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Beyond the spectral Standard Model: Pati-Salamunification
Walter van Suijlekom
(joint with Ali Chamseddine and Alain Connes)
22 September 2017
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Overview
• Motivation: NCG and HEP
• Noncommutative Riemannian spin manifolds (aka spectraltriples)
• Gauge theory from spectral triples: gauge group, gauge fields
• The spectral Standard Model and Beyond
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A fermion in a spacetime background
• Spacetime is a (pseudo)Riemannian manifold M:local coordinates xµ generatealgebra C∞(M).
• Propagator is described by Diracoperator DM , essentially a’square root’ of the Laplacian.
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A fermion in a spacetime background
• Spacetime is a (pseudo)Riemannian manifold M:local coordinates xµ generatealgebra C∞(M).
• Propagator is described by Diracoperator DM , essentially a’square root’ of the Laplacian.
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A fermion in a spacetime background
• Spacetime is a (pseudo)Riemannian manifold M:local coordinates xµ generatealgebra C∞(M).
• Propagator is described by Diracoperator DM , essentially a’square root’ of the Laplacian.
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The circle
• The Laplacian on the circle S1 is given by
∆S1 = − d2
dt2; (t ∈ [0, 2π))
• The Dirac operator on the circle is
DS1 = −id
dt
with square ∆S1 .
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The circle
• The Laplacian on the circle S1 is given by
∆S1 = − d2
dt2; (t ∈ [0, 2π))
• The Dirac operator on the circle is
DS1 = −id
dt
with square ∆S1 .
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The 2-dimensional torus
• Consider the two-dimensional torus T2 parametrized by twoangles t1, t2 ∈ [0, 2π).
• The Laplacian reads
∆T2 = − ∂2
∂t21
− ∂2
∂t22
.
• At first sight it seems difficult to construct a differentialoperator that squares to ∆T2 :(
a∂
∂t1+ b
∂
∂t2
)2
= a2 ∂2
∂t21
+ 2ab∂2
∂t1∂t2+ b2 ∂
2
∂t22
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The 2-dimensional torus
• Consider the two-dimensional torus T2 parametrized by twoangles t1, t2 ∈ [0, 2π).
• The Laplacian reads
∆T2 = − ∂2
∂t21
− ∂2
∂t22
.
• At first sight it seems difficult to construct a differentialoperator that squares to ∆T2 :(
a∂
∂t1+ b
∂
∂t2
)2
= a2 ∂2
∂t21
+ 2ab∂2
∂t1∂t2+ b2 ∂
2
∂t22
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• This puzzle was solved by Dirac who considered the possibilitythat a and b be complex matrices:
a =
(0 1−1 0
); b =
(0 ii 0
)then a2 = b2 = −1 and ab + ba = 0
• The Dirac operator on the torus is
DT2 =
0∂
∂t1+ i
∂
∂t2
− ∂
∂t1+ i
∂
∂t20
,
which satisfies (DT2)2 = − ∂2
∂t21
− ∂2
∂t22
.
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• This puzzle was solved by Dirac who considered the possibilitythat a and b be complex matrices:
a =
(0 1−1 0
); b =
(0 ii 0
)then a2 = b2 = −1 and ab + ba = 0
• The Dirac operator on the torus is
DT2 =
0∂
∂t1+ i
∂
∂t2
− ∂
∂t1+ i
∂
∂t20
,
which satisfies (DT2)2 = − ∂2
∂t21
− ∂2
∂t22
.
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The 4-dimensional torus
• Consider the 4-torus T4 parametrized by t1, t2, t3, t4 and theLaplacian is
∆T4 = − ∂2
∂t21
− ∂2
∂t22
− ∂2
∂t23
− ∂2
∂t24
.
• The search for a differential operator that squares to ∆T4
again involves matrices, but we also need quaternions:
i2 = j2 = k2 = ijk = −1.
• The Dirac operator on T4 is
DT4 =
(0 ∂
∂t1+i ∂
∂t2+j ∂
∂t3+k ∂
∂t4
− ∂∂t1
+i ∂∂t2
+j ∂∂t3
+k ∂∂t4
0
)• The relations ij = −ji , ik = −ki , et cetera imply that its
square coincides with ∆T4 .
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The 4-dimensional torus
• Consider the 4-torus T4 parametrized by t1, t2, t3, t4 and theLaplacian is
∆T4 = − ∂2
∂t21
− ∂2
∂t22
− ∂2
∂t23
− ∂2
∂t24
.
• The search for a differential operator that squares to ∆T4
again involves matrices, but we also need quaternions:
i2 = j2 = k2 = ijk = −1.
• The Dirac operator on T4 is
DT4 =
(0 ∂
∂t1+i ∂
∂t2+j ∂
∂t3+k ∂
∂t4
− ∂∂t1
+i ∂∂t2
+j ∂∂t3
+k ∂∂t4
0
)• The relations ij = −ji , ik = −ki , et cetera imply that its
square coincides with ∆T4 .
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The 4-dimensional torus
• Consider the 4-torus T4 parametrized by t1, t2, t3, t4 and theLaplacian is
∆T4 = − ∂2
∂t21
− ∂2
∂t22
− ∂2
∂t23
− ∂2
∂t24
.
• The search for a differential operator that squares to ∆T4
again involves matrices, but we also need quaternions:
i2 = j2 = k2 = ijk = −1.
• The Dirac operator on T4 is
DT4 =
(0 ∂
∂t1+i ∂
∂t2+j ∂
∂t3+k ∂
∂t4
− ∂∂t1
+i ∂∂t2
+j ∂∂t3
+k ∂∂t4
0
)
• The relations ij = −ji , ik = −ki , et cetera imply that itssquare coincides with ∆T4 .
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The 4-dimensional torus
• Consider the 4-torus T4 parametrized by t1, t2, t3, t4 and theLaplacian is
∆T4 = − ∂2
∂t21
− ∂2
∂t22
− ∂2
∂t23
− ∂2
∂t24
.
• The search for a differential operator that squares to ∆T4
again involves matrices, but we also need quaternions:
i2 = j2 = k2 = ijk = −1.
• The Dirac operator on T4 is
DT4 =
(0 ∂
∂t1+i ∂
∂t2+j ∂
∂t3+k ∂
∂t4
− ∂∂t1
+i ∂∂t2
+j ∂∂t3
+k ∂∂t4
0
)• The relations ij = −ji , ik = −ki , et cetera imply that its
square coincides with ∆T4 .
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Hearing the shape of a drum: motivation from mathKac (1966), Connes (1989)
• The geometry of M is not fully determined by spectrum ofDM .
• This is considerably improved by considering besides DM alsothe algebra C∞(M) of smooth (coordinate) functions on M
• In fact, the Riemannian distance function on M is equal to
d(x , y) = supf ∈C∞(M)
{|f (x)− f (y)| : gradient f ≤ 1}
x y x y
f
• The gradient of f is given by the commutator
[DM , f ] = DM f − fDM (e.g. [DS1 , f ] = −idf
dt)
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Hearing the shape of a drum: motivation from mathKac (1966), Connes (1989)
• The geometry of M is not fully determined by spectrum ofDM .
• This is considerably improved by considering besides DM alsothe algebra C∞(M) of smooth (coordinate) functions on M
• In fact, the Riemannian distance function on M is equal to
d(x , y) = supf ∈C∞(M)
{|f (x)− f (y)| : gradient f ≤ 1}
x y x y
f
• The gradient of f is given by the commutator
[DM , f ] = DM f − fDM (e.g. [DS1 , f ] = −idf
dt)
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Hearing the shape of a drum: motivation from mathKac (1966), Connes (1989)
• The geometry of M is not fully determined by spectrum ofDM .
