Quantum gravitational corrections to particle creation by black holesSérgio Sacani
We calculate quantum gravitational corrections to the amplitude for the emission of a Hawking particle
by a black hole. We show explicitly how the amplitudes depend on quantum corrections to the exterior
metric (quantum hair). This reveals the mechanism by which information escapes the black hole. The
quantum state of the black hole is reflected in the quantum state of the exterior metric, which in turn
influences the emission of Hawking quanta.
Lecture by prof. dr Neven Bilic from the Ruđer Bošković Institute (Zagreb, Croatia) at the Faculty of Science and Mathematics (Niš, Serbia) on October 29, 2014.
The visit took place in the frame of the ICTP – SEENET-MTP project PRJ-09 “Cosmology and Strings”.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus
modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy
eigen value and its associated total wave function . This potential with some suitable conditions reduces to
two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical
results for energy eigen value with different values of q as dimensionless parameter. The result shows that
the values of the energies for different quantum number(n) is negative(bound state condition) and increases
with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1)
shows the different energy levels for a particular quantum number.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy eigen value and its associated total wave function . This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter. The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1) shows the different energy levels for a particular quantum number.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
Marija Dimitrijević Ćirić "Matter Fields in SO(2,3)⋆ Model of Noncommutative Gravity"
1. BW2018 workshop ”Field Theory and the Early Universe”
June 10-14 , 2018, Niˇs, Serbia
Matter fields in SO(2, 3) model
of noncommutative gravity
Marija Dimitrijevi´c ´Ciri´c
University of Belgrade, Faculty of Physics,
Belgrade, Serbia
with: D. Goˇcanin, N. Konjik, V. Radovanovi´c;
arXiv:1612.00768, 1708.07437, 1804.00608
2. NC geometry and gravity
Early Universe =⇒ Quantum gravity =⇒ Quantum space-time
Noncommutative (NC) space-time =⇒ Gravity on NC spaces.
General Relativity is based on the diffeomorphism symmetry. This
concept (space-time symmetry) is difficult to generalize to NC
spaces. Different approaches:
NC spectral geometry [Chamseddine, Connes, Marcolli ’07; Chamseddine,
Connes, Mukhanov ’14].
Emergent gravity [Steinacker ’10, ’16].
Frame formalism, operator description [Buri´c, Madore ’14; Fritz,
Majid ’16].
Twist approach [Wess et al. ’05, ’06; Ohl, Schenckel ’09; Castellani, Aschieri
’09; Aschieri, Schenkel ’14].
NC gravity as a gauge theory of Lorentz/Poincar´e group
[Chamseddine ’01,’04, Cardela, Zanon ’03, Aschieri, Castellani ’09,’12; Dobrski ’16].
3. Overview
Review of SO(2, 3) NC gravity
General
Action
NC Minkowski space-time
Adding matter fields
Spinors
U(1) gauge field
NC Landau problem
Discussion
4. SO(2, 3) NC gravity: General
SO(2, 3) NC gravity is based on:
-NC space-time → Moyal-Weyl deformation with small, constant
NC parameter θαβ = −θβα; [ˆxµ, ˆxν] = iθµν.
-gravity → SO(2, 3) gauge theory with symmetry broken to
SO(1, 3), [Stelle, West ’80; Wilczek ’98].
- -product formalism: Moyal-Weyl -product.
-Seiberg-Witten (SW) map → relates NC fields to the
corresponding commutative fields.
Our goals:
-consistently construct NC gravity action, add matter fields,
calculate equations of motion and find NC gravity solutions;
investigate phenomenological consequences of the constructed
model.
-diffeomorphism symmetry broken by fixing constant θαβ, give
physical meaning to θαβ.
