1 Introduction

This article is a further step in a program of investigation of the infrared problems in electrodynamics. Among them, the long-time asymptotic behavior of the charged matter fields is one of the key issues. Such questions as understanding what is a charged particle, or how to define scattering operator in quantum electrodynamics, are correlated to this problem.

It is well known that the long-time asymptotics poses problems in theories with long-range interactions. Standard way to deal with that is to modify the asymptotic dynamics of charged particles or fields, by Dollard or similar methods, augmented by some ‘dressing’ of charged particles. The cost is the loss of a clear interpretation of the asymptotic dynamics. Moreover, in physically realistic quantum field theory models this procedure has not resulted, up to now, in a non-perturbational understanding of the issue.Footnote 1

In a series of articles I have put forward the idea, that the long-time asymptotics problem may be relieved by an appropriate choice of gauge of the electromagnetic potential. In a recent article [11] a Schrödinger (nonrelativistic) particle was considered in a time-dependent electromagnetic field, of the form typical for scattering situations. It was found that an appropriate choice of gauge allows the existence of the asymptotic dynamics, with no need for dressing.Footnote 2 We also refer the reader to this article for more extensive description of the context and our motivation.

Here we implement the idea in the case of the classical Dirac field evolving in the external electromagnetic field. The system is not fully self-interacting, but the form of the external electromagnetic field mimics the expected properties of this field in fully interacting system. We consider the evolution of the Dirac field as a unitary evolution in a Hilbert space. However, similarly as in the previously considered nonrelativistic case, the evolution does not take place in standard time, over flat pure space sheets. It turns out, that for our purpose it is convenient to consider a Cauchy foliation supplied by constant \(\tau \) surfaces, where \((\tau ,\mathbf {z})\) form the coordinate systemFootnote 3 defined by:

$$\begin{aligned} x^0=\tau (|\mathbf {z}|^2+1)^\frac{1}{2}\,,\qquad \mathbf {x}=(\tau ^2+1)^\frac{1}{2}\mathbf {z}\,. \end{aligned}$$
(1)

This poses technical complication even at the free Dirac equation level, as the unitary evolution is not a unitary one-parameter group (Hamiltonian depends on \(\tau \)).Footnote 4 This is to be contrasted with the nonrelativistic case considered earlier, where the Niederer transformation introducing suitable coordinates leads from the free particle Hamiltonian to that of the harmonic oscillator (a kind of beautiful miracle). Nevertheless, the Schrödinger operator case becomes steeply difficult when the potential with space part is introduced, as then the term \(\mathbf {A}\cdot \mathbf {p}\) prevents the application of the Dyson series method. To deal with that case, we have applied the Kato theorem. Here, for the free Dirac equation we proceed differently, and our method may be of independent interest. The addition of the interaction with the electromagnetic field may be then treated, in contrast to the nonrelativistic Schrödinger case, by a variation of the Dyson method combined with the relativistic causality. For this system we show that an appropriate choice of gauge removes the asymptotic problem. We show the existence and asymptotic completeness of the wave operators, with no need for any modification of the asymptotic free Dirac evolution.Footnote 5

This result will show that the choice of a gauge \(\mathcal {A}(x)\) in this classical field setting has a decisive importance for the asymptotic identification of the incoming/outgoing charged fields. The main qualitative feature of gauges in this appropriate class is that the product \(x\cdot \mathcal {A}(x)\) vanishes in timelike directions, see remarks in Sect. 8 (a property already identified in [8]). We would like to stress that the theory is formulated from the outset in such chosen gauge. Whether such formulation may be carried over to the quantum field theory is a subject for future research.Footnote 6 This prospective investigation should also find contact with the asymptotic algebra of fields in quantum electrodynamics postulated by the author [9] (see also [10]).

Here let us only mention that the use of a gauge in the class anticipated above could lead to some broadening of the scope of external classical, time-dependent electromagnetic fields for which the scattering operator for the classical Dirac field may be lifted to the case of the respective quantum field. As is well known, the mixing of electrons/positrons leads to rather severe restrictions in this respectFootnote 7 (see, e.g., the monograph [17], Sections 2.4 and 2.5). Precisely what gain would be possible is an open question.

The outline of the article is as follows. In Sect. 2 we formulate the Dirac evolution as a unitary evolution, with time-dependent self-adjoint generators, over a rather general family of Cauchy surfaces. Section 3 gives the formulation of the external field problem with the Dyson series method, but with the important use of the relativistic causality. In both free and interacting case the self-adjointness is achieved with the use of the commutator theorem (see Appendix B) and the harmonic oscillator Hamiltonian. In Sect. 4 we specify our choice of coordinates to those mentioned above and transform the evolution to a new ‘picture.’ In this picture the free Dirac evolution on our foliation has well-defined limits as unitary operators. Section 5 specifies the general external problem to the electromagnetic case and discusses the gauge transformation. Section 6 contains our central results formulated for a wide class of electromagnetic fields: the existence and asymptotic completeness of the wave operators. Section 7 gives a theorem showing that the electromagnetic fields typical for scattering contexts described above admit potentials in gauges satisfying the demands of the main theorems of Sect. 6. Section 8 offers some remarks on the implications of our results and on further physically motivated restriction in the class of the obtained electromagnetic gauges.

Large parts of the material are shifted to Appendix. In Appendix A we discuss a spinor transformation needed in Sect. 2. Appendix B describes our method to deal with a class of time-dependent Hamiltonians. A lemma needed for the application of this method to the system considered here is discussed in Appendix C. In Appendix D we gather geometrical facts and relations in our special coordinate system. Appendices E and F recapitulate some properties of the solutions of the free Dirac and wave equations, respectively. Appendix G contains some estimates of the decay of the advanced and retarded solutions of the inhomogeneous wave equation and their differences (radiation fields). The results of Appendices E–G are applied next in Appendix H to the case of the electromagnetic fields typical for scattering contexts. Finally, the necessary decay properties of the special gauge introduced in Sect. 7 are obtained in Appendix I.

2 Free Dirac Evolution

Throughout the article we set \(\hbar =1\), \(c=1\). We choose a reference point, and then the flat spacetime is identified with the Minkowski vector space. Let m be the mass parameter in the free Dirac equation. To simplify notation, we rescale Minkowski vectors by multiplying them by m, and denote the resulting space by M, and its dimensionless vectors by \(x^a\). The flat (covariant) derivative in the rescaled Minkowski space is denoted by \(\nabla _a\). Also, the electromagnetic interaction to appear later will be introduced by the interaction term \(\mathcal {A}_a\overline{\psi }\gamma ^a\psi \), so to recover the physical units and quantities one should replace \(\mathcal {A}\rightarrow (e/m)\mathcal {A}\), with e the elementary charge.

Let \(\tau :M\mapsto {\mathbb {R}}\) be a smooth surjective function such that the hypersurfaces \(\Sigma _\tau \) of constant \(\tau \) form a Cauchy foliation of the Minkowski spacetime, with \(\tau \) increasing into the future. Consider the Dirac equation, which we write in the form

$$\begin{aligned} (\tfrac{1}{2}[\gamma ^a,i\nabla _a]_+-1)\psi =0 \end{aligned}$$
(2)

in our dimensionless coordinates, where \(\gamma ^a\) are the Dirac matrices and \([.,.]_+\) symbolizes the anticommutator. As is well known, the Cauchy problem for this equation with the initial data \(\psi |_{\Sigma _{\tau _0}}=f\) is explicitly solved by the formula

$$\begin{aligned} \psi (x)=i^{-1}\int _{\Sigma _{\tau _0}} S(x-y)\gamma ^a f(y)\, \hbox {d}\sigma _a(y)\,, \end{aligned}$$

where \(\hbox {d}\sigma _a\) is the dual integration element on \(\Sigma _{\tau _0}\) and S(x) is the standard Green function of the free Dirac field, in the dimensionless coordinates

$$\begin{aligned} S(x)=(i\gamma \cdot \nabla +1)D_1(x)\,,\quad D_1(x)=\frac{i}{(2\pi )^3} \int {{\,\mathrm{sgn}\,}}(v^0)\delta (v^2-1)e^{-iv\cdot x}\hbox {d}v\,. \end{aligned}$$

If f is a smooth bi-spinor function on \(\Sigma _{\tau _0}\), with a compact support, then \(\psi (x)\) is a smooth function in M, with compact support on each Cauchy surface \(\Sigma _{\tau }\). Therefore, we obtain a bijective evolution mapping between the spaces of smooth, compactly supported bi-spinor functions on our family of Cauchy surfaces. Moreover, if on each \(\Sigma _\tau \) one defines the scalar product

$$\begin{aligned} (\psi _1,\psi _2)_\tau =\int _{\Sigma _\tau } \overline{\psi _1}\gamma ^a\psi _2\,\hbox {d}\sigma _a\,, \end{aligned}$$
(3)

then the evolution is isometric.

Let now \((\zeta ^\mu )=(\zeta ^0,\zeta ^i)=(\tau ,z^i)\equiv (\tau ,\mathbf {z})\in {\mathbb {R}}^4\) (\(i=1,2,3\)), with \(\tau \) described above, be a smooth curvilinear coordinate system, mapping M diffeomorphically onto \({\mathbb {R}}^4\). Denote by \(\eta _{ab}\) and \(C^{a\ldots }{}_{b\ldots }\) the Minkowski spacetime metric tensor, and any other tensor, respectively. Then the geometrical components in the coordinate system \((\xi ^\mu )\) will be denoted by

$$\begin{aligned} \begin{aligned}&\partial _\mu =\frac{\partial }{\partial \zeta ^\mu }=\frac{\partial x^a}{\partial \zeta ^\mu }\nabla _a\,,\quad \partial _\tau =\frac{\partial }{\partial \tau }\,,\quad \partial _i=\frac{\partial }{\partial z^i}\,,\\&\hat{\gamma }^\mu =(\nabla _a\zeta ^\mu )\gamma ^a\,,\quad \hat{C}^{\mu \ldots }{}_{\nu \ldots } =(\nabla _a\zeta ^\mu )\ldots C^{a\ldots }{}_{b\ldots }\frac{\partial x^b}{\partial \zeta ^\nu }\ldots \,,\\&g_{\mu \nu }=\frac{\partial x^a}{\partial \zeta ^\mu }\frac{\partial x^b}{\partial \zeta ^\nu }\eta _{ab}\,,\quad g=\det (g_{\mu \nu })\,,\quad g_{(z)}=\det [(g_{ij})_{i,j\le 3}]\,, \end{aligned} \end{aligned}$$
(4)

Coordinates restricted to the indices 1, 2, 3 will be written as \(\hat{\gamma }^i\), \(g_{ij}\) and \(\hat{C}^{i\ldots }{}_{j\ldots }\), and the zeroth coordinate will be indicated by \(\tau \); thus, for instance: \(\mathcal {A}_a\gamma ^a=\hat{\mathcal {A}}_\mu \hat{\gamma }^\mu =\hat{\mathcal {A}}_\tau \hat{\gamma }^\tau +\hat{\mathcal {A}}_i\hat{\gamma }^i\).

In these coordinates, the Dirac equation (2) and the product (3) take the form

$$\begin{aligned}&\Big (\tfrac{i}{2}\Big [|g|^{\frac{1}{2}}\hat{\gamma }^\mu ,|g|^{-\frac{1}{2}} \partial _\mu \Big ]_+ -1\Big )\psi =0\,, \end{aligned}$$
(5)
$$\begin{aligned}&(\psi _1,\psi _2)_\tau =\int \overline{\psi _1}\hat{\gamma }^\tau \psi _2 \, |g|^{\frac{1}{2}}d^3z\,. \end{aligned}$$
(6)

We now choose a Minkowski reference system \((e_0,\ldots ,e_3)\). Let us denote

$$\begin{aligned} n=[g^{\tau \tau }]^{-\frac{1}{2}}\nabla \tau \end{aligned}$$
(7)

and let K be the Lorentz rotation of Dirac spinors in the hyperplane spanned by the pair of timelike unit vectors \((e_0,n)\), such that

$$\begin{aligned} K^{-1}\gamma ^a n_a\,K=\gamma ^0\,,\qquad K^\dagger =\gamma ^0K^{-1}\gamma ^0\,, \end{aligned}$$
(8)

where dagger \(\dagger \) denotes the Hermitian conjugation of matrices. The form and further properties of K are discussed in Appendix A. We define the following transformation:

$$\begin{aligned} \psi =T_\tau \chi \,,\qquad T_\tau =(|g|g^{\tau \tau })^{-\frac{1}{4}}K=|g_{(z)}|^{-\frac{1}{4}}K\,. \end{aligned}$$

The product (6) then takes the standard \({\mathbb {C}}^4\otimes L^2({\mathbb {R}}^3)\) form

$$\begin{aligned} (\psi _1,\psi _2)_\tau =\int \chi _1^\dagger \chi _2 d^3z\,. \end{aligned}$$

Setting \(\psi \) into (5) and applying from the left the transformation \((g^{\tau \tau })^{-\frac{1}{2}}T_\tau ^{-1}\) we obtain an equivalent form of the Dirac equation

$$\begin{aligned}&\Big (\tfrac{i}{2}\Big [(g^{\tau \tau })^{-\frac{1}{2}}\hat{\gamma }_K^\mu ,\partial _\mu \Big ]_+ -(g^{\tau \tau })^{-\frac{1}{2}}\Big )\chi \nonumber \\&\quad +\tfrac{i}{2}(g^{\tau \tau })^{-\frac{1}{2}} \Big [\hat{\gamma }_K^\mu ,(K^{-1}\partial _\mu K)\Big ]+\chi =0\,, \end{aligned}$$
(9)

where we have introduced notation

$$\begin{aligned} \gamma _K=K^{-1}\gamma K\,. \end{aligned}$$
(10)

We now make an additional assumption: \(\Sigma _\tau \) is rotationally symmetric in the chosen Minkowski system, that is \(\tau =\tau (x^0,|\mathbf {x}|)\). We show in Appendix A that in this case the anticommutator in the second line of (9) vanishes. We also note that \((g^{\tau \tau })^{-\frac{1}{2}}\hat{\gamma }_K^\tau =\gamma ^0 \equiv \beta \). This allows us to write the Dirac equation in the form

$$\begin{aligned} i\partial _\tau \chi =\Big (-\tfrac{i}{2}\big [(g^{\tau \tau })^{-\frac{1}{2}}\beta \hat{\gamma }_K^i,\partial _i\big ]_+ +(g^{\tau \tau })^{-\frac{1}{2}}\beta \Big )\chi \,. \end{aligned}$$

We summarize the above discussion.

Theorem 1

(i) Let the smooth function on the Minkowski space \(\tau =\tau (x^0,|\mathbf {x}|)\) determine its foliation by Cauchy surfaces \(\Sigma _\tau \), with \(\tau \) increasing into the future. Denote by \(C_0^\infty (\Sigma _\tau , {\mathbb {C}}^4)\) the space of smooth, compactly supported functions on \(\Sigma _\tau \) with values in \({\mathbb {C}}^4\).