• This is considerably improved by considering besides DM alsothe algebra C∞(M) of smooth (coordinate) functions on M
• In fact, the Riemannian distance function on M is equal to
d(x , y) = supf ∈C∞(M)
{|f (x)− f (y)| : gradient f ≤ 1}
x y x y
f
• The gradient of f is given by the commutator
[DM , f ] = DM f − fDM (e.g. [DS1 , f ] = −idf
dt)
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Hearing the shape of a drum: motivation from mathKac (1966), Connes (1989)
• The geometry of M is not fully determined by spectrum ofDM .
• This is considerably improved by considering besides DM alsothe algebra C∞(M) of smooth (coordinate) functions on M
• In fact, the Riemannian distance function on M is equal to
d(x , y) = supf ∈C∞(M)
{|f (x)− f (y)| : gradient f ≤ 1}
x y x y
f
• The gradient of f is given by the commutator
[DM , f ] = DM f − fDM (e.g. [DS1 , f ] = −idf
dt)
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Hearing the shape of a drum: motivation from mathKac (1966), Connes (1989)
• The geometry of M is not fully determined by spectrum ofDM .
• This is considerably improved by considering besides DM alsothe algebra C∞(M) of smooth (coordinate) functions on M
• In fact, the Riemannian distance function on M is equal to
d(x , y) = supf ∈C∞(M)
{|f (x)− f (y)| : gradient f ≤ 1}
x y x y
f
• The gradient of f is given by the commutator
[DM , f ] = DM f − fDM (e.g. [DS1 , f ] = −idf
dt)
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NCG and HEP
Replace spacetime by spacetime × finite (nc) space:M × F
• F is considered as internal space (Kaluza–Klein like)
• F is described by a noncommutative algebra, such as M3(C),just as spacetime is described by coordinate functions xµ(p).
• ‘Propagation’ of particles in F is described by a Dirac-typeoperator DF which is actually simply a hermitian matrix.
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NCG and HEP
Replace spacetime by spacetime × finite (nc) space:M × F
• F is considered as internal space (Kaluza–Klein like)
• F is described by a noncommutative algebra, such as M3(C),just as spacetime is described by coordinate functions xµ(p).
• ‘Propagation’ of particles in F is described by a Dirac-typeoperator DF which is actually simply a hermitian matrix.
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NCG and HEP
Replace spacetime by spacetime × finite (nc) space:M × F
• F is considered as internal space (Kaluza–Klein like)
• F is described by a noncommutative algebra, such as M3(C),just as spacetime is described by coordinate functions xµ(p).
• ‘Propagation’ of particles in F is described by a Dirac-typeoperator DF which is actually simply a hermitian matrix.
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NCG and HEP
Replace spacetime by spacetime × finite (nc) space:M × F
• F is considered as internal space (Kaluza–Klein like)
• F is described by a noncommutative algebra, such as M3(C),just as spacetime is described by coordinate functions xµ(p).
• ‘Propagation’ of particles in F is described by a Dirac-typeoperator DF which is actually simply a hermitian matrix.
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Finite spaces
• Finite space F , discrete topology
F = 1 • 2 • · · · · · · N•
• Smooth functions on F are given by N-tuples in CN , and thecorresponding algebra C∞(F ) corresponds to diagonalmatrices
f (1) 0 · · · 00 f (2) · · · 0...
. . ....
0 0 . . . f (N)
• The finite Dirac operator is an arbitrary hermitian matrix DF ,
giving rise to a distance function on F as
d(p, q) = supf ∈C∞(F )
{|f (p)− f (q)| : ‖[DF , f ]‖ ≤ 1}
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Finite spaces
• Finite space F , discrete topology
F = 1 • 2 • · · · · · · N•
• Smooth functions on F are given by N-tuples in CN , and thecorresponding algebra C∞(F ) corresponds to diagonalmatrices
f (1) 0 · · · 00 f (2) · · · 0...
. . ....
0 0 . . . f (N)
• The finite Dirac operator is an arbitrary hermitian matrix DF ,giving rise to a distance function on F as
d(p, q) = supf ∈C∞(F )
{|f (p)− f (q)| : ‖[DF , f ]‖ ≤ 1}
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Finite spaces
• Finite space F , discrete topology
F = 1 • 2 • · · · · · · N•
• Smooth functions on F are given by N-tuples in CN , and thecorresponding algebra C∞(F ) corresponds to diagonalmatrices
f (1) 0 · · · 00 f (2) · · · 0...
. . ....
0 0 . . . f (N)
• The finite Dirac operator is an arbitrary hermitian matrix DF ,
giving rise to a distance function on F as
d(p, q) = supf ∈C∞(F )
{|f (p)− f (q)| : ‖[DF , f ]‖ ≤ 1}
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Example: two-point space
F = 1 • 2•
• Then the algebra of smooth functions
C∞(F ) :=
{(λ1 00 λ2
) ∣∣∣∣λ1, λ2 ∈ C}
• A finite Dirac operator is given by
DF =
(0 cc 0
); (c ∈ C)
• The distance formula then becomes
d(1, 2) =1
|c |
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Example: two-point space
F = 1 • 2•
• Then the algebra of smooth functions
C∞(F ) :=
{(λ1 00 λ2
) ∣∣∣∣λ1, λ2 ∈ C}
• A finite Dirac operator is given by
DF =
(0 cc 0
); (c ∈ C)
• The distance formula then becomes
d(1, 2) =1
|c |
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Example: two-point space
F = 1 • 2•
• Then the algebra of smooth functions
C∞(F ) :=
{(λ1 00 λ2
) ∣∣∣∣λ1, λ2 ∈ C}
• A finite Dirac operator is given by
DF =
(0 cc 0
); (c ∈ C)
• The distance formula then becomes
d(1, 2) =1
|c |
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Finite noncommutative spaces
The geometry of F gets much more interesting if we allow for anoncommutative structure at each point of F .
• Instead of diagonal matrices, we consider block diagonalmatrices
A =
a1 0 · · · 00 a2 · · · 0...
. . ....
0 0 . . . aN
,
where the a1, a2, . . . aN are square matrices of sizen1, n2, . . . , nN .
• Hence we will consider the matrix algebra
AF := Mn1(C)⊕Mn2(C)⊕ · · · ⊕MnN (C)
• A finite Dirac operator is still given by a hermitian matrix.
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Finite noncommutative spaces
The geometry of F gets much more interesting if we allow for anoncommutative structure at each point of F .
• Instead of diagonal matrices, we consider block diagonalmatrices
A =
a1 0 · · · 00 a2 · · · 0...
. . ....
0 0 . . . aN
,
where the a1, a2, . . . aN are square matrices of sizen1, n2, . . . , nN .
• Hence we will consider the matrix algebra
AF := Mn1(C)⊕Mn2(C)⊕ · · · ⊕MnN (C)
• A finite Dirac operator is still given by a hermitian matrix.
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Finite noncommutative spaces
The geometry of F gets much more interesting if we allow for anoncommutative structure at each point of F .
• Instead of diagonal matrices, we consider block diagonalmatrices
A =
a1 0 · · · 00 a2 · · · 0...
. . ....
0 0 . . . aN
,
where the a1, a2, . . . aN are square matrices of sizen1, n2, . . . , nN .
• Hence we will consider the matrix algebra
AF := Mn1(C)⊕Mn2(C)⊕ · · · ⊕MnN (C)
• A finite Dirac operator is still given by a hermitian matrix.
