Baker-Campbell-Hausdorff Formula in su(2)

The Problem of Composition

Let's begin by formalizing the problem. A Lie group element $g$ near the identity can be written as the exponential of a Lie algebra element $X$, $g=e^X$. The product of two group elements $g_1=e^X$ and $g_2=e^Y$ is another group element $g_3=g_1{g}_2$. Since $g_3$ is also in the group, it must also be representable as an exponential of some algebra element $Z$, so $e^X{e}^Y=e^Z$. The central question is: what is $Z$ in terms of $X$ and $Y$?

If $X$ and $Y$ commute, $[X,Y]=0$, the answer is trivial: $Z=X+Y$. But for non-abelian groups like SO(3) and SU(2), the generators do not commute. The BCH formula provides the answer, expressing $Z$ as an infinite series involving nested commutators of $X$ and $Y$.

The Baker-Campbell-Hausdorff Formula

To see how the commutator naturally arises, we can perform a direct expansion. The process involves taking the logarithm of the product of two exponentials. A full proof is quite involved, but a sketch of the first few terms provides excellent intuition.

The following video provides a clear, step-by-step calculation for the first few terms of the BCH formula by expanding the expressions for the exponential and the logarithm.

As the video demonstrates, by expanding $e^A$, $e^B$, and the logarithm $\log (1+x)$ as power series, we find that the product $e^A{e}^B$ can be written as $e^Z$ where:

The appearance of the commutator $[A,B]=AB-BA$ is the first-order correction that accounts for the non-commutativity of the algebra.

The full Baker-Campbell-Hausdorff formula gives the complete series for $Z$. While there are several ways to write it, a common form for the first few terms is:

The most remarkable feature of this formula, as hinted at in the video and stated in the resource below, is that every term in this infinite series can be expressed as a linear combination of $X$, $Y$, and their nested commutators.

This confirms the central principle we've been building towards: the structure of the Lie algebra, captured entirely by its commutation relations, completely determines the group's multiplication law in a neighborhood of the identity. The BCH formula is the explicit machine that translates the algebra's structure into the group's structure.

Application: Composition of SU(2) Rotations

Let's now apply this to our main example, SU(2). A general SU(2) transformation corresponding to a rotation by angle $\alpha$ about an axis $\hat{n}$ is given by a unitary operator:

The generators are $T_k=-i{\sigma }_k/2$, which satisfy the $\mathfrak{su}(2)$ Lie algebra $[T_i,T_j]={ϵ}_{ijk}{T}_k$.

Consider two successive rotations: first by ${\alpha }_1$ about ${\hat{n}}_1$, then by ${\alpha }_2$ about ${\hat{n}}_2$. The combined operation is $U_{tot}=U({\hat{n}}_2,{\alpha }_2)U({\hat{n}}_1,{\alpha }_1)$. We want to find the single equivalent rotation $U({\hat{n}}_{tot},{\alpha }_{tot})$. Using the BCH formula, we have $e^Y{e}^X=e^Z$, with $X=-i\frac{{\alpha }_1}{2}{\hat{n}}_1\cdot \vec{\sigma }$ and $Y=-i\frac{{\alpha }_2}{2}{\hat{n}}_2\cdot \vec{\sigma }$.

Let's look at the infinitesimal case, where ${\alpha }_1,{\alpha }_2\ll 1$. The BCH formula to first order gives $Z\approx X+Y$. The next correction is $\frac{1}{2}[Y,X]$. Let's compute this commutator:
\begin{align*}
[Y, X] &= \left[ -i\frac{\alpha_2}{2} (\hat{n}2 \cdot \vec{\sigma}), -i\frac{\alpha_1}{2} (\hat{n}1 \cdot \vec{\sigma}) \right] \
&= \left(-\frac{i}{2}\right)^2 \alpha_1 \alpha_2 [n{2j}\sigma_j, n{1k}\sigma_k] \
&= -\frac{\alpha_1 \alpha_2}{4} n_{2j}n_{1k} [\sigma_j, \sigma_k] \
&= -\frac{\alpha_1 \alpha_2}{4} n_{2j}n_{1k} (2i \epsilon_{jkl} \sigma_l) \
&= -i \frac{\alpha_1 \alpha_2}{2} (n_{2j}n_{1k} \epsilon_{jkl}) \sigma_l \
&= -i \frac{\alpha_1 \alpha_2}{2} (\hat{n}_2 \times \hat{n}_1) \cdot \vec{\sigma}
\end{align*}
So, $Z\approx X+Y+\frac{1}{2}[Y,X]=-i\frac{{\alpha }_1}{2}{\hat{n}}_1\cdot \vec{\sigma }-i\frac{{\alpha }_2}{2}{\hat{n}}_2\cdot \vec{\sigma }+i\frac{{\alpha }_1{\alpha }_2}{4}({\hat{n}}_1\times {\hat{n}}_2)\cdot \vec{\sigma }$.

This shows that for infinitesimal rotations, the resulting axis is not just the sum of the initial vectors, but includes a correction proportional to their cross product—a direct consequence of non-commutativity. This is the well-known result for the composition of infinitesimal rotations.

For finite rotations, the full BCH series for SU(2) can be summed into a closed form. While the general derivation is complex, we can achieve the same result by directly multiplying the SU(2) matrices, using the properties of the Pauli matrices. The following resource does exactly this.

From the derivation, the product of two rotations, $U_{tot}=U({\hat{n}}_2,{\beta }_2)U({\hat{n}}_1,{\beta }_1)$, is equivalent to a single rotation $U({\hat{n}}_{12},{\beta }_{12})$ where the new angle ${\beta }_{12}$ and axis ${\hat{n}}_{12}$ are given by:

These equations are the exact, closed-form result of the Baker-Campbell-Hausdorff formula applied to $\mathfrak{su}(2)$. They beautifully illustrate the core concept: the composition law for finite rotations is determined by the dot product (${\hat{n}}_1\cdot {\hat{n}}_2$) and the cross product (${\hat{n}}_1\times {\hat{n}}_2$), which are themselves related to the anticommutator and commutator of the underlying algebra generators, respectively.

Conclusion

In this lesson, we have formally bridged the gap between a Lie algebra and its corresponding Lie group's multiplication law.

Key Takeaways:

• The Baker-Campbell-Hausdorff (BCH) formula provides an explicit expression for $Z$ in $e^X{e}^Y=e^Z$, where $Z$ is an infinite series of nested commutators of the algebra elements $X$ and $Y$.
• The formula's leading terms are $Z=X+Y+\frac{1}{2}[X,Y]+\dots$. The commutator term is the essential correction for non-abelian groups.
• The BCH formula demonstrates that the Lie algebra's commutator structure fully determines the group's local multiplication properties.
• For the $\mathfrak{su}(2)$ algebra, which governs rotations, the BCH formula has a closed-form solution. We showed how to use this to find the single equivalent rotation axis and angle that results from composing two arbitrary finite rotations.

This concludes our exploration of the fundamental properties of Lie groups and Lie algebras. We have developed the tools to define these structures, relate them via the exponential map, and compose their elements.

In the next module, we will move on to one of the most powerful aspects of group theory in physics: Representation Theory. For SU(2), this will allow us to classify all possible types of angular momentum and spin allowed by quantum mechanics, providing the framework for understanding everything from atomic spectra to particle physics.