2.1 Disambiguating confinement from related notions

It is well known that the result of applying Noether’s theorem to gauge symmetries results in a conserved current that takes values in the Lie algebra associated to the symmetry group.Footnote ² In the case of electrodynamics, this reduces to the usual sense of conserved electric charge given in real numbers. The relevant symmetry group is $U(1)$ , whose Lie algebra is simply $\mathbb{R}$ . Thus, we may not immediately recognize that electric charge is in fact a Lie-algebra-valued quantity. In chromodynamics, the conserved Noether current takes values in the Lie algebra $\mathfrak{su}(3)$ because $\mathfrak{su}(3)$ is the Lie algebra of the symmetry group $SU(3)$ .

For the interpretation of charge in QCD, however, this $\mathfrak{su}(3)$ feature of color charge is obscured by the further recognition that color charge is confined. As is well known, quarks, anti-quarks, and gluons are not observable in isolation. Rather, nature only allows for them to appear in bound states with each other in such a way that the net color charge of the observed state is always neutral. In discussions of the conservation of color charge, most textbook discussions proceed at the level of specific quark and anti-quark color states (r, b, g; $\bar{r}$ , $\bar{b}$ , $\bar{g}$ ). Often in conjunction with Feynman diagrams, physicists speak of the conservation of color in terms of, separately, the conservation of each of these kinds of color at each vertex in a diagram (see figure 1). Because (r, b, g; $\bar{r}$ , $\bar{b}$ , $\bar{g}$ ) are not elements of $\mathfrak{su}(3)$ , this gives a very different way of thinking of the conservation of color charge than the sense of conservation given by Noether’s theorem.Footnote ³

Figure 1.

Color states r, b, and g accounted for at each vertex within a Feynman diagram, here given for the process of nucleon scattering via pion exchange. Straight lines with arrows depict quark states, and curly lines with double coloring depict gluon states.

It is valuable to disambiguate several ideas related to confinement. First, there is the as-yet-unproven result of quark/gluon confinement as a consequence of the dynamics of QCD. Confinement is the phenomenon that free quarks and gluons do not exist at low energies.Footnote ⁴ Second, there is the theoretical stipulation of color neutrality. For example, in the construction of proton states, p, we take the tensor product of the $SU(3)$ carrier spaces for each of the three valance quarks within the proton: $p \in \mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3 \cong \mathbb{C}^{1} \oplus \mathbb{C}^8 \oplus \mathbb{C}^8 \oplus \mathbb{C}^{10}$ . On the right-hand side, we have the four different mathematical possibilities for which representation of $SU(3)$ the proton could be in: $\mathbb{C}^1$ , two copies of $\mathbb{C}^8$ , or $\mathbb{C}^{10}$ . The $\mathbb{C}^1$ option is the only physical possibility, according to the stipulation of color neutrality: all observable states must transform according to the trivial representation (whose carrier space is $\mathbb{C}^1$ ) of the $SU(3)$ color symmetry, that is, be overall color-charge-neutral.

Research into the ultraviolet and infrared behavior of the strong coupling constant $\alpha _s$ in QCD aims to give a dynamical explanation for both quark/gluon confinement and color neutrality. Here, the term constant is something of a misnomer because $\alpha _s$ is a “running” coupling constant whose value depends on the energy scale. There is widespread consensus (although see Seiler [Reference Seiler2003] for a dissenting view) that applying perturbation theory to QCD proves that the theory has ultraviolet asymptotic freedom; that is, the effective QCD coupling goes to zero in the ultraviolet limit as distance goes to zero. QCD appears to have, as demonstrated numerically in lattice QCD, the opposite effect in its infrared behavior: in the infrared regime, the coupling increases as distance increases.Footnote ⁵ If the coupling in fact increases in this manner, as suggested by lattice QCD, then this would serve to provide a dynamical explanation of quark confinement. The increase of the coupling with increased distance would keep individual quarks and gluons from leaving the confines of their hadronic homes. These numerical results are forcefully suggestive of quark confinement, yet they fall short of an airtight proof. If a proof of this infrared behavior is found, it is expected that it would further show that color neutrality is a dynamical consequence of the theory. Another possibility is that rather than displaying this increasing infrared behavior, the effective coupling “freezes” at a definite value at a given energy and remains frozen for higher energies. This possibility is suggested by recent work within the functional renormalization group framework.Footnote ⁶ Our primary focus in the next subsection will be the interpretive difficulties posed by the phenomena of quark confinement and color neutrality, setting aside questions of how the ultraviolet and infrared behavior of the strong coupling might explain these phenomena.