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Finite noncommutative spaces
The geometry of F gets much more interesting if we allow for anoncommutative structure at each point of F .
• Instead of diagonal matrices, we consider block diagonalmatrices
A =
a1 0 · · · 00 a2 · · · 0...
. . ....
0 0 . . . aN
,
where the a1, a2, . . . aN are square matrices of sizen1, n2, . . . , nN .
• Hence we will consider the matrix algebra
AF := Mn1(C)⊕Mn2(C)⊕ · · · ⊕MnN (C)
• A finite Dirac operator is still given by a hermitian matrix.
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Example: noncommutative two-point space
The two-point space can be given a noncommutative structure byconsidering the algebra AF of 3× 3 block diagonal matrices of thefollowing form λ 0 0
0 a11 a12
0 a21 a22
A finite Dirac operator for this example is given by a hermitian3× 3 matrix, for example
DF =
0 c 0c 0 00 0 0
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Spectral triplesNoncommutative Riemannian spin manifolds
(A,H,D)
• Extended to real spectral triple:• J : H → H real structure (charge conjugation)
such thatJ2 = ±1; JD = ±DJ
• Right action of A on H: aop = Ja∗J−1 so that(ab)op = bopaop and
[aop, b] = 0; a, b ∈ A
• D is said to satisfy first-order condition if
[[D, a], bop] = 0
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Spectral triplesNoncommutative Riemannian spin manifolds
(A,H,D)
• Extended to real spectral triple:• J : H → H real structure (charge conjugation)
such thatJ2 = ±1; JD = ±DJ
• Right action of A on H: aop = Ja∗J−1 so that(ab)op = bopaop and
[aop, b] = 0; a, b ∈ A
• D is said to satisfy first-order condition if
[[D, a], bop] = 0
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Spectral triplesNoncommutative Riemannian spin manifolds
(A,H,D)
• Extended to real spectral triple:• J : H → H real structure (charge conjugation)
such thatJ2 = ±1; JD = ±DJ
• Right action of A on H: aop = Ja∗J−1 so that(ab)op = bopaop and
[aop, b] = 0; a, b ∈ A
• D is said to satisfy first-order condition if
[[D, a], bop] = 0
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Spectral triplesNoncommutative Riemannian spin manifolds
(A,H,D)
• Extended to real spectral triple:• J : H → H real structure (charge conjugation)
such thatJ2 = ±1; JD = ±DJ
• Right action of A on H: aop = Ja∗J−1 so that(ab)op = bopaop and
[aop, b] = 0; a, b ∈ A
• D is said to satisfy first-order condition if
[[D, a], bop] = 0
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Spectral invariantsChamseddine–Connes (1996, 1997)
Trace f (D/Λ) +1
2〈Jψ,Dψ〉
• Invariant under unitaries u ∈ U(A) acting as
D 7→ UDU∗; U = u(u∗)op
• Gauge group: G(A) := {u(u∗)op : u ∈ U(A)}.• Compute rhs:
D 7→ D + u[D, u∗]± Ju[D, u∗]J−1
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Spectral invariantsChamseddine–Connes (1996, 1997)
Trace f (D/Λ) +1
2〈Jψ,Dψ〉
• Invariant under unitaries u ∈ U(A) acting as
D 7→ UDU∗; U = u(u∗)op
• Gauge group: G(A) := {u(u∗)op : u ∈ U(A)}.
• Compute rhs:
D 7→ D + u[D, u∗]± Ju[D, u∗]J−1
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Spectral invariantsChamseddine–Connes (1996, 1997)
Trace f (D/Λ) +1
2〈Jψ,Dψ〉
• Invariant under unitaries u ∈ U(A) acting as
D 7→ UDU∗; U = u(u∗)op
• Gauge group: G(A) := {u(u∗)op : u ∈ U(A)}.• Compute rhs:
D 7→ D + u[D, u∗]± Ju[D, u∗]J−1
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Semigroup of inner perturbationsChamseddine–Connes-vS (2013)
Extend this to more general perturbations:
Pert(A) :=
∑j
aj ⊗ bopj ∈ A⊗A
op
∣∣∣∣∣∣∣∣∑j
ajbj = 1∑j
aj ⊗ bopj =
∑j
b∗j ⊗ a∗opj
with semi-group law inherited from product in A⊗Aop.
• U(A) maps to Pert(A) by sending u 7→ u ⊗ u∗op.
• Pert(A) acts on D:
D 7→∑j
ajDbj = D +∑j
aj [D, bj ]
and this also extends to real spectral triples via the map
Pert(A)→ Pert(A⊗ JAJ−1)
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Semigroup of inner perturbationsChamseddine–Connes-vS (2013)
Extend this to more general perturbations:
Pert(A) :=
∑j
aj ⊗ bopj ∈ A⊗A
op
∣∣∣∣∣∣∣∣∑j
ajbj = 1∑j
aj ⊗ bopj =
∑j
b∗j ⊗ a∗opj
with semi-group law inherited from product in A⊗Aop.
• U(A) maps to Pert(A) by sending u 7→ u ⊗ u∗op.
• Pert(A) acts on D:
D 7→∑j
ajDbj = D +∑j
aj [D, bj ]
and this also extends to real spectral triples via the map
Pert(A)→ Pert(A⊗ JAJ−1)
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Semigroup of inner perturbationsChamseddine–Connes-vS (2013)
Extend this to more general perturbations:
Pert(A) :=
∑j
aj ⊗ bopj ∈ A⊗A
op
∣∣∣∣∣∣∣∣∑j
ajbj = 1∑j
aj ⊗ bopj =
∑j
b∗j ⊗ a∗opj
with semi-group law inherited from product in A⊗Aop.
• U(A) maps to Pert(A) by sending u 7→ u ⊗ u∗op.
• Pert(A) acts on D:
D 7→∑j
ajDbj = D +∑j
aj [D, bj ]
and this also extends to real spectral triples via the map
Pert(A)→ Pert(A⊗ JAJ−1)
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Perturbation semigroup for matrix algebras
Proposition
Let AF be the algebra of block diagonal matrices (fixed size).Then the perturbation semigroup of AF is
Pert(AF ) '
∑j
Aj ⊗ Bj ∈ AF ⊗AF
∣∣∣∣∣∣∣∣∑j
Aj(Bj)t = I∑
j
Aj ⊗ Bj =∑j
Bj ⊗ Aj
The semigroup law in Pert(AF ) is given by the matrix product inAF ⊗AF :
(A⊗ B)(A′ ⊗ B ′) = (AA′)⊗ (BB ′).
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Perturbation semigroup for matrix algebras
Proposition
Let AF be the algebra of block diagonal matrices (fixed size).Then the perturbation semigroup of AF is
Pert(AF ) '
∑j
Aj ⊗ Bj ∈ AF ⊗AF
∣∣∣∣∣∣∣∣∑j
Aj(Bj)t = I∑
j
Aj ⊗ Bj =∑j
Bj ⊗ Aj
The semigroup law in Pert(AF ) is given by the matrix product inAF ⊗AF :
(A⊗ B)(A′ ⊗ B ′) = (AA′)⊗ (BB ′).
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Example: perturbation semigroup of two-point space
• Now AF = C2, the algebra of diagonal 2× 2 matrices.
• In terms of the standard basis of such matrices
e11 =
(1 00 0
), e22 =
(0 00 1
)we can write an arbitrary element of Pert(C2) as
z1e11 ⊗ e11 + z2e11 ⊗ e22 + z3e22 ⊗ e11 + z4e22 ⊗ e22
• Matrix multiplying e11 and e22 yields for the normalizationcondition:
z1 = 1 = z4.
