Some features of the k-Minkoski space-time: coproduct structures

Some features of the κ-Minkowski space-time:
coproduct structures
Università degli Studi di Milano
Dipartimento di Matematica
Programme: Coalgebra e applicazioni in fisica e matematica
O.M. Lecian
Kursk State University, Kursk, Russia.
23 Feb 2023
O.M. Lecian Kursk State University, Kursk, Russia. Some features of the κ-Minkowski space-time: coproduct struc

Abstract
The features of the κ-Minkowski space-time are analytically described as
a generalization of the Minkowski(-flat) space-time within the formalism
of General Relativity by means of the algebraic structures formulations.
The quantum space-time is described after a non-commutative algebra
with space-time uncertainty relations.
The distance measurements are achieved within the frameworks of both
Quantum Mechanics and General Relativity, where measurability bounds
compatible with structures in the κ-Poincaré Group are found after
modified uncertainty relations.
Coordinate-independent structures are obtained for non-commutative
space-times. The Lie-algebra non-commutative space-times in the case of
the κ-Minkowski non-commutative space-times are presented. The
κ-deformed Poincaré quantum algebra is demonstrated to follow after
semi-direct coalgebra, where the κ-Poincaré algebra acts covariantly on
the κ-Minkowski space-time.

In the case of a commutative momentum space, anti-Hermitian
generators are found.
The bi-cross product is studied to be suited for a reformulation of the
classical mechanics and of the quantum one.
The κ-Minkowski space-time is further analysed within the framework of
a deformed Dirac algebra of constraints in General Relativity.
The κ-Minkowski space-time is explained to be compatibility achieved
also after particular implementations of the modified Loop-deformed
Poincaré algebra. The interpretation of the κ-Minkowski space-time as a
generalization of the Minkowski space-time within the framework of
General Relativity is corroborated after the new calculation of non-trivial
Killing vectors in 1 + 1 space-time dimensions.

Summary
• Classical gravity and space quantization:
- limitation of the precision of localization in space-time;
- mathematical framework;
- space-time uncertainty relations ⇒ requests on the commutation
relations, Poincaré covariance, vanishing of commutators in the
large-scale limit;
- deviations manifested at Planck scale, Minkowski space recovered as
large-scale structure;
⇒ non-commutative algebra with space-time uncertainty relations.
• κ-Minkowski space:
- classical mechanics and quantum mechanics;
- algebra sector,
- coalgebra sector, and
- Casimir operators.

• κ-deformed Poincaré quantum algebra
- acts covariantly on a κ-Minkowski space;
- κ-Poincaré algebra, and
- coproducts;
- measurability bounds found after modified uncertainty relations.
• Canonical non-commutative space-times:
- κ-Minkowski non-commutative space-time:
-κ-deformed Poincaré quantum algebra ⇒ the coalgebra is semi-direct.
• Bicross-products.
• κ-deformed mass shell.
• κ-Minkowski space-time recovered from:
- Deformed general relativity (analysis in 1 + 1 dimensions of the
κ-deformed Heisenberg algebra and of the κ-deformed Poincaré algebra;
non-trivial Killing vector);
- deformation of the local Poincaré algebra:
⇒ first Casimir operator; and
- Geroch-compatible deformed hypersurface-deformation algebra
three-surfaces.

• Perspectives:
- the new roles of the series of the exponential functions in the
tensor products, and
- the new roles of the series of the exponential-related functions in
the tensor products.

Introduction
The modification of the Minkowski space-time at the Planck-scale level
can be achieved after the modification of the Poincaré group.
The non-trivially modified κ-Poincaré group is found to act on the
modified κ-Minkowski space.
The Dirac formulation of the theory of General Relativity can be
formulated as a constrained theory of convolution products of operators
as a surface-deformation algebra. The proper conditions have to be taken
onto account to reconduct the model to General-Relativity scheme
endowed with new structures.
A physical Minkowski limit is recovered, endowed with new non-trivial
structures.

The Geroch Theorem
Time foliation
In a globally-hyperbolic space-time
the time direction is irrotational.

