2026-10-01 · chapter Yang–Mills and the mass gapPart 1 Mathematical preparationlinear algebra

Vector spaces and linear maps — reading a matrix as "a rule that moves vectors", and lining up eigenvalues, the exponential, Hermitian and unitary

In Yang–Mills theory, matrices go where numbers would be expected. They have three roles.

If the size of the matrices used for gauge transformations is $N$ ($N \times N$ matrices), electromagnetism is the case $N = 1$, and a gauge transformation just multiplies by a number $e^{i\alpha}$ of absolute value 1 (2-03). Products of numbers do not depend on order, so in the absence of matter the equations of the gauge field itself (Maxwell's equations) remain linear. For $N \ge 2$ order matters, and products of gauge fields with each other appear in the equations even without matter (3-02). This chapter lays that foundation in finite dimensions.

A vector space is a collection in which addition and scalar multiplication can be done, and the same rules as for numbers hold, such as the associative, commutative and distributive laws. Taking scalars from $\mathbb{R}$ gives a real vector space, from $\mathbb{C}$ a complex vector space. When every vector can be written uniquely as $v = c_1v_1 + \cdots + c_nv_n$, $v_1, \dots, v_n$ is called a basis and the number $n$ the dimension. Examples are $\mathbb{R}^n$, $\mathbb{C}^n$, all $n \times n$ matrices (dimension $n^2$), and all continuous functions on an interval (infinite-dimensional, 1-10).

An example this book returns to again and again is the set of all traceless 2×2 Hermitian matrices. Hermitian means $H^\dagger = H$, where $\dagger$ is the operation of transposing and taking the complex conjugate of each entry (§05, §06). Sums and real multiples are Hermitian, but multiplying by $i$ gives $(iH)^\dagger = -iH$, which falls outside. So it is a real vector space even though its entries are complex, and its dimension is 3.

The three on the right reappear in 1-02 §05 and 1-04 as the Pauli matrices $\sigma_3, \sigma_1, \sigma_2$, and the values of the $\mathrm{SU}(2)$ gauge field lie in this space (times $i$).

A linear map is a map that preserves sums and scalar multiples ($L(u+v) = Lu + Lv$, $L(cv) = c\,Lv$). Once a basis is fixed it can be written as a matrix, and column $j$ is where the basis vector $e_j$ goes. Composition is the matrix product $(AB)v = A(Bv)$, and unlike numbers, swapping the order generally changes it.

The difference $[A, B] = AB - BA$ is called the commutator. In this example it is $[A, B] = \operatorname{diag}(1, -1)$, the first of the basis of §02. How commutators become the skeleton of Lie algebras is treated in 1-03.

The determinant $\det A$ is the signed scaling factor of area (volume), and the trace $\operatorname{tr} A$ is the sum of the diagonal entries. From $\det(AB) = \det A \det B$ and $\operatorname{tr}(AB) = \operatorname{tr}(BA)$, neither changes under a change of basis $A \mapsto P^{-1}AP$; they are properties of the map itself. Example: the rotation $R(\theta) = \left(\begin{smallmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{smallmatrix}\right)$ has $\det R = 1$, $\operatorname{tr} R = 2\cos\theta$.

A map changes the orientation of most vectors, but it may have directions that it only stretches or shrinks without changing their orientation. A vector in such a direction is an eigenvector, and the stretching factor is the eigenvalue.

For 2×2, $\lambda^2 - (\operatorname{tr} A)\lambda + \det A = 0$. For a real 2×2 matrix, if the discriminant $D = (\operatorname{tr} A)^2 - 4\det A$ is positive there are two real eigendirections, if negative there are none (the eigenvalues are a complex pair), and if 0 there is a double root (with a double root there is only one eigendirection, or, for a scalar matrix $\lambda I$, every direction). With $P$, made by lining up $n$ independent eigenvectors, $P^{-1}AP$ becomes a diagonal matrix (diagonalisation).

That $v$ and $Av$ lie on the same line is the same as the determinant $\det[v, Av]$ of the two side by side being 0. This can be checked in Figure 1.

ComputedThe numbers in this figure were computed on this machine (the table by NumPy's eigenvalue computation). Matrices separate rows with ";". Clicking a row of the table below puts that matrix into the figure.

For a symmetric matrix the eigendirections are orthogonal (the reason is in §06). The shear has only one eigendirection and cannot be diagonalised. The rotation has no real eigendirection, and $\det[v, Av]$ is $\sin 60^\circ \approx 0.866$ in every direction.

Detour: rotations have no real eigenvectors

The inner product measures lengths and angles. If we take $\sum_k u_kv_k$ directly with complex entries, the "squared length" of $v = (1, i)$ becomes $1 + i^2 = 0$. So we put a complex conjugate on one side ($\overline{z}\,z = |z|^2 \ge 0$).