• The self-adjointness condition reads
z2 = z3
leaving only one free complex parameter so thatPert(C2) ' C.
• More generally, Pert(CN) ' CN(N−1)/2 with componentwiseproduct.
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Example: perturbation semigroup of two-point space
• Now AF = C2, the algebra of diagonal 2× 2 matrices.• In terms of the standard basis of such matrices
e11 =
(1 00 0
), e22 =
(0 00 1
)we can write an arbitrary element of Pert(C2) as
z1e11 ⊗ e11 + z2e11 ⊗ e22 + z3e22 ⊗ e11 + z4e22 ⊗ e22
• Matrix multiplying e11 and e22 yields for the normalizationcondition:
z1 = 1 = z4.
• The self-adjointness condition reads
z2 = z3
leaving only one free complex parameter so thatPert(C2) ' C.
• More generally, Pert(CN) ' CN(N−1)/2 with componentwiseproduct.
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Example: perturbation semigroup of two-point space
• Now AF = C2, the algebra of diagonal 2× 2 matrices.• In terms of the standard basis of such matrices
e11 =
(1 00 0
), e22 =
(0 00 1
)we can write an arbitrary element of Pert(C2) as
z1e11 ⊗ e11 + z2e11 ⊗ e22 + z3e22 ⊗ e11 + z4e22 ⊗ e22
• Matrix multiplying e11 and e22 yields for the normalizationcondition:
z1 = 1 = z4.
• The self-adjointness condition reads
z2 = z3
leaving only one free complex parameter so thatPert(C2) ' C.
• More generally, Pert(CN) ' CN(N−1)/2 with componentwiseproduct.
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Example: perturbation semigroup of two-point space
• Now AF = C2, the algebra of diagonal 2× 2 matrices.• In terms of the standard basis of such matrices
e11 =
(1 00 0
), e22 =
(0 00 1
)we can write an arbitrary element of Pert(C2) as
z1e11 ⊗ e11 + z2e11 ⊗ e22 + z3e22 ⊗ e11 + z4e22 ⊗ e22
• Matrix multiplying e11 and e22 yields for the normalizationcondition:
z1 = 1 = z4.
• The self-adjointness condition reads
z2 = z3
leaving only one free complex parameter so thatPert(C2) ' C.
• More generally, Pert(CN) ' CN(N−1)/2 with componentwiseproduct.
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Example: perturbation semigroup of two-point space
• Now AF = C2, the algebra of diagonal 2× 2 matrices.• In terms of the standard basis of such matrices
e11 =
(1 00 0
), e22 =
(0 00 1
)we can write an arbitrary element of Pert(C2) as
z1e11 ⊗ e11 + z2e11 ⊗ e22 + z3e22 ⊗ e11 + z4e22 ⊗ e22
• Matrix multiplying e11 and e22 yields for the normalizationcondition:
z1 = 1 = z4.
• The self-adjointness condition reads
z2 = z3
leaving only one free complex parameter so thatPert(C2) ' C.
• More generally, Pert(CN) ' CN(N−1)/2 with componentwiseproduct.
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Example: perturbation semigroup of M2(C)
• Let us consider a noncommutative example, AF = M2(C).
• We can identify M2(C)⊗M2(C) with M4(C) so that elementsin Pert(M2(C) are 4× 4-matrices satisfying the normalizationand self-adjointness condition. In a suitable basis:
Pert(M2(C)) =
1 v1 v2 iv3
0 x1 x2 ix3
0 x4 x5 ix6
0 ix7 ix8 x9
∣∣∣∣ v1, v2, v3 ∈ Rx1, . . . x9 ∈ R
and one can show that
Pert(M2(C)) ' R3 o S .
• More generally (B.Sc. thesis Niels Neumann),
Pert(MN(C)) 'W o S ′.
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Example: perturbation semigroup of M2(C)
• Let us consider a noncommutative example, AF = M2(C).
• We can identify M2(C)⊗M2(C) with M4(C) so that elementsin Pert(M2(C) are 4× 4-matrices satisfying the normalizationand self-adjointness condition. In a suitable basis:
Pert(M2(C)) =
1 v1 v2 iv3
0 x1 x2 ix3
0 x4 x5 ix6
0 ix7 ix8 x9
∣∣∣∣ v1, v2, v3 ∈ Rx1, . . . x9 ∈ R
and one can show that
Pert(M2(C)) ' R3 o S .
• More generally (B.Sc. thesis Niels Neumann),
Pert(MN(C)) 'W o S ′.
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Example: perturbation semigroup of M2(C)
• Let us consider a noncommutative example, AF = M2(C).
• We can identify M2(C)⊗M2(C) with M4(C) so that elementsin Pert(M2(C) are 4× 4-matrices satisfying the normalizationand self-adjointness condition. In a suitable basis:
Pert(M2(C)) =
1 v1 v2 iv3
0 x1 x2 ix3
0 x4 x5 ix6
0 ix7 ix8 x9
∣∣∣∣ v1, v2, v3 ∈ Rx1, . . . x9 ∈ R
and one can show that
Pert(M2(C)) ' R3 o S .
• More generally (B.Sc. thesis Niels Neumann),
Pert(MN(C)) 'W o S ′.
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Example: noncommutative two-point space
• Consider noncommutative two-point space described byC⊕M2(C)
• It turns out that
Pert(C⊕M2(C)) ' M2(C)× Pert(M2(C))
• Only M2(C) ⊂ Pert(C⊕M2(C)) acts non-trivially on DF :
DF =
0 c 0c 0 00 0 0
7→ 0 cφ1 cφ2
cφ1 0 0cφ2 0 0
• Physicists call φ1 and φ2 the Higgs field.
• The group of unitary block diagonal matrices is nowU(1)× U(2) and an element (λ, u) therein acts as(
φ1
φ2
)7→ λu
(φ1
φ2
).
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Example: noncommutative two-point space
• Consider noncommutative two-point space described byC⊕M2(C)
• It turns out that
Pert(C⊕M2(C)) ' M2(C)× Pert(M2(C))
• Only M2(C) ⊂ Pert(C⊕M2(C)) acts non-trivially on DF :
DF =
0 c 0c 0 00 0 0
7→ 0 cφ1 cφ2
cφ1 0 0cφ2 0 0
• Physicists call φ1 and φ2 the Higgs field.
• The group of unitary block diagonal matrices is nowU(1)× U(2) and an element (λ, u) therein acts as(
φ1
φ2
)7→ λu
(φ1
φ2
).
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Example: noncommutative two-point space
• Consider noncommutative two-point space described byC⊕M2(C)
• It turns out that
Pert(C⊕M2(C)) ' M2(C)× Pert(M2(C))
• Only M2(C) ⊂ Pert(C⊕M2(C)) acts non-trivially on DF :
DF =
0 c 0c 0 00 0 0
7→ 0 cφ1 cφ2
cφ1 0 0cφ2 0 0
• Physicists call φ1 and φ2 the Higgs field.
• The group of unitary block diagonal matrices is nowU(1)× U(2) and an element (λ, u) therein acts as(
φ1
φ2
)7→ λu
(φ1
φ2
).
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Example: noncommutative two-point space
• Consider noncommutative two-point space described byC⊕M2(C)
• It turns out that
Pert(C⊕M2(C)) ' M2(C)× Pert(M2(C))
• Only M2(C) ⊂ Pert(C⊕M2(C)) acts non-trivially on DF :
DF =
0 c 0c 0 00 0 0
7→ 0 cφ1 cφ2
cφ1 0 0cφ2 0 0
• Physicists call φ1 and φ2 the Higgs field.
