D5 / D4 x U(1) Symmetric Space - SpaceTime plus Internal Symmetry Space
E6 / D5 x U(1) Symmetric Space - Fermion Particles and AntiParticles
The MacroSpace of Many-Worlds is related to Jordan-like Algebras and
Physics models, including the D4-D5-E6-E7-E8 VoDou Physics model, have two parts:
In the case of the D4-D5-E6-E7-E8 VoDou Physics model, the basic ingredients of M are:
The structure of Q is determined by how those M states (Worlds of the Many-Worlds) fit together.
The structure of Q has both Geometric and Algebraic Aspects:
The Geometric Structure of Q comes from a chain of Lie Algebras whose groups are the related to Automorphism Groups of Jordan-like Algebras describing the Algebraic Structure of Q:
Automorphism Groups ---- of -------------------- Jordan-like Algebras D4 ---------------- Gauge Bosons --------- Chevalley Algebra Chev3(O) D5 ------ SpaceTime + Internal Symmetry -- Freudenthal Algebra Fr2(O) E6 ---------------- Fermions ------------- Freudenthal Algebra Fr3(O) E7 --------------- Many-Worlds ----------------- Brown Algebra Br3(O) ----------------------- E8 ------------------------------------------
248-dim E8 is made up of 133-dim E7 plus 112-dim Br3(O) plus Quaternionic SU(2).
The lowest dimensional non-trivial representation of E8 is its 248-dim Adjoint representation, and the algebra E8 corresponds to its E8 group as Automorphism Group. Therefore
E8 is Self-Automorphic, and the sequence ends with E8,
which contains both of the
Geometric and Algebraic descriptions of the MacroSpace of Many-Worlds
McKay Correspondence
Lie Algebra Finite Group Interpretation
28-dim D4 8-dim Binary Dihedral Group 8 D4 vectors
4 D4 Cartan Subalgebra
16 D4 spinors
45-dim D5 12-dim Binary Dihedral Group 12 D3 root vectors
5 D5 Cartan Subalgebra
28 D4 generators
78-dim E6 24-dim Binary Tetrahedral Group 24 D4 root vectors
6 E6 Cartan Subalgebra
48 F4 root vectors
133-dim E7 48-dim Binary Octahedral Group 48 F4 root vectors
7 E7 Cartan Subalgebra
78 E6 generators
248-dim E8 120-dim Binary Icosahedral Group 120 +E8 root vectors
8 E8 Cartan Torus
120 -E8 root vectors
E8, the final Lie Algebra in the chain, is not only Self-Automorphic, and reflexive and self-referential with respect to the McKay Correspondence, but is also reflexive and self-referential with respect to Octonion Lattices
In the D4-D5-E6-E7-E8 VoDou Physics model, at high energies prior to dimensional reduction:
D5 / D4 x U(1) Symmetric Space - SpaceTime plus Internal Symmetry Space
E6 / D5 x U(1) Symmetric Space - Fermion Particles and AntiParticles
Since the Gauge Bosons can be represented as pairs of Fermion nearest-neighbors, the Histories can be represented by location on 8-dimensional SpaceTime plus Internal Symmetry Space of 8 types of Fermion Particles and 8 types of Fermion AntiParticles, so that MacroSpace must have at least the dimensionality of an 8 + 8 + 8 = 24-dimensional Tangent Space.
Since the full dimensionality of SpaceTime plus Internal Symmetry Space and the representation spaces of Fermion Particles and AntiParticles are Complex, the MacroSpace must have at least 24 Complex dimensions.The geometric structure of MacroSpace is the 27-Complex-dimensional Symmetric Space
MacroSpace has Conformal Light-Cone Structure, and a 26-dimensional sub-space that corresponds to Bosonic String Theory.
The last Lie algebra in the E series, 248-dim E8, corresponds to its own 248-dim Automorphism Group. By the fibration E8 / E7 x SU(2), E8 is made up of:
In hep-th/0008063, Murat Gunaydin describes "... a quasiconformal nonlinear realization of E8 on a space of 57 dimensions. This space may be viewed as the quotient of E8 by its maximal parabolic subgroup; there is no Jordan algebra directly associated with it, but it can be related to a certain Freudenthal triple system which itself is associated with the "split" exceptional Jordan algebra J3(OS) where OS denote the split real form of the octonions O .It furthermore admits an E7 invariant norm form N4 , which gets multiplied by a (coordinate dependent) factor under the nonlinearly realized "special conformal" transformations. Therefore the light cone, defined by the condition N4 = 0, is actually invariant under the full E8, which thus plays the role of a generalized conformal group. ... results are based on the following five graded decomposition of E8 with respect to its E7 x D subgroup ... with the one-dimensional group D consisting of dilatations ...
g(-2) g(-1) g(0) g(1) g(2) 1 56 133+1 56 1
... D itself is part of an SL(2; R ) group, and the above decomposition thus corresponds to the decomposition ... of E8 under its subgroup E7 x SL(2;R) ...".
