an exceptionally simple theory of everything, by garrett lisi (48 p.) -

8/14/2019 An Exceptionally Simple Theory of Everything, by Garrett Lisi (48 p.) - WWW.OLOSCIENCE.COM

1/48

Periodic table of the standard model

An Exceptionally Simple Theory of Everything Garrett Lisi

http://arxiv.org/abs/0711.0770 FQXi
http://arxiv.org/abs/0711.0770


2/48

A connection with everything

!!

W!

B!

g!

=

=

=

=

! ! dxk! 2

1

k

"#"#

W $ dxk! k%i2 %

B idxk! k

g & dxk! k

A i2 A

2

222

(3; )Cl!

2 1

(2)su!

(1)u!

(3)su!

(e ) ! (3; )e!

= dxk! k

"" 2 Cl! 1 1

!

'

+' 0

" !#

eLe L

"Y

u ; ;[ r ug ub]

2

666664e ^

L

e

_Le ^

R

e _R

3

777775

l

A!"

= ! ' # )21

!+ 4

1 e!

+ W!

+ B!

+ g!

+ (" e

+ e"

+ u"

+ d"

+ # ) # )( " " + "" + c" + s" + ( " ( + (" + t" + b"

F A A; ="

= d!!"

+2

1#

!"A!"

$


3/48


4/48


5/48

Strong G2

G2 V g gg 1 0

g 0

g

g

gg

q [1; ; ]

q [0; ; ]

q [0; ; ] 0

q [1; ; ]q [0; ; ]

q [0; ; ] 0

*

3 8

%& rg! (T T )2 ! i 1%& rg! (!T T )2 ! i 1 !1

%& rb! (T T )5 ! i 4 21 2p

3

%& rb! (!T T )5 ! i 4 ! 21 ! 2p

3

%&g b!

(!T T )7 ! i 6 21

! 2

p3

%& gb! (T T )7 ! i 6 ! 21 2p

3

NMr 0 0 2

1 12p

3

NMg 1 0 ! 2

1 12p

3

NMb 0 1 ! 1p

3

HO!

r

0 0 ! 21

!

1

2p3HO !

g 1 0 21 ! 1

2p

3

HO !b 0 1 1p

3


6/48

Exceptional Lie brackets

The Lie algebra elements of the smallest exceptional Lie group, :

Structure of implies Lie bracket equivalent to fundamental action,

corresponding to the strong interactions, such as

14 G2

g2 = u(3)s

g!

+

+

3

q"

+

+

3!

q!" 2 g2!"G2

[g; ] g T ; T q =#

A A qB B$

= g q =

2

6664g g i2

3+ i2p

3

8

g r g!

g r b!

g rg!

g g ! i2 3+ i2p3 8g gb!

g rb!

g gb!

! g ip3

8

3

77752

4

q r

q gq b

3

5

g ; #

rg! qg$

= qr


7/48


8/48

G2 in SO(7) ..

G2+ x y z g g B

g 1 0 1 0 0g 1 0 0 0

g 0 1 0

g 1 0 0

g 0 1 0

g 0 1 0q

q

q 0

q

qq 0

q " "

q " "

l 0 0

3 8 p3

p8

2

%&rg!

!1%& rg! !1 !1%& rb! !1

21 2

p3

%& rb! !1 ! 21 ! 2p

3

%& g b! !121 ! 2

p3

%&gb!

!1 ! 21

2

p3

NMIr ! 2

121

21

21 1

2p

3! 3

1

NMI

g21 ! 2

121 ! 2

1 12p

3! 3

1

NM bI 2

121 ! 2

1 ! 1p3

! 31

HO !rI 2

1 ! 21 ! 2

1 ! 21 ! 1

2p

3 31

HO!

g

I!

21

2

1 !2

1

2

1 !

1

2p3 31

HO !bI

! 21 ! 2

121 1p

3 31

HONM II '1 ( 32

NMHO III (1 (1 ' 34

HONM ' 21 ' 2

1 ' 21 (1


9/48

G2 in SO(7) ...

G2+ x y z g g B

g 1 0 1 0 0g 1 0 0 0

g 0 1 0

g 1 0 0

g 0 1 0

g 0 1 0q

q

q 0

q

qq 0

q " "

q " "

l 0 0

3 8 p3

p8

2

%&rg!

