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`falsifiable noncommutative field theories of NOT everything’
or
`noncommutative Planck-scale theories of NOT everything’
29.9.2008
Giovanni AMELINO-CAMELIA
Univ. of Rome “LA SAPIENZA”
& INFN sez. ROMA1
“theory of everything”, theories of not everything, and the Planck scale
criteria for developing noncommutative field theories that
can be “Planck-scale theories of not everything”
some examples of successful applications of a Noether analysis to
the Hopf-algebra spacetime symmetries of a noncommutative field theory
A possible “physicist intuition” for Hopf-algebra spacetime symmetries
last few decades of work in fundamental physics strongly affected by the
theory-of-everything utopia
theory-of-everything
Planck-scale theories of not everything
how did we get here?
The theory-of-everything utopia did have some
“reasonably good” supporters….
Einstein’s theory-of-everything utopia :
“I would like to state a theorem…: there are
no arbitrary constants ... that is to say, nature
is so constituted that it is possible logically
to lay down such strongly determined laws
that within these laws only rationally
completely determined constants occur”
but they were wrong
what did it cost us?
what could we do now?
“what did it cost us” and “what could we do now” are partly
linked ….let us start by looking back at the Planck scale:
we have robust (though mostly “conceptual”) evidence that something new must
be there at the Planck scale….some authors argue that the “quantum-gravity
scale” could be a lower scale, but nobody will argue against claim that something
new must be there by the time we get to the Planck scale….
so we better sharpen our tools for the Planck scale!!
Also because there may well be nothing (nothing quantumgravitie) between here
and the Planck scale:
ELEP;MW
EGUT EP
falsifiable Planck-scale theories of NOT everything
quantum gravity in modern times described in the language of electroweak
interactions in the Fermi era:
we have been looking for the full description of electroweak interactions, without
access to the electroweak scale and without even data on beta decay
result: we don’t even have a Fermi theory
Quantum Gravity likely to be a theory with much richer structure than, say, the
electroweak theory…so actually we might need guidance from many steppingstone theories, analogous to Fermi’s theory of weak interaction and Planck’s
description of black-body radiation (i.e. ugly, inconsistent, illogical, “theories”
that somehow work!):
*the Minkowski limit
*the DeSitter limit
*the Schwartzschild limit
* the “Newton-Planck limit” (c “going to” ; Lp,h finite)
*….. ……. ……… …….
for most of these possibilities no work at all….toooo busy doing the “theory of
everything”….
Another “victim”? String Theory!! probably could be the best Planck-scale
theory of not everything on the market…
c
but “walking the plank” is scary:
guessing about the Planck-scale realm (10-35 m) on the
basis of data from “LIGO/VIRGO+LEP/LHC”
(10-18, 10-19 m) should be roughly as difficult as guessing the
laws of physics at scales of the order of the Bohr radius if someone could only
build intuition by observing the motion of the Moon around the Earth
and notice that even when we were just a couple of orders of magnitude away
from the electroweak scale, our best description of weak interactions
was given by the simple-minded (logically/mathematically inconsistent, but very
valuable) Fermi effective theory
Depressing, isn’t it? Some hope: Einstein’s studies of Brownian motion, which
provided evidence for atoms, give a good illustrative example of how we manage
sometimes to gain insight on distance scales much shorter than the distance scales
at which we conduct the experiments
In which sense we have recently “proven sensitivity to
effects introduced genuinely at the Planck scale”?
 m  4
imagine spacetime is discretized with lattice scale the Planck length, then
naturally you end up with something like
see, e.g., t’Hooft, CQG(1996)
then compute the threshold energy requirement for photopion production
p +γCMBR => p+π
with this modified dispersion relation. One finds a shift of the threshold of the type
which can induce an observably large
shift of the “GZK scale” (109Ep)for
the cosmic-ray spectrum
GAC+Ellis+Mavromatos+Nanopoulos+Sarkar,
Nature(1998)
Kifune, Astr.Journ.Lett.(1999)
GAC+Piran, PhysRevD(2001)
GAC,Nature(2000)
kth,QG 
2m prot m

