Gauge transformation for twisted spectral triples and Lorentz - - PowerPoint PPT Presentation

Gauge transformation for twisted spectral triples and Lorentz signature

Pierre Martinetti Universit` a di Genova Cortona, 6th June 2018

Introduction Noncommutative geometry is an extension of Riemannian geometry, useful in mathematics to describe objects whose properties are not captured by usual geometry. Interesting for physics, for it provides a description of the standard model of elementary particles [SM] minimally coupled with (Euclidean) general relativity. Added value: the Higgs field comes out as a connection 1-form, like the other bosons, but a connection between the discrete and the continuum part of the geometry. Recent developments - e.g. twist [D, a] → Da − ρ(a)D - open the way to models beyond the SM, and might help to adress the question of the Lorentzian signature.

• 1. Noncommutative geometry in a nutshell
• 2. Gauge theory by Morita equivalence
• 3. The standard model: need for twist !
• 4. Twisted spectral triple
• 5. Shadow of the Lorentz signature

• 1. Noncommutative geometry ina nutshell

Spectral triple: algebra A acting on a Hilbert H together with selfadjoint D s.t. [D, a] is bounded ∀a ∈ A. Graded spectral triple: there exists Γ = Γ∗, Γ2 = I, such that {Γ, D} = 0, [Γ, a] = 0 ∀a ∈ A. Real spectral triple: there exists antilinear operator J such that J2 = ǫI, JD = ǫ′DJ, JΓ = ǫ′′ΓJ where ǫ, ǫ′, ǫ′′ = ±1 define the KO-dimension k ∈ [0, 7].

• 1. Noncommutative geometry ina nutshell

Spectral triple: algebra A acting on a Hilbert H together with selfadjoint D s.t. [D, a] is bounded ∀a ∈ A. Graded spectral triple: there exists Γ = Γ∗, Γ2 = I, such that {Γ, D} = 0, [Γ, a] = 0 ∀a ∈ A. Real spectral triple: there exists antilinear operator J such that J2 = ǫI, JD = ǫ′DJ, JΓ = ǫ′′ΓJ where ǫ, ǫ′, ǫ′′ = ±1 define the KO-dimension k ∈ [0, 7]. J implements a map a → a◦ := Ja∗J−1 from A to the opposite algebra A◦. This yields a right action of A on H, ψa := a◦ψ, which is asked to commute with the left action (order zero condition) [a, Jb∗J−1] = 0 ∀a, b ∈ A. As well, holds the first order condition [[D, a], Jb∗J−1] = 0 ∀a, b ∈ A.

Connes’ reconstruction theorem: with other extra-conditions one has the following spectral characterization of manifolds: Compact Riemannian manifold M = ⇒ spectral triple (C ∞ (M) , L2(M, S), ∂ /) M such that A=C ∞ (M) ⇐ = (A, H, D) with A commutative, unital commutative spectral triple → noncommutative spectral triple

• ↓

Riemannian geometry non-commutative geometry

• 2. Gauge theory by Morita equivalence

Gauge theory with gauge group G Fermionic fields = sections of a G-bundle E, Bosonic fields = connections on E. Sections of a bundle on a manifold M ⇐ ⇒ a finite projective C ∞(M)-module.

Serre-Swan

◮ Bundle in noncommutative geometry ⇐

⇒ finite projective A-module E.

• 2. Gauge theory by Morita equivalence

Gauge theory with gauge group G Fermionic fields = sections of a G-bundle E, Bosonic fields = connections on E. Sections of a bundle on a manifold M ⇐ ⇒ a finite projective C ∞(M)-module.

Serre-Swan

◮ Bundle in noncommutative geometry ⇐

⇒ finite projective A-module E.

◮ Connection on a (right) A-module E: application E → E ⊗A Ω such that

∇(ηa) = ∇(η)a + η ⊗ δ(a) ∀η ∈ E, a ∈ A, where Ω is a A-bimodule generated by a derivation δ of A.

Morita equivalence A, B Morita equivalent ⇐ ⇒ B = EndA(E), E a Hermitian finite proj. A-module. How to export a spectral triple (A, H, D) to a Morita equivalent algebra B ?

