The source of trouble in the radial gauge is the term \(\partial \cdot V(0,l)\) in the function \({\dot{V}}^\mathrm {rg}(s,l)\). Our plan is to modify the definition (33) so as to eliminate this term. This will only be possible with some smearing of the central, reference point. The result of this program is the construction of an almost radial gauge potential \(A_b^\mathrm {ar}(x)\) in Formulas (49), (50), (51) and (72) below.
The starting point for the construction is the following variant of the identity (32)
where \(\xi \) is a real parameter, and to abbreviate notation we introduced a usual affine space symbol
To obtain this identity, we shift the argument of A and F in (32) by vector a, and write it with x replaced by y:
$$\begin{aligned} \lambda y^a F_{ab}(a+\lambda y)=\frac{\partial }{\partial \lambda }\big [\lambda A_b(a+\lambda y)\big ]-\frac{\partial }{\partial y^b}(y\cdot A(a+\lambda y)). \end{aligned}$$
Now the substitutions
and \(\lambda =\xi +1\) give the result.
Potential of Fast Decay
We start discussion with the case of \(A_b(x)\) a \(C^1\)-function of fast decay, together with its derivative. For such function one could think of integrating identity (37) with respect to \(\xi \) on \([0,\infty )\), which would correspond to the \(\lambda \)-integration of (32) on \([1,\infty )\). However, for F—a free electromagnetic field, which will be our object of study later, this would not be convergent for spacelike
. In that case, one has to exploit the oddness (in the spacetime) of its spacelike tail (and evenness of the tail of A, see (14)) and define integrals as appropriate limits. Anticipating this, we multiply (37) by \((-\tfrac{1}{2}){{\,\mathrm{sgn}\,}}(\xi )\) and integrate with respect to \(\xi \) on \({\mathbb {R}}\), which results in
Using a technique similar to that applied in Appendix B to analyze function \(r^a(x,z)\) defined below, it is easy to show that S(a, x) is regular outside \(x=a\), bounded by a constant, while \(|\partial ^xS(a,x)|\le \mathrm {const}(1+|a|)|x-a|^{-1}\) (for that, use the second formula in (39)). In order to remove the singularity, we smear (38) with a real Schwartz function \(\rho (a)\) on Minkowski space, such that
$$\begin{aligned} \int _M\rho (a)\,\mathrm {d}a=1,\quad \int _M \rho (a)a^\alpha \mathrm {d}a=0\quad \text {for}\quad 1\le |\alpha |\le n, \end{aligned}$$
(40)
for some arbitrarily chosen (large) \(n\in {\mathbb {N}}\cup \{\infty \}\). The smearing of \(\partial ^xS(a,x)\) may be pulled under the differential sign, and we obtain
In the first step above we substituted S(a, x) (39) and changed integration variable a to \(y=x+\xi (x-a)\). In the second step we substituted \(\xi =u^{-1}\).
We now introduce the vector function
$$\begin{aligned} r^a(x,z)=\tfrac{1}{2}z^a\int _{\mathbb {R}}\rho (x+uz)|u|^3\,\mathrm {d}u, \end{aligned}$$
(41)
which together with its derivatives with respect to x is estimated as in (83) and satisfies the distributional equation (see Appendix B)
$$\begin{aligned} \partial ^z\cdot r(x,z)=\delta (z), \end{aligned}$$
(42)
and define the almost radial gauge by
$$\begin{aligned}&=A_b(x)-\partial _bS^\mathrm {ar}(x), \end{aligned}$$
(44)
$$\begin{aligned} S^\mathrm {ar}(x)&=\int _M r(x,x-y)\cdot A(y)\,\mathrm {d}y=\int _M r(x,z)\cdot A(x-z)\,\mathrm {d}z. \end{aligned}$$
(45)
Formulas (42), (44) and (45) together show that our almost radial gauge is in the class of generalized ‘gauge-invariant gauges’ first postulated by Dirac [7] in an attempt to obtain ‘physical,’ out of local potentials (see also [25]). The generalization consists in nontrivial dependence of \(r^a(x,x-y)\) on the first argument. Potential \(A^\mathrm {ar}(x)\) is a continuous function, decaying as \(|x|^{-1}\) for large |x|, thus
$$\begin{aligned} |A^\mathrm {ar}(x)|\le \frac{\mathrm {const}}{1+|x|}. \end{aligned}$$
To estimate the radial component of \(A^\mathrm {ar}\), assume for simplicity that the support of \(\rho (a)\) is contained in a small ball \(|a|\le \delta \), and note that
. Then the contraction of (43) with x gives
$$\begin{aligned} |x\cdot A^\mathrm {ar}(x)|\le \delta \frac{\mathrm {const}}{1+|x|}. \end{aligned}$$
In the present case of a fast decaying function A, the ‘almost radial’ property has only a global aspect, appearance of \(\delta \) on the rhs. It is the case of free field discussed below, where this term has a fuller justification.