Then the free Dirac evolution of initial data in \(C_0^\infty (\Sigma _\sigma , {\mathbb {C}}^4)\) is determined by a family of bijective linear evolution operators

$$\begin{aligned} U^\Sigma _0(\tau ,\sigma ): C_0^\infty (\Sigma _\sigma , {\mathbb {C}}^4) \mapsto C_0^\infty (\Sigma _\tau , {\mathbb {C}}^4) \end{aligned}$$

isometric with respect to the products (3). The size of the support in \(\Sigma _\tau \) is restricted by the size of the support in \(\Sigma _\sigma \) and the relativistic causality. By continuity, \(U_0^\Sigma (\tau ,\sigma )\) extend to unitary operators

$$\begin{aligned} U^\Sigma _0(\tau ,\sigma ): {\mathcal {H}}(\Sigma _\sigma )\mapsto {\mathcal {H}}(\Sigma _\tau )\,, \end{aligned}$$

where \({\mathcal {H}}(\Sigma _\tau )\) is the Hilbert space of spinor functions on \(\Sigma _\tau \) with the product (3).

(ii) Let \((\tau ,z^i):M\mapsto {\mathbb {R}}^4\) be a smooth diffeomorphism, with notation (4), such that \(\tau \) satisfies the assumptions of (i). Denote

$$\begin{aligned} {\mathcal {H}}={\mathbb {C}}^4&\otimes L^2({\mathbb {R}}^3,d^3z)\,,\nonumber \\ H_0(\tau )=\tfrac{1}{2}[\lambda ^i(\tau ),p_i]_+&+\mu (\tau )\,,\quad h_0=\tfrac{1}{2}(\mathbf {p}^2+\mathbf {z}^2)\,, \nonumber \\ p_i=-i\partial /\partial z^i\,,\qquad \mu (\tau )&=(g^{\tau \tau })^{-\frac{1}{2}}\beta \,, \qquad \lambda ^i(\tau )=\mu (\tau )\hat{\gamma }_K^i\,. \end{aligned}$$
(11)

Then the transformation

$$\begin{aligned} T_\tau :{\mathcal {H}}\mapsto {\mathcal {H}}(\Sigma _\tau )\,,\qquad T_\tau =|g_{(z)}|^{-1/4}K\,, \end{aligned}$$

with operator K discussed in Appendix A, is a unitary operator and the family

$$\begin{aligned} U_0(\tau ,\sigma )=T_\tau ^{-1}U_0^\Sigma (\tau ,\sigma )T_\sigma \,, \qquad U_0(\tau ,\sigma ):{\mathcal {H}}\mapsto {\mathcal {H}}\end{aligned}$$
(12)

forms a unitary, strongly continuous evolution system, such that the following holds:

  1. (A)

    \(U_0(\tau ,\sigma )C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)=C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\) and the relativistic causality is respected.

  2. (B)

    For \(\varphi \in C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\) the maps \((\tau ,\sigma )\mapsto H_0(\tau )U_0(\tau ,\sigma )\varphi \) and \((\tau ,\sigma )\mapsto h_0U_0(\tau ,\sigma )\varphi \) are strongly continuous, the map \((\tau ,\sigma )\mapsto U_0(\tau ,\sigma )\varphi \) is in the class \(C^1\) in the strong sense, and the following equations are satisfied

    $$\begin{aligned} i\partial _\tau U_0(\tau ,\sigma )\varphi&=H_0(\tau )U_0(\tau ,\sigma )\varphi \,,\\ i\partial _\sigma U_0(\tau ,\sigma )\varphi&=-U_0(\tau ,\sigma )H_0(\sigma )\varphi \,. \end{aligned}$$
  3. (C)

    The operators \(H_0(\tau )\) are symmetric on \(C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\).

Proof

The symmetry of \(H_0(\tau )\) follows from the symmetry of the operators \(\lambda ^i(\tau )\), \(\mu (\tau )\) and \(p_i\), and invariance of \(C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\) with respect to them. The other statements of the theorem follow easily from the discussion preceding the theorem. \(\square \)

We now add assumptions which will be satisfied in the context of our application, and which allow us to significantly extend the domains of validity of the results in the last theorem. We use the abbreviation

$$\begin{aligned} \langle z\rangle =(|\mathbf {z}|^2+1)^\frac{1}{2}\,. \end{aligned}$$

Also, the usual multi-index notation will be used.

Theorem 2

Let all the assumptions of Theorem 1 be satisfied. Suppose in addition that the following bounds of matrix norms hold:

$$\begin{aligned} \begin{aligned}&|\lambda ^i(\tau ,\mathbf {z})|\le C(\tau )\langle z\rangle \,,\quad |\partial _z^\alpha \lambda ^i(\tau ,\mathbf {z})|\le C(\tau )\,,\quad 1\le |\alpha |\le 3\\&|\partial _z^\beta \mu (\tau ,\mathbf {z})|\le C(\tau )\langle z\rangle ^{2-|\beta |}\,,\quad |\beta |\le 2\,, \end{aligned} \end{aligned}$$
(13)

where \(C(\tau )\) is a continuous function. Denote \(h_0=\tfrac{1}{2}(\mathbf {p}^2+\mathbf {z}^2\)), the self-adjoint harmonic oscillator Hamiltonian.

Then the following holds:

  1. (A)

    \(H_0(\tau )\) are essentially self-adjoint on \(C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\) and on each core of \(h_0\), and \({\mathcal {D}}(h_0)\subseteq {\mathcal {D}}(H_0(\tau ))\). Moreover, \(h_0^{-1}H_0(\tau )\) and \(H(\tau )h_0^{-1}\) extend to bounded operators, and as functions of \(\tau \) are strongly continuous.

  2. (B)

    The operator \(h_0U_0(\tau ,\sigma )h_0^{-1}\) is bounded, with the norm

    $$\begin{aligned} \Vert h_0U_0(\tau ,\sigma )h_0^{-1}\Vert \le \exp \left[ \mathrm {const}\int _\sigma ^\tau C(\rho )\hbox {d}\rho \right] \,, \end{aligned}$$

    so in particular

    $$\begin{aligned} U_0(\tau ,\sigma ){\mathcal {D}}(h_0)={\mathcal {D}}(h_0)\,, \end{aligned}$$

    and the map \((\tau ,\sigma )\mapsto h_0U(\tau ,\sigma )h_0^{-1}\) is strongly continuous.

  3. (C)

    For \(\varphi \in {\mathcal {D}}(h_0)\) the vector functions \(h_0U_0(\tau ,\sigma )\varphi \), \(H_0(\tau )U_0(\tau ,\sigma )\varphi \) and \(U_0(\tau ,\sigma )H_0(\sigma )\varphi \) are strongly continuous with respect to parameters, the vector function \(U_0(\tau ,\sigma )\varphi \) is of the class \(C^1\) with respect to parameters in the strong sense and the following equations hold

    $$\begin{aligned} i\partial _\tau U_0(\tau ,\sigma )\varphi&=H_0(\tau ) U_0(\tau ,\sigma )\varphi \,, \\ i\partial _\sigma U_0(\tau ,\sigma )\varphi&=-U_0(\tau ,\sigma )H_0(\sigma )\varphi \,. \end{aligned}$$
  4. (D)

    Relativistic causality is respected by \(U_0(\tau ,\sigma )\). Therefore, if we denote

    $$\begin{aligned}&{\mathcal {H}}\supset {\mathcal {H}}_c\ \text {--the subspace of functions with compact essential support}\,,\\&{\mathcal {D}}_c(\mathbf {p}^2)={\mathcal {H}}_c\cap {\mathcal {D}}(\mathbf {p}^2)\,, \end{aligned}$$

    then

    $$\begin{aligned} U_0(\tau ,\sigma ){\mathcal {H}}_c={\mathcal {H}}_c\,,\quad U_0(\tau ,\sigma ){\mathcal {D}}_c(\mathbf {p}^2)={\mathcal {D}}_c(\mathbf {p}^2)\,. \end{aligned}$$

Proof

It follows from the assumptions on \(\lambda ^i\) and \(\mu \) that \(H_0(\tau )\) fulfill all the conditions imposed on the operator h in Lemma 14 in Appendix C. In Theorems 12 and 13 in Appendix B, we now put

$$\begin{aligned} {\mathcal {D}}=C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\,,\quad h(\tau )=H_0(\tau )\,,\quad u(\tau ,\sigma )=U_0(\tau ,\sigma ) \end{aligned}$$

and \(h_0\) as defined in the present assumptions. Then, by the results of Theorem 1, all the assumptions of these theorems are satisfied and the present thesis follows. \(\square \)

3 Evolution in External Field

We now consider the Dirac equation in an external field, of the form

$$\begin{aligned} (\tfrac{1}{2}[\gamma ^a,i\nabla _a]_+-1-V)\psi =0\,, \end{aligned}$$

where V(x) is a matrix function satisfying the condition

$$\begin{aligned} V(x)^\dagger =\gamma ^0V(x)\gamma ^0\,, \end{aligned}$$
(14)

which guarantees the reality of the interaction term \(-\overline{\psi }V\psi \) in the Lagrangian and the conservation of the current \(\overline{\psi }\gamma ^a\psi \). The addition of this interaction term leads to the modification of the free curvilinear version to the equation of the form

$$\begin{aligned} i\partial _\tau \varphi =(\tfrac{1}{2}[\lambda ^i,p_i]_++\mu +W)\varphi \,, \end{aligned}$$
(15)

where

$$\begin{aligned} W(\tau ,\mathbf {z})=\mu K^{-1}V(x)K=W(\tau ,\mathbf {z})^\dagger \,, \end{aligned}$$

Hermicity being equivalent to the property (14) [see (8)].

Using the interaction picture technique, we shall obtain the evolution operators for which Eq. (15) is satisfied. However, thanks to the relativistic causality we can extend the applicability of the technique to the following setting, in which the operators \(W(\tau )\) do not need to be bounded.

Theorem 3

Let all the assumptions of Theorems 1 and 2 be satisfied. Suppose in addition that \(W(\tau ,\mathbf {z})\) is a Hermitian matrix function such that the mappings

$$\begin{aligned} \big \{{\mathbb {R}}\ni \tau \mapsto \partial _z^\alpha W(\tau ,\mathbf {z})\big \} \in C^0({\mathbb {R}}, L_\mathrm {loc}^\infty ({\mathbb {R}}^3))\,,\qquad |\alpha |\le 2\,, \end{aligned}$$
(16)

and

$$\begin{aligned} \Vert \langle z\rangle ^{-2+|\alpha |}\partial _z^\alpha W(\tau ,\mathbf {z})\Vert _\infty \le C(\tau )\,,\qquad |\alpha |\le 1\,, \end{aligned}$$
(17)

where \(C(\tau )\) is a continuous function. Denote

$$\begin{aligned} H(\tau )=H_0(\tau )+W(\tau )\,, \end{aligned}$$

with the initial domain \(C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\), and define the formal series

$$\begin{aligned} \begin{aligned} U(\tau ,\sigma )&=\sum _{n=0}^\infty U^{(n)}(\tau ,\sigma )\,, \qquad U^{(0)}(\tau ,\sigma )=\mathbf {1}\,,\\ U^{(n)}(\tau ,\sigma )&=(-i)^n \int \limits _{\tau \ge \tau _{n} \ge \ldots \ge \tau _1\ge \sigma } U_0(\tau ,\tau _n)W(\tau _n)U_0(\tau _n,\tau _{n-1})\ldots \\&\quad \ldots W(\tau _1)U_0(\tau _1,\sigma )\mathrm{d}\tau _n\ldots \mathrm{d}\tau _1\\&=-i\int \limits _\sigma ^\tau U_0(\tau ,\rho )W(\rho )U^{(n-1)}(\rho ,\sigma )\mathrm{d}\rho \\&=-i\int \limits _\sigma ^\tau U^{(n-1)}(\tau ,\rho )W(\rho )U_0(\rho ,\sigma )\mathrm{d}\rho \,, \qquad n\ge 1\,. \end{aligned} \end{aligned}$$
(18)

Then the following is true:

  1. (A)

    \(H(\tau )\) are essentially self-adjoint on \(C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\) and on each core of \(h_0\), and \({\mathcal {D}}(h_0)\subseteq {\mathcal {D}}(H(\tau ))\).

  2. (B)

    Series \(U(\tau ,\sigma )\) and its conjugation converge strongly on \({\mathcal {H}}_c\), and the limit operator extends to a unitary propagator on \({\mathcal {H}}\), strongly continuous in its parameters.

  3. (C)

    \(U(\tau ,\sigma )\) respects relativistic causality and

    $$\begin{aligned} U(\tau ,\sigma ){\mathcal {D}}_c(\mathbf {p}^2)={\mathcal {D}}_c(\mathbf {p}^2)\,. \end{aligned}$$
  4. (D)

    For \(\varphi \in {\mathcal {D}}_c(\mathbf {p}^2)\) the vector functions \(h_0U(\tau ,\sigma )\varphi \), \(H(\tau )U(\tau ,\sigma )\varphi \) and \(U(\tau ,\sigma )H(\sigma )\varphi \) are strongly continuous with respect to parameters, the function \(U(\tau ,\sigma )\varphi \) is of the class \(C^1\) with respect to parameters in the strong sense and the following equations hold

    $$\begin{aligned} \begin{aligned} i\partial _\tau U(\tau ,\sigma )\varphi&=H(\tau ) U(\tau ,\sigma )\varphi \,,\\ i\partial _\sigma U(\tau ,\sigma )\varphi&=-U(\tau ,\sigma )H(\sigma )\varphi \,. \end{aligned} \end{aligned}$$
    (19)
  5. (E)

    Unitary operators satisfying (D), with \(U(\sigma ,\sigma )=\mathbf {1}\), are unique.

Proof

The proof of (A) is similar as the proof in the free case, statement (A) in Theorem 2. One has to note that the assumption (17) ensures the validity of items (i) and (ii) in Lemma 14; the result of item (iii) is not needed.

Consider the series (18). As each of the operators \(U_0(\tau ,\tau _n)\), \(U_0(\tau _k,\tau _{k-1})\) and \(U_0(\tau _1,\sigma )\) in the definition of \(U^{(n)}(\tau ,\sigma )\) respects causality, and multiplication by \(W(\tau _k,\mathbf {z})\) does not enlarge the support of the function, each of the operators \(U^{(n)}(\tau ,\sigma )\) respects causality. Let, for chosen T and r,

$$\begin{aligned} \tau ,\sigma \in [-T,T]\,,\quad \varphi \in {\mathcal {H}}\,,\quad \mathop {\mathrm{ess}\,\mathrm{supp}}\limits \varphi \subseteq \{\mathbf {z}\mid |\mathbf {z}|\le r\}\,, \end{aligned}$$

which is assumed for the rest of this proof. Then there exists R(Tr) such that the essential support of all functions \(U^{(n)}(\tau ,\sigma )\varphi \) is contained in \(|\mathbf {z}|\le R(T,r)\) (uniformly with respect to \(\tau \), \(\sigma \) and \(\varphi \) in the assumed range). Let \(J(\mathbf {z})\) be a smooth function such that \(J(\mathbf {z})=1\) for \(|\mathbf {z}|\le R\) and \(J(\mathbf {z})=0\) for \(|\mathbf {z}|\ge R+1\). Denote by \(U_J(\tau ,\sigma )\) and \(U_J^{(n)}(\tau ,\sigma )\) the series (18) and its terms in which \(W(\rho ,\mathbf {z})\) have been replaced by \(W_J(\rho ,\mathbf {z})=J(\mathbf {z})W(\rho ,\mathbf {z})\); note that by assumption (16) \(W_J(\rho )\) is a bounded, strongly continuous operator, with \(\Vert W_J(\rho )\Vert \le d_1\) for some constant \(d_1\), uniformly on \(\rho \in [-T,T]\). Using the first of the recursive relations in (18), it is now easy to see that in the given setting

$$\begin{aligned} U^{(n)}(\tau ,\sigma )\varphi =U_J^{(n)}(\tau ,\sigma )\varphi \,. \end{aligned}$$

Arguing as in the proof of the Dyson expansion (see Thm.  X.69 in [15]), one shows that \(U_J(\tau ,\sigma )\) converges uniformly to a strongly continuous unitary propagator. As the space \({\mathcal {H}}_c\) is dense in \({\mathcal {H}}\), the statement (B) is proved. Also, the causality is respected.

