Causality in Quantum Field Theory with Classical Sources

Bo-Sture K. Skagerstam1,a), Karl-Erik Eriksson2,b), Per K. Rekdal3,c)

a)Department of Physics, NTNU, Norwegian University of Science and Technology, N-7491 Trondheim, Norway b) Department of Space, Earth and Environment, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden c)Molde University College, P.O. Box 2110, N-6402 Molde, Norway

Abstract

In an exact quantum-mechanical framework we show that space-time expectation values of the second-quantized electromagnetic fields in the Coulomb gauge, in the presence of a classical source, automatically lead to causal and properly retarded electromagnetic field strengths. The classical ~-independent and gauge invariant Maxwell’s equations then naturally emerge and are therefore also consistent with the classical special theory of relativity. The fundamental difference between interference phenomena due to the linear nature of the classical Maxwell theory as considered in, e.g., classical optics, and interference effects of quantum states is clarified. In addition to these issues, the framework outlined also provides for a simple approach to invariance under time-reversal, some spontaneous photon emission and/or absorption processes as well as an approach to Vavilov-Cherenkovˇ radiation. The inherent and necessary quantum uncertainty, limiting a precise space-time knowledge of expectation values of the quantum fields considered, is, finally, recalled. arXiv:1801.09947v2 [quant-ph] 30 Mar 2019

1Corresponding author. Email address: [email protected] 2Email address: [email protected] 3Email address: [email protected] 1. Introduction

The roles of causality and retardation in classical and quantum-mechanical versions of electrodynamics are issues that one encounters in various contexts (for recent discussions see, e.g., Refs.[1]-[14]). In electrodynamics it is natural to introduce gauge-dependent scalar and vector potentials. These potentials do not have to be local in space-time. It can then be a rather delicate issue to verify that gauge-independent observables obey the physical constraint of causality and that they also are properly retarded. Attention to this and related issues are often discussed in a classical framework where one explicitly shows how various choices of gauge give rise to the same electromagnetic field strengths (see, e.g., the excellent discussion in Ref.[8]). Even though issues related to causality in physics have been discussed for many years, we are still facing new insights regarding such fundamental concepts. In a recent investigation [12] the near-, intermediate-, and far-field causal properties of classical electromagnetic fields have been discussed in great detail. In terms of experimental and theoretical considerations, locally backward velocities and apparent super-luminal features of electromagnetic fields were demonstrated. Such observations do not challenge our un- derstanding of causality since they describe phenomena that occur behind the light front of electromagnetic signals (see, e.g., Refs.[7, 12, 14] and references cited therein). In the present paper, it is our goal to recall the problems mentioned above in a quantum- mechanical framework. Some of theses aspects were already considered a long time ago by Fermi [15]. We consider, in particular, the finite and exact time-evolution as dictated by quantum mechanics with second-quantized electromagnetic fields in the presence of a general classical conserved current. In terms of suitable and well-known optical quadratures (see, e.g., Ref.[16]), the corresponding ~-dependent dynamical equations can then be reduced to a system of decoupled harmonic oscillators with a space-time dependent external force. No pre-defined global causal order is assumed other than the deterministic time-evolution as prescribed by the Schrödinger equation. The classical ~-independent theory of Maxwell then naturally emerges in terms of properly causal and retarded expectation values of the second- quantized electromagnetic field for any initial quantum state. This is in line with more general S-matrix arguments due to Weinberg [17]. Furthermore, the fundamental role of interference in the sense of quantum mechanics as compared to classical interference effects due to the linearity of Maxwell’s equations, can then be clarified. The quantum-mechanical approach also leads to a deeper insight with regard to the role of unavoidable quantum uncertainty of average values of quantum fields. This presentation can, in a rather straightforward manner, be extended to gravitational quantum uncertainties around a flat Minkowski space-time, in the presence of a classical source in terms of a conserved energy-momentum tensor. As a result, the classical weak-