We note that a strict Dirac gauge with \(r(0,x-y)\) replacing \(r(x,x-y)\) in (45) is also possible, but it does not have the almost radial property.
Free Classical Field
We now assume that \(F_{ab}(x)\) is a free classical field obtained from the potential (4), where \({\dot{V}}(s,l)\) satisfies all restrictions formulated there. The definition (45) may not be directly applied in this case, but we shall find that an extension of this definition may be obtained as a limit of a regularized expression. We admit infrared singular fields, thus \(\Delta V(l)\) may be different from zero. For the sake of the construction of almost radial gauge at the classical level, it will prove convenient to use the convention \({\bar{V}}(s,l)\) (7). The choice (26) will be restored at the stage of quantization.
The following auxiliary functions will be needed:
$$\begin{aligned} \begin{aligned} \sigma (s,l)&=\tfrac{1}{2}\int _M{{\,\mathrm{sgn}\,}}(s-a\cdot l)\rho (a)\mathrm {d}a,\\ \zeta (s,l)&=\int _M\frac{\rho (a)}{s-a\cdot l}\,\mathrm {d}a =\int _{\mathbb {R}}\frac{{\dot{\sigma }}(\tau ,l)}{s-\tau }\,\mathrm {d}\tau ,\\ \eta (s,l)&=\int _M\frac{\rho (a)a}{s-a\cdot l}\,\mathrm {d}a =-\int _{\mathbb {R}}\frac{\partial \sigma (\tau ,l)}{s-\tau }\,\mathrm {d}\tau \end{aligned} \end{aligned}$$
(46)
(we assume \(\sigma (s,l)\) is defined by the above formula also in a neighborhood of the light cone, so \(\partial \sigma (s,l)\) is also unique in this case). Properties of these functions are discussed in Appendix C. Let \(\chi (x)\) be a Schwartz function, such that
$$\begin{aligned} \chi (-x)=\chi (x),\qquad \chi (0)=1, \end{aligned}$$
and for \(\delta >0\) denote (cf. (45))
$$\begin{aligned} S^\mathrm {ar}_\delta (x)=\int _M \chi (\delta z) r(x,z)\cdot A(x-z)\,\mathrm {d}z, \end{aligned}$$
(47)
with A(x) as described above. Then the following limit exists:
$$\begin{aligned} S^\mathrm {ar}(x)=\lim _{\delta \searrow 0}S^\mathrm {ar}_\delta (x) =\frac{1}{2\pi }\int \big [\eta (x\cdot l,l)-\zeta (x\cdot l,l)x\big ]\cdot {\bar{V}}(x\cdot l,l)\,\mathrm {d}^2l. \end{aligned}$$
(48)
We give the proof of this fact in Appendix D. Gradient of this function is easily obtained
$$\begin{aligned} \partial _bS^\mathrm {ar}(x) =\frac{1}{2\pi }\int \Big \{l_b\partial _s\big [(\eta (s,l)-\zeta (s,l)x)\cdot {\bar{V}}(s,l)\big ]-\zeta (s,l){\bar{V}}_b(s,l)\Big \}\Big |_{s=x\cdot l}\,\mathrm {d}^2l. \end{aligned}$$
We use an identity of the form (15) to transform
$$\begin{aligned} -\frac{1}{2\pi }x^a\int l_b \partial _s\big [{\bar{V}}_a(s,l)\zeta (s,l)\big ]\big |_{s=x\cdot l}\,\mathrm {d}^2l =\frac{1}{2\pi }\int \partial _a\big [l_b {\bar{V}}^a(s,l)\zeta (s,l)\big ]\big |_{s=x\cdot l}\,\mathrm {d}^2l. \end{aligned}$$
Taking into account that \(\partial _a[l_b{\bar{V}}^a\zeta ]={\bar{V}}_b\zeta +l_b\partial \cdot [{\bar{V}}\zeta ]\), we find
$$\begin{aligned} \partial _bS^\mathrm {ar}(x) =\frac{1}{2\pi }\int l_b\Big \{\partial _s\big [\eta (s,l)\cdot {\bar{V}}(s,l)\big ]+\partial \cdot \big [\zeta (s,l){\bar{V}}(s,l)\big ]\Big \}\Big |_{s=x\cdot l}\,\mathrm {d}^2l. \end{aligned}$$
We can now extend the definition of the almost radial gauge to the free field:
$$\begin{aligned} A^\mathrm {ar}_b(x)=A_b(x)-\partial _b S^\mathrm {ar}(x) =-\frac{1}{2\pi }\int {\dot{V}}^\mathrm {ar}_b(x\cdot l,l)\,\mathrm {d}^2l, \end{aligned}$$
(49)
where
$$\begin{aligned} {\dot{V}}^\mathrm {ar}(s,l)&=\dot{{\bar{V}}}(s,l)+l\,\Big \{\partial _s\big [\eta (s,l)\cdot {\bar{V}}(s,l)\big ] +\partial \cdot \big [\zeta (s,l){\bar{V}}(s,l)\big ]\Big \} \end{aligned}$$
(50)
$$\begin{aligned}&= \dot{{\bar{V}}}(s,l)+l\,\Big \{\eta (s,l)\cdot \dot{{\bar{V}}}(s,l) +\zeta (s,l)\partial \cdot {\bar{V}}(s,l)\Big \}, \end{aligned}$$
(51)
the second form (51) obtained with the use of identity (84). Our almost radial gauge is a functional of the smearing function \(\rho \) (40), which serves to define functions \(\zeta \) and \(\eta \) (46). Note that putting formally \(\rho (a)=\delta (a)\) (which, precisely, is not allowed), one has \(\zeta (s,l)=s^{-1}\), \(\eta (s,l)=0\) and (51) reproduces Formula (36) with the term \((\partial \cdot V)(0,l)\) omitted (which, again, would result in a singularity).