To prove (C) (the invariance of the subspace) and (D), we write

$$\begin{aligned} h_0W_J(\rho ) h_0^{-1}=[h_0,W_J(\rho )]h_0^{-1}+W_J(\rho ) \end{aligned}$$

and note that by Theorem 12 applied to \(h(\tau )=W_J(\tau )\) (with Lemma 14) the first term on the rhs is a bounded operator, strongly continuous by the assumption (16). Therefore, the operator \(h_0W_J(\rho )h_0^{-1}\) is also bounded and strongly continuous, and \(\Vert h_0W_J(\rho )h_0^{-1}\Vert \le d_2\) for some constant \(d_2\), uniformly on \(\rho \in [-T,T]\). Taking also into account statement (B) in Theorem 2, we conclude that \(h_0U_J^{(n)}(\tau ,\sigma )h_0^{-1}\) is bounded, strongly continuous, with the norm estimated by

$$\begin{aligned} \Vert h_0 U_J^{(n)}(\tau ,\sigma )h_0^{-1}\Vert \le (n!)^{-1}(d_1+d_2)^n \exp \left[ \mathrm {const}\int _\sigma ^\tau C(\rho )\hbox {d}\rho \right] \,, \end{aligned}$$

This implies that the series \(\sum _nh_0U_J^{(n)}(\tau ,\sigma )h_0^{-1}\) converges uniformly to the strongly continuous function \(h_0U_J(\tau ,\sigma )h_0^{-1}\). Let us now impose on \(\varphi \) a stronger assumption that \(\varphi \in {\mathcal {D}}_c(\mathbf {p}^2)\). Then

$$\begin{aligned} h_0U(\tau ,\sigma )\varphi =h_0U_J(\tau ,\sigma )\varphi =h_0U_J(\tau ,\sigma )h_0^{-1}h_0\varphi \,. \end{aligned}$$

This proves (C), and also the strong continuity of \(h_0U(\tau ,\sigma )\varphi \) in (D). The strong continuity of \(H(\tau )U(\tau ,\sigma )\varphi \) and \(U(\tau ,\sigma )H(\sigma )\varphi \) follows now from the strong continuity of \(H(\tau )h_0^{-1}\), similarly as in Theorem 2, basing on Theorem 12 and assumptions (16) and (17).

Next, we note that

$$\begin{aligned} U_0(0,\tau )U^{(n)}(\tau ,\sigma )\varphi =-i\int _\sigma ^\tau U_0(0,\rho )W(\rho )U^{(n-1)}(\rho ,\sigma )\varphi \, \hbox {d}\rho \,, \end{aligned}$$

so this vector function is continuously differentiable in \(\tau \) in the strong sense and

$$\begin{aligned} i\partial _\tau [U_0(0,\tau )U^{(n)}(\tau ,\sigma )]\varphi = U_0(0,\tau )W(\tau )U^{(n-1)}(\tau ,\sigma )\varphi \end{aligned}$$

(remember that on the rhs W may be replaced by \(W_J\)). Again, using the uniform convergence of \(U_J(\tau ,\sigma )\) we obtain

$$\begin{aligned} i\partial _\tau [U_0(0,\tau )U(\tau ,\sigma )]\varphi =U_0(0,\tau )W(\tau )U(\tau ,\sigma )\varphi \,, \end{aligned}$$

and the rhs is strongly continuous in \((\tau ,\sigma )\). Finally, differentiating

$$\begin{aligned} U(\tau ,\sigma )\varphi =U_0(\tau ,0)[U_0(0,\tau )U(\tau ,\sigma )]\varphi \end{aligned}$$

by the Leibnitz rule and using (C) from Theorem 2 we arrive at the first equation in (19). The uniqueness (E) follows easily from this equation. The proof of the second equation in (19) is similar, with the use of the second of the recursive relations in (18); we omit the details. \(\square \)

The most general matrix field V(x) satisfying condition (14) may be concisely represented in the form

$$\begin{aligned} V=\sum _{k=0}^4 i^{\frac{1}{2}k(k-1)}C^k_{a_1\ldots a_k}\gamma ^{a_1}\ldots \gamma ^{a_k}\,, \end{aligned}$$

where the tensor fields \(C^k_{a_1\ldots a_k}\) are real and antisymmetric. The corresponding form of the Hermitian matrix function W is

$$\begin{aligned} W=\mu \sum _{k=0}^4 i^{\frac{1}{2}k(k-1)} \hat{C}^k_{\mu _1\ldots \mu _k}\hat{\gamma }^{\mu _1}_K\ldots \hat{\gamma }^{\mu _k}_K\,. \end{aligned}$$

This form encompasses the scalar (\(k=0\)) and pseudoscalar (\(k=4\)) potentials, the electromagnetic vector potential (\(k=1\)) and the pseudovector potential (\(k=3\)), the interactions characteristic for anomalous magnetic and electric moments, as well as the linearized gravitation (\(k=2\)).

4 Special Foliation, New Picture and Free Asymptotics

We now choose the foliation \(\tau \) and the variables \(\mathbf {z}\) by

$$\begin{aligned} x^0=\tau \langle z\rangle \equiv \tau (|\mathbf {z}|^2+1)^{\frac{1}{2}}\,,\qquad \mathbf {x}=\langle \tau \rangle \mathbf {z}\equiv (\tau ^2+1)^{\frac{1}{2}}\mathbf {z}\,, \end{aligned}$$
(20)

and we also denote

$$\begin{aligned} \langle \tau |z\rangle =(\tau ^2+|\mathbf {z}|^2+1)^{\frac{1}{2}}\,. \end{aligned}$$

The idea behind this choice is that for \(\tau \) tending to \(\pm \infty \) the Cauchy surfaces should tend, for timelike directions in the spacetime, to the hyperboloids \(x^2=\tau ^2\). All geometrical facts on these curvilinear coordinates needed for our purposes are gathered in Appendix D.

From now on, we adopt the coordinate system (20). Using the properties of this system, it is then easy to see that all the conditions of Theorems 1 and 2 are satisfied. Also, a convenient property of these coordinates is that \(g_{\tau i}=0\). Therefore, we find

$$\begin{aligned} {[}\hat{\gamma }_K^i,\beta ]_+=(g^{\tau \tau })^{-\frac{1}{2}}K^{-1}[\hat{\gamma }^i,\hat{\gamma }^\tau ]_+ K =2(g^{\tau \tau })^{-\frac{1}{2}}g^{i\tau }=0\,, \end{aligned}$$

and then

$$\begin{aligned}&{[}\lambda ^i,\beta ]_+=0=[\lambda ^i,\mu ]_+\,, \end{aligned}$$
(21)
$$\begin{aligned}&\tfrac{1}{2}[\lambda ^i,\lambda ^j]_+=-g_{\tau \tau }g^{ij}\mathbf {1}\equiv \rho ^{ij}\mathbf {1}\,; \end{aligned}$$
(22)

the symmetric form \(\rho \) is positive definite.

The free evolution \(U_0(\tau ,\sigma )\) may be easily expressed in the coordinate system (20). For this purpose, it is sufficient to consider \(U_0(\tau ,0)\). Using the representation of the Dirac solution (103) in Appendix E and the definition (12), for \(\varphi \in {\mathcal {S}}({\mathbb {R}}^3,{\mathbb {C}}^4)\) we obtain

$$\begin{aligned} {[}U_0(\tau ,0)\varphi ](\mathbf {z})&=\left( \frac{\langle \tau |z\rangle }{\langle \tau \rangle \langle z\rangle }\right) ^{\frac{1}{2}}K(\tau ,\mathbf {z})^{-1}\nonumber \\&\quad \times \left( \frac{\langle \tau \rangle }{2\pi }\right) ^{\frac{3}{2}}\int \left[ e^{-ix(\tau ,\mathbf {z})\cdot v}P_+(v) {-}e^{ix(\tau ,\mathbf {z})\cdot v}P_-(v)\right] [{\mathcal {F}}^{-1}\varphi ](v)\hbox {d}\mu (v)\,, \end{aligned}$$
(23)

where \(x(\tau ,\mathbf {z})=(\tau \langle z\rangle ,\langle \tau \rangle \mathbf {z})\). We consider the asymptotic form of this evolution for \(\tau \rightarrow \pm \infty \). Let \(\varphi \in {\mathcal {F}}C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\) and restrict \(\mathbf {z}\) to a compact set. Then both \(\mathbf {v}\) (the space part of v) and \(\mathbf {z}\) are restricted to bounded sets and by the stationary phase method (see, e.g., [19]) the leading asymptotic behavior of the second line in (23) is given by

$$\begin{aligned} \mp i\big [e^{-i(\tau \pm \frac{\pi }{4})}P_+(z^0,\pm \mathbf {z}) +e^{i(\tau \pm \frac{\pi }{4})}P_-(z^0,\pm \mathbf {z})\big ] {[}{\mathcal {F}}^{-1}\varphi ](z^0,\pm \mathbf {z}) +O(\langle \tau \rangle ^{-1})\,, \end{aligned}$$
(24)

where \(z^0=\langle z\rangle \), the upper/lower signs \(\mp \) and \(\pm \) correspond to the limits \(\tau {\rightarrow } {\pm } {\infty }\), respectively, and the rest is bounded uniformly for \(\mathbf {z}\) in the given set. The limit of the term on the rhs of (23) in the first line is equal to \(\langle z\rangle ^{-\frac{1}{2}}K(\pm \infty ,\mathbf {z})^{-1}\), with

$$\begin{aligned} K(\pm \infty ,\mathbf {z})=2^{-\frac{1}{2}}\big [(1+\langle z\rangle )^{\frac{1}{2}} \pm (1+\langle z\rangle )^{-\frac{1}{2}}\gamma ^0\mathbf {z}\cdot \varvec{\gamma }\big ]=K(\infty ,\pm \mathbf {z})\,, \end{aligned}$$
(25)

and again the rest is bounded by \(\mathrm {const}\langle \tau \rangle ^{-1}\), uniformly in the given set. We now note two facts:

$$\begin{aligned}&K(\infty ,\mathbf {v})^{-1} v\cdot \gamma \, K(\infty ,\mathbf {v})=\beta \,,\\&e^{-i\sigma }\tfrac{1}{2}(\mathbf {1}+\beta ) +e^{i\sigma }\tfrac{1}{2}(\mathbf {1}-\beta ) =e^{-i\sigma \beta }\,. \end{aligned}$$

The first identity is the limit form of the first of relations in (8), but it may also be checked directly. The second identity is most easily evaluated on the two complementary eigenspaces of \(\beta \) (with eigenvalues \(\pm 1\)).

Setting the asymptotic forms (24) and (25) into formula (23), and using the above identities (the second one with \(\sigma =\tau \pm \frac{\pi }{4}\)), we obtain

$$\begin{aligned} {[}U_0(\tau ,0)\varphi ](\mathbf {z})&= \mp ie^{-i(\tau \pm \frac{\pi }{4})\beta } \langle z\rangle ^{-\frac{1}{2}}K(\infty ,\pm \mathbf {z})^{-1} [{\mathcal {F}}^{-1}\varphi ](z^0,\pm \mathbf {z})\nonumber \\&\quad +O(\langle \tau \rangle ^{-1})\,. \end{aligned}$$
(26)

The last formula suggests the definition of the following unitary transformation:

$$\begin{aligned} \Phi (\tau )=\exp [-i\tau \beta ]\,, \end{aligned}$$

and the associated change of the evolution ‘picture’ (also in the interacting case):

$$\begin{aligned} \begin{aligned}&U_{0\Phi }(\tau ,\sigma )=\Phi (\tau )^*U_0(\tau ,\sigma )\Phi (\sigma )\,, \quad U_\Phi (\tau ,\sigma )=\Phi (\tau )^*U(\tau ,\sigma )\Phi (\sigma )\,,\\&H_{0\Phi }(\tau )=\Phi (\tau )^*H_0(\tau )\Phi (\tau )-\beta \,,\\\&H_{\Phi }(\tau )=H_{0\phi }(\tau )+W_\Phi (\tau )\,, \quad W_\Phi (\tau )=\Phi (\tau )^* W(\tau )\Phi (\tau )\,. \end{aligned} \end{aligned}$$
(27)

Remark 4

Under the conditions of Theorem 3, all statements of its thesis remain valid with the replacements defined by Eqs. (27).

This follows quite trivially, as \(\Phi (\tau )\) acts only on the factor \({\mathbb {C}}^4\) in the Hilbert space \({\mathcal {H}}={\mathbb {C}}^4\otimes L^2({\mathbb {R}}^3)\).

After that, we go back to the asymptotics of the free evolution, where we shall need the parity operator

$$\begin{aligned} {[}{\mathcal {P}}\varphi ](\mathbf {z})=\varphi (-\mathbf {z})\,. \end{aligned}$$

Moreover, we observe that the map

$$\begin{aligned} {[}{\mathcal {K}}\varphi ](\mathbf {z})=\langle z\rangle ^{\frac{1}{2}}K(\infty ,\mathbf {z})\varphi (\mathbf {z}) \end{aligned}$$

is a unitary operator \({\mathcal {H}}\mapsto L^2_\gamma (H)\) (with the latter space defined in Appendix E).

