(2020) Space-Time-Resolved Quantum Electrodynamics Description Of

PHYSICAL REVIEW A 102, 062203 (2020)

Space-time-resolved quantum electrodynamics description of Compton scattering

Scott Glasgow* Mathematics Department, Brigham Young University, Provo, Utah 84602, USA

Michael J. Ware † Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA

(Received 16 December 2019; revised 29 October 2020; accepted 4 November 2020; published 3 December 2020)

The standard method for approaching quantum electrodynamic (QED) field theory uses a perturbative S-matrix approach. This approach is explicitly nondynamical and provides only a one-time, static map between an initial state to be evolved by the “full propagator” of a bona-fide interacting field theory and an asymptotically equivalent effective initial state to be evolved by the “free propagator” of the corresponding noninteracting field theory. We provide a detailed derivation of a nonperturbative and dynamical approach to QED that allows one to study the space-time dynamics of electron-photon interactions directly. As an example of this method, we compute the time-resolved dynamics of Compton scattering for a system with a nontrivial spatial structure in only one dimension while restricting to the case of a single electron and at most one photon. This approach retains the massless photon of quantum electrodynamics in contrast to previous approaches that resorted to using massive bosons [T. Cheng, E. R. Gospodarczyk, Q. Su, and R. Grobe, Ann. Phys. 325, 265 (2010)]torepresent the photon. The dynamics of Compton scattering are illustrated using joint probability distributions that evolve in time. This information is compared to that provided by the S matrix.

DOI: 10.1103/PhysRevA.102.062203

I. INTRODUCTION particles via the Coulomb, action-at-a-distance mechanism, which is often present in classical approximations to quantum Quantum electrodynamics (QED) successfully models the field theory. fundamental interactions between photons and electrons. The group of Grobe et al. have made great strides in These interactions are often studied using the perturbative qualitatively visualizing time-resolved particle dynamics in S-matrix approach [1–3], typically depicted using Feynman QED-like systems [7–10]. For example, Wagner et al. from diagrams. This approach is very successful in calculating this group looked at the time evolution of a one-dimensional time-independent quantities such as energies of atomic sub- simplified Hamiltonian based on a Yukawa interaction [10]. levels and scattering cross sections. However, it provides only This model uses a massive boson as an analog to the photon, indirect insight into the time-dependent dynamics of the inter- and it neglects spin and polarization. They used this simpler action, arguably even for the subset of initial states for the full, interaction in place of the full Dirac-Maxwell QED Hamilto- interacting-field evolution that nevertheless eventually leads nian to make numerical simulation in one spatial dimension to asymptotically free evolution. accessible. They produced animations of a bare fermion being Computing the space-time-resolved dynamics of a fully dressed by a boson field as well as scattering processes analo- second-quantized scattering process currently appears in- gous to Compton scattering. We build upon this previous work tractable for three-dimensional QED, especially for high- in developing our description. intensity radiation fields. However, under the approximation Our description is generated by projecting three- of semiclassical optical fields, Krebs et al. have modeled dimensional QED onto a single spatial dimension plus time. three-dimensional (3D) Compton scattering for intense x-ray This projection is accomplished by quantizing in a finite box photons [4] and compared their theoretical results with experi- laterally while allowing infinite expanse in the longitudinal mental measurements made by Fuchs et al. [5]. The success of direction. This results in discrete labels for lateral modes and this type of hybrid quantum-classical model derives from de- continuous labels for longitudinal modes. This combination liberately accounting for time-resolved quantum field theory, allows for essentially one-dimensional calculations, while as championed by Grobe et al. [6], and requires a nontrivial retaining three-dimensional notions of structure such as process to transition from a quantum field description to a polarization and spin. This projection of full QED allows the classical field description. An important result of this tran- force mediator to be identified with the massless photon and sition is removing the unphysical self-repulsion of charged the fermion to be identified with the (bare) electron. As a projection of full QED, the photon’s coupling-free dispersion relation is that of a relativistic massless particle, *[email protected] †[email protected] Eγ = h¯ωγ = chk¯ . (1)