The Thin-Sandwich Problem
Thin-Sandwich Conjecture:
- initial data on a Riemannian metric g of a Riemannian manifold M,
- tangent vector of (M, g);
solve the constraints after the data
and assure of existence and uniqueness of geodesics on a spacelike
three-surface of solutions.
Failure of the Thin-Sandwich Conjecture on flat space-time:
- after the behaviour of the linearised theory of quantum gravitation.
Relevance in:
- directional derivatives,
- Lie derivatives, and
- Killing vectors.
C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, W. H. Freeman and Company, San Francisco, USA (1973);
D. Christodoulou, M. Francaviglia, Remarks about the thin sandwich conjecture, Rep. on Math. Phys. 11, 377
(1977).

Canonical formulation of General Relativity
Covariant derivatives on the space slices are defined as
Dj Vk = Vk , j − Γ
l
jk Vl =j Vk −
1
2
h
li
hji,k + hik,j − hjk,i

Vl
extrinsic curvature
Kjk = −
1
2N
(t hjk − Dj Nk − Dk Nj ),
LEH =
c3
16πG
Z
d
3
x N
√
h (
3
R + Kjk K
jk
− (Kjk h
jk
)
2
).
Conjugated kinetic canonical momenta
π =
δL
δṄ
= 0, π
j
=
δL
δṄj
= 0, π
jk
=
δL
δḣjk
= −
c3
√
h
16πG

K
jk
− Kh
jk

.
Constraints Hamiltonian
H =
Z
d
3
x

πṄ + π
k
Ṅk + π
jk
ḣjk

− L
the lapse function and the shift vector act as Lagrange multipliers:
the following constraints are found
Cgrav =
16πG
c3
√
h

πjk π
jk
−
1
2
(π
l
l )
2

−
c3
√
h
16πG
!
3
R = 0, C
k
grav = −2Dj π
jk
= 0.

Canonical quantization of General Relativity
In the Schroedinger representation, the following operators are defined
ĥij ψ(hmn) = hij ψ(hmn), π̂
ij
ψ(hmn) = −i
δ
δhij
ψ(hmn).
which act on a Hilbert space of functionals of the 3-metric.
The theory is complete, as the four constraints are found
δĤ
δN
ψ(hmn) = 0
δĤ
δNk
ψ(hmn) = 0,
for instance
−
8π
√
h
hik hjl + hjk hil − hij hkl
δ2
ψ(hmn)
δhij (x)δhkl (x)
−
√
h
16π
!
3
R(x)ψ(hmn) = 0.
The solutions of the constraints in not straightforward.
B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
B. S. DeWitt, Phys. Rev. 162, 1195 (1967) .
B. S. DeWitt, Phys. Rev. 162, 1239 (1967) .

Classical gravity and space-time quantization
Limitations on the precision of localization in space-time:
- approaches to gravity
D. Amati, M. Cialfaloni and G. Veneziano, Nucl. Phys. B 347 (1990) 551;
D. Amati, On Spacetime at Small Distances, in: Sakharov Memorial Lectures, eds.
L.V. Kaldysh and V.Ya. Fainberg (Nova Science Publishers Inc., 1992).
J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, A Liouville String 7000/93.
A. Ashtekar, Quantum Gravity: a Mathematical Physics Perspective, Preprint
CGPG-93 / 12-2
. - mathematical framework
A. Kempf, Uncertainty Relations in Quantum Mechanics with Quantum Group
Symmetry, Preprint DAMT/93-65.

Quantum nature manifesting itself in commutation relations
postulated on the basis of:
- space-time uncertainty relations,
- Poincaré covariance,
- commutators should vanish in the large-scale limit.
Expected theory as
- deviations should be manifest at the Planck scale only, while
- the large-scale structure of quantum space-time should be the
same as for the usual Minkowski space

Space-time uncertainty relations suggested after
- the Heisenberg uncertainty principle, and
- the Einstein classical theory of gravity.
⇒ commutation relations:
- between self-adjoint coordinate operators qµ, µ = 0, ..., 3,
- acting on a Hilbert space H
- Poincaré covariant

Quantum space-time described by a non-commutative algebra with
space-time uncertainty relations, and
Einstein’s theory of classical gravity
applications in QFT
applications in QST

[qµ, qν] = iQµν
Qµν anti-symmetric tensor
fundamental invariants constructed after Qµν
• QµνQµν, and
• Qµν(∗Qµν = 1
2ϵµνλρQµνQλρ
should not be both vanishing
further condition
• [qµ, Qλρ] = 0

choose the quantum conditions
• QµνQµν
= 0
• [1
4 QµνQµν
(∗
Q)µν
]2
= I
• [qµ, [qλ, qν]] = 0
i.e. symmetry of the space-time under
- time reversal, and
- space reflections
- specific interactions could break the discrete symmetry, but
- the basic geometry should remain unchanged
Qµν assumed
• Hermitian, and
• self-adjoint
the centre of the algebra implies that the spectral resolution of qµ
and Qλρ commute