Morita equivalence A, B Morita equivalent ⇐ ⇒ B = EndA(E), E a Hermitian finite proj. A-module. How to export a spectral triple (A, H, D) to a Morita equivalent algebra B ? Assume E = ER is a right A-module. The algebra B acts on HR := ER ⊗A H as b(η ⊗ ψ) = bη ⊗ ψ ∀b ∈ B, η ∈ ER, ψ ∈ H. The “natural” action of D on HR, DR(η ⊗ ψ) := η ⊗ Dψ, is ill-defined since ∀a ∈ A one gets DR(ηa ⊗ ψ) − DR(η ⊗ aψ) = −η ⊗ [D, a]ψ.

Morita equivalence A, B Morita equivalent ⇐ ⇒ B = EndA(E), E a Hermitian finite proj. A-module. How to export a spectral triple (A, H, D) to a Morita equivalent algebra B ? Assume E = ER is a right A-module. The algebra B acts on HR := ER ⊗A H as b(η ⊗ ψ) = bη ⊗ ψ ∀b ∈ B, η ∈ ER, ψ ∈ H. The “natural” action of D on HR, DR(η ⊗ ψ) := η ⊗ Dψ, is ill-defined since ∀a ∈ A one gets DR(ηa ⊗ ψ) − DR(η ⊗ aψ) = −η ⊗ [D, a]ψ. Take a connection ∇ on ER with value in the A-bimodule generated by the derivation δ(.) = [D, .]: Ω1

D(A) := {ai[D, bi], ai, bi ∈ A}. ◮ The covariant derivative DR(η ⊗ ψ) := η ⊗ Dψ + (∇η)ψ is well defined.

Morita equivalence A, B Morita equivalent ⇐ ⇒ B = EndA(E), E a Hermitian finite proj. A-module. How to export a spectral triple (A, H, D) to a Morita equivalent algebra B ? Assume E = ER is a right A-module. The algebra B acts on HR := ER ⊗A H as b(η ⊗ ψ) = bη ⊗ ψ ∀b ∈ B, η ∈ ER, ψ ∈ H. The “natural” action of D on HR, DR(η ⊗ ψ) := η ⊗ Dψ, is ill-defined since ∀a ∈ A one gets DR(ηa ⊗ ψ) − DR(η ⊗ aψ) = −η ⊗ [D, a]ψ. Take a connection ∇ on ER with value in the A-bimodule generated by the derivation δ(.) = [D, .]: Ω1

D(A) := {ai[D, bi], ai, bi ∈ A}. ◮ The covariant derivative DR(η ⊗ ψ) := η ⊗ Dψ + (∇η)ψ is well defined.

Morita self-equivalence: B = ER = A, then DR = D + AR with AR ∈ Ω1

D(A).

Same construction with left module EL, HL = H ⊗A EL, ∇◦ a connection on EL with value in the bimodule Ω1

D(A◦) =

• i

a◦

i [D, b◦ i ],

a◦

i , b◦ i ∈ A◦

• .

The covariant derivative DL(ψ ⊗ η) := Dψ ⊗ η + (∇◦η)ψ is well defined, Morita self-eq.: DL = D + A◦ = D + ǫ′J AL J−1 for A◦, AL ∈ Ω1

D(A◦), Ω1 D(A).

Same construction with left module EL, HL = H ⊗A EL, ∇◦ a connection on EL with value in the bimodule Ω1

D(A◦) =

• i

a◦

i [D, b◦ i ],

a◦

i , b◦ i ∈ A◦

• .

The covariant derivative DL(ψ ⊗ η) := Dψ ⊗ η + (∇◦η)ψ is well defined, Morita self-eq.: DL = D + A◦ = D + ǫ′J AL J−1 for A◦, AL ∈ Ω1

D(A◦), Ω1 D(A).

Assume E is a A-bimodule. By combining the two constructions one gets D′ = D + AR + ǫ′J ALJ−1. One has that D′J = ǫ′JD′ if and only if there exists A ∈ Ω1

D(A) such that

D′ = DA := D + A + J AJ−1

Same construction with left module EL, HL = H ⊗A EL, ∇◦ a connection on EL with value in the bimodule Ω1

D(A◦) =

• i

a◦

i [D, b◦ i ],

a◦

i , b◦ i ∈ A◦

• .

The covariant derivative DL(ψ ⊗ η) := Dψ ⊗ η + (∇◦η)ψ is well defined, Morita self-eq.: DL = D + A◦ = D + ǫ′J AL J−1 for A◦, AL ∈ Ω1

D(A◦), Ω1 D(A).