The electron’s coupling-free dispersion relation is that of a study the scattering of a photon from an electron, within relativistic massive particle, several approximations. Formally, the terms can be written as = ω = 2 2 + 2, 3 † 2 Ee h¯ e (mec ) (chk¯ ) (2) HD = d r (r) βmc − ihc¯ α · ∇ (r), where m is the mass of an electron andhk ¯ is the electron’s e ρ(r)ρ(r ) momentum. H = d3r d3r , C 8π r − r In an earlier article [11] we described this projection 0 process in a manner where the dimensional reduction was H = d3kh¯ω a†(k)a (k) + a†(k)a (k) , accomplished using an analysis of units. In this article, we R k 1 1 2 2 provide a more robust derivation wherein the discrete lat- 3 † eral quantization process gives rise to an explicit connection HI =−qc d r (r)[α · A⊥(r)](r). (5) between the quantization length and the strength of the electron-photon interaction. To allow computational tractabil- We discuss each of these terms below, starting with the ity, we restrict our attention to the lowest lateral mode wherein Dirac term. there is no lateral variation of the wave functions. This re- The Dirac term HD is most easily described in momentum sults in something like a plane-wave approximation in that all space. The standard Fourier transformation is given by nontrivial dynamics occur in a single dimension. However, it 1 = 3 ˜ ik·r, differs from a true plane-wave description in that the lateral (r) / d k (k)e (2π )3 2 dimension is taken as finite so that the electron has a finite 1 probability of being found in a given spatial volume. ˜ (k) = d3r(r)e−ik·r, (6) We illustrate how this description can be used to cal- (2π )3/2 culate the dynamics of Compton scattering in such a where the field is a four-component column vector. In one-dimensional system. We compare these computational momentum space, the Dirac term becomes results to the standard S-matrix approach and find that the information provided by the S-matrix is useful but incomplete = 3 ˜ † β 2 + α · ˜ . and is at least as numerically expensive to extract as the full HD d k (k)[ mc hc¯ k] (k) (7) gamut of information that the time-resolved dynamics imme- The matrix β is defined by diately gives. 1 × 0 × β = 2 2 2 2 , (8) II. DERIVATION 02×2 −12×2 The dynamical form of QED derives from the abstract and α is a vector of 4 × 4 matrices with components Schrödinger equation 02×2 σx ∂ α1 = , ih¯ |(t ) = H |(t ), (3) σx 02×2 ∂t QED 02×2 σy where the Hamiltonian HQED describes the second-quantized α2 = , (9) σy 02×2 interaction of radiation and matter. Following Cohen- Tannoudji [12] et al.,wehave 02×2 σz α3 = , σ 0 × HQED = HD + HC + HR + HI , (4) z 2 2 where the Dirac term H gives the bare energy of electrons and D and positrons, the Coulomb term HC gives the energy as- 01 0 −i 10 σ = ,σ= ,σ= . (10) sociated with charged particles repelling and attracting one x 10 y i 0 z 0 −1 another, the radiation term HR gives the energy operator for photons, and the interaction term HI describes the interaction With these definitions, the bracketed term in Eq. (7) can be of matter and radiation. We use this canonical operator to explicitly written as

⎡ ⎤ 2 mc 0¯hck3 hc¯ (k1 − ik2 ) ⎢ 2 + − ⎥ β 2 + α · = ⎢ 0 mc hc¯ (k1 ik2 ) hck¯ 3 ⎥. mc hc¯ k ⎣ 2 ⎦ (11) hck¯ 3 hc¯ (k1 − ik2 ) −mc 0 2 hc¯ (k1 + ik2 ) −hck¯ 3 0 −mc

We can express the matrix (11) as a unitary transformation of where the diagonal matrix β as follows: E(k) = (mc2 )2 + (hc¯ k)2, (13)

2 † βmc + hc¯ α · k = E(k) U (k)βU (k), (12) U (k) = cos(θk/2)I − sin(θk/2)βα · kˆ , (14)

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I is the identity matrix, and the fact that, in quantum ﬁeld theory, at least two charged particles must be present for a Coulomb interaction to occur. h¯k k θ = , ˆ = . Thus, the Coulomb term makes no contribution to the result k arctan k (15) mc k after our one-electron projection is applied.

