Effectiveness and naturalness of the spectral action for ... · measure theory via general...

Effectiveness and naturalness of the spectral action for theories on

foliated space-times

Aleksandr Pinzul

Universidade de Brasília

Strings at Dunes, Natal, 04-15 July 2016

Measure theory and topology

• Commutative Von Neumann algebra σ-finite measure space

• This motivates to define the non-commutative measure theory via general (non-commutative) von Neumann algebras

• Commutative C*-algebra locally compact topological space

• This motivates to define the non-commutative topology via general (non-commutative) C*-algebras


Measure theory and topologyTopology Algebra

C0(X) C*-algebra A

proper map morphism

homeomorphism automorphism

measure positive functional

compact unital

σ-compact σ-unital

open subset ideal

open dense subset essential ideal

closed subset quotient

compactification unitization

C(αX) A~

C(βX) M(A)

… ...


Differential Calculus


Classical Quantum

Complex variable Operator in H

Real variable Selfadjoint operator in H

Infinitesimal Compact operator in H

Infinitesimal of order of α Compact operator in H whose characteristic values behave as μn=O(n-α) when n→∞

Differential of real or complex variable

df=i[F,f]

Integral of infinitesimal of order 1

Dixmier trace, Trω(T)

Geometry

Commutative case (reconstruction theorem, Connes 2008)A spectral triple: (A,H,D), A is commutative andi) μn(R(D))=O(n-1/p)ii) [[D,a],b]=0 for any a,b2Aiii) For any a2A, a and [D,a] belong to Dom(δm) where δm(T)=

[|D|m,T] is a derivationiv) Exists a Hochschild p-cycle c: πD(c)=1 for p odd or πD(c)=g –

Z2 grading for p evenv) A-module H∞=XDom(Dm) is finite and projective and the

Hermitian structure (.|.) defined by <x,ah>=:$a(x|h)|D|-p

Then (A,H,D) is in one-to-one correspondence with smooth oriented compact manifold.


Geometry


2

0

;

4

2

4

2

2

0

4

0

4

022

0

2

||

1

20

11

10

1

20

3

1248

log

)(lim where,

)2(

2)||(

)()2(

2)(

1],[),(|sup

mORRRCCgxdf

RgxdfmgxdfmN

m

DTr

N

AATrf

nDfTr

MVoln

N

fDMC:ff(x)-f(y)|d(x,y)

N

NM

gn

n

mn

n

n

n

m

D

A=C∞(M), H=L2(M,S), D=γμ(∂μ+ωμ)

Geometry

We take the generalized spectral triple (A,D,H) as

the definition of non-commutative geometry.


Dirac operator

Geometry

Space-time

Gauge

Matter


Examples. I – AC geometry and SM


Usual geometry: A=C∞(M), H=L2(M,S), D=γμ(∂μ+ωμ); JM , gM

Finite (matrix) geometry: AF , HF , DF ; JF , gF

_________________________________________________________

Almost commutative (AC) geometry: MxF=(C∞(M,AF), H=L2(S5(MxHF)),D51+ gM5DF; JM5JF, gM5gF )


Take the finite (matrix) geometry as follows: AF =C4H4M3(C)HF =( Hl 4 Hl* 4 Hq 4 Hq* )43

DF =‘Yukawa mass matrices’JF =‘charge conjugation’gF =left-handed particles are eigenvectors with +1, right-

handed with -1

Gauge fields come from the fluctuation of the full Diracoperator: DA=D+JAJ-1 , where A=Saj[D, bj]

Then the spectral action produces the full

action of the Standard Model!

2

0

2

m

DTr

Why do we “deform” geometry?

1) Problems with the quantization of gravity


0only when ability renormalizexpect can We

verticesofnumber -

legs external theof numberf -

dimension time-space -

divergence of degree lsuperficia -

)12/(

units momentumin ][

n

E

d

D

nEddD



As the result, the effective dimensionless constant is given by

p

P

P

ME

GeVG

cM

M

EGE

when i.e.

101.22 where: 19

2

2

02 ,4for

2][ 16

1 4

d

dGRgxdG

SEH



i) (Super)string theory: contains a spin-2 massles mode => has to describe gravity. GR is recovered in long-wave regime. But, the predictive power is quite poor: the string theory landscape has 10500 vacua.ii) Loop quantum gravity: one can perform non-perturbative quantization. Among problems, the difficulty of the recovery quasiclassical space.iii) Some other approaches treat gravity as an emergent phenomenon (e.g., entropic gravity).