Theorem 5

The following strong limits exist as unitary operators in \({\mathcal {H}}\):

$$\begin{aligned}&U_{0\Phi }(+\infty ,0)=\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow +\infty }U_{0\Phi }(\tau ,0) =-ie^{-i\frac{\pi }{4}\beta }{\mathcal {K}}^{-1}{\mathcal {F}}^{-1}\,,\\&U_{0\Phi }(-\infty ,0)=\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow -\infty }U_{0\Phi }(\tau ,0) =ie^{+i\frac{\pi }{4}\beta }{\mathcal {P}} {\mathcal {K}}^{-1}{\mathcal {F}}^{-1}\,,\\&U_{0\Phi }(0,+\infty )=\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow +\infty }U_{0\Phi }(0,\tau ) =i{\mathcal {F}}{\mathcal {K}}e^{+i\frac{\pi }{4}\beta }\,,\\&U_{0\Phi }(0,-\infty )=\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow -\infty }U_{0\Phi }(0,\tau ) =-i{\mathcal {F}}{\mathcal {K}}{\mathcal {P}}e^{-i\frac{\pi }{4}\beta }\,,\\&U_{0\Phi }(0,\pm \infty )=U_{0\Phi }(\pm \infty ,0)^*\,. \end{aligned}$$

Therefore,Footnote 8

$$\begin{aligned} U_{0\Phi }(+\infty ,-\infty )=i\beta {\mathcal {P}}\,. \end{aligned}$$

Proof

For \(\psi \in C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\) and \(\varphi \in {\mathcal {F}}C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\) formula (26) gives

$$\begin{aligned}&\lim _{\tau \rightarrow +\infty }(\psi ,U_{0\Phi }(\tau ,0)\varphi ) =(\psi ,(-i)e^{-i\frac{\pi }{4}\beta }{\mathcal {K}}^{-1}{\mathcal {F}}^{-1}\varphi )\,,\\&\lim _{\tau \rightarrow -\infty }(\psi ,U_{0\Phi }(\tau ,0)\varphi ) =(\psi ,ie^{+i\frac{\pi }{4}\beta }{\mathcal {P}} {\mathcal {K}}^{-1}{\mathcal {F}}^{-1}\varphi )\,. \end{aligned}$$

Both subspaces are dense in \({\mathcal {H}}\), so the weak operator limits result. But the limit operators are evidently unitary, so the weak limits imply the strong limits of \(U_{0\Phi }(\tau ,0)\), as well as its conjugate. \(\square \)

5 Electromagnetic Interaction and Gauge Transformation

In the rest of this article we are interested in the standard, minimal coupling electromagnetic interaction. For the electromagnetic field \(F_{ab}\), we reserve notation \(A_a\) for the Lorenz gauge potential (fully specified in what follows). We write \(\mathcal {A}_a\) for the potential in a general gauge to be used in the Dirac equation. Also, we recall our conventions defined in formulas (4) and the following remarks, so \(\hat{F}_{\mu \nu }\), \(\hat{A}_\mu \) and \(\hat{\mathcal {A}}_\mu \) are components of these fields in our coordinate system, and for \(\mu ,\nu =i,j\), etc. the range is restricted to values 1, 2, 3.

Therefore, in the electromagnetic case the field V and its transformed version W are

$$\begin{aligned} V(x)=\mathcal {A}_a(x)\gamma ^a\,,\qquad W(\tau ,\mathbf {z}) =\hat{\mathcal {A}}_\tau (\tau ,\mathbf {z})+\hat{\mathcal {A}}_i(\tau ,\mathbf {z})\lambda ^i(\tau ,\mathbf {z})\,, \end{aligned}$$
(28)

and we write the Hamiltonian as

$$\begin{aligned} H=\tfrac{1}{2}[\lambda ^i,\pi _i]_++\mu +\hat{\mathcal {A}}_\tau \,,\qquad \pi _i=p_i+\hat{\mathcal {A}}_i\,. \end{aligned}$$

Theorem 6

The coordinate system \((\tau ,\mathbf {z})\) given by (20) is assumed.

(i) Let the electromagnetic potential \(\mathcal {A}(x)\) have components \(\hat{\mathcal {A}}_\mu (\tau ,\mathbf {z})\) such that for all indices \(\mu =\tau ,1,2,3\) the mappings

$$\begin{aligned} \big \{{\mathbb {R}}\ni \tau \mapsto \partial _z^\alpha \hat{\mathcal {A}}_\mu (\tau ,\mathbf {z})\big \} \in C^0({\mathbb {R}},L^\infty _\mathrm {loc}({\mathbb {R}}^3))\,,\quad |\alpha |\le 2\,, \end{aligned}$$
(29)

and

$$\begin{aligned} \left. \begin{array}{l} \Vert \langle z\rangle ^{-2+|\alpha |}\partial _z^\alpha \hat{\mathcal {A}}_\tau (\tau ,\mathbf {z})\Vert _\infty \\ \Vert \langle z\rangle ^{-1+|\alpha |}\partial _z^\alpha \hat{\mathcal {A}}_i(\tau ,\mathbf {z})\Vert _\infty \end{array}\right\} \le C(\tau )\,,\quad |\alpha |\le 1\,, \end{aligned}$$
(30)

where \(C(\tau )\) is a continuous function. Then the conditions of Theorem 3 are fulfilled and the unitary propagator \(U(\tau ,\sigma )\) with the listed properties is obtained.

(ii) Let the potential \(\mathcal {A}(x)\) and the corresponding propagator \(U(\tau ,\sigma )\) be as in (i). Define a new gauge

$$\begin{aligned} \mathcal {A}^{\mathcal {G}}=\mathcal {A}-\nabla {\mathcal {G}}\,, \end{aligned}$$

where \({\mathcal {G}}(x)\) is a gauge function such that for all indices \(\mu =\tau ,1,2,3\) the mappings

$$\begin{aligned} \big \{{\mathbb {R}}\ni \tau \mapsto \partial _z^\alpha \partial _\mu {\mathcal {G}}(\tau ,\mathbf {z})\big \} \in C^0({\mathbb {R}},L^\infty _\mathrm {loc}({\mathbb {R}}^3))\,,\quad |\alpha |\le 1\,, \end{aligned}$$
(31)

and

$$\begin{aligned} \left. \begin{array}{lll} \Vert \langle z\rangle ^{-2+|\alpha |}\partial _z^\alpha \partial _\tau {\mathcal {G}}(\tau ,\mathbf {z})\Vert _\infty \\ \Vert \langle z\rangle ^{-1+|\alpha |}\partial _z^\alpha \partial _i{\mathcal {G}}(\tau ,\mathbf {z})\Vert _\infty \end{array} \right\} \le C(\tau )\,,\quad |\alpha |\le 1\,, \end{aligned}$$
(32)

where \(C(\tau )\) is a continuous function. Denote by \(H^{\mathcal {G}}(\tau )\) the interacting Hamiltonian with \(\hat{\mathcal {A}}\) replaced by \(\hat{\mathcal {A}}^{\mathcal {G}}\), and

$$\begin{aligned} U^{\mathcal {G}}(\tau ,\sigma )= e^{i {\mathcal {G}}(\tau )}U(\tau ,\sigma )e^{-i {\mathcal {G}}(\sigma )}\,. \end{aligned}$$

Then for such modified operators the statements (A) and (C)–(E) of Theorem 3 are satisfied.

Proof

  1. (i)

    The bounds (13) are satisfied in our coordinate system. Together with the assumed properties of \(\hat{\mathcal {A}}_\mu \), this ensures that the interaction term (28) satisfies the assumptions of Theorem 3, so the thesis follows.

  2. (ii)

    For (A) we note that the assumption (32) implies that the interaction term \(W^{\mathcal {G}}=\hat{\mathcal {A}}^{\mathcal {G}}_\tau +\hat{\mathcal {A}}^{\mathcal {G}}_i\lambda ^i\) satisfies assumptions imposed on M in Lemma 14 (i) and (ii) (not necessarily (iii)). This is sufficient for the conclusion on self-adjointness of \(H^{\mathcal {G}}\) obtained as in Theorem 3. Moreover, it follows from (31) that \(e^{i{\mathcal {G}}(\tau )}{\mathcal {D}}_c(\mathbf {p}^2)={\mathcal {D}}_c(\mathbf {p}^2)\), so (C) is satisfied. Finally, for \(\varphi \in {\mathcal {D}}_c(\mathbf {p}^2)\) we have

    $$\begin{aligned}&i\partial _\tau e^{i{\mathcal {G}}(\tau )}\varphi =-e^{i{\mathcal {G}}(\tau )}\partial _\tau {\mathcal {G}}(\tau )\varphi \,,\\&e^{i{\mathcal {G}}(\tau )}H(\tau )e^{-i{\mathcal {G}}(\tau )}\varphi =[H(\tau )-\lambda ^i(\tau )\partial _i{\mathcal {G}}(\tau )]\varphi \,, \end{aligned}$$

    so the remaining statements easily follow with the use of (31).

\(\square \)

6 Scattering

We come here to our main objective in this article, scattering of the Dirac field in an external time-dependent electromagnetic field. With the assumptions of Theorem 6, augmented by some decay conditions formulated in the two assumptions below, we shall obtain the complete description of scattering in terms of the Cauchy surfaces of constant \(\tau \), as to be found in Theorem 7. The existence of the wave operators needs only a rather simple additional Assumption I, but their unitarity and completeness are more demanding, and they follow from Assumption II. It is with regard to this latter question that we need to discuss some further notation and properties.

It is easy to see that the operators

$$\begin{aligned} \tilde{H}=\tfrac{1}{2}[\lambda ^i,\pi _i]_+=H-\mu -\hat{\mathcal {A}}_\tau \,, \quad \tilde{H}_0=\tfrac{1}{2}[\lambda ^i,p_i]_+=H_0-\mu \end{aligned}$$
(33)

have similar properties as H and \(H_0\), in particular they are essentially self-adjoint on \(C_{0}^{\infty }({\mathbb {R}}^3, {\mathbb {C}}^4)\), so \({\mathcal {D}}_c(\mathbf {p})={\mathcal {H}}_c\cap {\mathcal {D}}(\mathbf {p})\) is contained in their domains. Therefore, \({\mathcal {D}}_c(\mathbf {p}^2)\) is contained in the domains of \(\tilde{H}^2\) and \(\tilde{H}_0^2\) [as well as \(H^2\) and \(H_0^2\)]. Moreover, we note for later use that

$$\begin{aligned} {[}\tilde{H},\beta ]_+=[\tilde{H}_0,\beta ]_+=0\,, \end{aligned}$$
(34)

which is a consequence of (21).

We shall need a more explicit form of \(\tilde{H}^2\) below. We calculate (with \(\rho \) defined in (22))

$$\begin{aligned} \tilde{H}^2&=(\pi _i\lambda ^i+\tfrac{i}{2}\partial \cdot \lambda )(\lambda ^j\pi _j-\tfrac{i}{2}{}\partial \cdot \lambda )\nonumber \\&=\pi _i\rho ^{ij}\pi _j+\tfrac{1}{2}\pi _i[\lambda ^i,\lambda ^j]\pi _j +\tfrac{i}{2}(\partial \cdot \lambda )\lambda ^j\pi _j-\tfrac{i}{2}\pi _j\lambda ^j(\partial \cdot \lambda ) +\tfrac{1}{4}(\partial \cdot \lambda )^2\,. \end{aligned}$$
(35)

The second term on the rhs above may be written in two alternative ways:

$$\begin{aligned} \begin{aligned} \tfrac{1}{2}\pi _i[\lambda ^i,\lambda ^j]\pi _j=-\tfrac{i}{2}(\partial _i[\lambda ^i,\lambda ^j])\pi _j +\tfrac{1}{2}[\lambda ^i,\lambda ^j]\pi _i\pi _j\\ =\tfrac{i}{2}\pi _i(\partial _j[\lambda ^i,\lambda ^j]) +\tfrac{1}{2}\pi _i\pi _j[\lambda ^i,\lambda ^j]\,. \end{aligned} \end{aligned}$$
(36)

Taking into account that \([\lambda ^i,\lambda ^j]\) is antisymmetric in the indices, one can replace the product \(\pi _i\pi _j\) by \(\frac{1}{2}[\pi _i,\pi _j]=-\frac{i}{2}\hat{F}_{ij}\), and then replace \([\lambda ^i,\lambda ^j]\) multiplying this expression by \(2\lambda ^i\lambda ^j\). Taking now one half of the sum of the two expressions in (36), we find

$$\begin{aligned} \tfrac{1}{2}\pi _i[\lambda ^i,\lambda ^j]\pi _j=\tfrac{i}{4}\big [\pi _j, \partial _i[\lambda ^j,\lambda ^i]\big ]_+ -\tfrac{i}{2}\lambda ^i\lambda ^j\hat{F}_{ij}\,. \end{aligned}$$
(37)

For the next two terms on the rhs of (35) we note

$$\begin{aligned} \tfrac{i}{2}(\partial \cdot \lambda )\lambda ^j\pi _j-\tfrac{i}{2}\pi _j\lambda ^j(\partial \cdot \lambda ) =-\tfrac{i}{4}[\pi _j,[\lambda ^j,\partial \cdot \lambda ]]_+-\tfrac{1}{4}\partial _j[\partial \cdot \lambda ,\lambda ^j]_+\,, \end{aligned}$$
(38)

which is shown in a somewhat similar way as the former identity. The sum of the first terms on the rhs of (37) and (38) gives \(\tfrac{i}{4}[\pi _j,[\partial _i\lambda ^j,\lambda ^i]]_+\). In this way we obtain

$$\begin{aligned} \tilde{H}^2&=\pi _i\rho ^{ij}\pi _j+\tfrac{i}{4}[\partial _i\lambda ^j,\lambda ^i]\pi _j +\tfrac{i}{4}\pi _j[\partial _i\lambda ^j,\lambda ^i]\\&\quad -\tfrac{1}{4}\partial _j([\lambda ^j,\partial \cdot \lambda ]_+)+\tfrac{1}{4}(\partial \cdot \lambda )^2 -\tfrac{i}{2}\lambda ^i\lambda ^j\hat{F}_{ij}\,. \end{aligned}$$

With the use of further notation

$$\begin{aligned}&\Lambda _j=\tfrac{i}{4}(\rho ^{-1})_{jk}[\partial _i\lambda ^k,\lambda ^i]\,, \quad \pi _{\Lambda i}=\pi _i+\Lambda _i\,, \end{aligned}$$
(39)
$$\begin{aligned}&Q=\tfrac{1}{4}\partial _j([\lambda ^j,\partial \cdot \lambda ]_+)-\tfrac{1}{4}(\partial \cdot \lambda )^2\,, \quad N=Q+\Lambda _i\rho ^{ij}\Lambda _j\,,\\&{\mathcal {B}}=\tfrac{i}{2}\lambda ^i\lambda ^j\hat{F}_{ij}\,,\nonumber \end{aligned}$$
(40)

we can write

$$\begin{aligned} \tilde{H}^2=\pi _{\Lambda }\rho \pi _{\Lambda }-N-{\mathcal {B}}\,, \end{aligned}$$
(41)

where in the first term on the rhs a symbolic notation for summation over indices is used. A straightforward calculation shows that both Q and N are positive numerical functions (times the unit matrix; see Appendix D, formula (96) for Q, and then for N this is obvious). Therefore, if we further denote

$$\begin{aligned} s=\sqrt{\rho }\,, \end{aligned}$$
(42)

then for \(\varphi \in {\mathcal {D}}_c(\mathbf {p}^2)\) we have

$$\begin{aligned} \Vert \tilde{H}\varphi \Vert \le \big (\Vert s\pi _\Lambda \varphi \Vert ^2-(\varphi ,{\mathcal {B}}\varphi )\big )^{\frac{1}{2}} \le \Vert s\pi _\Lambda \varphi \Vert +|(\varphi ,{\mathcal {B}}\varphi )|^{\frac{1}{2}} \end{aligned}$$
(43)

Next denote

$$\begin{aligned} X=\tfrac{1}{2}\mu ^{-1} \hat{\mathcal {A}}_i\lambda ^i=\tfrac{1}{2}\beta a_i\lambda ^i\,, \qquad a_i=(g^{\tau \tau })^{\frac{1}{2}}\hat{\mathcal {A}}_i\,. \end{aligned}$$
(44)

Below we shall need the following identity valid, with our assumptions on \(\hat{\mathcal {A}}_i\), on \({\mathcal {D}}_c(\mathbf {p})\):

$$\begin{aligned} \beta [X,\tilde{H}]=a_i\rho ^{ij}\pi _{\Lambda j}-\tfrac{i}{2}\lambda ^i\lambda ^j\partial _i a_j -\tfrac{i}{2}(\partial _i\rho ^{ij})a_j\,. \end{aligned}$$
(45)

To show this, we note that \([X,\tilde{H}]=\tfrac{1}{2} \beta [\lambda ^ia_i,\tilde{H}]_+\), write \(\tilde{H}=\lambda ^j\pi _j -\tfrac{i}{2}\partial \cdot \lambda \), and then the lhs of (45) takes the form

$$\begin{aligned}&\tfrac{1}{2}a_i[\lambda ^i,\lambda ^j]_+\pi _j-\tfrac{i}{2}\lambda ^j\partial _j(\lambda ^i a_i) -\tfrac{i}{4}a_i [\lambda ^i,\partial _j\lambda ^j]_{+}\\&\quad =a_i\rho ^{ij}\pi _{\Lambda j}-\tfrac{i}{2}\lambda ^j\lambda ^i\partial _j a_i -\tfrac{i}{2}a_i\big (\lambda ^j\partial _j\lambda ^i+\tfrac{1}{2}[\lambda ^i,\partial _j\lambda ^j]_+ +\tfrac{1}{2}[\partial _j\lambda ^i,\lambda ^j]\big )\,, \end{aligned}$$

where after the equality sign we have added and subtracted the term \(a_i\rho ^{ij}\Lambda _j\). It is now easy to show that the terms in parentheses multiplying \(a_i\) sum up to \(\tfrac{1}{2}\partial _j[\lambda ^j,\lambda ^i]_+\), which ends the proof of (45).