• The group of unitary block diagonal matrices is nowU(1)× U(2) and an element (λ, u) therein acts as(
φ1
φ2
)7→ λu
(φ1
φ2
).
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Example: noncommutative two-point space
• Consider noncommutative two-point space described byC⊕M2(C)
• It turns out that
Pert(C⊕M2(C)) ' M2(C)× Pert(M2(C))
• Only M2(C) ⊂ Pert(C⊕M2(C)) acts non-trivially on DF :
DF =
0 c 0c 0 00 0 0
7→ 0 cφ1 cφ2
cφ1 0 0cφ2 0 0
• Physicists call φ1 and φ2 the Higgs field.
• The group of unitary block diagonal matrices is nowU(1)× U(2) and an element (λ, u) therein acts as(
φ1
φ2
)7→ λu
(φ1
φ2
).
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Example: perturbation semigroup of a manifold
Recall, for any involutive algebra A
Pert(A) :=
∑j
aj ⊗ bopj ∈ A⊗A
op
∣∣∣∣∣∣∣∣∑j
ajbj = 1∑j
aj ⊗ bopj =
∑j
b∗j ⊗ a∗opj
• We can consider functions in C∞(M)⊗ C∞(M) as functions
of two variables in C∞(M ×M).
• The normalization and self-adjointness condition inPert(C∞(M)) translate accordingly and yield
Pert(C∞(M)) =
{f ∈ C∞(M ×M)
∣∣∣∣ f (x , x) = 1
f (x , y) = f (y , x)
}• The action of Pert(C∞(M)) on the partial derivatives
appearing in a Dirac operator DM is given by
∂
∂xµ7→ ∂
∂xµ+
∂
∂yµf (x , y)
∣∣∣∣y=x
=: ∂µ + Aµ
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Example: perturbation semigroup of a manifold
Recall, for any involutive algebra A
Pert(A) :=
∑j
aj ⊗ bopj ∈ A⊗A
op
∣∣∣∣∣∣∣∣∑j
ajbj = 1∑j
aj ⊗ bopj =
∑j
b∗j ⊗ a∗opj
• We can consider functions in C∞(M)⊗ C∞(M) as functions
of two variables in C∞(M ×M).• The normalization and self-adjointness condition in
Pert(C∞(M)) translate accordingly and yield
Pert(C∞(M)) =
{f ∈ C∞(M ×M)
∣∣∣∣ f (x , x) = 1
f (x , y) = f (y , x)
}
• The action of Pert(C∞(M)) on the partial derivativesappearing in a Dirac operator DM is given by
∂
∂xµ7→ ∂
∂xµ+
∂
∂yµf (x , y)
∣∣∣∣y=x
=: ∂µ + Aµ
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Example: perturbation semigroup of a manifold
Recall, for any involutive algebra A
Pert(A) :=
∑j
aj ⊗ bopj ∈ A⊗A
op
∣∣∣∣∣∣∣∣∑j
ajbj = 1∑j
aj ⊗ bopj =
∑j
b∗j ⊗ a∗opj
• We can consider functions in C∞(M)⊗ C∞(M) as functions
of two variables in C∞(M ×M).• The normalization and self-adjointness condition in
Pert(C∞(M)) translate accordingly and yield
Pert(C∞(M)) =
{f ∈ C∞(M ×M)
∣∣∣∣ f (x , x) = 1
f (x , y) = f (y , x)
}• The action of Pert(C∞(M)) on the partial derivatives
appearing in a Dirac operator DM is given by
∂
∂xµ7→ ∂
∂xµ+
∂
∂yµf (x , y)
∣∣∣∣y=x
=: ∂µ + Aµ
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Applications to particle physics
• Combine (4d) Riemannian spin manifold M with finitenoncommutative space F :
M × F
• F is internal space at each point of M
• Described by matrix-valued functions on M: algebraC∞(M,AF )
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Applications to particle physics
• Combine (4d) Riemannian spin manifold M with finitenoncommutative space F :
M × F
• F is internal space at each point of M
• Described by matrix-valued functions on M: algebraC∞(M,AF )
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Applications to particle physics
• Combine (4d) Riemannian spin manifold M with finitenoncommutative space F :
M × F
• F is internal space at each point of M
• Described by matrix-valued functions on M: algebraC∞(M,AF )
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Dirac operator on M × F
• Recall the form of DM :
DM =
(0 D+
MD−M 0
).
• Dirac operator on M × F is the combination
DM×F = DM + γ5DF =
(DF D+
MD−M −DF
).
• The crucial property of this specific form is that it squares tothe sum of the two Laplacians on M and F :
D2M×F = D2
M + D2F
• Using this, we can expand the heat trace:
Trace e−D2M×F /Λ2
=Vol(M)Λ4
(4π)2Trace
(1−
D2F
Λ2+
D4F
2Λ4
)+O(Λ−1).
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Dirac operator on M × F
• Recall the form of DM :
DM =
(0 D+
MD−M 0
).
• Dirac operator on M × F is the combination
DM×F = DM + γ5DF =
(DF D+
MD−M −DF
).
• The crucial property of this specific form is that it squares tothe sum of the two Laplacians on M and F :
D2M×F = D2
M + D2F
• Using this, we can expand the heat trace:
Trace e−D2M×F /Λ2
=Vol(M)Λ4
(4π)2Trace
(1−
D2F
Λ2+
D4F
2Λ4
)+O(Λ−1).
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Dirac operator on M × F
• Recall the form of DM :
DM =
(0 D+
MD−M 0
).
• Dirac operator on M × F is the combination
DM×F = DM + γ5DF =
(DF D+
MD−M −DF
).
• The crucial property of this specific form is that it squares tothe sum of the two Laplacians on M and F :
D2M×F = D2
M + D2F
• Using this, we can expand the heat trace:
Trace e−D2M×F /Λ2
=Vol(M)Λ4
(4π)2Trace
(1−
D2F
Λ2+
D4F
2Λ4
)+O(Λ−1).
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Dirac operator on M × F
• Recall the form of DM :
DM =
(0 D+
MD−M 0
).
• Dirac operator on M × F is the combination
DM×F = DM + γ5DF =
(DF D+
MD−M −DF
).
• The crucial property of this specific form is that it squares tothe sum of the two Laplacians on M and F :
D2M×F = D2
M + D2F
• Using this, we can expand the heat trace:
Trace e−D2M×F /Λ2
=Vol(M)Λ4
(4π)2Trace
(1−
D2F
Λ2+
D4F
2Λ4
)+O(Λ−1).
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The Higgs mechanism
We apply this to the noncommutative two-point space describedbefore
• Algebra AF = C⊕M2(C)
• Perturbation of Dirac operator DM parametrized by gaugebosons for U(1)× U(2).
• Perturbation of finite Dirac operator DF parametrized byφ1, φ2.
• Spectral action for the perturbed Dirac operator induces apotential:
V (φ) = −2Λ2(|φ1|2 + |φ2|2) + (|φ1|2 + |φ2|2)2
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The Higgs mechanism
We apply this to the noncommutative two-point space describedbefore
• Algebra AF = C⊕M2(C)• Perturbation of Dirac operator DM parametrized by gauge
bosons for U(1)× U(2).
• Perturbation of finite Dirac operator DF parametrized byφ1, φ2.
