Intuitive

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Concrete

Gauss' Law for Yang-Mills Theories

Gauss law is the $\nu=0$ component of the Yang-Mills equation

$$ (\partial_\mu F_{\mu \nu})^a = g j_\nu^a $$ $$ \rightarrow (\partial_i F_{i 0})^a = g j_0^a $$ which is exactly analogous to the inhomogeneous Maxwell equation in the presence of matter fields.

It does not contain second order time derivatives is therefore not an equation that governs the time development (=an equation of motion), but rather a constraint on the initial conditions.

(Source: "Classical Theory of Gauge Fields" by Rubakov. See also the discussion there for more details.)

We can rewrite it in terms of the gauge potentials as

$$ I(x) \equiv D_i G^{0i} =\partial_i E^i + [A_i,E^i]=0 .$$

This equation contradicts the commutator relations and thus is not an operator equation. Instead, we use it as a condition for physical states, which have to satisfy:

$$ D_i E^i |\Psi\rangle _{phys}=0. $$

If we restrict ourselves to those functions $\Lambda (x)$ that become zero at infinity, we can integrate by parts to see that operators which generate gauge transformations can be written as

$$ U = exp\left( \frac{-2i}{e} \int_{-\infty}^\infty dx^3 Tr(\tilde \Lambda(x) D_i E^i \right)$$

(see page 340 in Solitons by Rajaraman)

According to Gauss' law, we therefore have

$$ U_{\tilde \Lambda} | \psi\rangle_{phys} =| \psi\rangle_{phys}.$$

The thing is now, that we can see this way that Gauss' law only forces gauge equivalence under gauge functions with a gauge function that satisfies $\tilde \Lambda (\pm \infty) =0$.

States that are connected by a gauge transformation that does not satisfy this condition, can actually be physically distinct!

Source: page 340 in Solitons by Rajaraman

Recommended Reading

http://ethesis.helsinki.fi/julkaisut/mat/fysik/vk/salmela/gausssla.pdf

Abstract

Gauss law generates gauge transformations

Those gauge transformations that approach identity at infinity and are contractible to the identity are generated by Gauss' law $((\vec D\cdot \vec E) -\rho) |\text{phys}\rangle =0$, where $\rho$ describes a possible contribution from matter. Source

The choice $A_0= 0$ does not exhaust all gauge freedom; time-independent gauge transformations are still permitted. We must examine the precise content auf gauge invariance under such transformations, in the quantum field theoretical context. In the canonical Hamiltonian procedure in the presence of constraints, this freedom due to time-independent gauge transformations can be traced to Gauss's law:

\begin{align} I(x) &= Div E(x) -j_0(x) \notag \\ &= dE_x/dx-\frac{1}{2}ie(\phi^\star D_0\Phi-\Phi(D_0\Phi)^\star)=0. \end{align}

Recall that this equation is one of the Euler-Lagrange equations arising from the Lagrangian (10.31). But it involves no time derivatives of the electromagnetic field and is a constraint equation rather than a genuine equation of motion. In the quantized theory, it cannot be considered an operator equation since it conflicts with canonical commutation rules. This difficulty is well known (see for example Bjorken and Drell 1965). Equal time commutation rules give

\begin{align} &[E_x(x_1),A_x(x_2)]_{t=0}=i \delta(x_1-x_2); \notag \\ &[\phi,A_\mu]_{t=0}=[D_0\Phi,A_\mu]_{t=0}=0. \end{align}

Hence

$$[I(x_1),A_x(x_2)]_{t=0}=i \partial/\partial_{x_1} [ \delta(x_1-x_2)]\neq 0 $$

and $I(x_1)$, as an operator equation, cannot vanish. On therefore imposes Gauss's law as a constraint on 'physical' states. These are defined as that subset of states which obey

$$ I(x) |\Psi\rangle _{phys}=0. $$ page 321 in Solitons by Rajaraman

Next, consider the operator

$$ U_\Lambda = exp\left( \frac{i}{e} \int_{-\infty}^\infty \Lambda(x) I(x)dx \right), $$ where \Lambda(x) is any c-number function. Then

$$ U_\Lambda |\Psi\rangle _{phys} = |\Psi\rangle _{phys} \tag{10.43} $$

Now, suppose we restrict ourselves to that subset of functions, labelled $\tilde \Lambda(x)$, which satisfies $\tilde \Lambda (x) \to 0$ as $x\to \infty$. […]

We see that $U_{\tilde \Lambda}$ is just the operator which executes time-independent gauge transformations, in the quantum context […]

Equation (10.43) thus tells us that physical states must be invariant under such transformations. This is hardly an unfamiliar result, but in deriving it this way we notice the important restriction that $\tilde \Lambda( \pm \infty) =0$. Gauge transformations certainly exist for a general $\Lambda (x)$ not satisfying $\Lambda (\pm \infty)=0$ and they do leave the Lagrangian and the Hamiltonian invariant. But Gauss's law does not necessarily force all wavefunctionals to be invariant under them. Gauge transformations by the restricted subset of functions $\tilde \Lambda (x)$ may be called, for a want of a better name, "small gauge transformation". The name implies not that $\tilde \Lambda (x)$ is small everywhere, but that $\Lambda (\pm \infty)=0$. Gauge transformations that are not "small", will be called large.

page 323 in Solitons by Rajaraman

Additional complications arise when defining quantised gauge field theories. This is because the restriction of a theory to be invariant under a gauge group symmetry $\mathcal{G}$, which corresponds to local invariance under some global symmetry group $G$, leads to a strengthened form of the Noether current conservation condition called the local Gauss law [11]: \begin{align} J^{a}_{\mu} = \partial^{\nu}G^{a}_{\mu\nu}, \hspace{6mm} G^{a}_{\mu\nu} = -G^{a}_{\nu\mu} \tag{(1.5)} \end{align} where $J^{a}_{\mu}$ is the Noether current associated with invariance under the global group $G$. Because of this condition it turns out that there are essentially two quantisation strategies [11]:

