Variational perturbation theory
Encyclopedia
In mathematics
, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say
,
into a convergent series in powers
,
where
is a critical exponent
(the so-called index of "approach to scaling" introduced by Franz Wegner
). This is possible with the help of variational parameters
, which are determined by optimization order by order in
. The partial sums are converted to convergent
partial sums by a method developed in 1992.
Most perturbation expansions in quantum mechanics
are divergent for any small coupling strength
$g$. They can be made convergent by VPT (for details see the first textbook cited below). The convergence
is exponentially fast
.
After its success in quantum mechanics, VPT has been developed further to become an important mathematical tool in
quantum field theory with its anomalous dimensions
. Applications focus on the theory
of critical phenomena
. It has led to the most accurate predictions of critical exponent
s.
More details can be read
here.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say
,into a convergent series in powers
,where
is a critical exponentCritical exponent
Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...
(the so-called index of "approach to scaling" introduced by Franz Wegner
Franz Wegner
Franz Wegner is emeritus professor for theoretical physics at the University of Heidelberg. Franz Wegner attained a doctorate in 1968 with thesis advisor Wilhelm Brenig at Technical University Munich with the thesis "Zum Heisenberg-Modell im paramagnetischen Bereich und am kritischen Punkt" ...
). This is possible with the help of variational parameters
Variational method (quantum mechanics)
In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states...
, which are determined by optimization order by order in
. The partial sums are converted to convergentpartial sums by a method developed in 1992.
Most perturbation expansions in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
are divergent for any small coupling strength
$g$. They can be made convergent by VPT (for details see the first textbook cited below). The convergence
is exponentially fast
.
After its success in quantum mechanics, VPT has been developed further to become an important mathematical tool in
quantum field theory with its anomalous dimensions
Anomalous scaling dimension
In theoretical physics, by anomaly one usually means that the symmetry remains broken when the symmetry-breaking factor goes to zero.When the symmetry which is broken is scale invariance, then true power laws usually cannot be found from dimensional reasoning like in turbulence or quantum field...
. Applications focus on the theory
of critical phenomena
Critical phenomena
In physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...
. It has led to the most accurate predictions of critical exponent
Critical exponent
Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...
s.
More details can be read
here.