Decomposition of Time-Ordered Products and Path-Ordered Exponentials

The solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data. Accurate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schr\"odinger equation, it is immaterial whether or not the decomposition coefficients are positive. In fact, most symplectic algorithms for solving classical dynamics contain some negativ...

How to calculate the exponential of matrices in an explicit manner is one of fundamental problems in almost all subjects in Science. Especially in Mathematical Physics or Quantum Optics many problems are reduced to this calculation by making use of some approximations whether they are appropriate or not. However, it is in general not easy. In this paper we give a very useful formula which is both elementary and getting on with computer.

In this work we provide a new approach for approximating an ordered operator exponential using an ordinary operator exponential that acts on the Hilbert space of the simulation as well as a finite-dimensional clock register. This approach allows us to translate results for simulating time-independent systems to the time-dependent case. Our result solves two open problems in simulation. It first provides a rigorous way of using discrete time-displacement operators to generate ...

Relations between integrals of time-ordered product of operators, and their representation in terms of energy-ordered products are studied. Both can be decomposed into irreducible factors and these relations are discussed as well. The energy-ordered representation was invented to separate various infrared contributions in gauge theories. It is shown that the irreducible time-ordered expressions can be used to accomplish the same purpose. Besides, it has the added advantage of...

In this report the emphasis is on an alternative representation of the Magnus series by proper operator (matrix) exponential solutions to differential equations (systems), both linear and nonlinear ODEs and PDEs. The main idea here is in \emph{exact} \emph{linear} representations of the \emph{nonlinear} DEs. We proceeded from Dyson's time-ordered solutions, and using only generalizations of the well-known Baker-Campbell- Hausdorff (BCH) and Zassenhaus formulae for $t$-depende...

We propose and analyze a symmetric version of the Zassenhaus formula for disentangling the exponential of two non-commuting operators. A recursive procedure for generating the expansion up to any order is presented which also allows one to get an enlarged domain of convergence when it is formulated for matrices. It is shown that the approximations obtained by truncating the infinite expansion considerably improve those arising from the standard Zassenhaus formula.

By using methods of umbral nature, we discuss new rules concerning the operator ordering. We apply the technique of formal power series to take advantage from the wealth of properties of the exponential operators. The usefulness of the obtained results in quantum field theory is discussed.

We study $n$-point functions at finite temperature in the closed time path formalism. With the help of two basic column vectors and their dual partners we derive a compact decomposition of the time-ordered $n$-point functions with $2^n$ components in terms of $2^{n-1} -1$ independent retarded/advanced $n$-point functions. This representation greatly simplifies calculations in the real-time formalism.

This paper contains an implicit assumption that the summand in eq. (4.2) for different $n$'s commute. This paper should be replaced by hep-th/9804181 where this assumption is removedand the result generalized.

We show that there are {\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\exp(X)\exp(Y)\exp(Z)=\exp({AX+BZ+CY+DI}) \ , $$ derived in arXiv:1502.06589, JHEP {\bf 1505} (2015) 113. This includes, as a particular case, $\exp(X) \exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\exp(X)\exp(Y)\...