![Page 1: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/1.jpg)
Kappa-deformed space-time: Field Theory
and Twisted Symmetry
E. Harikumar1
School of PhysicsUniversity of Hyderabad
Hyderabad
December-2008
![Page 2: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/2.jpg)
Motivations/Introduction
Non-commutative space and twisted symmetry
k-spacetime and k-Poincare algebra
Realisation of kappa spacetime and its Symmetry Algebra
Conclusion
![Page 3: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/3.jpg)
Motivations/Introduction
Non-commutative space and twisted symmetry
k-spacetime and k-Poincare algebra
Realisation of kappa spacetime and its Symmetry Algebra
Conclusion
![Page 4: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/4.jpg)
Motivations....
Quantum gravity can be, possibly modeled usingnon-commutative space-time
lPlanck =√
hGc3
may have a significant role to play in
q-gravity.(a) String theory models predict existence of minimumlength scale(b) Area and volume operators in certain loop gravitymodels have discrete spectra with minimal values. Theseminimal values are proportional to l2p and lp3 respectively.
lp sets a minimum length scale lmin in models describingquantum gravity
![Page 5: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/5.jpg)
Motivations....
Quantum gravity can be, possibly modeled usingnon-commutative space-time
lPlanck =√
hGc3
may have a significant role to play in
q-gravity.(a) String theory models predict existence of minimumlength scale(b) Area and volume operators in certain loop gravitymodels have discrete spectra with minimal values. Theseminimal values are proportional to l2p and lp3 respectively.
lp sets a minimum length scale lmin in models describingquantum gravity
![Page 6: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/6.jpg)
Motivations....
Quantum gravity can be, possibly modeled usingnon-commutative space-time
lPlanck =√
hGc3
may have a significant role to play in
q-gravity.(a) String theory models predict existence of minimumlength scale(b) Area and volume operators in certain loop gravitymodels have discrete spectra with minimal values. Theseminimal values are proportional to l2p and lp3 respectively.
lp sets a minimum length scale lmin in models describingquantum gravity
![Page 7: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/7.jpg)
Special Theory of Relativity: Laws of physics must besame in all inertial frames
If ls ≥ lmin, ls′ ≥ lmin.
But this is not guaranteed(!) due to Lorentz-Fitsgeraldlength contraction
Modify STR Space-time structure is governed not onlyby a fundamental velocity scale c, but also by afundamental length scale lp. Doubly Special Relativity
DSR introduces a minimum length scale without singlingout any preferred frame
The Energy-Momentum relation get a length scaledependent modification.
Ex: E2 = p2c2 +m2c4 + αlpE3 + βl2pE
4 + ......
![Page 8: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/8.jpg)
Special Theory of Relativity: Laws of physics must besame in all inertial frames
If ls ≥ lmin, ls′ ≥ lmin.
But this is not guaranteed(!) due to Lorentz-Fitsgeraldlength contraction
Modify STR Space-time structure is governed not onlyby a fundamental velocity scale c, but also by afundamental length scale lp. Doubly Special Relativity
DSR introduces a minimum length scale without singlingout any preferred frame
The Energy-Momentum relation get a length scaledependent modification.
Ex: E2 = p2c2 +m2c4 + αlpE3 + βl2pE
4 + ......
![Page 9: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/9.jpg)
Special Theory of Relativity: Laws of physics must besame in all inertial frames
If ls ≥ lmin, ls′ ≥ lmin.
But this is not guaranteed(!) due to Lorentz-Fitsgeraldlength contraction
Modify STR Space-time structure is governed not onlyby a fundamental velocity scale c, but also by afundamental length scale lp. Doubly Special Relativity
DSR introduces a minimum length scale without singlingout any preferred frame
The Energy-Momentum relation get a length scaledependent modification.
Ex: E2 = p2c2 +m2c4 + αlpE3 + βl2pE
4 + ......
![Page 10: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/10.jpg)
Special Theory of Relativity: Laws of physics must besame in all inertial frames
If ls ≥ lmin, ls′ ≥ lmin.
But this is not guaranteed(!) due to Lorentz-Fitsgeraldlength contraction
Modify STR Space-time structure is governed not onlyby a fundamental velocity scale c, but also by afundamental length scale lp. Doubly Special Relativity
DSR introduces a minimum length scale without singlingout any preferred frame
The Energy-Momentum relation get a length scaledependent modification.
Ex: E2 = p2c2 +m2c4 + αlpE3 + βl2pE
4 + ......
![Page 11: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/11.jpg)
Special Theory of Relativity: Laws of physics must besame in all inertial frames
If ls ≥ lmin, ls′ ≥ lmin.
But this is not guaranteed(!) due to Lorentz-Fitsgeraldlength contraction
Modify STR Space-time structure is governed not onlyby a fundamental velocity scale c, but also by afundamental length scale lp. Doubly Special Relativity
DSR introduces a minimum length scale without singlingout any preferred frame
The Energy-Momentum relation get a length scaledependent modification.
Ex: E2 = p2c2 +m2c4 + αlpE3 + βl2pE
4 + ......
![Page 12: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/12.jpg)
Special Theory of Relativity: Laws of physics must besame in all inertial frames
If ls ≥ lmin, ls′ ≥ lmin.
But this is not guaranteed(!) due to Lorentz-Fitsgeraldlength contraction
Modify STR Space-time structure is governed not onlyby a fundamental velocity scale c, but also by afundamental length scale lp. Doubly Special Relativity
DSR introduces a minimum length scale without singlingout any preferred frame
The Energy-Momentum relation get a length scaledependent modification.
Ex: E2 = p2c2 +m2c4 + αlpE3 + βl2pE
4 + ......
![Page 13: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/13.jpg)
Special Theory of Relativity: Laws of physics must besame in all inertial frames
If ls ≥ lmin, ls′ ≥ lmin.
But this is not guaranteed(!) due to Lorentz-Fitsgeraldlength contraction
Modify STR Space-time structure is governed not onlyby a fundamental velocity scale c, but also by afundamental length scale lp. Doubly Special Relativity
DSR introduces a minimum length scale without singlingout any preferred frame
The Energy-Momentum relation get a length scaledependent modification.
Ex: E2 = p2c2 +m2c4 + αlpE3 + βl2pE
4 + ......
![Page 14: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/14.jpg)
Modified dispersion relations
Many Q-gravity models do give modifiedEnergy-Momentum relations
Observations of ultra high energy cosmic ray scatteringcontradicts standard notions of astroparticle physics.
These observations can be explained if the thresholdenergies required for these processes are not dictated byusual Energy-Momentum relations but by modified onesinvolving a length scale!
DSR: Two seemingly different models were constructedrecently.
Are they related? Equivalent?
IS DSR UNIQUE?
We will come back to this.
![Page 15: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/15.jpg)
Modified dispersion relations
Many Q-gravity models do give modifiedEnergy-Momentum relations
Observations of ultra high energy cosmic ray scatteringcontradicts standard notions of astroparticle physics.
These observations can be explained if the thresholdenergies required for these processes are not dictated byusual Energy-Momentum relations but by modified onesinvolving a length scale!
