Stochastic Quantization of an Abelian Gauge Theory

Abstract

We study the Langevin dynamics of a U(1) lattice gauge theory on the two-dimensional torus, and prove that they converge for short time in a suitable gauge to a system of stochastic PDEs driven by space-time white noises. We fix gauge via a DeTurck trick. This also yields convergence of some gauge invariant observables on a short time interval. The proof relies on a discrete version of the theory of regularity structures.

Appendices

Appendix A Discussions and possible extensions

1.1 A.1 Review of blackbox theorems and possible extension to three dimensions

The model under consideration is also defined in three spatial dimensions, with fields $A=(A_1,A_2,A_3)= \sum _{j=1}^3 A_j \text{ d }x_j$ and $\Phi : \mathbf {T}^3 \rightarrow \mathbf{C}$:

$$\begin{aligned} \mathcal {H}(A,\Phi ) {\mathop {=}\limits ^{ \text{ def }}}\frac{1}{2} \int _{\mathbf {T}^3} \Big ( \sum _{j=1}^3 F_{A,j}(x)^2 + \sum _{j=1}^3 |D^A_j \Phi (x)|^2 \Big ) \, \text{ d}^3 x . \end{aligned}$$

(A.1)

Here, the curvature field $F_A = dA$ with components $F_{A,j}$ defined by

$$\begin{aligned}&F_{A,3}\, \text{ d }x_1 \wedge \text{ d }x_2 + F_{A,1} \, \text{ d }x_2 \wedge \text{ d }x_3 + F_{A,2} \, \text{ d }x_1 \wedge \text{ d }x_3 \\&\quad = ( \partial _1 A_2 -\partial _2 A_1) \, \text{ d }x_1 \wedge \text{ d }x_2 +( \partial _2 A_3 -\partial _3 A_2) \, \text{ d }x_2 \wedge \text{ d }x_3 +( \partial _1 A_3 -\partial _3 A_1) \, \text{ d }x_1 \wedge \text{ d }x_3 \end{aligned}$$

and $D^A_j \Phi$ is defined the same way. The SPDE is a system for $(A_1,A_2,A_3,\Phi )$ driven by independent white noises $\xi _1,\xi _2,\xi _3$ and a complex valued white noise $\zeta$. The system is again not parabolic, for instance, the equation for $A_1$ is of the form

$$\begin{aligned} \partial _t A_1 = \partial _2^2 A_1 -\partial _1\partial _2 A_2 + \partial _3^2 A_1 -\partial _1\partial _3 A_3+ \big ( \cdots \big ) + \xi _1 \end{aligned}$$

where $\big ( \cdots \big )$ denotes the nonlinearities. Using the same gauge tuning trick, one obtains a parabolic system, and the regularity of its solutions is expected to be $\alpha$ for $\alpha < -\tfrac{1}{2}$. Since it is more singular than the situation in two spatial dimensions, to obtain a local solution theory via lattice approximations, certain “general tools” should then be in place.

We thus take this chance to review the fast-growing literature in the theory of regularity structures—particularly focusing on the “blackbox” theorems, and discuss the possibility and challenge of applying them to the three dimensional convergence problem of (Abelian) lattice gauge theory.

The key idea of the theory is to lift the noise input of an SPDE to a space of models denoted by $\mathscr {M}$. The seminal work [Hai14] constructed this space $\mathscr {M}$, and proved continuity of the solution map from $\mathscr {M}$ to the space of distributions. Discrete analogues of these results are then developed by [HM18] on which the present article crucially relies; soon later [EH19] developed a more general discrete framework. These results would constitute the analytical foundation if one were to study the 3D lattice gauge theory (A.1). In parallel, “universality results” in regularity structures are proved in a series of papers [HQ18, HX19, HX18, SX18]; one possible challenge for 3D Abelian lattice gauge theory would be to see whether the techniques developed in these papers—suitably implemented on lattice—are sufficient to control the “remainder terms”. In 2D, in some sense, the “remainder terms” (4.23) are in a situation that is similar as in the “intermediate disorder” regime in [HQ18].