• Spectral action for the perturbed Dirac operator induces apotential:
V (φ) = −2Λ2(|φ1|2 + |φ2|2) + (|φ1|2 + |φ2|2)2
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The Higgs mechanism
We apply this to the noncommutative two-point space describedbefore
• Algebra AF = C⊕M2(C)• Perturbation of Dirac operator DM parametrized by gauge
bosons for U(1)× U(2).• Perturbation of finite Dirac operator DF parametrized byφ1, φ2.
• Spectral action for the perturbed Dirac operator induces apotential:
V (φ) = −2Λ2(|φ1|2 + |φ2|2) + (|φ1|2 + |φ2|2)2
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The Higgs mechanism
We apply this to the noncommutative two-point space describedbefore
• Algebra AF = C⊕M2(C)• Perturbation of Dirac operator DM parametrized by gauge
bosons for U(1)× U(2).• Perturbation of finite Dirac operator DF parametrized byφ1, φ2.
• Spectral action for the perturbed Dirac operator induces apotential:
V (φ) = −2Λ2(|φ1|2 + |φ2|2) + (|φ1|2 + |φ2|2)2
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The spectral Standard Model
Describe M × FSM by [CCM 2007]
• Coordinates: xµ(p) ∈ C⊕H⊕M3(C) (with unimodularunitaries U(1)Y × SU(2)L × SU(3)).
• Dirac operator DM×F = DM + γ5DF where
DF =
(S T ∗
T S
)is a 96× 96-dimensional hermitian matrix where 96 is:
3 × 2 ×( 2⊗ 1 + 1⊗ 1 + 1⊗ 1 + 2⊗ 3 + 1⊗ 3 + 1⊗ 3 )
families
anti-particles
(νL, eL) νR eR (uL, dL) uR dR
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The spectral Standard Model
Describe M × FSM by [CCM 2007]
• Coordinates: xµ(p) ∈ C⊕H⊕M3(C) (with unimodularunitaries U(1)Y × SU(2)L × SU(3)).
• Dirac operator DM×F = DM + γ5DF where
DF =
(S T ∗
T S
)is a 96× 96-dimensional hermitian matrix where 96 is:
3 × 2 ×( 2⊗ 1 + 1⊗ 1 + 1⊗ 1 + 2⊗ 3 + 1⊗ 3 + 1⊗ 3 )
families
anti-particles
(νL, eL) νR eR (uL, dL) uR dR
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The spectral Standard Model
Describe M × FSM by [CCM 2007]
• Coordinates: xµ(p) ∈ C⊕H⊕M3(C) (with unimodularunitaries U(1)Y × SU(2)L × SU(3)).
• Dirac operator DM×F = DM + γ5DF where
DF =
(S T ∗
T S
)is a 96× 96-dimensional hermitian matrix where 96 is:
3 × 2 ×( 2⊗ 1 + 1⊗ 1 + 1⊗ 1 + 2⊗ 3 + 1⊗ 3 + 1⊗ 3 )
families
anti-particles
(νL, eL) νR eR (uL, dL) uR dR
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The spectral Standard Model
Describe M × FSM by [CCM 2007]
• Coordinates: xµ(p) ∈ C⊕H⊕M3(C) (with unimodularunitaries U(1)Y × SU(2)L × SU(3)).
• Dirac operator DM×F = DM + γ5DF where
DF =
(S T ∗
T S
)is a 96× 96-dimensional hermitian matrix where 96 is:
3 × 2 ×( 2⊗ 1 + 1⊗ 1 + 1⊗ 1 + 2⊗ 3 + 1⊗ 3 + 1⊗ 3 )
families
anti-particles
(νL, eL) νR eR (uL, dL) uR dR
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The spectral Standard Model
Describe M × FSM by [CCM 2007]
• Coordinates: xµ(p) ∈ C⊕H⊕M3(C) (with unimodularunitaries U(1)Y × SU(2)L × SU(3)).
• Dirac operator DM×F = DM + γ5DF where
DF =
(S T ∗
T S
)is a 96× 96-dimensional hermitian matrix where 96 is:
3 × 2 ×( 2⊗ 1 + 1⊗ 1 + 1⊗ 1 + 2⊗ 3 + 1⊗ 3 + 1⊗ 3 )
families
anti-particles
(νL, eL) νR eR (uL, dL) uR dR
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The Dirac operator on FSM
DF =
(S T ∗
T S
)
• The operator S is given by
Sl :=
0 0 Yν 00 0 0 Ye
Y ∗ν 0 0 00 Y ∗e 0 0
, Sq ⊗ 13 =
0 0 Yu 00 0 0 Yd
Y ∗u 0 0 00 Y ∗d 0 0
⊗ 13,
where Yν , Ye , Yu and Yd are 3× 3 mass matrices acting onthe three generations.
• The symmetric operator T only acts on the right-handed(anti)neutrinos, TνR = YRνR for a 3× 3 symmetric Majoranamass matrix YR , and Tf = 0 for all other fermions f 6= νR .
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The Dirac operator on FSM
DF =
(S T ∗
T S
)
• The operator S is given by
Sl :=
0 0 Yν 00 0 0 Ye
Y ∗ν 0 0 00 Y ∗e 0 0
, Sq ⊗ 13 =
0 0 Yu 00 0 0 Yd
Y ∗u 0 0 00 Y ∗d 0 0
⊗ 13,
where Yν , Ye , Yu and Yd are 3× 3 mass matrices acting onthe three generations.
• The symmetric operator T only acts on the right-handed(anti)neutrinos, TνR = YRνR for a 3× 3 symmetric Majoranamass matrix YR , and Tf = 0 for all other fermions f 6= νR .
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The Dirac operator on FSM
DF =
(S T ∗
T S
)
• The operator S is given by
Sl :=
0 0 Yν 00 0 0 Ye
Y ∗ν 0 0 00 Y ∗e 0 0
, Sq ⊗ 13 =
0 0 Yu 00 0 0 Yd
Y ∗u 0 0 00 Y ∗d 0 0
⊗ 13,
where Yν , Ye , Yu and Yd are 3× 3 mass matrices acting onthe three generations.
• The symmetric operator T only acts on the right-handed(anti)neutrinos, TνR = YRνR for a 3× 3 symmetric Majoranamass matrix YR , and Tf = 0 for all other fermions f 6= νR .
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Inner perturbations
• Inner perturbations of DM give a matrix
Aµ =
Bµ 0 0 0
0 W 3µ W +
µ 0
0 W−µ −W 3
µ 00 0 0 (G a
µ)
corresponding to hypercharge, weak and strong interaction.
• Inner perturbations of DF give(Yν 00 Ye
)
(Yνφ1 −Yeφ2
Yνφ2 Yeφ1
)corresponding to SM-Higgs field. Similarly for Yu,Yd .
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Inner perturbations
• Inner perturbations of DM give a matrix
Aµ =
Bµ 0 0 0
0 W 3µ W +
µ 0
0 W−µ −W 3
µ 00 0 0 (G a
µ)
corresponding to hypercharge, weak and strong interaction.
• Inner perturbations of DF give(Yν 00 Ye
)
(Yνφ1 −Yeφ2
Yνφ2 Yeφ1
)corresponding to SM-Higgs field. Similarly for Yu,Yd .
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Dynamics and interactions
If we consider the spectral action:
Trace f (DM/Λ) ∼ c0
∫FµνFµν−c ′2|φ|2 + c ′0|φ|4 + · · ·
we observe [CCM 2007]:
• The coupling constants of hypercharge, weak and stronginteraction are expressed in terms of the single constant c0
which implies
g 23 = g 2
2 =5
3g 2
1
In other words, there should be grand unification.
