Overview An Exceptionally Simple Theory of Everything

electrons , quarks, electric (q) , color (g) charges, make color-neutral protons (with total electric charge q=+1) , neutrons (with electric charge q=0), make atoms.



the pattern of weak isospin, t3, , weak hypercharge, yw, , color charge of known elementary particles, rotated weak mixing angle show electric charge, q, along vertical. neutral higgs field (gray square) breaks electroweak symmetry , interacts other particles give them mass.



the pattern of weak isospin, w, weaker isospin, w , strong g3 , g8, , baryon minus lepton, b, charges particles in so(10) model, rotated show embedding of georgi-glashow model , standard model, electric charge along vertical. in addition standard model particles, theory includes thirty colored x bosons, responsible proton decay, , 3 w , z bosons.



the pattern of weak isospin, w, weaker isospin, w , strong g3 , g8, , baryon minus lepton, b, charges particles in so(10) grand unified theory, rotated show embedding in e6.

the goal of e8 theory describe elementary particles , interactions, including gravitation, quantum excitations of single lie group geometry—specifically, excitations of noncompact quaternionic real form of largest simple exceptional lie group, e8. lie group, such one-dimensional circle, may understood smooth manifold fixed, highly symmetric geometry. larger lie groups, higher-dimensional manifolds, may imagined smooth surfaces composed of many circles (and hyperbolas) twisting around 1 another. @ each point in n-dimensional lie group there can n different orthogonal circles, tangent n different orthogonal directions in lie group, spanning n-dimensional lie algebra of lie group. lie group of rank r, 1 can choose @ r orthogonal circles not twist around each other, , form maximal torus within lie group, corresponding collection of r mutually-commuting lie algebra generators, spanning cartan subalgebra. each elementary particle state can thought of different orthogonal direction, having integral number of twists around each of r directions of chosen maximal torus. these r twist numbers (each multiplied scaling factor) r different kinds of elementary charge each particle has. mathematically, these charges eigenvalues of cartan subalgebra generators, , called roots or weights of representation.

in standard model of particle physics, each different kind of elementary particle has 4 different charges, corresponding twists along directions of four-dimensional maximal torus in twelve-dimensional standard model lie group, su(3)×su(2)×u(1). 2 strong “color” charges, g , g, correspond twists along directions in two-dimensional maximal torus of eight-dimensional su(3) lie group of strong interaction. weak isospin, t3 (or w), , weak hypercharge, yw (or y), correspond twists along directions in two-dimensional maximal torus of four-dimensional su(2)×u(1) lie group of electroweak interaction, w , y combining electric charge, q. whenever interaction occurs between elementary particles, 2 coming , becoming third, or 1 particle becoming two, each type of charge must conserved. example, red quark, having charges (g






=



1
2





{\displaystyle {}={\tfrac {1}{2}}}

, g






=



1

2


3








{\displaystyle {}={\tfrac {1}{2{\sqrt {3}}}}}

, w



=



1
2





{\displaystyle ={\tfrac {1}{2}}}

, y






=



1
3





{\displaystyle {}={\tfrac {1}{3}}}

) can interact weak boson, w, having charges (g = 0, g = 0, w = −1, y = 0), produce red down quark, having charges (g






=



1
2





{\displaystyle {}={\tfrac {1}{2}}}

, g






=



1

2


3








{\displaystyle {}={\tfrac {1}{2{\sqrt {3}}}}}

, w






=




−
1

2





{\displaystyle {}={\tfrac {-1}{2}}}

, y






=



1
3





{\displaystyle {}={\tfrac {1}{3}}}

). complete pattern of standard model particle charges in 4 dimensions may projected down 2 dimensions , plotted in charge diagram.