The algebraic step was then developed in [BHZ19], which has a systematic description of a group action on the space $\mathscr {M}$, which allows for the transformation from the canonical model to the renormalized model. This algebraic “blackbox” result is very general and applies to 2D (as in the starting part of Sect. 5.1) and 3D gauge theories as well.

A probabilistic step was then achieved by [CH16] which gives us—in an automatic way under very general conditions—convergence in probability of the renormalized models (in continuous regularizations) to a limit model. In 3D, there would be much more elements in the regularity structure, and explicit moment calculations as done in Sect. 5.2 would be a formidable task, thus it would require a discrete version of the BPHZ theorem in [CH16] to prove convergence of the discrete models. Although it seems to me that the arguments in [CH16] should be adapted to discrete settings given substantial effort, a complete formulation and proof for such a discrete theorem is not available yet at the moment.

Finally, another component of the “blackbox” which is also algebraic (or combinatoric), is established in [BCCH17] to identify the renormalized equations satisfied by the sequence of solutions given by the solution map for the renormalized models.

Regarding observables, the loop or string observables would be more difficult (if possible) to construct since one has to integrate a more singular gauge field along a curve.

Remark A.1

It would also be interesting to construct long time solutions as done by [MW17b, MW20, MW18] for the dynamical $\Phi ^4$ equation. To obtain some bound uniform in time, it might be helpful to have some damping terms, for instance adding more gauge invariant terms $\mu |\Phi ^2| - |\Phi |^4$ to (1.1). Let’s mention a very simple trick that the equation for B in for instance (6.6) can obtain a term $-\lambda ^2 c^2 B$ by considering the equations for B and the shifted field $\Psi \mapsto \Psi +c$. This would be reminiscent to the “Higgs mechanism” for the gauge field to acquire a mass by shifting the scalar field to a point in the bottom of a mexican-hat potential. These heuristics still seem to be far from achieving any rigorous results.

1.2 A.2 Non-Abelian case

As hinted in Remark 1.2, the gauge tuning method being exploited in the present paper is very likely to be generalized into the non-Abelian case (whereas a gauge fixing with a linear decomposition by (1.12) seems not generalizable). We would like to have some further discussion on this point (informally), and also make a comparison between gauge fixing in stochastic PDE formulation and that in functional integral formulation of quantum gauge theories.

In general gauge theories, with only gauge field A, one has $F_A {\mathop {=}\limits ^{ \text{ def }}}dA + A\wedge A$, where A is a 1-form taking values in an Lie algebra—for instance $\mathfrak u(N)$ for some $N\ge 1$. In the Abelian case, $N=1$, and $A\wedge A=0$ which is studied in this paper. In the functional integral approach, one is interested in the formal measure $e^{-\mathcal {H}(A)} {\mathcal D}A$ where $\mathcal {H}(A)= \frac{1}{2} \int _{\mathbf {T}^d} \hbox {tr}(F\wedge *F)$ is invariant under the gauge transformation $A\rightarrow g^*A {\mathop {=}\limits ^{ \text{ def }}}g^{-1}Ag+g^{-1}dg$ for $g: \mathbf {T}^d\rightarrow U(N)$. Note that in Abelian case $g=e^{i f}$ and this is precisely the first transformation in (1.2). To make the “measure” normalizable, a Fadeev–Popov trick based on the identity $1= \int \delta (g(x)) \,\text{ det }(\frac{\partial g}{\partial x})\, dx$ is often used. In Abelian lattice gauge theory, writing $A_f = A+df$ so that $d^* A_f = d^* A + \Delta f$, one then writes the formal partition function as

$$\begin{aligned} \int e^{-\mathcal {H}(A)} {\mathcal D}A = \int e^{-\mathcal {H}(A)} \delta (d^* A_f - \omega )\, \text{ det }(\frac{\delta (d^* A_f-\omega )}{\delta f}) \, {\mathcal D}f{\mathcal D}A . \end{aligned}$$

The Jacobian factor $\text{ det }(\frac{\delta (d^* A_f-\omega )}{\delta f})=\text{ det }(\Delta )$ can be factorized out, and using gauge invariance one can also replace $A_f$ by A and factor out the “infinite integral” $\int {\mathcal D}f$. If $\omega =0$ this simply amounts to imposing the divergence free condition as mentioned in Remark 1.2. (The only reason one usually introduces the field $\omega$ in the Abelian case being considered here is that upon integrating it against a Gaussian weight one obtains a factor $e^{-\frac{(d^* A)^2}{2}}$ that is slightly more convenient to analyze.)