DSR: Two seemingly different models were constructedrecently.
Are they related? Equivalent?
IS DSR UNIQUE?
We will come back to this.
![Page 16: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/16.jpg)
Modified dispersion relations
Many Q-gravity models do give modifiedEnergy-Momentum relations
Observations of ultra high energy cosmic ray scatteringcontradicts standard notions of astroparticle physics.
These observations can be explained if the thresholdenergies required for these processes are not dictated byusual Energy-Momentum relations but by modified onesinvolving a length scale!
DSR: Two seemingly different models were constructedrecently.
Are they related? Equivalent?
IS DSR UNIQUE?
We will come back to this.
![Page 17: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/17.jpg)
Modified dispersion relations
Many Q-gravity models do give modifiedEnergy-Momentum relations
Observations of ultra high energy cosmic ray scatteringcontradicts standard notions of astroparticle physics.
These observations can be explained if the thresholdenergies required for these processes are not dictated byusual Energy-Momentum relations but by modified onesinvolving a length scale!
DSR: Two seemingly different models were constructedrecently.
Are they related? Equivalent?
IS DSR UNIQUE?
We will come back to this.
![Page 18: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/18.jpg)
Modified dispersion relations
Many Q-gravity models do give modifiedEnergy-Momentum relations
Observations of ultra high energy cosmic ray scatteringcontradicts standard notions of astroparticle physics.
These observations can be explained if the thresholdenergies required for these processes are not dictated byusual Energy-Momentum relations but by modified onesinvolving a length scale!
DSR: Two seemingly different models were constructedrecently.
Are they related? Equivalent?
IS DSR UNIQUE?
We will come back to this.
![Page 19: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/19.jpg)
Modified dispersion relations
Many Q-gravity models do give modifiedEnergy-Momentum relations
Observations of ultra high energy cosmic ray scatteringcontradicts standard notions of astroparticle physics.
These observations can be explained if the thresholdenergies required for these processes are not dictated byusual Energy-Momentum relations but by modified onesinvolving a length scale!
DSR: Two seemingly different models were constructedrecently.
Are they related? Equivalent?
IS DSR UNIQUE?
We will come back to this.
![Page 20: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/20.jpg)
Modified dispersion relations
Many Q-gravity models do give modifiedEnergy-Momentum relations
Observations of ultra high energy cosmic ray scatteringcontradicts standard notions of astroparticle physics.
These observations can be explained if the thresholdenergies required for these processes are not dictated byusual Energy-Momentum relations but by modified onesinvolving a length scale!
DSR: Two seemingly different models were constructedrecently.
Are they related? Equivalent?
IS DSR UNIQUE?
We will come back to this.
![Page 21: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/21.jpg)
DSR and k-deformed space-time
There are certain q-gravity models whose low energylimit shows modified Energy-Momentum relations as inDSR.
These q-gravity models with Λ > 0 ( and goes over toΛ = 0 limit smoothly) are shown to have deformed deSitter group as the symmetry group. The deformationparameter q here is related to lp as in q = lp
√Λ.
In the Λ → 0 limit, the symmetry group reduces tok-Poincare group and NOT Poincare group.
Algebraic structure governing the deformation ofEnergy-Momentum relation in these models at Planckscale is k-Poincare algebra
![Page 22: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/22.jpg)
DSR and k-deformed space-time
There are certain q-gravity models whose low energylimit shows modified Energy-Momentum relations as inDSR.
These q-gravity models with Λ > 0 ( and goes over toΛ = 0 limit smoothly) are shown to have deformed deSitter group as the symmetry group. The deformationparameter q here is related to lp as in q = lp
√Λ.
In the Λ → 0 limit, the symmetry group reduces tok-Poincare group and NOT Poincare group.
Algebraic structure governing the deformation ofEnergy-Momentum relation in these models at Planckscale is k-Poincare algebra
![Page 23: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/23.jpg)
DSR and k-deformed space-time
There are certain q-gravity models whose low energylimit shows modified Energy-Momentum relations as inDSR.
These q-gravity models with Λ > 0 ( and goes over toΛ = 0 limit smoothly) are shown to have deformed deSitter group as the symmetry group. The deformationparameter q here is related to lp as in q = lp
√Λ.
In the Λ → 0 limit, the symmetry group reduces tok-Poincare group and NOT Poincare group.
Algebraic structure governing the deformation ofEnergy-Momentum relation in these models at Planckscale is k-Poincare algebra
![Page 24: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/24.jpg)
DSR and k-deformed space-time
There are certain q-gravity models whose low energylimit shows modified Energy-Momentum relations as inDSR.
These q-gravity models with Λ > 0 ( and goes over toΛ = 0 limit smoothly) are shown to have deformed deSitter group as the symmetry group. The deformationparameter q here is related to lp as in q = lp
√Λ.
In the Λ → 0 limit, the symmetry group reduces tok-Poincare group and NOT Poincare group.
Algebraic structure governing the deformation ofEnergy-Momentum relation in these models at Planckscale is k-Poincare algebra
![Page 25: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/25.jpg)
Motivations/Introduction
Non-commutative space and twisted symmetry
k-spacetime and k-Poincare algebra
Realisation of kappa spacetime and its Symmetry Algebra
Conclusion
![Page 26: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/26.jpg)
Moyal space: summary of essential results
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
Moyal space is the one where θ λµν , θ
λσµν , ..... all are set
to ZERO.[Xµ, Xν ] = iθµν
Weyl-Moyal map:
f =
∫
dkdxf(x)eik·(X−x)
![Page 27: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/27.jpg)
Moyal space: summary of essential results
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
Moyal space is the one where θ λµν , θ
λσµν , ..... all are set
to ZERO.[Xµ, Xν ] = iθµν
Weyl-Moyal map:
f =
∫
dkdxf(x)eik·(X−x)
![Page 28: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/28.jpg)
Moyal space: summary of essential results
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
Moyal space is the one where θ λµν , θ
λσµν , ..... all are set
to ZERO.[Xµ, Xν ] = iθµν
Weyl-Moyal map:
f =
∫
dkdxf(x)eik·(X−x)
![Page 29: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/29.jpg)
Moyal space: summary of essential results
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
Moyal space is the one where θ λµν , θ
λσµν , ..... all are set
to ZERO.[Xµ, Xν ] = iθµν
Weyl-Moyal map:
f =
∫
dkdxf(x)eik·(X−x)
![Page 30: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/30.jpg)
Moyal space:......
f g induces a modified product rule:Moyal Star Product f ∗ g
f ∗ g = f(x)ei2∂x
µθµν∂yν g(y)|x=y
1. ∗ product is associative2.
∫
dxf ∗ g =∫
dxfg
3.∫
dx(f ∗ g ∗ h) =∫
dx(g ∗ h ∗ f) =∫
dx(h ∗ f ∗ g)4. (f ∗ g)cc = gcc ∗ f cc
Quadratic part of the NC action is same as thecommutative one
Propagator is not modified: no change in dispersionrelations
Interactions are modified
![Page 31: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/31.jpg)
Moyal space:......
f g induces a modified product rule:Moyal Star Product f ∗ g
f ∗ g = f(x)ei2∂x
µθµν∂yν g(y)|x=y
1. ∗ product is associative2.
