2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationcanonical quantisation, the Gauss constraint

Canonical quantisation and the Hamiltonian — Gauss's law becomes a condition on states

The problem statement defines the mass gap by the spectrum of the Hamiltonian $H$ (the vacuum is $H\Omega=0$): “A quantum field theory has a mass gap $\Delta$ if $H$ has no spectrum in the interval $(0,\Delta)$ for some $\Delta\gt0$.” (Jaffe–Witten §4) Known. One route to writing down that $H$ from the classical field is canonical quantisation (the other is 4-02).

With finitely many degrees of freedom there are three steps (2-05). (i) Make the momenta $p_i=\partial L/\partial\dot q_i$. (ii) Write $H=\sum_ip_i\dot q_i-L$ in terms of $q,p$. (iii) Replace $q,p$ by operators on a Hilbert space and impose the following.

On wave functions, $p_j=-i\,\partial/\partial q_j$. For fields the index becomes a point $\mathbf x$, and $\delta_{ij}$ becomes $\delta^3(\mathbf x-\mathbf y)$. Example. A scalar field of mass $m$ splits into an oscillator for each wave number, and the gap was $m$ (2-06 §08). In Yang–Mills, at step (i) a variable appears for which no momentum can be made.

The conventions are the same as in 3-03 §01 ($\hbar=c=1$, $D_\mu=\partial_\mu-igA_\mu$, $(D_iX)^a=\partial_iX^a+gf^{abc}A^b_iX^c$ on adjoint fields). The colour electric field is

and $L=\tfrac12\int\sum_a(|\mathbf E^a|^2-|\mathbf B^a|^2)\,d^3x$ (the magnetic field contains no time derivative). The momentum of $A^a_i$ is $\Pi^a_i=-E^a_i$, but $L$ does not contain $\partial_tA_0$, so the momentum of $A_0$ is identically 0.

We write step (ii). What corresponds to $\sum_ip_i\dot q_i$ is $\int\Pi^a_i\,\partial_tA^a_i\,d^3x$. Reading (2) as $\partial_tA_i=-E_i+D_iA_0$ and substituting,

follows. Subtracting $L$ from this leaves $\tfrac12(|\mathbf E|^2+|\mathbf B|^2)$ and $-\int\mathbf E\cdot\mathbf DA_0$. In the last term, integration by parts moves $D$ onto $\mathbf E$ (we take the fields to be 0 far away). The derivative part $\partial_i$ changes sign by ordinary integration by parts. For the $gf^{abc}A^b_i$ part, $f^{abc}$ changes sign under exchange of indices (it is antisymmetric), so relabelling the indices likewise changes the sign once. So $-\int\mathbf E\cdot\mathbf DA_0\,d^3x=\int A^a_0(\mathbf D\cdot\mathbf E)^a\,d^3x$, and

follows ($|\mathbf E|^2=\sum_a|\mathbf E^a|^2$) Known. $A_0$ is an undetermined multiplier that appears only linearly (the $\lambda$ of the toy in 2-01 §08), and its equation $(\mathbf D\cdot\mathbf E)^a=0$ is Gauss's law ((3) of 3-03). Such a condition is called a constraint, and its general treatment is said to begin with Dirac (1950).

We set $A_0=0$ by a gauge transformation (the temporal gauge; it is enough to solve $\partial_t\Omega=-ig\,\Omega A_0$ at each point). The remaining freedom is time-independent gauge transformations. The coordinates are $A^a_i(\mathbf x)$ and the momenta are the $\Pi^a_i=-E^a_i$ of §02. Since $\Pi=-E$, the field version of (1), $[A,\Pi]=i\delta$, is

(the sign is negative because $E=-\Pi$). We represent states by functions of configurations (wave functionals) $\psi[A]$. For small $\epsilon$ and a function $h^a_i(\mathbf x)$,

the function $\delta\psi/\delta A^a_i(\mathbf x)$ for which this holds is called the functional derivative. It is the finite-dimensional $\psi(q+\epsilon h)-\psi(q)\approx\epsilon\sum_ih_i\,\partial\psi/\partial q_i$ with the index $i$ replaced by the point $\mathbf x$ and the sum by an integral. Following the $p=-i\,\partial/\partial q$ of (1), $\Pi^a_i(\mathbf x)=-i\,\delta/\delta A^a_i(\mathbf x)$, so it acts as $E^a_i(\mathbf x)=i\,\delta/\delta A^a_i(\mathbf x)$, and (3) is formally

