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The Structure of theMacroSpace ofMany-Worlds

 has both Geometric andAlgebraic Aspects,

and Sum-Over-HistoriesPaths are related to PrimeNumbers.

Here is the relevance of the McKayCorrespondence.

The MacroSpace of Many-Worlds is related to Jordan-likeAlgebras and

Bosonic ClosedStrings

Hilbert Space, Octonions,and Cl(8) Clifford Algebra

von Neumann algebras


 What is MacroSpace and Where does it Come From?

Physics models, including the D4-D5-E6-E7-E8VoDou Physics model, have two parts:

 

In the case of the D4-D5-E6-E7-E8 VoDouPhysics model, the basic ingredients of M are:

The quantum theory MacroSpace Q is then made up of all thepossible M states.

The structure of Q is determined by how those M states (Worldsof the Many-Worlds) fit together.

The structure of Q  has both Geometricand 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 AlgebrasD4 ---------------- 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) plusQuaternionic SU(2).

The lowest dimensional non-trivial representation of E8 is its248-dim Adjoint representation, and the algebra E8 corresponds to itsE8 group as Automorphism Group. Therefore

E8 is Self-Automorphic, and thesequence ends with E8,

which contains both of the

Geometric and Algebraicdescriptions of the MacroSpace of Many-Worlds

 

 E8, the final LieAlgebra in the chain used in the D4-D5-E6-E7-E8VoDou Physics model, is not only Self-Automorphic but is alsoreflexive and self-referential with respect to theMcKay Correspondence:
    McKay CorrespondenceLie 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 thechain, is not only Self-Automorphic, and reflexive andself-referential with respect to the McKayCorrespondence, but is also reflexive and self-referential withrespect to OctonionLattices

 


Geometric Aspects:

In the D4-D5-E6-E7-E8 VoDou Physicsmodel, at high energies prior to dimensional reduction:

D4 Lie Algebra - GaugeBosons

D5 / D4 x U(1) Symmetric Space - SpaceTimeplus Internal Symmetry Space

E6 / D5 x U(1) Symmetric Space - FermionParticles and AntiParticles

Since the Gauge Bosons can berepresented as pairs of Fermion nearest-neighbors, the Historiescan be represented by location on 8-dimensional SpaceTime plusInternal Symmetry Space of 8 types of Fermion Particles and 8 typesof Fermion AntiParticles, so that MacroSpacemust have at least the dimensionality of an 8 + 8 + 8 =24-dimensional Tangent Space.

Since the full dimensionality of SpaceTime plus Internal SymmetrySpace and the representation spaces of Fermion Particles andAntiParticles are Complex, the MacroSpace must have at least 24Complex dimensions.The geometric structure of MacroSpace is the27-Complex-dimensional Symmetric Space

E7 / E6 x U(1) SymmetricSpace - MacroSpaceof Many-Worlds

MacroSpace has Conformal Light-ConeStructure, and a 26-dimensionalsub-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 fibrationE8 / E7 x SU(2), E8 is made up of:

 

Therefore, E8 carries in itself both the Geometricand Algebraic structures of the MacroSpace ofMany-Worlds, and the series ends with E8.

In hep-th/0008063,Murat Gunaydin describes "... a quasiconformal nonlinear realizationof E8 on a space of 57 dimensions. This space may be viewed as thequotient of E8 by its maximal parabolic subgroup; there is no Jordanalgebra directly associated with it, but it can be related to acertain Freudenthal triple system which itself is associated with the"split" exceptional Jordan algebra J3(OS) where OS denote the splitreal form of the octonions O .It furthermore admits an E7 invariantnorm form N4 , which gets multiplied by a (coordinate dependent)factor under the nonlinearly realized "special conformal"transformations. Therefore the light cone, defined by the conditionN4 = 0, is actually invariant under the full E8, which thus plays therole of a generalized conformal group. ... results are based on thefollowing five graded decomposition of E8 with respect to its E7 x Dsubgroup ... with the one-dimensional group D consisting ofdilatations ...

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 abovedecomposition thus corresponds to the decomposition ... of E8 underits 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 theJordan algebra J3(O)and that J3(O) has a 26-dim traceless subalgebra J3(O)oandthat world-lines in Q (lines of states that could form a world-linesuccession of states)look sort of like bosonic strings in 26-dim J3(O)o.Then use techniques of bosonic string theoryto construct a concrete model of a Bohmian landscape,thusunifying Bohm theory and Deutsch's Many-Worlds.