Assume E is a A-bimodule. By combining the two constructions one gets D′ = D + AR + ǫ′J ALJ−1. One has that D′J = ǫ′JD′ if and only if there exists A ∈ Ω1

D(A) such that

D′ = DA := D + A + J AJ−1

◮ The substitution of D with a DA is called a fluctuation of the metric. ◮ When DA is selfadjoint, it is called a covariant Dirac operator.

Gauge transformation Group U(E) of unitary endomorphisms of E: u ∈ EndA(E) such that u∗u = I. Its adjoint action on Ω-value connections on E, ∇u := u∇u∗ ∀u ∈ U(E), yields a new Ω-value connection ∇u.

Gauge transformation Group U(E) of unitary endomorphisms of E: u ∈ EndA(E) such that u∗u = I. Its adjoint action on Ω-value connections on E, ∇u := u∇u∗ ∀u ∈ U(E), yields a new Ω-value connection ∇u. Both connections decompose as ∇ = ∇Gr + A, ∇u = ∇Gr + Au, where ∇Gr is the Grassmann connection, while the gauge potentials A and Au are A-linear maps E → E ⊗A Ω (right module case) or E → Ω ⊗A E (left module).

◮ A gauge transformation is the map A → Au.

Consider a covariant operator obtained by Morita self-equivalence: DA = D + A + J A J−1 with A ∈ Ω1

D(A).

Substituting the connections ∇, ∇◦ with ∇u, (∇◦)u for some u ∈ U(A) yields DAu = D + Au + J Au J−1 with Au := u[D, u∗] + uAu∗.

Consider a covariant operator obtained by Morita self-equivalence: DA = D + A + J A J−1 with A ∈ Ω1

D(A).

Substituting the connections ∇, ∇◦ with ∇u, (∇◦)u for some u ∈ U(A) yields DAu = D + Au + J Au J−1 with Au := u[D, u∗] + uAu∗. Furthermore, DAu is also obtained by the conjugate action of Ad(u) : ψ → uψu∗ = u(u∗)◦ψ = uJuJ−1ψ, namely Ad(u) DA Ad(u)−1 = DAu.

◮ Gauge transformations are implemented by inner automorphisms of A.

They preserve the selfadjointness of the covariant Dirac operator.

• 3. The standard model

A = C ∞ (M) ⊗ Asm, H = L2(M, S) ⊗ Hsm, D = ∂ / ⊗ I32 + γ5 ⊗ Dsm where Asm = C ⊕ H ⊕ M3(C), Hsm = C32=2×2×8 = HR ⊕ HL ⊕ Hc

R ⊕ HC L ,

Dsm =     08 M 08 08 M† 08 08 08 08 08 08 ¯ M 08 08 MT 08    

• D0

+     08 08 MR 08 08 08 08 08 M†

R

08 08 08 08 08 08 08    

• DR

.

◮ M contains the Yukawa couplings of the electron, the quarks up and down,

and the (Dirac) mass of the electronic neutrino. MR contains only one non-zero entry kR (Majorana mass of the electronic neutrino).

• 3. The standard model

A = C ∞ (M) ⊗ Asm, H = L2(M, S) ⊗ Hsm, D = ∂ / ⊗ I32 + γ5 ⊗ Dsm where Asm = C ⊕ H ⊕ M3(C), Hsm = C32=2×2×8 = HR ⊕ HL ⊕ Hc

R ⊕ HC L ,

Dsm =     08 M 08 08 M† 08 08 08 08 08 08 ¯ M 08 08 MT 08    

• D0

+     08 08 MR 08 08 08 08 08 M†

R

08 08 08 08 08 08 08    

• DR

.

◮ M contains the Yukawa couplings of the electron, the quarks up and down,

and the (Dirac) mass of the electronic neutrino. MR contains only one non-zero entry kR (Majorana mass of the electronic neutrino). One also needs Γ = γ5 ⊗ γsm and J = J ⊗ Jsm with J the charge conjugation and γsm =     I8 −I8 −I8 I8     , Jsm =

• 016

I16 I16 016

• .