($\delta^2/\delta A^a_i(\mathbf x)^2$ is summed over $a$ and $i$) Known. It can be read as a particle with infinitely many coordinates moving in the potential energy $\tfrac12\int|\mathbf B|^2$. Strictly, (5) is form only (§09). Rescaling as $\hat A=gA$,

and when $g$ is large the electric term wins, when small the magnetic term. The coefficients $g^2/2a$ and $2N/g^2a$ in (16) of 5-03 are the lattice version of these two terms. Example. Of the $3(N^2-1)$ components $A^a_i$ at each point, the condition of the next section removes $N^2-1$ gauge directions, leaving $2(N^2-1)$ (the same number as in 3-03 §03). The one component that the classical theory removed with two things, the constraint and the remaining gauge transformations, is removed at once by (10) in the quantum theory.

Since $A_0$ has been removed, Gauss's law is no longer among the equations of motion. The Gauss operator

has three properties Known. (a) It generates gauge transformations. Setting $G(\alpha)=\int\alpha^aG^a\,d^3x$, integration by parts and (4) give

($A\mapsto A+\epsilon D\alpha$ is (4) of 3-02 with $\alpha$ taken as $g\epsilon\alpha$, that is, an infinitesimal gauge transformation). (b) It commutes with $H$. The two terms of (5) are gauge invariant, so $[H,G^a(\mathbf x)]=0$. (c) It forms a closed algebra.

$G^a=0$ cannot be made an operator identity, because the commutator of $G(\alpha)$ and $A_j$ is $-i\,D_j\alpha\ne0$. Instead it is imposed on states.

The $\Omega$ on the right are those continuously connected to the identity transformation. States satisfying (10) are the physical states. By (b) time evolution preserves this subspace, and by (c) the conditions do not contradict one another. Observables are restricted to gauge-invariant ones, and the mass gap is the gap of $H$ on the physical states (4-11 §09).

Example. For a uniform field of $SU(2)$ (a real matrix $A^a_i$ of $3\times3$; uniform fields are in 3-02 §07), $G^a=ig\,\varepsilon^{abc}A^b_i\,\partial/\partial A^c_i=-gL^a$. $L^a=-i\varepsilon^{abc}A^b_i\,\partial/\partial A^c_i$ is the angular momentum that turns the colour index, and satisfies $[L^a,L^b]=i\varepsilon^{abc}L^c$. This is the same commutation relation as that of the generators $[T^a,T^b]=if^{abc}T^c$ (3-02 §02), so $G=-gL$ gives the $-ig$ of (9). Gauge-invariant functions are functions of $A^{\mathsf T}A$ and of the sign of $\det A$. With differences we checked that $G^a$ annihilates $\tfrac12|\mathbf B|^2$ and $\det A$ to within $2\times10^{-10}$ but does not annihilate $A^1_1$, and that (9) holds to relative $2\times10^{-11}$ Computed.

Charges, and large gauge transformations

If there are static colour charges, the condition is $G^a\psi=\rho^a\psi$. The abstract of Kogut–Susskind (1975) describes the gauge-invariant configurations on the lattice as “a collection of strings with quarks at their ends. The strings are lines of non-Abelian electric flux.” Known (Table 3). Also, what (8) generates is only transformations connected to the identity. A transformation $T$ that approaches 1 far away and has winding number 1 from $\mathbb R^3\cup\{\infty\}=S^3$ to $SU(2)$ (3-05 §07) is not connected to the identity. So $\psi[A^T]=e^{-i\theta}\psi[A]$ does not contradict (10) (with the convention $(T\psi)[A]=\psi[A^T]$ it has the same direction as (12) of 3-05; with the opposite convention, $e^{+i\theta}$). This is the $\theta$ of θ vacua Physics.