∫
dxf ∗ g =∫
dxfg
3.∫
dx(f ∗ g ∗ h) =∫
dx(g ∗ h ∗ f) =∫
dx(h ∗ f ∗ g)4. (f ∗ g)cc = gcc ∗ f cc
Quadratic part of the NC action is same as thecommutative one
Propagator is not modified: no change in dispersionrelations
Interactions are modified
![Page 32: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/32.jpg)
Moyal space:......
f g induces a modified product rule:Moyal Star Product f ∗ g
f ∗ g = f(x)ei2∂x
µθµν∂yν g(y)|x=y
1. ∗ product is associative2.
∫
dxf ∗ g =∫
dxfg
3.∫
dx(f ∗ g ∗ h) =∫
dx(g ∗ h ∗ f) =∫
dx(h ∗ f ∗ g)4. (f ∗ g)cc = gcc ∗ f cc
Quadratic part of the NC action is same as thecommutative one
Propagator is not modified: no change in dispersionrelations
Interactions are modified
![Page 33: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/33.jpg)
Moyal space:......
f g induces a modified product rule:Moyal Star Product f ∗ g
f ∗ g = f(x)ei2∂x
µθµν∂yν g(y)|x=y
1. ∗ product is associative2.
∫
dxf ∗ g =∫
dxfg
3.∫
dx(f ∗ g ∗ h) =∫
dx(g ∗ h ∗ f) =∫
dx(h ∗ f ∗ g)4. (f ∗ g)cc = gcc ∗ f cc
Quadratic part of the NC action is same as thecommutative one
Propagator is not modified: no change in dispersionrelations
Interactions are modified
![Page 34: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/34.jpg)
Twisted symmetry
[Xµ, Xν ] = iθµν breaks the Lorentz invariance of Moyalplane.
The notion of fields transforming under representations ofPoincare group is in trouble -Can not view field quanta asparticles with definite spin and mass
![Page 35: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/35.jpg)
Twisted symmetry
[Xµ, Xν ] = iθµν breaks the Lorentz invariance of Moyalplane.
The notion of fields transforming under representations ofPoincare group is in trouble -Can not view field quanta asparticles with definite spin and mass
![Page 36: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/36.jpg)
in commutative space fρσ = (xρxσ) transform as a rank-2tensor,
Mµνfρσ = i(fµσηνρ − fνσηµρ + fρνηµσ − fρνηµσ − fρµηνσ)
Chaichian and co workers showed that the symmetryalgebra of Moyal spacetime is realised by the twisted
Poincare-Hopf algebra and not by the Poincare algebra
fρσ = 12(xρ ∗ xσ + xσ ∗ xρ) transform as a rank-2 tensor
under twisted action, i.e.,
M tµνfρσ = i(fµσηνρ − fνσηµρ + fρνηµσ − fρνηµσ − fρµηνσ).
M tµν([xρ, xσ]∗) = 0 = M t
µνθρσ
.
![Page 37: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/37.jpg)
in commutative space fρσ = (xρxσ) transform as a rank-2tensor,
Mµνfρσ = i(fµσηνρ − fνσηµρ + fρνηµσ − fρνηµσ − fρµηνσ)
Chaichian and co workers showed that the symmetryalgebra of Moyal spacetime is realised by the twisted
Poincare-Hopf algebra and not by the Poincare algebra
fρσ = 12(xρ ∗ xσ + xσ ∗ xρ) transform as a rank-2 tensor
under twisted action, i.e.,
M tµνfρσ = i(fµσηνρ − fνσηµρ + fρνηµσ − fρνηµσ − fρµηνσ).
M tµν([xρ, xσ]∗) = 0 = M t
µνθρσ
.
![Page 38: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/38.jpg)
in commutative space fρσ = (xρxσ) transform as a rank-2tensor,
Mµνfρσ = i(fµσηνρ − fνσηµρ + fρνηµσ − fρνηµσ − fρµηνσ)
Chaichian and co workers showed that the symmetryalgebra of Moyal spacetime is realised by the twisted
Poincare-Hopf algebra and not by the Poincare algebra
fρσ = 12(xρ ∗ xσ + xσ ∗ xρ) transform as a rank-2 tensor
under twisted action, i.e.,
M tµνfρσ = i(fµσηνρ − fνσηµρ + fρνηµσ − fρνηµσ − fρµηνσ).
M tµν([xρ, xσ]∗) = 0 = M t
µνθρσ
.
![Page 39: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/39.jpg)
in commutative space fρσ = (xρxσ) transform as a rank-2tensor,
Mµνfρσ = i(fµσηνρ − fνσηµρ + fρνηµσ − fρνηµσ − fρµηνσ)
Chaichian and co workers showed that the symmetryalgebra of Moyal spacetime is realised by the twisted
Poincare-Hopf algebra and not by the Poincare algebra
fρσ = 12(xρ ∗ xσ + xσ ∗ xρ) transform as a rank-2 tensor
under twisted action, i.e.,
M tµνfρσ = i(fµσηνρ − fνσηµρ + fρνηµσ − fρνηµσ − fρµηνσ).
M tµν([xρ, xσ]∗) = 0 = M t
µνθρσ
.
![Page 40: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/40.jpg)
Attempts to construct NC gravity by demanding acompatibility between ∗ product and the action ofdeformed generators led to the twisted Leibnitz rule forthe symmetry generators.
α⊗ β −→ (ρ⊗ ρ)∆(g)α⊗ β
m ↓ ↓ m
m(α⊗ β) −→ ρ(g)m(α⊗ β)
It was argued that the twisted Hopf structure of thesymmetries have interesting implications in field theory
We study the k-Poincare algebra which is the symmetryalgebra of k-deformed spacetime, construction of fieldtheory on k-spacetime and some of the interestingproperties of this theory.
![Page 41: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/41.jpg)
Attempts to construct NC gravity by demanding acompatibility between ∗ product and the action ofdeformed generators led to the twisted Leibnitz rule forthe symmetry generators.
α⊗ β −→ (ρ⊗ ρ)∆(g)α⊗ β
m ↓ ↓ m
m(α⊗ β) −→ ρ(g)m(α⊗ β)
It was argued that the twisted Hopf structure of thesymmetries have interesting implications in field theory
We study the k-Poincare algebra which is the symmetryalgebra of k-deformed spacetime, construction of fieldtheory on k-spacetime and some of the interestingproperties of this theory.
![Page 42: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/42.jpg)
Attempts to construct NC gravity by demanding acompatibility between ∗ product and the action ofdeformed generators led to the twisted Leibnitz rule forthe symmetry generators.
α⊗ β −→ (ρ⊗ ρ)∆(g)α⊗ β
m ↓ ↓ m
m(α⊗ β) −→ ρ(g)m(α⊗ β)
It was argued that the twisted Hopf structure of thesymmetries have interesting implications in field theory
We study the k-Poincare algebra which is the symmetryalgebra of k-deformed spacetime, construction of fieldtheory on k-spacetime and some of the interestingproperties of this theory.
