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Asymptotic Extended Uncertainty Principle

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Abstract

The formalism presented on this poster allows for the perturbative derivation of the Extended Uncertainty Principle using the non-relativistic limit of the 3+1 formalism. The leading curvature induced correction depends to the Ricci scalar of the effective 3-metric and the corresponding co-variant derivative of the shift vector. This method can be equivalently applied in curved momentum space allowing for a generalized uncertainty principle or curved momentum space quantum mechanics.
ASYMPTOTIC EXTENDED UNCERTAINTY PRINCIPLE
FABIAN WAGNER, UNIVERSITY OF SZCZECIN
ABSTRACT
The formalism presented on this poster allows for
the perturbative derivation of the Extended Un-
certainty Principle using the non-relativistic limit
of the 3+1 formalism. The leading curvature in-
duced correction depends to the Ricci scalar of
the effective 3-metric and the corresponding co-
variant derivative of the shift vector. This method
can be equivalently applied in curved momen-
tum space allowing for a generalized uncertainty
principle or curved momentum space quantum
mechanics.
Keywords: phenomenology, GUP, EUP.
1. INTRO:LOADS OF ADJECTIVES
Extended (uncertainty principle - EUP)
phenomenologically expected uncertainty
in the presence of curvature
xp/2(1 + βx2
l2
C)(1)
Asymptotic
expand metric in Riemann Normal Coordi-
nates (xi) around p0(cf. fig.1)
equivalent to assumption x/lC1
gij δij 1
3Rikjl|p0xkxj(2)
Figure 1: Cartoon of
space approximated by
Riemann Normal Coordi-
nates constructed around
p0
2. BACKGROUND
work on spacelike 3D hypersurfaces of
space-time (cf. fig. 2)
observer (slicing) dependent
QM on curved 3D manifold
Figure 2: Slicing defined by observer
generalize Heisenberg’s slit (cf. fig. 3)
to domain in curved 3D backgrounds (cf.
Geodesic Balls) and confine wave function ψ
(cf. fig. 2) to it
Figure 3: Heisenberg microscope with slit width x
4. MOMENTUM UNCERTAINTY
standard deviation with Dirichlet boundary
conditions at ball surface (ψ|Bρ= 0):
pˆp2⟩−⟨pi⟩ ⟨pi⟩ ≥ p(ψ0)(3)
ψ0denotes ground state of negative
Laplace-Beltrami operator 2
Extended Uncertainty Relation:
ρpρp(ψ0)(4)
3. GEODESIC BALLS
diffeomorphism invariant domains Bρ
geodesic distance (σ) from center (p0)
σ(p0, x)ρwith radius ρ
Schwarzschild example: see fig. 3
generalise Heisenberg’s slits to curved 3D
backgrounds
ρanalogous to Heisenberg’s x(see fig. 1,2)
Figure 4: 3 geodesic balls (blue) in a spatial section
of the Schwarzschild static patch (horizons coloured
black) with differing distances to the horizon.
5. RESULT [1, 3]
ADM (shift Ni,lapse N, ind. metric hij )
effective 3-metric Gij =hij /N and gravito-
magnetic vector field Ni
ρpπ(1
(3)R
12π2ρ2+ξρ4
λ2iNjiNi)
p0
(5)
Rand iRicci scalar and covariant deriva-
tive derived from Gij respectively
in coordinate notation pi
0=ˆ
Xi
Compton wavelength λ=/m
mathematical constant ξ= (2 32)/9
6. CONCLUSION
generic spatial curvature/ non-vanishing shift implies EUP
even in Minkowski (appropriately sliced)
can even vanish if manifold is closed
classical curvature effects have to be taken account for in QG phenomenology
usually stronger than QG motivated Generalised Uncertainty Principle
by Born reciprocity (invariance of QM under exchange xp, p → −x)
assume curvature in momentum space in low-energy limit pi0: Generalised Uncertainty
Relation
approach well-suited for application to more complex problems with curved position/ momen-
tum space
Figure 5: Exemplary tra-
jectory (left) and devia-
tion from flat-space uncer-
tainty relation measured
by the static observer at in-
finity (right) of a particle
orbiting a rotating black
hole described by the Kerr
space-time. The colour
changes with increasing
affine parameter τ.
REFERENCES
[1] Dabrowski, M. P. & FW. EPJC, 80(7):676, 2020.
[2] Dabrowski, M. P. & FW. EPJC, 79(8):716, 2019.
[3] Petruzziello, L. & FW. arXiv:2101.05552 [gr-qc].
FUTURE RESEARCH
Can this formalism be extended to curvature in both position and momentum space simultaneously?
Is it possible to formulate uncertainty principles in a relativistically covariant way?
CONTACT INFORMATION
Email fabian.wagner@usz.edu.pl
Web https://cosmo.usz.edu.pl/fwagner
ResearchGate has not been able to resolve any citations for this publication.
  • L Petruzziello
  • Fw
Petruzziello, L. & FW. arXiv:2101.05552 [gr-qc].