#
kth3 , 0 kth, 0
O(1)
correction important already when k4  Ep2 (mprotm )
E p2 
*following this line of analysis (and data recently gathered at the Pierre Auger
cosmic-ray observatory) we are now close to establishing as a scientific fact
that certain rigid Planck-scale discretizations of spacetime are not allowed
(this was “expected” but not “known”)
*so we do have, at least in some cases, a chance to probe effects introduced
genuinely at the Planck scale
* and these “threshold-anomaly analyses” provide examples of how wrong
the naive “Yang-Mills integrated-out-gauge-bosons intuition”
is about the magnitude of Planck-scale-induced effects can be:
“must require Planckian energies”
requires large boosts (even at relatively “low” energies)!!
*and there are several other examples of phenomenological analyses
leading to the same conclusions (GAC, arXiv:0806.0339)
what changes for the strategy?
*not uncomfortable when “failing to explain parameters”;
*not uncomfortable when not providing full solution to QGproblem;
*shoot for the Planck scale
An example of the implications of this strategy:
in the “Minkowski limit of quantum gravity” (clearly a theory of not everything)
one might have something like
One perspective/interpretation of this
formula (with θ endowed of nontrivial
algebraic properties) has been studied for
more than a decade….may well lead to a
“theory of everything”, but can also inspire
much work on theories of not everything
version1
Doplicher+Fredenhagen+Roberts,
CommMathPhys(1995)
PhysLettB(1994)
rediscovered more recently…hundreds of recent papers inspired by role of this
formula in description of strings in a
e.g.Douglas+Nekrasov, RevModPhys(2001)
Calmet+Jurco+Schupp+Wess+Wohlgenannt,
B-tensor background, adopting
version2
EurPhysJ.C(2002)
Bichl+Grimstrup+Grosse+Popp+Schweda+
Wulkenhaar, JHEP(2001)
Matusis+Susskind+Toumbas,JHEP(2000)
with θ as a Lorentz tensor (observer-dependent matrix)
- Lorentz-symmetry breakdown
Hewett+Petriello+Rizzo,PhysRevD(2001)
Carlson+Carone+ Lebed, PhysLett (2001)
- birefringence
GAC+Nam+Seo, PhysRevD(2003)
- IR/UV mixing
GAC+Mandanici+Yoshida, JHEP(2004)
BUT if θ is observer-dependent then θLp2 could only be implemented
for one class of observers….for the purest type of “noncommutative
Planck-scale microscope” one should have θLp2 for all observers:
version3
with observer-independent θ,
of order Lp2
(same matrix for all observers)
Chaichian+Kulish+Nishijima+Tureanu,PhysLettB(2004)
Fiore+Wess,ClassQuantGrav(2004)
Balachandran+Mangano+Pinzul+Vaidya,PhysRevD(2007)
STnoncommutativity unique (presently) opportunity to probe Planck scale in this
purest sense!!! (effects introduced at the Planck scale for all observers)
focus today on version 3
for description of symmetry transformations it is useful (though not necessary) to
introduce a “Weyl map” connecting noncommutative fields to auxiliary
commutative fields
(see, e.g., Madore+Schraml+Schupp+Wess, EPJC16,161)
then translations and Lorentz-sector generators can be described in terms of:
These leave commutators of coordinates invariant.
Chaichian+Kulish+Nishijima+Tureanu,PhysLettB(2004)
Fiore+Wess,ClassQuantGrav(2004)
Balachandran+Mangano+Pinzul+Vaidya,PhysRevD(2007)
It is a particular manifestation of the
nontrivial coproduct in the Lorentz sector
These P,M operators generate a Hopf (not a Lie) Algebra (nontrivial coproduct)
[Note that, as stressed most elegantly by Wess, the “Hopf direction” is to be viewed from
the same perspective of the “SUSY direction” (in relation to Coleman-Mandula).]
Many observations were suggesting that these operators would generate the
symmetries of the noncomm spacetime
Chaichian+Kulish+Nishijima+Tureanu,PhysLettB604,98
For example
Fiore+Wess,PhysRevD75,105022
Balachandran+Mangano+Pinzul+Vaidya,PhysRevD75,045009
(where □=PνPν) is formally invariant under the P,M transformations, as a result
of the nontrivial coproduct.
Is this really a (candidate) physical symmetry or just a mathematical curiosity?
If these are genuine symmetries they might provide us an explicit realization of
the idea of spacetime symmetries that are deformed
GAC, grqc0012051IJMPD11,35
but not broken at the Planck scale
GAC,hepth0012238PLB510,255
(in the “Minkowski limit” of QG)
GAC,Nature418,34(2002)
BUT nobody had managed to obtain conserved charges even for the simplest
actions. Some authors had explicitly argued that the conserved charges would not
exist and formal invariance under the Hopf transformations
would be a contentless mathematical oddity [e.g.Gonera+Kosinski+Giller,PhysLettB622,192]
reported failures of Noether analysis clearly shared a common feature:
the transformation was written in terms of
With commutative (numeric) transformation parameters.
Proposal: in a noncommutative spacetime it might make sense for some
transformations to be described by “noncommutative parameters”,
since in x →x’ not only x but also x’ should be element of a certain algebra
With these the Noether analysis works flawlessly!!
GAC+Briscese+Gubitosi+Marciano
+Martinetti+Mercati,
PhysRevD (2008)
….currents
and charges
GAC+Briscese+Gubitosi+Marciano
+Martinetti+Mercati,
PhysRevD (2008)
Not a peculiarity of the simplicity of non-commutativity:
the kappaPOINCARE/kappaMINKOWSKI Lukierski+Nowicki+Ruegg+Tolstoy,PLB(1991)
Majid+Ruegg,PLB (1994)
case
Lukierski+Ruegg+Zakrzewski, AnnPhys(1995)
here I denote the “Weyl map” by W
W (g f )  f

f ( x, t )   d 4 k  ( k )e ikx e ik0t

g f ( , )   d 4 k  (k )e ik ik0
and the (generalized) * product
W ( g f1 * g f 2 )  f1 f 2