2.2 Obscuring ${\mathfrak {su}}(3)$

Because the fundamental entities that are the bearers of color charge are confined within color-neutral hadrons, the metaphysical complexities of how color charge is conserved are far from obvious. Prima facie, conserving a quantity by never allowing it to divert from zero in observable systems is a particularly uninteresting conservation law. Just below the surface, however, there are two ways in which an overall color-neutral state can be found. First, equal amounts of r, b, and g combine to form a “white” or color-neutral state. Second, components of these three colors can be compensated for with equal amounts of their corresponding anti-colors, $\bar{r}$ $\bar{b}$ , and $\bar{g}$ . This suggests (as does figure 1) that what is conserved is red, and separately blue, and separately green. But that is not what Noether’s theorem gives us. By focusing on the (r, b, g; $\bar{r}$ , $\bar{b}$ , $\bar{g}$ ) properties separately, the Feynman-diagram approach to color charge conservation obscures the role of $\mathfrak{su}(3)$ in this conservation law.

I argue in the next section that we can, instead, make sense of the conservation of the $\mathfrak{su}(3)$ Noether charge in QCD by investigating color charge in a classical version of quantum chromodynamics. In that classical theory, the formulation of the Wong force law (the non-Abelian generalization of the Lorentz force law) leads to the ascription of an $\mathfrak{su}(3)$ -valued quantity of charge for matter fields, as well as for the gauge field. This gives us a vantage point from which to see clearly the $\mathfrak{su}(3)$ character of charge for the matter fields themselves and how they therefore contribute to the overall conserved $\mathfrak{su}(3)$ Noether charge. Moreover, by choosing to investigate charge in this classical version of chromodynamics, we have set aside the physically important but conceptually confounding issues of color neutrality, quark/gluon confinement, and ultraviolet and infrared behavior. By looking at this classical version of the theory with unconfined matter fields, we can more clearly see the $\mathfrak{su}(3)$ character of quantum color charge.

3.1 The Wong force law

We may formulate the basic features of a classical version of chromodynamics as follows.Footnote ⁸ We will adopt the abstract index notion developed by Wald (Reference Wald1984) with the further notational conventions of Weatherall (Reference Weatherall2016). Vectors and tensors tangent to M have lowercase Latin indices a, b, c; vectors and tensors tangent to the total space P have lowercase Greek indices; and uppercase Fraktur indices are used for vectors with a Lie-algebra structure. In addition, indices i, j, and so on will be used to label vectors in the carrier space V of a representation of $SU(3)$ used to construct an associated bundle. Using these conventions, fix a relativistic spacetime $(M, g_{ab}$ ) and an $SU(3)$ principal bundle $P \overset{\wp} {\rightarrow}M$ over M. In addition, fix a principal connection $\omega ^{\mathfrak{A}}_{\alpha }$ on P and an inner product $k_{\mathfrak{AB}}$ on the Lie-algebra $\mathfrak{su(3)}$ associated to $SU(3)$ .

The curvature of the principal connection is interpreted as the chromodynamic field strength. This curvature is

(1) $$\Omega _{\;\;\alpha \beta }^{\mathfrak{A}} = {d_\alpha }\omega _{\;\;\beta }^{\mathfrak{A}} + {1 \over 2}[\omega _{\;\;\alpha }^{\mathfrak{A}},\omega _{\;\;\beta }^{\mathfrak{A}}],$$

where $d_\alpha$ is the exterior derivative on P, and the bracket $[\cdot,\cdot ]$ is the Lie bracket on $\mathfrak{su}(3)$ .