Using this unitary transform, we can rewrite HD as The third term in the QED Hamiltonian is the radiation term H = d3k E(k)˜ †(k)U (k)βU †(k)˜ (k). (16) D = 3 ω † + † , HR d k h¯ k[a1(k)a1(k) a2(k)a2(k)] (21) The quantum field nature of ˜ is made explicit by writing it ω = † in terms of creation and annihilation operators as follows: where k ck. The creation operator a j (k) and the annihi- ⎡ ⎤ ⎡ ⎤ lation operator a j (k) create and destroy photons with linear c↑(k) c↑(k) polarization j (k) and definite momentumh ¯k. ⎢ ⎥ ⎢ ⎥ ⎢c↓(k)⎥ ⎢c↓(k)⎥ The final term in HQED is the interaction term † ˜ = ⎢ ⎥ ⇔ ˜ = ⎢ ⎥, U (k) (k) ⎣ † ⎦ (k) U (k)⎣ † ⎦ d↑(k) d↑(k) 3 † H =−qc d r (r)[α · A⊥(r)](r). (22) † † I d↓(k) d↓(k) In three spatial dimensions the vector potential has the follow- where c†(k) and c(k) create and annihilate bare electrons and ing second-quantized form: d†(k) and d(k) create and annihilate bare positrons of definite momentumh ¯k. The up- and down-arrow subscripts indicate 3 h¯ the spin of the relevant particle. Using this notation, we arrive A⊥(r) = d k 2 ω (2π )3 at the well-known form j=1,2 0 k ik·r † −ik·r 3 † † × [a j (k) j (k)e + a (k) j (k)e ]. (23) HD = d k E(k)[c↑(k)c↑(k) + c↓(k)c↓(k) j Enforcing spatial periodicity in the x and y directions † † − d↑(k)d↑(k) − d↓(k)d↓(k)]. (17) with arbitrary lateral dimensions × , the interaction term becomes Requiring anticommutation relations, we then have 1 h¯ 2 = 3 E † + † H =−cq dk dk hˆ (k , k), HD d k (k)[c↑(k)c↑(k) c↓(k)c↓(k) I 2 c2π2 z z I 0 n,m,n,m + d†(k)d (k) + d†(k)d (k) − 2δ(k − k)I]. (18) (24) ↑ ↑ ↓ ↓ where δ As usual, we neglect the infinite function term since it is = = π /, π /, proportional to the identity and an overall energy shift does k knm (2 n 2 m kz ) (25) not affect the dynamics of measurables. Equivalently, one can and n, m, n, and m take on all integer values from −∞ to enforce normal ordering from the onset. +∞ in the indicated sums. The momentum-resolved energy The next term in HQED is the Coulomb term which de- density operator hÎ (k , k)is scribes interactions between charged particles: ⎡ ⎤ ⎡ ⎤ T c† (k ) c↑(k) ρ(r)ρ(r ) ↑ 3 3 ⎢ † ⎥ ⎢c↓(k)⎥ HC = d r d r , (19) ⎢c (k )⎥ † ⎢ ⎥ ˆ , = ↓ A † 8π 0r − r hI (k k) : ⎣ ⎦ U (k ) U (k)⎣ ⎦ : (26) d↑(k ) d↑(k) † with d↓(k ) d↓(k) ρ(r) = q†(r)(r). (20) where the colons indicate normal ordering and ρ − α · − + † − α · − The product of terms in Eq. (19) results in all (normal- a j (k k) j (k k) a j (k k ) j (k k ) ordered) charge-conserving permutations of the product of A= √ . k − k four creation and annihilation operators for the various elec- j=1,2 trons and positrons. However, for the current paper, we project (27) Under the spatial periodicity requirement, the anticommuta- the HQED operator onto the smallest set of occupation numbers that still allows for nontrivial scattering between matter and tion and commutation relations include both Kronecker and radiation. As we find below, this can be accomplished by Dirac δ functions, for example, retaining just two types of states: those giving rise to the † {c↓(k ), c (k)}=Iδ , δ , δ(k − k ), (28) measurement of exactly one (bare) electron and exactly zero ↓ n n m m z z photons; and those giving rise to the measurement of exactly † [a (k ), a (k)] = Iδ , δ , δ , δ(k − k ). (29) one (bare) electron and exactly one photon. Under this pro- i j i j n n m m z z jection all matrix elements between allowed states have zero In the interest of computational tractability, we now project contribution from the Coulomb term because the combination the full HQED operator onto one preferred spatial dimension, of four creation and annihilation operators always results in z, in which nontrivial dynamics are allowed. In the other an annihilation operator killing a vacuum ket or a creation two spatial dimensions, x and y, we make the assumption of operator (acting to the left) killing a vacuum bra. This reflects periodicity and project onto only the lowest (constant) mode.