Notice that we would have introduced a DIFFERENT Weyl map if, instead of

f ( x, t )   d k  ( k ) e e
4
ikx
ik0 t

we adopted one of these other possibilities

f ( x, t )   d k  ( 2 ) ( k ) e
4

f ( x, t )   d k  ( 3 ) ( k )e
4
ikx ik0 t
ik0 t / 2

ikx ik0 t / 2
e e

(not an ambiguity!!!)
Translation generators in kappa-Minkowski:

ik0t
ikx
P e e
then
  k e

ikx
ik0t
e

e e   P e
 k  e
K e e e e 
e
 P e e e e   e
ikx
P e e


ik0 t
iKx
 k 0
ikx
ik0 t
iK0 t


ikx
iKx
ik0 t
iK0 t
i ( k  e k0 K ) x
iKx

classical action
e i ( k0  K 0 ) t

Hausdorff
iK0 t
 P0
Baker
Campbell
ikx
 
e ik0t P e iKx e iK0t
Nontrivial coproduct!!
Translations are not classical in kappa-Minkowski

Rotation generators in kappa-Minkowski:



R j eikx eiEt  i jlm ( xl Pm  xm Pl ) eikx eiEt
then

 R e
ikx
Rj e e e
j
  R e
e e  e
iEt ix
ikx
e
iEt
e
i ( k  e E  ) x
it
j
ix
it
ikx
e
iEt

ei ( E   ) t
R e
j
ix
classical action

e
it

Trivial coproduct!!
Rotations are really classical in k-M
Boosts in kappa-Minkowski:

N j eikx eiEt
(1  e 2P0 )

 xj
 x0 Pj  x j Pl Pl  xl Pl Pj (eikxeiEt )
2
2
 

Modified action needed for consistency with Hopf algebra structure….
IF one adopted unmodified (classical) action then the would-be coproduct
requires operators external to the algebra…
Modification of boosts was expected since commutators involve a length scale…
With this modified action the coproduct is OK (can be expressed in terms of P,R,N)
Note that:
2 P0
(1  e

N j , Pl   iPl Pj  i jl  Pm Pm 
2
2

and the “mass Casimir” for these deformed transformations is
coshm  coshE  

2
2
E
e P
“inviting” a heuristic argument for a modified dispersion relation
2
)


But changing the Weyl map one could for example go from
the “classical action”

ik0t
  k e
ik0t / 2
  k e
ikx
P e e
ikx

ik0t
e

to this alternative

ik0t / 2 ikx
P' e
e e

ik0t / 2 ikx
ik0t / 2
e e
Different action of translation generators!!!!
Two different concepts of translation transformations??
….leads to different deformed Casimir
coshm  coshE ' 
2
2
e 2E ' P'2
Does this mean that nonlinear redefinitions change the physics?
does this mean that the dispersion relation is not unambiguously
determined in this theory?
Which form of disp rel should be tested?

Agostini+GAC+Arzano+Marciano’+Tacchi,
Noether analysis goes through easily (!!)
Mod.Phys.Lett.A22:1779-1786,2007
if we simply insist that Leibnitz rule
be valid for “d”
d ( fg )  (df ) g  f (dg)
For example in the time-to-the-right Weyl-map conventions this takes the form
df ( x)  i  P f ( x)
if the noncommutative transformation parameters have the following properties
[ 0 , x ]  0;[ j , xl ]  0;[ j , x0 ]  i j
For example from the action
one arrives at charges
whose t-independence (on solutions of the EoM) is easily verified
Properties of “noncommutative transf parameters”
linked to properties of generators….
using generators of our other choice of ordering convention gives same charges!!!
TWO PROBLEMS SOLVED AT ONCE!!!
Encouraging but incomplete (e.g. must consider other ordering prescriptions and
especially the peculiar kappaMinkowski possibility of 5parameter
translation transformations Freidel,Kowalski-Glikman,Nowak,hepth0706.3658 )
For θ-noncommutativity:
-same story but much more general proof of independence on the choice
of “basis” (choice of ordering prescription)
-no apparent alternatives to the differential calculus we adopted
- Casimir not modified (and apparently dispersion relation not modified)
Closing remarks
on the anthropic “principle”
“our Universe is what it is (e.g. it hosts humans, and other such things)”
many realizations (some rather significant from a philosophy
perspective), but evidently all with no science
-many universes? (?????)
-on its own it makes of course no predictions
- natural temptation to think that
“anthropic principle”+”principle of mediocrity”
does make predictions, but they are not predictions
“principle of mediocrity”: actual values of
parameters should be not very different
from the mean of the values suitable for life
(of course no reason why this should be
the case, but if it led to some predictions
then it could be either right or wrong)
Take for example one of the most famous “predictions” obtained using
“anthropic principle”+”principle of mediocrity”:
Weinberg’s “prediction” of the cosmological term
Now that we have data on the cosmological term :
the computed principle-of-mediocrity probability (under the assumption that
the a priori probability distribution is flat) to observe a cosmological
term of the size we do observe turns out to be roughly 10% ….
only way to judge if this 10% is encouraging or disappointing would be
to visit a “statistically significant number of universes” and gain insight on the
correct form of the statistical distribution….
here we go again….not even wrong….