![Page 43: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/43.jpg)
Motivations/Introduction
Non-commutative space and twisted symmetry
k-spacetime and k-Poincare algebra
Realisation of kappa spacetime and its Symmetry Algebra
Conclusion
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k-spacetime
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
where Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
with θ0µν = 0, θ λσ
µν = 0, ...
Only non-vanishing term θ λµν
Thus we have [xµ, xν ] = iCλµν xλ Lie algebraic type NC
choice: Cλµν = aµδνλ − aνδµλ, aµ, µ = 0, 1, ...., n− 1 are
real
choice: a0 = a = 1k, ai = 0, i = 1, 2, ......, n− 1
![Page 45: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/45.jpg)
k-spacetime
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
where Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
with θ0µν = 0, θ λσ
µν = 0, ...
Only non-vanishing term θ λµν
Thus we have [xµ, xν ] = iCλµν xλ Lie algebraic type NC
choice: Cλµν = aµδνλ − aνδµλ, aµ, µ = 0, 1, ...., n− 1 are
real
choice: a0 = a = 1k, ai = 0, i = 1, 2, ......, n− 1
![Page 46: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/46.jpg)
k-spacetime
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
where Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
with θ0µν = 0, θ λσ
µν = 0, ...
Only non-vanishing term θ λµν
Thus we have [xµ, xν ] = iCλµν xλ Lie algebraic type NC
choice: Cλµν = aµδνλ − aνδµλ, aµ, µ = 0, 1, ...., n− 1 are
real
choice: a0 = a = 1k, ai = 0, i = 1, 2, ......, n− 1
![Page 47: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/47.jpg)
k-spacetime
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
where Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
with θ0µν = 0, θ λσ
µν = 0, ...
Only non-vanishing term θ λµν
Thus we have [xµ, xν ] = iCλµν xλ Lie algebraic type NC
choice: Cλµν = aµδνλ − aνδµλ, aµ, µ = 0, 1, ...., n− 1 are
real
choice: a0 = a = 1k, ai = 0, i = 1, 2, ......, n− 1
![Page 48: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/48.jpg)
k-spacetime
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
where Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
with θ0µν = 0, θ λσ
µν = 0, ...
Only non-vanishing term θ λµν
Thus we have [xµ, xν ] = iCλµν xλ Lie algebraic type NC
choice: Cλµν = aµδνλ − aνδµλ, aµ, µ = 0, 1, ...., n− 1 are
real
choice: a0 = a = 1k, ai = 0, i = 1, 2, ......, n− 1
![Page 49: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/49.jpg)
k-spacetime
Generic NC spaces are defined with co-ordinates obeying
[Xµ, Xν ] =i
k2Θµν(kx)
where Θµν(kx) = θ0µν + θ λ
µν xλ + θ λσµν xλxσ + ..........
with θ0µν = 0, θ λσ
µν = 0, ...
Only non-vanishing term θ λµν
Thus we have [xµ, xν ] = iCλµν xλ Lie algebraic type NC
choice: Cλµν = aµδνλ − aνδµλ, aµ, µ = 0, 1, ...., n− 1 are
real
choice: a0 = a = 1k, ai = 0, i = 1, 2, ......, n− 1
![Page 50: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/50.jpg)
k-spacetime.......
k-spacetime co-ordinates satisfy:
[xi, xj] = 0, [x0, xi] = iaxi
The symmetry algebra of this spacetime is k-Poincarealgebra
[Mµν ,Mαβ] = i(ηµβMνα − ηµαMνβ + ηναMµβ − ηνβMµα)
[Mi, Pµ] = iǫiµjPj, [Pµ, Pν ] = 0, [Ni, P0] = iPi
[Ni, Pj] = iδij
(
1
2a(1 − e−2aP0) +
a
2~P 2
)
− iaPiPj
with Casimir m2 = ( 2asinh(aPo
2))2 − ~P 2eaP0
![Page 51: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/51.jpg)
k-spacetime.......
k-spacetime co-ordinates satisfy:
[xi, xj] = 0, [x0, xi] = iaxi
The symmetry algebra of this spacetime is k-Poincarealgebra
[Mµν ,Mαβ] = i(ηµβMνα − ηµαMνβ + ηναMµβ − ηνβMµα)
[Mi, Pµ] = iǫiµjPj, [Pµ, Pν ] = 0, [Ni, P0] = iPi
[Ni, Pj] = iδij
(
1
2a(1 − e−2aP0) +
a
2~P 2
)
− iaPiPj
with Casimir m2 = ( 2asinh(aPo
2))2 − ~P 2eaP0
![Page 52: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/52.jpg)
Another k-deformed algebra studied is one where
[Ni, Pj ] = i(δijP0 − aPiPj), [Ni, P0] = i(1 − aP0)Pi
with Casimir M2 = (P 20 − ~P 2)(1 − aP0)
−2
The two DSR models constructed have energy-momentumrelations given by these two Casimirs respectively.
The non-commutative structure of the underlyingspace-time of both DSRs are same, showing theequivalence of the physical models.
![Page 53: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/53.jpg)
Another k-deformed algebra studied is one where
[Ni, Pj ] = i(δijP0 − aPiPj), [Ni, P0] = i(1 − aP0)Pi
with Casimir M2 = (P 20 − ~P 2)(1 − aP0)
−2
The two DSR models constructed have energy-momentumrelations given by these two Casimirs respectively.
The non-commutative structure of the underlyingspace-time of both DSRs are same, showing theequivalence of the physical models.
![Page 54: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/54.jpg)
Another k-deformed algebra studied is one where
[Ni, Pj ] = i(δijP0 − aPiPj), [Ni, P0] = i(1 − aP0)Pi
with Casimir M2 = (P 20 − ~P 2)(1 − aP0)
−2
The two DSR models constructed have energy-momentumrelations given by these two Casimirs respectively.
The non-commutative structure of the underlyingspace-time of both DSRs are same, showing theequivalence of the physical models.
![Page 55: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/55.jpg)
Symmetry algebra of k-spacetime
There are different approaches to construct field theoryon k-spacetime.
Using fields which are functions of xµ and defining theaction which is invariant under k-Poincare algebra.
Map kappa-spacetime co-ordinates and their functions tocommutative ones and work with these commutativefunctions.
We take the second approach
![Page 56: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/56.jpg)
Symmetry algebra of k-spacetime
There are different approaches to construct field theoryon k-spacetime.
Using fields which are functions of xµ and defining theaction which is invariant under k-Poincare algebra.
Map kappa-spacetime co-ordinates and their functions tocommutative ones and work with these commutativefunctions.
We take the second approach
![Page 57: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/57.jpg)
Symmetry algebra of k-spacetime
There are different approaches to construct field theoryon k-spacetime.
Using fields which are functions of xµ and defining theaction which is invariant under k-Poincare algebra.
Map kappa-spacetime co-ordinates and their functions tocommutative ones and work with these commutativefunctions.