5. SO(2, 3) NC gravity: Action
SO(2, 3) gauge theory: gauge field ωµ and field strength tensor
Fµν of the SO(2, 3) gauge group:
ωµ =
1
2
ωAB
µ MAB =
1
4
ωab
σab +
1
2
ea
µγa , (1)
Fµν =
1
2
FAB
µν MAB = ∂µων − ∂νωµ − i[ωµ, ων]
=
1
4
Rab
µν −
1
l2
(ea
µeb
ν − eb
µea
ν ) σab +
1
2
Fa5
µνγa , (2)
with
[MAB , MCD] = i(ηADMBC + ηBC MAD − ηAC MBD − ηBDMAC ) , (3)
ηAB = (+, −, −, −, +), A, B = 0, . . . , 3, 5
MAB → (Mab, Ma5) = (
i
4
[γa, γb],
1
2
γa), a, b = 0, . . . 3
Rab
µν = ∂µωab
ν − ∂νωab
µ + ωa
µc ωcb
ν − ωb
µc ωca
ν ,
Fa5
µν =
1
l
µe a
ν − νe a
µ , µe a
ν = ∂µe a
ν + ωa
µbe b
ν .
6. The decomposition in (1, 2) very much resembles the definitions of
curvature and torsion in Einstein-Cartan gravity leading to General
Relativity. Indeed, after introducing proper action(s) and the
breaking of SO(2, 3) symmetry (gauge fixing) down to the
SO(1, 3) symmetry one obtains this result [Stelle, West ’80].
To fix the gauge: the field φ transforming in the adjoint
representation:
φ = φA
ΓA, δ φ = i[ , φ] ,
with ΓA = (iγaγ5, γ5) and γa and γ5 are the usual Dirac gamma
matrices in four dimensions.
7. Inspired by [Stelle, West ’80; Wilczek ’98] we define the NC gravity
action as
SNC = c1S1NC + c2S2NC + c3S3NC , (4)
with
S1NC =
il
64πGN
Tr d4
x µνρσ ˆFµν
ˆFρσ
ˆφ ,
S2NC =
1
64πGN l
Tr d4
x µνρσ ˆφ ˆFµν
ˆDρ
ˆφ ˆDσ
ˆφ + c.c. ,
S3NC = −
i
128πGN l
Tr d4
x εµνρσ
Dµ
ˆφ Dν
ˆφ ˆDρ
ˆφ ˆDσ
ˆφ ˆφ .
The action is written in the 4-dimensional Minkowski space-time,
as an ordianry NC gauge theory. It is invariant under the NC
SO(2, 3) gauge symmetry and the SW map guarantees that after
the expansion it will be invariant under the commutative SO(2, 3)
gauge symmetry.
8. Using the SW map solutions for the fields ˆFµν and ˆφ and the
Moyal-Weyl -product, we expand the action (4) in the orders of
NC parameter θαβ. In the commutative limit θαβ → 0 and after
the gauge fixing: φa = 0, φ5 = l, these actions reduce to
S
(0)
1NC → −
1
16πGN
d4
x
l2
16
µνρσ
abcd R ab
µν R cd
ρσ + e(R −
6
l2
) ,
S
(0)
2NC → −
1
16πGN
d4
xe R −
12
l2
,
S
(0)
3NC → −
1
16πGN
d4
xe −
12
l2
,
with e a
µ = 1
l ωa5
µ , e = det ea
µ, R = R ab
µν e µ
a e ν
b . The constants
c1, c2 and c3 are arbitrary and can be determined from some
consistency conditions.
9. Comments:
-main adventage of SO(2, 3) approach: basic fields are not metric
and/or vielbeins but gauge fields of (A)dS group; consequences
also in the NC setting.
-after the symmetry breaking: spin connection ωµ and vielbeins eµ.
They are independent, 1st order formalsim.
-varying (4) with respect to ωµ and vielbeins eµ gives equations of
motion for these fields. The spin connection in not dynamical (the
equation of motion is algebraic, the zero-torsion condition) and can
be expressed in terms of vielbeins, 2nd order formalism.
-(4) written in the 2nd order formalism has three terms:
Gauss-Bonnet topological term, Einstein-Hilbert term and the
cosmological constant term.
