18,058 research outputs found
All Order Running Coupling BFKL Evolution from GLAP (and vice-versa)
We present a systematic formalism for the derivation of the kernel of the
BFKL equation from that of the GLAP equation and conversely to any given order,
with full inclusion of the running of the coupling. The running coupling is
treated as an operator, reducing the inclusion of running coupling effects and
their factorization to a purely algebraic problem. We show how the GLAP
anomalous dimensions which resum large logs of x can be derived from the
running-coupling BFKL kernel order by order, thereby obtaining a constructive
all-order proof of small x factorization. We check this result by explicitly
calculating the running coupling contributions to GLAP anomalous dimensions up
to next-to-next-to leading order. We finally derive an explicit expression for
BFKL kernels which resum large logs of Q^2 up to next-to-leading order from the
corresponding GLAP kernels, thus making possible a consistent collinear
improvement of the BFKL equation up to the same order
Exactly solvable one-qubit driving fields generated via non-linear equations
Using the Hubbard representation for we write the time-evolution
operator of a two-level system in the disentangled form. This allows us to map
the corresponding dynamical law into a set of non-linear coupled equations. In
order to find exact solutions, we use an inverse approach and find families of
time-dependent Hamiltonians whose off-diagonal elements are connected with the
Ermakov equation. The physical meaning of the so-obtained Hamiltonians is
discussed in the context of the nuclear magnetic resonance phenomeno
Time-ordering and a generalized Magnus expansion
Both the classical time-ordering and the Magnus expansion are well-known in
the context of linear initial value problems. Motivated by the noncommutativity
between time-ordering and time derivation, and related problems raised recently
in statistical physics, we introduce a generalization of the Magnus expansion.
Whereas the classical expansion computes the logarithm of the evolution
operator of a linear differential equation, our generalization addresses the
same problem, including however directly a non-trivial initial condition. As a
by-product we recover a variant of the time ordering operation, known as
T*-ordering. Eventually, placing our results in the general context of
Rota-Baxter algebras permits us to present them in a more natural algebraic
setting. It encompasses, for example, the case where one considers linear
difference equations instead of linear differential equations
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