The asymptotic expansion Λ → ∞ of the spectral action Tr f (D2

A

Λ2 ) (f be a smooth approximation of the characteristic function of [0, 1]) yields the bosonic Lagrangian of the SM coupled with the Einstein-Hilbert action

The asymptotic expansion Λ → ∞ of the spectral action Tr f (D2

A

Λ2 ) (f be a smooth approximation of the characteristic function of [0, 1]) yields the bosonic Lagrangian of the SM coupled with the Einstein-Hilbert action

• M

√gd4x ( 1 κ2 R + α0CµνρσC µνρσ + γ0 + τ0R∗R∗ + 1 4G i

µν ¯

G µν

i

+ 1 4F α

µν ¯

F µν

α

+ 1 4Bµν ¯ Bµν + 1 2|DµH|2 − µ2

0|H|2− 1

12R|H|2 + λ0|H|4 ) where λ0, α0, τ0, κ0, γ0 are functions of Λ and the momenta fβ =

∞ f (v)v β−1dv,

and we assume a unique unification scale g 2

3 f0

2π2 = 1 4, g 2

3 = g 2 2 = 5

3g 2

1 .

The spectral action provides initial conditions at a putative unification scale. Physical predictions by running down the parameters of the theory under the renormalization group equation. Assuming there is no new physics between the unification scale and our scale, one finds mH ≃ 170 GeV = 125, 1GeV

Chamseddine, Connes, Marcolli

The spectral action provides initial conditions at a putative unification scale. Physical predictions by running down the parameters of the theory under the renormalization group equation. Assuming there is no new physics between the unification scale and our scale, one finds mH ≃ 170 GeV = 125, 1GeV

Chamseddine, Connes, Marcolli

but...

The spectral action provides initial conditions at a putative unification scale. Physical predictions by running down the parameters of the theory under the renormalization group equation. Assuming there is no new physics between the unification scale and our scale, one finds mH ≃ 170 GeV = 125, 1GeV

Chamseddine, Connes, Marcolli

but... for a Higgs boson with mass mH ≤ 130 Gev, the quartic coupling of the Higgs field becomes negative at high energy, meaning the electroweak vacuum is meta-stable rather than stable.

V (H) = −µ 2 H2 + λ 4 H4

ll ,

• l-

, i-

• ,

100 105 108 1011 1014 1017 1020 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08 Energy GeV Higgs SelfCoupling Λ

• FIG. 2:

Higgs self-coupling λ as a function of energy, for different values of the Higgs mass from 2-loop RG evolu-

• tion. Lower curve is for mH = 116 GeV, middle curve is for

mH = 126 GeV, and upper curve is for mH = 130 GeV. All

• ther Standard Model couplings have been fixed in this plot,

including the top mass at mt = 173.1 GeV. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 3 2 1 1 2 3 4 Higgs Field h Effective Potential Veff

vEW E MPl MetaStable Unstable

• FIG. 3: Schematic of the effective potential Veff as a function
• f the Higgs field h. This is not drawn to scale; for a Higgs

mass in the range indicated by LHC data, the heirarchy is vEW E∗ MPl, where each of these 3 energy scales is separated by several orders of magnitude.

• M. P. Hertzberg, A correlation between the Higgs mass and dark matter, arXiv:1210.3624

50 100 150 200 50 100 150 200 Higgs mass Mh in GeV Top mass Mt in GeV Instability Non-perturbativity Stability Meta-stability Instability 107 109 1010 1012 115 120 125 130 135 165 170 175 180 Higgs mass Mh in GeV Pole top mass Mt in GeV 1,2,3 s Instability Stability Meta-stability Degrassi, Di Vita, Elias-Miro, Espinosa, Guidice, Isidori and A. Sturmia, Higgs mass and Vacuum Stability in the SM at NNLO, arXiv:1205.6497

The instability of the electroweak vacuum can be cured by a new scalar field σ: V (H, σ) = 1 4(λH4 + λσσ4 + 2λHσH2σ2).

The instability of the electroweak vacuum can be cured by a new scalar field σ: V (H, σ) = 1 4(λH4 + λσσ4 + 2λHσH2σ2). In the spectral triple of the standard model, turning into a field the neutrino Majorana mass, kR → kRσ, yields the required field, and alters the running of the parameters so that to make the computation of mH compatible with 125 Gev.

Chamseddine, Connes 2012

The instability of the electroweak vacuum can be cured by a new scalar field σ: V (H, σ) = 1 4(λH4 + λσσ4 + 2λHσH2σ2). In the spectral triple of the standard model, turning into a field the neutrino Majorana mass, kR → kRσ, yields the required field, and alters the running of the parameters so that to make the computation of mH compatible with 125 Gev.