We take the second approach
![Page 58: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/58.jpg)
Symmetry algebra of k-spacetime
There are different approaches to construct field theoryon k-spacetime.
Using fields which are functions of xµ and defining theaction which is invariant under k-Poincare algebra.
Map kappa-spacetime co-ordinates and their functions tocommutative ones and work with these commutativefunctions.
We take the second approach
![Page 59: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/59.jpg)
Symmetry......
We derive the action of Lorentz algebra on k-spacetimeco-ordinates and also obtain their derivative operators.
These operators satisfy usual Poincare algebra relations,but have modified Casimirs
We obtain different possible invariant actions for scalartheory.
We derive the modified Leibnitz rule( twisted co-products)of these generators and compatible flip operator.
Using this, we derive the deformed commutation rulesbetween A,A†/twisted statistics.
![Page 60: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/60.jpg)
Symmetry......
We derive the action of Lorentz algebra on k-spacetimeco-ordinates and also obtain their derivative operators.
These operators satisfy usual Poincare algebra relations,but have modified Casimirs
We obtain different possible invariant actions for scalartheory.
We derive the modified Leibnitz rule( twisted co-products)of these generators and compatible flip operator.
Using this, we derive the deformed commutation rulesbetween A,A†/twisted statistics.
![Page 61: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/61.jpg)
Symmetry......
We derive the action of Lorentz algebra on k-spacetimeco-ordinates and also obtain their derivative operators.
These operators satisfy usual Poincare algebra relations,but have modified Casimirs
We obtain different possible invariant actions for scalartheory.
We derive the modified Leibnitz rule( twisted co-products)of these generators and compatible flip operator.
Using this, we derive the deformed commutation rulesbetween A,A†/twisted statistics.
![Page 62: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/62.jpg)
Symmetry......
We derive the action of Lorentz algebra on k-spacetimeco-ordinates and also obtain their derivative operators.
These operators satisfy usual Poincare algebra relations,but have modified Casimirs
We obtain different possible invariant actions for scalartheory.
We derive the modified Leibnitz rule( twisted co-products)of these generators and compatible flip operator.
Using this, we derive the deformed commutation rulesbetween A,A†/twisted statistics.
![Page 63: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/63.jpg)
Symmetry......
We derive the action of Lorentz algebra on k-spacetimeco-ordinates and also obtain their derivative operators.
These operators satisfy usual Poincare algebra relations,but have modified Casimirs
We obtain different possible invariant actions for scalartheory.
We derive the modified Leibnitz rule( twisted co-products)of these generators and compatible flip operator.
Using this, we derive the deformed commutation rulesbetween A,A†/twisted statistics.
![Page 64: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/64.jpg)
Motivations/Introduction
Non-commutative space and twisted symmetry
k-spacetime and k-Poincare algebra
Realisation of kappa spacetime and its Symmetry Algebra
Conclusion
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K-spacetime, ordering, Leibnitz rules
We have [x0, xi] = iaxi, [xi, xj] = 0
xµ = xαΦαµ(∂) This defines a unique mapping offunctions on k-spacetime to that on commutative spacetime
F (xϕ)|0 >= Fϕ(x)
Any M(x) can be expanded as a power series in xµ.M(x) can be written as LC of monomials ofx0, x1, ....., xn−1 with m0,m1, ....mn−1 as powers andpolynomials of lower order P (x). Thus
[M(x) − P (x)] |0 >= M(x)
Natural ordering:x0 to the right/left of xi
x0 and xi treated symmetrically.
![Page 66: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/66.jpg)
K-spacetime, ordering, Leibnitz rules
We have [x0, xi] = iaxi, [xi, xj] = 0
xµ = xαΦαµ(∂) This defines a unique mapping offunctions on k-spacetime to that on commutative spacetime
F (xϕ)|0 >= Fϕ(x)
Any M(x) can be expanded as a power series in xµ.M(x) can be written as LC of monomials ofx0, x1, ....., xn−1 with m0,m1, ....mn−1 as powers andpolynomials of lower order P (x). Thus
[M(x) − P (x)] |0 >= M(x)
Natural ordering:x0 to the right/left of xi
x0 and xi treated symmetrically.
![Page 67: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/67.jpg)
K-spacetime, ordering, Leibnitz rules
We have [x0, xi] = iaxi, [xi, xj] = 0
xµ = xαΦαµ(∂) This defines a unique mapping offunctions on k-spacetime to that on commutative spacetime
F (xϕ)|0 >= Fϕ(x)
Any M(x) can be expanded as a power series in xµ.M(x) can be written as LC of monomials ofx0, x1, ....., xn−1 with m0,m1, ....mn−1 as powers andpolynomials of lower order P (x). Thus
[M(x) − P (x)] |0 >= M(x)
Natural ordering:x0 to the right/left of xi
x0 and xi treated symmetrically.
![Page 68: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/68.jpg)
K-spacetime, ordering, Leibnitz rules
We have [x0, xi] = iaxi, [xi, xj] = 0
xµ = xαΦαµ(∂) This defines a unique mapping offunctions on k-spacetime to that on commutative spacetime
F (xϕ)|0 >= Fϕ(x)
Any M(x) can be expanded as a power series in xµ.M(x) can be written as LC of monomials ofx0, x1, ....., xn−1 with m0,m1, ....mn−1 as powers andpolynomials of lower order P (x). Thus
[M(x) − P (x)] |0 >= M(x)
Natural ordering:x0 to the right/left of xi
x0 and xi treated symmetrically.