 


 

Algebraic Aspects:

The Algebraic Structure of MacroSpace Q comes from a chain ofJordan-like Algebras:

 Graded Lie Algebras


 

The Automorphism Group of the D4 Lie Algebra of GaugeBosons is S3 x Spin(8), where S3, the permutation group on 3elements, corresponds to the Triality Outer Automorphism. SinceSpin(8) is a Lie Group of type D4, you can also say that theAutomorphism Group of D4 is S3 x D4. If you look only at theInner Automorphisms, you get the Derivation Group of D4, which is D4itself.

D4 is the Derivation Group of the subalgebra of the27-dimensional Jordan algebra of 3x3Hermitian Octonionic matrices thatleaves invariant the primitiveidempotents (real diagonal elements), that is, the24-dimensional Chevalley Algebra Chev3(O) of 3x3 matrices of theform:

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 AlgebraJ3(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 OuterAutomorphism. If you look only at the Inner Automorphisms, you getthe Derivation Group of D5, which is D5 itself.

D5 of the D5 Lie Algebra of D5 / D4 x U(1) SpaceTimeplus Internal Symmetry Space is the Derivation Group of the22-dimensional Freudenthal Algebra Fr2(O) of 2x2vector-matrices

a XY b

where a and b are real numbers and X and Y are elements of the10-dimensional Jordan Algebra J2(O) of 2x2Hermitian Octonionic matrices

d V V* e

where d and are real numbers; V is Octonion; and * denotesconjugation.

Fr2(O) is the Complexification of J2(O), so that theVector SpaceTime plus Internal Symmetry Representation Space has 8Complex Dimensions and a corresponding Bounded Complex Domain with8-real-dimensional Shilov Boundary S7 x RP1.

Restriction to the real J2(O) would have producedan Automorphism Group B4 and a real 8-dimensional Spinor spacecorresponding 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 E6Lie Algebra of E6 / D5 x U(1) FermionParticles and AntiParticles is the Automorphism Group of the56-dimensional Freudenthal Algebra Fr3(O) of 2x2 Zorn-typevector-matrices

a XY b

where a and b are real numbers and X and Y are elements of the27-dimensional Jordan Algebra J3(O) of3x3 Hermitian Octonionicmatrices

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 thateach Half-Spinor Fermion Representation Space has 8 ComplexDimensions and a corresponding Bounded Complex Domain with8-real-dimensional Shilov Boundary S7 x RP1.

 

Restriction to the real J3(O) would have producedan Automorphism Group F4 and a real 16-dimensional Spinor spacecorresponding to OP2 = F4 / B4.

Complex Structure is useful because the math ofHermitian symmetric spaces and bounded complex domains can be used incalculations of force strength constants and particle masses. (ArmandWyler was the first to do this, as far as I know, but he did not doit entirely correctly and his physical interpretations (beingpre-standard model) were not very clear or convincing.)

Further, the Freudenthal algebra Fr3(O) structurehas advantages over such things as simple tensor products ofOctonions. For example, in Freudenthal algebras X(XX) = (XX)X whichis 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 / F4correspond to a 26-dimensionalbosonic String Theory that could represent theBohm Potential of the Many-Worlds Quantum Theory.

 


 

The Lie Group E7 of the E7Lie Algebra of E7 / E6 x U(1) is the Automorphism Group of the112-dimensional Brown Algebra Br3(O).

Br3(O) is a complexification of the 56-dimensional Fr3(O),and so includes a complexification of a complexification (effectivelya 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 QuaternifiedOctonions QxO; and * denotes Octonion conjugation, for 4x27 = 108 ofits 112 real dimensions. The other 4 dimensions come fromcomplexification of the 2 real entries in the 2x2 Zorn-typematrices

a XY b

of Fr3(O). John Baez, in his week193, says, about the above construction of Fr3(O), "... I suspectthey'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 8Complex Dimensions and a corresponding Bounded Complex Domain with8-real-dimensional Shilov Boundary S7 x RP1.

 

Note that, unlike Fr3(O), Br3(O) is not a binary algebra, butis 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-dimensionalbosonic string M-theory that could represent TimelikeBrane Universes.

