured in seconds, f = 1/∆t frequency s inverse time interval measured in Hz. The six color charges are red r on +x as octonian coordinate 1, green g on +y as 2 , blue b on -z as 6, turquoise on -x as 5, magenta on -y as 3, yellow on +z as 4..
Some finite subspace models L are presented for quantum structures which replace the use of countable infinite Hilbert space H dimensions. A maximal Boolean sublattice, called block, is 24, where its four atoms directly above 0εL, base vectors of H in 24 are drawn as four points on an interval. Blocks can overlap in one or two atoms. Different kinds of operators can map one block onto another and interpretations are given such that subspaces can carry on their base vector tuple real, complex or quaternionic numbers, energies, symmetries and generate coordinate lines. Describing states of physical systems is done using L and its applications for dynamical modelling. They don‘t need the infinte dimensional vectors of H. L has in the first model 11 blocks and 24 atoms (figure 1). They correspond to the 24 elements of the tetrahedral S4 symmetry. S4 arises from a spin-line rgb-graviton whirl operator with center at the tip of a tetrahedron and a nucleon triangle base with three quarks as vertices. The triangles factor group D3 of S4 is due to the CPT Klein normal subgroup Z2 x Z2 of S4 . It has a strong interaction SI rotor for the nucleons inner dynamics which is used for integrating functions, exchanging energies of nucleon with its environment and setting barycentrical coordinates in the triangle. At their intersection B as barycenter sets a Higgs boson or field the rescaled quark mass of a nucleon. Each factor class of one element from D3 assigns to it a color charge, a coordinate, an energy vector and a symmetry. Symmetries attached can be different according to interactions involved. Every atom of L has then a specific character with different properties.Three characters are added to octonian base vectors, listed by their indices as n = 0,1,…,7, and named for the atoms of L as na, nb, nc. The structure and element attributes of the finite subspace lattices L are desribed in many examples and models which technical constructed run macroscopically. Several models are described below. Example, the color charge whirl as rgb-graviton projection operator maps the block 2c3b5a6a to 0a1a2a3a. The symmetries change dimension from 3x3- to 2x2-matrices. From SU(3) are λ1 on 3b mapped to the SU(2) x-coordinate Pauli matrix σ1, from λ2 on 5a to σ2 y-coordinate and from λ3 on 6a to σ3 z-coordinate of real Euclidean space R³. The SU(3) matrices have complex w3 = z +ict, w2 = (iy,f), w1 = (x,m) coordinates. In figure 3 is shown how a rotation of two proton tetrahedrons for fusion changes the two linearly independent wj vectors to the 1-dimensional x,y,z base vectors. In deuteron then on one coordinate line sit with Cooper paire u-d-quarks at the ends the Heisenberg coupled energy or space vector rays 15 (x,m), m mass measured in kg, x in meter, 23 (iy,E(rot)), E(rot) rotational energy measured in Joule J, y in meter, 46 (ict,f), t time measured in seconds, f = 1/∆t frequency s inverse time interval measured in Hz. The six color charges are red r on +x as octonian coordinate 1, green g on +y as 2 , blue b on -z as 6, turquoise on -x as 5, magenta on -y as 3, yellow on +z as 4.
Topological color codes are among the stabilizer codes with remarkable properties from the quantum information perspective. In this paper, we construct a lattice, the so-called ruby lattice, with coordination number 4 governed by a two-body Hamiltonian. In a particular regime of coupling constants, in a strong coupling limit, degenerate perturbation theory implies that the low-energy spectrum of the model can be described by a many-body effective Hamiltonian, which encodes the color code as its ground state subspace. Ground state subspace corresponds to a vortex-free sector. The gauge symmetry Z2×Z2 of the color code could already be realized by identifying three distinct plaquette operators on the ruby lattice. All plaquette operators commute with each other and with the Hamiltonian being integrals of motion. Plaquettes are extended to closed strings or string-net structures. Non-contractible closed strings winding the space commute with Hamiltonian but not always with each other. This gives rise to exact topological degeneracy of the model. A connection to 2-colexes can be established via the coloring of the strings. We discuss it at the non-perturbative level. The particular structure of the two-body Hamiltonian provides a fruitful interpretation in terms of mapping onto bosons coupled to effective spins. We show that high-energy excitations of the model have fermionic statistics. They form three families of high-energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. The emergence of invisible charges is related to the string-net structure of the model. The emerging fermions are coupled to nontrivial gauge fields. We show that for particular 2-colexes, the fermions can see the background fluxes in the ground state. Also, we use the Jordan–Wigner transformation in order to test the integrability of the model via introducing Majorana fermions. The four-valent structure of the lattice prevents the fermionized Hamiltonian from being reduced to a quadratic form owing to interacting gauge fields. We also propose another construction for the two-body Hamiltonian based on the connection between color codes and cluster states. The corresponding two-body Hamiltonian encodes a cluster state defined on a bipartite lattice as its low-energy spectrum, and subsequent selective measurements give rise to the color code model. We discuss this latter approach along with the construction based on the ruby lattice.