![Page 69: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/69.jpg)
K-spacetime, ordering, Leibnitz rules
Imposing
[∂i, xj] = δijϕ(A), [∂i, x0] = ia∂iγ(A)
[∂0, xi] = 0, [∂0, x0] = 1,
with A = ia∂0, we get from xµ = xαΦαµ(∂)
xi = xiϕ(A)
x0 = x0ψ(A) + ia∂iγ(A)
from the commutators we get ϕ′
ϕψ = γ − 1
( ϕ(0) = 1, ψ(0) = 1, γ(0) = ϕ′(0) + 1) Leibnitz rule for ∂i is modified
∆ϕ(∂i) = ∂xi
ϕ(Ax + Ay)
ϕ(Ax)+ ∂
yi
ϕ(Ax + Ay)
ϕ(Ay)
∆ϕ(∂0) = ∂0 ⊗ I + I ⊗ ∂0 = ∂x0 I
y + Ix∂y0
![Page 70: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/70.jpg)
K-spacetime, ordering, Leibnitz rules
Imposing
[∂i, xj] = δijϕ(A), [∂i, x0] = ia∂iγ(A)
[∂0, xi] = 0, [∂0, x0] = 1,
with A = ia∂0, we get from xµ = xαΦαµ(∂)
xi = xiϕ(A)
x0 = x0ψ(A) + ia∂iγ(A)
from the commutators we get ϕ′
ϕψ = γ − 1
( ϕ(0) = 1, ψ(0) = 1, γ(0) = ϕ′(0) + 1) Leibnitz rule for ∂i is modified
∆ϕ(∂i) = ∂xi
ϕ(Ax + Ay)
ϕ(Ax)+ ∂
yi
ϕ(Ax + Ay)
ϕ(Ay)
∆ϕ(∂0) = ∂0 ⊗ I + I ⊗ ∂0 = ∂x0 I
y + Ix∂y0
![Page 71: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/71.jpg)
K-spacetime, ordering, Leibnitz rules
Imposing
[∂i, xj] = δijϕ(A), [∂i, x0] = ia∂iγ(A)
[∂0, xi] = 0, [∂0, x0] = 1,
with A = ia∂0, we get from xµ = xαΦαµ(∂)
xi = xiϕ(A)
x0 = x0ψ(A) + ia∂iγ(A)
from the commutators we get ϕ′
ϕψ = γ − 1
( ϕ(0) = 1, ψ(0) = 1, γ(0) = ϕ′(0) + 1) Leibnitz rule for ∂i is modified
∆ϕ(∂i) = ∂xi
ϕ(Ax + Ay)
ϕ(Ax)+ ∂
yi
ϕ(Ax + Ay)
ϕ(Ay)
∆ϕ(∂0) = ∂0 ⊗ I + I ⊗ ∂0 = ∂x0 I
y + Ix∂y0
![Page 72: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/72.jpg)
K-spacetime, ordering, Leibnitz rules
Imposing
[∂i, xj] = δijϕ(A), [∂i, x0] = ia∂iγ(A)
[∂0, xi] = 0, [∂0, x0] = 1,
with A = ia∂0, we get from xµ = xαΦαµ(∂)
xi = xiϕ(A)
x0 = x0ψ(A) + ia∂iγ(A)
from the commutators we get ϕ′
ϕψ = γ − 1
( ϕ(0) = 1, ψ(0) = 1, γ(0) = ϕ′(0) + 1) Leibnitz rule for ∂i is modified
∆ϕ(∂i) = ∂xi
ϕ(Ax + Ay)
ϕ(Ax)+ ∂
yi
ϕ(Ax + Ay)
ϕ(Ay)
∆ϕ(∂0) = ∂0 ⊗ I + I ⊗ ∂0 = ∂x0 I
y + Ix∂y0
![Page 73: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/73.jpg)
K-spacetime, ordering, Leibnitz rules
Imposing
[∂i, xj] = δijϕ(A), [∂i, x0] = ia∂iγ(A)
[∂0, xi] = 0, [∂0, x0] = 1,
with A = ia∂0, we get from xµ = xαΦαµ(∂)
xi = xiϕ(A)
x0 = x0ψ(A) + ia∂iγ(A)
from the commutators we get ϕ′
ϕψ = γ − 1
( ϕ(0) = 1, ψ(0) = 1, γ(0) = ϕ′(0) + 1) Leibnitz rule for ∂i is modified
∆ϕ(∂i) = ∂xi
ϕ(Ax + Ay)
ϕ(Ax)+ ∂
yi
ϕ(Ax + Ay)
ϕ(Ay)
∆ϕ(∂0) = ∂0 ⊗ I + I ⊗ ∂0 = ∂x0 I
y + Ix∂y0
![Page 74: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/74.jpg)
k-Poincare algebra, Casimir and Dispersion relation
No modification in the Lorentz algebra
Demand Mµν and xµ close linearly, satisfy Jacobi identity,smooth commutative limit
[Mi0, x0] = xi + iaMi0
[Mi0, xj] = −δijx0 − iaMij
Leibnitz rule
∆ϕ(Mij) = Mij ⊗ I + I ⊗Mij
∆ϕ(Mi0) = Mi0 ⊗ I + eA ⊗Mi0 + ia∂j
1
ϕ(A)⊗Mij
![Page 75: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/75.jpg)
k-Poincare algebra, Casimir and Dispersion relation
No modification in the Lorentz algebra
Demand Mµν and xµ close linearly, satisfy Jacobi identity,smooth commutative limit
[Mi0, x0] = xi + iaMi0
[Mi0, xj] = −δijx0 − iaMij
Leibnitz rule
∆ϕ(Mij) = Mij ⊗ I + I ⊗Mij
∆ϕ(Mi0) = Mi0 ⊗ I + eA ⊗Mi0 + ia∂j
1
ϕ(A)⊗Mij
![Page 76: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/76.jpg)
k-Poincare algebra, Casimir and Dispersion relation
No modification in the Lorentz algebra
Demand Mµν and xµ close linearly, satisfy Jacobi identity,smooth commutative limit
[Mi0, x0] = xi + iaMi0
[Mi0, xj] = −δijx0 − iaMij
Leibnitz rule
∆ϕ(Mij) = Mij ⊗ I + I ⊗Mij
∆ϕ(Mi0) = Mi0 ⊗ I + eA ⊗Mi0 + ia∂j
1
ϕ(A)⊗Mij
![Page 77: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/77.jpg)
k-Poincare algebra, Casimir and Dispersion relation
No modification in the Lorentz algebra
Demand Mµν and xµ close linearly, satisfy Jacobi identity,smooth commutative limit
[Mi0, x0] = xi + iaMi0
[Mi0, xj] = −δijx0 − iaMij
Leibnitz rule
∆ϕ(Mij) = Mij ⊗ I + I ⊗Mij
∆ϕ(Mi0) = Mi0 ⊗ I + eA ⊗Mi0 + ia∂j
1
ϕ(A)⊗Mij
![Page 78: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/78.jpg)
k-Poincare algebra, Casimir and Dispersion relation
Enlarge the algebra:
[Mµν , Dλ] = δνλDµ − δµλDν
[Dµ, Dν ] = 0
[Dµ, xν ] = δµν
√
1 − a2DαDα + ia0(δµ0Dν − δµνD0)
D0 = −i∂0sinhA
A− ia
e−A
2ϕ2; Di = ∂i
e−A
ϕ
[Mµν ,] = 0, [, xµ] = 2Dµ
=e−A
ϕ2+ 2∂2
0(1 − coshA)A2
![Page 79: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/79.jpg)
k-Poincare algebra, Casimir and Dispersion relation
Enlarge the algebra:
[Mµν , Dλ] = δνλDµ − δµλDν
[Dµ, Dν ] = 0
[Dµ, xν ] = δµν
√
1 − a2DαDα + ia0(δµ0Dν − δµνD0)
D0 = −i∂0sinhA
A− ia
e−A
2ϕ2; Di = ∂i
e−A
ϕ
[Mµν ,] = 0, [, xµ] = 2Dµ
=e−A
ϕ2+ 2∂2
0(1 − coshA)A2
![Page 80: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/80.jpg)
k-Poincare algebra, Casimir and Dispersion relation
The Casimir
DµDµ = (1 − a2
4) quartic
is quadratic in space derivatives.
((1 − a2
4) −m2)Φ(x) = 0
A−m2 − a2
4A2 = 0
with A = a2
4Sinh2(ap0
2) − p2
ie−aP0
ϕ2(ap0)
( −m2)Φ(x) = 0
4
a2Sinh2(
ap0
2) − p2
i
e−ap0
ϕ(ap0)2−m2 = 0
![Page 81: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/81.jpg)
k-Poincare algebra, Casimir and Dispersion relation
The Casimir
DµDµ = (1 − a2
4) quartic
is quadratic in space derivatives.