in grand unified theories (guts), 12-dimensional standard model lie group, su(3)×su(2)×u(1) (modded z6), considered subgroup of higher-dimensional lie group, such of 24-dimensional su(5) in georgi–glashow model or of 45-dimensional spin(10) in so(10) model (spin(10) being double cover of so(10), , having same lie algebra). since there different elementary particle each dimension of lie group, in addition 12 standard model gauge bosons there 12 x , y bosons in su(5) model , 18 more x bosons , 3 w , z bosons in spin(10). in spin(10) there five-dimensional maximal torus, , standard model hypercharge, y, combination of 2 new spin(10) charges: “weaker charge”, w , , baryon minus lepton number, b. in spin(10) model, 1 generation of 16 fermions (including left-handed electrons, neutrinos, 3 colors of quarks, 3 colors of down quarks, , anti-particles) lives neatly in 16-complex-dimensional spinor representation space of spin(10). combination of these 32 real fermions , 45 bosons, along u(1) lie group (corresponding peccei–quinn symmetry), constitute 78-dimensional real compact exceptional lie group, e6. (this unusual algebraic structure, reminiscent of supersymmetry, of gauge fields , spinors combined in simple lie group, characteristic of exceptional groups.)

as being in representation space of standard model or grand unified theory lie group, each physical fermion spinor under gravitational noncompact spin(1,3) lie group of rotations , boosts. six-dimensional lie group has two-dimensional maximal torus (technically hyperboloid) , 2 kinds of charge, spin, sz, , boost, st. dirac fermion (consisting of fermion , anti-fermion) has 8 real degrees of freedom corresponding real vs. imaginary parts, left or right chirality, , being spin or down. using lie group equivalence of spin(1,3) , sl(2,c), , chirality of standard model weak force fermion interactions, each fermion (and each anti-fermion) can described two-complex-dimensional left-chiral weyl spinor under gravitational sl(2,c). accounting or down spin each of 16 left-chiral fermions of 1 generation (or 15 fermions if neutrinos majorana), each fermion generation corresponds 64 (or 60) real degrees of freedom.

in gravigut unification, gravitational spin(1,3) , spin(10) gut lie groups combined (modded z2) parts of spin(11,3) lie group, acting on each generation of fermions in real 64-dimensional spinor representation. remaining parts of spin(11,3) include 4-dimensional spacetime frame , higgs field transforming 10 under spin(10). resulting gauge theory of gravity, higgs, , gauge bosons extension of macdowell-mansouri formalism higher dimensions. several physicists objected apparent violation of coleman-mandula theorem, states impossibility of mixing gravity , gauge fields in unified lie group on spacetime, given reasonable assumptions. proponents of gravigut unification , e8 theory claim coleman-mandula theorem not violated because assumptions not met.

in e8 theory, observed gravigut algebra of spin(11,3) acting on 1 generation of fermions in real positive-chiral 64-spinor, 64+, can part of 248-dimensional real quaternionic e8 lie algebra,

e8 = spin(12,4) + 128+

the strongest criticism of e8 theory, stated distler, garibaldi, , others, including lisi in original paper, given embedding of gravitational spin(1,3) in spin(12,4) subalgebra of e8, 128+ includes not 64+ of generation of fermions, 64− “anti-generation” of mirror fermions non-physical chirality. since not see mirror fermions in nature, distler , garibaldi consider disproof of e8 theory. lisi has voiced 2 responses criticism. first response these mirror fermions might exist , have large masses. second response, stated in original paper , in latest work, there not single embedding of gravitational spin(1,3) in e8, 3 embeddings related triality, respect 64− contains second generation of physical fermions, , third generation of fermions contained within spin(12,4).

the algebraic breakdown of 248-dimensional e8 lie algebra relevant e8 theory is

e8 = spin(4,4) + spin(8) + 8v ⊗ 8v + 8+ ⊗ 8+ + 8− ⊗ 8−

this decomposition, attributed bertram kostant, relies on triality isomorphism between eight-dimensional vectors, 8v, positive-chiral spinors, 8+, , negative-chiral spinors, 8−, relating division algebra of octonions. within decomposition, strong force su(3) embeds in spin(8), 3 triality-related gravitational spin(1,3)’s embed in spin(4,4), 3 generations of 60 fermions embed in 8v ⊗ 8v + 8+ ⊗ 8+ + 8− ⊗ 8−, , gravitational frame, higgs, , electroweak bosons embed throughout, 18 colored x bosons remaining new predicted particles.

in e8 theory’s current state, not possible calculate masses existing or predicted particles. lisi states theory young , incomplete, requiring better understanding of 3 fermion generations , masses, , places low confidence in predictions. however, discovery of new particles not fit in lisi s classification, such superpartners or new fermions, fall outside model , falsify theory.