The almost radial gauge is again a Lorenz gauge satisfying the wave equation. It is also gauge invariant with respect to gauge transformations
$$\begin{aligned} A_b(x)\rightarrow A_b(x)+\partial _b\Lambda (x),\quad \Lambda (x)=-\frac{1}{2\pi }\int {\bar{\alpha }}(x\cdot l,l)\,\mathrm {d}^2l, \end{aligned}$$
which correspond to the replacement \({\bar{V}}_b(s,l)\rightarrow {\bar{V}}_b(s,l)+l_b{\bar{\alpha }}(s,l)\). This replacement leaves \({\dot{V}}_b^\mathrm {ar}(s,l)\) unchanged. To show this, we put \(l{\bar{\alpha }}(s,l)\) in place of \({\bar{V}}(s,l)\) in (51). Recall that to calculate \(\partial \cdot [l{\bar{\alpha }}(s,l)]\) one has to extend \(l{\bar{\alpha }}(s,l)\) to the neighborhood of the light cone respecting orthogonality to l, e.g., by
$$\begin{aligned} l_a{\bar{\alpha }}(s,l)\rightarrow [l_a{\bar{\alpha }}(s,l)]_\mathrm {ext} =[l_a-(l^0)^{-1}l^2\delta _a^0]\,{\bar{\alpha }}(s,l), \end{aligned}$$
and then on the light cone \(\partial \cdot [l{\bar{\alpha }}(s,l)]_\mathrm {ext} =(l\cdot \partial +2){\bar{\alpha }}(s,l)\). In this way the change in (51) becomes
$$\begin{aligned} l\zeta (s,l)\big (s\partial _s+l\cdot \partial +2\big ){\bar{\alpha }}(s,l)=0, \end{aligned}$$
where we used identity (85) and the homogeneity of degree \(-2\) of \({\bar{\alpha }}(s,l)\).
Similarly as in the radial gauge case, \({\dot{V}}^\mathrm {ar}(s,l)\) decays only as 1/s for \(|s|\rightarrow \infty \), so \(A^\mathrm {ar}(x)\) has no null asymptotes and decays as
$$\begin{aligned} |A^\mathrm {ar}(x)|\le \frac{\mathrm {const}}{1+|x^0|+|{\mathbf {x}}|}\log (2+|{\mathbf {x}}|). \end{aligned}$$
(52)
On the other hand, using identity (85) we find
$$\begin{aligned}&x\cdot {\dot{V}}^\mathrm {ar}(x\cdot l,l)\\ =&x\cdot \dot{{\bar{V}}}(x\cdot l,l)+(\partial \cdot {\bar{V}})(x\cdot l,l) +\eta (x\cdot l,l)\cdot \big [l\,\partial \cdot {\bar{V}}(x\cdot l,l)+x\cdot l\,\dot{{\bar{V}}}(x\cdot l,l)\big ], \end{aligned}$$
and by identity (15) we obtain
$$\begin{aligned} x\cdot A^\mathrm {ar}(x)= -\frac{1}{2\pi }\int \eta (s,l)\cdot \big [l\,\partial \cdot {\bar{V}}(s,l)+s\dot{{\bar{V}}}(s,l)\big ]\big |_{s=x\cdot l}\,\mathrm {d}^2l. \end{aligned}$$
(53)
The expression in square brackets in (53) is bounded by a constant, while \(\eta (s,l)\) satisfies (86), so by the use of estimate (82) we find
$$\begin{aligned} \begin{aligned} |x\cdot A^\mathrm {ar}(x)|&\le \mathrm {const}\int \frac{\mathrm {d}\Omega (l)}{(1+|x\cdot l|)^n}\\&\le \frac{\mathrm {const}}{1+|x^0|+|{\mathbf {x}}|} \bigg \{\theta (-x^2)+\frac{\theta (x^2)}{(1+|x^0|-|{\mathbf {x}}|)^{n-1}}\bigg \}. \end{aligned} \end{aligned}$$
(54)