With $g=0$, split the component of wave number $\mathbf k$ into the longitudinal part $a_L$ parallel to $\mathbf k$ and the transverse part $a_T$ perpendicular to it. The magnetic field contains only the transverse part, so writing just one transverse direction,

and $G_{\mathbf k}\propto\mathbf k\cdot\mathbf E_{\mathbf k}\propto p_L$. The longitudinal direction is the direction of the gauge transformation $\mathbf A\mapsto\mathbf A+\nabla\chi$ and has no restoring force, so in the whole $H$ the energy of longitudinal motion piles up continuously on top of each level. Gauss's law $p_L\psi=0$ removes this continuum, and the Fock space of the transverse oscillators (photons; 2-06 §07) remains Known. Figure 1 closes the longitudinal direction into a circle of length $2R$ ($|\mathbf k|=1$) and shows how, of the levels $E(n,m)=n+\tfrac12(\pi m/R)^2$ ($m\in\mathbb Z$), $m=0$ remain.

ComputedThe numbers in this figure were computed on this machine (closed formulas computed on the spot; agreeing with Table 1).

The total gap $\min(1,\pi^2/2R^2)$ is 0 for $R\to\infty$: the gap of $H$ itself in the temporal gauge is filled by motion in gauge directions. The physical gap $|\mathbf k|$ also tends to 0 as the box is enlarged, and the free field has no mass gap (for a field of mass $m$ the gap was $m$; 2-06 §08).

When space is a circle of length $L$ ($1+1$ dimension) there is no magnetic field, and $H=\tfrac12\int_0^LE^aE^a\,dx$. Gauss's law says that $E$ is carried by parallel transport (1-07), and parallel transport in the adjoint preserves length, so $H=\tfrac L2E^aE^a(0)$. All that remains for gauge-invariant functions is the conjugacy class of the holonomy round the circle, $U=P\exp(ig\oint A_1dx)$. A function unchanged by conjugation $U\mapsto gUg^{-1}$ is called a class function, and the trace of a representation matrix a character (1-02). Witten (1992) §4 writes “Ψ must be a class function of U, invariant under conjugation, and so must have an expansion in characters”, and expresses $H$ by the Casimir operator Known. $E^a(0)$ acts as $g$ times the derivative that turns $U$ in the direction $T^a$, and

follows ($\chi_r$ is the character of the irreducible representation $r$, and $C_2$ the value of $\sum_aT^aT^a$; 1-04). The eigenvalues were found exactly by Rajeev (1988), and the route from the transfer matrix is 6-02 §07. Example. For $SU(2)$, writing the eigenvalues of $U$ as $e^{\pm i\theta}$, $\chi_j=\sin((2j+1)\theta)/\sin\theta$, and the levels are $g^2L/2$ times $j(j+1)$. The gap $\tfrac38g^2L$ can be read as the energy of a loop of electric flux running once round the circle (tension $\tfrac38g^2$).

The volume of the orbit fixes the levels

The inner product of class functions is taken with Haar measure $\tfrac2\pi\sin^2\theta\,d\theta$. Taking the density $\tfrac2\pi\sqrt{1-a_0^2}\,da_0$ (a semicircle) of $a_0$ in 1-08 §03 and setting $a_0=\cos\theta$ gives this form. $\sin^2\theta$ is the area of the conjugacy class (a sphere in $S^3$), that is, the volume of the gauge orbit (4-02 §07). Writing (12) with $\theta$,

and on $u=\sin\theta\,f$, $\frac{g^2L}{8}(-u''-u)$, that is, a particle in a box on the interval $[0,\pi]$ ($u=\sin n\theta$, $n=2j+1$) minus 1. If the measure is forgotten and the flat Laplacian is used, the levels change to $k^2/4$ (Table 2). In the three-dimensional Coulomb gauge this volume factor appears as the Faddeev–Popov determinant (§08).

ComputedThe numbers in this figure were computed on this machine (characters and the closed formula of (12) computed on the spot; agreeing with Table 2).

