2026-10-01 · chapter Yang–Mills and the mass gapPart 1 Mathematical preparationLie groups and Lie algebras

Lie groups and Lie algebras — reading a curved group through the space of tangents at the identity

The SO(3) and SU(2) of 1-02 are groups whose elements move continuously with real parameters. As spaces they are curved (SU(2) is the three-dimensional sphere), so we take a first-order approximation only very near the identity. As a metaphor, even on the curved surface of the earth, the map at your feet can be drawn on a plane. To be exact, the Lie algebra is the flat vector space that collects all velocities $g'(0)$ of curves $g(t)$ in the group with $g(0)=I$.

The Lie algebra of the plane rotations SO(2) is the real multiples of $J$ (1-01 §08). The order of multiplication survives in the second-order term (§04), and from that information alone the multiplication near the identity can be recovered (§07). In Yang–Mills theory, gauge transformations take values in the group, and the gauge field in the Lie algebra (3-02).

Definition A subgroup $G$ of $\mathrm{GL}(n,\mathbb{C})$ (the group of invertible complex $n\times n$ matrices) that is closed (if a sequence of elements converges to an invertible matrix, the limit is also an element of $G$) is called a matrix Lie group Known. Its Lie algebra is

and $X\mapsto e^X$ is called the exponential map. $X$ is the velocity of the curve $e^{tX}$ starting at the identity ((a) of 1-01 §08). The condition is "for all $t$", and $e^X\in G$ alone is not enough. For example, for the group $G=\{I\}$ consisting only of the identity and $X=2\pi J$, $e^X=I\in G$, but $e^{tX}$ for $t=\tfrac14$ is a 90° rotation and is not in $G$.

Example: U(n). Differentiating the condition $e^{tX^\dagger}e^{tX}=I$ in $t=0$ gives $X^\dagger=-X$. Conversely, in that case $e^{tX}$ is unitary (1-01 §09). From $\det e^{tX}=e^{t\operatorname{tr}X}$, the condition of determinant 1 becomes $\operatorname{tr}X=0$.

The Lie algebra is a real vector space (multiplying by $i$ makes it Hermitian, which falls outside), and its dimension is called the dimension of the group (the number of "real parameters" in the table of 1-02). The same computation gives the following table Known.

Near the identity the exponential map is one-to-one with a continuous inverse Known. The inverse map is the logarithm series $\log(I+A)=A-\tfrac{A^2}{2}+\tfrac{A^3}{3}-\cdots$ ($\lVert A\rVert\lt 1$), and a nearby element can be written uniquely as $e^X$ with a small $X$ (exponential coordinates).

Far away, SU(2) reaches every element by the $U(\theta,n)=\exp(-\tfrac{i\theta}{2}\,n\cdot\sigma)$ of 1-02 (surjective), and shifting θ by 4π gives the same element, so it is not one-to-one. For a compact connected group the exponential map is surjective Known.

Detour: SL(2, ℝ) has elements the exponential does not reach

Multiplying four $e^{tX}=I+tX+\tfrac{t^2}{2}X^2+O(t^3)$ together, the first-order terms cancel and the second order remains.

The left-hand side is called the group commutator, and $[X,Y]$ the commutator. If $[X,Y]\ne0$, a path that ought to close after four moves fails to close by an amount of order $t^2$.

Example: parallel parking. Write the group SE(2) of "points with an orientation" in the plane (position $(x,y)$, orientation $\alpha$) as 3×3 matrices.

$J$ is the rotation with the $J$ of §01 in the upper left, and $P$, $P_\perp$ are the translation matrices with only the (1, 3) and (2, 3) entries equal to 1. Multiplying $e^{tP}$ on the right means "move forward $t$ in the direction you face", and $e^{tJ}$ means "turn left by $t$ on the spot". As a metaphor, it is a trolley that can turn on the spot but cannot move sideways. To be exact, by (1), "forward → turn left → back → turn right" equals $e^{t^2[P,J]}=e^{-t^2P_\perp}$ (a translation of $t^2$ to the right) up to third order, and the exact displacement is $(t(1-\cos t),\,-t\sin t)$.

The reason for the θ² of 1-02. In 1-02 §04, the mismatch between rotations about the x and y axes applied in opposite orders had an angle close to θ² and an axis almost along −z. With the generators $(L_a)_{bc}=-\varepsilon_{abc}$ ($\varepsilon_{123}=1$ the totally antisymmetric symbol), $R_x=e^{\theta L_1}$, $R_y=e^{\theta L_2}$, $[L_1,L_2]=L_3$. The mismatch $D=R_yR_xR_y^{-1}R_x^{-1}$ has the form of (1), so $D=I-\theta^2L_3+O(\theta^3)$, a rotation by the angle θ² about the −z axis. ComputedThe difference between $\log D/\theta^2$ and $-L_3$ was 0.12 at θ = 10° and 0.012 at 1°.