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This spatial projection makes all wave functions arising from In terms of the linearly polarized photon creation and an- the reduced theory independent of x and y. For example, nihilation operators, the circularly polarized photon creation a wave function that describes the probability of finding a and annihilation operators are given by bare electron and a photon at time t in the neighborhood of † + † , , , , † a1(kγ zˆ) isign(kγ )a2(kγ zˆ) spatial coordinates (xe ye ze) for the electron and (xγ yγ zγ ) a (kγ ):= √ , (34) for the photon, and which would therefore have been of the 2 ψ , , , , a kγ z − i kγ a kγ z form eγ (xe ye ze; xγ yγ zγ ; t ) in the full 3D theory of QED, = 1( ˆ) sign(√ ) 2( ˆ). ψ a(kγ ): (35) will now be of the simpler form eγ (ze; zγ ; t ). While the 2 suppressed dimensions x and y still exist after the projection and are necessary to describe three-dimensional concepts such Note that in the projection (33) the bare electron state † | as polarization and spin, there is no remaining spatial variation c↑(kezˆ) is spin up, whereas the bare electron state in the † † in those transverse dimensions after the projection. correlated pair state c↓(kezˆ)a (kγ zˆ)| is spin down. This is Applying this projection, the interaction term of the QED important because if we keep the spins the same, for each Hamiltonian becomes linear polarization j we have † 1/2 a j (kγ zˆ)c↑(kezˆ)|HI c↑(kzzˆ)| h¯ HI =−cq dk dkzhÎ (k , k00 ), (30) π2 z 00 δ(k − k − kγ ) 2 0c2 ∝ z e † α · / [U (kezˆ) j (kγ zˆ)U (kzzˆ)]1,1 |kγ |1 2 the radiation term becomes = , 0 (36) = ω † + † , where the subscripts on the bracketed 4 × 4 matrix give the HR dkzh¯ k00 [a1(k00 )a1(k00 ) a2(k00 )a2(k00 )] (31) row and column indices. On the other hand, one finds that for the opposite output spin we get the nontrivial results and the Dirac term becomes | † | a j (kγ zˆ)c↓(kezˆ)HI c↑(kzzˆ) † † HD = dkz E(k00 )[c↑(k00 )c↑(k00 ) + c↓(k00 )c↓(k00 ) δ − − (kz ke kγ ) † ∝ α · γ , 1/2 U (kezˆ) j (k zˆ)U (kzzˆ) 2,1 |kγ | + d†(k )d (k ) + d†(k )d (k )]. (32) ↑ 00 ↑ 00 ↓ 00 ↓ 00 (37) where The vector k00 indicates a wave vector with components † (0, 0, kz ) as per Eq. (25). [U (kezˆ)α · j (kγ zˆ)U (kzzˆ)]2,1 = [iδ j1 − sign(kγ )δ j2] Even after projecting onto a preferred spatial direction, the hk¯ hk¯ hk¯ hk¯ model described by Eqs. (30)–(32) is still computationally ×i S e C z − S z C e . (38) intractable, since it allows for arbitrarily large numbers of mc mc mc mc photons and any number of charged particles. To reduce the The first term in Eq. (38) demonstrates the appropriateness of computational space, we choose an initial state with a single the circular polarization basis described in Eqs. (34) and (35). electron and project onto a subspace that allows for at most The terms C and S are defined by one photon. This projection is given by 1 1 C(p) = 1 + , P = dk c† (k zˆ)| |c (k zˆ) 2 he(p) e ↑ e ↑ e 1 1 † † S(p) = sign(p) 1 − , (39) + dkγ c↓(kezˆ)a (kγ zˆ)| |a(kγ zˆ)c↓(kezˆ) . (33) 2 he(p) with In the last term in Eq. (33) we have employed photonic 2 creation and annihilation operators without subscripts; they he(p) = 1 + p , (40) create or destroy right- and left-hand circularly polarized photons. This basis is more natural for describing photons in Here we have defined the unitless momentum p as = / Compton scattering than the linear basis because the described p hk¯ mc. photons carry definite angular momentum. Mathematically, The projection (33) allows us to study a much simpler the two circular polarizations separately give the smallest model comprised of a single electron and at most one photon H invariant subspaces in which angular momentum is well- described by a unitless reduced Hamiltonian given by defined, and investigation of one of the subspaces provides all 1 H = PH P = H + Hγ + H ,γ . (41) the information extractable from the other. Equivalently, the mc2 QED e e spin state for the bare electron we chose to consider couples to, and only to, one of these circularly polarized photons, not With this projection, the Schrödinger equation (3) can be both. The spin state for the bare electron that we chose not written in a unitless form, to consider would couple to, and only to, photons with the ∂ i |ψ = H|ψ, opposite circular polarization. ∂τ (42)

062203-4 SPACE-TIME-RESOLVED QUANTUM ELECTRODYNAMICS … PHYSICAL REVIEW A 102, 062203 (2020) where τ = tmc2/h¯. We deﬁne unitless state basis vectors as and follows: w p2w2 φ(p, w, x) = √ exp − − ipx . (50) π 4 mc † mcpe 2 |pe = c zˆ |, (43) h¯ ↑ h¯ The unitless scaled position x is related to the physical po- X x = Xm c/h mc † mcpe † mcpγ sition by e ¯. The electron portion of this initial |p , pγ = c zˆ a zˆ |. (44) e h¯ ↓ h¯ h¯ state is not a fully dressed electron. Such a dressed initial state would require more photons than are in this simpliﬁed system, With these unitless vectors, we can then write the reduced both to dress the electron and to scatter from it. Nevertheless, Hamiltonian terms as the initial state (48) gives the proper projection of the more complicated initial state (dressed with additional photons) H = | | + | , , | , e dpe he(pe) pe pe dpγ pe pγ pe pγ onto the states available in our reduced system. Importantly this projection onto our current state space

