Simple definition for the generator of an infinitesimal transformation

Question

Studying quantum mechanics, or QFT, the concept of generator $G$ of an infinitesimal transformation $T$ keeps showing up. My problem is that I don't have in mind a solid (dare I say "rigorous") definition for it. Sometimes I have seen it defined as this:

$$T=\mathbb{I}+G\varepsilon$$

sometimes instead like this:

$$T=\mathbb{I}+iG\varepsilon$$

and sometimes like this:

$$T=\mathbb{I}-\frac{i}{\hbar}G\varepsilon$$

or like this:

$$T=e^{iG}$$

or this

$$T=e^{iG\varepsilon}$$

and so on.

On the other hand I know that the concept of generator is deeply connected with Lie groups and Lie algebra, of which I know something about, but not nearly enough to understand the rigorous definitions of generator that I may find on a math textbook. I don't want to spend a week studing Lie algebra right now, a definition of generator limited to the physics context will do just fine, unfortunately I saw that often knowledge of this definition is taken for granted.

I would like a clear definition of the concept of generator of an infinitesimal transformation that doesn't relay on previous knowledge of Lie theory, possibily also citing a bibliographical source.

Answer 1

There is no unique definition in the sense that all the above are valid but absorb various factors so the resulting exponential is “simpler” or the generators satisfy some specific condition.

Physicists tend to prefer the second definition because it makes the generator hermitian when $\epsilon\in\mathbb{R}$, and we like hermitian operators.

In quantum mechanics people will often absorb the $\hbar$ in the definition, especially when dealing with angular momentum problem, so the commutators don’t have $\hbar$’s floating around. The leave the $\hbar$ in so the eigenvalues are proportional to $\hbar$. For the same reason one often has the $\epsilon$ explicit so that it doesn’t appear in commutation relations. These factors amount to just rescaling the generators and have no conceptual consequences. (There are exceptions, such as algebra contractions, where scaling are important but that’s a niche application.)

Mathematicians prefer the first definition because they often work with transformations over specific fields, and they don’t want the imaginary unit to confuse things. It means that many of their generators - when used in applications - are anti-hermitian. This doesn’t seem very natural in physics until you deal with real forms that are non-compact (v.g. the Lorentz group).

So various authors will have various definitions, depending on the target audience. I rather like the books by Cornwell, Group theory in physics, which look rather formal but do a good job of bridging the mathematical and physics perspectives. I believe vol II is on continuous groups and it’s all there, including a discussion of real forms where the inclusion or not of an “i” in the definition makes a difference.

As an example, consider the group of (real) orthogonal matrices with determinant $1$ (real rotations) in $3d$. A rotation about $\hat z$ is given by the matrix $$ R_z(\theta)=\left(\begin{array}{ccc} \cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{array}\right) $$ If we take the definition $R_z(\theta)=e^{\theta L_z}=\mathbb{1}+\theta L_z+\ldots$, we have $$ L_z=\frac{d}{d\theta}R_z(\theta)\Bigl\vert_{\theta=0}= \left(\begin{array}{ccc} 0&-1&0\\ 1&0&0\\ 0&0&0\end{array}\right) $$ which is antihermitian. Its eigenvalue are $\pm i,0$ and the operator cannot be diagonalized over the reals, i.e. not all eigenvectors have only real coefficients. The other generators $L_x$ and $L_y$ are likewise antisymmetric matrices and in fact representing generators of rotation by antisymmetric matrices is common in classical mechanics. (This is related to how one may think of angular frequency as a vector $\vec\omega$.)

If we use instead $R_z(\theta)=e^{i\theta L_z}$ then $L_z$ becomes hermitian, with eigenvalues $\pm 1,0$. Dividing by $\hbar$ in the definition will give eigenvalues $\pm \hbar,0\hbar$. It is possible to diagonalize $L_z$ over the complex field but at the cost of using complex functions as eigenfunctions. (This is why spherical harmonics are complex.) The operators $\hat L_\pm$ are complex linear combinations of $L_x$ and $L_y$ and a lot of representation theory is based on the laddering action of $L_\pm$.

SO(3) is compact and (as discussed in Cornwell) it's not an issue to do all the theory using the complex extension, where $L_\pm$ are well-defined, and then come back to real matrices (we're excluding $SU(2)$ here so only representations with integer angular momentum are possible.)

The situation changes for $SO(3,1)$ (or the Lorentz group). If you allow for complex combinations of the generators, then the resulting algebra $so(3,1)\sim su(2)\oplus su(2)$ but because $SO(3,1)$ is non-compact there's all kinds of issues in going between real matrices of $SO(3,1)$ and matrices in $SU(2)\otimes SU(2)$. If anything, the finite dimensional representations of $SO(3,1)$ are not unitary, which is not so good in physics. In particular, the adjoint representation (i.e. the representation spanned by the generators under the operation of commutation) is irreducible under the reals but reducible as a sum of two $su(2)$ irreps over the complex field. There's an obscure paper which discusses this distinction as some length and shows how the "i"'s come or not the case of finite or infinite-dimensional representations.

My explanation is a bit butchered from a formal perspective, but it should be enough to give you a feel for situation.