The quark and anti-quark matter fields correspond to sections of different associated bundles $P \times _G V \ \overset{\pi \rho} {\rightarrow } M$ , where V is the carrier space for an irreducible representation $\rho$ of $SU(3)$ . Thus, a matter field $\Psi : M \rightarrow P \times _G V$ maps points $x \in M$ to equivalence classes $[p, v^i]$ for some $p \in P$ and $v^i \in V$ . Here $[p, v^i] \cong [p', v^j]$ just in case there exists a $g \in SU(3)$ such that $(pg, \rho (g^{-1})v^i) = (p', v^j)$ . These vectors $v^k$ describe state vectors in the internal charge space for an elementary particle of this kind of matter field (either a quark, anti-quark, or gluon).

The sections $\Psi : M \rightarrow P \times _G V$ are in one-to-one correspondence with V-valued, G-equivariant maps $\psi ^i$ on P. Given a section evaluated at a point x in M, $\Psi (x) = [p, v]$ , we define $\psi ^i : P \rightarrow V$ by $\psi ^i(p) \mapsto \lambda ^{-1}_p(\Psi (\wp (p)))$ , where the map $\lambda _p : V \rightarrow \pi _\rho ^{-1}(\wp (p))$ is defined by $\lambda _p(v) = [p, v]$ . In different contexts, it is sometimes more convenient to use $\psi ^i$ on P rather than $\Psi$ on M. In particular, it is technically simpler to define the action of the covariant derivative induced by the connection $\omega ^{\mathfrak{A}}_{\alpha }$ on $\psi ^i$ than on $\Psi$ . For a field $\psi ^i: P \rightarrow V$ , its covariant derivative is the ordinary exterior derivative $d_\alpha$ on P following the vertical projection. That is, for any vector $\xi ^\alpha$ on P, the covariant derivative induced by the connection is ${\overset {\omega }{D}}{_\alpha }\psi ^i\xi ^\alpha = d_\alpha \psi ^i\xi ^{\alpha |} $ , where $\xi ^{\alpha |} = \omega ^{\mathfrak{A}}\ _{\alpha }\xi ^\alpha$ .

The generalization of the Lorentz force law to non-Abelian gauge theory is known as the Wong force law, first given by Wong (Reference Wong1970). It can be mathematically derived as follows, although the status of this derivation as a physical argument is unclear.Footnote ⁹ Using the inner product $k_{\mathfrak{AB}}$ , the metric $g_{ab}$ on M, and our principal connection $\omega ^{\mathfrak{A}}_{\alpha }$ , we can construct a metric on the total space known as the bundle metric. First, we take $g_{ab}$ on the base space M, and we pull this back along the projection to get a symmetric rank 2 tensor $(\wp ^*g)_{\alpha \beta }$ on the total space P. That is,

(2) $${({\wp ^*}g)_{\alpha \beta }}{\sigma ^\alpha }{\eta ^\beta } = {g_{ab}}({\wp _*}{\sigma ^\alpha })({\wp _*}{\eta ^\beta })$$

for all vectors $\sigma ^\alpha, \eta ^\alpha$ at a point p in P. There is another symmetric rank 2 tensor on P, denoted $k_{\alpha \beta }$ , defined in terms of the connection $\omega ^{\mathfrak{A}}_{\alpha }$ . It is given by

(3) $$k_{\alpha \beta }\sigma ^\alpha \eta ^\beta = k_{\mathfrak{AB}} \omega ^{\mathfrak{A}}\ _{\alpha }\sigma ^\alpha \omega ^{\mathfrak{B}}\ _{\beta }\eta ^\beta. $$

For any two vectors $\sigma ^\alpha$ and $\eta ^\beta$ at a point p in P, $k_{\alpha \beta }$ gives the inner product of their vertical projections. Adding these two tensors gives us our bundle metric $h_{\alpha \beta } = (\wp ^*g)_{\alpha \beta } + k_{\alpha \beta }$ .