Hγ = dpe dpγ hγ (pγ )|pe, pγ pe, pγ |, does not yield a state with any bare electron component in it. This might seem strange at first if one only briefly considers that a dressed electron in this model is such a bare elec- H = , | , − | + . . , e,γ dpe dpeg(pe pe )( pe pe pe pe H c ) tron state superposed with correlated electron-photon pairs. (45) However, it becomes immediately understandable once one where considers that what we want here is the tensor product of such a dressed state with yet another photon: one gets the hγ (p) =|p|, described correlated electron-photon pairs that we use here, S(p)C(p) − S(p)C(p ) and, in addition, gets states with a bare electron correlated g(p, p) = η √ , (46) |p − p| with two photons, and a bare electron never arises in any such superposition. and η is the coupling strength between matter and radiation, To model dynamics, we discretize the unitless momentum defined by p and use a finite number of the corresponding discretized momentum states to create a matrix representation of the reduced h¯ √ η = α, Hamiltonian H in momentum space. In our previous paper (47) mec [11], we discuss in detail the considerations needed when where α is the fine-structure constant. choosing a discretization grid. We then spectrally resolve the Note that the denominator in the coupling g(p, p) goes matrix representation to find the states of well-defined en- to zero when the input momentum p and the output mo- ergy within this discretized model which have straightforward mentum p of the electron are the same, which corresponds dynamics, and then we use these states to generate the time to no change in the electron’s momentum whence the pho- propagator for the Schrödinger equation as usual. With this ton’s momentum being zero. Fortunately, the numerator in g propagator in hand, we can choose an initial state and compute goes to zero faster than the denominator in this case, so that its time-resolved evolution. g(p, p) = 0 and the cross section for “scattering” with no ac- To compute Compton scattering, we choose an initial state = . / tual change in the fermion’s momentum is zero. We emphasize with the electron located at x0e 0 94¯h mec and the pho- =− . / that this fortunate structure is encoded in full 3D QED and that ton located at x0γ 0 94¯h mec, with equal spatial widths = = . / = our projection down onto a single nontrivial space dimension we wγ 0 25¯h mec and opposite central momenta p0e − . = . η = . does not lose this benign feature. The exact details of this 0 03mec and p0γ 0 03mec. We use a coupling of 0 4, structure are those needed in the minimal coupling approach which corresponds to a transverse discretization length scale to preserve the Lorentz covariance of the theory. of ≈ 0.1 pm. Figure 1 plots the evolution of the joint prob- 2 ability distribution |ψ(pe, pγ )| for these parameters as time progresses. At early times, the joint probability distribution III. COMPTON SCATTERING resembles the initial condition with electron and photon mo- We can use this reduced Hamiltonian to model Compton menta centered on their initial values. During the scattering scattering under this simplifying projection. To do this, we process, the probability shifts into the scattered states with choose an initial state |ψ0 as a tensor product state such that opposite momenta. the electron and photon portions are individually Gaussian. To visualize the evolution of this initial state in the spatial These distributions are, respectively, centered at positions x0e domain, we compute the Fourier transform of the joint proba- and x γ , with spatial widths w and wγ , and central momenta bility amplitude 0 e p0e and p0γ , so that 1 ip x ipγ xγ ψ(x , xγ ) = dp dpγ ψ(p , pγ )e e e e . (51) e 2π e e |ψ0 = dpe dpγ ψ(pe, pγ )|pe, pγ , (48) 2 The evolution of |ψ(xe, xγ )| is shown for various times in where Fig. 2, where the dashed line represents the collection of points where the electron and photon positions are the same. ψ(pe, pγ ) = φ(pe − p0e, we, x0e )φ(pγ − p0γ , wγ , x0γ ) The scattering is spatially local, as can be seen by noting (49) that scattering does not occur until the distribution crosses

