Post on 08-Oct-2020
transcript
Standard model and gravity
http://deferentialgeometry.org Garrett Lisi FQXi
Everything as a principal bundle connection
1918;
1954;
1967;
1977;
2002;
2005;
now;
Weyl
Y:M:
F:P:
M:M:
Y:T:
Y:T:
Y:T:
:
:
:
:
:
:
:
A (G) +2 Lie
+
A (G) (1) (2) (3) += B
++W
++G
+2 Lie
+= su
++ su
++ su
+
(G) A!+
= A++ g
!2 Lie
!+
A (G) (1; ) += !
++ e
+2 Lie
+= so
+4
ï != g
!
! A!+
= 21!++ 4
1 e+
+B++W
++G
++ "e
!+ e
!+ u
!+ d
!
(G) (1; ) 2 Lie+
= Cl+
7
! A!+
= 21!++ 4
1 e+
+B++W
++G
++ "e
!+ e
!+ u
!+ d
!
+ "#!+ #
!+ c
!+ s
!+ "$
!+ $
!+ t
!+ b
!
(G) ? 2 Lie+
= e8+
Standard model and gravity in a matrix
Correct interactions and charges from curvature:
(1; ) (8) (8 ) A!+
= H++G
++ ï
!=
! H ++
ï !
"
G "+
"2 so
+7 + so
++ C
!# 8
=
2
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
! W 21
L++i 3
+
iW 1+"W 2
+
" e ! 41
L+
0
e ! 41
L+
$
+
W i 1+
+W 2+
! W 21
L+"i 3
+
e ! 41
L+
+
e ! 41
L+
$
0
e ! " 41
R+
$
0
e ! 41
R+
$
+
! B 21
R++i
+
e ! 41
R+
+
e ! 41
R+
0
! B 21
R+"i
+
"! L
e! L
"! R
e!R
B i+
u!
r
L
d!
r
L
u!
r
R
d!
r
R
G 3"iB++i 3+8
+
G i 1++G2
+
G i 4++G5
+
u!
g
L
d!
g
L
u!
g
R
d!
g
R
G i 1+"G2
+
G 3"iB+"i 3+8
+
G i 6++G7
+
u!
b
L
d!
b
L
u!
b
R
d!
b
R
G i 4+"G5
+
G i 6+"G7
+
G 3"iB+"
2ip3
8+
3
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
F!+!+
=
=
d+A!+
+ A!+A!+
H H) G G) ï ï G) (d++
+H++
+ (d++
+G++
+ (d+ !
+H+ !
+ ï! +
Gravitational part of the connection
Using chiral (Weyl) representation of Cl(1,3) Dirac matrices:
Spacetime frame and spin connection:
Note algebraic equivalence: Cl (1; ) l (1; ) o(1; )
C(4 ) # 4
% 0
% 0"
=
=
&1 % 1 =
! 1
1 "
% %0 " =
! "& "
& "
" %'
%"'
=
=
& i 2 % &' =
! "& '
& '"
% %" ' =
! "i( & "'$ $
"i( & "'$ $
"
!++ e
+=
=
=
dx ! % (e ) % a+ 2
1a#"
#" + dxa+ a
##
" ("! & ! ( & ) 0"
+ " "i2
"'+ "'$ $
(e & ) 0+" e'
+ '
(e & ) 0++ e'
+ '
(! & ! ( & ) 0"+ " "
i2
"'+ "'$ $
#
(1; )
2
4
! L+
e L+
e R+
! R+
3
5 2 Cl+
1+2 3
1+2 3 = C 2 4 = s 4
Bosonic part of the connection
Clifford bivector parts:
indices:
H +
=
=
! 21!++ 4
1 e+
+B++W
+=
2
6 6 6 4
! W 21
L++i 3
+
iW 1+ "W2
+
" e ! 41
L+
0
e ! 41
L+
$+
iW 1+ +W2
+
! W 21
L+"i 3
+
e ! 41
L+
+
e ! 41
L+
$
0
" e ! 41
R+
$
0
e ! 41
R+
$+
! B 21
R++i+
e ! 41
R+
+
e ! 41
R+
0
! B 21
R+"i+
3
7 7 7 5
h % (1; ) (1; ) (8 ) dxa+ 2
1a)*
)* 2 so+
7 = Cl+
2 7 & C+
# 8
!+
! e+
=
=
! % dxa+ 2
1a#"
#" ï spin connection
% dxa+
e( a)#!!
#!
8 > <
> :
e+
!
=
=
dxa+
e( a)#%# ï frame (vierbein)
% !!!