The spreading of the past and future is characterized in our coordinate system by Lemma 15 in Appendix D. Denote

$$\begin{aligned} {\mathcal {D}}(r,\mathbf {p}^2)=\{\psi \in {\mathcal {D}}(\mathbf {p}^2)\mid \psi (\mathbf {z})=0\ \text {for}\ |\mathbf {z}|\ge r\}\,, \end{aligned}$$

so that

$$\begin{aligned} {\mathcal {D}}_c(\mathbf {p}^2)=\bigcup _{r>0}{\mathcal {D}}(r,\mathbf {p}^2)\,. \end{aligned}$$

We set \(\tau _0=0\) in this lemma, and replace \(r_0\) and r by r and \(r(\tau )\), respectively, so that

$$\begin{aligned} r(\tau )=\langle r \rangle |\tau |+r\langle \tau \rangle \,. \end{aligned}$$

Then, according to this lemma, and statement (C) of Theorem 3, we have

$$\begin{aligned} \psi \in {\mathcal {D}}(r,\mathbf {p}^2)\quad \Longrightarrow \quad U(\tau ,0)\psi \in {\mathcal {D}}(r(\tau ),\mathbf {p}^2)\,. \end{aligned}$$
(46)

For a measurable function \(f(\tau ,\mathbf {z})\) we define a semi-norm function

$$\begin{aligned} \tau \mapsto \Vert f\Vert _{r,\tau }=\mathop {\mathrm{ess}\,\mathrm{sup}}\limits _{|\mathbf {z}|\le r(\tau )}|f(\tau ,\mathbf {z})|\,, \end{aligned}$$

and then for \(\psi \) as in (46) we find

$$\begin{aligned} \Vert f(\tau ,.)U(\tau ,0)\psi \Vert \le \Vert f\Vert _{r,\tau }\Vert \psi \Vert \,. \end{aligned}$$
(47)

Note that according to Lemma 15 we have

$$\begin{aligned} |\mathbf {z}|\le r(\tau )\ \Longrightarrow \ \langle z\rangle \le \langle r \rangle \langle \tau \rangle +r|\tau |\,, \end{aligned}$$

so in that case

$$\begin{aligned} |\mathbf {z}|\le \langle z\rangle \le 2\langle r \rangle \langle \tau \rangle \,. \end{aligned}$$

For any measurable function \(k(\mathbf {z})\) and \(r>0\) we shall denote

$$\begin{aligned} \Vert k\Vert _r=\mathop {\mathrm{ess}\,\mathrm{sup}}\limits _{|\mathbf {z}|\le r}|k(\mathbf {z})|\,. \end{aligned}$$

For a function \(f(\tau ,\mathbf {z})\) we obviously have \(\Vert f(\tau ,.)\Vert _r\le \Vert f(\tau ,.)\Vert _{r,\tau }\).

Our scattering theorem will apply to potentials satisfying the following conditions of increasing restrictiveness.

Assumption I. Potential \(\hat{\mathcal {A}}_\mu (\tau ,\mathbf {z})\) satisfies the assumptions of Theorem 6, and in addition the mapping \(\tau \mapsto \hat{\mathcal {A}}_i(\tau ,\mathbf {z}) \) is in \(C^1({\mathbb {R}},L^\infty _\mathrm {loc}({\mathbb {R}}^3))\). For each \(r>0\) the following expressions are integrable on \({\mathbb {R}}\) with respect to \(\tau \):

$$\begin{aligned} \Vert \hat{\mathcal {A}}_\tau \Vert _r\,,\quad \langle \tau \rangle ^{-2}\Vert \hat{\mathcal {A}}_i\Vert _r^2\,,\quad \langle \tau \rangle ^{-2}\Vert \partial _i\hat{\mathcal {A}}_j\Vert _r\,,\quad \langle \tau \rangle ^{-1}\Vert \partial _\tau \hat{\mathcal {A}}_i\Vert _r\,. \end{aligned}$$

Assumption II. Potential \(\hat{\mathcal {A}}_\mu (\tau ,\mathbf {z})\) satisfies the assumptions of Theorem 6, and in addition the mapping \( \tau \mapsto \hat{\mathcal {A}}_i(\tau ,\mathbf {z})\) is in \(C^1({\mathbb {R}},L^\infty _\mathrm {loc}({\mathbb {R}}^3))\). For each \(r>0\) the following expressions are integrable on \({\mathbb {R}}\) with respect to \(\tau \):

$$\begin{aligned}&\Vert \hat{\mathcal {A}}_\tau \Vert _{r,\tau }\,,\\&\langle \tau \rangle ^{-2}\big (\Vert \hat{\mathcal {A}}_i\Vert _{r,\tau }^2+\Vert z^i\hat{\mathcal {A}}_i\Vert _{r,\tau }^2\big )\,,\\&\langle \tau \rangle ^{-2}\big (\Vert \partial _i\hat{\mathcal {A}}_j\Vert _{r,\tau }+\Vert z^i\partial _i\hat{\mathcal {A}}_j\Vert _{r,\tau } +\Vert z^j\partial _i\hat{\mathcal {A}}_j\Vert _{r,\tau }+\Vert z^iz^j\partial _i\hat{\mathcal {A}}_j\Vert _{r,\tau }\big )\,,\\&\langle \tau \rangle ^{-1}\big (\Vert \partial _\tau \hat{\mathcal {A}}_i\Vert _{r,\tau }+\Vert z^i\partial _\tau \hat{\mathcal {A}}_i\Vert _{r,\tau }\big )\,. \end{aligned}$$

Moreover, let \(\xi :[1,\infty )\mapsto {\mathbb {R}}\) be an appropriately chosen smooth, positive, nondecreasing function, such that \(\xi (1)=1\) and

$$\begin{aligned} \xi '(u)\le \frac{\kappa }{u}\xi (u)\,,\ \kappa \in (0,\tfrac{1}{2})\,, \qquad |\xi ''(u)|\le \frac{\mathrm {const}}{u}\xi (u)\,, \end{aligned}$$
(48)

where in the second bound the constant is arbitrary. The following bounds are satisfied

$$\begin{aligned} \Vert \hat{F}_{ij}\Vert _{r,\tau }+\Vert z^i\hat{F}_{ij}\Vert _{r,\tau }&\le \mathrm {const}(r)\frac{\langle \tau \rangle }{\xi (\langle \tau \rangle )}\,,\\ \Vert \hat{F}_{i\tau }\Vert _{r,\tau }+\Vert z^i\hat{F}_{i\tau }\Vert _{r,\tau }&\le \frac{\mathrm {const}(r)}{\xi (\langle \tau \rangle )}\,, \end{aligned}$$

and for each \(r>0\) the following expressions are integrable on \({\mathbb {R}}\)

$$\begin{aligned} \frac{\Vert \xi (\langle z\rangle )\hat{\mathcal {A}}_i\Vert _{r,\tau }}{\langle \tau \rangle \xi (\langle \tau \rangle )}\,, \qquad \frac{\Vert \xi (\langle z\rangle )z^i\hat{\mathcal {A}}_i\Vert _{r,\tau }}{\langle \tau \rangle \xi (\langle \tau \rangle )}\,. \end{aligned}$$

Before stating the theorem, we make a comment on the function \(\xi \) and consider some additional consequences of Assumption II.

If one sets \(\xi (u)=u^\kappa \xi _0(u)\), then the first bound in (48) is equivalent to \(\xi _0'(u)\le 0\). It follows that

$$\begin{aligned} \xi (u)\le u^\kappa \,. \end{aligned}$$
(49)

Therefore, \(\xi \) is a slowly increasing, non-oscillating function. Examples include:

$$\begin{aligned}&\xi _1(u)=u^\kappa \,,\\&\xi _2(u)=\Big [1+\frac{\kappa }{m}\log (u)\Big ]^m\,,\quad m>0\,, \end{aligned}$$

where \(\kappa \) is as assumed in (48). A particular choice of \(\xi \) must guarantee the validity of the assumptions (if this is possible).

By the Schwarz inequality, also the following integral is finite:

$$\begin{aligned} \int _{{\mathbb {R}}}\frac{\Vert \hat{\mathcal {A}}_i\Vert _{r,\tau }}{\langle \tau \rangle ^2}\hbox {d}\tau \le \sqrt{\pi }\left( \int _{{\mathbb {R}}}\frac{\Vert \hat{\mathcal {A}}_i\Vert _{r,\tau }^2}{\langle \tau \rangle ^2}d \tau \right) ^{\frac{1}{2}}<\infty \end{aligned}$$

and similarly for \(z^i\hat{\mathcal {A}}_i\). Moreover, for \(\tau _2\ge \tau _1\) we have

$$\begin{aligned} \frac{\hat{\mathcal {A}}_i(\tau _2,\mathbf {z})}{\langle \tau _2\rangle } -\frac{\hat{\mathcal {A}}_i(\tau _1,\mathbf {z})}{\langle \tau _1\rangle }= \int _{\tau _1}^{\tau _2}\left( \frac{\partial _\sigma \hat{\mathcal {A}}_i(\sigma ,\mathbf {z})}{\langle \sigma \rangle } -\frac{\sigma \hat{\mathcal {A}}_i(\sigma ,\mathbf {z})}{\langle \sigma \rangle ^3}\right) \hbox {d}\sigma \,, \end{aligned}$$

and similarly for \(z^i\hat{\mathcal {A}}_i\). Therefore, by Assumption II the functions

$$\begin{aligned} \langle \tau \rangle ^{-1}\Vert \hat{\mathcal {A}}_i\Vert _{r,\tau }\quad \text {and} \quad \langle \tau \rangle ^{-1}\Vert z^i\hat{\mathcal {A}}_i\Vert _{r,\tau } \end{aligned}$$

have limits for \(\tau \rightarrow \pm \infty \). Limits different from zero would contradict other assumptions, so

$$\begin{aligned} \lim _{\tau \rightarrow \pm \infty }\frac{\Vert \hat{\mathcal {A}}_i\Vert _{r,\tau }}{\langle \tau \rangle } =\lim _{\tau \rightarrow \pm \infty }\frac{\Vert z^i\hat{\mathcal {A}}_i\Vert _{r,\tau }}{\langle \tau \rangle }=0\,. \end{aligned}$$

For the sake of the proof of the coming theorem, we note the following estimates easily obtained with the use of formula (89):

$$\begin{aligned}&|\lambda ^i\hat{\mathcal {A}}_i|\le \frac{\mathrm {const}}{\langle \tau \rangle } \bigg (\frac{\langle \tau |z\rangle }{\langle \tau \rangle }|\hat{\mathcal {A}}_i|+|z^i\hat{\mathcal {A}}_i|\bigg )\,, \end{aligned}$$
(50)
$$\begin{aligned}&|\lambda ^i\lambda ^j\partial _i\hat{\mathcal {A}}_j| \le \frac{\mathrm {const}}{\langle \tau \rangle ^2}\Big [\frac{\langle \tau |z\rangle ^2}{\langle \tau \rangle ^2}|\partial _i\hat{\mathcal {A}}_j| +\frac{\langle \tau |z\rangle }{\langle \tau \rangle }\big (|z^i\partial _i\hat{\mathcal {A}}_j|+|z^j\partial _i\hat{\mathcal {A}}_j|\big ) +|z^iz^j\partial _i\hat{\mathcal {A}}_j|\Big ]\,. \end{aligned}$$
(51)

Theorem 7

  1. (i)

    Let \(\hat{\mathcal {A}}_\mu \) satisfy Assumption I. Then the following strong limits exist:

    $$\begin{aligned}&\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow \pm \infty }U(0,\tau )U_0(\tau ,0)=\Omega _\mp \,, \end{aligned}$$
    (52)
    $$\begin{aligned}&\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow \pm \infty }U_\Phi (0,\tau )=U_\Phi (0,\pm \infty ) =\Omega _{\mp }U_{0\Phi }(0,\pm \infty )\,. \end{aligned}$$
    (53)
  2. (ii)

    Let \(\hat{\mathcal {A}}_\mu \) satisfy Assumption II. Then the operators \(\Omega _\mp \) are unitary and also the following strong limits exist:

    $$\begin{aligned}&\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow \pm \infty }U_0(0,\tau )U(\tau ,0)=\Omega _\mp ^*\,, \end{aligned}$$
    (54)
    $$\begin{aligned}&\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow \pm \infty }U_\Phi (\tau ,0)=U_\Phi (\pm \infty ,0) =U_{0\Phi }(\pm \infty ,0)\Omega ^*_{\mp }\,. \end{aligned}$$
    (55)

Proof

For the sake of the whole proof we assume that \(\psi \in {\mathcal {D}}(r,\mathbf {p}^2)\) and \(\Vert \psi \Vert =1\).

(i) For \(\tau \rightarrow \infty \), we prove the existence of the limit (53), from which the limit (52) follows with the use of Theorem 5. The case \(\tau \rightarrow -\infty \) is analogous.