The Fadeev–Popov trick gets much more complicated in non-Abelian case, because the determinant will generally depend on A and thus does not factor out. This determinant however can be expressed by an integral over anti-commuting variables, which are called ghost fields. One then studies the model for gauge field A coupled with ghost fields. Furthermore, fixing the gauge globally is not always possible in non-Abelian theories, due to topological obstructions, a phenomena usually referred to as Gribov ambiguity.

The ghost fields would not show up in the stochastic PDE approach with gauge fixed by DeTurck trick; this seems to be already observed by physicists [Zwa81, Sad87, BHST87]. In fact, for the corresponding stochastic quantization equation

$$\begin{aligned} \frac{\partial A}{\partial t} + d_{A}^*F_{A} = \xi \end{aligned}$$

(A.2)

with a Lie algebra valued d-component space-time white noise $\xi$, where $d_A$ is the gauge covariant derivative, one can check by straightforward computation that $B{\mathop {=}\limits ^{ \text{ def }}}g^* A$ satisifes $\frac{\partial B}{\partial t} = g^{-1}\frac{\partial A}{\partial t}g + d_{B}\left( g^{-1} \frac{\partial g}{\partial t}\right)$, so by solving g from the ODE $g^{-1}\frac{\partial g}{\partial t} = -d_B^*B$ and invoking gauge invariance of $d_{A}^*F_{A}$ one obtains a parabolic equation $\frac{\partial B}{\partial t} = -d_{B}^*F_{B} - d_{B}d_B^*B + g^{-1}\xi g$ where $g^{-1}\xi g {\mathop {=}\limits ^{law}} \xi$. This is well-know when $\xi =0$, see [DK90, Section 6.3]; and in the presence of the noise $\xi$, the aforementioned physics literature simply put a term $- d_{A}d_A^*A$ into the Eq. (A.2) and call this a gauge-fixing term.

To obtain a local solution theory to (A.2) via lattice approximations, one again needs some general tools as discussed in the three dimensional Abelian case.

Appendix B Ward identity

We derive an important identity that will be useful for cancellation of renormalization in Sect. 5.2 as well as showing convergence of certain observables in Sect. 6.1.

A reader familiar with renormalization would imagine that the equations (3.11) or (3.15) would need a mass renormalization for $B_j^\varepsilon$ (i.e. a term $\tilde{C}^\varepsilon B^\varepsilon$ for some constant $\tilde{C}^\varepsilon$). However, a mass renormalization for $B_j^\varepsilon$ would break gauge symmetry so many proofs such as Lemma 3.2 would break down. We will see that actually there will be several contributions to a mass renormalization for $B_j^\varepsilon$ and these contributions cancel. This cancellation is due to gauge symmetry. One can prove such cancellation by some elementary tricks (see Remark 5.7), but here we derive a version of “Ward identity” which seems more systematic, and could be more useful when certain tricks are not available (i.e. in $d=3$). The idea is straightforward: renormalization constants arise from expansion of nonlinearities [see (B.4)] in $\lambda$ and taking expectations, and we can make use of gauge invariance of such nonlinearities to obtain useful identities.