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FIG. 1. Time evolution of the joint probability distribution 2 |ψ(pe, pγ )| as time progresses for the initial state described in the text. this equal-position line where the electron and photon states spatially overlap. As the system evolves, the vertical (electron) component of the joint probability distribution spreads while the horizontal (photon) component does not, resulting in a vertically elongated joint probability distribution for the unscattered portion FIG. 2. Time evolution of the joint probability distribution |ψ , |2 of the state. Thus, during the scattering process, the narrower (xe xγ ) , the Fourier transform of the distribution plotted photon scatters over a wider range of spatial locations where in Fig. 1. the electron probability is present, resulting in the scattered portion of the distribution being horizontally elongated. Since effective, scattered data it generates in order for interference the electron is spreading throughout the interaction, the scat- phenomena encoded in momentum space to show itself as the tered portion of the distribution at large times is asymmetric. corresponding interference phenomena noticed at early times Note the interference of the unscattered and the scattered in physical space under the full evolution. 2 portions of the distribution at t = 40¯h/mec when these two portions overlap. We see evidence of this spatial interference IV. COMPARISON TO THE S-MATRIX APPROACH in the S-matrix results below, albeit only in momentum space. Presumably this is because S-matrix theory only yields, for We now compare the fully time-resolved results above to any given initial data to be propagated by the full propagator the standard second-order scattering theory. The usual devel- for the interacting theory, the corresponding initial data to be opment of that theory involves, roughly speaking, comparing evolved freely that nevertheless gives the same asymptotic a state |ψ−∞ evolved from t =−∞ to time t = 0 with a dynamics. Thus to visualize the spatial interference evident co-state ψ+∞| evolved from t =+∞ back to time t = 0, at early times during the full propagation, it appears that while necessarily removing the effects of free evolution of evolving freely the effective data that S-matrix theory yields is the bare particles to get convergence. Therefore, the scattering crucial, here turning the interference phenomena evident only operator is often defined by considering the product of Möller in momentum space (for the S-matrix results) into the corre- wave operators, essentially given by sponding interference phenomenon evident only in physical H − H † − H H space and only at early times in the full dynamics. Sˆ = lim eit e it 0 lim e it eit 0 , (52) →∞ →∞ We emphasize that Figs. 1 and 2 illustrate the scattering t t dynamics of a fully second-quantized theory, projected down where H0 = He + Hγ [3]. to one dimension. The model represents an electron moving S-matrix theory provides a mathematical map between data and interacting with the photon in a self-consistent way. The to be evolved according to the full propagator and effective scattering dynamics evident in these plots, including the elec- data to be evolved freely, the correspondence made by de- tron spreading and the interference, are distinctively quantum manding the long-time dynamics be asymptotically the same. mechanical features which are only indirectly available in the In the Möller wave operator approach the S operator is the corresponding S-matrix theory. Below, we show evidence that composition of two maps. The first map is from the final data in S-matrix theory one seems to need to propagate (freely) the for the free problem to actual final data for the full one (each

062203-6 SPACE-TIME-RESOLVED QUANTUM ELECTRODYNAMICS … PHYSICAL REVIEW A 102, 062203 (2020) to be evolved backwards in time, whence “final”). The second important test suggested here, namely that our model allows map is from the actual initial data for the full problem onto the particles to be asymptotically free, i.e., that our model is equivalent effective initial data for the free problem (each purely spatially local. Given the way we derived our model, to be evolved forwards in time, whence “initial”). Then one there is no guarantee of this apriori—wehavethrownaway thinks of what is here the intermediate actual data for the antimatter, which is mathematically crucial to getting perfect full problem as both the relevant final and relevant initial spatial locality. However, the nonlocality that our model must data. Because of this precise definition, the following issues necessarily exhibit seems to be made arbitrarily small in effect arise: does the first map from the effective final data to the as the interaction is made weak. In practice, we have never ob- actual final data exist? Less loosely, on what subspace of the served particles interacting before their spatial wave functions Hilbert space does it exist? And for that map one wonders overlap in our numerical simulations. whether, even when it exists (i.e., on which subspace), it will With the initial data swapped in this way, the only re- map onto the actual initial data for the full problem we would maining issue is that of existence, for both maps indicated in like to consider. The latter is the matter of asymptotic com- Eq. (52), the second of which (the one on the left) actually pleteness. Note that the latter issue arises because we consider being the inverse of the operator suggested in the Möller the inverses of what might be reasonably considered the more wave operator approach (which we have not described here natural mappings: One could argue that it is more natural to fully). For the latter the issue of asymptotic completeness directly try to develop a map from the actual (initial or final) arises because we hope that the indicated Hermitian conjugate data (to be evolved fully) to the corresponding effective data to is the same as the suggested inverse, meaning defined on a be evolved freely. We use this idea in order to circumvent the large enough space to include states that we want to scatter. If issue of completeness and replace it with only a consideration these maps exist, the first with data swapped so that it acts on of existence, as follows. only initial data of the full problem, the overall map arising Instead of thinking of the first map suggested in Eq. (52)as from their composition does too, and one might hope that that acting on data for the free problem, we think of it as acting on composition is directly computable: We find it is, exactly so. data intended for the full problem. It should be the case that In the Appendix we show that if one defines the Sˆ operator the distinction can be made as small as one likes by making more directly via sure the data for the full problem includes only particles well- H − H H Sˆ = lim eit 0 e 2it eit 0 , (53) separated whence not interacting. Here we do this by making t→∞ sure that in our time-resolved dynamics of the full problem its matrix elements can be computed explicitly, without any no interactions appear to occur until well into the evolution perturbative approximation, assuming the operator exists on when the particles start to spatially overlap. Thus there is an the relevant subspace of our Hilbert space. The result is