This book follows the physicists' convention and takes it linear in the second argument ($\langle cu, v\rangle = \overline{c}\langle u, v\rangle$). Many mathematics textbooks use the opposite convention (for example Axler's definition 6.2), and the conclusions do not change. If $\langle u, v\rangle = 0$ the vectors are orthogonal, and a basis of mutually orthogonal vectors of length 1 is an orthonormal basis.

The adjoint $A^\dagger$ of $A$ is the matrix obtained by transposing and taking the complex conjugate of the entries (in mathematics $A^*$). It moves $A$ to the other side inside the inner product.

A matrix with $H^\dagger = H$ is a Hermitian matrix; with real entries it is a symmetric matrix. In quantum mechanics measured values (energy and so on) are represented by Hermitian operators (2-05). The key is two properties.

The eigenvalues are real: if $Hv = \lambda v$ and $\lVert v\rVert = 1$, then $\lambda = \langle v, Hv\rangle = \langle Hv, v\rangle = \overline{\lambda}$. Eigenvectors of different eigenvalues are orthogonal: if $Hu = \mu u$, then $\lambda\langle u, v\rangle = \langle u, Hv\rangle = \langle Hu, v\rangle = \overline{\mu}\langle u, v\rangle = \mu\langle u, v\rangle$ ($\mu$ is real), and since $\mu \ne \lambda$, $\langle u, v\rangle = 0$.

Pushing this further gives the spectral theorem Known. A Hermitian matrix has an orthonormal basis of eigenvectors, and the eigenvalues $\lambda_1 \le \cdots \le \lambda_n$ are real. With the unitary matrix $U$ (§07) that lines up the basis, $U^\dagger HU$ becomes real diagonal.

The difference between the smallest eigenvalue and the next is called the gap. In quantum mechanics the smallest eigenvalue $E_0$ is the energy of the ground state, and $E_1 - E_0$ the energy needed for the first excitation. In finite dimensions the gap can always be computed. The problem is in infinite dimensions: whether the gap stays positive after the limit (1-10, 4-11).

Example: the matrix of §02 has $\operatorname{tr} H = 0$ and $\det H = -(x^2 + y^2 + z^2)$, so its eigenvalues are $\pm\sqrt{x^2 + y^2 + z^2}$, and the gap is 0 only for $x = y = z = 0$. Figure 2 is the case $y = c$, $z = 0$.

ComputedThe curves are drawn on the spot from $\pm\sqrt{x^2 + c^2}$, and the table gives the values of NumPy's Hermitian eigenvalue computation (the two agree to the digits shown). Taking $x$ to $-x$ swaps the weights. Clicking a row of the table puts that $x$ and $c = 0.3$ into the figure.

At $c = 0$ the lines $\pm x$ cross. If $c \ne 0$ the curves only come close, a minimal gap $2c$ remains, and past that point the content of the lower eigenvector swaps.

Detour: levels avoid each other

A matrix that preserves the inner product is a unitary matrix. It is a transformation that changes neither lengths nor angles; with real entries it is an orthogonal matrix (rotations and reflections).

Left to right follows from $\langle Uu, Uv\rangle = \langle u, U^\dagger Uv\rangle$. It is the same as the columns forming an orthonormal basis, and $U^{-1} = U^\dagger$. The eigenvalues are $|\lambda| = 1$, $|\det U| = 1$. Products and inverses are unitary too, so the whole set $\mathrm{U}(n)$ and the part $\mathrm{SU}(n)$ with $\det U = 1$ form "groups" (1-02).

Example: $\mathrm{U}(1) = \{e^{i\alpha}\}$ is the unit circle. $\left(\begin{smallmatrix} a & -\overline{b} \\ b & \overline{a} \end{smallmatrix}\right)$, made from complex numbers with $|a|^2 + |b|^2 = 1$, is an element of $\mathrm{SU}(2)$ (that every element has this form is in 1-02 §05).

Unitary matrices can also be diagonalised. The basis is the complex spectral theorem Known. A matrix satisfying $AA^\dagger = A^\dagger A$ (a normal matrix) can be diagonalised by a unitary matrix. A unitary matrix has $U^\dagger U = UU^\dagger = I$, so it is one (Hermitian matrices are too), and it can be written $U = V\operatorname{diag}(e^{i\theta_1}, \dots, e^{i\theta_n})V^\dagger$ ($V$ unitary, $\theta_k$ real).

Unitary matrices appear in gauge theory because matter fields take values in $\mathbb{C}^N$ and the squared length $\psi^\dagger\psi$ appears in the action. Gauge transformations are chosen to preserve it (3-01). On the lattice, the gauge field itself is a unitary matrix on each edge (5-01).

The series of the exponential uses only multiplication and addition, so it can be used for matrices as it is.