In timelike directions the radial product \(x\cdot A^\mathrm {ar}(x)\) vanishes arbitrarily fast (depending on the choice of the function \(\rho (x)\), see (40)). Moreover, let us introduce the scaling \(\rho _\delta (a)=\delta ^{-4}\rho (a/\delta )\), which for \(\delta \searrow 0\) contracts the function to the central point, and denote by \(A^\mathrm {ar}_\delta \) the corresponding potential. Then \(\eta _\delta (s,l)=\eta (s/\delta ,l)\) and for \(x\cdot A^\mathrm {ar}_\delta (x)\) the bound (54) holds with the replacement \(x\rightarrow x/\delta \). Thus, outside \(x=0\) this quantity vanishes with \(\delta \) as
$$\begin{aligned} |x\cdot A^\mathrm {ar}_\delta (x)|\le \frac{\mathrm {const}\,\delta }{\delta +|x^0|+|{\mathbf {x}}|} \bigg \{\theta (-x^2)+\frac{\theta (x^2)\,\delta ^{n-1}}{(\delta +|x^0|-|{\mathbf {x}}|)^{n-1}}\bigg \}. \end{aligned}$$
Smearing with a Test Function
Let \(K^b(x)\) be a vector Schwartz test function (no conservation condition) and denote
$$\begin{aligned} W(s,l)=\int _M K(x)\delta (s-x\cdot l)\,\mathrm {d}x, \end{aligned}$$
(55)
which is a \(C^\infty \)-function, fast decaying in s, together with all its derivatives. Then
$$\begin{aligned} A^\mathrm {ar}(K)=\int _M A^\mathrm {ar}(x)\cdot K(x)\,\mathrm {d}x =-\frac{1}{2\pi }\int {\dot{V}}^\mathrm {ar}(s,l)\cdot W(s,l)\,\mathrm {d}s\,\mathrm {d}^2l. \end{aligned}$$
Setting here (50) and integrating by parts (use (2) to transfer the derivative \(\partial \)), one finds
$$\begin{aligned} \begin{aligned} A^\mathrm {ar}(K)&=\frac{1}{2\pi }\int {\bar{V}}(s,l)\cdot {\dot{V}}_K(s,l)\,\mathrm {d}s\,\mathrm {d}^2l\\&=-\frac{1}{2\pi }\int {\dot{V}}(s,l)\cdot {\bar{V}}_K(s,l)\,\mathrm {d}s\,\mathrm {d}^2l =\{{\bar{V}}_K,{\bar{V}}\}, \end{aligned} \end{aligned}$$
(56)
where
$$\begin{aligned} \begin{aligned} {\dot{V}}_K(s,l)&={\dot{W}}(s,l)+\eta (s,l)l\cdot {\dot{W}}(s,l)+\zeta (s,l)\partial (l\cdot W(s,l)),\\&={\dot{W}}(s,l)+\partial _s[\eta (s,l)l\cdot W(s,l)]+\partial [\zeta (s,l) l\cdot W(s,l)],\\ {\bar{V}}_K(s,l)&=\tfrac{1}{2}\int _{\mathbb {R}}{{\,\mathrm{sgn}\,}}(s-\tau ){\dot{V}}_K(\tau ,l)\,\mathrm {d}\tau . \end{aligned} \end{aligned}$$
(57)
It follows that
$$\begin{aligned} l\cdot {\bar{V}}_K(s,l)&=0, \end{aligned}$$
(58)
$$\begin{aligned} \Delta V_K(l)&=\partial \int _{\mathbb {R}}\zeta (s,l) l\cdot W(s,l)\,\mathrm {d}s. \end{aligned}$$
(59)
Orthogonality (58) is showed with the use of identity (85) as follows:
$$\begin{aligned} l\cdot {\dot{V}}_K(s,l)&=l\cdot {\dot{W}}(s,l)+(s\zeta (s,l)-1)l\cdot {\dot{W}}(s,l) +\zeta (s,l)l\cdot \partial (l\cdot W(s,l))\\&=\zeta (s,l)\big (s\partial _s+l\cdot \partial \big )(l\cdot W(s,l))=0, \end{aligned}$$
the last equality by the homogeneity of \(l\cdot W(s,l)\). Equality (59) is immediate from the second form of \({\dot{V}}_K\) in (57).