Counting on the lattice — states that violate Gauss's law

Cut the circle into $N$ links ($a=L/N$) and set $H=\frac{g^2a}{2}\sum_\ell E_\ell^2$ (the two-dimensional version of 5-03 §07; there are no plaquettes). As a metaphor, one link is a top spinning on $SU(2)$ (5-03 §06), with $(2j+1)\times(2j+1)$ states of angular momentum $j$ for left and right rotations. Precisely, this is the Peter–Weyl theorem, which splits the state space $L^2(SU(2))$ of a link by representation Known. So the energy is $\frac{g^2a}{2}\sum_\ell j_\ell(j_\ell+1)$ and the multiplicity $\prod_\ell(2j_\ell+1)^2$. At a vertex the ends of two links meet, and Gauss's law requires that the combination of their two representations be unchanged by rotation (be a scalar). $0$ appears in the coupling of angular momenta $j\otimes j'=|j-j'|\oplus\cdots\oplus(j+j')$ only when $j=j'$, and then exactly once (Schur's lemma) Known. So neighbouring $j$ are equal, there is one physical state for each $j$, and the levels are (12) regardless of $N$.

The lowest excitation of the whole space is a single link with $j=\tfrac12$, that is, a state with a pair of charges at its two ends (§04), and its energy $\tfrac38g^2a$ is the tension times the distance $a$. It tends to 0 as $a\to0$, so $\tfrac38g^2L$ remains only once we restrict to physical states.

There is also the route of fixing the gauge further and solving the constraint first. Splitting as $\mathbf E=\mathbf E_T-\nabla\phi$ ($\nabla\cdot\mathbf E_T=0$) in the Coulomb gauge $\nabla\cdot\mathbf A=0$, Gauss's law is a linear equation for $\phi$.

$\rho^a=gf^{abc}A^b_iE^c_{T,i}$ is the colour charge carried by the transverse field itself. If $M$ can be inverted, then with $K=M^{-1}(-\nabla^2)M^{-1}$,

(the integral is $d^3x$) Known. The abstract of Christ–Lee (1980) writes that the change of variables from the temporal gauge produces “an unconventional operator ordering”, and that “new nonlocal interactions, even in the familiar Coulomb gauge” appear Known. The volume factor of the orbit corresponding to the $\sin^2\theta$ of §06 is $\det M$ Known. This is the Jacobian of changing variables to the gauge condition (coordinates on the slice $\nabla\cdot\mathbf A=0$), and is called the Faddeev–Popov determinant (details in 4-08). The difficulty is that when $A$ is large, $M$ has zero eigenvalues. The abstract of Gribov (1978) writes “the fixing of the divergence of the potential in non-Abelian theories does not fix its gauge” Known. That a slice cannot be chosen globally is Singer's theorem in 3-04. The (10) of the temporal gauge avoids this difficulty.

In four dimensions (5) lacks meaning in three respects. (i) $E^a_i(\mathbf x)E^a_i(\mathbf x)$ is a product of distributions at the same point (1-11 §09, 1-09 §09). (ii) The sum of zero-point energies $\tfrac12|\mathbf k|$ diverges (2-06 §09), and with interactions this is paired with renormalisation (4-04). (iii) In the inner product $\int|\psi|^2\mathcal DA$, the absence of Lebesgue measure in infinite dimensions (1-09 §02) overlaps with the divergence of the orbit volume (the fact that an interacting $H$ cannot be placed on the Fock space of the free field, that is, Haag's theorem, is also involved; 4-09).

On the lattice the state space is fixed, and Gauss's law can be imposed exactly at each vertex (Table 3). What remains is whether the gap stays positive in the limit where the lattice spacing goes to 0 and the box to infinity. The problem statement writes “Establishing a quantum mechanical Hilbert space is part of the solution to this Millennium problem.” (§6.5) Known. §6.1 states that the Osterwalder–Schrader construction links the route of Hilbert spaces and the route of functional integrals. §6.5 writes that much of the mathematical progress so far has come from understanding functional integration (“Much of the mathematical progress reviewed above results from understanding functional integration”). On this route $H$ is reconstructed from the measure in imaginary time (4-02 §08, 4-10 §09). This chapter shows nothing about four-dimensional Yang–Mills.

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Next chapter to read: 4-02 The path integral and Euclideanisation. The lattice Hamiltonian is in 5-03, gauge fixing in 4-08.

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