12,872 research outputs found
Invariance of Spooky Action at a Distance in Quantum Entanglement under Lorentz Transformation
We study the mechanism by which the particle-antiparticle entangled state
collapses instantaneously at a distance. By making two key assumptions, we are
able to show not only that instantaneous collapse of a wave function at a
distance is possible but also that it is an invariant quantity under Lorentz
transformation and compatible with relativity. In addition, we will be able to
detect in which situation a many-body entangled system exhibits the maximum
collapse speed among its entangled particles. Finally we suggest that every
force in nature acts via entanglement
The product of operators with closed range in Hilbert C*-modules
Suppose and are bounded adjointable operators with close range
between Hilbert C*-modules, then has closed range if and only if
is an orthogonal summand, if and only if is
an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between
and is positive and is an orthogonal summand then has closed range.Comment: 12 pages, abstract was changed, accepte
EP modular operators and their products
We study first EP modular operators on Hilbert C*-modules and then we provide
necessary and sufficient conditions for the product of two EP modular operators
to be EP. These enable us to extend some results of Koliha [{\it Studia Math.}
{\bf 139} (2000), 81--90.] for an arbitrary C*-algebra and the C*-algebras of
compact operators.Comment: 10 pages, accepte
Iwasawa theory and the Eisenstein ideal
In this paper, we relate three objects. The first is a particular value of a
cup product in the cohomology of the Galois group of the maximal unramified
outside p extension of a cyclotomic field containing the pth roots of unity.
The second is an Iwasawa module over a nonabelian extension of the rationals, a
subquotient of the maximal pro-p abelian unramified completely split at p
extension of a certain pro-p Kummer extension of a cyclotomic field that
contains all p-power roots of unity. The third is the quotient of an Eisenstein
ideal in an ordinary Hecke algebra of Hida by the square of the Eisenstein
ideal and the element given by the pth Hecke operator minus one. For the
relationship between the latter two objects, we employ the work of Ohta, in
which he considered a certain Galois action on an inverse limit of cohomology
groups to reestablish the Main Conjecture (for p at least 5) in the spirit of
the Mazur-Wiles proof. For the relationship between the former two objects, we
construct an analogue to the global reciprocity map for extensions with
restricted ramification. These relationships, and a computation in the Hecke
algebra, allow us to prove an earlier conjecture of McCallum and the author on
the surjectivity of a pairing formed from the cup product for p < 1000. We give
one other application, determining the structure of Selmer groups of the
modular representation considered by Ohta modulo the Eisenstein ideal.Comment: 37 page
A reciprocity map and the two variable p-adic L-function
For primes p greater than 3, we propose a conjecture that relates the values
of cup products in the Galois cohomology of the maximal unramified outside p
extension of a cyclotomic field on cyclotomic p-units to the values of p-adic
L-functions of cuspidal eigenforms that satisfy mod p congruences with
Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an
isomorphism of certain spaces that allows us to compare the value of a
reciprocity map on a particular norm compatible system of p-units to what is
essentially the two-variable p-adic L-function of Mazur and Kitagawa.Comment: 55 page
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