We note that the function \({\bar{V}}_K(s,l)\) satisfies all demands (5) with \(N=\infty \), (11) and (24) on a test function in the extended symplectic form (21), but in general—for non-conserved K—is outside the original test functions space.
More General Smearing
Now we want to extend the smearing \(A^\mathrm {ar}(K)\) to \(C^1\)-vector functions K(x), with the spacetime asymptotic behavior characteristic for currents of scattered free particles or fields. We shall show that with appropriately specific conditions on K, the function W(s, l) (55) is of class \(C^1\) and satisfies the bounds
$$\begin{aligned} |{\dot{W}}(s,l)|\le \frac{\mathrm {const}}{(1+|s|)^{1+\varepsilon }},\quad \big |\partial [l\cdot W(s,l)]\big |\le \frac{\mathrm {const}}{(1+|s|)^\varepsilon }. \end{aligned}$$
(60)
Therefore, the integrals in (56), with the definitions (57), are absolutely convergent for this extension. Moreover, \({\bar{V}}_K(s,l)\) thus formed satisfies conditions (5) with \(N=0\), (11) and (24). In addition, let g(x) be a Schwartz function, with \(g(0)=1\) and denote \(K_\delta (x)=g(\delta x)K(x)\). Then we shall find that irrespective of the shape of g the following extension of the identity (56) holds true
$$\begin{aligned} \begin{aligned} A^\mathrm {ar}(K)\equiv \lim _{\delta \searrow 0}A^\mathrm {ar}(K_\delta )&=\frac{1}{2\pi }\int {\bar{V}}(s,l)\cdot {\dot{V}}_K(s,l)\,\mathrm {d}s\,\mathrm {d}^2l\\&=-\frac{1}{2\pi }\int {\dot{V}}(s,l)\cdot {\bar{V}}_K(s,l)\,\mathrm {d}s\,\mathrm {d}^2l =\{{\bar{V}}_K,{\bar{V}}\}. \end{aligned} \end{aligned}$$
(61)
We first note that once the existence of W(s, l) with the properties (60) is established, the existence of \({\bar{V}}_K(s,l)\) with the property (5) with \(N=0\) and orthogonality (58) are obviously satisfied.
To specify the class for which the above results hold, it is convenient to characterize such class by splitting the function K into three \(C^1\)-contributions: \(K=K_1+K_2+K_3\), with the following properties:
$$\begin{aligned}&K_1(x)=x\, \kappa (x),\quad {{\,\mathrm{supp}\,}}\kappa \subseteq \{x^2\ge 0,\ |x^0|\ge \tfrac{1}{2}\}, \nonumber \\&\kappa (\lambda x)=\lambda ^{-4}\kappa (x)\qquad \text {for}\qquad \lambda \ge 1,\ |x^0|\ge 1,\\&|\kappa (x)|\le \frac{\mathrm {const}}{(1+|x|)^4},\quad |\partial \kappa (x)|\le \frac{\mathrm {const}}{(1+|x|)^5}, \nonumber \end{aligned}$$
(62)
$$\begin{aligned}&{{\,\mathrm{supp}\,}}K_2\subseteq \{x^2\ge 0\},\quad |K_2(x)|\le \frac{\mathrm {const}}{(1+|x|)^3},\quad \text {oscillatory part}, \end{aligned}$$
(63)
$$\begin{aligned}&|K_3(x)|\le \frac{\mathrm {const}}{(1+|x|)^{3+\varepsilon }},\quad |\partial K_3(x)|\le \frac{\mathrm {const}}{(1+|x|)^{4+\varepsilon }}. \end{aligned}$$
(64)
Condition (62) characterizes the dominant asymptotic behavior of particles, and the non-oscillatory part of asymptotic behavior of fields. The oscillatory contribution in the case of fields is represented by \(K_2\), bounded by (63), and we shall characterize it more precisely below. The rest is represented by \(K_3\), conditions (64) characterizing the next to leading behavior inside the light cone, and decay of K outside.Footnote 9
We prove our claims separately for each contribution \(K_i\), except for property (11), which will be discussed at the end.