δ(p + p − p − pγ )g(p + p , p )g(p + pγ , p ) , | ˆ − | , =− π δ , − , e γ e e γ e e e , pγ pe S I pe pγ 2 i (E(pe pγ ) E(pe pγ )) 2 (54) g(pe+pγ ,pe+pγ −p) E(pe, pγ ) − E(p + pγ ) + i + dp e E(pe+pγ −p,p)−E(pe,pγ )−i where I is the identity operator, the energy of a bare electron standard time-dependent perturbation theory, truncating the state is relevant Neumann series at second order. Applying the here-developed version of Sˆ to the specific E(p ) = h (p ), (55) e e e initial state used to create Figs. 1 and 2,wehave the bare energy of an electron-photon state is |ψ+∞ = Sˆ|ψ−∞ = Sˆ|ψ0. (58) E(p , pγ ) = h (p ) + hγ (pγ ), (56) e e e Note here our mapping of the effective initial data for the free and is a positive infinitesimal. To second order in perturba- system (to be propagated backwards in time in the Möller op- tion theory, Eq. (54)gives erator approach) to the actual initial data of the full problem. − iπδ(E p , p − E p , pγ ) Again, this assumes that our model allows, at least approxi- (2) 2 ( e γ ) ( e ) p , p |Sˆ |p , pγ = mately, for our particles to be asymptotically free. Using the γ e e , − + + E(pe pγ ) E(pe pγ ) i second-order perturbation representation of Sˆ allows us to × δ + − − calculate the projection of the output state |ψ+∞ onto the pair (pe pγ pe pγ ) states as × g(p + pγ , p )g(pe + pγ , pe ). (57) e e (2) ψ (pe, pγ ) = pe, pγ |ψ+∞. (59) The first δ function in Eq. (54) describes conservation of energy, and the second δ function describes conservation of Figure 3 plots ψ (2) compared to the initial data. Figure 3(a) momentum. The perturbative approximation described above shows the initial spatial distribution and Fig. 3(b) plots the makes errors on the order of η3 in general, but for terms such initial momentum distribution. After applying the scatter- as Eq. (54), which are traditionally used to describe Compton ing theory above, we plot the scattered spatial distribution 4 (2) 2 scattering, the error is on the order of η , as can be seen by |ψ (xe, xγ )| in Fig. 3(c) and the corresponding momen- (2) 2 comparison with the exact formula. Importantly, this version tum distribution |ψ (pe, pγ )| in Fig. 3(d). The scattering of Sˆ agrees to the indicated order with that obtained by the theory distribution plotted in Fig. 3 agrees with the overall

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rated portion of that same photon. This is incorrect for several reasons, mainly the misinterpretation of correlated states as not being the same as tensor product states. For example, one must realize that this other photon is necessarily correlated to another electron in the charge-1 subspace, so that this would not be the scenario of pure Compton scattering, but rather would include the scattering of two electrons as well. Alter- natively, if it is actually a pure photon state, not correlated with an additional electron, because of charge conservation, the evolution of the latter exists in another invariant subspace of our Hilbert space, and any overall wave function describing the two separated subspaces simply describes the usual Schrödinger cat superposition of the two kinds of uncorrelated states. To better understand why the simplified system in this paper does not allow the electron to locally dress itself before scattering, as suggested earlier, consider a state with a dressed | electron, denoted by e d , correlated with an additional “scat- 2 |γ FIG. 3. (a) The initial spatial distribution |ψ(xe, xγ )| as tered” photon state, denoted by . We can write this sort of 2 in Fig. 2. (b) The initial momentum distribution |ψ(p , pγ )| tensor product state in the form e as in Fig. 1. (c) The scattered joint probability in space distribution (2) 2 |ψ (x , xγ )| . (d) The scattered joint probability distribution in mo- e |e |γ = C(e)|e + C(e,γ)|eγ |γ |γ |ψ (2) , |2 d b b mentum (pe pγ ) . γ = C | |γ + C ,γ | |γ |γ , (e) e b (e ) eγ b (60) magnitude of scattering shown in our time-resolved plots in γ Fig. 1. Recall that all frames of Fig. 3 represent data at C the initial time; the latter two frames represent the effective where represents the probability amplitudes for the various data to be propagated freely. Figures 3(a) and 3(b) propagate basis states. Note that the final form of this state contains forward with interactions to eventually arrive at a condition three-particle states which are outside the scope of the sim- where all further dynamical interactions appear finished and plified system. Projecting Eq. (60) back onto states with no further propagation is asymptotically free. Figures 3(c) and more than one photon as in this paper, we find 3(d) represent initial conditions that, if propagated freely at P0,1γ |ed |γ all times (i.e., with no interaction, but only the trivial standard quantum spreading effects), would result eventually in the = C | |γ + C ,γ | |γ |γ very same dynamics as for the previous case in the long-time P0,1γ (e) e b (e ) eγ b γ limit. Presumably the structure evident in the momentum plot in Fig. 3(d) encodes the interference and dynamical behavior = C | |γ + C ,γ | |γ |γ (e)P0,1γ e b (e )P0,1γ eγ b discussed above in relation to the spatial version of the full γ dynamics animations. = C · | |γ + C ,γ | |γ |γ (e) 1 e b (e )0 eγ b γ V. CONCLUSION = C(e)|eb|γ , (61) It might be somewhat surprising that we can, in a qualitatively correct way, model electron-photon scattering with such that is, few photons. Previous work regarding the minimal dressing of P , γ |e |γ = C(e)|e |γ . (62) bare electrons to become full-fledged electrons required dress- 0 1 d b ing of a bare electron with at least one photon. This allows Thus, in our minimal model with no “extra” scattering photon, the dressed object to acquire spatial stability, meaning that we encode the initial data of a dressed electron correlated with the dressed object’s spatial structure becomes time invariant a scattering photon onto just the bare correlated pair states | |γ aside from the usual quantum spreading. So, apriori, it might e b . This mapping makes an error on the order of the not have been surprising to have learned that one needed size of the vector of dressing coefficient values C(e,γ), which multiple photons to describe the process: at least one to dress becomes arbitrarily small for arbitrarily small couplings be- the electron and another to scatter. tween radiation and matter. This observation is consistent Various ways of misthinking this scenario exist. In addition with standard S-matrix theory where one typically considers to the error described previously, another possible misinter- scattering between bare particle states and not dressed states. pretation would be to think that to observe Compton scattering The time-resolved approach described above yields re- one needs to start with a bare electron and a spatially sepa- sults comparable to standard scattering theory and provides rated single photon, expecting the bare electron to dress itself a more direct access to the time-resolved dynamics of the locally with the photon in addition to scattering from the sepa- interaction. Moreover, to observe these same dynamical