Fix $\varepsilon >0$. Consider Eq. (3.11), but now on $\Lambda _\varepsilon ^M$ which is the discrete torus of length size $M\ge 1$ and lattice spacing $\varepsilon$, and with initial condition $B^\varepsilon (-T)=0$ and $\Psi ^\varepsilon (-T)=0$ at time $-T$ for $T>0$. Let ${\mathcal E}^{M,j}_\varepsilon$ for $j\in \{1,2\}$ denote the sets of horizontal and vertical edges of $\Lambda _\varepsilon ^M$. With a slight abuse of notation we write $P_M^\varepsilon$ for the transition probability of continuous time random walk either on the vertices $\Lambda _\varepsilon ^M$, or the edges ${\mathcal E}^{M,j}_\varepsilon$ for some $j\in \{1,2\}$; they are essentially the same kernel since $\Lambda _\varepsilon ^M$ and ${\mathcal E}^{M,j}_\varepsilon$ only differ by a small translation, and it will be clear which case is under consideration in the context.

Fix a function $f^\varepsilon$ on $\Lambda _\varepsilon ^M$ which is independent of time. Let

$$\begin{aligned} B_f^\varepsilon = B^\varepsilon + \nabla ^\varepsilon f^\varepsilon , \qquad \hat{\Psi }_f^\varepsilon = \Psi ^\varepsilon e^{i\lambda f^\varepsilon }, \end{aligned}$$

where $(B^\varepsilon ,\Psi ^\varepsilon )$ is the solution to the above initial value problem. One then has $B_f^\varepsilon (-T)=\nabla ^\varepsilon f^\varepsilon$ and $\hat{\Psi }_f^\varepsilon (-T)=0$. By gauge invariance of the nonlinearity as shown in (3.8), one has

$$\begin{aligned} B^\varepsilon _{f,j} (t,e)&= B^\varepsilon _j (t,e) + \nabla _j^\varepsilon f^\varepsilon (e) \\&= P_M^\varepsilon *_{\varepsilon ,T} \Big ( \varepsilon ^{-1}\lambda \,\text {Im}\Big ( e^{-i\varepsilon \lambda B_{f,j}^\varepsilon ({\varvec{\cdot }})} \hat{\Psi }_f^\varepsilon ({\varvec{\cdot }}_+)\bar{\hat{\Psi }}_f^\varepsilon ({\varvec{\cdot }}_-)\Big ) +\xi ^\varepsilon _j ({\varvec{\cdot }}) \Big ) (t,e) + \nabla ^\varepsilon _j f^\varepsilon (e) . \end{aligned}$$

Here, for $\Lambda \in \{\Lambda ^M_\varepsilon ,{\mathcal E}^{M,1}_\varepsilon ,{\mathcal E}^{M,2}_\varepsilon \}$, $x\in \Lambda$, and any space-time function $g^\varepsilon \in C(\mathbf{R},\mathbf{R}^\Lambda )$ we have introduced the convolution notation

$$\begin{aligned} P_M^\varepsilon *_{\varepsilon ,T} g^\varepsilon (t,x) {\mathop {=}\limits ^{ \text{ def }}}\int _{-T}^\infty \sum _{y\in \Lambda } P_M^\varepsilon (t-s,x-y) g^\varepsilon (s,y) \,ds . \end{aligned}$$

By the covariance property (3.9) of the covariant Laplacian one also has

$$\begin{aligned} \partial _t \hat{\Psi }_f^\varepsilon (x) = e^{i\lambda f^\varepsilon (x)} \partial _t \Psi ^\varepsilon (x) = \Delta _{B_f^\varepsilon }^\varepsilon \hat{\Psi }_f^\varepsilon (x) +i\lambda \text {div}^\varepsilon (B_f^\varepsilon - \nabla ^\varepsilon f^\varepsilon )(x) \,\hat{\Psi }_f^\varepsilon (x) + \hat{\zeta }^\varepsilon (x), \end{aligned}$$

where $\hat{\zeta }^\varepsilon (x) {\mathop {=}\limits ^{ \text{ def }}}e^{i\lambda f^\varepsilon (x)} \zeta ^\varepsilon (x)$. Let $\Psi _f^\varepsilon$ solve

$$\begin{aligned} \partial _t \Psi _f^\varepsilon (x) = \Delta _{B_f^\varepsilon }^\varepsilon \Psi _f^\varepsilon (x) +i\lambda \text {div}^\varepsilon (B_f^\varepsilon - \nabla ^\varepsilon f^\varepsilon )(x) \, \Psi _f^\varepsilon (x) + \zeta ^\varepsilon (x), \end{aligned}$$