The Heisenberg Relations in Quantum Mechanics might have singular
realizations:
• (q, p) unbounded, and
• the same is requested for the quantum conditions
eiαµqµ
eiβµqµ
= e
1
2
αµQµν βν
ei(αµ+βµ)qµ
with
αµ real 4-vector, and
eiαµqµ
unitary operator
Their representation corresponds to non-degenerate representations π of
a C∗
algebra C generated after the continuous functions F with F
vanishing at infinity
π(F) = g(Q)
R
f (α)eiαµqµ
d4
α
• f ∈ L1
(R4
)
commutative C∗
algebra replaces the C0(R4
) of the classical Minkowski
space-time
S. Doplicher, K. Fredenhagen, J. E. Roberts, Spacetime quantization induced by
classical gravity, Physics Letters B 331 (1994) 39-44.

κ-Minkowski space
The κ-Minkowski space is defined as
[xi , xj ] = 0
[xi , x0] = xi
κ
∆xµ = xµ ⊗ 1 + 1 ⊗ xµ

Classical Mechanics and Quantum Mechanics of free
κ-Relativistic Systems
- Hamiltonian formalism, and
- Lagrangian formalism:
free -relativistic particles
with four-momenta constrained to the κ-deformed mass shell:
modifications arising from
- from the introduction of space coordinates with non-vanishing Poisson
brackets, and
- from the new definition of the energy operator.
Quantum mechanics
- of free κ-relativistic particles, and
- of the free κ-relativistic oscillator (application.)
relation with a κ-deformed Schroedinger quantum mechanics in
which the time derivative is replaced by a finite-difference
derivative.

quantum deformations
of the D = 4 Poincaré Algebra
V. G. Drienfel’d, Proceedings of the XX International Math. Congress Berkeley, 1, 798
(1896);
L. D. Faddeedv, N. Yu. Reshetikin, L. A. Takhtajan, Algebra Anal. 178, 1 (1986);
S. L. Worowowicz, Comm. Math. Phys. 613, 111 (1987);
S. L. Worowowicz, Comm. Math. Phys. 125, 122 (1989).
to obtain the D = 4 Poincaré group
κ-deformation of the D = 4 Poincaré Algebra
modifications of the relativistic symmetries
limκ→∞: undeformed space
κ-deformation of the mass-shell condition
p2
0 − ⃗
p2 = m2 → 2κsinh p0
2κ
2
− ⃗
p2 = m2

quantum contractions
basis obtained after the q-deformed de-Sitter algebra
J. Lukierski, A. Ruegg, H. Nowicki, V. N. Tolstoy, q-deformation of Poincaré algebra,
Phys. Let. B 331, 264 (1991);
J. Lukierski, A Nowicki, H. Ruegg, New quantum Poincaré algebra and κ-deformed
field theory, Physics Letters B 344, 293 (1992).
κ real mass-like parameter

algebra sector
[Mi , Mj ] = iϵijk mk ,
[Ni , Mj] = iϵijk Nk ,
[Mi , Pj ] = iϵijk Pk ,
[Ni , Pj ] = iκδij sinhP0
k ,
[Ni , P0] = iPi ,
[Ni , Nj ] = iϵijk

Mk cos P0
κ − 1
4κ2 Pk (⃗
P ⃗
M)

coalgebra sector
∆(Mi ) = Mi ⊗ I + I ⊗ Mi
∆(Ni ) = N − i ⊗ eP0/2κ
+ e−P0/2κ
⊗ Ni +
+ 1
2κ ϵijk Pj ⊗ Mk eP0/2κ
+ e−P0/2κ
Mj ⊗ Pk

∆(P0) = P0 ⊗ I + I ⊗ P0
antipoles
S(Mi ) = −Mi
S(Pµ) = −Pµ
S(Ni ) = −Ni + 3i
2κ Pi

deformed bilinear mass square Casimir of the Poincaré Algebra
C1 → C1 = ⃗
p2 + 2κ2

1 − coshP0
κ

= ⃗
p2 −

2κsinhP0
2κ
2
κ Poincaré Algebra can be realised after vector fields on a
commuting 4-momentum space
J. Lukierski, H. Ruegg, W. J. Zarzenwski, Classical and Quantum Mechanics of Free κ
Relativistic systems, Annals of Phys 243, 90 (1995).