# ! +! 0
=
=
"! ! ) ( 5+i 6
! ! ) ( 7+i 8
$
ï Higgs
!! M = " 2
B +
W +
=
=
B % "dxa+ 2
1a
%56" %78
&ï # electroweak gauge fields
W % W W % " 21 1+
%67 + %58
&" 2
1 2+
%" %57 + %68
&" 2
1 3+
%56 + %78
&
; ; ; 0 ' a b ' 3 0 ' # " ' 3 5 ' ! ï ' 8
Curvature of bosonic part
Modified BF action over 4D base manifold:
F ++
=
=
=
=
H H ! d++
+H++
H+= 2
1!++ 4
1 e+
+B++W
+
( ) M
'21 d+!++ 2
1!+!+
+ 116
2 e+e+
(
[ ; ] ! ! B ; ] +'
41%d+e++ 2
1 !+
e+
&" 4
1 e+
%d+
+ [++W
+!&(
B W W +'d++
+ d+ +
+W+ +
(
R M T! D! F 21%+++ 8
1 2 e+e+
&+ 4
1%++
" e++
&+%
B++
+ FW++
&
Fs++
+ Fm++
+ Fh++
ï spacetime %#"
ï mixed %#!
ï higher %!ï
2 + 67 58 2 + 57 68 2 + 56 78
S =
=
B (H; ) B B B % $B $B
Z )++F+++ !
+B++
*=
Z )++F++"
4
1s
++
s++
+Bm++
m++
+Bh++
h++
*
F % F $F F $F
Z )s
++
Fs++
" +4
1m++
m++
+4
1h++
h++
*
Gravitational action
S s
S s
S s
=
=
=
B F (B ) B R M B B %
Z )s
++
s++
+ !s s++
*=
Z )s
++
2
1%+++
8
1 2 e+e+
&"
4
1s
++
s++
*
+B R M % s++
! Bs++
=%+++
8
1 2 e+e+
&" pseudoscalar: % = % % % %0 1 2 3
R M R M % F F % 4
1
Z )%+++
8
1 2 e+e+
&%+++
8
1 2 e+e+
&"*=
Z )s
++
s++
"*
RR% % )++++
"*= d
+
)%!+d+!++
3
1!+!+!+
&"*
ï Chern-Simons
% 1
4!
)e+e+e+e+
"*= e
"ï volume element
R R )e+e+++%"
*= e
"ï curvature scalar
M "
12
Ze"R "( + 2 ) cosmological constant: " =
4
3 2
F % F $F F $F s++
Fs++
" +4
1m++
m++
+4
1h++
h++
Action for everything
Modified BF action for everything, using B :
Fermionic part, using anti-ghost Grassmann 3-form, B ï :
H H G G ï ï G F!+!+
= d+A!+
+ A!+A!+
=%d++
+H++
&+%d++
+G++
&+%d+ !
+H+ !
+ ï! +
&
M "
12e"R "( + 2 ) cosmological constant: " =
4
3 2
:
++= B
+++B
:
"
S =
=
B (H; ; )
Z ) :
++F!+!+
+ !" +
G+B++
*
B ï ï G F B B % $B
Z ) :
"
%d+ !
+H+ !
+ ï! +
&+B
++++"
4
1s
++
s++
+Bm;h;G++
m;h;G++
*
:
"= e
"
:
e*
S f =
=
=
B ï ï G
Z ) :
"
%d+ !
+H+ !
+ ï! +
&*
ï ï ï !ï ï ï G
Z )e"
:
e*%d+ !
+2
1!+ !
+4
1e+ !
+B+ !
+W+ !
+ ï! +
&*
jej ï% (e ) @ ï ! % ï ï ï G
Zd x4"
) :#
#i%
i!+
4
1
i
#"#"
!+Bi
!+Wi
!" ï
!i
&+ ï
:
!ï!
*
Why this Lie algebra
Note: Only one generation, and fermion masses not quite right.
For three generations:
BIG Lie algebra:
jej ï% (e ) @ ï ! % ï ï ï G d x4"
) :#
#i%
i!+
4
1
i
#"#"
!+Bi
!+Wi
!" ï
!i
&+ ï
:
!ï!
*
A!+
= H++G
++ ï
!=
! H ++
ï !
"
G "+
"
=
2
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
! W 21
L++i 3
+
iW 1+"W 2
+
" e ! 41
L+
0
e ! 41
L+
$
+
W i 1+
+W 2+
! W 21
L+"i 3
+
e ! 41
L+
+
e ! 41
L+
$
0
e ! " 41
R+
$
0
e ! 41
R+
$
+
! B 21
R++i
+
e ! 41
R+
+
e ! 41
R+
0
! B 21
R+"i
+
"! L
e! L
"! R
e!R
B i+
u!
r
L
d!
r
L
u!
r
R
d!
r
R
G 3"iB++i 3+8
+
G i 1++G2
+
G i 4++G5
+
u!
g
L
d!
g
L
u!
g
R
d!
g
R
G i 1+"G2
+
G 3"iB+"i 3+8
+
G i 6++G7
+
u!
b
L
d!
b
L
u!
b
R
d!
b
R
G i 4+"G5
+
G i 6+"G7
+
G 3"iB+"
2ip3
8+
3
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
(1; ) (8) (8 ) A!+
2 so+
7 + so+
+ 3 $ R!