-arbitrary constants c1, c2 and c3: EH term requires c1 + c2 = 1,
while the absence of the cosmological constant is provided with
c1 + 2c2 + 2c3 = 0. Applying both constraints leaves one free
parameter (can be used later in the NC generalization).
10. Calculations show that the first order correction S
(1)
NC = 0. Already
known result [Chamseddine ’01,’04, Cardela, Zanon ’03, Aschieri, Castellani ’09].
The first non-vanishing correction is of the second order in the NC
parameter; it is long and difficult to calculate. However, the second
order corrections can be anayzed sector by sector: high/low energy,
high/low/zero cosmological constant, zero/non-zero torsion.
In the low energy sector, i.e., keeping only terms of the zeroth, the
first and the second order in the derivatives of vierbeins (linear in
Rαβγδ, quadratic in Ta
αβ), we calculate NC induced corrections to
Minkowski space-time. This calculation cen be generalized to other
solutions of vacuum Einstein equations.
11. SO(2, 3) NC gravity: NC Minkowski space-time
A solution of the form:
g00 = 1 − R0m0nxm
xn
,
g0i = −
2
3
R0minxm
xn
,
gij = −δij −
1
3
Rimjnxm
xn
, (5)
where Rµνρσ ∼ θαβθγδ: the Reimann tensor for this solution. The
coordinates xµ we started with, are Fermi normal coordinates:
inertial coordinates of a local observer moving along a geodesic;
can be constructed in a small neighborhood along the geodesic
(cylinder), [Manasse, Misner’63; Chicone, Mashoon’06; Klein, Randles ’11].
The measurements performed by the local observer moving along
the geodesic are described in the Fermi normal coordinates. He/she
is the one that measures θαβ to be constant! In any other reference
frame, observers will measure θαβ different from constant.
Fixed NC background: gauge fixed diffeomorphism symmetry,
prefered reference frame given by Fermi normal coordinates.
12. Adding matter fields: spinors
Spinors are naturally coupled to gravity in the first order formalism.
NC generalization of an action for Dirac spinor coupled to gravity
in SO(2, 3) model:
Sψ =
i
12
d4
x µνρσ ¯ψ (Dµφ) (Dν φ) (Dρφ) (Dσψ)
− (Dσ
¯ψ) (Dµφ) (Dν φ) (Dρφ) ψ (6)
+
i
144
m
l
−
2
l2
d4
x µνρσ ¯ψ Dµφ Dν φ Dρφ Dσφ φ
− Dµφ Dν φ Dρφ φ Dσφ + Dµφ Dν φ φ Dρφ Dσφ ψ + h.c. ,
where Dσψ = ∂σψ − iω ψ is the SO(2, 3) covariant derivative in
the defining representation. Expanging the action (6) ( -product,
SW-map) gives a nontivial first order correction for a Dirac fermion
coupled to gravity.
13. Phenomenological consequences: in the flat space-time limit we
find a deformed propagator:
iSF (p) =
i
/p − m + i
+
i
/p − m + i
(iθαβ
Dαβ)
i
/p − m + i
+ . . . , (7)
Dαβ =
1
2l
σ σ
α pβpσ +
7
24l2
ε ρσ
αβ γργ5pσ −
m
4l2
+
1
6l3
σαβ . (8)
A spinor moving along the z-axis and with only θ12 = θ = 0 has a
deformed dispersion relation (analogue to the birefringence effect):
v1,2 =
∂E
∂p
=
p
Ep
1 ±
m2
12l2
−
m
3l3
θ
E2
p
+ O(θ2
) , (9)
with Ep = m2 + p2
z . These results are different from the usual
NC free fermion action/propagator in flat space-time:
Sψ = d4
x ¯ψ i /∂ψ − mψ = d4
x ¯ψ i /∂ψ − mψ .
14. Adding matter fields: U(1) gauge field
Metric tensor in SO(2, 3) gravity is an emergent quantity.
Therefeore, it is not possible to define the Hodge dual ∗H.
