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Scientific Report 2007-2009
Theoretical physics
T6. Theory and phenomenology of quantum-spacetime symmetries
The last century of physics has been primarily characterized
by a long list of successes of the “quantumtheory
paradigm”. In a significant part of the literature
on the search of a “quantum gravity”, a theory providing
a unified description of both quantum theory and general
relativity, researchers are looking for ways to apply this
quantum paradigm also to the description of spacetime.
This effort is faced by significant conceptual challenges,
and perhaps even more sizeable are the experimental
challenges, since it is expected that the “spacetime quantization”
should be characterized by a ultrasmall length
scale, roughly given by the Planck length ∼ 10 −35 m.
One of the most popular attempted formalizations of
spacetime quantization is “spacetime noncommutativity”,
a formalism that endows the spacetime coordinates
of particles with intrinsically nontrivial algebraic properties,
whose most studied examples introduce two modeldependent
“noncommutativity matrices” θ µν , ξ α µν:
[x µ , x ν ] = iθ µν + iξ α µνx α .
Amelino-Camelia was one the first advocates of an
approach to the study of noncommutative spacetimes
which is centered on symmetry analysis, searching for
both a suitable formalization and an associated phenomenology
programme. Of particular interest are cases
in which the symmetries of a noncommutative spacetime
require a Hopf-algebra description. The core feature of
this novel concept of a Hopf-algebra description of spacetime
symmetries resides in the way in which the generators
of the symmetries act on states of two of more particles,
states which are therefore formalized as elements of
a tensor product of multiple copies of the single-particle
Hilbert space. For some of the most compelling choices
of the noncommutativity matrices one finds an incompatibility
between the noncommutativity of spacetime
coordinates and the imposition of Leibniz law for the
action of the generators T α of spacetime symmetries on
elements of the relevant tensor products,
T α [Φ(x)Ψ(x)] ≠ T α [Ψ(x)]Φ(x) + Ψ(x)T α [Φ(x)] .
Our most significant recent theory result [4] provides a
generalization of the Noether theorem that is applicable
to the Hopf-algebra symmetries of some noncommutative
spacetimes. This had been a long-standing open issue for
physical applications of Hopf-algebra spacetime symmetries,
in which of course the conserved charges derived
in the Noether analysis should play a key role.
Some of our recent studies on the phenomenology side
have used in part this Noether-theorem result. In particular,
there is strong interest in the community in the possibility
to use observations of gamma-ray bursts, bursts
of high-energy photons emitted by sources at cosmological
distances, as an opportunity to gather indirect evidence
on the short-distance quantum structure of spacetime
and its symmetries. In most other contexts the new
effects are too small to be observed, but some gamma-ray
bursts have a rich structure of space/time/energy correlations
and the fact that they travel cosmological distances
allows for the minute quantum-spacetime/Hopfsymmetry
effects to have in some cases a nonnegligible
cumulative effect [1,3].
While for this gamma-ray-burst opportunity our recent
results contribute to an established phenomenology
programme, we also opened recently a completely new
direction for quantum-spacetime phenomenology. This
was inspired by theory results establishing that for some
choices of the noncommutativity matrices one finds the
novel effect of “infrared-ultraviolet mixing”. This new
scenario, which in just a few years was investigated in
several hundred publications, is such that the effects induced
by the short-distance quantum structure of spacetime,
besides the normally expected implications for the
ultraviolet sector of the theory, have implications which
are significant in a dual infrared regime. Our proposal
has been [2] to use the high accuracy of intereferometric
techniques applied on “cold” (ultraslow) atoms as a
way to look for signatures of these infrared manifestations
of spacetime quantization. Our main result concerns
measurements of the “recoil frequency” of atoms,
and is summarized by the formula [2]
∆ν ≃ 2hν2 ∗
m
(1 + λ m2
2hν ∗
)
, (1)
where ∆ν is the frequency difference of a pair of lasers
used to induce the recoil, hν ∗ is the energy of an excited
level that plays a role in the recoil process, m is the
mass of the atoms, and λ is a length scale characterizing
the noncommutativity matrix. This relationship can
be tested presently with accuracy of roughly 1 part in
10 9 , and, also thanks to the fact that in the relevant
experiments m/(hν ∗ ) is very large, allowed us to set
a bound of λ 10 −34 m. And planned improvements
of these atom-recoil experiments should comfortably
provide sensitivity to values of λ as small as ∼ 10 −35 m,
thereby reaching the desired “Planck length sensitivity”.
References
1. G. Amelino-Camelia, Nature 462, 291(2009)
2. G. Amelino-Camelia et al., Phys. Rev. Lett. 103, 171302
(2009).
3. G. Amelino-Camelia et al., Phys. Rev. D 80, 084017
(2009).
4. G. Amelino-Camelia et al., Phys. Rev. D 78, 025005
(2008).
Authors
G. Amelino-Camelia, G. Gubitosi, P. Martinetti, F. Mercati
http://www.roma1.infn.it/∼amelino/gacResearch.html
Sapienza Università di Roma 29 Dipartimento di Fisica