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Decomposition of time-ordered products and path-ordered exponentials

C. S. Lam

Published Jan 1, 1998• 28 citations
OpenAlex

Abstract

We present a decomposition formula for Un, an integral of time-ordered products of operators, in terms of sums of products of the more primitive quantities Cm, which are the integrals of time-ordered commutators of the same operators. The resulting factorization enables a summation over n to be carried out to yield an explicit expression for the time-ordered exponential, an expression which turns out to be an exponential function of Cm. The Campbell-Baker-Hausdorff formula and the nonabelian eikonal formula obtained previously are both special cases of this result. Typeset using REVTEX 1 I.

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We present a decomposition formula for Un, an integral of time-ordered products of operators, in terms of sums of products of the more primitive quantities Cm, which are the integrals of time-ordered commutators of the same operators. The resulting factorization enables a summation over n to be carried out to yield an explicit expression for the time-ordered exponential, an expression which turns out to be an exponential function of Cm. The Campbell-Baker-Hausdorff formula and the nonabelian eikonal formula obtained previously are both special cases of this result. Typeset using REVTEX 1 I.
Abstract The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators X and Y, according to the Campbell-Baker-Hausdorff-Dynkin theorem, e X+Y is not equivalent to e X e Y , but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators {X, Y, Z}, such that [X, Y] = κ Z for scalar κ and cyclic permutations, here it is proven that e a X+b Y is equivalent to e p Z e q X e−p Z for scalar p and q. Extensions for e a X+b Y+c Z are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields.
We propose a high order numerical decomposition of exponentials of Hermitian operators in terms of a product of exponentials of simple terms, following an idea which has been pioneered by M Suzuki, implementing it for complex coefficients. We outline a convenient fourth-order formula which can be written compactly for an arbitrary number of non-commuting terms in the Hamiltonian and which is superior to the optimal formula with real coefficients, both in complexity and accuracy. We show asymptotic stability of our method for a sufficiently small time step and demonstrate its efficiency and accuracy in different numerical models.
The solution of many physical evolution equations can be expressed as an exponential of two or more operators. Approximate 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 the decomposition coefficients are positive or negative. For time-irreversible systems, such as the Fokker-Planck equation or the quantum statistical propagator, only positive-coefficient decompositions which respect the time-irreversibility of the diffusion kernel can yield practical algorithms. These positive time step, or forward decompositions, are a highly restrictive class of factorization algorithms. This work proves a fundamental theorem: in order for a 2n2n2nth order forward algorithm to become (2n+2)(2n+2)(2n+2)th order, one must include a new, higher order commutator in the decomposition process. Since these higher order commuatators are highly complex, it seems difficult to produce practical forward algorithms beyond fourth order. This proof generalize the Sheng-Suzuki theorem for the case of n=1n=1n=1. In particular, this work shows that it is not possible to have a sixth order forward algorithm by including just the [V,[T,V]][V,[T,V]][V,[T,V]] commutator.
We propose a high order numerical decomposition of exponentials of hermitean operators in terms of a product of exponentials of simple terms, following an idea which has been pioneered by M. Suzuki, however implementing it for complex coefficients. We outline a convenient fourth order formula which can be written compactly for arbitrary number of noncommuting terms in the Hamiltonian and which is superiour to the optimal formula with real coefficients, both in complexity and accuracy. We show asymptotic stability of our method for sufficiently small time step and demonstrate its efficiency and accuracy in different numerical models.
An expansion equivalent to Taylor's series is derived for functions of two-operators of the form F(A + B) where B isa shift operator with regard to A; the result is obtained as a special case of a theorem due to Kermack and McCrea. The development in powers of Binvolves finite differences of the function F(A) instead of derivatives as in the case of commuting variables. Exponential functions can be factorized into two terms, each depending on one operator only. The result allows a Gaussian potential energy to be expressed as a product of exponentials involving only creation and destruction operators. Explicit results are derived for the matrix elements of such a Gaussian function in the representations which diagonalize the Hamiltoniana of the harmonic oscillator in one and two dimensions; they can be expressed in terms of Jacobi polynomials (finite hypergeometric series).