The outline of why $\mathfrak{g}$ is a vector space closed under commutators Known. Sums: the right-hand side of the Lie product formula $e^{t(X+Y)}=\lim_{m\to\infty}(e^{tX/m}e^{tY/m})^m$ (§07) is a sequence of elements of $G$, and $G$ is closed. Commutators: from $ge^{sY}g^{-1}=e^{s\,gYg^{-1}}$, $e^{tX}Ye^{-tX}\in\mathfrak{g}$; differentiating this in $t=0$ gives $XY-YX$.

In the definition that forgets matrices, a Lie algebra is a pair of a real vector space and a bilinear operation $[\cdot,\cdot]$ satisfying the following (the second is the Jacobi identity).

The cross product in three dimensions is also a Lie algebra; $L(v)=\sum_av_aL_a$ satisfies $L(v)w=v\times w$, $[L(a),L(b)]=L(a\times b)$.

The Lie algebra sees only the local. Differentiating the double cover $\rho:\mathrm{SU}(2)\to\mathrm{SO}(3)$ of 1-02 at the identity gives a one-to-one linear map $\mathfrak{su}(2)\to\mathfrak{so}(3)$ that preserves commutators (in §06 we see that the structure constants agree). The kernel $\{\pm I\}$ is two separated points and does not appear near the identity; the difference between 720° and 360° is global and is not reflected in the Lie algebra.

The adjoint. The group acts on the Lie algebra by $\operatorname{Ad}_gX=gXg^{-1}$, and its derivative is $\operatorname{ad}_XY=[X,Y]$. $\operatorname{Ad}_{e^X}=e^{\operatorname{ad}_X}$ holds Known. The gauge field changes under gauge transformations through Ad (3-02). As a representation it is treated in 1-04.

Choosing a basis $X_1,\dots,X_d$, one can write $[X_a,X_b]=\sum_cf_{ab}{}^{c}X_c$, and the coefficients $f$ are called the structure constants. Antisymmetry and the Jacobi identity become conditions on $f$ Known.

• $\mathfrak{so}(3)$: $[L_a,L_b]=\sum_c\varepsilon_{abc}L_c$.
• $\mathfrak{su}(2)$: with $X_a=-\tfrac{i}{2}\sigma_a$ (the Pauli matrices, 1-02 §05), $[X_a,X_b]=\sum_c\varepsilon_{abc}X_c$ Computed. The constants are the same, so $X_a\mapsto L_a$ is the isomorphism of §05, and the $U(\theta,n)$ of 1-02 is $\exp(\theta\sum_an_aX_a)$.

In the physicists' convention (written $e^{-itT}$), one writes $[T^a,T^b]=if^{abc}T^c$ with Hermitian $T^a$, omitting the summation sign over the repeated index $c$. Setting $X_a=-iT^a$ gives $[X_a,X_b]=f^{abc}X_c$, so the real $f$ is the same in the two conventions. Normalising with $\operatorname{tr}(T^aT^b)=\tfrac12\delta^{ab}$ for $\mathfrak{su}(N)$, $f^{abc}$ is totally antisymmetric (the sign changes when any two indices are swapped). Multiplying the commutator formula by $T^c$ and taking the trace gives $\operatorname{tr}([T^a,T^b]T^c)=\tfrac{i}{2}f^{abc}$, so

The trace is cyclic ($\operatorname{tr}(ABC)=\operatorname{tr}(CAB)$ from $\operatorname{tr}(PQ)=\operatorname{tr}(QP)$), so the right-hand side does not change under the cyclic permutation of indices $a\to b\to c\to a$, and changes sign under swapping $a$ and $b$. Combining cyclic permutations with one swap produces every permutation, so it is totally antisymmetric. The $f^{abc}$ of $\mathfrak{su}(3)$ is derived in 1-04.

This is the formula that writes the difference of $e^Ae^B\ne e^{A+B}$ seen in 1-01 §08 in terms of commutators. When $X,Y$ is small enough, $e^Xe^Y=e^Z$ gives

The terms up to third order are in Hall and Kirillov Known. The fourth-order term appears in the paper of Casas and Murua in the form $\tfrac1{24}[X,[Y,[Y,X]]]$, and by the Jacobi identity it equals the form of (2) Known. The fourth-order coefficient was also checked numerically on this machine (Figure 3 and the table below) Computed.

Everything beyond $X+Y$ is nested commutators, and the coefficients do not depend on the group Known. So the multiplication near the identity is determined by the Lie algebra. A connected group is generated by products of elements near the identity, so if the group is known as a space, the operation is determined too Known. However, the Lie algebra alone does not determine the group itself. SU(2) and SO(3) have the same Lie algebra but are different as groups. The difference lies in the global part seen in §05 (the kernel $\{\pm I\}$).

For $X=L(1,0,0.3)$, $Y=L(0.5,1,0)$ of $\mathfrak{so}(3)$ multiplied by $t$, we measured the error of (2) truncated at order $k$. If the formula is right, the error is proportional to $t^{k+1}$.

The values 0.5274, 0.1017 and 0.0220 in the last row agree with the size of the next term (the $\tfrac12[X,Y]$, third-order and fourth-order terms), and the coefficients fitted by a polynomial in $t$ also agreed with (2) to within $5\times10^{-9}$ Computed.