Convergence. The operator norm $\lVert X\rVert = \max_{\lVert v\rVert = 1}\lVert Xv\rVert$ (the largest factor by which it multiplies a vector of length 1) satisfies $\lVert XY\rVert \le \lVert X\rVert\lVert Y\rVert$, so each term is at most $\lVert X\rVert^k/k!$ and the sum at most $e^{\lVert X\rVert}$. The error of the partial sum $S_N(X)$ up to $k = N$ is as follows.

Properties. (a) $\frac{d}{dt}e^{tX} = Xe^{tX}$. Hence $x(t) = e^{tX}x_0$ is the solution of $x' = Xx$, $x(0) = x_0$. (b) $e^{(s+t)X} = e^{sX}e^{tX}$, in particular $(e^X)^{-1} = e^{-X}$. (c) $Pe^XP^{-1} = e^{PXP^{-1}}$. (d) $\det e^X = e^{\operatorname{tr} X}$. If diagonalisable, $\prod_k e^{\lambda_k} = e^{\sum_k \lambda_k}$ follows from (c); in general it can be shown by triangularisation Known. Every complex square matrix $X$ can be brought to upper triangular form $V^\dagger XV$ by a unitary matrix $V$ (Schur's theorem), and the exponential of an upper triangular matrix is upper triangular with diagonal entries $e^{\lambda_k}$.

Caution. If $XY = YX$, then $e^{X+Y} = e^Xe^Y$, but not in general. The $A, B$ of §03 have $A^2 = B^2 = 0$, so $e^A = I + A$, $e^B = I + B$, and

ComputedAll three differ. A formula that writes the difference in terms of commutators is the subject of 1-03. (d) held to within a relative error of $5.7\times 10^{-14}$ for 1000 random 4×4 complex matrices.

Example: $J = \left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$ has $J^2 = -I$, so splitting into even and odd orders gives $e^{tJ} = \cos t\,I + \sin t\,J = R(t)$, that is, a rotation. In general, if $X^2 = -\omega^2 I$ then $e^{tX} = \cos(\omega t)\,I + \frac{\sin(\omega t)}{\omega}X$.

ComputedThe partial sums in the figure add up the series on the spot. The table gives the difference, measured in the operator norm, between the double-precision partial sums and the rotation matrix $e^{tJ}$ ($a = 0$). Clicking a number cell of the table puts that $N$ and $t$ ($a = 0$) into the figure.

When $t$ is large the terms $t^k/k!$ swell for a while, so the $N$ needed grows with $t$. At $t = \pi$, $N = 16$, the actual error is $7.8\times 10^{-7}$ against the error bound $1.8\times 10^{-5}$ above.

Detour: an example where summing the series as it is fails

Hermitian matrices and unitary matrices are connected by the exponential. As a metaphor, a Hermitian matrix is "how to turn", and a unitary matrix is "the result of turning". To be exact, it is as follows.

Taking the adjoint term by term gives $(e^{iH})^\dagger = e^{-iH}$. $iH$ and $-iH$ commute, so $U^\dagger U = e^0 = I$ follows from §08 (b), and the determinant from (d). If $\operatorname{tr} H = 0$, then $\det U = 1$ and $U \in \mathrm{SU}(n)$. Conversely, every unitary matrix can be written as $e^{iH}$ by setting $H = V\operatorname{diag}(\theta_k)V^\dagger$ from the diagonalisation of §07 Known ($\theta_k$ can be chosen only up to integer multiples of $2\pi$, so $H$ is not unique).

Example: $H = \left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right)$ has $H^2 = I$, so $e^{i\theta H} = \cos\theta\,I + i\sin\theta\,H$, with determinant $\cos^2\theta + \sin^2\theta = 1$. ComputedFor 1000 random 3×3 Hermitian matrices, $\lVert U^\dagger U - I\rVert \le 1.6\times 10^{-15}$, and with trace 0, $|\det U - 1| \le 1.1\times 10^{-15}$.

This correspondence is used in two places. Time evolution in quantum mechanics $e^{-itH/\hbar}$ is unitary and preserves total probability (2-05). In gauge theory, sending the space of §02 widened to $N \times N$ (traceless $N \times N$ Hermitian matrices, times $i$) through the exponential lands in $\mathrm{SU}(N)$; the former is the space of values of the gauge field, and the latter the space of gauge transformations (1-03, 1-04). Restricted to real entries, it takes the form "the $e^K$ of an antisymmetric matrix $K$ is a rotation".

Attaching "times $i$" to the values of the gauge field is the geometers' way of writing (1-07). The physicists' way writes the gauge field as a Hermitian matrix as it is, and the two are carried into each other by $A_{\text{geom}}=-iA_{\text{phys}}$ (the form in chapters that write the coupling constant separately is in the table in 3-02).

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Next chapter: 1-02 What a group is — it starts from the fact that the set of unitary matrices forms a "group", and goes on to the relation between $\mathrm{SO}(3)$ and $\mathrm{SU}(2)$.

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