Global interference phenomena start to manifest in quantum transitions involving at least two projection filters which cannot be realized simultaneously, like in the case of the double-slit ...experiment. In terms of the 3-vertex invariants of the projective space of rays, thought of as a symplectic phase space, we encounter products involving the complex-valued
I
3
(
Ψ
in
,
Ψ
b
,
Ψ
fin
)
with the complex conjugate
I
3
∗
(
Ψ
in
,
Ψ
b
′
,
Ψ
fin
)
, where the projection operators
|
ψ
b
⟩
⟨
ψ
b
|
and
|
ψ
b
′
⟩
⟨
ψ
b
′
|
are not simultaneously realizable. In this way, interference can be expressed in terms of products of this form defined over squares in the space of rays, which can be triangulated. Triangles in this space encode the invariant information of the geometric phase factor. From this perspective, we qualify the proposal of Aharonov and collaborators pertaining to the consideration of commutative modular variables evaluated in
R
/
Z
in deciphering the quantum interference pattern of the double-slit experiment. The quantum modular variables pertaining to conjugate observables are encoded in terms of one-parameter unitary groups acting jointly on the phase space, thus modeled through the continuous group action of
R
2
. We show that
h
2
expresses the minimal indistinguishable invariant area of the 2D symplectic Abelian shadow of the symplectic ball of radius
R
=
ħ
in the 2
n
-phase space of the conjugate position and momenta. The above conclusion leads to a re-estimation of Weyl’s view of the quantum kinematical space in terms of an Abelian group of unitary ray rotations, and in particular the role that the discrete Heisenberg group plays in this conundrum.
In this paper, we prove an asymptotic mean value formula of sub-p-harmonic functions in the viscosity sense on the Heisenberg group. As an application, we give a new proof of the Harnack inequality ...for sub-p-harmonic functions on the Heisenberg group.
We propose a direct method to control the first-order fractional difference quotients of solutions to quasilinear subelliptic equations in the Heisenberg group. In this way we implement iteration ...methods on fractional difference quotients to obtain weak differentiability in the
T-direction and then second-order weak differentiability in the horizontal directions.
The Radon transform on the Heisenberg group was introduced by R. Strichartz. We regard it as a particular case of a more general transversal Radon transform that integrates functions on
R
m
over ...hyperplanes meeting the last coordinate axis. The paper contains new boundedness results and explicit inversion formulas for both transforms of
L
p
functions in the full range of the parameter
p. We also show that these transforms are isomorphisms of the corresponding Semyanistyi–Lizorkin spaces of smooth functions. In the framework of these spaces we obtain inversion formulas, which are pointwise analogues of the corresponding formulas by R. Strichartz.
Let Hn=R2n×R be the n-dimensional Heisenberg group, ∇Hn be its sub-elliptic gradient operator, and ρ(ξ)=(|z|4+t2)1/4 for ξ=(z,t)∈Hn be the distance function in Hn. Denote Q=2n+2 and Q′=Q/(Q−1). It is ...proved in this paper that there exists a positive constant α∗ such that for any pair β and α satisfying 0≤β<Q and αα∗+βQ≤1, sup‖u‖W1,Q(Hn)≤1∫Hn1ρ(ξ)β{eα|u|Q′−∑k=0Q−2αk|u|kQ′k!}dξ<∞, where W1,Q(Hn) is the Sobolev space on Hn. When αα∗+βQ>1, the above integral is still finite for any u∈W1,Q(Hn). Furthermore the supremum is infinite if α/αQ+β/Q>1, where αQ=QσQ1/(Q−1), σQ=∫ρ(z,t)=1|z|Qdμ. Actually if we replace Hn and W1,Q(Hn) by unbounded domain Ω and W01,Q(Ω) respectively, the above inequality still holds. As an application of this inequality, a sub-elliptic equation with exponential growth is considered.
The goal of this note is to explore the relationship between the Folland-Kohn basic estimate and the Z(q)-condition. In particular, on unbounded domains, we prove that the Folland-Kohn basic estimate ...is equivalent to a uniform Z(q) condition. As a corollary, we observe that despite the Siegel upper half space being strictly pseudoconvex and biholomorphic to the unit ball, it fails to satisfy uniform strict pseudoconvexity and hence the Folland-Kohn basic estimate fails.
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