Note that the MacroSpace Q of Many-Worlds is 27-complex-dimensional, and that the 27-dim of Q are related to the 27-dim of the Jordan algebra J3(O) and that J3(O) has a 26-dim traceless subalgebra J3(O)o and that world-lines in Q (lines of states that could form a world-line succession of states) look sort of like bosonic strings in 26-dim J3(O)o. Then use techniques of bosonic string theory to construct a concrete model of a Bohmian landscape, thus unifying Bohm theory and Deutsch's Many-Worlds.
The Algebraic Structure of MacroSpace Q comes from a chain of Jordan-like Algebras:
The Automorphism Group of the D4 Lie Algebra of Gauge Bosons is S3 x Spin(8), where S3, the permutation group on 3 elements, corresponds to the Triality Outer Automorphism. Since Spin(8) is a Lie Group of type D4, you can also say that the Automorphism Group of D4 is S3 x D4. If you look only at the Inner Automorphisms, you get the Derivation Group of D4, which is D4 itself.
D4 is the Derivation Group of the subalgebra of the 27-dimensional Jordan algebra of 3x3 Hermitian Octonionic matrices that leaves invariant the primitive idempotents (real diagonal elements), that is, the 24-dimensional Chevalley Algebra Chev3(O) of 3x3 matrices of the form:
0 S+ V S+* 0 S- V* S-* 0
where S+, V, and S- are Octonions; and * denotes conjugation.
Note that the full 27-dimensional Jordan Algebra J3(O) has Automorphism Group F4.
The 24-dimensional Chevalley Algebra Chev3(O) is described by Jorg Schray and Corinne Manogue in Chapter VI (An Octonionic Description of the Chevalley Algebra and Triality) of their paper Octonionic Representations of Clifford Algebras and Triality, hep-th/9407179.
The Automorphism Group of the D5 Lie Algebra is S2 x D5, where S2, the permutation group on 2 elements, corresponds to the Outer Automorphism. If you look only at the Inner Automorphisms, you get the Derivation Group of D5, which is D5 itself.
D5 of the D5 Lie Algebra of D5 / D4 x U(1) SpaceTime plus Internal Symmetry Space is the Derivation Group of the 22-dimensional Freudenthal Algebra Fr2(O) of 2x2 vector-matrices
a X Y b
where a and b are real numbers and X and Y are elements of the 10-dimensional Jordan Algebra J2(O) of 2x2 Hermitian Octonionic matrices
d V V* e
where d and are real numbers; V is Octonion; and * denotes conjugation.
Fr2(O) is the Complexification of J2(O), so that the Vector SpaceTime plus Internal Symmetry Representation Space has 8 Complex Dimensions and a corresponding Bounded Complex Domain with 8-real-dimensional Shilov Boundary S7 x RP1.
Restriction to the real J2(O) would have produced an Automorphism Group B4 and a real 8-dimensional Spinor space corresponding to OP1 = B4 / D4 = S8.
Derivations of Fr2(O) and J2(O) are described by A. Sudbery in his paper Division Algebras, (Pseudo) Orthogonal Groups, and Spinors (J. Phys. A: Math. Gen. 17 (1984) 939-955).
The Lie Group E6 of the E6 Lie Algebra of E6 / D5 x U(1) Fermion Particles and AntiParticles is the Automorphism Group of the 56-dimensional Freudenthal Algebra Fr3(O) of 2x2 Zorn-type vector-matrices
a X Y b
where a and b are real numbers and X and Y are elements of the 27-dimensional Jordan Algebra J3(O) of 3x3 Hermitian Octonionic matrices
d S+ V S+* e S- V* S-* f
where d, e, and f are real numbers; S+, V, and S- are Octonions; and * denotes conjugation.
Fr3(O) includes a complexification of J3(O), so that each Half-Spinor Fermion Representation Space has 8 Complex Dimensions and a corresponding Bounded Complex Domain with 8-real-dimensional Shilov Boundary S7 x RP1.
Restriction to the real J3(O) would have produced an Automorphism Group F4 and a real 16-dimensional Spinor space corresponding to OP2 = F4 / B4.
Complex Structure is useful because the math of Hermitian symmetric spaces and bounded complex domains can be used in calculations of force strength constants and particle masses. (Armand Wyler was the first to do this, as far as I know, but he did not do it entirely correctly and his physical interpretations (being pre-standard model) were not very clear or convincing.)