 

Some TERNARY ALGEBRAIC STRUCTURES AND THEIR APPLICATIONS INPHYSICS are described by Richard Kerner in math-ph/0011023.As he says: "... We discuss certain ternary algebraic structuresappearing more or less naturally in various domains of theoreticaland mathematical physics. Far from being exhaustive, this article isintended above all to draw attention to these algebras ...".

 


 

The Lie Group E8 ofthe E8 Lie Algebra of E8 / E7 xSU(2) is the Automorphism Group of itself.

 

E8 is 248-dimensional, and ismade up of:

Therefore,

E8 carries in itself both the Geometricand Algebraic structures of theMacroSpace of Many-Worlds

and the series ends with E8,which is not only Self-Automorphic, but is also substantiallyreflexive and self-referential with respect to OctonionLattices and with respect to the McKayCorrespondence.

The 112 = 4x28 dimensions of E8 / E7x SU(2) correspond to a quaternification of a 28-dimensionalbosonic string F-theory that could represent SpacelikeBrane Universes.

 

Click here to read about a 7-gradingstructure for E8 noticed by Thomas Larsson on reading a paper byPeter West, which structure has Vector, Lie,Freudenthal (Jordan),and Clifford parts.

 


Graphs and PrimeNumbers

 

As Kuratowski showed in 1930, there are Graphs ( those thatcontain no subgraphs contractible to the pentagonal graph K5 or thehexagonal graph K(3,3) ) that cannot be represented in a2-dimensional plane without edges crossing. A 3-dimensional space isnecessary for all Graphs to be embedded without edges crossing.

Since the Sum-Over-Histories containsHistories with complicated non-planar Graph structure, MacroSpaceQ should be represented by a space of at least 3 dimensions greaterthan the minimal 8+8+8 = 24 dimensions neededto describe SpaceTime plus Internal Symmetry Space and FermionParticles and AntiParticles (Gauge Bosons being implicitly describedin terms of Fermion Particle-AntiParticle Pairs and Internal SymmetrySpace), which condition is fulfilled by Q = E7 / E6 x U(1) with 27Complex dimensions.

Since the Smooth PoincareConjecture is true in dimension 3, the 3+24 = 27-dimensionalMacroSpace representation Q = E7 / E6 x U(1) is useful for SmoothStructure.

However, the Topological PoincareConjecture is unknown in dimension 3, so that you must go todimension 4 to know that the TopologicalPoincare Conjecture is true.

Therefore,

to get a MacroSpace representation that is useful forTopological Structure, you need touse the 4+24 = 28-dimensional Symmetric Space E8 / E7xS3

 


To see that the Sum-Over-Histories does contain complicatednon-planar Graph structure, consider its description by RichardFeynman, in QED (Princeton University Press, 1985,1988):

and what John and Mary Gribbin say in theirbiography Richard Feynman (Penguin 1997):

"The insight Feynman had, while lying in bed one night,unable to sleep, was that you had to consider every possible way inwhich a particle could go from A to B - every possible 'history'. Theinteraction between A and B is conceived as involving a summade up of contributions from all of the possible paths thatconnect the two events.".

 

At the first level ( order 1 ), Sum-Over-Histories meansconsidering 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 electrongoes along for a while and suddenly emits a photon; then ... itabsorbs its own photon ...". These and other higher-order processesinvolve 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 orany 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 youneed to introduce a new set of loops with pitchforks instead ofbinary bifurcations:

  __ / \ -----*----*---- \__/

 

Now you can, by nesting, make loops of any order that isrepresentable as 2^p 3^q.

By introducing loops whose orders are prime numbers, and includingall the prime numbers, you can complete the Sum-Over-Histories withall the Histories.

Therefore:

Sum-Over-HistoriesQuantum Systems are related to PrimeNumbers and the zeroes of theRiemann Zeta Function

as shown on aweb page of mwatkins@maths.ex.ac.uk:

 

 

Further:

Since period-3 implies chaos, the need to include order-3loops sheds light on the relationship between BernoulliSchemes of Chaos Theory and Sum-Over-Histories QuantumSystems.

  

Alain Connes and Dirk Kreimer have written interestingpapers about the mathematics of Feynman diagrams, perturbationtheory, and renormalization.

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....".

 


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