maxwell-weyl gauge theory of gravityqssg16.ipb.ac.rs/slides/student-presentations-kibaroglu.pdf ·...

MAXWELL-WEYL GAUGE THEORY OF GRAVITY

Salih KibaroğluKocaeli University and University of Wrocław

This presentation based on «O. Cebecioğlu and S. Kibaroğlu PHYSICAL REVIEW D 90, 084053 (2014)»

Contents

• Motivation– Maxwell group

– Summary of Literature

– Influence area of Maxwell group

• General overview– Aim and method

• Maxwell-Weyl group– Tensor extension of Weyl algebra

– Gauge theory of Maxwell-Weyl group

26.08.2016Salih Kibaroğlu - Maxwell-Weyl Gauge

Theory of Gravity2

MOTIVATION


Theory of Gravity3

1. Maxwell group2. Summary of Literature3. Influence area of Maxwell group

What is the Maxwell Algebra?


Theory of Gravity4

G.W. Gibbons, J. Gomis and C. N. Pope, PHYSICAL REVIEW D 82, 065002 (2010)

• The Maxwell algebra is a non-central extension of the Poincare algebra, in which the momentum generators no longer commute, but satisfy 𝑃𝑎, 𝑃𝑏 = 𝑍𝑎𝑏. The charges 𝑍𝑎𝑏 commute with the momenta, and transform tensorially under the action of the angular momentum generators.

• If one constructs an action for a massive particle, invariant under thesesymmetries, one finds that it satisfies the equations of motion of a charged particle interacting with a constant electromagnetic field via the Lorentz force.

a b a b abP ,P 0 P ,P iZ

Maxwell Algebra


Theory of Gravity5

a b

ab cd ad bc bc ad ac bd bd ac

ab c bc a ac b


ab d

ab c

a

c

b

M ,M i M M M M

M , i

M ,F i F F F F

F ,F 0

F , 0

, iF

a a ai eA x a

ai m x,f 0

Maxwell algebra New momentum op.

Where, 𝐴𝑎: Electromagnetic potential, e: Electric charge𝑓𝑎𝑏: Electromagnetic field tensor

b1a a b2

A x f x ab a b b af A x A x

• If we enlarge the Poincare algebra by six additional tensorial abelian generators as showed below then we get Maxwell algebra.

This antisymmetric generator 𝐹[𝑎𝑏] can be used to describe the motion of a relativistic particle in a

constant electromagnetic field.

H. Bacry, P. Combe, and J.L. Richard, Nuovo Cim. 67, (1970), 267R. Schrader, Fortsch.Phys. 20, (1972), 701-734

If we select 𝐹[𝑎𝑏] = 0 then we get well-known Poincare algebra which describes flat Minkowski space-

time.

Maxwell Algebra


Theory of Gravity6

a b


ab c bc a ac b


ab

a

d

b c

b

c

a

P ,P i

M ,M i M M M M

M ,P i P P

M ,Z i Z Z Z Z

Z ,Z 0

Z ,P 0

Z

b1ab2

ab ab

c c

a bc b

a a

ab a b b a ac

P i( )

M i(

x

Z i

x x ) 2i( )

Maxwell group Differential realisationof generators:

• In 2005 D. V. Soroka, V. A. Soroka shows us the Maxwell algebra can be found with new

additional tensorial coordinate 𝜃𝑎𝑏 and corresponding tensorial derivative 𝜕𝑎𝑏 =𝜕

𝜕𝜃𝑎𝑏.

D. V. Soroka, V. A. Soroka, Physics Letters B 607 (2005) 302–305.

Some relations of tensorial coordinates and its derivatives:

cd c d c d

ab a b b a

1

2

c

abx 0 ab c, 0 ab cd, 0

A Brief Summary of Litarature


Theory of Gravity7

Gauge Theories of Some Important Groups

Lorentz group

R. Utiyama – 1956

Poincare group

T.W.B. Kibble – 1961

Weyl group

J.M. Charap and W. Tait – 1974

De Sitter group

T. Kawai and H. Yoshida – 1979

Affine group

A.B. Borisov and V. I. Ogievetskii – 1974

Gauge Theories of Maxwell Groups

Simple Maxwell Group

J.A. Azcarraga, K. Kamimura, J. Lukierski

- 2010

SemisimpleMaxwell Group

D.V. Soroka, V.A. Soroka - 2011

Maxwell-WeylGroup

O. Cebecioğlu, S. Kibaroğlu -2014

AdS-MaxwellGroup

R. Durka, J. Kowalski-Glikman, M.

Szczachor – 2011

Maxwell-AffineGroup

O. Cebecioğlu, S. Kibaroğlu -2015

Influence Area of Maxwell Group


Theory of Gravity8

New symmetries

Cosmologicalconstant

Dark energy

Extendedgravity

HS fields

Supergravity

GENERAL OVERVIEW


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1. Aim and method

Aim


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Our fundamental aim is to going beyond the

basic gravity.