((1 − a2
4) −m2)Φ(x) = 0
A−m2 − a2
4A2 = 0
with A = a2
4Sinh2(ap0
2) − p2
ie−aP0
ϕ2(ap0)
( −m2)Φ(x) = 0
4
a2Sinh2(
ap0
2) − p2
i
e−ap0
ϕ(ap0)2−m2 = 0
![Page 82: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/82.jpg)
k-Poincare algebra, Casimir and Dispersion relation
The Casimir
DµDµ = (1 − a2
4) quartic
is quadratic in space derivatives.
((1 − a2
4) −m2)Φ(x) = 0
A−m2 − a2
4A2 = 0
with A = a2
4Sinh2(ap0
2) − p2
ie−aP0
ϕ2(ap0)
( −m2)Φ(x) = 0
4
a2Sinh2(
ap0
2) − p2
i
e−ap0
ϕ(ap0)2−m2 = 0
![Page 83: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/83.jpg)
k-Poincare algebra, Casimir and Dispersion relation
The Casimir
DµDµ = (1 − a2
4) quartic
is quadratic in space derivatives.
((1 − a2
4) −m2)Φ(x) = 0
A−m2 − a2
4A2 = 0
with A = a2
4Sinh2(ap0
2) − p2
ie−aP0
ϕ2(ap0)
( −m2)Φ(x) = 0
4
a2Sinh2(
ap0
2) − p2
i
e−ap0
ϕ(ap0)2−m2 = 0
![Page 84: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/84.jpg)
Star Product
xµ = xαΦαµ(∂)
Fϕ(xϕ)Gϕ(xϕ)|0 >= Fϕ∗ϕGϕ
(f ∗ϕ g)(x) = m0[exi(ϕ−0)∂if(u)g(v)]|u=t=xi
ϕ is the twisted co-product of ∂i
Fϕ = eNxln
ϕ(Ax+Ay)
ϕ(Ax)+Ny(Ax+ln
ϕ(Ax+Ay)
ϕ(Ay))
Twist element FΛ = e−ΛN⊗A+(1−Λ)A⊗N , Λ = 1, 0 forL/R ordering. Here N = xi∂i.
![Page 85: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/85.jpg)
Star Product
xµ = xαΦαµ(∂)
Fϕ(xϕ)Gϕ(xϕ)|0 >= Fϕ∗ϕGϕ
(f ∗ϕ g)(x) = m0[exi(ϕ−0)∂if(u)g(v)]|u=t=xi
ϕ is the twisted co-product of ∂i
Fϕ = eNxln
ϕ(Ax+Ay)
ϕ(Ax)+Ny(Ax+ln
ϕ(Ax+Ay)
ϕ(Ay))
Twist element FΛ = e−ΛN⊗A+(1−Λ)A⊗N , Λ = 1, 0 forL/R ordering. Here N = xi∂i.
![Page 86: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/86.jpg)
Star Product
xµ = xαΦαµ(∂)
Fϕ(xϕ)Gϕ(xϕ)|0 >= Fϕ∗ϕGϕ
(f ∗ϕ g)(x) = m0[exi(ϕ−0)∂if(u)g(v)]|u=t=xi
ϕ is the twisted co-product of ∂i
Fϕ = eNxln
ϕ(Ax+Ay)
ϕ(Ax)+Ny(Ax+ln
ϕ(Ax+Ay)
ϕ(Ay))
Twist element FΛ = e−ΛN⊗A+(1−Λ)A⊗N , Λ = 1, 0 forL/R ordering. Here N = xi∂i.
![Page 87: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/87.jpg)
Star Product
xµ = xαΦαµ(∂)
Fϕ(xϕ)Gϕ(xϕ)|0 >= Fϕ∗ϕGϕ
(f ∗ϕ g)(x) = m0[exi(ϕ−0)∂if(u)g(v)]|u=t=xi
ϕ is the twisted co-product of ∂i
Fϕ = eNxln
ϕ(Ax+Ay)
ϕ(Ax)+Ny(Ax+ln
ϕ(Ax+Ay)
ϕ(Ay))
Twist element FΛ = e−ΛN⊗A+(1−Λ)A⊗N , Λ = 1, 0 forL/R ordering. Here N = xi∂i.
![Page 88: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/88.jpg)
Twisted Flip Operator
(anti)Symmetric states of the physical Hilbert space areprojected from the tensor product state
1
2(1 ± τ0)(f ⊗ g) =
1
2(f ⊗ g ± g ⊗ f).
g : f ⊗ g = (D ⊗D) (g)f ⊗ g, g ∈ symm. algebra
[ (g), τ0] = 0
for the NC case [∆ϕ, τϕ] = 0
ϕ= F−1ϕ ∆0Fϕ
τϕ = F−1ϕ τ0Fϕ = ei(xiPi⊗A−A⊗xiPi)τ0
![Page 89: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/89.jpg)
Twisted Flip Operator
(anti)Symmetric states of the physical Hilbert space areprojected from the tensor product state
1
2(1 ± τ0)(f ⊗ g) =
1
2(f ⊗ g ± g ⊗ f).
g : f ⊗ g = (D ⊗D) (g)f ⊗ g, g ∈ symm. algebra
[ (g), τ0] = 0
for the NC case [∆ϕ, τϕ] = 0
ϕ= F−1ϕ ∆0Fϕ
τϕ = F−1ϕ τ0Fϕ = ei(xiPi⊗A−A⊗xiPi)τ0
![Page 90: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/90.jpg)
Twisted Flip Operator
(anti)Symmetric states of the physical Hilbert space areprojected from the tensor product state
1
2(1 ± τ0)(f ⊗ g) =
1
2(f ⊗ g ± g ⊗ f).
g : f ⊗ g = (D ⊗D) (g)f ⊗ g, g ∈ symm. algebra
[ (g), τ0] = 0
for the NC case [∆ϕ, τϕ] = 0
ϕ= F−1ϕ ∆0Fϕ
τϕ = F−1ϕ τ0Fϕ = ei(xiPi⊗A−A⊗xiPi)τ0
![Page 91: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/91.jpg)
Twisted Flip Operator
(anti)Symmetric states of the physical Hilbert space areprojected from the tensor product state
1
2(1 ± τ0)(f ⊗ g) =
1
2(f ⊗ g ± g ⊗ f).
g : f ⊗ g = (D ⊗D) (g)f ⊗ g, g ∈ symm. algebra
[ (g), τ0] = 0
for the NC case [∆ϕ, τϕ] = 0
ϕ= F−1ϕ ∆0Fϕ
τϕ = F−1ϕ τ0Fϕ = ei(xiPi⊗A−A⊗xiPi)τ0
![Page 92: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/92.jpg)
Twisted Flip Operator
for bosons f ⊗ g = τϕ(f ⊗ g)
φ(x) ⊗ φ(y) − e−(A⊗N−N⊗A)φ(y)φ(x) = 0
( −m2)Φ(x) = 0 with ϕ = e−A2 = e−
ia∂02 is
[
∂2i +
4
a2Sinh2(
ia∂0
2) −m2
]
Φ = 0
Φ(x) =∫
d4p
2Ωk(p)
[
A(ωk, ~p)e−ip·x + A†(ωk, ~p)e
ip·x]
A†(±ωk, ~p) = A†(∓ωk, ~p).