(B.1)

with initial condition $\Psi _f^\varepsilon (-T)=0$. We have $(B_f^\varepsilon , \Psi _f^\varepsilon ) {\mathop {=}\limits ^{law}} (B_f^\varepsilon , \hat{\Psi }_f^\varepsilon )$. Define

$$\begin{aligned} b_j^\varepsilon = b_j^\varepsilon (f^\varepsilon )&{\mathop {=}\limits ^{ \text{ def }}}B^\varepsilon _{f,j} |_{\lambda =0} = P_M^\varepsilon *_{\varepsilon ,T} \xi ^\varepsilon _j + \nabla ^\varepsilon _j f^\varepsilon , \nonumber \\ \psi ^\varepsilon&{\mathop {=}\limits ^{ \text{ def }}}\Psi _f^\varepsilon |_{\lambda =0} = P_M^\varepsilon *_{\varepsilon ,T} \zeta ^\varepsilon . \end{aligned}$$

(B.2)

(We drop the M dependence in the notation $b_j^\varepsilon$ and $\psi ^\varepsilon$ for simplicity.) Also, differentiating (B.1) with respect to $\lambda$ and by similar computation as in proof of Lemma 3.5, one can check that

$$\begin{aligned}&\partial _t \left( \partial _\lambda \Psi _f^\varepsilon (x) |_{\lambda =0}\right) \\&\quad = \Delta ^\varepsilon \partial _\lambda \Psi _f^\varepsilon (x) |_{\lambda =0} -i \sum \nolimits _{\mathbf {e}} B_f^\varepsilon (x,x+\mathbf {e}) \nabla ^\varepsilon _\mathbf {e}\Psi _f^\varepsilon (x) |_{\lambda =0} - i\Delta ^\varepsilon f^\varepsilon (x) \, \Psi _f^\varepsilon (x) |_{\lambda =0} \end{aligned}$$

where $\mathbf {e}\in \{\pm \mathbf {e}_1,\pm \mathbf {e}_2\}$, namely

$$\begin{aligned} \partial _\lambda \Psi _f^\varepsilon |_{\lambda =0} = -i P_M^\varepsilon *_{\varepsilon ,T} \Big ( \sum \nolimits _{\mathbf {e}} b^\varepsilon (\cdot ,\cdot +\mathbf {e}) \nabla ^\varepsilon _\mathbf {e}\psi ^\varepsilon +\Delta ^\varepsilon f^\varepsilon \, \psi ^\varepsilon \Big ). \end{aligned}$$

(B.3)

Note that this quantity depends on $f^\varepsilon$ via $b^\varepsilon$ and $\Delta f^\varepsilon$.

For any $x\in \Lambda ^M_\varepsilon$ and $t\ge -T$ consider the following observable

$$\begin{aligned} Z^\varepsilon _j (t,x) {\mathop {=}\limits ^{ \text{ def }}}\varepsilon ^{-1} \,\text {Im}\Big ( e^{-i\varepsilon \lambda B_{f,j}^\varepsilon (t,e)} \Psi _f^\varepsilon (t,x+\mathbf {e}_j)\bar{\Psi }_f^\varepsilon (t,x)\Big ) \end{aligned}$$