062203-8 SPACE-TIME-RESOLVED QUANTUM ELECTRODYNAMICS … PHYSICAL REVIEW A 102, 062203 (2020) effects in scattering theory, one must first calculate the scat- Inserting Eq. (A2) into the right-hand side of Eq. (A1), ac- tered “initial data” and then use free propagation to observe complishing the time integral, and then taking matrix elements the ensuing dynamics. We have found that, at least for the spe- with respect to eigenstates of H0, represented by β| and |α, cific problem studied here, analyzing dynamics directly using gives a triple integral over the λ spectral variables that can the approach introduced in this work is computationally less be accomplished using Cauchy’s theorem. The result of this demanding than generating even the analogous information triple integration is from scattering theory. Our method directly produces visual- izations of the scattering process in both space and momentum + − + β| A − lim e itH0 e 2itH e itH0 |α representations. While we have restricted to the case of an t→∞ electron-photon interaction here, this time-resolved approach = β| i |α, can be extended to the study of electron-electron interactions lim β + α (A3) ↓0 E0 ( ) E0 ( ) + − and to include multiple dimensions. For electron-electron in- 2 i H teractions, the Coulomb term will become important in the second-quantized Hamiltonian. where E0(α) gives the eigenvalue for |α and likewise for the state β|. This result is completely general, applicable to many ACKNOWLEDGMENTS problems in addition to the one studied here. The last fact is that the operator between β| and |α in This work is supported by the National Science Foundation Eq. (A3) is a resolvent for H and for the particular model under Grant No. 1708185. considered in this paper, can be derived exactly. For Compton scattering, we are interested in |α=|p, q and β|= p, q|, APPENDIX and one finds that When it exists, we can compute the matrix elements of Sˆ defined in Eq. (53) exactly because of three facts. The first fact 1 |p, q |p, q= is that, when this limit exists, it is the same as that which is z − H z − E0(p, q) g(p+q,p) given by the “Abel limit”: − , + z E0 (p q) + − + + , + − 2 ≡ − itH0 2itH itH0 g(p q p q p˜γ ) Sˆ A lim e e e z − E0(p + q) + d p˜γ →∞ E0 (p+q−p˜γ ,p˜γ )−z t ∞ + , + − − + − + g(p q p q pγ ) ≡ lim dt e t e itH0 e 2itH e itH0 . (A1) × |p + q+ dpγ ↓ − + − , 0 0 z E0(p q pγ pγ ) The second fact is that for any self-adjoint operator H, and for ×|p+ q− pγ , pγ ) > any time t 0, we have the following spectral resolutions: (A4) −iλt − = 1 λ e , exp( itH ) lim d ↓0 −2πi λ + i − H Inserting Eq. (A4) into Eq. (A3), using z = [E(p , q ) + , / + → 1 eiλt E(p q)] 2 i , and taking the limit 0 gives the result exp(itH ) = lim dλ . (A2) in Eq. (54). ↓0 2πi λ − i − H

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