κ-deformed Poincaré quantum algebra
The κ-deformed Poincaré quantum algebra has the structure of a Hopf
algebra bicross product.
The algebra is a semidirect product of the classical Lorentz group so(1, 3)
acting in a deformed way on the momentum sector T.
The coalgebra is also semidirect
with a backreaction of the momentum sector on the Lorentz
rotations.
⇒ the κ-Poincaré acts covariantly on a κ-Minkowski space

commutative momentum space
[Pµ, Pν] = 0
κ-Poincaré Algebra Pk
[Pµ, Pν] = 0
[mi , Mj ] = ϵijk Mk
[Mi , Pi ] = ϵijk Pk
[Mi , P0] = 0
[Mi , N̄j ] = ϵijk N̄k
[N̄i , P0] = Pi
[N̄i , Pj ]=δij κsinhP0
κ
[N̄i , N̄j ] − ϵijk

Mk coshP0
κ − 1
4κ2 Pk
⃗
P ⃗
M

coproducts
∆P0 = P0 ⊗ I + I ⊗ P0
∆Mi = Mi ⊗ I + I ⊗ Mi
∆Pi = Pi ⊗ eP0/2κ
+ e−P0/2κ⊗Pi
∆N̄i = N̄i ⊗eP0/2κ+e−P0/2κ
⊗Mi
+ 1
2κ ϵijk Pj ⊗ Mk eP0/2κ
+ e−P0/2κ
Mj ⊗ Pk

S. Majid, H. Ruegg, Bicrossproduct structure of κ-Poincaré group and
non-commutative geometry, Phys.Lett. B 334, 348 (1994).

maeasurability bound after the κ-Poincaré group from κ-deformed
Minkowski space
[xj , xk ] = 0
[xj , t] =
xj
κ
interpreted as implying the uncertainties
δxj δt ≥
kj
|κ|
analysis of distance measurements
- quantum-mechanical features
- general-relativistic features
measurability bound compatible with structures on the Quantum
κ-Poincaré Group
for a quantum theory containing the gravitational interaction, the lower
measurability bound of a length L should coincide with the Planck length
LP

-can be obtained after the modified uncertainty relations
δxδP ≥ ℏ +
L2
P
ℏ δP2
G. Veneziano, Europhys. Lett. 2, 199 (1986);
D.J. Gross, P.F. Mende, Nucl. Phys. B 303, 407 (1988);
D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B 40, 216 (1989)1;
K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B 234, 276 (1990).
and its developments.
G. Amelino-Camelia, Enlarged Bound on the Measurability of Distances and Quantum
κ-Poincaré Group, Phys.Lett. B 392,283 (1997) 283.

Canonical non-commutative space-times
Canonical non-commutative space-times
[xµ, xν] = iθµ,ν
θµ,ν coordinate-independent
S. Doplicher, K. Fredenhagen and J.E. Roberts, Phys. Lett. B 331, 39 (1994).
’Lie-algebra form’ of non-commutativity
[xµ, xν] = iζσ
µ,νxσ
ζσ
µ,ν coordinate-independent
G. Amelino-Camelia, Phys. Lett. B 392, 283 (1997).

Action functional for Lie-algebra non-commutative
space-times in the case of the κ-Minkowski
non-commutative space-time
[xj , x0] = iλxj , [xj , xk] = 0
gives rise to
- a time-to-the right star product,
- a symmetrized star-product, and
- a symmetric star product.
A. Agostini, G. Amelino-Camelia, M. Arzano, Francesco D’Andrea, Action functional
for κ-Minkowski Noncommutative Spacetime arXiv:hep-th/0407227.

The k-deformed Poincaré quantum algebra
- semi-direct product of a classical Lorentz group so(1, 3) acting in
deformed way on the momentum sector;
⇒ the coalgebra is semidirect
- the k-Poincaré algebra acts covariantly on the k-Minkowski
space, and
- connected to approaches of the Planck-scale Physics.

commutative momentum space P
[Pµ, Pν]=0,
k-Poincaré algebra Pκ
- antihermitian generators of translations, rotations and
boosts
- allows for a physical interpretation of bicross products
[xi , x0] =
xi
κ
[xi , xj ] = 0
S. Majid and H. Ruegg, Phys. Lett. B 334, 348 (1994).