Chamseddine, Connes 2012

However the field σ cannot be obtained by a fluctuation of the Dirac operator, for [γ5 ⊗ DR, a] = 0 ∀a, b ∈ A = C ∞ (M) ⊗ Asm.

The instability of the electroweak vacuum can be cured by a new scalar field σ: V (H, σ) = 1 4(λH4 + λσσ4 + 2λHσH2σ2). In the spectral triple of the standard model, turning into a field the neutrino Majorana mass, kR → kRσ, yields the required field, and alters the running of the parameters so that to make the computation of mH compatible with 125 Gev.

Chamseddine, Connes 2012

However the field σ cannot be obtained by a fluctuation of the Dirac operator, for [γ5 ⊗ DR, a] = 0 ∀a, b ∈ A = C ∞ (M) ⊗ Asm. But it can be obtained by a twisted fluctuation [γ5 ⊗ DR, a]ρ = 0.

• f the twisted spectral triple

(C ∞ (M) ⊗ Asm) ⊗ C2, H = L2(M, S) ⊗ Hsm, D = ∂ / ⊗ I32 + γ5 ⊗ Dsm where the automorphism ρ is the flip ρ((f , g) ⊗ m) = (g, f ) ⊗ m f , g ∈ C ∞ (M) , m ∈ Asm.

Devastato, Lizzi, P.M. 2014

• 4. Twisted spectral triples

Given a triple (A, H, D), instead of asking the commutators [D, a] to be bounded, one asks the boundedness of the twisted commutators

Connes, Moscovici 2008

[D, a]ρ := Da − ρ(a)D for some fixed ρ ∈ Aut(A).

• 4. Twisted spectral triples

Given a triple (A, H, D), instead of asking the commutators [D, a] to be bounded, one asks the boundedness of the twisted commutators

Connes, Moscovici 2008

[D, a]ρ := Da − ρ(a)D for some fixed ρ ∈ Aut(A).

◮ Makes sense mathematically. Relevant to deal with type III algebras.

• 4. Twisted spectral triples

Given a triple (A, H, D), instead of asking the commutators [D, a] to be bounded, one asks the boundedness of the twisted commutators

Connes, Moscovici 2008

[D, a]ρ := Da − ρ(a)D for some fixed ρ ∈ Aut(A).

◮ Makes sense mathematically. Relevant to deal with type III algebras.

Twisted spectral triples are compatible with the real structure. A twisted fluctuation of D is defined as

Devastato, Landi, PM 2016/17

DAρ := D + Aρ + J Aρ J−1 where Aρ is an element of the set of twisted 1-forms Ω1

D(A, ρ) := {ai[D, bi]ρ, ai, bi ∈ A}.

Such twisted fluctuations arise by Morita equivalence, in the same way as non twisted ones. The only difference is that the “natural action” of D on HR,L, η ⊗ ψ → η ⊗ Dψ, ψ ⊗ η → Dψ ⊗ η needs to be twisted by ρ, η ⊗ ψ → ρ(η) ⊗ Dψ, ψ ⊗ η → Dψ ⊗ ρ(η) where ρ(η) := p   ρ(η1) . . . ρ(ηN)   ∀η =   η1 . . . ηN   ∈ ER, ηi ∈ A, for ER = pAN (and similarly for a left module).

Under the adjoint action of a unitary on a connection, the twisted covariant Dirac

• perator

DAρ = D + Aρ + J Aρ J−1 is mapped to DAu

ρ = D + Au

ρ + J Au ρJ−1

where Au

ρ := ρ(u)[D, u∗]ρ + ρ(u)Au∗.

Under the adjoint action of a unitary on a connection, the twisted covariant Dirac

• perator

DAρ = D + Aρ + J Aρ J−1 is mapped to DAu

ρ = D + Au

ρ + J Au ρJ−1

where Au

ρ := ρ(u)[D, u∗]ρ + ρ(u)Au∗.

Such a twisted gauge transformation is also obtained throug the twisted adjoint action of U = Ad(u) = uJuJ−1: DAu

ρ = ρ(U) DAρ U−1.

where ρ(U) = ρ(u)Jρ(u)J−1.

Under the adjoint action of a unitary on a connection, the twisted covariant Dirac

• perator

DAρ = D + Aρ + J Aρ J−1 is mapped to DAu

ρ = D + Au

ρ + J Au ρJ−1

where Au

ρ := ρ(u)[D, u∗]ρ + ρ(u)Au∗.