!1%& rg! !1 !1%& rb! !1

21 2

p3

%& rb! !1 ! 21 ! 2p

3

%& g b! !121 ! 2

p3

%&gb!

!1 ! 21

2

p3

NMIr ! 2

121

21

21 1

2p

3! 3

1

NMI

g21 ! 2

121 ! 2

1 12p

3! 3

1

NM bI 2

121 ! 2

1 ! 1p3

! 31

HO !rI 2

1 ! 21 ! 2

1 ! 21 ! 1

2p

3 31

HO!

g

I!

21

2

1 !2

1

2

1 !

1

2p3 31

HO !bI

! 21 ! 2

121 1p

3 31

HONM II '1 ( 32

NMHO III (1 (1 ' 34

HONM ' 21 ' 2

1 ' 21 (1


10/48

G2 in SO(7) ....

G2+ x y z g g B

g 1 0 1 0 0g 1 0 0 0

g 0 1 0

g 1 0 0

g 0 1 0

g 0 1 0q

q

q 0

q

qq 0

q " "

q " "

l 0 0

3 8 p3

p8

2

%&rg!

!1%& rg! !1 !1%& rb! !1

21 2

p3

%& rb! !1 ! 21 ! 2p

3

%& g b! !121 ! 2

p3

%&gb!

!1 ! 21

2

p3

NMIr ! 2

121

21

21 1

2p

3! 3

1

NMI

g21 ! 2

121 ! 2

1 12p

3! 3

1

NM bI 2

121 ! 2

1 ! 1p3

! 31

HO !rI 2

1 ! 21 ! 2

1 ! 21 ! 1

2p

3 31

HO !

g

I!

2

1

2

1 ! 2

1

2

1 ! 12p3 3

1

HO !bI

! 21 ! 2

121 1p

3 31

HONM II '1 ( 32

NMHO III (1 (1 ' 34

HONM ' 21 ' 2

1 ' 21 (1


11/48


12/48

Pati-Salam model plus gravity

SO(3; ) U(2) U(2) U(1) U(3)

)1 + 4 ) 4 + S L + S R

*+

)+ S

*


13/48

Gravitational SO(3,1)

SO(3; ) ! !1 0

0

0 1

0

!"

!"

!"

!"

0

00

0

! ! ! = 21 "#

"# = ! ! L

! R"

!L=R =

"i!

3L=R

! _L=R

! ^L=R

!i! 3L=R

#

e ! = e" " =!

e L

e R"

eL=R = " e

T

^=_

'e S_

'e S^

e T_=

^#f =

!f Lf R

"fL=R =

"f ^L=R

f _L=R

#

1 21 3

L 21 3

R

%& ! L^

%& ! L_ !1

%& ! ^R

%& ! _R

!1

e S 21

21

e S_ ! 21 ! 21

e T^ ! 21 21

e T_

21 ! 2

1

NM f L^

21

NM f L_ ! 21

NM f R^

21

NM f R_ ! 21

El k SU(2) d U(1)


14/48

Electroweak SU(2) and U(1)

SO(4) W B B Y Q

W 1 0 0 0 10 0 0

B 0 1 0 1 1

0 0

!" ' 0 1

0

!" ' 0 0

' 0 0

# 0 0

e 0

# 0 0 0

e 0

u 0

d 0

u 0

d 0

W =!

W i23

W !

W +

! W i2 3"

B1 ="

B i2 13

B 1!

B 1+

! B i2 13#

# !

W

B 1

";

!' W

' B" $

'W=B ="

!' 0=1

' !

' +

' 1=0

#

!

W

B 1"

2

6664# eLe L

# eRe R

3

7775 2

6664u Ld Lu Rd R

3

7775

B B ) Y )p

5

p3

13 ! p

5

p2

2

*= (p

5

p3

21 ! g1 = p5

p3

313 p

3

p2

2 21

%&+

%& W ! !1 !1%&

1+

%& B 1! !1 !1!1

+ 21

21

21

#$ ' ! ! 21 ! 2

1 ! 21 !1

0 ! 21

21

21

#$ 1 21 ! 2

1 ! 21

NM eL 21

21 ! 2

1

NM L ! 21

21 ! 2

1 !1

NM eR 21

21

NM R ! 21

21 !1!1

NMNMNM L 21 ! 6

161

32

NMNMNM L ! 21 ! 6

161 ! 3

1

NMNMNM R 21 ! 6

132

32

NMNMNM R ! 21 ! 6

1 ! 31 ! 3

1

G i k SO(7 1)


15/48

Graviweak SO(7,1)

D4 ! ! W B

0 0 0

0 0 0

0 0 0

0 0 0

!"