We first note that all \(W_i(s,l)\), defined by (55) with K replaced by \(K_i\), are absolutely convergent and bounded. Moreover, in case \(i=3\) one has
$$\begin{aligned} |W_3(s,l)|,\ |L_{ab} W_3(s,l)|\le \frac{\mathrm {const}}{(1+|s|)^\varepsilon },\quad |{\dot{W}}_3(s,l)|\le \frac{\mathrm {const}}{(1+|s|)^{1+\varepsilon }}\,; \end{aligned}$$
all these statements are proved as in Lemma 20 and Theorem 21 in ref. [15]. Therefore, bounds (60) are satisfied for \(W_3\). Next, we observe that both
$$\begin{aligned} A^\mathrm {ar}(x)\cdot K_3(x)\quad \text {and}\quad A^\mathrm {ar}(x)\cdot K_1(x)=x\cdot A^\mathrm {ar}(x)\,\kappa (x) \end{aligned}$$
are absolutely integrable (see (52) and (54)), so the \(\delta \)-regularization in (61) is not needed for these contributions. Thus in case \(i=3\) we obtain
$$\begin{aligned} A(K_3)&=-\frac{1}{2\pi }\int _M\int {\dot{V}}^\mathrm {ar}(x\cdot l,l)\,\mathrm {d}^2l\, K_3(x)\,\mathrm {d}x\\&=-\frac{1}{2\pi }\int {\dot{V}}^\mathrm {ar}(s,l)\cdot W_3(s,l)\,\mathrm {d}s\,\mathrm {d}^2l, \end{aligned}$$
so the thesis for this contribution follows.
In case \(i=1\) we denote
$$\begin{aligned} U(s,l)=\int _M \kappa (x)\delta (s-x\cdot l)\,\mathrm {d}x,\qquad |U(s,l)|\le \frac{\mathrm {const}}{1+|s|} \end{aligned}$$
(the bound again by Lemma 20 in [15]), and observe that
$$\begin{aligned} \partial U(s,l)=-{\dot{W}}_1(s,l),\quad sU(s,l)=l\cdot W_1(s,l). \end{aligned}$$
(65)
Moreover, we note that the interior of the light cone (past and future) may be parametrized by \(x=\lambda v\), \(\lambda \in {\mathbb {R}}\) and v on the future hyperboloid \(v^2=1\), \(v^0\ge 0\), and then \(\mathrm {d}x=|\lambda |^3\mathrm {d}\lambda \,\mathrm {d}\mu (v)\), where \(\mathrm {d}\mu (v)=\mathrm {d}^3v/v^0\). It follows then from assumptions (62) that for \(|s|\ge l^0\) one has
$$\begin{aligned}&W_1(s,l)=\int {{\,\mathrm{sgn}\,}}(\lambda )\,v\,\kappa \big ({{\,\mathrm{sgn}\,}}(\lambda )v\big )\delta (s-\lambda v\cdot l)\,\mathrm {d}\lambda \,\mathrm {d}\mu (v)\nonumber \\&={{\,\mathrm{sgn}\,}}(s)\int \kappa \big ({{\,\mathrm{sgn}\,}}(s)v\big )\frac{v}{v\cdot l}\,\mathrm {d}\mu (v),\\&l\cdot W_1(s,l)={{\,\mathrm{sgn}\,}}(s)\int \kappa \big ({{\,\mathrm{sgn}\,}}(s)v\big )\,\mathrm {d}\mu (v).\nonumber \end{aligned}$$
(66)
Therefore,
$$\begin{aligned} {\dot{W}}_1(s,l)=0,\quad \partial [l\cdot W_1(s,l)]=0\quad \text {for}\quad |s|\ge l^0. \end{aligned}$$
We can now use (53) to obtain (we omit the arguments (s, l) for the sake of clarity)
$$\begin{aligned} \begin{aligned} A(K_1)&=-\frac{1}{2\pi }\int \eta \cdot \big [l\,\partial \cdot {\bar{V}}+s\dot{{\bar{V}}}\big ]U\,\mathrm {d}s\,\mathrm {d}^2l\\&=-\frac{1}{2\pi }\int \big [sU\partial _s({\bar{V}}\cdot \eta )+(1-s\zeta ){\bar{V}}\cdot \partial U\big ]\,\mathrm {d}s\,\mathrm {d}^2l, \end{aligned} \end{aligned}$$
where for the second equality we integrated \(\partial \) by parts and used the equality \(\partial (l\cdot \eta )=-s{\dot{\zeta }}\), which follows from identities (84) and (85). Performing now the substitutions (65) (for sU using the second of these relations in both places where it appears) and integrating \(\partial _s\) by parts, we obtain
$$\begin{aligned} A(K_1)=\frac{1}{2\pi }\int {\bar{V}}\cdot \big [{\dot{W}}_1 +\eta l\cdot {\dot{W}}_1+\zeta \partial (l\cdot W_1)\big ]\,\mathrm {d}s\,\mathrm {d}^2l, \end{aligned}$$
which is the required result.
