Cohl Furey - Academia.edu (original) (raw)

Papers by Cohl Furey

Research paper thumbnail of Charge quantization from a number operator

Physics Letters B

We explain how an unexpected algebraic structure, the division algebras, can be seen to underlie ... more We explain how an unexpected algebraic structure, the division algebras, can be seen to underlie a generation of quarks and leptons. From this new vantage point, electrons and quarks are simply excitations from the neutrino, which formally plays the role of a vacuum state. Using the ladder operators which exist within the system, we build a number operator in the usual way. It turns out that this number operator, divided by 3, mirrors the behaviour of electric charge. As a result, we see that electric charge is quantized because number operators can only take on integer values. Finally, we show that a simple hermitian form, built from these ladder operators, results uniquely in the nine generators of SU (3)c and U (1)em. This gives a direct route to the two unbroken gauge symmetries of the standard model.

Research paper thumbnail of Towards a unified theory of ideals

Phys.Rev.D

Unified field theories act to merge the internal symmetries of the standard model into a single g... more Unified field theories act to merge the internal symmetries of the standard model into a single group. Here we lay out something different. That is, instead of aiming to unify the internal symmetries, we demonstrate a sense in which the group transformations may be unified with the quarks and leptons that they act on. Similarly, the (3+1) Lorentz transformations may be united with the scalars, spinors, four-vectors and field strength tensors that they act on. These simplifications occur because the representations can be found in the form of an algebra acting on itself. The approach described in this paper is meant to tie everything into the Dixon algebra: R ⊗ C ⊗ H ⊗ O, the tensor product of the only four normed division algebras over R. Here we demonstrate that the standard model's Lorentz representations may be cast as a special set of generalized ideals within the algebra C ⊗ H. We then make an early attempt at extending this idea to one generation of quarks and leptons.

Research paper thumbnail of Standard model physics from an algebra

PhD Thesis, University of Waterloo

This thesis constitutes a first attempt to derive aspects of standard model particle physics from... more This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra. Here, we argue that physical concepts such as particles, causality, and irreversible time may result from the algebra acting on itself. We then focus on a special case by considering the algebra ℝ⊗ℂ⊗ℍ⊗𝕆. Using nothing more than ℝ⊗ℂ⊗ℍ⊗𝕆 acting on itself, we set out to find standard model particle representations.
From the complex quaternionic portion of the algebra, we find generalized ideals, and show that they describe concisely each of the Lorentz representations of the standard model. From the complex octonionic portion of the algebra, we find minimal left ideals, and show that they mirror the behaviour of a generation of quarks and leptons under su(3)_c and u(1)_em. We then demonstrate a rudimentary electroweak model which yields a straightforward explanation as to why SU(2)_L acts only on left-handed states. This holds in the case of leptons.

Finally, we demonstrate how ℂ⊗𝕆 can generate a 64-ℂ-dimensional algebra, wherein we find the SU(3)_c irreducible representations corresponding to three generations of quarks and leptons. We then conclude by showing how to arrive at all 48 electric charges.

Research paper thumbnail of Wormhole throats in R^m gravity

Class.Quant.Grav.

We consider wormhole geometries subject to a gravitational action consisting of non-linear powers... more We consider wormhole geometries subject to a gravitational action consisting of non-linear powers of the Ricci scalar. Specifically, wormhole throats are studied in the case where Einstein gravity is supplemented with a Ricci-squared and inverse Ricci term. In this modified theory it is found that static wormhole throats respecting the weak energy condition can exist. The analysis is done locally in the vicinity of the throat, which eliminates certain restrictions on the models introduced by considering the global topology.

Research paper thumbnail of SU(3) C × SU(2) L × U(1) Y (×U(1) X ) as a symmetry of division algebraic ladder operators

European Journal of Physics C

We demonstrate a model which captures certain attractive features of SU(5) theory, while providin... more We demonstrate a model which captures certain attractive features of SU(5) theory, while providing a possible escape from proton decay. In this paper we show how ladder operators arise from the division algebras R, C, H, and O. From the SU(n) symmetry of these ladder operators, we then demonstrate a model which has much structural similarity to Georgi and Glashow's SU(5) grand unified theory. However, in this case, the transitions leading to proton decay are expected to be blocked, given that they coincide with presumably forbidden transformations which would incorrectly mix distinct algebraic actions. As a result, we find that we are left with G_sm = SU(3)_C × SU(2)_L × U(1)_Y /Z_6. Finally, we point out that if U(n) ladder symmetries are used in place of SU(n), it may then be possible to find this same G_sm = SU(3)_C × SU(2)_L × U(1)_Y /Z_6 , together with an extra U(1)_X symmetry, related to B − L.