Possible solutions


2) General arguments that the notion of a space-time as a classical manifold should be abandoned Doplicher, Fredenhagen and Roberts 1995


Examples. II – Horava-Lifshits models

• Lifshitz model (Lifshitz 1941)

4222

222

1),(

:form thehas propagator The

1[c] ,2][ ,1][

)(

kgkckG

tx

cgxdtdS n


...1111

:IR

...1111

:UV

222

42

22222242222

422

22

42242242222

kckg

kckckgkc

kgkc

kgkgkgkc

I.e. we have two fixed points: UV, which corresponds to z=2 and has significantly improved behavior and IR, in which by the time rescaling we can set c=1 and restore relativistic invariance, z=1


Why to break Lorentz invariance?

Let us consider the same type of the modification, but when the higher derivatives are added in the Lorentz invariant way.

gkkgkkgkkkG

gxdS

/1

11

)1(

11),(

:form the takespropagator The

)(

222242

24


form lfundamenta second - 2

1 where

16

1

)())((

323

22

ijjiijij

ij

ijEH

jjii

ij

NNgN

K

RKKKgxNdtdG

S

Ncdt - dt + Ndxdt + Ndx = gds

ADM


We take ADM slicing as fundamental, i.e. instead of considering just a manifold, we endow it with the foliation structure:

FDiffsor diffeos preserving-foliation are These

)(~~

),,(~~ ttttxxx ii

Also, we introduce anisotropic scaling between x and t:

zGzNgNzc

ztxttxx

iij

z

3][1][ , 0][][ , 1][

:dimensions following thegprescribin toequivalent is This

][ , 1][or ,


• Projectable FDiff gravity (Horava 2009)

• Non-projectable FDiff gravity (Blas et al. 2010)

3

54321

4

2

2

1

2

232

2

2

~ ),(

RBRRRBRRBRRRBRRBM

RRARAMRV

VKKKNgxdtdM

S

t

tNNtNN

jk

jkjki

jki

i

k

jk

ij

ij

ijP

P

ij

ijP

ij

ji

k

k

i

i

i

i

ij

ji

i

i

i

i

i

iPNP

NP

ij

ijP

ii

RaaaaDaaDaaDM

RaaCaaCaaCMaaVV

VKKKNgxdtdM

S

NNaxtNN

3

3

2

2

1

4

3

2

21

2

232

1

)(

)(

2

: ),,(


Some properties

• Broken 4d diffeos => Lorentz violation

• Extra scalar mode in addition to two graviton polarizations

• In general the scalar mode does not decouple in IR, this can endanger the renormalizability

• The model with the detailed balance condition does not pass the Solar system tests

• The healthy extension (with ai) has A LOT of free parameters and some of them still require fine tuning

• …


IR limit

С

i

i

ij

ijP

IR

ii

RaaKKKNgxdtdM

S

NNaxtNN

)3(232

1

2

: ),,(

Keeping only the terms with lowest derivatives, we

arrive at IR limit

This action is used to study the gravitational equations

of motion (Barausse & Sotiriou 2013)


Spectral dimension

(AP 2010)

• The choice of the Dirac operator in the form D=γμ(∂μ+ωμ) is not natural anymore

• The foliation structure dictates the following (schematic) form for D (for z=3)D= ∂t+σμ∂μΔ+M*Δ+M*

2σμ∂μ

• This D should be used to obtain “physical” geometry instead of auxiliary 3+1 dimensional

(AP 2010, Gregory & AP 2012)


Model calculation

• M=S1×T3 , D2=∂t2+Δ3+M*

2Δ2+M*4 Δ

• sp(D2)={n2+(n12+n2

2+n32)3+M*

2(n12+n2

2+n32)2

+M*4(n1

2+n22+n3

2) , ni ϵ Z}

• N|D|(λ)={# eigenvalues < λ}

• when λ<<M*6 the last term dominates:

4 4)(0

)(

0

42

||

2/122

dddnN

n

D

2 4)(0

)(

0

22

||

6/122

dddnN

n

D

when λ>>M*6 the first term dominates:


One can do better and go beyond the flat case.