Method


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Space-Time Symmetry

Extension withantisymmetric

tensor generator

ExtendedSpace-Time (Maxwell) Symmetry

Gauge Theory• Differential Geometry

Extended Gravity

(Maxwell Gravity)

Supersymmetricextension of the

extendedsymmetry

Super MaxwellGravity

MAXWELL-WEYL GROUP (MW(1,3))


Theory of Gravity12

1. Tensor extension of Weyl algebra2. Gauge theory of Maxwell-Weyl group

Tensor Extension of Weyl Algebra


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ab c bc a ac b

a b

a a

ab

M ,M i M M M M

M ,P i P P

P ,P 0

D,D 0

P ,D iP

M ,D 0

Weyl algebra:

Weyl algebra contains threegenerators (𝑀𝑎𝑏, 𝑃𝑎, 𝐷). Thesegenrators correspond to Lorentz, momentum and scale symmetryrespectively.

a a

ab a b b a

P i

D i x

M i x x

Differential realisation of thegenerators:



Theory of Gravity14

a b ab

ab cd ad bc bc ad ac bd bd


ab c bc a a

ac

ab cd

ab c

ab a

c b

a

b

a

ab

M ,M i M M M M

M ,P i P P

D,D

P ,P iZ

M ,Z i Z Z Z

0

P ,D iP

M

Z

Z ,Z 0

Z ,P 0

Z ,D 2 Z

,D

i

0

Weyl algebra can be extended as in the box.

This algebra satisfy all JacobiIdentities.

Where 𝑍𝑎𝑏 are antisymmetricgenerators.

One can find this algebra with different notation in following paper:S. Bonanos, J. Gomis, K. Kamimura, and J. Lukierski, Journal of Math. Phys. 51, 102301 (2010)

Maxwell-Weyl algebra:



Theory of Gravity15

In order to find differential realisation of generators, we use following coset transformation;

i2

i2

x Mix x P i x D

u Mi

i x Z

i Z

i x Z

a P i D

ix x P i x D

g x, , , e e e

g a, , ,u e e e

K x , e

e

e

e

e,

K x , , h g a, , ,u K x, ,

Where,

ab ab [a cb] ab

a a

[a b

a

]1

b a

b

ab

c 4

ab

u 2 a

x a u

x

x x

u



Theory of Gravity16

If we use δx and δθ on the trasformation law of scalar field and comparefollowing equation;

ab aba ba aax , x x ,

i2

i Zx, ia P i D u M x,

Trasformation of scalar field is defined by;

Then we get;

a a

a

a

b1ab2

ab

ab

ab

a b b a

ab

ab

c c

a bc b ac

P i( )

D i x

M i(x

x

Z

x )

i

2

2i( )

Gauge Theory of Maxwell-Weyl Group


Theory of Gravity17

Defining gauge field ( 𝔸 ) by;

A

A

a ab1a ab2

ab

abB Z

X

e P D M

a a ab a aba bbe e dx , dx , dx , B B dx

a ab1a ab2

a ab1a ab2

ab

ab

ab

ab

e P B Z

B Z

D M

e P D M

Where XA represents group generatorsand ea, Bab, χ, ωab are associated gaugefields.

If we going over the space-time indices ea, Bab, χ and ωab take the following form;

Also gauge field and its variation can be written by;



Theory of Gravity18

Under a local gauge transformation with the values in Maxwell algebra 𝜁 𝑥 ;

A

A

a ab12

a

ab b

b

aa x

x x X

y x P x DZ x M

ab ab [a cb] ab [a cb] ab [a b

a a a c a c a a

c c

ab ab [a c

]

c

b]

c

c

1B 2 B 2 B e y

e y y y e

2

e

We get following transformations;

i ,

To find variation of gauge field, we can use the following formula;



Theory of Gravity19

The curvature forms can be found with the use of following equations;

i2

d , Where: A a ab1A a ab2

ab

abF ZX F P fD R M

ab ab [a cb] ab a b1c 2

a a a b a

b

ab ab a cb

c

F d

F dB

e e e

f d

R d

B 2 B e e

One gets;