We can use the bundle metric to classify curves through the total space. Let $\gamma : [0, 1] \rightarrow P$ be a geodesic relative to $h_{\alpha \beta }$ with tangent field $\xi ^\alpha (t)$ . It follows that $\omega ^{\mathfrak{A}}\ _{\alpha }\xi ^\alpha (t) = Q^{\mathfrak{A}} \in \mathfrak{su}(3)$ is independent of t (see Bleecker Reference Bleecker2013, theorem 10.1.5). Let $\tilde{\gamma } = \wp \circ \gamma$ be the projection of $\gamma$ down to the base space with tangent field $\xi ^a$ . It follows that, relative to a choice of section $\sigma$ , the acceleration of this curve $\tilde{\gamma }$ on spacetime obeys the Wong force law:

(4) $$\xi ^n\nabla _n\xi ^b = k_{\mathfrak{AB}}g^{cb}Q^{\mathfrak{A}}\Omega ^{\mathfrak{B}} _{\,\,\,\,ac}\xi ^a = Q^{\mathfrak{A}}\Omega _{\mathfrak{A}a}^{\,\,\,\,\,\,b}\xi ^a, $$

where $\Omega ^{\mathfrak{A}}_{\,\,\,ab} = \sigma ^*(\Omega ^{\mathfrak{A}}_{\,\,\,\,\alpha \beta })$ is the field strength (see Bleecker Reference Bleecker2013, theorem 10.1.6). We interpret $\tilde{\gamma }$ as the world line for a particle of mass m and charge $q^{\mathfrak{A}} = Q^{\mathfrak{A}}/m$ . If $G = U(1)$ , this reduces to the familiar Lorentz force law for electromagnetism. In that case, we interpret $q^{\mathfrak{A}} \in \mathbb{R}$ as the amount of electric charge carried by the particle whose world line is $\tilde{\gamma }$ . Thus, in classical chromodynamics, $q^{\mathfrak{A}}$ is the non-Abelian charge property analog of electric charge.

3.2 Interpretation

How may we interpret $q^{\mathfrak{A}}$ in this formulation of classical chromodynamics? We usually predicate the color charge of a particle in the sense of a basis vector in a representation of $SU(3)$ , that is, a specific color state (r, b, g; $\bar{r}$ , $\bar{b}$ , $\bar{g}$ ). Using the electromagnetic interpretation as a guide, one might then expect that $q^{\mathfrak{A}}$ gives the color state of the particle under the influence of the Wong force. And because gluons are the only type of particle with Lie-algebra-valued color states, one might conclude that $\tilde{\gamma }$ is the world line of a gluon.

However, such reasoning is mistaken. The color-charged matter fields subject to this Wong force law couple to the gauge field via the charge-current density. This charge-current density is itself a Lie-algebra-valued 1-form $J^{\mathfrak{A}}_{\alpha }$ on the total space P. The charge $q^{\mathfrak{A}}$ is Lie-algebra-valued because the charge current density is Lie-algebra-valued.

The definition of $J^{\mathfrak{A}}_{a}$ relies on the inner product $k_{\mathfrak{A}\mathfrak{B}}$ on $\mathfrak{su}(3)$ , as well as an inner product $h_{ij}$ on the carrier space V. Fix a basis $\{e^{\mathfrak{A}}\}$ of the Lie algebra $\mathfrak{su}(3)$ . Then, following Bleecker (Reference Bleecker2013, theorem 5.1.2), the current $J^{\mathfrak{A}}_{a}$ is given by

(5) $$J^{\mathfrak{A}}_{\alpha } = k^{\mathfrak{AB}}e_{\mathfrak{B}}h_{ij}\tilde{\psi }^j\overset{\omega }{D}_\alpha \psi ^i, $$

where $\tilde{\psi }^j = \rho _*(e_{\mathfrak{A}})\vartriangleright \psi ^j$ . That is, $\tilde{\psi }^j$ is the result of transforming $\psi ^j$ under the representation $\rho _*$ of $\mathfrak{su}(3)$ on V induced by the representation $\rho$ of G on V.Footnote ¹⁰ The definition of the current in equation 5 gives us mathematical reason to acknowledge that the charge values $q^{\mathfrak{A}}$ for charged matter fields that contribute to $J^{\mathfrak{A}}_{\alpha }$ are Lie-algebra-valued.