Bicross products
Bicross products:
reformulation of the classical mechanics and of the quantum
description within the framework of non-commutative geometry:
non-commutative geometry to reformulate the classical and
particle moving on a homogeneous space-time
symmetry between observables and states expressed as a Hopf
algebra
S. Majid. Hopf algebras for physics at the Planck scale, Classical and Quantum
Gravity 5, 1587 (1988).

Hamiltonian formalism and Lagrangian one to describe free
κ-relativistic particles with 4-momenta constrained to a
κ-deformed mass shell
- space coordinates with non-vanishing Poisson brackets, and
- new definitions of the energy operator:
⇒ κ-deformed Schroedinger quantum mechanics:
time derivative is replaced by a finite-difference derivative.
J. Lukierski, H. Ruegg, W. J. Zakrzewski, Ann. Phys. 243, 90 (1995).

Dirac quantization:
Hamiltonian formulation of General Relativity
Dirac quantization of the ADM Hamiltonian formulation of
General Relativity
Poisson brackets of the
- three-metric hij (x), and
- its conjugated momentum πij (x)
{hij (x), hkl (y)} = 0
n
πij
(x), πkl
(y)
o
= 0
n
hij (x), πkl
(y)
o
=
1
2
(δk
i δl
j + δk
j δl
i )δ3
(x − y)
where the arguments of each Poisson brackets must be intended to be
taken at the same time

⇒ upgrade them to commutators replacing the classical variables
with operators:
hij → ĥij ,
πij
→ π̂ij
.
P. A. M. Dirac. Generalized Hamiltonian dynamics, Can. J. Math. 2, 129 (1950);
P. A. M. Dirac, The Theory of gravitation in Hamiltonian form, Proc. Roy. Soc. Lond.
A 246, 333 (1958);
P. A. M. Dirac, Lectures On Quantum Mechanics, Belfer Graduate School Of Science,
Yeshiva University Press (1964).

Classical Hamiltonian
classical Hamiltonian can be rewritten as
H =
Z
d3
xNCgrav
+ Nk Ck
grav =
Z
d3
xN
δH
δN
+ Nk
δH
δNk
where
H[N] =
Z
d3
x N(x)Cgrav (x) =
Z
d3
xN
16π
√
h

πjk πjk
−
1
2
(πl
l )2

−
√
h
16π
!
3
R
!
,
D[Nk
] =
Z
d3
x Nk
(x)Cgrav
k (x) = −2
Z
d3
xNk
hki Dl πli
.
F. Cainfrani, OML, M. Lulli, G. Montani, Canonical Quantum Gravity: Fundamentals
And Recent Developments, World Scientific Publishing, Singapore (2014).

General Relativity:
Dirac algebra of the constraints
n
D[Mã], D[N
e
b]
o
= D[LNb̃ Mã]

H[M], D[Nã] = H[LNe
b M]
{H[M], H[N]} = D[hãb̃(M∂b̃N − N∂ãM)]
with
- L Lie derivative,
- hab
the spatial metric,

Deformed General Relativity
Most general modification to the constraints which does not
modify the structure of the Hamiltonian constraint
{ D[Ma
], D[Nb
]} = D[LNb Ma
]
{ H[M], D[Na
]} = H[La
N]
{ H[M], H[N]} = D[βhab
(M∂bN − N∂bM)]
where β can in principle be a generic function; the most general choice
for a comparison with GR resulting as a model that does not violate the
Geroch foliation theorem is β depending at most on the extrinsic
curvature.
M. Bojowald, G. M. Paily, Phys. Rev. D (87) 044044.

k-deformed Heisenberg algebra and
k-deformed Poincaré algebra
in 1 + 1 dimensions
• ’Spherical dimensional reduction’
[Br , P0] = i cos(λPr )Pr , [Br , Pr ] = iP0, [P0, Pr ] = 0.
• Invent a Cartesian algebra because the spherical coordinates of
k-Minkowski are not treatable
[B1, P0] = i cos(λP1)P1, [B1, P1] = iP0, [P0, P1] = 0.
- compatibility k-Minkowski space-time checked after
h
X̂0, X̂j
i
= iλX̂j
h
X̂j , X̂k
i
= 0
con j, k = 1.
M. Bojowald and G. M. Paily, Phys. Rev. D 87, 044044 (2013) arXiv: 1212.4773.