!"

!"

!"

0 0

0 0

0 0

0 0

H ! e' )1

= (2

1 +4

1 + W + B1

o(3; ) su(2) u(2) 2 s 1 + 4 ) 4 + ) + s *l (7; ) o(7; ) 4= C 2 1 = s 1 = d

8 # )S+ ! H1 ( e + e=

2664

! W21 L + i2 3

W !

! e ' 41

L 0

e ' 41

L !

W +

! W21

L !i2

3

e ' 41

L +

e ' 41

L 1

! e ' 41 R 1e ' 4

1R !

! B21

R+i2 1

3

B 1!

e ' 41 R +e ' 4

1R 0

B 1+

! B 21

R !i2 1

3

3

7752

664# eLe L# eRe R

3

775

21 3

L 21 3

R3

13

%& ! L

^=_(1

%& ! R

^=_(1

%& W ( (1%& B

1( (1

e ' T

^=_+ ' 2

1 ( 21

21

21

e ' S

^=_+ ( 2

1 ( 21

21

21

#$e ' T

^=_! ' 2

1 ( 21 ! 2

1 ! 21

#$e ' S

^=_! ( 2

1 ( 21 ! 2

1 ! 21

e ' T

^=_0 ' 2

1 ( 21 ! 2

121

e ' S

^=_0 ( 2

1 ( 21 ! 2

121

#$e ' T

^=_1 ' 2

1 ( 21

21 ! 2

1

#$e ' S

^=_1 ( 2

1 ( 21

21 ! 2

1

NM # eL

^=_( 2

121

NM e L

^=_( 2

1 ! 21

NM # eR

^=_( 2

121

NM e R

^=_( 2

1 ! 21

G i k F4


16/48

Graviweak F4

A triality rotation, , of :

Roots invariant under this :

Rotations to triality-equivalent vector and

negative chiral spinor representation spaces:

Three generations, related by triality:

8 ! ! W B

tri ! B W !

0 0

0 0

0 0

0 0

! ! W B

tri B ! W !

0 00 0

0 0

0 0

T D4

2664!

2

1 03L

! 21 03

R

W 03

B 031

3775=26400

0

1

100

0

001

0

010

0

3752664

! 21 3

L! 2

1 3R

W 3

B 13

3775=2664

! 21 3

RB

13

W 3

! 21 3

L

3775T T T !^

R

= T T !L

^ = T B1

+ = !^R

T

fW ; ; ' ; ' ; ' ; ' g+ W! eS^

+ eS^

0 eS_

! eS_

1

T 8S+ = 8V T 8V = 8S! T 8S! = 8S+

T e^L

= "^L

T "^L

= (L^ T (

L^ = e^

L

V 21 3

L21 3

R

3

1

3

21 3

R 13 3

21 3

L

NM# "L

^=_21 ( 2

1

NM" L

^=_! 2

1 ( 21

NM# "R

^=_( 2

121

NM" R=

_ ( 21

! 21

8 S! 21 3

L 21 3

R3

13

13

21 3

L3

21 3

R

NM # (L=

_ ( 21

21

NM ( L

^=_( 2

1 ! 21

NM # (R

^=_21 ( 2

1

NM ( R

^=_! 2

1 ( 21


17/48

F4 and G2 together


18/48

F4 and G2 together

Breakdown of E8 to the standard model and gravity:

Two new quantum numbers and some non-standard particles:

F4

G2

E8

:

:

:

( ! ; ! ; ; ) 21 3

L 21 3

RW3 B

13

(B ; ; )2 g3 g8

( ! ; ! ; ; ; ; ; ; )21 3

L 21 3

RW3 B

13 w B2 g

3 g8

+graviweak interactions

three generations+strong interactionsanti-particles

f everything

e8

A

=

=

=

f4 2 6 + g + 2 )7

so(7; ) u(3) 8 ) 1 ) 3 ) 1 + s + ( S++8V +8S! )( +1+3+3! + 3)( +3! + 2

! e' " )

21 + 41 + W + B1

*+ g + 3)# + x + B2 + w

f B ) " " w (13+B2 B1

( x1=2=3r=g=b x1=2=3

r=g=b! ! ! g

E8 root system


19/48

E8 root system .