We turn to the case \(i=2\). The decay of \(K_2\) is not sufficient to apply the proof presented in case \(i=3\). However, the oscillatory behavior of \(K_2\) damps the integrals, and we now add further assumption that \(W_2\) satisfies bounds as in case \(W_3\):
$$\begin{aligned} |W_2(s,l)|,\ |L_{ab} W_2(s,l)|\le \frac{\mathrm {const}}{(1+|s|)^\varepsilon },\quad |{\dot{W}}_2(s,l)|\le \frac{\mathrm {const}}{(1+|s|)^{1+\varepsilon }}. \end{aligned}$$
(67)
We denote
$$\begin{aligned} \begin{aligned} W_{2\delta }(s,l)=\int _M g(\delta x)K_2(x)\delta (s-x\cdot l)\,\mathrm {d}x, \end{aligned} \end{aligned}$$
(68)
and assume further that
$$\begin{aligned} |W_{2\delta }(s,l)|\le \frac{\mathrm {const}}{(1+|s|)^\varepsilon }. \end{aligned}$$
(69)
With these assumptions, we have
$$\begin{aligned} A(K_{2\delta })&=\int {\dot{V}}^\mathrm {ar}(s,l)\cdot W_{2\delta }(s,l)\,\mathrm {d}s\,\mathrm {d}^2l\\&\rightarrow \int {\dot{V}}^\mathrm {ar}(s,l)\cdot W_2(s,l)\,\mathrm {d}s\,\mathrm {d}^2l\quad \text {for}\quad \delta \searrow 0, \end{aligned}$$
the limit by the dominated convergence theorem. This, together with the estimates (67), leads to the thesis. We show in Appendix E that our assumptions on \(W_2\) are satisfied for a term of the type characteristic for the Dirac field.
Finally, we shall close the proof of our claims by showing (11), namely
$$\begin{aligned} \Delta V_K(l)=-\partial \Phi _K(l), \end{aligned}$$
(70)
with
$$\begin{aligned} \begin{aligned} \Phi _K(l)&= \int [\kappa (v)+\kappa (-v)]\rho (a)\log \frac{|a\cdot l|}{v\cdot l}\,\mathrm {d}\mu (v)\,\mathrm {d}a\\&-\int \zeta (s,l)l\cdot \big [W(s,l)-W({{\,\mathrm{sgn}\,}}(s)\infty ,l)\big ]\,\mathrm {d}s. \end{aligned} \end{aligned}$$
(71)
First, we observe that it is only \(W_1\) that contributes to \(W(\pm \infty ,l)\), so by (66) we have
$$\begin{aligned} W(\pm \infty ,l)=\pm \int \kappa (\pm v)\frac{v}{v\cdot l}\,\mathrm {d}\mu (v),\quad l\cdot W(\pm \infty ,l)=\pm \int \kappa (\pm v)\,\mathrm {d}\mu (v)\equiv Q_\pm . \end{aligned}$$
Differentiating the first line in (71) we obtain
$$\begin{aligned} \int [\kappa (v)+\kappa (-v)]\rho (a)\Big [\frac{a}{a\cdot l}&-\frac{v}{v\cdot l}\Big ]\,\mathrm {d}\mu (v)\,\mathrm {d}a\\&=\eta (0,l)(Q_- -Q_+)-W(+\infty ,l)+W(-\infty ,l). \end{aligned}$$
Differentiation of the second line in (71) gives
$$\begin{aligned} -\int \partial [\zeta (s,l) l\cdot W(s,l)]\,\mathrm {d}s+\int \partial \zeta (s,l)\,Q_{{{\,\mathrm{sgn}\,}}(s)}\,\mathrm {d}s. \end{aligned}$$
Using (84) in the second integral and summing the contributions, we obtain
$$\begin{aligned} \partial \Phi _K(l)=-W(+\infty ,l)+W(-\infty ,l)-\int \partial [\zeta (s,l) l\cdot W(s,l)]\,\mathrm {d}s, \end{aligned}$$
which substituted into (70) gives the correct value. Note that the differential \(\partial \) cannot be extracted outside the integral in the last formula, as without it the integrand is not integrable.
Quantum Field
We have shown in the classical case that \(A^\mathrm {ar}(K)=\{{\bar{V}}_K,{\bar{V}}\}\). However, recall the identity (31), which holds in each of the cases (26), that is both for ‘in’ and for ‘out’ algebra. Thus, on the algebraic level, in each of these two cases we propose to define the quantum almost radial gauge of the potential byFootnote 10
$$\begin{aligned} A^\mathrm {ar}(K)=\{V_K,V^\mathrm {q}\}, \end{aligned}$$
(72)
whenever \(V_K(s,l)\) defined by (57) and (26) falls in our extended symplectic space defined after (27). This condition demands stronger regularity properties of K than those assumed in Section 5.4, leading to classical version of (72). One can show that a class of such functions may be obtained by a modification of the currents J considered in [13], where the modification consists in demanding the continuity equation to be satisfied only asymptotically in time (in the form of an appropriately regular analogue of (62)). On the other hand, in concrete representations the scope of admitted test functions is wider, and the definition of the almost radial gauge may be also appropriately extended at the level of representation. We do not go into details here and leave the question to be decided in applications.