• Moreover, the quartic Higgs coupling λ is related via
λ ≈ 243 + ρ4
(3 + ρ2)2g 2
2 ; ρ =mν
mtop
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Dynamics and interactions
If we consider the spectral action:
Trace f (DM/Λ) ∼ c0
∫FµνFµν−c ′2|φ|2 + c ′0|φ|4 + · · ·
we observe [CCM 2007]:
• The coupling constants of hypercharge, weak and stronginteraction are expressed in terms of the single constant c0
which implies
g 23 = g 2
2 =5
3g 2
1
In other words, there should be grand unification.
• Moreover, the quartic Higgs coupling λ is related via
λ ≈ 243 + ρ4
(3 + ρ2)2g 2
2 ; ρ =mν
mtop
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Dynamics and interactions
If we consider the spectral action:
Trace f (DM/Λ) ∼ c0
∫FµνFµν−c ′2|φ|2 + c ′0|φ|4 + · · ·
we observe [CCM 2007]:
• The coupling constants of hypercharge, weak and stronginteraction are expressed in terms of the single constant c0
which implies
g 23 = g 2
2 =5
3g 2
1
In other words, there should be grand unification.
• Moreover, the quartic Higgs coupling λ is related via
λ ≈ 243 + ρ4
(3 + ρ2)2g 2
2 ; ρ =mν
mtop
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Phenomenology of the spectral Standard Model
This can be used to derive predictions as follows:
• Interpret the spectral action as an effective field theory atΛGUT ≈ 1013 − 1016 GeV.
• Run the quartic coupling constant λ to SM-energies to predict
m2h =
4λM2W
3g 22
2 4 6 8 10 12 14 16
1.1
1.2
1.3
1.4
1.5
1.6
log10 H�GeVL
Λ
This gives [CCM 2007]
167 GeV ≤ mh ≤ 176 GeV
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Phenomenology of the spectral Standard Model
This can be used to derive predictions as follows:
• Interpret the spectral action as an effective field theory atΛGUT ≈ 1013 − 1016 GeV.
• Run the quartic coupling constant λ to SM-energies to predict
m2h =
4λM2W
3g 22
2 4 6 8 10 12 14 16
1.1
1.2
1.3
1.4
1.5
1.6
log10 H�GeVL
Λ
This gives [CCM 2007]
167 GeV ≤ mh ≤ 176 GeV
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Phenomenology of the spectral Standard Model
This can be used to derive predictions as follows:
• Interpret the spectral action as an effective field theory atΛGUT ≈ 1013 − 1016 GeV.
• Run the quartic coupling constant λ to SM-energies to predict
m2h =
4λM2W
3g 22
2 4 6 8 10 12 14 16
1.1
1.2
1.3
1.4
1.5
1.6
log10 H�GeVL
Λ
This gives [CCM 2007]
167 GeV ≤ mh ≤ 176 GeV
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Three problems
1 This prediction is falsified by thenow measured value.
2 In the Standard Model there isnot the presumed grandunification.
3 There is a problem with the lowvalue of mh, making the Higgsvacuum un/metastable[Elias-Miro et al. 2011].
5 10 15
0.5
0.6
0.7
0.8
log10 H�GeVL
gaug
eco
uplin
gs
g3
g2
5�3 g1
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Three problems
1 This prediction is falsified by thenow measured value.
2 In the Standard Model there isnot the presumed grandunification.
3 There is a problem with the lowvalue of mh, making the Higgsvacuum un/metastable[Elias-Miro et al. 2011].
5 10 15
0.5
0.6
0.7
0.8
log10 H�GeVL
gaug
eco
uplin
gs
g3
g2
5�3 g1
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Three problems
1 This prediction is falsified by thenow measured value.
2 In the Standard Model there isnot the presumed grandunification.
3 There is a problem with the lowvalue of mh, making the Higgsvacuum un/metastable[Elias-Miro et al. 2011].
5 10 15
0.5
0.6
0.7
0.8
log10 H�GeVL
gaug
eco
uplin
gs
g3
g2
5�3 g1
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Beyond the SM with noncommutative geometryA solution to the above three problems?
• The matrix coordinates of the Standard Model arise naturallyas a restriction of the following coordinates
xµ(p) =(qµR(p), qµL (p),mµ(p)
)∈ HR ⊕HL ⊕M4(C)
corresponding to a Pati–Salam unification:
U(1)Y × SU(2)L × SU(3)→ SU(2)R × SU(2)L × SU(4)
• The 96 fermionic degrees of freedom are structured as(νR uiR νL uiL
eR diR eL diL
)(i = 1, 2, 3)
• Again the finite Dirac operator is a 96× 96-dimensionalmatrix (details in [CCS 2013]).
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Beyond the SM with noncommutative geometryA solution to the above three problems?
• The matrix coordinates of the Standard Model arise naturallyas a restriction of the following coordinates
xµ(p) =(qµR(p), qµL (p),mµ(p)
)∈ HR ⊕HL ⊕M4(C)
corresponding to a Pati–Salam unification:
U(1)Y × SU(2)L × SU(3)→ SU(2)R × SU(2)L × SU(4)
• The 96 fermionic degrees of freedom are structured as(νR uiR νL uiL
eR diR eL diL
)(i = 1, 2, 3)
• Again the finite Dirac operator is a 96× 96-dimensionalmatrix (details in [CCS 2013]).
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Beyond the SM with noncommutative geometryA solution to the above three problems?
• The matrix coordinates of the Standard Model arise naturallyas a restriction of the following coordinates
xµ(p) =(qµR(p), qµL (p),mµ(p)
)∈ HR ⊕HL ⊕M4(C)
corresponding to a Pati–Salam unification:
U(1)Y × SU(2)L × SU(3)→ SU(2)R × SU(2)L × SU(4)
• The 96 fermionic degrees of freedom are structured as(νR uiR νL uiL
eR diR eL diL
)(i = 1, 2, 3)
• Again the finite Dirac operator is a 96× 96-dimensionalmatrix (details in [CCS 2013]).
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Inner perturbations
• Inner perturbations of DM now give three gauge bosons:
W µR , W µ
L , V µ
corresponding to SU(2)R × SU(2)L × SU(4).
• For the inner perturbations of DF we distinguish two cases,depending on the initial form of DF :
I The Standard Model DF =
(S T ∗
T S
)II A more general DF with zero f L − fL-interactions.
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Inner perturbations
• Inner perturbations of DM now give three gauge bosons:
W µR , W µ
L , V µ
corresponding to SU(2)R × SU(2)L × SU(4).
• For the inner perturbations of DF we distinguish two cases,depending on the initial form of DF :
I The Standard Model DF =
(S T ∗
T S
)II A more general DF with zero f L − fL-interactions.
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Inner perturbations
• Inner perturbations of DM now give three gauge bosons:
W µR , W µ
L , V µ
corresponding to SU(2)R × SU(2)L × SU(4).
• For the inner perturbations of DF we distinguish two cases,depending on the initial form of DF :
I The Standard Model DF =
(S T ∗
T S
)
II A more general DF with zero f L − fL-interactions.
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Inner perturbations
• Inner perturbations of DM now give three gauge bosons:
W µR , W µ
L , V µ
corresponding to SU(2)R × SU(2)L × SU(4).
• For the inner perturbations of DF we distinguish two cases,depending on the initial form of DF :
I The Standard Model DF =
(S T ∗
T S
)II A more general DF with zero f L − fL-interactions.
