Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Refined formulation of quantum analysis,q-derivative and exponential splitting

Masuo Suzuki

Published Apr 24, 2006• 5 citations
Publisher

Abstract

Quantum analysis is reformulated to clarify its essence, namely the invariance of quantum derivative for any choice of definitions of the differential d f ( A ) satisfying the Leibniz rule. This formulation with use of the inner derivation δ A is convenient to study quantum corrections in contrast to the Feynman operator calculus. The present analysis can also be used to find a general scheme of constructing exponential product formulae of higher order. General recursive schemes are also reviewed with an emphasis to standard symmetric splitting formulae. Multiple integral representations of q -derivatives are derived using such general integral formulae of quantum derivatives as are expressed by hyperoperators. A simple explanation of the connection between quantum derivatives and q -derivative is also given.

DOI
10.1088/0305-4470/39/19/s17

Related papers

Exponential Operators and Parameter Differentiation in Quantum Physics

Ralph M. Wilcox

Elementary parameter-differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker-Campbell-Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equation dY(t)/dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated-commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed-form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed-form single-exponential solutions to the differential equation dY(t)/dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple-exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincaré which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincaré formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic-oscillator canonical-ensemble averages of general operators defined by the Weyl prescription. Solutions of the density-matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian.

Journal of Mathematical Physics•Apr 1, 1967•1,207 citations

Quantum Stochastic Generators

John Edward Gough

We discuss stochastic derivations, stochastic Hamiltonians and the flows that they generate, algebraic fluctuaion-dissipation theorems, etc., in a language common to both classical and quantum algebras. It is convenient to define distinct notions of time-ordered exponentials to take account of the breakdown of the Leibniz rule in the Ito calculus. We introduce a notion of quantum Stratonovich calculus and show how it relates to Stratonovich-Dyson time ordered exponentials. We then use it to demonstrate a natural way to add stochastic derivations.

arXiv:quant-ph/0309102•Sep 12, 2003•1 citation

Physics analysis with Leibniz’s differential operators dn

U-Rae Kim, Sungwoong Cho, Jungil Lee

Abstract We introduce a systematic approach to represent Leibniz’s nth-order differential operator $$d^n$$ d n as the ratio of an infinite product of infinitesimal difference operators to an infinitesimal parameter. Because every difference operator can be expressed as a difference of two shift operators that translate the argument of a function by finite amounts, Leibniz’s differential operator $$d^n$$ d n is eventually expressed as the infinite product of infinitesimal binomial operators consisting of the shift operators. We apply this strategy to demonstrate the derivation of the translation or time-evolution operators in quantum mechanics. This fills the logical gap in most textbooks on quantum mechanics that usually omit explicit derivations. Our approach could be employed in general physics or classical mechanics classes with which one can solve the equation of motion without prior knowledge of differential equations.

Journal of the Korean Physical Society•Aug 29, 2023•0 citations

Operatorcalculus

Yuri M Makeenko

Operator calculusThe operator calculus developed by Feynman [Fey51] makes it possible to represent functions of (noncommuting) operators as path integrals, with the integrand being the path-ordered exponential of operators, the order of which is controlled by a parameter that varies along the trajectory.This procedure is termed Feynman disentangling.It is also applicable to functions of matrices (say, γ-matrices which are associated with a spinor particle).When applied to the evolution operator, this procedure results in the standard path-integral representation of quantum mechanics.In this chapter we first demonstrate the general technique using the simplest example, a free propagator in Euclidean space, and then consider the path-integral representation of quantum mechanics, as well as propagators in an external electromagnetic field. Free propagatorLet us first consider the simplest propagator of a free scalar field which is given in the operator formalism by the vacuum expectation value of thewith ϕ being the field-operator.The T -product (1.1) obeys the equationwhere d = 4 is the dimension of space-time, however the formulas are applicable at any value of d.In the operator formalism, Eq. (1.2) is a * The ordered products of operators were introduced by Dyson [Dys49].This paper and other classical papers on quantum electrodynamics are collected in the book edited by Schwinger [Sch58].

Cambridge University Press eBooks•Jul 13, 2023•0 citations

Formal means

Helmut Hofmann

Abstract This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.