(B.4)

where $e=\{x,x+\mathbf {e}_j\}$. Applying (B.2) (B.3) (dropping t for simplicity of notation)

$$\begin{aligned}&\partial _\lambda Z^\varepsilon _j (t,x) |_{\lambda =0} = \text {Re}\Big ( b_j^\varepsilon (e) \psi ^\varepsilon (x+\mathbf {e}_j) \bar{\psi }^\varepsilon (x) \Big ) \\&\qquad - \varepsilon ^{-1} \text {Re}\Big ( P_M^\varepsilon *_{\varepsilon ,T} \Big ( \sum \nolimits _{\mathbf {e}} b^\varepsilon (\cdot ,\cdot +\mathbf {e}) \nabla ^\varepsilon _\mathbf {e}\psi ^\varepsilon +\Delta ^\varepsilon f^\varepsilon \, \psi ^\varepsilon \Big )(x+\mathbf {e}_j) \cdot \bar{\psi }^\varepsilon (x)\Big ) \\&\qquad + \varepsilon ^{-1} \text {Re}\Big ( \psi ^\varepsilon (x+\mathbf {e}_j) \cdot P_M^\varepsilon *_{\varepsilon ,T} \Big ( \sum \nolimits _{\mathbf {e}} b^\varepsilon (\cdot ,\cdot +\mathbf {e}) \nabla ^\varepsilon _\mathbf {e}\bar{\psi }^\varepsilon +\Delta ^\varepsilon f^\varepsilon \, \bar{\psi }^\varepsilon \Big )(x)\Big ) \\&\quad = - \sum _{k} b_j^\varepsilon (e) \psi ^\varepsilon _k (x+\mathbf {e}_j) \psi _k^\varepsilon (x) \\&\qquad - \sum _{k} \nabla ^\varepsilon _j P_M^\varepsilon *_{\varepsilon ,T} \Big ( \sum \nolimits _{\mathbf {e}} b^\varepsilon (\cdot ,\cdot +\mathbf {e}) \nabla ^\varepsilon _\mathbf {e}\psi _k^\varepsilon +\Delta ^\varepsilon f^\varepsilon \, \psi _k^\varepsilon \Big )(x) \cdot \psi _k^\varepsilon (x) \\&\qquad + \sum _{k} \nabla ^\varepsilon _j \psi _k^\varepsilon (x) \cdot P_M^\varepsilon *_{\varepsilon ,T} \Big ( \sum \nolimits _{\mathbf {e}} b^\varepsilon (\cdot ,\cdot +\mathbf {e}) \nabla ^\varepsilon _\mathbf {e}\psi _k^\varepsilon +\Delta ^\varepsilon f^\varepsilon \, \psi _k^\varepsilon \Big )(x) \end{aligned}$$

where k sums over $\{1,2\}$, $\psi ^\varepsilon = \psi ^\varepsilon _1+ i \psi ^\varepsilon _2$. Here, we have expanded a factor in the third line as $\psi ^\varepsilon (x+\mathbf {e}_j) =\psi ^\varepsilon (x) +\varepsilon \nabla ^\varepsilon _j \psi ^\varepsilon (x)$, and expanded a factor in the second line as $P_M^\varepsilon *_{\varepsilon ,T} ( \cdots )(x+\mathbf {e}_j) = P_M^\varepsilon *_{\varepsilon ,T} ( \cdots )(x) + \varepsilon \nabla ^\varepsilon _j P_M^\varepsilon *_{\varepsilon ,T} ( \cdots )(x)$; the two terms of order $\varepsilon ^{-1}$ cancel each other, therefore the factor $\varepsilon ^{-1}$ does not show up in the last two lines of the above equation.

It is important to note that $Z^\varepsilon _j (t,x)$ is gauge-invariant in law, in the sense that although $( B_f^\varepsilon , \Psi _f^\varepsilon )$ depends on $f^\varepsilon$, the law of the quantity $Z^\varepsilon _j (t,x)$ is independent of $f^\varepsilon$. Indeed, $Z^\varepsilon _j (t,x)$ would be deterministically gauge invariant by the calculation (3.8) if $\Psi _f^\varepsilon$ was replaced by $\hat{\Psi }_f^\varepsilon$; on the other hand $(B_f^\varepsilon , \Psi _f^\varepsilon ) {\mathop {=}\limits ^{law}} (B_f^\varepsilon , \hat{\Psi }_f^\varepsilon )$. Therefore the expectation of $\partial _\lambda Z^\varepsilon _j (t,x) |_{\lambda =0}$ is independent of $f^\varepsilon$.