# 8 = ?
n 8 8 4 48 = 2 + 2 + 3 $ 6 = 2
Real simple compact Lie groups
rank group a.k.a. dim name
special unitary group
odd special orthogonal group
symplectic group
even special orthogonal group
G2
F4
E6
E7
E8
"E8 is perhaps the most beautiful structure in all of mathematics, but it's
very complex."
— Hermann Nicolai
r A r SU (r ) + 1 r(r ) + 2
r B r SO(2r ) + 1 r(2r ) + 1
r C r Sp(2r) r(2r ) + 1
r > 2 D r SO(2r) r(2r ) " 1
2 G 2 14
4 F 4 52
6 E 6 78
7 E 7 133
8 E 8 248
Triality decomposition of E8
John Baez in TWF90:
we now look at the vector space
...Since has a representation as linear transformations
of , it has two representations on , corresponding to
left and right matrix multiplication; glomming these two
together we get a representation of on .
Similarly we have representations of on
and . Putting all this stuff together we get a
Lie algebra, if we do it right - and it's .
so(8) o(8) nd(V ) nd(S ) nd(S ) + s + e + e + + e "
so(8)
V end(V )
so(8) o(8) + s end(V )
so(8) o(8) + s
end(S ) + end(S ) "
E8
E (E8) = H +G+#I +#II +#III 2 Lie
[H; ] #I
[G; ] #I
=
=
H #I
#I G
[H; ] #II
[G; ] #II
=
=
H+#II
#II G+
[H; ] #III
[G; ] #III
=
=
H"#III
#III G"
E8 T.O.E.
Build a real form of complex E8 by using instead of
. Then E8 T.O.E. connection is:
Cl (1; ) o(1; ) 2 7 = s 7Cl (8) o(8) 2 = s
A!+
= H++G
++#
! I+#
! II+#
! III=
something like
2
6 6 6 6 4
! W 21
L++i 3
+
iW 1+"W 2
+
" e ! 41
L+
0
e ! 41
L+
$
+
W i 1+
+W 2+
! W 21
L+"i 3
+
e ! 41
L+
+
e ! 41
L+
$
0
e ! " 41
R+
$
0
e ! 41
R+
$
+
! B 21
R++i
+
e ! 41
R+
+
e ! 41
R+
0
! B 21
R+"i
+
3
7 7 7 7 5 +
2
6 6 6 4
iB +
G 3"iB++i 3+8
+
G i 1++G2
+
G i 4++G5
+
G i 1+"G2
+
G 3"iB+"i 3+8
+
G i 6++G7
+
G i 4+"G5
+
G i 6+"G7
+
G 3"iB+"
2ip3
8+
3
7 7 7 5
+
2
6 6 6 4
" !
e
L
e ! L
" !
e
R
e !R
u !
r
L
d !
r
L
u !
r
R
d !
r
R
u !
g
L
d !
g
L
u !
g
R
d !
g
R
u !b
L
d !
b
L
u !
b
R
d !
b
R
3
7 7 7 5 +
2
6 6 6 6 4
" !
#
L
# ! L
" !
#
R
# ! R
c !
r
L
s !
r
L
c !
r
R
s !
r
R
c !
g
L
s !
g
L
c !
g
R
s !
g
R
c !b
L
s !
b
L
c !
b
R
s !
b
R
3
7 7 7 7 5 +
2
6 6 6 4
" !
$
L
$ ! L
" !
$
R
$ !R
t !
r
L
b !
r
L
t !
r
R
b !
r
R
t !
g
L
b !
g
L
t !
g
R
b !
g
R
t !
b
L
b !
b
L
t !
b
R
b !
b
R
3
7 7 7 5
Geometry of Yang-Mills theory
Start with a Lie group manifold (torsor), , coordinatized by .
Two sets of invariant vector fields (symmetries, Killing vector fields):
, (y) g (y) (y) g (y)
Lie derivative: [, ; ] ,
Lie bracket: T ; T
Killing form (Minkowski metric): g C
Maurer-Cartan form (frame): (, ) T
Entire space of a principal bundle:
Ehresmann principal bundle connection over patches of :
(x; ) A (x) (y) @
Gauge field connection over :
G y p
L
A
*
d+
= TA g ,A
R*
d+
= g TA
AR*
,BR*
= CAB
CCR*
[ A TB] = CAB
CC
AB = CAC
DBD
C
I+= dyp
+ pR A
A
E (M #G
E
E+
*y = dxi
+ iB ,L
B
*
+ dyp+
p
*
M
A(x) A (x) +
= &0$E+
*I+= dxi
+ iB TB
Cartan subalgebra and charges
Mutually commuting set of Lie algebra generators:
Cartan subalgebra:
Eigenvalues, , and eigenvectors, , using the Lie bracket:
Unique eigenvalue for each of the eigenvectors, corresponding
to roots, ) , in dimensional vector space.
