quantum algorithms:
In quantum mechanics what is called the Dyson formula is what in mathematics is called the iterated integral-expression for parallel transport: It an expression for the solution to a differential equation of the form
where $t \mapsto \psi(t) \in \mathcal{H}$ is a one-parameter family of elements of some Hilbert space and $t \mapsto H(t)$ is a one-parameter family of linear operators on this Hilbert space.
This appears prominently in the expression of the Schrödinger equation in the interaction picture, in which case $H(t)$ is the interaction Hamiltonian in the Heisenberg picture of the free theory. In this case the Dyson series gives the S-matrix. This is the context in which the term “Dyson formula” orginates. But of course also the plain Schrödinger equation (“in the Schrödinger picture”) is already of this form if the Hamiltonian is time-dependent.
The idea is to think of the solution as given by the limit
and to think of this as the result of forming the exponential expression $\exp(\int_{[0,t] } H(t)\,d t)$ and then re-ordering in the resulting sum of products all the factors of $H(t)$ such that they are time ordered with larger values of $t$ ordered to the left of smaller values.
Accordingly, in the physics literature solutions to this equation are written
with $T(-)$ the time ordering operator producing time-ordered products. The corresponding series of iterated integrals
is called the Dyson series.
The idea generalizes to relativistic field theory perturbative quantum field theory on Lorentzian spacetimes if due care is exercised (including adiabatic switching and point extension of operator-valued distributions). Here the “time ordering” is generalized to a causal ordering, this is the content of the construction of the S-matrix and the interacting field algebra in causal perturbation theory.
See at time-ordered product and S-matrix for details.
For the Definition of the Knizhnik-Zamolodchikov connection we need the following notation:
configuration spaces of points
For $N_{\mathrm{f}} \in \mathbb{N}$ write
for the ordered configuration space of n points in the plane, regarded as a smooth manifold.
Identifying the plane with the complex plane $\mathbb{C}$, we have canonical holomorphic coordinate functions
for the quotient vector space of the linear span of horizontal chord diagrams on $n$ strands by the 4T relations (infinitesimal braid relations), regarded as an associative algebra under concatenation of strands (here).
The universal Knizhnik-Zamolodchikov form is the horizontal chord diagram-algebra valued differential form (3) on the configuration space of points (1)
given in the canonical coordinates (2) by:
where
is the horizontal chord diagram with exactly one chord, which stretches between the $i$th and the $j$th strand.
Regarded as a connection form for a connection on a vector bundle, this defines the universal Knizhnik-Zamolodchikov connection $\nabla_{KZ}$, with covariant derivative
for any smooth function
with values in modules over the algebra of horizontal chord diagrams modulo 4T relations.
The condition of covariant constancy
is called the Knizhnik-Zamolodchikov equation.
Finally, given a metric Lie algebra $\mathfrak{g}$ and a tuple of Lie algebra representations
the corresponding endomorphism-valued Lie algebra weight system
turns the universal Knizhnik-Zamolodchikov form (4) into a endomorphism ring-valued differential form
The universal formulation (4) is highlighted for instance in Bat-Natan 95, Section 4.2, Lescop 00, p. 7. Most authors state the version after evaluation in a Lie algebra weight system, e.g. Kohno 14, Section 5.
(Knizhnik-Zamolodchikov connection is flat)
The Knizhnik-Zamolodchikov connection $\omega_{ZK}$ (Def. ) is flat:
(Kontsevich integral for braids)
The Dyson formula for the holonomy of the Knizhnik-Zamolodchikov connection (Def. ) is called the Kontsevich integral on braids.
(e.g. Lescop 00, side-remark 1.14)
Last revised on June 4, 2022 at 11:17:05. See the history of this page for a list of all contributions to it.