• Choose the Heisenberg algebra
h
X̂0, X̂j
i
= iλX̂j [Pµ, Pν] = 0
[Pµ, X̂ν] = −iηµν + iλP0ηµν − iβ1λ2
P2
0 ηµν + i2β1λ2
PµPν + iβ1λ2
P2
1 ηµν,
µ, ν = 0, 1 e j, k = 1.
The ansatz on
[Pµ, X̂ν]
has already been reduced after the Jacobi identity of the
coordinates and that of the momenta.

• Investigation based on commutators
[B1, P0] = i cos(λP1)P1, [B1, P1] = iP0.
- at second order in λ the following hypothesis is made
[B1, P0] = i(1 −
1
2
λ2
P2
1 )P1
• constraint
[B1, P1] = iP0.
to be verified.

• The commutator
[B1, P1] = iP0
is considered as unknown;
- Ansaetze of the actions of the boost on P1, X̂1, X̂0
[B1, P1] = iP0 + iγ1λP2
0 + iγ2λP2
1 + iγ3λP1P0 + iρ1λ2
P3
0 + iρ2λ2
P3
1
+iρ3λ2
P1P2
0 + iρ4λ2
P2
1 P0
[B1, X̂1] = iX̂0 + iϕ1λX̂1P1 + iϕ2λX̂0P0 + iϕ3λX̂0P1 + iϕ4λ2
X̂0P2
1
+iϕ5λ2
X̂0P2
0 + iϕ6λ2
X̂1P0P1 + iϕ7λ2
X̂1P2
1 + iϕ8λ2
X̂0P1P0 + iϕ9λ2
X̂1P2
1
[B1, X̂0] = +iX̂1 + iψ1λX̂0P1 + iψ2λX̂1P0 + iψ3λX̂1P1 + iλ2
ψ4X̂0P0P1 + iψ5λ2
X̂1P0P1
+iψ6λ2
X̂1P2
0 + iψ7λ2
X̂1P2
1 .

• Jacobi identities of boosts, coordinates and momenta:
- the following algebra is obtained
[B1, P0] = i(1 −
1
2
λ2
P2
1 )P1,
[B1, P1] = iP0 + iγ1λP2
0 − i
1
2
λP2
1 + iρ1λ2
P3
0 + iρ2λ2
P3
1 + iρ4λ2
P2
1 P0,
[B1, X̂1] = iX̂0 − i
3
2
λ2
X̂0P2
1 − i(2ρ4 + 1)λ2
X̂1P0P1 − i3ρ2λ2
X̂1P2
1 ,
[B1, X̂0] = +iX̂1 − iλX̂0P1 + i2γ1λX̂1P0 − iλ2
X̂0P0P1
+i3ρ1λ2
X̂1P2
0 + iρ4λ2
X̂1P2
1 .
⇒ incompatibility among the assumptions of the k-Minkowski space-time
[B1, P0] = i cos(λP1)P1 and [B1, P1] = iP0.
G. Amelino-Camelia, M. M. da Silva, M. Ronco, L. Cesarini, OML, Spacetime-noncommutativity regime of Loop
Quantum Gravity, Phys. Rev. D 95, 024028 (2017).

Further proposals
• k-Minkowski most studied, but
other models can be analysed;
⇒ analyse the compatibility wit other non-commutative space-times
[X̂n, X̂ν] = iλX̂ν, [X̂µ, X̂ν] = 0, µ, ν = 0, 1, ..., n − 1, n + 1, ..., D
(in D space-time dimensions) which includes the k-Minkowski space-time
as a special case.
M. Bojowald and G. M. Paily, Deformed General Relativity, Phys. Rev. D 87, 044044 (2013) arXiv:1212.4773;
D. Kovacevic, S. Meljanac, A. Pachol and R. Strajn, Generalized Poincaré algebras, Hopf algebras and
kappa-Minkowski spacetime, Phys. Lett. B 711, 122 (2012) arXiv:1202.3305;
J. Mielczarek, Loop-deformed Poincaré algebra, arXiv:1304.2208.

It is crucial to remark that the implementation of the 1 + 1
dimensional case, within the Geroch ADM foliation, admits
non-vanishing killing vectors.
L. Cesarini, Deformed Relativistic Transformations for Loop Quantum Gravity, Laurea Thesis, Sapienza, Rome,
Italy (2014).
The implementation of the composition of Hamiltonian constraints
is therefore in order.