•Apr 17, 2008•0 citations

A Combinatorial Proof Of The Time-Ordered Exponential Using The Heaviside Representation Of Time-Ordering Operator

Abdelwahab Amr

We present a novel combinatorial proof of the \textbf{Time-Ordered Exponential} formula for the \textbf{Time-Evolution Operator} based on the \textbf{Heaviside Formulation Of The Time-Ordering Operator}. Unlike conventional derivations that rely on induction our approach decomposes the \textbf{Symmetric Group} into permutations that produce decreasing time sequences and those that do not using a simple discriminant function. This decomposition together with an \textbf{Interchange Rule } between time-ordering and multiple integrals yields the \textbf{Dyson–Chen Identity} directly. The proof is self-contained and avoids recursion entirely. It appears in the literature as a problem in Advanced Quantum Mechanics (Springer, 2012, Problem 13.1).

•Apr 13, 2026•0 citations
Quantum analysis is reformulated to clarify its essence, namely the invariance of quantum derivative for any choice of definitions of the differential d f ( A ) satisfying the Leibniz rule. This formulation with use of the inner derivation δ A is convenient to study quantum corrections in contrast to the Feynman operator calculus. The present analysis can also be used to find a general scheme of constructing exponential product formulae of higher order. General recursive schemes are also reviewed with an emphasis to standard symmetric splitting formulae. Multiple integral representations of q -derivatives are derived using such general integral formulae of quantum derivatives as are expressed by hyperoperators. A simple explanation of the connection between quantum derivatives and q -derivative is also given.
Elementary parameter-differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker-Campbell-Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equation dY(t)/dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated-commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed-form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed-form single-exponential solutions to the differential equation dY(t)/dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple-exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincaré which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincaré formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic-oscillator canonical-ensemble averages of general operators defined by the Weyl prescription. Solutions of the density-matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian.
We discuss stochastic derivations, stochastic Hamiltonians and the flows that they generate, algebraic fluctuaion-dissipation theorems, etc., in a language common to both classical and quantum algebras. It is convenient to define distinct notions of time-ordered exponentials to take account of the breakdown of the Leibniz rule in the Ito calculus. We introduce a notion of quantum Stratonovich calculus and show how it relates to Stratonovich-Dyson time ordered exponentials. We then use it to demonstrate a natural way to add stochastic derivations.
Abstract We introduce a systematic approach to represent Leibniz’s nth-order differential operator dnd^ndn d n as the ratio of an infinite product of infinitesimal difference operators to an infinitesimal parameter. Because every difference operator can be expressed as a difference of two shift operators that translate the argument of a function by finite amounts, Leibniz’s differential operator dnd^ndn d n is eventually expressed as the infinite product of infinitesimal binomial operators consisting of the shift operators. We apply this strategy to demonstrate the derivation of the translation or time-evolution operators in quantum mechanics. This fills the logical gap in most textbooks on quantum mechanics that usually omit explicit derivations. Our approach could be employed in general physics or classical mechanics classes with which one can solve the equation of motion without prior knowledge of differential equations.
Operator calculusThe operator calculus developed by Feynman [Fey51] makes it possible to represent functions of (noncommuting) operators as path integrals, with the integrand being the path-ordered exponential of operators, the order of which is controlled by a parameter that varies along the trajectory.This procedure is termed Feynman disentangling.It is also applicable to functions of matrices (say, γ-matrices which are associated with a spinor particle).When applied to the evolution operator, this procedure results in the standard path-integral representation of quantum mechanics.In this chapter we first demonstrate the general technique using the simplest example, a free propagator in Euclidean space, and then consider the path-integral representation of quantum mechanics, as well as propagators in an external electromagnetic field. Free propagatorLet us first consider the simplest propagator of a free scalar field which is given in the operator formalism by the vacuum expectation value of thewith ϕ being the field-operator.The T -product (1.1) obeys the equationwhere d = 4 is the dimension of space-time, however the formulas are applicable at any value of d.In the operator formalism, Eq. (1.2) is a * The ordered products of operators were introduced by Dyson [Dys49].This paper and other classical papers on quantum electrodynamics are collected in the book edited by Schwinger [Sch58].
Abstract This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.
We present a novel combinatorial proof of the \textbf{Time-Ordered Exponential} formula for the \textbf{Time-Evolution Operator} based on the \textbf{Heaviside Formulation Of The Time-Ordering Operator}. Unlike conventional derivations that rely on induction our approach decomposes the \textbf{Symmetric Group} into permutations that produce decreasing time sequences and those that do not using a simple discriminant function. This decomposition together with an \textbf{Interchange Rule } between time-ordering and multiple integrals yields the \textbf{Dyson–Chen Identity} directly. The proof is self-contained and avoids recursion entirely. It appears in the literature as a problem in Advanced Quantum Mechanics (Springer, 2012, Problem 13.1).