Cartan subalgebra of the standard model and gravity:
C ! % ! % i$ iY i- i-
Eigenvectors are elementary particles, roots are their charges:
)(e ) ) ; ; 1; 1; ; )
r
T ; ; ::; T ; ] f 1 T2 : Trg [ i Tj = 0
T C = ci i 2 Lie(G)
) a V a 2 Lie(G)
[C; ] V ) V Va = )aa =
X
i
ciia
a
(n ) " r
(n ) " ria r
=2
1 0101 + 2
1 1212 +W 3
3 +B +G33 +G8
8
L = (2
1*
2
1" " 0 0
E8 roots
has roots in 8D space — vertices of P 4 :
T.O.E.: Each vertex corresponds to an elementary particle.
Lie(E8) (248 ) 40 " 8 = 2 2;1
E8
Reducing E8 to the standard model
One particularly interesting way can be broken down:
How does this breakdown relate to e8 triality decomposition?
e8
e8 =
=
=
!
e6 u(3) 4 + s + 5 #3
so(1; ) (1) 2 u(3) 4 9 + u + 3 + s + 5 #3
so(1; ) u(2) u(2) (1) (1) 2 u(3) 4 3 + s + s + u + 4#8 + u + 3 + s + 5 #3
! e! ? 2
1 +W +B +4
1 +G+ 3#ï +X
e8
e8 =
=
=
so(1; ) o(8) 7 + s + 3#8#8
so(1; ) o(4) o(6) o(2) 3 + s + 4#4 + s + s + 6#2 + 3#8#8
so(1; ) u(2) u(2) u(4) (1) 3 + s + s + 4#4 + s + u + 6#2 + 3#8#8
Discussion
What is done:
All gauge fields, gravity, and Higgs in one connection, with
fermions as BRST ghosts.
To do:
Will particle assignments work with E8? (Get the mass matrix?)
Why is the action what it is? (How's symmetry breaking happen?)
Is a four dimensional base manifold emergent?
How does this theory get quantized? (LQG methods should apply.)
Natural explanation for QM as a bonus?
What this theory will mean, if it all works:
Gravitational frame and Higgs are intimately related.
Naturally combines standard model with gravity — so it's a T.O.E.
(It's also a U.F.T., but I don't like to call it that.)
Our universe is a very pretty shape!
Gar@Lisi.com
http://deferentialgeometry.org
BRST gauge fixing
+L under gauge transformation:
Account for gauge part of A by introducing Grassmann valued ghosts,
, anti-ghosts, B, partners, -, and BRST transformation:
This satisfies +L and ++ .
Choose a BRST potential, , to get new Lagrangian:
BRST partners act as Lagrange multipliers; effective Lagrangian:
"= 0 +A rC C A;
+= "
+= "d
+"++C,
+
C (G) !2 Lie
! g
:
" "
+A !+
+B !++
+- ! "
=
=
=
"rC + !
B; +++C!
,
0
+C ! !
+B !
:
"
=
=
" C; 21+!C!
,
- "
! "= 0
! != 0
# BA :
"=) :
"+
*
L # -A BrC 0"= L
"+ +
!
:
"= L
"+)" g+
*+) :
" + !
*
L [B ; ] Br C eff"
= L"
0++
A0+
+) :
"
0+ !
*
BRST extended connection
Replace pure gauge part of connection with ghosts:
BRST extended curvature:
Effective Lagrangian, with B :
Crazy idea: Fermions are gauge ghosts
L [B ; ] Br C eff"
= L"
0++
A0+
+)"
0+ !
*
A!+
= A0++ C
!
F!+!+
=
=
; C C; d+A!+
+2
1+A!+
A!+
,= F 0
+++r0
+ !+
2
1+!C!
,
A A C A ; C; %d+
0++A0
+0
+
&+%d+ !
++ 0+
C!
,&+
2
1+!C!
,
0:
"= B0
"+B
:
"
L B (A ; ) eff"
=) 0
:
"F!+!+
+ !"
0+
B0"
*
A 0+
C !
=
=
H ! ++G
+=%2
1!++
4
1e+
+B++W
+
&+G
+
ï " !=%!+ e
!+ u
!
r;b;g + d!
r;b;g&
ï " !=%!+ e
!+ u
!
r;b;g + d!
r;b;g&