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arXiv:2001.11909 · International Journal of Geometric Methods in Modern Physics

A Baker–Campbell–Hausdorff formula for the logarithm of permutations

Hans-Thomas Elze

Published Feb 11, 2020• 4 citations
arXivHTMLPDF

Abstract

The dynamics-from-permutations of classical Ising spins are studied for a chain of four spins. We obtain the Hamiltonian operator which is equivalent to the unitary permutation matrix that encodes assumed pairwise exchange interactions. It is shown how this can be summarized by an exact terminating Baker–Campbell–Hausdorff formula, which relates the Hamiltonian to a product of exponentiated two-spin exchange permutations. We briefly comment upon physical motivation and implications of this study.

DOI
10.1142/s0219887820500528

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The dynamics-from-permutations of classical Ising spins are studied for a chain of four spins. We obtain the Hamiltonian operator which is equivalent to the unitary permutation matrix that encodes assumed pairwise exchange interactions. It is shown how this can be summarized by an exact terminating Baker–Campbell–Hausdorff formula, which relates the Hamiltonian to a product of exponentiated two-spin exchange permutations. We briefly comment upon physical motivation and implications of this study.
In order to prepare for the introduction of dynamical many-body and, eventually, field theoretical models, we show here that quantum mechanical exchange interactions in a three-spin chain can emerge from the deterministic dynamics of three classical Ising spins. States of the latter form an ontological basis, which will be discussed with reference to the ontology proposed in the Cellular Automaton Interpretation of Quantum Mechanics by ’t[Formula: see text]Hooft. Our result illustrates a new Baker–Campbell–Hausdorff (BCH) formula with terminating series expansion.
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.
Multiplication of two elements of the special unitary group SU(N) determines uniquely a third group element. A Baker - Campbell - Hausdorff relation is derived which expresses the group parameters of the product (written as an exponential) in terms of the parameters of the exponential factors. This requires the eigenvalues of three matrices. Consequently, the relation can be stated analytically up to N = 4, in principle. Similarity transformations encoding the time evolution of quantum-mechanical observables, for example, can be obtained by the same means.
The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.