E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.

E8 root system


20/48

E8 root system ..


E8 root system


21/48

E8 root system ...



22/48

E8 root system x


23/48

E8 root system ....x


E8 root system x


24/48

E8 root system ....x.



25/48

E8 root system ....x.x.


26/48

E8 root system ....x.x.


E8 root system ....x.x..


27/48

y


E8 root system ....x.x...


28/48

y


E8 root system ....x.x....


29/48

y


E8 root system ....x.x....x


30/48

y


E8 root system ....x.x....x.


31/48


E8 root system ....x.x....x..


32/48


E8 root system ....x.x....x..x


33/48



34/48

E8 root system ....x.x....x..x..


35/48


E8 root system ....x.x....x..x...


36/48



37/48

E8 periodic table


38/48

"E8 is perhaps the most beautiful structure in all of mathematics, but it's

very complex." Hermann Nicolai


39/48

E8 curvature


40/48

F A A # ="

= d!!"

+ A!"!"

= F= 1

+ F= 2

+ D!

)" I

+ #" II

+ #" III

* 2 e8="

F= 1

F= 2

= R ' T' D' F 21)

= ! 81 e!e! 2*

+ 41)

= ! e! !*

+)

= B1 + F= W*

R ! !!=

= d! !

+ 21

! !

T' D' d [!; ] ' d' B ; ] =

!e! !

=)

!e!

+ 21

!e!

*! e

!

)!

+[! 1

+W!

'*

F dB B ) dW W)= B1 + F= W = (! ! 1 + B! 1 ! 1 + (! ! + W! !

= F "x" (Dx)" D" )

= w+ F

= B2+ x

! !

*+

)! !

! x! !

*+ F

= g

F w B = w

+ F= B2

= d! !

+ d! ! 2

(Dx)" D" dx w ;x] " d" ;"]! !

!x! !

=)! !

+[!

+B!

2!* !x

!)

!

+[g!

*F = g

= d!

g!

+ g!

g!

222

222

2

2

so(7; )! 1

so(3; )!

1

2 )4!

) ( + 2!

su(2) (2)! +su!

so(8)!

u(1) (1)!

+ u!

3 )3!

) ( + 3!

su(3)!

D# d ! ' # # # w " ! "

=

)!+

2

1!

+4

1 e! * "

+ W! " L

+ B! 1 " R

! #" )!

+ B! 2

+ x! *

! #" q

g!

Action for everything: :


41/48

Modified BF action, using :

Cosmological constant from the Higgs VEV:

Implies frame VEV is de Sitter:

Vacuum expectation value of the curvature vanishes:

B :

== B

=+ B

:

*

S=

=

=

BF (H ; ; ) Z, :

= =" + "+ ! 1 H! 2 B=-

BD# F B B ! ,B

Z, :* ! "

+ B= =

+4

%G

= G = G+ B0

=0

=

-

BD# ' R ' F ,F Z, :

* ! " + e+ 116%G 2)

! 23 2*

+ 41 0= 0=-

' $ = 43 2

$R= = 6$

e!e! R = 4

F="

= 0


42/48

Fermionic part of the action: :

*


43/48

Choosing the anti-Grassmann 3-form to be gives the

massive Dirac action in curved spacetime:

The is the standard Higgs mass term.

The term... I don't understand yet promising for CKM.

B # *

= e+

e*

S f =

=

=

=

BF BD# Z, :

* ="

-=Z, :

* ! "

-

# d# # H

Z,e+

:e*)

! "+ H

! 1 "! #

" ! 2

*-

# (d ! ' )# (w " ) Z,

e+

:

e*)

! + 21 ! + 41 e! + W! + B! 1 " ! #" ! + B! 2 + x! + g!*-

ej #! (e ) @ # ! ! # # #Z

d x4+

j , : " " i) i " + 41 i"# "# " + Wi " + B1i "+ w B x

"

g#

" i +#

" 2i +#

" i +#

" i*

+#

:' #

"-

# :

' #"

#! #x ":

"" "

E8 Theory summary

E thi i i i l b dl tiE8


44/48

Everything in an principal bundle connection,

Periodic table of interactions (Feynman vertices) from curvature,

described by the root polytope. Three generations through triality,

Pati-Salam GUT and MM gravity together,

No free parameters masses from Higgs VEV's,

Everything is pure geometry, and it's very beautiful.