p±0 = ±ωk(p) = ± 2asinh−1(a
2
√
p2i +m2),
Ωk(p) = 1aSinh(aωk(p))
![Page 93: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/93.jpg)
Twisted Flip Operator
for bosons f ⊗ g = τϕ(f ⊗ g)
φ(x) ⊗ φ(y) − e−(A⊗N−N⊗A)φ(y)φ(x) = 0
( −m2)Φ(x) = 0 with ϕ = e−A2 = e−
ia∂02 is
[
∂2i +
4
a2Sinh2(
ia∂0
2) −m2
]
Φ = 0
Φ(x) =∫
d4p
2Ωk(p)
[
A(ωk, ~p)e−ip·x + A†(ωk, ~p)e
ip·x]
A†(±ωk, ~p) = A†(∓ωk, ~p).
p±0 = ±ωk(p) = ± 2asinh−1(a
2
√
p2i +m2),
Ωk(p) = 1aSinh(aωk(p))
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Twisted Flip Operator
for bosons f ⊗ g = τϕ(f ⊗ g)
φ(x) ⊗ φ(y) − e−(A⊗N−N⊗A)φ(y)φ(x) = 0
( −m2)Φ(x) = 0 with ϕ = e−A2 = e−
ia∂02 is
[
∂2i +
4
a2Sinh2(
ia∂0
2) −m2
]
Φ = 0
Φ(x) =∫
d4p
2Ωk(p)
[
A(ωk, ~p)e−ip·x + A†(ωk, ~p)e
ip·x]
A†(±ωk, ~p) = A†(∓ωk, ~p).
p±0 = ±ωk(p) = ± 2asinh−1(a
2
√
p2i +m2),
Ωk(p) = 1aSinh(aωk(p))
![Page 95: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/95.jpg)
Twisted Flip Operator
for bosons f ⊗ g = τϕ(f ⊗ g)
φ(x) ⊗ φ(y) − e−(A⊗N−N⊗A)φ(y)φ(x) = 0
( −m2)Φ(x) = 0 with ϕ = e−A2 = e−
ia∂02 is
[
∂2i +
4
a2Sinh2(
ia∂0
2) −m2
]
Φ = 0
Φ(x) =∫
d4p
2Ωk(p)
[
A(ωk, ~p)e−ip·x + A†(ωk, ~p)e
ip·x]
A†(±ωk, ~p) = A†(∓ωk, ~p).
p±0 = ±ωk(p) = ± 2asinh−1(a
2
√
p2i +m2),
Ωk(p) = 1aSinh(aωk(p))
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Twisted commutators
A†(p)A(q) − e−a(q0∂pipi+∂qi
qip0)A(q)A†(p) = −δ3(~p− ~q)
A†(p0, ~p)A†(q0, ~q)−e−a(−q0∂pi
pi+∂qiqip0)A†(q0, ~q)A
†(p0, ~p) = 0
A(p0, ~p)A(q0, ~q) − e−a(q0∂pipi−∂qi
qip0)A(q0, ~q)A(p0, ~p) = 0
p0, q0 as given above
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Deformed Product
A(p) A(q) = e−3a2
(p0−q0)A(p0, eaq02 ~p)A(q0, e
−ap02 ~q)
A†(p) A†(q) = e3a2
(p0−q0)A†(p0, e−
aq02 ~p)A(q0, e
ap02 ~q)
A†(p) A(q) = e3a2
(p0+q0)A†(p0, eaq02 ~p) A(q0, e
ap02 ~q)
A(p) A†(q) = e−3a2
(p0+q0)A(p0, e−
aq02 ~p) A†(q0, e
−ap02 ~q).
Using this, we can re-express commutators as in thecommutative case
[A(p0, ~p), A(q0, ~q)] = 0, [A†(p0, ~p), A†(q0, ~q)] = 0,
[A(p0, ~p), A†(q0, ~q)] = δ3(~p− ~q)
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Deformed Product
A(p) A(q) = e−3a2
(p0−q0)A(p0, eaq02 ~p)A(q0, e
−ap02 ~q)
A†(p) A†(q) = e3a2
(p0−q0)A†(p0, e−
aq02 ~p)A(q0, e
ap02 ~q)
A†(p) A(q) = e3a2
(p0+q0)A†(p0, eaq02 ~p) A(q0, e
ap02 ~q)
A(p) A†(q) = e−3a2
(p0+q0)A(p0, e−
aq02 ~p) A†(q0, e
−ap02 ~q).
Using this, we can re-express commutators as in thecommutative case
[A(p0, ~p), A(q0, ~q)] = 0, [A†(p0, ~p), A†(q0, ~q)] = 0,
[A(p0, ~p), A†(q0, ~q)] = δ3(~p− ~q)
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Motivations/Introduction
Non-commutative space and twisted symmetry
k-spacetime and k-Poincare algebra
Realisation of kappa spacetime and its Symmetry Algebra
Conclusion
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Conclusion
We have obtained the twisted co-product for thesymmetry algebra of kappa-space time.
Using the casimirs, we have shown that more than oneinvariant action for scalar field is possible ( having correctcommutative limit).
Flip operator compatible with the twisted co-product isderived.
Twisted commutators between creation and annihilationoperators are obtained.In collaboration with: S. Meljanac, D. Meljanac, K. S. Gupta,T. R.
Govindarajan
Phys.Rev.D77:105010,2008
![Page 101: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/101.jpg)
Conclusion
We have obtained the twisted co-product for thesymmetry algebra of kappa-space time.
Using the casimirs, we have shown that more than oneinvariant action for scalar field is possible ( having correctcommutative limit).
Flip operator compatible with the twisted co-product isderived.
Twisted commutators between creation and annihilationoperators are obtained.In collaboration with: S. Meljanac, D. Meljanac, K. S. Gupta,T. R.
Govindarajan
Phys.Rev.D77:105010,2008
![Page 102: Kappa-deformed space-time: Field Theory and Twisted Symmetry › ~trg › Hari.pdf · Modify STR Space-time structure is governed not only by a fundamental velocity scale c, but also](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d10a17e708231d438813a/html5/thumbnails/102.jpg)
Conclusion
We have obtained the twisted co-product for thesymmetry algebra of kappa-space time.
Using the casimirs, we have shown that more than oneinvariant action for scalar field is possible ( having correctcommutative limit).
Flip operator compatible with the twisted co-product isderived.
Twisted commutators between creation and annihilationoperators are obtained.In collaboration with: S. Meljanac, D. Meljanac, K. S. Gupta,T. R.
Govindarajan
Phys.Rev.D77:105010,2008