1. One demands that equation (1.5) holds as an operator equation, which implies that the algebra of fields $\mathcal{F}$ is no longer local. In particular, if a field transforms non-trivially with respect to the group $G$ (i.e. has a non-zero $G$-charge), the field must be non-local.
2. One adopts a local gauge quantisation in which the local Gauss law is modified. This modification ensures that the field algebra remains local (even for charged fields), but necessitates the introduction of an indefinite inner product on the space of states $\mathcal{V}$, and a condition: $\langle \Psi |J^{a}_{\mu} - \partial^{\nu}G^{a}_{\mu\nu} | \Psi \rangle = 0$ for identifying the physical states $|\Psi\rangle \in \mathcal{V}_{\text{phys}} \subset \mathcal{V}$ (the weak Gauss law).

The advantage with the latter approach is that it allows one keep a local field algebra, and so all of the results from local field theory remain applicable. For the purposes of discussion in this paper we will only consider local quantisations, and in particular we will focus on the local BRST quantisation of Yang-Mills theory. A key feature of BRST quantisation is that a gauge-fixing term $\mathcal{L}_{GF}$ is added to the Lagrangian density $\mathcal{L}$. The modified Lagrangian $\mathcal{L}+\mathcal{L}_{GF}$ is no longer gauge-invariant, but remains invariant under a residual BRST symmetry with a conserved charge $Q_{B}$. By defining the physical space of states $\mathcal{V}_{\text{phys}} \subset \mathcal{V}$ to be the states which satisfy the subsidiary condition: $Q_{B}\mathcal{V}_{\text{phys}}=0$, this ensures that the weak Gauss law is satisfied and that the field algebra $\mathcal{F}$ is local. The introduction of an indefinite inner product on $\mathcal{V}$ also leads to unphysical negative norm states, which are generated by the Faddeev-Popov ghost degrees of freedom in $\mathcal{L}_{GF}$. In terms of these extended state spaces, the physical Hilbert space is a quotient space of the form $\mathcal{H}:=\overline{\mathcal{V}_{\text{phys}}/ \mathcal{V}_{0}}$, where $\mathcal{V}_{0}$ contains the zero norm states in $\mathcal{V}$ [12]. (The bar denotes the completion of $\mathcal{V}_{\text{phys}} / \mathcal{V}_{0}$ to include the limiting states of Cauchy sequences in $\mathcal{V}_{\text{phys}} / \mathcal{V}_{0}$.)

Boundary terms in quantum field theory and the spin structure of QCD by Peter Lowdon

Why is it interesting?

Gauss law in an important constraint on the initial data in gauge theories. It is particularly important if we want to understand how we can fix the gauge consistently.

The main output of this analysis is therefore the suggestion that Gauss law is the basic and primary feature which characterized elementary particle interactions, rather than gauge invariance, a concept which is more difficult to grasp on physical grounds since it can be given a meaning only by introducing unobservable quantities. Gauge Invariance can therefore be regarded as a technical tool for constructing Lagrangian functions or equations of motion which guarantee the validity of Gauss' law. This may be the right track to get an insight into the structure of GQFT and possibly understand why nature seems to choose gauge theories for elementary particle interactions.

Gauss’ Law in Local Quantum Field Theory by F. Strocchi

The recognition that local Gauss laws are the characteristic features of gauge quantum field theories has been argued and stressed in view of quantum theories in [20] [16] [5] and later reproposed, without quoting the above references, by Karatas and Kowalski (1990) [21], Al-Kuwari and Taha (1990) [22], Brading and Brown (2000) [23]. Actually, such papers confine the discussion to the derivation of local Gauss laws from local gauge invariance (second Noether theorem at the classical level, with no gauge fixing), missing the crucial fact that at the quantum level local gauge invariance of the Lagrangian has to be broken by the gauge fixing and it is devoid of any empirical (and philosophical) significance, whereas the validity of local Gauss laws keeps being satisfied by the physical states, and it explains the interesting (revolutionary) properties of gauge theories (as explained in Section 4). In contrast with global gauge symmetries, local gauge symmetries are only useful tricks used in intermediate steps (which use an auxiliary unphysical field algebra, initially a Lagrangian which has local gauge invariance, to be next broken by the gauge fixing, a redundant space of vector "states", only a subspace of which describes physical states, on which local gauge symmetries reduce to the identity). The final emerging picture is a description of the physical system characterized by conserved (actually superselected) quantum numbers, provided by the generators of the global gauge symmetry, and by the validity of local Gauss laws (no trace remaining of local gauge invariance). In my opinion, from a philosophical point of view, one should invest in the meaning of local Gauss laws rather than on local gauge invariance (or on the so-called Gauge Principle).

https://arxiv.org/pdf/1502.06540.pdf

The existence of anomalies associated with global currents does not necessarily mean difficulties for the theory. On the contrary, as we saw in the case of the axial anomaly, its existence provides a solution of the Sutherland–Veltman paradox and an explanation of the electromagnetic decay of the pion. The situation is very different when we deal with local symmetries. A quantum mechanical violation of gauge symmetry leads to many problems, from lack of renormalizability to nondecoupling of negative norm states. This is because the presence of an anomaly in the theory implies that the Gauss’ law constraint $D · E_A = ρ A$ cannot be consistently implemented in the quantum theory. As a consequence, states that classically were eliminated by the gauge symmetry become propagating in the quantum theory, thus spoiling the consistency of the theory.

page 189 in Invitation to Quantum Field Theory by Alvarez-Gaume et. al.