• Define a generalized ζ-function

ζΔ(s) := Tr(∆-s)

• Now ∆ can be any generalized elliptic operator.

• ζ-function can be extended to a meromorphic function on the whole complex plane with the only poles given by

,...2

,...,2

1,

2 z

kzppn

z

zppn

z

zppn

The first pole is related to the analytic dimension

22

an

z

zppn

n=D+1, p=1 (co-dimension) we have z

Dna 1


Spectral Action

• Dirac operator is very complicated:

gravity? Lifshitz-Horava 2

0

2

m

DTr

j

ij

ix

x

ggNgNN

g

gN

fD

1 and

1 where

, )(2

To calculate the trace of this operator one has to find the heat kernel

)',()0;',(

0);',()( 2

xxxxK

sxxKDs

Part I


Even the flat case is not trivial (Mamiya & AP 2013)

This allows to perform a completely analytical study of the spectral dimension flow:


Part II Matter

DSmatter

The operator D is the same that was used for the gravity part!

The matter coupling to geometry is restricted only by FPDiff.

This permits inclusion of the higher spatial derivatives in Smatter

There is no guiding principle on how to proceed except the control over the amount of Lorentz violation (Pospelov&Shang2010, Kimpton&Padilla 2013)

The spectral action approach has the second part (Chamsedinne&Connes 1996)


• What happens to the geodesic motion?

motion geodesic 0

T

(Dixon 1970, Hawking&Ellis 1973)

Now we DO NOT have

Instead we do have

0

T

g

STTh matt where,0

Alternative way to get geodesics:

Write a field theory

Find field equations

Restrict to the 1-particle sector

Do quasi-classical analysis

Hamilton-Jacobi => geodesic motion


Immediate result is that “geodesics” changeStrings at Dunes, Natal, 04-15 July 2016

Applications: IR limit

C

i

i

ij

ijP

IR RaaKKKNgxdtdM

S )3(232

2

acKcacKcDcDD n

0

543

0

2

)3(

1

0

The most general FPDiff covariant generalized operator in IR

limit takes the form (AP 2014)

0,2

1,

2

1,1 54321 ccccc

To the Diff covariant case correspond the following values:


i)Geodesic motion

• The approach based on Hamilton-Jacobi equation

or

• Calculating the spectral distance based on the

deformed Dirac operator lead to the same result:

The geodesic motion of a point test particle is the

same as for a (pseudo)Riemannian manifold with

the effective metric

hc

nng2

1

1~


ii) Spectral action (Lopes, Mamiya & AP 2015)

Using the heat kernel expansion for the deformed Dirac

operator

one arrives at

C

i

i

ij

ijP

IR RaaKKKNhxdtdM

S )3(232

2

where

2

2

02

5

2

41

2

2

2 12 ,12 , 361 , ,

12

Tr

4

1

2 f

fcccf

MC

P

1

acKcacKcDcDD n

0

543

0

2

)3(

1

0


iii) Matter coupling (Lopes, Mamiya & AP 2015)

DSmatter

acKcac

KcDcDNhxdtdS nmatter

0

543

0

2

)3(

1

03

This should be compared with (Kostelecky 2004)

ab

ba

b

b

b

b

bc

cb

aaab

b

ab

b

aaa

a

aLV

eeHebeaimmM

eeegefieeeedeec

MDegxdS

~

2

1~~

~

2

1~~~~

where,

55

5

55

4


Using our Dirac operator we can express the Lorentz violating parameters as some combinations of ci

This, in principle, could allow to put the bounds on the parameters of the gravity action that will be much more restricting that the ones coming from gravitational experiments.

15-

-3

10|| matter LV

10|| nalGravitatio

1 Here


Conclusions/Discussions

• Horava-Lifshitz could provide a UV completion of GR

• For this the original proposal should be modified (“healthy” extension?)

• It would be good to have a more geometrical approach to construct the theory

• At least in IR, the geodesic motion is still in some effective commutative geometry

• The spectral action allows to calculate both, gravity and matter parts


• A lot of fine tuning happens automatically due to the fact that both parts are defined by the same Dirac operator

• Bounds on LV parameters on the matter side could be used to bound the gravity action (and vice versa)

• Gauge sector, matter content (it is more natural now to have fields in reps of SO(3))

• Methods of spectral geometry plus spectral action principle have proven to be useful though we do expect much more complicated situation for the fully deformed theory


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