This general expression for the current associated with a charged matter field $\psi ^i$ in a non-Abelian gauge theory reduces to the more familiar current of electromagnetism as follows. Suppose now that $G = U(1)$ . Further, choose $i = \sqrt{-1}$ as a basis for $U(1)$ ’s associated Lie algebra $\mathfrak{g}= \mathbb{R}$ , and choose $k_{\mathfrak{A}\mathfrak{B}}$ such that $k_{\mathfrak{A}\mathfrak{B}}e^{\mathfrak{A}}e^{\mathfrak{B}} = 1$ . Then the current $J^{\mathfrak{A}}_{\alpha }$ becomes $J^{\mathfrak{A}}_{\alpha } = ih_{ij}(i\psi )^j\overset{\omega }{D}_\alpha \psi ^i.$ Now fix a choice of local section $\sigma : U \rightarrow P$ . The vector potential is $A_a = \sigma ^*(\omega ^{\mathfrak{A}}_{\alpha })$ . Similarly, the complex scalar field on M is $\Psi = \sigma ^*(\psi ^i)$ . With this notation, a local representation of the current on M is

(6) $$ J_a = i((i\Psi )^*\nabla _a\Psi + i\Psi (\nabla _a\Psi )^*) $$

(7) $$ = \Psi ^*\nabla _a\Psi -\Psi (\nabla _a\Psi )^*, $$

where $\nabla _a = \partial _a -iA_a$ is the covariant derivative on M, and the star ${}^*$ indicates complex conjugation.

The foregoing discussion shows how color charge has a distinctly Lie-algebra-valued character as a result of the structure of the field theory that is captured in the geometry and not from the quantum character of QCD. Electric charge in QED is also Lie-algebra-valued, but because the Lie algebra of $U(1)$ is $\mathbb{R}$ , we do not immediately recognize the role of the Lie algebra for electric charge. The field strength (equation 1) is Lie-algebra-valued from the start. This fits with what we expect of the gauge field for chromodynamics in relation to gluons, which transform according to the adjoint representation. Moreover, the charge $q^{\mathfrak{A}} = Q^{\mathfrak{A}}/m$ in equation 4 is Lie-algebra-valued, and it gives the charge of the particle whose trajectory obeys the Wong force law. Thus, charge for quarks and anti-quarks is not only a matter of states within the fundamental representations of $SU(3)$ (wherein we find (r, b, g; $\bar{r}$ , $\bar{b}$ , $\bar{g}$ )) but also of Lie-algebra-valued contributions to the charge current density $J^{\mathfrak{A}}_{a}$ .

Because there is this sense in which charged particles have $\mathfrak{su}(3)$ charge, this gives us another vantage point from which to consider the conversation of the current $J^{\mathfrak{A}}_{\alpha }$ (see Bleecker Reference Bleecker2013, theorem 5.1.5). Rather than thinking of conservation of color charge in terms of (r, b, g; $\bar{r}$ , $\bar{b}$ , $\bar{g}$ ) values in Feynman diagrams, we can think of it in terms of $\mathfrak{su}(3)$ contributions to $J^{\mathfrak{A}}_{\alpha }$ . Although there is merit to the Feynman-diagram viewpoint of charge conservation, the benefit of this classical vantage point is that the Lie-algebra-valued character of $J^{\mathfrak{A}}_{\alpha }$ is manifest.

The point is not that we can only see these features of charge from the classical vantage point or that they are not available from the quantum-field-theory vantage point. Indeed, these features are present in the quantum version of the theory precisely because they are, first of all, present with the quantum theory’s “classical bones.” And by dissecting the “bones” apart from the quantum “flesh,” we are better poised to see the role the bones play in the final, finished quantum body of knowledge.