Replacing $f^\varepsilon$ by $\alpha f^\varepsilon$ in the above arguments, the expectation of $\partial _\alpha \partial _\lambda Z^\varepsilon _j (t,x) |_{\lambda =0}$ must then be zero, namely,

$$\begin{aligned}&\mathbf {E}\Big [ - \Big \langle \nabla _j^\varepsilon f^\varepsilon ,\, \sum \nolimits _{k} \delta (\cdot -x) \psi ^\varepsilon _k (\cdot +\mathbf {e}_j) \psi _k^\varepsilon (\cdot ) \Big \rangle _{\Lambda _\varepsilon ^M} \\&\quad - \sum _\mathbf {e}\Big \langle \nabla ^\varepsilon _\mathbf {e}f^\varepsilon ,\, \sum _{k} \psi _k^\varepsilon (x) \int _{-T}^\infty \nabla ^\varepsilon _j P_M^\varepsilon (t-s,x-\cdot ) \nabla ^\varepsilon _\mathbf {e}\psi _k^\varepsilon (s,\cdot )\,ds \Big \rangle _{\Lambda _\varepsilon ^M} \\&\quad + \sum _\mathbf {e}\Big \langle \nabla ^\varepsilon _\mathbf {e}f^\varepsilon ,\, \sum _{k} \nabla ^\varepsilon _j \psi _k^\varepsilon (x) \int _{-T}^\infty P_M^\varepsilon (t-s,x-\cdot ) \nabla ^\varepsilon _\mathbf {e}\psi _k^\varepsilon (s,\cdot )\,ds \Big \rangle _{\Lambda _\varepsilon ^M} \Big ] \\&\quad + \Big \langle \Delta ^\varepsilon f^\varepsilon ,\,{\mathcal U}_j^\varepsilon \Big \rangle _{\Lambda _\varepsilon ^M} =0 \end{aligned}$$

where all the functions without time variables are evaluated at t, and the function ${\mathcal U}_j^\varepsilon (y)$, whose precise form will not actually matter much in the sequel, is given by

$$\begin{aligned}&\sum _{k} \int _{-T}^\infty \Big ( P_M^\varepsilon (t-s,x-y) \mathbf {E}\Big [ \psi ^\varepsilon _k (s,y) \,\nabla ^\varepsilon _j\psi _k^\varepsilon (x) \Big ]\\&\quad -\nabla ^\varepsilon _j P_M^\varepsilon (t-s,x-y) \mathbf {E}\Big [ \psi ^\varepsilon _k (s,y) \,\psi _k^\varepsilon (x)\Big ] \Big )\,ds . \end{aligned}$$

Summing over $j\in \{1,2\}$ and integrating by parts, noting that $f^\varepsilon$ is arbitrary, one obtains

$$\begin{aligned} (\text {div}^\varepsilon {\mathcal V}^\varepsilon )(t,y) + \Delta ^\varepsilon {\mathcal U}^\varepsilon (t,y) =0 \qquad \forall y\in \Lambda ^M_\varepsilon \end{aligned}$$

(B.5)

where the vector field ${\mathcal V}^\varepsilon = ({\mathcal V}^\varepsilon _1,{\mathcal V}^\varepsilon _2)$ is defined as

$$\begin{aligned}&{\mathcal V}^\varepsilon (y,y+\mathbf {e}_\ell ) {\mathop {=}\limits ^{ \text{ def }}}- \sum _{k} \delta (y-x) \mathbf {E}\Big [ \psi ^\varepsilon _k (t,y +\mathbf {e}_\ell ) \psi _k^\varepsilon (t,y)\Big ] \nonumber \\&\quad - \sum _{\mathbf {e}\in \{0,\mathbf {e}_\ell \}} \sum _{k,j} \int _{-T}^\infty \nabla ^\varepsilon _j P_M^\varepsilon (t-s,x-y-\mathbf {e}) \, \mathbf {E}\Big [ \nabla ^\varepsilon _\ell \psi _k^\varepsilon (s,y) \psi _k^\varepsilon (t,x) \Big ]\,ds \nonumber \\&\quad + \sum _{\mathbf {e}\in \{0,\mathbf {e}_\ell \}} \sum _{k,j} \int _{-T}^\infty P_M^\varepsilon (t-s,x-y-\mathbf {e}) \,\mathbf {E}\Big [ \nabla ^\varepsilon _\ell \psi _k^\varepsilon (s,y) \nabla ^\varepsilon _j \psi _k^\varepsilon (t,x)\Big ]\,ds \end{aligned}$$