The almost radial gauge is a Lorenz gauge. For \(K(x)=\partial F(x)\) one finds that \(V_K(s,l)=W(s,l)\propto \,l\), which is in the zero equivalence class. Thus
$$\begin{aligned} A^\mathrm {ar}(\partial F)=0. \end{aligned}$$
Moreover, for \(K(x)=\Box G(x)\) we have \(W(s,l)=0\), so
$$\begin{aligned} A^\mathrm {ar}(\Box G)=0. \end{aligned}$$
For test functions K(x) in the class of conserved currents admitted in A(K) in one of the cases ‘in’ or ‘out,’ one has \(l\cdot W(s,l)=0\), so that \({\dot{V}}_K(s,l)={\dot{W}}(s,l)\) and \(V_K(s,l)=W(s,l)\). Therefore, in this case
$$\begin{aligned} A^\mathrm {ar}(K)=A(K), \end{aligned}$$
which reproduces the extended free algebra.
Finally, let us note the following important corollary. If K is a conserved, non-radiating current, such as the current of a free, charged classical particle, or of the free (classical or quantum) Dirac field, then \(W(s,l)=W(l)\) is constant in s and \(l\cdot W(l)=Q\), the charge of the current (see ref. [11], discussion starting with Eq. (2.70)). Therefore, for such currents \(V_K=0\) and
$$\begin{aligned} A^\mathrm {ar}(K)=0. \end{aligned}$$
Transformation and Commutation Properties
We recall that almost radial gauge is a functional of the function \(\rho \) serving to smear the point a from which the radial line to x is drawn. Let us make this dependence explicit by writing \(A^\mathrm {ar}(K;\rho )\). With Poincaré transformations of the extended algebra defined in (28), these elements have the covariance property given by
$$\begin{aligned} \alpha _{z,\Lambda }\big [A^\mathrm {ar}(K;\rho )\big ]=A^\mathrm {ar}\big (T_{z,\Lambda }K;T_{z,\Lambda }\rho \big ). \end{aligned}$$
(73)
In this sense \(A(K;\rho )\) may be regarded algebraically covariant, with unitarily implementable translations. For the proof of the covariance relation one notes that
$$\begin{aligned} W[T_{z,\Lambda }K](s,l)&=\Lambda W[K](s-z\cdot l,\Lambda ^{-1}l),\\ \zeta [T_{z,\Lambda }\rho ](s,l)&=\zeta [\rho ](s-z\cdot l,\Lambda ^{-1}l),\\ \eta [T_{z,\Lambda }\rho ](s,l)&=\Lambda \eta [\rho ](s-z\cdot l,\Lambda ^{-1}l)+z\zeta [\rho ](s-z\cdot l,\Lambda ^{-1}l). \end{aligned}$$
Setting these relations into Formula (57) for \(V_K\), and then using the result in (29), one arrives at the thesis.
Elements \(A^\mathrm {ar}(K;\rho )\) do not have compact localization even if the support of K is compact. However, the following remnant timelike locality with respect to A holds. Let the support of an admissible test vector function K and the support of a conserved test current J be timelike separated, and let V(s, l) for current J (16) have \(\Delta V(l)=0\) (in particular, this is always true for compactly supported J). Then
$$\begin{aligned}{}[A^\mathrm {ar}(K;\rho ),A(J)]=0, \end{aligned}$$
(74)
irrespective of the choice of \(\rho \). To show this, we first note that it is sufficient to show this when the two supports are placed inside the future and the past parts of the same light cone, with any vertex point b. Next, we apply to the commutator translation automorphism \(\alpha _{-b,{{\,\mathrm{\mathbb {1}}\,}}}\). The commutator, without changing its value (being proportional to identity), takes now the form \([A^\mathrm {ar}(K';\rho '),A(J')]\), where \(K'\) and \(J'\) are similarly separated as K and J, but with the vertex of the light cone in the origin. In consequence, the functions \(W'(s,l)\) for \(K'\) (55), and \(V'(s,l)\) for \(J'\) (16), are supported in \(s\in (0,+\infty )\) or \(s\in (-\infty ,0)\), each in a different of the two sets. Now, it follows from (57) that the support of \({\dot{V}}_{K'}(s,l)\) is not larger than the support of \(W'(s,l)\), so the same conclusion is valid for the pair \({\dot{V}}_{K'}(s,l)\) and \(V'(s,l)\). But from \(\Delta V'(l)=0\), together with (26), it follows that
$$\begin{aligned} \{V_{K'},V'\}=\frac{1}{2\pi }\int _{\mathbb {R}}{\dot{V}}_{K'}(s,l)\cdot V'(s,l)\,\mathrm {d}s\,\mathrm {d}^2l, \end{aligned}$$
so the thesis follows.