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 eigen- values of three (N-by-N) 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 worked out by the same means.
The dynamics-from-permutations of classical Ising spins is 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 this work we introduce the contact Heisenberg algebra which is the restriction of the Jacobi algebra on contact manifolds to the linear and constant functions. We give the exact expression of its corresponding Baker-Campbell-Hausdorff formula. We argue that this result is relevant to the quantization of contact systems.
This short paper presents an efficient implementation of Baker-Campbell-Hausdorff formula for calculating the logarithm of product of two possibly non-commutative Lie group elements using only Lie algebra terms.
In a previous article, [arXiv:1501.02506, JPhysA {\bf48} (2015) 225207], we demonstrated that whenever $[X,Y] = u X + vY + cI$ the Baker-Campbell-Hausdorff formula reduces to the tractable closed-form expression \[ Z(X,Y)=\ln( e^X e^Y ) = X+Y+ f(u,v) \; [X,Y], \] where $f(u,v)=f(v,u)$ is explicitly given by \[ f(u,v) = {(u-v)e^{u+v}-(ue^u-ve^v)\over u v (e^u - e^v)} = {(u-v)-(ue^{-v}-ve^{-u})\over u v (e^{-v} - e^{-u})}. \] This is much more general than the results usually presented for either the Heisenberg commutator $[P,Q]=-i\hbar I$, or the creation-destruction commutator $[a,a^\dagger]=I…
The Baker-Campbell-Hausdorff formula is a general result for the quantity $Z(X,Y)=\ln( e^X e^Y )$, where $X$ and $Y$ are not necessarily commuting. For completely general commutation relations between $X$ and $Y$, (the free Lie algebra), the general result is somewhat unwieldy. However in specific physics applications the commutator $[X,Y]$, while non-zero, might often be relatively simple, which sometimes leads to explicit closed form results. We consider the special case $[X,Y] = u X + vY + cI$, and show that in this case the general result reduces to \[ Z(X,Y)=\ln( e^X e^Y ) = X+Y+ f(u,v) \…
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 $\exp(X)\exp(Y)\exp(Z)=\exp(X)\exp({αY}) \exp({(1-α) Y}) \exp(Z)$, with $α$ fixed in such a way that it reduces to $\exp({\tilde X})\exp…
The Baker-Campbell-Hausdorff series computes the quantity \begin{equation*} Z(X,Y)=\ln\left( e^X e^Y \right) = \sum_{n=1}^\infty z_n(X,Y), \end{equation*} where $X$ and $Y$ are not necessarily commuting, in terms of homogeneous multinomials $z_n(X,Y)$ of degree $n$. (This is essentially equivalent to computing the so-called Goldberg coefficients.) The Baker-Campbell-Hausdorff series is a general purpose tool of wide applicability in mathematical physics, quantum physics, and many other fields. The Reinsch algorithm for the truncated series permits one to calculate up to some fixed order $N$ by…
Application of current and near-term quantum hardware to the electronic structure problem is highly limited by qubit counts, coherence times, and gate fidelities. To address these restrictions within the variational quantum eigensolver (VQE) framework, many recent contributions have suggested dressing the electronic Hamiltonian to include a part of electron correlation, leaving the rest to be accounted by VQE state preparation. We present a new dressing scheme that combines preservation of the Hamiltonian hermiticity and an exact quadratic truncation of the Baker-Campbell-Hausdorff expansion. …
A simple algorithm, which exploits the associativity of the BCH formula, and that can be generalized by iteration, extends the remarkable simplification of the Baker-Campbell-Hausdorff (BCH) formula, recently derived by Van-Brunt and Visser. We show that if $[X,Y]=uX+vY+cI$, $[Y,Z]=wY+zZ+dI$, and, consistently with the Jacobi identity, $[X,Z]=mX+nY+pZ+eI$, then $$ \exp(X)\exp(Y)\exp(Z)=\exp({aX+bY+cZ+dI}) $$ where $a$, $b$, $c$ and $d$ are solutions of four equations. In particular, the Van-Brunt and Visser formula $$\exp(X)\exp(Z)=\exp({aX+bZ+c[X,Z]+dI}) $$ extends to cases when $[X,Z]$ conta…
We developed a novel perturbative expansion based on the replica trick for the Floquet Hamiltonian governing the dynamics of periodically kicked systems where the kick strength is the small parameter. The expansion is formally equivalent to an infinite resummation of the Baker-Campbell-Hausdorff series in the un-driven (non-perturbed) Hamiltonian, while considering terms up to a finite order in the kick strength. As an application of the replica expansion, we analyze an Ising spin 1/2 chain periodically kicked with magnetic field of strength $h$, which has both longitudinal and transverse comp…
Recently it has been introduced an algorithm Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are {\it 13 types} of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, $X$, $Y$ of the Cartan-Weyl basis, we find $W$, linear combination of $X$, $Y$, such that $$ \exp(X) \exp(Y)=\exp(W) $$ The derivation of such closed forms follows, in part, …
Lie-Algebraic Bounds on Quantum Control Time via the Baker-Campbell-Hausdorff Formula
NICT（情报通信研究机构）；Faculty of Informatics, Shizuoka University（静冈大学信息学部）；Faculty of Engineering, University of Toyama（富山大学工学部）；Department of Chemistry and Materials Science, Osaka Metropolitan University（大阪公立大学化学材料科学系）
A universal disentangling formula (such as the Baker-Campbell-Hausdorff one) for coherent states of Perelomov's type ($ |z \ra = \exp (z\Adag -\bar{z}A) |0 \ra $) which are defined for generalized oscillator algebras is given.
Using Baker-Campbell-Hausdorff relations, the squeeze and harmonic-oscillator time-displacement operators are given in the form $\exp[δI] \exp[α(x^2)]\exp[β(x\partial)] \exp[γ(\partial)^2]$, where $α$, $β$, $γ$, and $δ$ are explicitly determined. Applications are discussed.
A closed expression to the Baker-Campbell-Hausdorff (B-C-H) formula in SO(4) is given by making use of the magic matrix by Makhlin. As far as we know this is the {\bf first nontrivial example} on (semi-) simple Lie groups summing up all terms in the B-C-H expansion.
Using the so(2,1) Lie algebra and the Baker, Campbell and Hausdorff formulas, the Green's function for the class of the confluent Natanzon potentials is constructed straightforwardly. The bound-state energy spectrum is then determined. Eventually, the three-dimensional harmonic potential, the three-dimensional Coulomb potential and the Morse potential may all be considered as particular cases.
It is shown that a SU(1,1) algebra may be used to provide a unified description of the simple hamonic oscillator and the angular momentum algebras and a class of other semi-infinite algebras. A normal ordered representation of a Unitary operator $U$ constructed from the generators of a SU(1,1) algebra, which is a generalisation of the Baker- Campbell - Hausdorff relation for Lie algebras, is given. It is shown that the normal ordered representatiion of $U$ may be used to calculate expectation values which are functions of the parameters used to construct the operator. The functions so construc…