Yang-Mills action S ∼ F ∧ (∗HF) cannot be defined. A method
of auxiliary field f [Aschieri, Castellani ’12], with δε f = i[Λε , f ]. An
action for NC U(1) gauge field coupled to gravity in SO(2, 3)
model:
SA = −
1
16l
Tr d4
x µνρσ ˆf ˆFµν Dρ
ˆφ Dσ
ˆφ ˆφ
+
i
3!
ˆf ˆf Dµ
ˆφ Dν
ˆφ Dρ
ˆφ Dσ
ˆφ ˆφ + h.c. . (10)
After the expansion ( -product, SW-map) and on the equations of
motion fa5 = 0, fab = −eµ
a eν
b Fµν the zeroth order of the action
reduces to
SA = −
1
4
d4
x e gµρ
gνσ
FµνFρσ . (11)
describing U(1) gauge field minimally coupled to gravity.
15. NC Landau problem
Phenomenological consequences of our model and NC in general:
the NC Landau problem: an electron moving in the x-y plane in
the constant magnetic field B = Bez. Our model in the flat
space-time limit gives
i /∂ − m + /A + θαβ
Mαβ ψ = 0 , (12)
where θαβMαβ is given by
θαβ
Mαβ = θαβ
−
1
2l
σ σ
α DβDσ +
7i
24l2
ρσ
αβ γργ5Dσ −
m
4l2
+
1
6l3
σαβ
+
3i
4
Fαβ /D −
i
2
Fαµγµ
Dβ −
3m
4
−
1
4l
Fαβ . (13)
For simplicity: θ12 = θ = 0 and Aµ = (0, By, 0, 0). Assume
ψ =
ϕ(y)
χ(y)
e−iEt+ipx x+ipz z
. (14)
with ϕ, χ and E represented as powers series in θ.
16. Deformed energy levels, i.e., NC Landau levels are given by
En,s =E
(0)
n,s + E
(1)
n,s , (15)
E
(0)
n,s = p2
z + m2 + (2n + s + 1)B ,
E
(1)
n,s = −
θs
E
(0)
n,s
m2
12l2
−
m
3l3
1 +
B
(E
(0)
n,s + m)
(2n + s + 1) +
θB2
2E
(0)
n,s
(2n + s + 1) .
Here s = ±1 is the projection of electron spin. In the
nonrelativistic limit and with pz = 0, (15) reduces to
En,s = m − sθ
m
12l2
−
1
3l3
+
2n + s + 1
2m
Beff −
(2n + s + 1)2
8m3
B2
eff + O(θ2
) , (16)
Beff = (B + θB2
) .
Consistent with string theory interpretation of noncommutativity
as a Neveu-Schwarz B-field.
17. In addition, the induced magnetic dipole moment of an electon is
given by
µn,s = −
∂En,s
∂B
= −µB (2n + s + 1)(1 + θB) , (17)
where µB = e
2mc is the Bohr magneton.
Some numbers:
-θ =
2c2
Λ2
NC
and ΛNC ∼ 10TeV ,
-accuracy of magnetic moment measurements δµn,s ∼ 10−13,
-for observable effects in µn,s, B ∼ 1011T needed. This is the
magnetic field of some neutron stars (magnetars), in laboratory
B ∼ 103T.
18. Discussion
A consistend model of NC gravity coupled to matter fields.
Pure gravity:
-NC as a source of curvature and torsion.
-The breaking of diffeomorphism invariance is understood as
gauge fixing: a prefered refernce system is defined by the
Fermi normal coordinates and the NC parameter θµν is
constant in that particular reference system.
Coupling of matter fields: spinors and U(1) gauge field.
-deformed propagator and dispersion relations for free
fermions in the flat space-time limit.
-nonstandard NC Electrodynamics (QED), new terms
compared with the standard QED on the Moyal-Weyl NC
space-time: phenomenological consequences, renormalizability.
Study: corrections to GR solutions (cosmological,
Reissner-Nordstr¨om black hole,. . . ); fermions in curved
space-time (cosmological neutrinos); . . .