Research paper thumbnail of Generations: Three Prints, in Colour

JHEP

We point out a somewhat mysterious appearance of SU_c(3) representations, which exhibit the behav... more We point out a somewhat mysterious appearance of SU_c(3) representations, which exhibit the behaviour of three full generations of standard model particles. These representations are found in the Clifford algebra Cl(6), arising from the complex octonions. In this paper, we explain how this 64-complex-dimensional space comes about. With the algebra in place, we then identify generators of SU (3) within it. These SU(3) generators then act to partition the remaining part of the 64-dimensional Clifford algebra into six triplets, six singlets, and their antiparticles. That is, the algebra mirrors the chromodynamic structure of exactly three generations of the standard model's fermions. Passing from particle to antiparticle, or vice versa, requires nothing more than effecting the complex conjugate, * : i → −i. The entire result is achieved using only the eight-dimensional complex octonions as a single ingredient.

Research paper thumbnail of Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra

Physics Letters B

A considerable amount of the standard model's three-generation structure can be realised from jus... more A considerable amount of the standard model's three-generation structure can be realised from just the 8C-dimensional algebra of the complex octonions. Indeed, it is a little-known fact that the complex octonions can generate on their own a 64C-dimensional space. Here we identify an su(3) ⊕ u(1) action which splits this 64C-dimensional space into complexified generators of SU(3), together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries. This article builds on a previous one, [1], by incorporating electric charge.

Finally, we close this discussion by outlining a proposal for how the standard model's full set of states might be identified within the left action maps of R ⊗ C ⊗ H ⊗ O (the Clifford algebra Cl(8)). Our aim is to include not only the standard model's three generations of quarks and leptons, but also its gauge bosons.

Research paper thumbnail of Charge quantization from a number operator

Physics Letters B

We explain how an unexpected algebraic structure, the division algebras, can be seen to underlie ... more We explain how an unexpected algebraic structure, the division algebras, can be seen to underlie a generation of quarks and leptons. From this new vantage point, electrons and quarks are simply excitations from the neutrino, which formally plays the role of a vacuum state. Using the ladder operators which exist within the system, we build a number operator in the usual way. It turns out that this number operator, divided by 3, mirrors the behaviour of electric charge. As a result, we see that electric charge is quantized because number operators can only take on integer values. Finally, we show that a simple hermitian form, built from these ladder operators, results uniquely in the nine generators of SU (3)c and U (1)em. This gives a direct route to the two unbroken gauge symmetries of the standard model.

Research paper thumbnail of Towards a unified theory of ideals

Phys.Rev.D

Unified field theories act to merge the internal symmetries of the standard model into a single g... more Unified field theories act to merge the internal symmetries of the standard model into a single group. Here we lay out something different. That is, instead of aiming to unify the internal symmetries, we demonstrate a sense in which the group transformations may be unified with the quarks and leptons that they act on. Similarly, the (3+1) Lorentz transformations may be united with the scalars, spinors, four-vectors and field strength tensors that they act on. These simplifications occur because the representations can be found in the form of an algebra acting on itself. The approach described in this paper is meant to tie everything into the Dixon algebra: R ⊗ C ⊗ H ⊗ O, the tensor product of the only four normed division algebras over R. Here we demonstrate that the standard model's Lorentz representations may be cast as a special set of generalized ideals within the algebra C ⊗ H. We then make an early attempt at extending this idea to one generation of quarks and leptons.