We note the identity

$$\begin{aligned} \big (1-\tfrac{1}{2}\beta \tilde{H}\big )\beta -H\big (1-\tfrac{1}{2}\beta \tilde{H}\big ) =(\mu -\beta +\hat{\mathcal {A}}_\tau )\big (\tfrac{1}{2}\beta \tilde{H}-1\big )-\tfrac{1}{2}\beta \tilde{H}^2\,, \end{aligned}$$

where \(\tilde{H}\) is as defined in (33) (to show this one eliminates H with the use of (33) and takes into account the anticommutation relation (34)). Using it, we obtain the evolution equation

$$\begin{aligned}&i\partial _\tau \big [U(0,\tau )\big (1-\tfrac{1}{2}\beta \tilde{H}\big )\Phi (\tau )\big ]\psi \\&\quad =U(0,\tau )\big [-\tfrac{i}{2}\beta (\partial _\tau \tilde{H}) +(\mu -\beta +\hat{\mathcal {A}}_\tau )\big (\tfrac{1}{2}\beta \tilde{H}-1\big ) -\tfrac{1}{2}\beta \tilde{H}^2\big ]\Phi (\tau )\psi \,. \end{aligned}$$

Taking into account the anticommutation relation (34), we can write this in the form

$$\begin{aligned}&i\partial _\tau \big [U_\Phi (0,\tau )-\tfrac{1}{2}U(0,\tau )\Phi (\tau )^*\beta \tilde{H}\big ]\psi \nonumber \\&\quad =-U(0,\tau )\Phi (\tau )\big [\mu -\beta +\hat{\mathcal {A}}_\tau +\tfrac{1}{2}\beta \tilde{H}^2\big ]\psi \nonumber \\&\qquad +\tfrac{1}{2} U(0,\tau )\Phi (\tau )^*\beta \big [-i\dot{\tilde{H}}+(\mu -\beta +\hat{\mathcal {A}}_\tau )\tilde{H}\big ]\psi \,. \end{aligned}$$
(56)

We estimate the terms in this equation, starting with the second term in brackets on the lhs, and then going to the successive terms on the rhs. As \(|\mathbf {z}|\le r\) on the support of \(\psi \), each \(\lambda ^i\), \(\partial _i\lambda ^j\) and \(\partial _i\partial _j\lambda ^k\) give a bounding factor \(\mathrm {const}(r)\langle \tau \rangle ^{-1}\), and each \(\partial _\tau \lambda ^i\) a factor \(\mathrm {const}(r)\langle \tau \rangle ^{-2}\), which leads to an easy straightforward estimation:

$$\begin{aligned}&\Vert \tilde{H}\psi \Vert \le \frac{\mathrm {const}(r)}{\langle \tau \rangle }\big (\Vert p_i\psi \Vert +\Vert \hat{\mathcal {A}}_i\Vert _r+1\big )\,,\\&\Vert (\mu -\beta )\psi \Vert \le \frac{r^2}{2\langle \tau \rangle ^2}\,,\qquad \Vert \hat{\mathcal {A}}_\tau \psi \Vert \le \Vert \hat{\mathcal {A}}_\tau \Vert _r\,,\\&\Vert \tilde{H}^2\psi \Vert \le \frac{\mathrm {const}(r)}{\langle \tau \rangle ^2}\Big [\Vert p_ip_j\psi \Vert +\big (\Vert \hat{\mathcal {A}}_i\Vert _r+1\big )\big (\Vert \hat{\mathcal {A}}_i\Vert _r+1+\Vert p_i\psi \Vert \big )\Big ]\,,\\&\Vert \dot{\tilde{H}}\psi \Vert \le \frac{\mathrm {const}(r)}{\langle \tau \rangle ^2}\big (\Vert p_i\psi \Vert +\Vert \hat{\mathcal {A}}_i\Vert _r+1\big ) +\frac{\mathrm {const}(r)}{\langle \tau \rangle }\Vert \partial _\tau \hat{\mathcal {A}}_i\Vert _r\,,\\&\Vert (\mu -\beta +\hat{\mathcal {A}}_\tau )\tilde{H}\psi \Vert \le \frac{\mathrm {const}(r)}{\langle \tau \rangle }\Big (\frac{r^2}{\langle \tau \rangle ^2} +\Vert \hat{\mathcal {A}}_\tau \Vert _r\Big )\big (\Vert p_i\psi \Vert +\Vert \hat{\mathcal {A}}_i\Vert _r+1\big )\,. \end{aligned}$$

Therefore, with the conditions of Assumption I all the terms on the rhs of (56) are integrable on \({\mathbb {R}}\), so the strong limit of \(U_\Phi (0,\tau )\psi -\tfrac{1}{2}U(0,\tau ) \Phi (\tau )^*\beta \tilde{H}\psi \) exists. But the second term vanishes in the limit, so the thesis follows for \(\psi \in {\mathcal {D}}_c(\mathbf {p}^2)\), and then by isometry for all \(\psi \in {\mathcal {H}}\). (For the integrability of \(\langle \tau \rangle ^{-2}\Vert \hat{\mathcal {A}}_i\Vert _r\) and for vanishing of \(\langle \tau \rangle ^{-1}\Vert \hat{\mathcal {A}}_i\Vert _r\) one argues similarly as in the remarks following Assumption II.)

(ii) We prove the existence of the limit (54) for \(\tau \rightarrow \infty \), from which the limit (55) follows. Combined with the existence of the limit (52), this also leads to unitarity and the conjugation relation. The case \(\tau \rightarrow -\infty \) is similar.

We note the identity

$$\begin{aligned} (1+X)H-H_0(1+X)=\hat{\mathcal {A}}_\tau +WX +[X,\tilde{H}]\,, \end{aligned}$$

where W and X are defined in (28) and (44), respectively, and we used the fact that

$$\begin{aligned} {[}X,H-\tilde{H}]=[X,\mu ]=-2\mu X=-\hat{\mathcal {A}}_i\lambda ^i\,. \end{aligned}$$

Thus

$$\begin{aligned}&i\partial _\tau \big [U_0(0,\tau )(1+X)U(\tau ,0)\big ]\psi \nonumber \\&\quad =U_0(0,\tau )\Big (i\partial _\tau X+\hat{\mathcal {A}}_\tau +WX +[X,\tilde{H}]\Big )U(\tau ,0)\psi \,. \end{aligned}$$
(57)

If we can show that the norm of the rhs of (57) is integrable over \([0,+\infty )\), then the strong limit of \(U_0(0,\tau )(1+X)U(\tau ,0)\psi \) for \(\tau \rightarrow \infty \) exists. But with the use of formula (47), and taking into account (50) and the value of \(g^{\tau \tau }\) to be found in Appendix D, we obtain

$$\begin{aligned} \Vert X(\tau )U(\tau ,0)\psi \Vert \le \Vert X\Vert _{r,\tau } \le \frac{\mathrm {const}}{\langle \tau \rangle }\Big (\Vert \hat{\mathcal {A}}_i\Vert _{r,\tau }+\Vert z^i\hat{\mathcal {A}}_i\Vert _{r,\tau }\Big )\,, \end{aligned}$$
(58)

so this norm vanishes in the limit, which then implies the desired result.

We estimate the norms of the successive terms on the rhs of (57), again with the use of (47). Differentiating formula (90) with respect to \(\tau \), we find

$$\begin{aligned} \Vert (\partial _\tau X)U(\tau ,0)\psi \Vert&\le \Vert \partial _\tau X\Vert _{r,\tau } \le \frac{\mathrm {const}}{\langle \tau \rangle ^2}\Big (\Vert \hat{\mathcal {A}}_i\Vert _{r,\tau }+\Vert z^i\hat{\mathcal {A}}_i\Vert _{r,\tau }\Big )\\&\quad +\frac{\mathrm {const}}{\langle \tau \rangle }\Big (\Vert \partial _\tau \hat{\mathcal {A}}_i\Vert _{r,\tau }+\Vert z^i\partial _\tau \hat{\mathcal {A}}_i\Vert _{r,\tau }\Big )\,, \end{aligned}$$

which is integrable. The norm of the second term is bounded by \(\Vert \hat{\mathcal {A}}_\tau (\tau )\Vert _{r,\tau }\), which is integrable by assumption. Next we note that

$$\begin{aligned} \beta WX=\hat{\mathcal {A}}_\tau \beta X-\tfrac{1}{2} (g^{\tau \tau })^{\frac{1}{2}}\hat{\mathcal {A}}_i\rho ^{ij}\hat{\mathcal {A}}_j\,, \end{aligned}$$

so using the explicit form of \((g^{\tau \tau })^{\frac{1}{2}}\rho ^{ij}\), see (91), we estimate the norm of the third term on the rhs of (57) by

$$\begin{aligned} \Vert WX\Vert _{r,\tau }\le \mathrm {const}\Vert \hat{\mathcal {A}}_\tau \Vert _{r,\tau }\Vert X\Vert _{r,\tau } +\frac{\mathrm {const}\langle r \rangle }{\langle \tau \rangle ^2}\Big (\Vert \hat{\mathcal {A}}_i\Vert ^2_{r,\tau }+\Vert z^i\hat{\mathcal {A}}_i\Vert ^2_{r,\tau }\Big )\,, \end{aligned}$$

which ensures integrability.

To estimate the norm of the fourth term, we use (45), (51) and (93) to find

$$\begin{aligned}&\Vert \lambda ^i\lambda ^j\partial _i a_j+(\partial _i\rho ^{ij})a_j\Vert _{r,\tau } \le \mathrm {const}\frac{\langle r \rangle }{\langle \tau \rangle ^2}\Vert \partial _i\hat{\mathcal {A}}_j\Vert _{r,\tau }\\&\quad +\frac{\mathrm {const}}{\langle \tau \rangle ^2}\Big (\Vert \hat{\mathcal {A}}_i\Vert _{r,\tau }+\Vert z^i\hat{\mathcal {A}}_i\Vert _{r,\tau } +\Vert z^i\partial _i\hat{\mathcal {A}}_j\Vert _{r,\tau }+\Vert z^j\partial _i\hat{\mathcal {A}}_j\Vert _{r,\tau } +\Vert z^iz^j\partial _i\hat{\mathcal {A}}_j\Vert _{r,\tau }\Big )\,, \end{aligned}$$

which again is integrable.

We are now left with the single term \(a\rho \pi _\Lambda U(\tau ,0)\psi \). As it turns out, in this case the methods applied up to now are insufficient and it is here that we make use of the function \(\xi \). We write \(\psi _\tau = U(\tau ,0)\psi \) and note that

$$\begin{aligned} \Vert a\rho \pi _{\Lambda }\psi _\tau \Vert&\le \Vert sa\xi (\langle z\rangle )\Vert _{r,\tau } \Vert \xi (\langle z\rangle )^{-1}s\pi _\Lambda \psi _\tau \Vert \\&\le \langle \tau \rangle ^{-1}\Big (\Vert \xi (\langle z\rangle )\hat{\mathcal {A}}_i\Vert _{r,\tau } +\Vert \xi (\langle z\rangle ) z^i\hat{\mathcal {A}}_i\Vert _{r,\tau }\Big )\Vert \xi (\langle z\rangle )^{-1}s\pi _\Lambda \psi _\tau \Vert \,. \end{aligned}$$

Now, the estimation of \(\Vert \xi (\langle z\rangle )^{-1}s\pi _\Lambda \psi _\tau \Vert \) is the most difficult part of the proof and we shift it to the lemma below. Substituting its result in the above estimate and using Assumption II, one completes the proof of the existence of the limit \(\tau \rightarrow +\infty \). \(\square \)

Lemma 8

Under the conditions of Assumption II, for \(\psi \in {\mathcal {D}}_c(\mathbf {p}^2)\) the following estimate holds

$$\begin{aligned} \Vert \xi (\langle z\rangle )^{-1}s\pi _\Lambda U(\tau ,0)\psi \Vert \le \mathrm {const}(\psi )\xi (\langle \tau \rangle )^{-1}\,. \end{aligned}$$

Proof

We assume again that \(\psi \in {\mathcal {D}}(r,\mathbf {p}^2)\) and \(\Vert \psi \Vert =1\). We observe that the lhs of the inequality may be equivalently replaced by \(\Vert s\pi _\Lambda \xi (\langle z\rangle )^{-1}U(\tau ,0)\psi \Vert \). Indeed, we have

$$\begin{aligned} \big \Vert \big [s\pi _\Lambda ,\xi (\langle z\rangle )^{-1}\big ]\big \Vert _\infty = \Big \Vert \frac{\mathbf {z}\xi '(\langle z\rangle )}{\langle \tau \rangle \xi (\langle z\rangle )^2}\Big \Vert _\infty \le \Big \Vert \frac{\kappa \,\mathbf {z}}{\langle \tau \rangle \langle z\rangle \xi (\langle z\rangle )}\Big \Vert _\infty \le \frac{\kappa }{\langle \tau \rangle }\,. \end{aligned}$$

Now we shall estimate the norm squared

$$\begin{aligned} \Vert s\pi _\Lambda \xi (\langle z\rangle )^{-1}\psi _\tau \Vert ^2 =(\pi _\Lambda \xi (\langle z\rangle )^{-1}\psi _\tau ,\rho \pi _\Lambda \xi (\langle z\rangle )^{-1}\psi _\tau ) \end{aligned}$$

by first finding a differential inequality, and then integrating. Preparing for that, we denote

$$\begin{aligned} \nu =i\xi (\langle z\rangle )^{-1}[H,\xi (\langle z\rangle )]=\frac{\lambda ^i\partial _i\xi (\langle z\rangle )}{\xi (\langle z\rangle )} =\frac{\xi '(\langle z\rangle )}{\langle \tau \rangle \xi (\langle z\rangle )}z^i\alpha ^i\,, \end{aligned}$$
(59)

with the standard notation \(\alpha ^i=\beta \gamma ^i\), and observe that

$$\begin{aligned} \xi (\langle z\rangle )^{-1} H=\tilde{H}\xi (\langle z\rangle )^{-1}+(\hat{\mathcal {A}}_\tau +\mu -i\nu )\xi (\langle z\rangle )^{-1}\,. \end{aligned}$$
(60)

To shorten notation, we shall write \(\psi _\tau ^\xi =\xi (\langle z\rangle )^{-1}\psi _\tau \). Looking at the explicit form of \(\Lambda _i\) (94), we note that

$$\begin{aligned} {[}\Lambda _i,\mu ]=0\,,\qquad [\Lambda _i,\nu ]_+=0\,. \end{aligned}$$

Now calculate

$$\begin{aligned} \partial _\tau [\pi _\Lambda \psi _\tau ^\xi ]&=-i\pi _\Lambda \xi (\langle z\rangle )^{-1} H\psi _\tau +(\partial _\tau \Lambda +\partial _\tau \hat{\mathcal {A}})\psi _\tau ^\xi \\&=-i\pi _\Lambda \tilde{H}\psi _\tau ^\xi +(\dot{\Lambda }+\hat{F}_{\tau .}+\partial (i\nu -\mu )+2\nu \Lambda )\psi _\tau ^\xi -(\nu +i\hat{\mathcal {A}}_\tau +i\mu )\pi _\Lambda \psi _\tau ^\xi \,, \end{aligned}$$

and

$$\begin{aligned}&\partial _\tau (\pi _\Lambda \psi _\tau ^\xi ,\rho \pi _\Lambda \psi _\tau ^\xi ) -(\pi _\Lambda \psi _\tau ^\xi ,\dot{\rho }\pi _\Lambda \psi _\tau ^\xi ) =-2(\pi _\Lambda \psi _\tau ^\xi ,\rho \nu \pi _\Lambda \psi _\tau ^\xi )\nonumber \\&\quad -i(\pi _\Lambda \rho \pi _\Lambda \psi _\tau ^\xi ,\tilde{H}\psi _\tau ^\xi ) +i(\tilde{H}\psi _\tau ^\xi ,\pi _\Lambda \rho \pi _\Lambda \psi _\tau ^\xi )\nonumber \\&\quad +2{{\,\mathrm{Re}\,}}(\pi _\Lambda \psi _\tau ^\xi ,\rho [\dot{\Lambda } +\hat{F}_{\tau .}+\partial (i\nu -\mu )+2\nu \Lambda ]\psi _\tau ^\xi )\,, \end{aligned}$$
(61)

where in both identities \(\hat{F}_{\tau .}\) denotes \(\hat{F}_{\tau i}\) with the index i suppressed (in the second identity summation over this index is implied). In the first identity we have used (60) and commuted \(\pi _\Lambda \) with the term \((\hat{\mathcal {A}}_\tau +\mu -i\nu )\).