If we going over space-time indices;

ab ab12

a a12

12

ab ab12

F F dx dx

f f dx dx

R R d

F F dx

x

x

dx

d

ab ab [a cb] ab a b

[ ] [ |c ] [ ] [

a a a b a

[ ] [ b ] [ ]

[ ]

ab ab a cb

[ ] [ c ]

]

|

1F B B

F e

2 B e e2

e e

f

R



Theory of Gravity20

To find the variation of curvatures under local gauge transformation one uses;

i ,

We get;

ab ab [a cb] [a b

a a a b a a b

b b

ab [a

] ab [a cb]

c c

cb]

c

1F 2 F R

F y f R y F F

f 0

R

y F 2 F F2

R



Theory of Gravity21

To find an invariant Lagrangian under gauge transformation, we have to take intoconsideration scale(dilatation) symmetry.

In our case, variation of metric tensor does not vanish. This issue relates to Weyl gauge theory.

g x 2 x g x

This is major difficulty of constructing of an invariant Lagrangian.

4

f fS d xe The free gravitational action requires Weyl weight zero. This condition implies that the Lagrangian density must have𝑤(𝐿𝑓) = −4.

w g 2

w g 2

w g w e 4

Describing a weyl weight by ‘w(f)’ then onegets expressions about metric.



Theory of Gravity22

Weyl weights of curvatures and gauge fields written as;

a

ab

ab

w F 1

w F 2

w f 0

w R 0

w R 2

a

ab

ab

w e 1

w B 2

w 0

w 0

w e 4

One can easily see that the Einstein-Hilbert actiondoes not invariant under the scale transformation.

4

E H

1S d xeR

2

To overcome this difficulty, if we multiply it by a compensating scalar field 𝜙 introduced by Brans-Dicke (1961) and elaborated by Dirac (1973), we can form a Weyl invariant action linear in R.

In our approach we will follow Dirac’s idea. The scalar field 𝜙 with 𝑤 𝜙 = −1 lets 𝜙2𝑅 be a regular part of 𝐿𝑓, and hence the

combination is invariant under scaletransformation.

C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961)P. A. M. Dirac, Proc. R. Soc. Lon. A. 333, 403-418 (1973)

4 21S d xe R

2



Theory of Gravity23

Covariant derivative can be written following form;

a a b a

b

ab [a cb] ab [a b]

c

ab

F R e f e ,

1F R B 2f B F e ,

2

R 0, f 0.

d w

From here one can write Bianchi identities;

Now we can start to construct Lagrangian.

ab

a a

a

ab a b

b b

12

a

F B e

F e

d

e

f

R

The curvatures take form;



Theory of Gravity24

It is easy to see that following equation has zero weyl weight. We can start to constructLagrangian with this shifted curvature;

ab ab 2 abR 2 F

With this combination we have an Einstein Lagrangian that involves the curvature scalarlinearly. Therefore we consider the following Lagrangian density 4-form as our starting point for the free gravitational part:

ab cd

f abcd

ab cd 2 ab cd 4

abc

ab cd

abcdd abcd

1 1J J J J

2 4

1R

4FR

1R F F

Where 𝛾 and 𝜅 are constants and the first term can be ignored because it is a closed form.



Theory of Gravity25

The introduction of a compensating field forces us to add its kinetic term to theLagrangian. We then get the total action for vacuum as follows:

4

0

1f f 1

2 4

Where λ is another constant. Our complete action is the sum of the free gravityaction and the vacuum action,

2 ab cd 4 ab cd

abcd abc

ab cd

abcd d

4

1 1R F FR R

4F

S1

f f 12 4



Theory of Gravity26

Then we get following equations of motion;

2 ab [a cb]

c

2 ab d a

abcd c a c

eb ab d 4

cb abcd c

2 ab

B

2ab cd

abcd

ab cd

abcd

S 0 F B 0

1 1e e e

2S 0 0

1 1f e f f e f e

4 2 4

S 0 0

2 1S 0 B f 0

2

2S 0 F

3 1 0



Theory of Gravity27

One can find following expression thanks to using second equation of previous page;

2 a a a a c 4 ac a cd

b b b b c cb b cd

1 1 1f f f f

2 2 2 2 4

If we swich from tangent space indices to space-time indices then one gets the field equation with a cosmological term depending on the dilaton field;

2 23 2 T B1

R R T T f2 2

ab ab

a b [ ] a b [ ]

1T B e e B e e B

2

41T

2 2

1

T f f f f f4

The energy-momentum tensors:

O. Cebecioğlu and S. Kibaroğlu PHYSICAL REVIEW D 90, 084053 (2014)

Thank you


Theory of Gravity28

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