Topological color codes are among the stabilizer codes with remarkable properties from the quantum information perspective. In this paper, we construct a lattice, the so-called ruby lattice, with coordination number 4 governed by a two-body Hamiltonian. In a particular regime of coupling constants, in a strong coupling limit, degenerate perturbation theory implies that the low-energy spectrum of the model can be described by a many-body effective Hamiltonian, which encodes the color code as its ground state subspace. Ground state subspace corresponds to a vortex-free sector. The gauge symmetry Z 2 × Z 2 of the color code could already be realized by identifying three distinct plaquette operators on the ruby lattice. All plaquette operators commute with each other and with the Hamiltonian being integrals of motion. Plaquettes are extended to closed strings or string-net structures. Non-contractible closed strings winding the space commute with Hamiltonian but not always with each other. This gives rise to exact topological degeneracy of the model. A connection to 2-colexes can be established via the coloring of the strings. We discuss it at the non-perturbative level. The particular structure of the two-body Hamiltonian provides a fruitful interpretation in terms of mapping onto bosons coupled to effective spins. We show that high-energy excitations of the model have fermionic statistics. They form three families of high-energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. The emergence of invisible charges is related to the string-net structure of the model. The emerging fermions are coupled to nontrivial gauge fields. We show that for particular 2-colexes, the fermions can see the background fluxes in the ground state. Also, we use the Jordan–Wigner transformation in order to test the integrability of the model via introducing Majorana fermions. The four-valent structure of the lattice prevents the fermionized Hamiltonian from being reduced to a quadratic form owing to interacting gauge fields. We also propose another construction for the two-body Hamiltonian based on the connection between color codes and cluster states. The corresponding two-body Hamiltonian encodes a cluster state defined on a bipartite lattice as its low-energy spectrum, and subsequent selective measurements give rise to the color code model. We discuss this latter approach along with the construction based on the ruby lattice.
The postulate in the Copenhagen interpretation is for quantum measures that in experiments from a set of operators only a subset of commuting operators can be measured simultaneously, the rest remains undetermined. As example the Stern-Gerlach measure shows for spin a 90 degree change and measures for instance spin up and down of the outcome particles only in the z-direction. The spacial xy-coordinates remain undetermined.Using octonion coordinates [1,2] for the quantum range this view can be applied to all octonion base GF triples which use the three noncommuting Pauli matrices. The coordinates are enumerated by indices 0,1,2,…,7. Entanglements of different GF are possible. Two examples are the spin-lepton cases 123 (for xyz) and 145 (1 for electrical EM charge, 4 for magnetism, 5 for leptonic mass) either in the gyromagnetic relation (EM) or the helicty (neutral leptons).A in the second case the Copenhagen interpretation is extended to a quantum interpretation for measurements of the GF. The systems and energies involved can be different. The neutrino N oscillations show that also the Heisenberg uncertainties HU play a role. Spin is aligned with the space coordinate 1 and the momentum p = mv on 6. The HU means that 1,6 cannot be measured sharp. Hence the leptonic kg measure 5 changes for p along the world line of N stochasticaly, observed as oscillation. The kg GF is 257 and has 6 possible values for leptons. The weights of the three GF coordinates are mostly positive real or complex numbers. There are seven octonion GF 123, 145, 167 (for electromagnetic interaction EMI), 246 (for heat, acoustics), 257, 347 (for rotational energy), 356 (for a nucleon inner dynamics). Beside these are for the strong interaction SI three more GF 126 as rgb-graviton , 345 for its dual Dg and 037 for a newly postulated color charge force cc.The new cc force has a different symmetry than the Pauli quaternions.The six complex cross ratios invariant under the Moebius transformations of the Riemannian sphere are discussed in relation to quantum measures.
By considering the one-parameter family of inequivalent typical four-dimensional irreps of U-q[gl(2 1)], we may construct the anisotropic supersymmetric U model. This is a model for correlated electrons acting on the unrestricted 4(L)-dimensional electronic Hilbert space x(n=1)(L)C(4), where L is the lattice length. An important characteristic of the model is that it contains two symmetry-preserving free real parameters. Eigenstates of the model are determined by the algebraic Bethe ansatz on a one-dimensional periodic lattice.
The author establishes relationships between Bose, Fermi and spin-1/2 systems on lattices of arbitrary dimension and arbitrary but finite size which are defined to live in the (common) state space of the boson system. The underlying systems display an essential affinity by being described by precisely the same form of Hamiltonians, which are the polynomial functions of canonical operator variables: boson, fermion or spin-1/2 variables, respectively. For large enough, densely populated finite volume lattices it entails the computation of partition and correlation functions for lattice fermions in terms of bosons only. Albeit an approximation, its accuracy increases as the continuum limit is approached.