Such a twisted gauge transformation is also obtained throug the twisted adjoint action of U = Ad(u) = uJuJ−1: DAu

ρ = ρ(U) DAρ U−1.

where ρ(U) = ρ(u)Jρ(u)J−1.

◮ A gauge fluctuation DA → UDAU−1 preserved selfadjointness. ◮ A twisted fluctuation DAρ → ρ(U)DAρU−1 has no reason to preserve it.

• 4. Shadow of Lorentz signature

H an Hilbert space with inner product ·, ·, and ρ an automorphism of B(H).

Definition

A ρ-twisted inner product ·, ·ρ is an inner product on H such that Ψ, OΦρ = ρ(O)†Ψ, Φρ ∀O ∈ B(H), Ψ, Φ ∈ H, where † is the adjoint with respect to the initial inner product. We denote O+ := ρ(O)†. the ρ-adjoint of O.

◮ The ρ-twisted inner product is non necessarily definite positive.

• 4. Shadow of Lorentz signature

H an Hilbert space with inner product ·, ·, and ρ an automorphism of B(H).

Definition

A ρ-twisted inner product ·, ·ρ is an inner product on H such that Ψ, OΦρ = ρ(O)†Ψ, Φρ ∀O ∈ B(H), Ψ, Φ ∈ H, where † is the adjoint with respect to the initial inner product. We denote O+ := ρ(O)†. the ρ-adjoint of O.

◮ The ρ-twisted inner product is non necessarily definite positive.

If ρ an inner automorphism of B(H), ρ(O) = ROR† ∀O ∈ B(H) for a unitary operator R on H, then a natural ρ-product is Ψ, Φρ = Ψ, RΦ.

In the twisted spectral triple of the Standard Model, the flip ρ is an inner automorphism of B(L2(M, S)), with R = γ0 the first Dirac matrix.

◮ The ρ-twisted inner product is the Krein product for the space of spinors on

a Lorentzian manifold.

In the twisted spectral triple of the Standard Model, the flip ρ is an inner automorphism of B(L2(M, S)), with R = γ0 the first Dirac matrix.

◮ The ρ-twisted inner product is the Krein product for the space of spinors on

a Lorentzian manifold.

◮ Furthermore, extending ρ to the whole of B(L2(M, S)), one finds

ρ(γ0) = γ0, ρ(γj) = −γj for j = 1, 2, 3. The flip is the square of the Wick rotation W (γ0) = γ0, W (γj) = iγj. that is ρ = W 2.

◮ Krein selfadjointness is preserved by twisted fluctuations.

Devastato, Lizzi, Farnsworth, PM 2017

Conclusions

◮ Fluctuations by Morita equivalence straightforwardly adapt to twisted

spectral triples.

◮ By minimally twisting the spectral triple of the Standard Model (i.e. the

fermionic sector - H and D - is untouched), one gets a model beyond the SM, that spontaneously breaks to the SM. Similar result as the one obtained by Chamseddine, Connes and van Suijlekom who considered “fluctuations without first-order condition”.

Conclusions

◮ Fluctuations by Morita equivalence straightforwardly adapt to twisted

spectral triples.

◮ By minimally twisting the spectral triple of the Standard Model (i.e. the

fermionic sector - H and D - is untouched), one gets a model beyond the SM, that spontaneously breaks to the SM. Similar result as the one obtained by Chamseddine, Connes and van Suijlekom who considered “fluctuations without first-order condition”.

◮ Minimal twist are highly constrained:

(A, H, D) a spectral triple (A, H, D, ρ) a twisted spectral triple

• =

⇒ a − ρ(a) is compact ∀a ∈ A. For (almost)-commutative geometry, this forces to change the algebra: any minimal twist of (A, H, D) has to be (A′, H, D; ρ). In addition, assuming A′ = A ⊗ B, the boundedness of [∂ / ⊗ I, a′]ρ implies B = C2 with ρ the flip.

◮ Selfadjointness is lost by twisted gauge transformations. What is naturally

preserved is Krein adjointness with respect to the Lorentz structure.

◮ But the twist ρ does not implement the Wick rotation W of the Dirac

• perator: ρ = W 2.

◮ In Connes-Moscovici, the automorphism comes from modular theory:

ρ = σt=−i for the modular group σt associated with a twisted trace. Possible that flip = σi, so that Wick rotation would be σ− i

2 ? Would be a realisation of

Connes-Rovelli thermal time hypothesis directly in the Standard Model.