The Lie product formula. From (2), $(e^{X/m}e^{Y/m})^m=\exp\bigl(X+Y+\tfrac1{2m}[X,Y]+O(m^{-2})\bigr)$, and the error decreases roughly as $1/m$. ComputedFor the $X,Y$ above, the errors at $m$ = 10, 100, 1000 were 4.57×10⁻², 4.57×10⁻³, 4.57×10⁻⁴ (multiplied by $m$, all are about 0.457). This formula has an extension to the case where $X,Y$ are unbounded operators (on an infinite-dimensional Hilbert space, 1-10), called the Trotter product formula Known. That slicing time finely in the path integral takes this form is seen in 2-07.

The Yang–Mills action needs a "squared length" on the Lie algebra that does not change under $\operatorname{Ad}_g$ (§09). The candidate that can be made from the Lie algebra alone is the Killing form.

$B$ is symmetric, and invariant in the sense $B(\operatorname{Ad}_gX,\operatorname{Ad}_gY)=B(X,Y)$, $B([Z,X],Y)+B(X,[Z,Y])=0$ Known. For $\mathfrak{su}(2)$, $\operatorname{ad}_{X_a}=L_a$, so $B(X_a,X_b)=\operatorname{tr}(L_aL_b)=-2\delta_{ab}$: it is negative definite.

A subspace $\mathfrak{i}$ is an ideal if $[\mathfrak{g},\mathfrak{i}]\subset\mathfrak{i}$. Simple means non-commutative with no ideals other than 0 and $\mathfrak{g}$, and semisimple means a direct sum of simple ones (the standard definition takes another form, known to be equivalent to this: Kirillov Definition 6.19, Corollary 6.43). The following are known Known. (i) Cartan's criterion: semisimple ⇔ $B$ is non-degenerate. (ii) For the Lie algebra of a compact group, $B$ is negative semidefinite, and the degenerate directions are exactly the centre (the elements that commute with everything). Conversely, if it is semisimple and $B$ is negative definite, it is the Lie algebra of a compact group.

ComputedThe matrices of ad were built from the structure constants and the signs of the eigenvalues of $B$ counted, confirming $B=2n\operatorname{tr}(XY)$ ($\mathfrak{su}(2)$–$\mathfrak{su}(4)$) and $B=(n-2)\operatorname{tr}(XY)$ ($\mathfrak{so}(3)$–$\mathfrak{so}(5)$). The deviation from invariance was at most 1.3×10⁻¹². That $\mathfrak{so}(4)$ is not simple cannot be read from the signs.

The symbols $d$ (exterior derivative), $\wedge$ (wedge product), $*$ (Hodge dual) and the integration of forms are defined in 1-06. Here we only look at where Tr appears.

Jaffe and Witten's description of the problem replaces U(1) by a compact group G, takes the curvature to be $F=dA+A\wedge A$, and writes of the Tr in the expression $L=\tfrac{1}{4g^2}\int\operatorname{Tr}F\wedge{*F}$, which it calls the Lagrangian (integrated over spacetime, so in this book's words it is the action, 2-01), that it is “an invariant quadratic form on the Lie algebra of G”. In this chapter's words:

• Values of the field: the gauge field has $A_\mu=\sum_aA_\mu^aX_a\in\mathfrak{g}$ at each point and in each direction ($\mu=0,1,2,3$ is the number of a direction of spacetime). The components of $A\wedge A$ can be written with the commutator $[A_\mu,A_\nu]$, which vanishes for U(1). The coefficient in front depends on the convention for components ($F=\tfrac12F_{\mu\nu}dx^\mu\wedge dx^\nu$ and so on), so the exact form is left to 3-02.
• Invariant: gauge transformations act through Ad, so the action needs a form invariant in the sense of §08.
• Compact: for $\mathfrak{su}(N)$, if $X\ne0$ then $-\operatorname{tr}X^2=\sum_{ij}|X_{ij}|^2\gt 0$. The energy is bounded below as a sum of squares; if signs are mixed, directions of negative energy appear Physics (3-03).
• Simple: on a compact simple Lie algebra ($\mathfrak{su}(N)$ and so on), the invariant symmetric bilinear form is unique up to a constant multiple Known. This is because it stays simple under complexification (extending real coefficients to complex; for $\mathfrak{su}(N)$, $\mathfrak{sl}(N,\mathbb{C})$), and on a complex simple Lie algebra uniqueness follows from Schur's lemma. "Simple" alone is not enough. $\mathfrak{sl}(2,\mathbb{C})$ regarded as a real Lie algebra is simple, but the real and imaginary parts of its Killing form are separate invariant forms. The constant of the unique form is absorbed into the coupling constant $g$ (as is the difference in the normalisation of Tr). For $G_1\times G_2$ a constant enters for each factor, and the Standard Model SU(3)×SU(2)×U(1) has three coupling constants (3-07).

Next is 1-04 SU(2) and SU(3): the Pauli matrices and Gell-Mann matrices, the adjoint representation, the list of compact simple groups.

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