κ-Minkowski space-time from Deformed General Relativity
It is possible, within the analysis of the phase space and of the definition
of the Poisson brackets, to outline the pertinent degree of freedom within
the (local) modifications of the ADM foliation. For this, the modifications
of the ADM foliation imply modifications of the Poisson brackets.
Such modifications are local, i.e. they depend on the foliation chosen for
the measure, and vanish at (space) infinity in the case of asymptotic
flatness.

Deformation of the local Poincaré algebra
Deformation of the local Poincaré algebra as a consequence of
quantum modification of effective off-shell hypersurface
deformation algebra:
- particular realization of the Loop-deformed Poincaré algebra, and
- can be related with curved momentum space,
- which indicates the relationship with relative locality;
G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Phys. Rev. D 84 , (2011) 84010;
G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Gen. Rel. Grav. 43, 2547 (2011);
G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Int. J. Mod. Phys. D 20, 2867 (2011).
- useful to study Loopy effects.

gµν = diag(−β¶0, 1, 1, 1)
Classical generators of rotations
deformed local Poincaré algebra
[Ji , Jj ] = iϵijkJk
[Ji , Kj ] = iϵijkKk
[Ji , Pj ] = iϵijkKk
[Ki , Pj ] = iϵijkP0
[Ji , P0] = 0
[Pi , P0] = 0
[Pi , P0] = 0
[Ki , K0] = −iβ(P0)ϵijkJk
[Ki , P0] = iβ(P0)
first Casimir operator of the deformed Poincaré algebra
C1 = −
R P0
0
2y
β(P0) dy + P2
i

Implications of the deformed Poincaré algebra
The possible modifications of General Relativity due to the
quantum features of the gravitational interaction, i.e. those due to
the quantum modifications of the space-time geometry, those due to the
quantum interaction between matter and space-time, as well as those due
to semiclassical effects should be matched with the possible
modifications to the Minkowskian limit of quantum field theory on
curved space-time.

After expressing the length element according to the perspective of
the metric depends on the generators of translations Pρ, i.e. on
both the generators of time translations P0 and space translations
Pi :
gµν ≡ gµν(P0; Pi ).
Differently from
M. Bojowald, G. M. Paily, Phys. Rev. D (87) 044044.
the simplified assumption
that the parameter β depends on the generator of time translations
only allows one to write the following deformed Heisenberg algebra
[Xµ, Xν] = 0; [Pµ, Pν] = 0; [Xµ, Pν] = igµν(β)

A metric depending on the generators of time translation only is
obtained from M. Bojowald, G. M. Paily, Phys. Rev. D 87, 044044 (2013).
the simplified assumption
as the proper limit of the metric in the case of the generators of
space translation tending to zero, i.e.
lim
Pi →0
gµν(Pρ) = gµν(P0)
The mechanisms generated by non-geometrical non-Einsteinian
contributions to the EFE’s are modified but not discarded. This is
analyzed within the enumeration of the corresponding degrees of
freedom, which holds also in the cosmological solution.

It is consistent to express the commutator between the generators Xµ
and Pµ of the deformed Heisenberg algebra
[Xi P0 − βX0Pi ] = (Xµ, Pν).
by the Poisson brackets:
⇒ symplectic structure for the manifold M,whose algebraic and
geometrical properties are expressed by the physical relevance of the
deformation parameter β.
The mechanisms generated by non-geometrical
non-Einsteinian contributions to the EFE’s are modified but
not discarded. This is analyzed within the enumerations of the
corresponding degrees of freedom, which holds also in the
cosmological solution.
OML, in preparation.

Composition of constraints within the Geroch theorem
Composition of Hamiltonian constraints
The most relevant investigation cases, i.e. the results for which the most
focused insight is achieved, are those for which the arbitrary function β is
a function of the (extrinsic) curvature, i.e.
{ H[M]H[N], H[P]} ∼ βH[Q].
with M ∈Cn
, N ∈Cn
, Q ∈Cn−1
.
The diffeomorphisms constraint results as
{ D[βC], D[S]} = D[LSb Ca
] + D[LΩb Ca
]
where Ω ≡ βS. For the most rigorous implementation of the
diffeomorphism constraint, the results of factor ordering in the
functional (faltung) integral has been take into account; the
hypotheses on the vanishing of the boundary terms are the strictest
possible, as β can be considered a function of the extrinsic curvature also.