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Scalar sector of the spectral Pati–Salam model
Case I For a SM DF , the resulting scalar fields are composite fields,expressed in scalar fields whose representations are:
SU(2)R SU(2)L SU(4)
φba 2 2 1∆aI 2 1 4
ΣIJ 1 1 15
Case II For a more general finite Dirac operator, we have fundamentalscalar fields:
particle SU(2)R SU(2)L SU(4)
ΣbJaJ 2 2 1 + 15
HaI bJ
{31
11
106
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Scalar sector of the spectral Pati–Salam model
Case I For a SM DF , the resulting scalar fields are composite fields,expressed in scalar fields whose representations are:
SU(2)R SU(2)L SU(4)
φba 2 2 1∆aI 2 1 4
ΣIJ 1 1 15
Case II For a more general finite Dirac operator, we have fundamentalscalar fields:
particle SU(2)R SU(2)L SU(4)
ΣbJaJ 2 2 1 + 15
HaI bJ
{31
11
106
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Action functional
As for the Standard Model, we can compute the spectral actionwhich describes the usual Pati–Salam model with
• unification of the gauge couplings
gR = gL = g .
• A rather involved, fixed scalar potential, still subject to furtherstudy
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Phenomenology of the spectral Pati–Salam model
However, independently from the spectral action, we can analyzethe running at one loop of the gauge couplings [CCS 2015]:
1 We run the Standard Model gauge couplings up to apresumed PS → SM symmetry breaking scale mR
2 We take their values as boundary conditions to thePati–Salam gauge couplings gR , gL, g at this scale via
1
g 21
=2
3
1
g 2+
1
g 2R
,1
g 22
=1
g 2L
,1
g 23
=1
g 2,
3 Vary mR in a search for a unification scale Λ where
gR = gL = g
which is where the spectral action is valid as an effectivetheory.
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Phenomenology of the spectral Pati–Salam model
However, independently from the spectral action, we can analyzethe running at one loop of the gauge couplings [CCS 2015]:
1 We run the Standard Model gauge couplings up to apresumed PS → SM symmetry breaking scale mR
2 We take their values as boundary conditions to thePati–Salam gauge couplings gR , gL, g at this scale via
1
g 21
=2
3
1
g 2+
1
g 2R
,1
g 22
=1
g 2L
,1
g 23
=1
g 2,
3 Vary mR in a search for a unification scale Λ where
gR = gL = g
which is where the spectral action is valid as an effectivetheory.
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Phenomenology of the spectral Pati–Salam model
However, independently from the spectral action, we can analyzethe running at one loop of the gauge couplings [CCS 2015]:
1 We run the Standard Model gauge couplings up to apresumed PS → SM symmetry breaking scale mR
2 We take their values as boundary conditions to thePati–Salam gauge couplings gR , gL, g at this scale via
1
g 21
=2
3
1
g 2+
1
g 2R
,1
g 22
=1
g 2L
,1
g 23
=1
g 2,
3 Vary mR in a search for a unification scale Λ where
gR = gL = g
which is where the spectral action is valid as an effectivetheory.
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Phenomenology of the spectral Pati–Salam model
However, independently from the spectral action, we can analyzethe running at one loop of the gauge couplings [CCS 2015]:
1 We run the Standard Model gauge couplings up to apresumed PS → SM symmetry breaking scale mR
2 We take their values as boundary conditions to thePati–Salam gauge couplings gR , gL, g at this scale via
1
g 21
=2
3
1
g 2+
1
g 2R
,1
g 22
=1
g 2L
,1
g 23
=1
g 2,
3 Vary mR in a search for a unification scale Λ where
gR = gL = g
which is where the spectral action is valid as an effectivetheory.
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Phenomenology of the spectral Pati–Salam modelCase I: Standard Model DF
For the Standard Model Dirac operator, we have found that withmR ≈ 4.25× 1013 GeV there is unification at Λ ≈ 2.5× 1015 GeV:
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Phenomenology of the spectral Pati–Salam modelCase I: Standard Model DF
In this case, we can also say something about the scalar particlesthat remain after SSB:
U(1)Y SU(2)L SU(3)(φ0
1
φ+1
)=
(φ1
1φ2
1
)1 2 1(
φ−2φ0
2
)=
(φ1
2φ2
2
)−1 2 1
σ 0 1 1
η −2
31 3
• It turns out that these scalar fields have a little influence onthe running of the SM-gauge couplings (at one loop).
• However, this sector contains the real scalar singlet σ thatallowed for a realistic Higgs mass and that stabilizes the Higgsvacuum [CC 2012].
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Phenomenology of the spectral Pati–Salam modelCase I: Standard Model DF
In this case, we can also say something about the scalar particlesthat remain after SSB:
U(1)Y SU(2)L SU(3)(φ0
1
φ+1
)=
(φ1
1φ2
1
)1 2 1(
φ−2φ0
2
)=
(φ1
2φ2
2
)−1 2 1
σ 0 1 1
η −2
31 3
• It turns out that these scalar fields have a little influence onthe running of the SM-gauge couplings (at one loop).
• However, this sector contains the real scalar singlet σ thatallowed for a realistic Higgs mass and that stabilizes the Higgsvacuum [CC 2012].
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![Page 111: Beyond the spectral Standard Model: Pati-Salam …math_ru_nl...Beyond the spectral Standard Model: Pati-Salam uni cation Walter van Suijlekom (joint with Ali Chamseddine and Alain](https://reader033.vdocuments.mx/reader033/viewer/2022043016/5f3891f277d6a40281376a0d/html5/thumbnails/111.jpg)
Phenomenology of the spectral Pati–Salam modelCase I: Standard Model DF
In this case, we can also say something about the scalar particlesthat remain after SSB:
U(1)Y SU(2)L SU(3)(φ0
1
φ+1
)=
(φ1
1φ2
1
)1 2 1(
φ−2φ0
2
)=
(φ1
2φ2
2
)−1 2 1
σ 0 1 1
η −2
31 3
• It turns out that these scalar fields have a little influence onthe running of the SM-gauge couplings (at one loop).
• However, this sector contains the real scalar singlet σ thatallowed for a realistic Higgs mass and that stabilizes the Higgsvacuum [CC 2012].
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![Page 112: Beyond the spectral Standard Model: Pati-Salam …math_ru_nl...Beyond the spectral Standard Model: Pati-Salam uni cation Walter van Suijlekom (joint with Ali Chamseddine and Alain](https://reader033.vdocuments.mx/reader033/viewer/2022043016/5f3891f277d6a40281376a0d/html5/thumbnails/112.jpg)
Phenomenology of the spectral Pati–Salam modelCase II: General Dirac
For the more general case, we have found that withmR ≈ 1.5× 1011 GeV there is unification at Λ ≈ 6.3× 1016 GeV:
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Conclusion
We have arrived at a spectral Pati–Salam model that
• goes beyond the Standard Model
• has a fixed scalar sector once the finite Dirac operator hasbeen fixed (only a few scenarios)
• exhibits grand unification for all of these scenarios (confirmedby [Aydemir–Minic–Sun–Takeuchi 2015])
• the scalar sector has the potential to stabilize the Higgsvacuum and allow for a realistic Higgs mass.
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Further reading
A. Chamseddine, A. Connes, WvS.
Beyond the Spectral Standard Model: Emergence ofPati-Salam Unification. JHEP 11 (2013) 132.[arXiv:1304.8050]
Grand Unification in the Spectral Pati-Salam Model. JHEP 11(2015) 011. [arXiv:1507.08161]
WvS.
Noncommutative Geometry and Particle Physics.Mathematical Physics Studies, Springer, 2015.
and also: http://www.noncommutativegeometry.nl
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