Further, the Freudenthal algebra Fr3(O) structure has advantages over such things as simple tensor products of Octonions. For example, in Freudenthal algebras X(XX) = (XX)X which is not true for the tensor product OxO of octonions.
The Freudenthal Algebra Fr3(O) is described by R. Skip Garibaldi in his papers Structurable Algebras and Groups of Types E6 and E7, math.RA/9811035 and Groups of Type E7 Over Arbitrary Fields, math.AG/9811056.Also, see the books
- The Book of Involutions, by M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol (with a preface by J. Tits), American Mathematical Society Colloquium Publications, vol. 44 (American Mathematical Society,Providence, RI, 1998);
- Einstein Manifolds, by Arthur L. Besse (a pseudonym for a group of French mathematicians) (Springer-Verlag 1987);
- Geometry of Lie Groups, by Boris Rosenfeld (Kluwer 1997); and
- On the Role of Division, Jordan and Related Algebras in Particle Physics, by Feza Gursey and Chia-Hsiung Tze (World 1996).
The 26 dimensions of E6 / F4 correspond to a 26-dimensional bosonic String Theory that could represent the Bohm Potential of the Many-Worlds Quantum Theory.
The Lie Group E7 of the E7 Lie Algebra of E7 / E6 x U(1) is the Automorphism Group of the 112-dimensional Brown Algebra Br3(O).
Br3(O) is a complexification of the 56-dimensional Fr3(O), and so includes a complexification of a complexification (effectively a quaternization) of J3(O):
d S+ V S+* e S- V* S-* f
where d, e, and f are Quaternions; S+, V, and S- are Quaternified Octonions QxO; and * denotes Octonion conjugation, for 4x27 = 108 of its 112 real dimensions. The other 4 dimensions come from complexification of the 2 real entries in the 2x2 Zorn-type matrices
a X Y b
of Fr3(O). John Baez, in his week 193, says, about the above construction of Fr3(O), "... I suspect they're "cheating" a bit and identifying h_3(O) with its dual. ...", and in my opinion he has a point.
Each Half-Spinor Fermion Representation Space S+ and S- has 8 Complex Dimensions and a corresponding Bounded Complex Domain with 8-real-dimensional Shilov Boundary S7 x RP1.
Note that, unlike Fr3(O), Br3(O) is not a binary algebra, but is a ternary algebra.
The Brown Algebra is described by R. Skip Garibaldi in his papers Structurable Algebras and Groups of Types E6 and E7, math.RA/9811035 and Groups of Type E7 Over Arbitrary Fields, math.AG/9811056.Also, see the books
- The Book of Involutions, by M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol (with a preface by J. Tits), American Mathematical Society Colloquium Publications, vol. 44 (American Mathematical Society,Providence, RI, 1998);
- Einstein Manifolds, by Arthur L. Besse (a pseudonym for a group of French mathematicians) (Springer-Verlag 1987);
- Geometry of Lie Groups, by Boris Rosenfeld (Kluwer 1997); and
- On the Role of Division, Jordan and Related Algebras in Particle Physics, by Feza Gursey and Chia-Hsiung Tze (World 1996).
The 54 = 2x27 dimensions of E7 / E6 x U(1) correspond to a complexification of a 27-dimensional bosonic string M-theory that could represent Timelike Brane Universes.
Some TERNARY ALGEBRAIC STRUCTURES AND THEIR APPLICATIONS IN PHYSICS are described by Richard Kerner in math-ph/0011023. As he says: "... We discuss certain ternary algebraic structures appearing more or less naturally in various domains of theoretical and mathematical physics. Far from being exhaustive, this article is intended above all to draw attention to these algebras ...".
E8 is 248-dimensional, and is made up of:
Therefore,
and the series ends with E8, which is not only Self-Automorphic, but is also substantially reflexive and self-referential with respect to Octonion Lattices and with respect to the McKay Correspondence.
The 112 = 4x28 dimensions of E8 / E7 x SU(2) correspond to a quaternification of a 28-dimensional bosonic string F-theory that could represent Spacelike Brane Universes.
Click here to read about a 7-grading structure for E8 noticed by Thomas Larsson on reading a paper by Peter West, which structure has Vector, Lie, Freudenthal (Jordan), and Clifford parts.
As Kuratowski showed in 1930, there are Graphs ( those that contain no subgraphs contractible to the pentagonal graph K5 or the hexagonal graph K(3,3) ) that cannot be represented in a 2-dimensional plane without edges crossing. A 3-dimensional space is necessary for all Graphs to be embedded without edges crossing.
