Download Lorentz Invaiance Violation and Granularity of space time

A Different Approach (?)
Collaboration with D Sudarsky
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Let us take up the notion that space-time
contains some granular/discrete aspect with
characteristic scale given by MPlanck
The lesson from the previous studies is that
such structure, if exists, can not lead to
breakdown of Lorentz Invariance.
It is of course hard to envision something like
that while thinking classically about a spacetime. But what is space-time in a quantum
world? We do not know!
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When faced with total
Ignorance proceed by analogy
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Consider a physicist trying to get information
about the granular structure of a crystal.
Assume that the fundamental crystal lattice
has some symmetry, say cubic.
We know that the fundamental granular
structure will not become manifest through a
breakdown of the symmetry if one studies a
macroscopic, but similarly cubic crystal.
However, if the macroscopic crystal does not
share the lattice symmetry, we might be able
to detect it.
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Phenomenology
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The results so far indicate that the granular
structure, if any, would have to respect the
Lorentz symmetry.
This would explain why we have not seen the
“expected” violation of L.I. : There is none.
So what would be the signature of the
discrete structure of space time?
In analogy with the crystal, we consider a
macroscopic space-time that is Not Fully
Lorentz Invariant.
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There lies the hope of seeing
something
A space-time that is not Minkowski on
an extended realm could exhibit the
mismatch between the symmetry of the
fundamental structure and that of the
macroscopic domain.
 The departure from Minkowski spacetime is characterized by the Riemman.
tensor:
R .
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The effects in question should represent a
coupling of Riemman to ordinary matter.
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Furthermore: part of Riemman is determined
by the local T :
R - (R/2 )g =8G T
and thus would look like a self coupling. That is
not the most interesting.
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We need to consider couplings of the Weyl
tensor W (Riemman without R or R).
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The direct approach indicates that all
such couplings are highly suppressed
A more open minded approach exists:
Extract from Weyl some aspects and couple
them to matter:
One possibility: Find eigenforms and
eigenvalues of Weyl:
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W  X =  X
And use these objects to couple to matter fields.
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Consider the coupling to
fermions
The least suppressed term is :
L =  (1/MPlanck) ∑a a (a)    
Can this ideas be experimentally explored?
In the non-relativistic regime the dominant new term in
the Hamiltonian for the spin 1/2 particle
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H=  (1/MPlanck) ∑a a0i(a) i
This looks like a coupling of the spin with magnetic
field. Can it be distinguished from that ?
Could the suppression be 1/M instead of 1/Mplanck ? 7
Comments:
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Magnitude:  is of the order of the
gravitational gradients GM/r3.
A non zero 0i implies a special direction
and a time asymmetry (rotation).
On the other hand the effect should be
associated with the gravitational sources and
thus susceptible of control.
It should affect even particles with no
electromagnetic couplings like Neutrinos.
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Other comments
Something like this could originate from
some QG aspects of holonomic degrees
of freedom (i.e. LQG (?)), or from QG
bounds in the sectional curvatures (?)...
 It is worth noting that no real tests of
Quantum Mechanics in the presence of
curvature exists up to date!
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THE CHALLENGE
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According to the GR view, COW experiment & the
experiments with neutrons at Grenoble are ``just”
tests of QM in a non inertial frame… but gravity lies
only in the curvature, i.e. in the fact that inertial
frames at different events do not coincide.
Other tests, such as those using Quantum Gravity
gradiometers, rely only on quantum aspects that are
purely local ( no quantum description needed in the
domain covered by different Local Inertial frames).
The challenge, from my point of view, is to detect
curvature (i.e. tidal forces) using quantum aspects of
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matter.
Further Ideas
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Point particles move along geodesics, but how
do extended/quantum objects move?
What is the operational definition of geodesic,
when we test the world with extended objects?
The path of the center of mass? NO!
What are in principle the geodesics of the
geometry, or the geometry itself, when all we
have to explore it, are quantum particles ( in
fact, quantum fields)?
There is too much we do not understand! We
need to be open minded when we explore!
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