Indeed, the Geroch-deformed diffeomorphisms constraint reduces to
the usual Diffeomorphism constraint, plus a modification term
D[LΩb Ca
], where the directional Lie derivative is performed in the b
direction of the newly-defined surface Ω; such surface is a deformation of
the surface S, which was supposed as undeformed.
The deformation term for the D operator defined as acting o the
manifold C acts as deforming the manifold S by the same
deformation.
Here, C ∈Cn
, βC ∈Cn
, S ∈Cn
, βS ∈Cn
.

Diffeomorphism constraint:
infinitesimal case
In the infinitesimal case, the diffeomorphisms constraint results as
{ D[βC(M)], D[βS(N)]} = D[LSb Ca
] + D[LΩb Ca
]
for which the expansion is consistent with the Geroch theorem only for
the case, for which the expansion
β = 1 + ϵ
holds.
In this case, the following expansion holds:
{ D[βC(M)], D[βS(N)]} = D[LN M] + D[LN M] + D[LN M] + D[LN M]
with M = βM.

The first-order correction in β is
D[LNM] + D[LN M];
the second-order correction in β is D[LN M].
After comparison with the constraint, it is important to note that
the two derivatives LAB and LA(B′), for some A′, (B)′(where the
index ′
is not intended as a derivation symbol),with C ∈Cn, S ∈Cn, are
the same order in β.

Diffeomorphism constraint: deformations of both manifolds
In the case both manifolds are deformed, the following equality holds:
{ D[βC(M)], D[βS(N)]} =
{ D[C], D[S]} + { D[ϵC], D[S]} + { D[C], D[ϵS]} + { D[ϵC], D[ϵC]}
for which the deformations are identified as
{ D[ϵC], D[S]} + { D[C], D[ϵS]} ≡ D[LN M] + D[LN M]
and the second order terms
{ D[ϵC], D[ϵC]} ≡ D[LN M]
Composition of constraints
The composition of the constraints
{ H[M], H[Q]D[Na
]} = D[V ] + H[Q]H[P] + D[LN R] + D[LN R]
holds from the analysis of the composition of diffeomorphism constraints.
Here, R ≡ D[h(M∇N − N∇M)]; if M ∈Cn
, N ∈Cn
, then R ∈Cn−1
,
V ∈Cn−1
.

The directional Lie derivatives with respect to one of the directions of a
modified manifold βM, i.e. LM coincides with the ordinary Lie derivative
along the directions individuated by M itself, i.e.
lim
⃗
x→∞
| LMN − LM N |= 0 (3)
with
D[LN R] ≡ −βD[U] − UD[β] − ϵD[T]D[Na
] − TD[ϵ]D[Na
] (4)

The undeformed time evolution of an undeformed three-surface.

The deformed time evolution of an undeformed three-surface.

The deformed time evolution of a deformed three-surface.
OML, in preparation.

Outlook and perspectives
Within the framework of the functional analysis
- second order in l
1 + 1 space-time dimensions
[x0, x1] = ilx1
2-D κ-Poincaré Hopf algebra;
S. Majid, H. Ruegg, Phys. Lett. B 334, 348 (1994), arXiv:hep-th/9405107.

- second order in l
1 + 2 space-time dimensions
[x2, x1] = ix1
⇒ deformation of the Euclidean algebra:
[P2, P1] = 0
[R1, P2] = −iP1
[R, P1] = i
2l 1 − e−2lP2

+ i l
2P2
1
with R rotation generator of the 2 − D κ-Minkowski space-time
⇒ deformation of the P2
1 + P2
2 Casimir of the euclidean algebra
C(P2
1 + P2
2 ) = 4
l2 sinh2

lP2
2

+ elP2 P2
1
G. Amelino-Camelia, Entropy 19, 400 (2017) arXiv:1407.7891.

Hamiltonian formulation:
- Noether symmetries, and
- compatible with deSitter space-times.
G. Amelino-Camelia, G. Fabiano, D. Frattulillo, Total momentum and other Noether charges for particles
interacting in a quantum spacetime, arXiv:2302.08569.
→ consider the l-expansion at further orders, and
OML, G. Gubbiotti et al., in preparation.
→ consider the sum of exponential-related functions.
OML, G. Gubbiotti et al., in preparation.

Acknowledgments
The Programme Education in Russia for Foreign Nationals of the
Ministry of Science and of Higher Education of the Russian
Federation is thanked.

Thank You for Your attention.

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