Since the Sum-Over-Histories contains Histories with complicated non-planar Graph structure, MacroSpace Q should be represented by a space of at least 3 dimensions greater than the minimal 8+8+8 = 24 dimensions needed to describe SpaceTime plus Internal Symmetry Space and Fermion Particles and AntiParticles (Gauge Bosons being implicitly described in terms of Fermion Particle-AntiParticle Pairs and Internal Symmetry Space), which condition is fulfilled by Q = E7 / E6 x U(1) with 27 Complex dimensions.
However, the Topological Poincare Conjecture is unknown in dimension 3, so that you must go to dimension 4 to know that the Topological Poincare Conjecture is true.
Therefore,
To see that the Sum-Over-Histories does contain complicated non-planar Graph structure, consider its description by Richard Feynman, in QED (Princeton University Press, 1985, 1988):
and what John and Mary Gribbin say in their biography Richard Feynman (Penguin 1997):
At the first level ( order 1 ), Sum-Over-Histories means considering all paths that look like lines:
--------------
However, as Feynman points out in QED, you also have to consider ".. an alternative way the electron can go from place to place: instead of going directly from one point to another, the electron goes along for a while and suddenly emits a photon; then ... it absorbs its own photon ...". These and other higher-order processes involve introducing loops into the paths.
The second level ( order 2 ) involves single loops:
__ / \ -----* *---- \__/
By nesting single loops, you can make loops of order 2^2 = 4 or any other power of 2, so you get all the orders 2^p for any p.
__ / \ --* *-- / \__/ \ / \ / \ -----* *----- \ / \ __ / \ / \ / --* *-- \__/
However, since 3 is not a power of 2, to get order-3 processes you need to introduce a new set of loops with pitchforks instead of binary bifurcations:
__ / \ -----*----*---- \__/
Now you can, by nesting, make loops of any order that is representable as 2^p 3^q.
By introducing loops whose orders are prime numbers, and including all the prime numbers, you can complete the Sum-Over-Histories with all the Histories.
Therefore:
as shown on a web page of mwatkins@maths.ex.ac.uk:
Further:
In hep-th/9909126, they say:
"... It has become increasingly clear ... that the nitty-gritty of the perturbative expansion in quantum field theory is hiding a beautiful underlying algebraic structure which does not meet the eye at first sight. As is well known most of the terms in the perturbative expansion are given by divergent integrals which require renormalization. ... the renormalization technique ....[has been]... shown to give rise to a Hopf algebra whose antipode S delivers the same terms as those involved in the subtraction procedure before the renormalization map R is applied. ... the group G associated to this Hopf algebra by the Milnor-Moore theorem was computed by exhibiting a basis and computing Lie brackets for its Lie algebra. It was shown that the collection of all bare amplitudes indexed by Feynman diagrams in dimensionally regularized perturbative quantum field theory is just a point ° in the group GK, where K = C[z^(-1), [z]] is the field of Laurent series. Though this made it clear that the Hopf algebra and its antipode are providing the correct framework to understand renormalization, some of the mystery was still around because of the somewhat ad hoc manner, in which the antipode S had to be twisted by the renormalization map R in order to fully account for the physical computations. ... We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann-Hilbert problem. Given a loop y(z), |z| = 1 of elements of a complex Lie group G the general procedure is given by evaluation of y+(z) at z = 0 after performing the Birkhoff decomposition y(z) = y-(z)^(-1) y+(z) where y+/-(z) in G are loops holomorphic in the inner and outer domains of the Riemann sphere (with y-(infinity) = 1). We show that, using dimensional regularization, the bare data in quantum field theory delivers a loop (where z is now the deviation from 4 of the complex dimension) of elements of the decorated Butcher group (obtained using the Milnor-Moore theorem from the Kreimer Hopf algebra of renormalization) and that the above general procedure delivers the renormalized physical theory in the minimal substraction scheme. ...".
In hep-th/0201157, they say:
"... The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine. ... The algebraic structure of perturbative QFT gives rise to commutative Hopf algebras H and corresponding Lie-algebras L, with H being the dual of the universal enveloping algebra of L. L can be represented by derivations of H, and two representations are most natural in this respect: elimination or insertion of subgraphs. Perturbation theory is indeed governed by a series over one-particle irreducible graphs. It is then a straightforward question how the basic operations of inserting or eliminating subgraphs act. These are the basic operations which are needed to construct the formal series over graphs which solve the Dyson-Schwinger equations. We give an account of these actions here as a further tool in the mathematician's toolkit for a comprehensible description of QFT....".
......