(B.6)

for $\ell \in \{1,2\}$, and the function ${\mathcal U}^\varepsilon {\mathop {=}\limits ^{ \text{ def }}}\sum _{j=1}^2{\mathcal U}^\varepsilon _j$; we suppressed their dependence on M, T for cleaner notation. Note that the left hand side of (B.5) is analytic in $t>-T$ so once we have obtained (B.5) for small $t-(-T)$ we have it for any $t>-T$. We can now take $T\rightarrow \infty$, because the quantities ${\mathcal U}^\varepsilon$, ${\mathcal V}^\varepsilon$ are nothing but combinations of heat kernels, and are analytic in T so once (B.5) holds for a small range of T it can be extend for all $T>0$. (In particular this is just a property of heat kernels and we are not using any long-time existence of the SPDE.) Also since (B.5) holds for any $M\ge 1$ we can send $M\rightarrow \infty$. So (B.5) holds with $P^\varepsilon$ in place of $P^\varepsilon _M$ and the time integrals over entire $\mathbf{R}$. We then have that there exists a function ${\mathcal W}^\varepsilon$ such that¹⁹

$$\begin{aligned} {\mathcal V}^\varepsilon + \nabla ^\varepsilon {\mathcal U}^\varepsilon = \text {curl}^*{\mathcal W}^\varepsilon \quad \big (= (-\nabla ^\varepsilon _2 {\mathcal W}^\varepsilon , \nabla ^\varepsilon _1 {\mathcal W}^\varepsilon )\big ) . \end{aligned}$$

It is easy to see that ${\mathcal U}^\varepsilon$ and ${\mathcal V}^\varepsilon$ both decay to zero at infinity, so ${\mathcal W}^\varepsilon$ must go to a constant at infinity; we can well shift ${\mathcal W}^\varepsilon$ by constant so that ${\mathcal W}^\varepsilon$ also decays to zero at infinity. Summing the above identity over horizontal or vertical edges of $\mathbf{Z}^2_\varepsilon$, one has

$$\begin{aligned} \sum _{e\in {\mathcal E}^\ell (\mathbf{Z}^2_\varepsilon )} {\mathcal V}^\varepsilon (e)=0, \qquad \ell \in \{1,2\}. \end{aligned}$$

(B.7)

This is precisely what we use for cancellation of mass renormalization of the gauge field as well as the convergence of the composite field observable (see Lemma 6.6). (Alternatively this can be also done using Remark 5.7.)

Remark B.1

Identities of the type (B.5) are usually called Ward–Takahashi identities in QFT, see [BFS80, Eq. (A.3)] and [Bal83, Eq. (2.27)] similar identities in the context of Abelian gauge theory (i.e. Higgs model). Note that the “useless” term of the form $\Delta ^\varepsilon {\mathcal U}^\varepsilon$ did not appear in these references; this is probably because their gauge fixing condition amounts to imposing the gauge field to be divergence free, and without the div term in (B.1) the term $\Delta ^\varepsilon {\mathcal U}^\varepsilon$ would not appear. For a reader who is familiar with quantum electrodynamics (QED), (B.5) (without the term $\Delta ^\varepsilon {\mathcal U}^\varepsilon$) is reminiscent to the QED Ward identity $\sum _j k_j {\mathcal M}^j (k)=0$ in Fourier space where ${\mathcal M}(k)$ is an amplitude for some QED process involving an external photon (i.e. gauge field) with momentum k.