From now on we continue the proof for \(\tau \ge 0\); for \(\tau \le 0\) the proof is analogous, but equation (61) has to be multiplied by \(-1\) before continuing. The first term on the rhs of (61) is bounded in absolute value by

$$\begin{aligned} 2\Vert \nu \Vert _\infty (\pi _\Lambda \psi _\tau ^\xi ,\rho \pi _\Lambda \psi _\tau ^\xi ) \le 2\kappa \langle \tau \rangle ^{-1}\Vert s\pi _\Lambda \psi _\tau ^\xi \Vert ^2\,, \end{aligned}$$

where we used the estimate given in (99). With the use of formula (41), the second line of equation (61) takes the form \(2{{\,\mathrm{Re}\,}}i(\tilde{H}\psi _\tau ^\xi ,(N+{\mathcal {B}})\psi _\tau ^\xi )\), and thanks to the estimate (43) is bounded in absolute value by

$$\begin{aligned} 2\big (\Vert s\pi _\Lambda \psi _\tau ^\xi \Vert +|(\psi _\tau ^\xi , {\mathcal {B}}\psi _\tau ^\xi )|^{\frac{1}{2}}\big )\Vert (N+{\mathcal {B}})\psi _\tau ^\xi \Vert \,. \end{aligned}$$

The third line in (61) is bounded by

$$\begin{aligned} 2\Vert s\pi _\Lambda \psi _\tau ^\xi \Vert \Vert s(\dot{\Lambda }+\hat{F}_{\tau \,.}+\partial (i\nu -\mu )+2\nu \Lambda )\psi _\tau ^\xi \Vert \,. \end{aligned}$$

Finally, we observe that for \(\tau >0\) we have [see (92)]

$$\begin{aligned} \dot{\rho }\le -\frac{2\tau }{\langle \tau \rangle ^2}\rho \,, \end{aligned}$$

which allows us to use (61) for the following estimate:

$$\begin{aligned} \partial _\tau \Vert s\pi _\Lambda \psi _\tau ^\xi \Vert ^2\le -b\Vert s\pi _\Lambda \psi _\tau ^\xi \Vert ^2 +2c\Vert s\pi _\Lambda \psi _\tau ^\xi \Vert +d\,, \end{aligned}$$
(62)

where

$$\begin{aligned}&b=\frac{2\tau }{\langle \tau \rangle ^2}-\frac{2\kappa }{\langle \tau \rangle }\,,\nonumber \\&c=\Vert (N+{\mathcal {B}})\psi _\tau ^\xi \Vert +\Vert s(\dot{\Lambda }+\hat{F}_{\tau \,.}+\partial (i\nu -\mu ) +2\nu \Lambda )\psi _\tau ^\xi \Vert \,,\\&d=2|(\psi _\tau ^\xi ,{\mathcal {B}}\psi _\tau ^\xi )|^{\frac{1}{2}}\Vert (N+{\mathcal {B}})\psi _\tau ^\xi \Vert \,.\nonumber \end{aligned}$$
(63)

The second term on the rhs of (62) may be estimated as follows

$$\begin{aligned} 2c\Vert s\pi _\Lambda \psi _\tau ^\xi \Vert&=2\Big [\frac{1-2\kappa }{\langle \tau \rangle }\Big ]^{\frac{1}{2}} \Vert s\pi _\Lambda \psi _\tau ^\xi \Vert \ \Big [\frac{\langle \tau \rangle }{1-2\kappa }\Big ]^{\frac{1}{2}}c\\&\le \frac{1-2\kappa }{\langle \tau \rangle }\Vert s\pi _\Lambda \psi _\tau ^\xi \Vert ^2+\frac{\langle \tau \rangle c^2}{1-2\kappa }\,, \end{aligned}$$

which results in the inequality

$$\begin{aligned} \partial _\tau \Vert s\pi _\Lambda \psi _\tau ^\xi \Vert ^2\le -b_0\Vert s\pi _\Lambda \psi _\tau ^\xi \Vert ^2+d_0\,, \end{aligned}$$
(64)

where

$$\begin{aligned} b_0=\frac{2\tau }{\langle \tau \rangle ^2}-\frac{1}{\langle \tau \rangle }\,,\quad d_0=\frac{\langle \tau \rangle c^2}{1-2\kappa }+d\,. \end{aligned}$$
(65)

We set

$$\begin{aligned} \Vert s\pi _\Lambda \psi _\tau ^\xi \Vert ^2=\exp \Big (-\int _0^\tau b_0(\sigma )\hbox {d}\sigma \Big )f(\tau ) =\frac{\tau +\langle \tau \rangle }{\langle \tau \rangle ^2}f(\tau ) \end{aligned}$$

and then (64) takes the form

$$\begin{aligned} \partial _\tau f\le \frac{\langle \tau \rangle ^2}{\tau +\langle \tau \rangle }d_0(\tau )\le \langle \tau \rangle d_0(\tau )\,. \end{aligned}$$

We note that \(f(0)=\Vert [s\pi _\Lambda \psi _\tau ^\xi ]_{\tau =0}\Vert ^2\) and find

$$\begin{aligned} \Vert s\pi _\Lambda \psi _\tau ^\xi \Vert ^2 \le \frac{2}{\langle \tau \rangle }\bigg [\Vert [s\pi _\Lambda \psi _\tau ^\xi ]_{\tau =0}\Vert ^2 +\int _0^\tau \langle \sigma \rangle d_0(\sigma )\hbox {d}\sigma \bigg ]. \end{aligned}$$
(66)

We have to estimate \(d_0\) defined in (65). We start by estimating c. For the terms depending on the electromagnetic field, we have (we use the form of s and estimates of \(\lambda ^i\) given in Appendix D)

$$\begin{aligned}&\Vert {\mathcal {B}}\Vert _{r,\tau }\le \frac{\mathrm {const}\langle r \rangle ^2}{\langle \tau \rangle ^2} \big (\Vert \hat{F}_{ij}\Vert _{r,\tau }+\Vert z^i\hat{F}_{ij}\Vert _{r,\tau }\big ) \le \frac{\mathrm {const}(r)}{\langle \tau \rangle \xi (\langle \tau \rangle )}\,,\\&\Vert s\hat{F}_{\tau \,.}\Vert _{r,\tau } \le \frac{\mathrm {const}\langle r \rangle }{\langle \tau \rangle } \big (\Vert \hat{F}_{\tau i}\Vert _{r,\tau }+\Vert z^i\hat{F}_{\tau i}\Vert _{r,\tau }\big ) \le \frac{\mathrm {const}(r)}{\langle \tau \rangle \xi (\langle \tau \rangle )}\,.\nonumber \end{aligned}$$
(67)

For the term \(s^{ij}\partial _j(i\nu -\mu )\), using the estimates (98) and (100) in Appendix D we findFootnote 9

$$\begin{aligned} \Vert s\partial (i\nu -\mu )\psi _\tau ^\xi \Vert \le \Big \Vert \frac{s\partial (i\nu -\mu )}{\xi (\langle z\rangle )}\Big \Vert _{r,\tau } \le \frac{\mathrm {const}}{\langle \tau \rangle ^2}\Big \Vert \frac{\langle z\rangle }{\xi (\langle z\rangle )}\Big \Vert _{r,\tau } \le \frac{\mathrm {const}(r)}{\langle \tau \rangle \xi (\langle \tau \rangle )}\,, \end{aligned}$$
(68)

where we used the fact, that both \(u/\xi (u)\) as well as \(\xi (u)\) are increasing, so

$$\begin{aligned} \frac{\langle z\rangle }{\xi (\langle z\rangle )}\le \frac{2\langle r \rangle \langle \tau \rangle }{\xi (2\langle r \rangle \langle \tau \rangle )} \le 2\langle r \rangle \frac{\langle \tau \rangle }{\xi (\langle \tau \rangle )}\,. \end{aligned}$$

The estimation of the other terms in (63) uses the bounds (95), (97) and (99) and gives

$$\begin{aligned} \Vert N\psi _\tau ^\xi \Vert +\Vert s\dot{\Lambda }\psi _\tau ^\xi \Vert +\Vert 2\nu \Lambda \psi _\tau ^\xi \Vert \le \frac{\mathrm {const}}{\langle \tau \rangle ^2}\,, \end{aligned}$$
(69)

so summing up, we have

$$\begin{aligned} c\le \frac{\mathrm {const}(r)}{\langle \tau \rangle \xi (\langle \tau \rangle )}\,,\qquad \langle \tau \rangle \, c^2\le \frac{\mathrm {const}(r)}{\langle \tau \rangle \xi (\langle \tau \rangle )^2}\,. \end{aligned}$$

The use of (67) and (69) shows that also

$$\begin{aligned} d(\tau )\le \frac{\mathrm {const}(r)}{[\langle \tau \rangle \xi (\langle \tau \rangle )]^{\frac{3}{2}}} \le \frac{\mathrm {const}(r)}{\langle \tau \rangle \xi (\langle \tau \rangle )^2}\,, \end{aligned}$$

where we used the bound \(u^\frac{1}{2}\ge \xi (u)\ge 1\), see (49). Summing up, we obtain

$$\begin{aligned} d_0(\tau )\le \frac{\mathrm {const}(r)}{\langle \tau \rangle \xi (\langle \tau \rangle )^2}\,. \end{aligned}$$

Now, it follows from (48) that \(u^\kappa /\xi (u)\) is an increasing function. Therefore,

$$\begin{aligned} \int _0^\tau \langle \sigma \rangle d_0(\sigma )\hbox {d}\sigma&\le \mathrm {const}(r)\int _0^\tau \frac{\hbox {d}\sigma }{\xi (\langle \sigma \rangle )^2} \le \mathrm {const}(r) \frac{\langle \tau \rangle ^{2\kappa }}{\xi (\langle \tau \rangle )^2}\int _0^\tau \frac{\hbox {d}\sigma }{\langle \sigma \rangle ^{2\kappa }}\\&\le \mathrm {const}(r)\frac{\langle \tau \rangle }{\xi (\langle \tau \rangle )^2}\,. \end{aligned}$$

This, when used in (66), gives

$$\begin{aligned} \Vert s\pi _\Lambda \psi _\tau ^\xi \Vert ^2 \le \frac{2\Vert [s\pi _\Lambda \psi _\tau ^\xi ]_{\tau =0}\Vert ^2}{\langle \tau \rangle } +\frac{\mathrm {const}(r)}{\xi (\langle \tau \rangle )^2} \le \frac{\mathrm {const}(\psi )}{\xi (\langle \tau \rangle )^2}\,, \end{aligned}$$

where for the second inequality we used (49). \(\square \)

7 Typical Electromagnetic Field and Its Special Gauges

Assumption II, on which our main theorem on scattering 7 is based, is rather technical and not easy for interpretation. Here we formulate a rather typical situation met in scattering processes.Footnote 10 We show that the electromagnetic field thus identified admits a gauge in which Assumption II is satisfied. Moreover, there is a class of gauges which need not satisfy this assumption, but still assure a similar asymptotic structure.

The retarded and advanced potentials are defined in terms of the source current J in standard way (as \(\varphi _{\mathrm{ret}/\mathrm{adv}}\) is defined in terms of \(\rho \) in (118) in Appendix G). Also, the radiated field of the current J is defined in standard way, \(A_\mathrm{rad}=A_\mathrm{ret}-A_\mathrm{adv}\). The Heaviside step function is denoted by \(\theta \).

Assumption III. The Lorenz potential \(A_a\) of the electromagnetic field \(F_{ab}\) is given by

$$\begin{aligned} A=A_\mathrm{ret}+A_\mathrm{in}=A_\mathrm{adv}+A_\mathrm{out}\,, \end{aligned}$$

where \(A_\mathrm{ret}\) is the retarded potential of a current J satisfying the assumptions listed below, and \(A_\mathrm{in}\) is the radiated potential of another current \(J_\mathrm{in}\) with similar properties as J. Then also \(A_\mathrm{adv}\) is the advanced potential of the current J, and \(A_\mathrm{out}\) is the radiated potential of the current \(J_\mathrm{out}=J+J_\mathrm{in}\), which has similar properties as J and \(J_\mathrm{in}\).

The conserved current J(x) is of class \(C^3\) and for some \(0< \varepsilon < \frac{1}{2}\) satisfies the following estimates:

$$\begin{aligned}&|\nabla ^\alpha J(x)|\le \frac{\mathrm {const}}{(|x|+1)^{3+|\alpha |}}\bigg [\theta (x^2)+\frac{1}{(|x|+1)^\varepsilon } \bigg ]\,,\qquad \text {for}\quad |\alpha |\le 3, \end{aligned}$$
(70)
$$\begin{aligned}&|\nabla ^\alpha (x\cdot \nabla +3) J(x)|\le \frac{\mathrm {const}}{(|x|+1)^{3+|\alpha |+\varepsilon }}\,, \qquad \qquad \quad \,\, \text {for}\quad |\alpha |\le 2\,. \end{aligned}$$
(71)

The same is assumed for \(J_\mathrm{in}\), and then the same follows for \(J_\mathrm{out}\). The potential A is then of class \(C^3\) on the Minkowski spacetime.

Note that A is a linear combination of retarded and advanced potentials of currents satisfying (70). Therefore, the last statement of Assumption III is a consequence of Lemma 20 (i) in Appendix G.

Theorem 9

Let the electromagnetic field F and its Lorenz potential A satisfy Assumption III. Define a new gauge by

$$\begin{aligned} \mathcal {A}(x)=A(x)-\nabla S(x)\,. \end{aligned}$$
(72)

Then the following holds.

(i) For the choice

$$\begin{aligned} S(x)=S_\ell (x)\equiv \log (\langle \tau \rangle \langle z\rangle )\,x\cdot A(x)\,, \end{aligned}$$
(73)

the potential (72) fulfills Assumption II, so Theorems 6 and 7 are satisfied.

(ii) Let S(x) be another gauge function, such that the difference

$$\begin{aligned} {\mathcal {G}}(x)=S(x)-S_\ell (x) \end{aligned}$$

satisfies the assumptions of Theorem 6 (ii), so that the thesis of this theorem is true. Suppose, in addition, that there exist point-wise limits

$$\begin{aligned} \lim _{\tau \rightarrow \pm \infty }{\mathcal {G}}(\tau ,\mathbf {z})\equiv {\mathcal {G}}_\pm (\mathbf {z})\,. \end{aligned}$$

Then the potential (72) satisfies the thesis of Theorem 7 (but not necessarily the assumption of this theorem).