E8

A !"2 e8

!"

F A A; ="

= d!!"

+2

1#

!"A!"

$E8

T e = " T " = ( T ( = e

SU(2) U(2) U(4)L ) S R ) S

S BF B B ! ,B =

Z, := ="

+4

%

= G = G+ B0

=0

=

-

g ' ; ; 1 =

r5

3g2 = 1 g3 = 1 $ = 4

3 2 '0 '1 " : : :

E8 Theory discussion

Quantization


45/48

Quantization

Coupling constants run.

Large compatible with UV fixed point.

Just a connection amenable to LQG, spin foams, etc.Understand triality-generation relationship better

Possible collapse or mixing to graviweak .

The role of and symmetry breaking.

Getting the CKMPMNS matrix would be nice.

Why is the action what it is?Pulling out and putting it into and seems weird.

Why simple?

Four dimensional base manifold emergent?

What this theory will mean, if it all works:

Combines standard model with gravity with LQG, it's a T.o.E.

Our universe is very pretty.

http://deferentialgeometry.org Garrett Lisi

$

SL(2; )Cw "!

+ x!

e!

F,F= =

B:

*

'e!

BRST extended connection

Start with principal bundle connection and its curvatureE8
http://deferentialgeometry.org/


46/48

Start with principal bundle connection and its curvature,

Action such that part is pure gauge,

BRST: Replace part with ghosts, , in extended connection,

Effective action for gauge fields, ghosts, and anti-ghosts:

E8

A dH H #) d# # H)!

= H!

+ #!

F=

= (! !

+ H! !

+ #! !

+ (! !

+ H! !

+ #! !

#!

S BF B B ! ,B =

Z,= =

+4

%G

= G = G+ B0

=0

=

-

#!

#"

A dH H d# H ; ] #!"

= H!

+ #"

F="

=)

! !+ H

! !

*+

)! "

+ [!

#"

*= F

= H+ D

! "

S=

=

BF B B ! ,B Z,:

= ="+

4

%G

= G = G+ B0

=0

= -BD# ' R ' F ,F

Z, :* ! "

+ e+

1

16%G2)

!2

3 2*

+4

1 0=

0=

-

Geometry of Yang-Mills theory

Start with a Lie group manifold (torsor) coordinatized byG yp


47/48

Start with a Lie group manifold (torsor), , coordinatized by .

Two sets of invariant vector fields (symmetries, Killing vectorfields):

Lie derivative: [, ; ] ,

Lie bracket: T ; T

Killing form (Minkowski metric): g C

Maurer-Cartan form (frame):

Entire space of a principal bundle:

Ehresmann principal bundle connection over patches of :

Gauge field connection over :

G y p

, (y) g (y) (y) g (y) L

A

*

d! = TA g ,AR

*

d! = g TA

AR

*

,BR

*

= CAB

CCR

*

[ A TB] = CABC

C

AB = CAC

D

BD

C

I (, ) T !

= dyp!

pR A

A

E + M ) GE

(x; ) A (x) (y) @ E!* y = dxi

! iB ,L

B

*

+ dyp!

p*

M

A(x) I A (x) !

= $0,E

!

*

!= dxi

! iB TB

The Coleman-Mandula theorem


48/48

Let G be a connected symmetry group of the S matrix, and

let the following five conditions hold: (1) G contains a

subgroup locally isomorphic to the Poincar group. (2)...Then, we show that G is necessarily locally isomorphic to the

direct product of an internal symmetry group and the

Poincar group.

E8 Theory does not allow a subgroup locally isomorphic to the Poincar

group. The S matrix only exists as an approximation, in which the

theorem is satisfied.

K. Cahill, "On the unification of the gravitational and electronuclear

forces," Phys. Rev. D 26, 1916 (1982).T. Love, "Geometry of Unification," Int. J. Th. Phys., 801 (1984).

...

F. Nesti and R. Percacci, "GraviWeak Unification" arxiv{0706.3307}

S. Alexander, "Isogravity: Toward an Electroweak and Gravitational

Unification " arxiv 0706.4481 .
http://arxiv.org/abs/0706.4481http://arxiv.org/abs/0706.3307http://users/aglisi/Sites/dg/papers/Cahill%20-%20On%20the%20unification%20of%20the%20gravitational%20and%20electronuclear%20forces.pdf

an exceptionally simple theory of everything, by garrett lisi (48 p.) -

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