Research paper thumbnail of Standard model physics from an algebra

PhD Thesis, University of Waterloo

This thesis constitutes a first attempt to derive aspects of standard model particle physics from... more This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra. Here, we argue that physical concepts such as particles, causality, and irreversible time may result from the algebra acting on itself. We then focus on a special case by considering the algebra ℝ⊗ℂ⊗ℍ⊗𝕆. Using nothing more than ℝ⊗ℂ⊗ℍ⊗𝕆 acting on itself, we set out to find standard model particle representations.
From the complex quaternionic portion of the algebra, we find generalized ideals, and show that they describe concisely each of the Lorentz representations of the standard model. From the complex octonionic portion of the algebra, we find minimal left ideals, and show that they mirror the behaviour of a generation of quarks and leptons under su(3)_c and u(1)_em. We then demonstrate a rudimentary electroweak model which yields a straightforward explanation as to why SU(2)_L acts only on left-handed states. This holds in the case of leptons.

Finally, we demonstrate how ℂ⊗𝕆 can generate a 64-ℂ-dimensional algebra, wherein we find the SU(3)_c irreducible representations corresponding to three generations of quarks and leptons. We then conclude by showing how to arrive at all 48 electric charges.

Research paper thumbnail of Wormhole throats in R^m gravity

Class.Quant.Grav.

We consider wormhole geometries subject to a gravitational action consisting of non-linear powers... more We consider wormhole geometries subject to a gravitational action consisting of non-linear powers of the Ricci scalar. Specifically, wormhole throats are studied in the case where Einstein gravity is supplemented with a Ricci-squared and inverse Ricci term. In this modified theory it is found that static wormhole throats respecting the weak energy condition can exist. The analysis is done locally in the vicinity of the throat, which eliminates certain restrictions on the models introduced by considering the global topology.

Research paper thumbnail of SU(3) C × SU(2) L × U(1) Y (×U(1) X ) as a symmetry of division algebraic ladder operators

European Journal of Physics C

We demonstrate a model which captures certain attractive features of SU(5) theory, while providin... more We demonstrate a model which captures certain attractive features of SU(5) theory, while providing a possible escape from proton decay. In this paper we show how ladder operators arise from the division algebras R, C, H, and O. From the SU(n) symmetry of these ladder operators, we then demonstrate a model which has much structural similarity to Georgi and Glashow's SU(5) grand unified theory. However, in this case, the transitions leading to proton decay are expected to be blocked, given that they coincide with presumably forbidden transformations which would incorrectly mix distinct algebraic actions. As a result, we find that we are left with G_sm = SU(3)_C × SU(2)_L × U(1)_Y /Z_6. Finally, we point out that if U(n) ladder symmetries are used in place of SU(n), it may then be possible to find this same G_sm = SU(3)_C × SU(2)_L × U(1)_Y /Z_6 , together with an extra U(1)_X symmetry, related to B − L.

Research paper thumbnail of Generations: Three Prints, in Colour

JHEP

We point out a somewhat mysterious appearance of SU_c(3) representations, which exhibit the behav... more We point out a somewhat mysterious appearance of SU_c(3) representations, which exhibit the behaviour of three full generations of standard model particles. These representations are found in the Clifford algebra Cl(6), arising from the complex octonions. In this paper, we explain how this 64-complex-dimensional space comes about. With the algebra in place, we then identify generators of SU (3) within it. These SU(3) generators then act to partition the remaining part of the 64-dimensional Clifford algebra into six triplets, six singlets, and their antiparticles. That is, the algebra mirrors the chromodynamic structure of exactly three generations of the standard model's fermions. Passing from particle to antiparticle, or vice versa, requires nothing more than effecting the complex conjugate, * : i → −i. The entire result is achieved using only the eight-dimensional complex octonions as a single ingredient.

Research paper thumbnail of Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra

Physics Letters B

A considerable amount of the standard model's three-generation structure can be realised from jus... more A considerable amount of the standard model's three-generation structure can be realised from just the 8C-dimensional algebra of the complex octonions. Indeed, it is a little-known fact that the complex octonions can generate on their own a 64C-dimensional space. Here we identify an su(3) ⊕ u(1) action which splits this 64C-dimensional space into complexified generators of SU(3), together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries. This article builds on a previous one, [1], by incorporating electric charge.

Finally, we close this discussion by outlining a proposal for how the standard model's full set of states might be identified within the left action maps of R ⊗ C ⊗ H ⊗ O (the Clifford algebra Cl(8)). Our aim is to include not only the standard model's three generations of quarks and leptons, but also its gauge bosons.