(iii) In particular, the gauges defined by:

$$\begin{aligned}&\text {(a)}\qquad S_{\log }(x)=\log \langle \tau \rangle \, x\cdot A(x)\,, \\&\text {(b)}\qquad S_\mathrm{tr}(x)=\int _0^\tau \partial _\sigma x(\sigma ,\mathbf {z})\cdot A(\sigma ,\mathbf {z})\mathrm{d}\sigma \,, \end{aligned}$$

are in the class defined in (ii). In case (b) one has \(\hat{\mathcal {A}}_\tau =0\) and

$$\begin{aligned} \hat{\mathcal {A}}_i(\tau ,\mathbf {z})=\hat{A}_i(0,\mathbf {z})-\int _0^\tau \hat{F}_{i\sigma }(\sigma ,\mathbf {z})\mathrm{d}\sigma \,. \end{aligned}$$

Remark 10

Assumption III, and consequently the validity of Theorem 9, is independent of the choice of the time axis for the definition of the foliation (20).

Proof of Theorem 9

(i) This potential obeys Theorem 23 in Appendix I. As \(\mathcal {A}\) is of class \(C^2\) on the Minkowski spacetime, so the assumption (29) of Theorem 6 and the assumption on continuous differentiability of the mapping \(\tau \mapsto \hat{\mathcal {A}}_i(\tau ,\mathbf {z})\) in Assumption II are clearly satisfied. Denote

$$\begin{aligned} \xi (u)=u^\kappa \,,\qquad \kappa<\varepsilon <\tfrac{1}{2}\,. \end{aligned}$$

The estimates of Theorem 23 imply then the following norm bounds:

$$\begin{aligned}&\displaystyle \Vert \hat{F}_{i\tau }\Vert _\infty \,,\ \Vert z^i\hat{F}_{i\tau }\Vert _\infty \le \frac{\mathrm {const}}{\langle \tau \rangle }\,,\\&\displaystyle \Vert \hat{F}_{ij}\Vert _\infty \,,\ \Vert z^i\hat{F}_{ij}\Vert _\infty \le \mathrm {const}\,,\\&\displaystyle \Vert \hat{\mathcal {A}}_\tau \Vert _{r,\tau }\le \mathrm {const}\Big (\frac{1+\log \langle \tau \rangle }{\langle \tau \rangle ^{1+\varepsilon }} +\frac{\log \langle r \rangle }{\langle \tau \rangle ^3}\Big )\,,\\&\displaystyle \Vert (1+\log \langle z\rangle )^{-1}\hat{\mathcal {A}}_\tau \Vert _\infty +\Vert \partial _i\hat{\mathcal {A}}_\tau \Vert _\infty +\Vert z^i\partial _i\hat{\mathcal {A}}_\tau \Vert _\infty \le \mathrm {const}\frac{1+\log \langle \tau \rangle }{\langle \tau \rangle ^{1+\varepsilon }}\,,\\&\displaystyle \Vert \partial _i\partial _j\hat{\mathcal {A}}_\tau \Vert _\infty \le \mathrm {const}(1+\log \langle \tau \rangle )\,,\\&\displaystyle \Vert \hat{\mathcal {A}}_i\Vert _\infty +\Vert z^i\hat{\mathcal {A}}_i\Vert _\infty \le \Vert \xi (\langle z\rangle )\hat{\mathcal {A}}_i\Vert _\infty +\Vert \xi (\langle z\rangle )z^i\hat{\mathcal {A}}_i\Vert _\infty \le \mathrm {const}(1+\log \langle \tau \rangle )\,,\\&\displaystyle \Vert \partial _\tau \hat{\mathcal {A}}_i\Vert _\infty +\Vert z^i\partial _\tau \hat{\mathcal {A}}_i\Vert _\infty \le \frac{\mathrm {const}}{\langle \tau \rangle }\,,\\&\displaystyle \Vert \partial _i\hat{\mathcal {A}}_j\Vert _\infty +\Vert z^i\partial _i\hat{\mathcal {A}}_j\Vert _\infty +\Vert z^j\partial _i\hat{\mathcal {A}}_j\Vert _\infty +\Vert z^iz^j\partial _i\hat{\mathcal {A}}_j\Vert _\infty \le \mathrm {const}(1+\log \langle \tau \rangle )\,. \end{aligned}$$

It is now easily checked that the potential satisfies assumption (30) in Theorem 6 and all the remaining bounds in Assumption II, from which the thesis follows. (Note that in this case all expressions are bounded in \(L^\infty \)-norm, except for \(\hat{\mathcal {A}}_\tau \).)

(ii) Let the evolution operator \(U(\tau ,\sigma )\) refer to the potential defined in (i), and denote the new potential now considered by \(\hat{\mathcal {A}}^{\mathcal {G}}\). Following further notation used in Theorem 6, we have

$$\begin{aligned}&\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow \pm \infty }U^{\mathcal {G}}_\Phi (\tau ,0) =\mathop {\mathrm{s{-}lim}}\limits _{\tau \pm \infty }\Phi ^*(\tau )e^{i {\mathcal {G}}(\tau )}U(\tau ,0)e^{-i{\mathcal {G}}(0)}\\&\quad =\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow \pm \infty }e^{i{\mathcal {G}}(\tau )}U_\Phi (\tau ,0)e^{-i{\mathcal {G}}(0)} =e^{i{\mathcal {G}}_\pm }U_\Phi (\pm \infty ,0)e^{-i{\mathcal {G}}(0)}\,, \end{aligned}$$

so

$$\begin{aligned}&\mathop {\mathrm{s{-}lim}}\limits _{\tau \rightarrow \pm \infty }U_0(0,\tau )U^{\mathcal {G}}(\tau ,0) =U_{0\Phi }(0,\pm \infty )U^{\mathcal {G}}_\Phi (\pm \infty ,0)\\&\quad =U_{0\Phi }(0,\pm \infty )e^{i{\mathcal {G}}_\pm }U_\Phi (\pm \infty ,0) \Omega _\mp e^{-i{\mathcal {G}}(0)} \equiv \Omega ^{\mathcal {G}}_\mp \,. \end{aligned}$$

Similarly for the limits of the conjugated operators.

(iii) Both gauges are easily seen to satisfy condition (31) of Theorem 6; we turn to the estimates (32). In the case (a), with \(C(x)=x\cdot A(x)\), we have

$$\begin{aligned} {\mathcal {G}}(\tau ,\mathbf {z})=-\log \langle z\rangle \,C(\tau ,\mathbf {z})\,, \end{aligned}$$

and the estimates are easily checked with the use of the results of the proof of Theorem 23. Also, it follows from the estimate of \(|\partial _\tau C|\) given there that \(C(\tau ,\mathbf {z})\) has limits for \(\tau \rightarrow \pm \infty \).

In the case (b), it is now sufficient to investigate the difference of the gauge function \(S_\mathrm{tr}(x)\) as compared to the gauge function of case (a):

$$\begin{aligned} {\mathcal {G}}'(x)=S_\mathrm{tr}(x)-S_{\log }(x)\,. \end{aligned}$$

Differentiating and using the form of \(\partial _\tau x\) given in the proof of Theorem 23, we find

$$\begin{aligned}&\partial _\tau {\mathcal {G}}'(\tau ,\mathbf {z})=\langle \tau \rangle ^{-2}\langle z\rangle A_0-\log \langle \tau \rangle \partial _\tau C\,,\\&\partial _i\partial _\tau {\mathcal {G}}'(\tau ,\mathbf {z})=\frac{1}{\langle \tau \rangle ^2}\Big [\frac{z^i}{\langle z\rangle }A_0+\langle z\rangle \partial _i A_0\Big ] -\log \langle \tau \rangle \partial _i\partial _\tau C\,, \end{aligned}$$

Now noting that \(\partial _i{\mathcal {G}}(0,\mathbf {z})=0\), integrating (the last term by parts) and applying the derivative \(\partial _j\), we obtain (fields in the integrand depend on \((\sigma ,\mathbf {z})\))

$$\begin{aligned} \partial _i\partial _j{\mathcal {G}}'(\tau ,\mathbf {z})&=\int _0^\tau \left[ \frac{d^{ij}}{\langle z\rangle }A_0 +\frac{1}{\langle z\rangle }(z^i\partial _j A_0+z^j\partial _i A_0) +\langle z\rangle \partial _i\partial _jA_0+\sigma \partial _i\partial _j C\right] \frac{\hbox {d}\sigma }{\langle \sigma \rangle ^2}\\&\quad -\log \langle \tau \rangle \partial _i\partial _j C\,, \end{aligned}$$

with \(d^{ij}\) defined in (133). With the use of the estimates listed in the proof of Theorem 23, one finds

$$\begin{aligned}&|\partial _\tau {\mathcal {G}}'|\le \mathrm {const}\frac{1+\log \langle \tau \rangle }{\langle \tau \rangle ^{1+\varepsilon }}\,, \qquad |\partial _i\partial _\tau {\mathcal {G}}'|\le \mathrm {const}\frac{1+\log \langle \tau \rangle }{\langle \tau \rangle ^{1+\varepsilon }}\,,\\&|\partial _i\partial _j{\mathcal {G}}'|\le \mathrm {const}(1+\log \langle \tau \rangle )\,,\qquad |\partial _i{\mathcal {G}}'|\le \mathrm {const}\,, \end{aligned}$$

the fourth estimate by the integration of the second one. Thus, the estimates (32) are satisfied. Finally, \(\partial _\tau {\mathcal {G}}'\) is integrable on \({\mathbb {R}}\), so the thesis follows. \(\square \)

8 Discussion

There are three questions we want to address in this section:

  1. (i)

    How far is the present analysis from a complete treatment of the Maxwell-Dirac system?

  2. (ii)

    Is there a further physical selection criterion to choose a gauge from the class of gauges obtained in Theorem 9?

  3. (iii)

    Open problems.

With regard to the first of these questions, we note that the form of the charged currents producing electromagnetic fields in Assumption III mimics what one should expect in fully interacting theory. A possible shortage of this assumption rests in the estimates of derivatives of the currents, which would not be satisfied for oscillating terms in asymptotic behavior. The Dirac field current does have such asymptotic terms, but it is quite plausible to predict that oscillations dump the asymptotic behavior of fields produced by them. What would be needed is an appropriately fast vanishing of the leading oscillating asymptotic terms in the neighborhood of the lightcone (which is a quite reasonable prediction). This would presumably lead to similar behavior of electromagnetic potentials as that following from Assumption III. Moreover, we note that our Assumption II, on which our analysis is based, leaves much more room for the types of potentials than Assumption III.

Let us once more at this point recall the work by Flato et al. [7]. These authors do have a theorem on the evolution of the complete system, but in a rather restricted setting and with not much control over the range of validity. The theorem states that in the space of smooth (i.e. \(C^\infty \)) initial data there exists a neighborhood of zero, which gives rise to the Cauchy evolution and completeness. Our analysis is not fully developed to the full interacting case, but gives complete results for the Dirac part of the system, with the electromagnetic fields plausibly guessed.

Our main motivation for the present work was the expectation that the choice of gauge does matter for asymptotic behavior and its interpretation. We have shown that for a class of gauges the asymptotic behavior of the Dirac field approaches that of a free field, without the need for any dynamical corrections.

This brings us to the second question mentioned at the beginning. Theorem 9 identifies a class of gauges for which the situation described above takes place. The concrete gauge defined by (73) was used for technical reasons: this gauge satisfies Assumption II, and it is the only gauge with that property among the gauges explicitly mentioned in Theorem 9. However, we want to argue now, in less precise terms then those of the preceding sections, that a different choice of gauges in the given class has a definite physical interpretation when an extension, as mentioned above, to fully interacting system is considered. Such extension, partly based on conjectures, was considered in [8], and we have to briefly recall a few results of this analysis. It was found that if one uses inside the lightcone the gauge related to the Lorenz gauge A (of similar properties as those of Assumption III) by

$$\begin{aligned} \mathcal {A}_\mathrm {cone}(x)=A(x)-\nabla S_\mathrm{cone}(x)\,, \qquad S_\mathrm{cone}(x)=\log \sqrt{x^2}\,x\cdot A(x)\,, \end{aligned}$$

or another with similar timelike asymptotic behavior, then the asymptotic total four-momentum and angular momentum taken away into timelike infinity have the same functional form as those for the free Dirac field. Moreover, the energy–momentum radiated into null infinity is fully due to the free outgoing electromagnetic field. However, the angular momentum going out into the null infinity, in addition to the free radiated contribution, has mixed adv-out electromagnetic terms. These latter terms may be incorporated into the free Dirac field by a change of the asymptotic limit addition to the gauge function. As a result, both energy–momentum as well as angular momentum are clearly separated into electromagnetic and Dirac parts.Footnote 11 Now we would like to identify in our class of global gauges those for which this separation may be expected. It is not difficult to show that for points inside the future lightcone, represented by \(\lambda v\), with \(\lambda >0\) and v on the future unit hyperboloid, we have

$$\begin{aligned} \lim _{\lambda \rightarrow \infty }[S_{\log }(\lambda v)-S_\mathrm{cone}(\lambda v)]=0 \end{aligned}$$

for the gauge function \(S_{\log }\) defined in Theorem 9 (iii). Therefore, we put forward the selection criterion for the gauge functions S to be used in (72) for the definition of \(\mathcal {A}\):

$$\begin{aligned} S(\tau ,\mathbf {z})=S_{\log }(\tau ,\mathbf {z})+\Delta S(\tau ,\mathbf {z})\,, \end{aligned}$$

where \(\Delta S\) has the limit

$$\begin{aligned} \Delta S(\infty ,\mathbf {z})=\lim _{\tau \rightarrow \infty }\Delta S(\tau ,\mathbf {z}) \end{aligned}$$

which satisfies the condition of the separation of angular momentum as described above (it is easy to see that for \(\Delta S\) sufficiently regular the rhs is equal to \(\lim _{\lambda \rightarrow \infty }\Delta S(\lambda (\langle z\rangle ,\mathbf {z}))\)). Similar conditions should be applied for past infinity.

Gauges in the class thus selected have the property announced in the introduction: \(x\cdot \mathcal {A}(x)\) vanishes asymptotically in timelike directions. This may be checked easily for \(S_{\log }\), and if \(\Delta S(\tau ,\mathbf {z})\) is not oscillating in \(\tau \), the same is true for this part.

The existence and completeness of the wave operators, as indicated in Remark 10, do not depend on the choice of the time axis for the definition of our spacetime foliation. On the other hand, and this is the first of open problems, the precise transformation law from one inertial observer to another needs further investigation. More generally, one can ask what is a general class of Cauchy foliations for which the results could be repeated, and how the results would depend on the choice in the class.

Let us end this discussion by expressing the belief that our scheme could be applied to analogous problems on at least some of the smooth curved spacetime backgrounds. Whether, as inquired by one of the Referees, further extension to black hole type spacetimes would be possible, is a more speculative question.