1 field limit of Einstein’s theory of general relativity emerges. This is discussed in a separate publication [18]. The paper is organized as follows. In Section 2 we recall, for reasons of completeness, the classical version of electrodynamics in vacuum and the corresponding issues of causality and retardation in the presence of a space-time dependent source, and the extraction of a proper set of physical but non-local degrees of freedom. The exact quantum-mechanical framework approach is illustrated in terms of a second-quantized single-mode electromagnetic field in the presence of a time-dependent classical source in Section 3, where emergence of the classical ~-independent physics is also made explicit. In Section 4, the analysis of Section 3 is extended to multi-modes and to a general space-time dependent classical source. The issues of causality, retardation, and time-reversal are then discussed in Section 5. The framework also provides for a discussion of some radiative processes, and in Section 6 we consider dipole radiation, and the fameous classical Vavilov-Cherenkovˇ radiation is reproduced in a straightforward and exact manner. In Section 7, we briefly discuss the role of the intrinsic quantum uncertainty of expectation values considered. Finally, and in Section 8, we present conclusions and final remarks. Some multi-mode considerations as referred to in the main text are presented in an Appendix.

2. Maxwell’s Equations with a Classical Source

Unless stated explicitly, we often make use of the notation E ≡ E(x, t) for the electric ﬁeld and similarly for other ﬁelds. The microscopic classical Maxwell’s equations in vacuum are then (see, e.g., Ref.[19]): ρ ∇ · E = , (2.1) 0

∇ · B = 0 , (2.2)

∂B ∇ × E = − , (2.3) ∂t ∂E ∇ × B = µ j + µ , (2.4) 0 0 0 ∂t √ with the velocity of light in vacuum as given by c = 1/ 0µ0. Eqs.(2.1) and (2.4) imply current conservation, i.e., ∂ρ + ∇ · j = 0 . (2.5) ∂t The classical Maxwell’s equations can, of course, be written in a form that is explicitly covariant under Lorentz transformations but this will not be of importance here.

2 The general vector identity

∇ × (∇ × F) = ∇(∇ · F) − ∇2F , (2.6)

applied to the electric field E and making use of Maxwell’s equations (2.3) and (2.4) implies that 2 1 ∂ E 2 ∂ j ρ 2 2 − ∇ E = −µ0 − ∇ , (2.7) c ∂t ∂t 0 with retarded as well as advanced solutions. By physical arguments one selects the retarded solution, even though Maxwell’s equations are invariant under time-reversal as, e.g., discussed by Rohrlich [6]. We now write the electric field E and the magnetic field B in terms of the vector potential A and the scalar potential φ, i.e., ∂A E = − − ∇φ , (2.8) ∂t and B = ∇ × A . (2.9) The Coulomb (or radiation) gauge, which, of course, is not Lorentz covariant, is defined by the requirement ∇ · A = 0 , (2.10) and therefore leads to at most two physical degrees of freedom of the electromagnetic field.

A deﬁned with this gauge-choice restriction is denoted by AT . By making use of the vector identity Eq.(2.6) with F = AT , Amp`ere’slaw, i.e., Eq.(2.3), may then be written in the form

2 ∂ AT 2 2 jT 2 − c ∇ AT = , (2.11) ∂t 0

where we have introduced a transverse current jT according to ∂ j ≡ j − ∇φ . (2.12) T 0 ∂t

Eq.(2.11) is, of course, the well-known wave-equation for the vector potential AT in the

Coulomb gauge. The transversality condition ∇ · jT = 0 follows from charge conservation and ρ ∇ · E = −∇2φ = , (2.13) 0 in the Coulomb gauge. Eq.(2.13) is, therefore, not dynamical but should rather be regarded as a constraint on the physical degrees of freedom in the Coulomb gauge enforcing current conservation. The instantaneous scalar potential φ degree of freedom can therefore

3 be eliminated entirely in terms of the physical charge density ρ (in this context see, e.g., Refs.[20, 21]). In passing we also recall that in the Coulomb gauge, the scalar potential φ is, according to Eq.(2.13), given by Z 0 1 3 0 ρ(x , t) φ(x, t) = d x 0 . (2.14) 4π0 |x − x | Due to the conservation of the current, i.e., Eq.(2.5), the time derivative of φ may be written in the form Z 0 0 ∂φ(x, t) 1 3 0 ∇ · j(x , t) = − d x 0 . (2.15) ∂t 4π0 |x − x |

According to the well-known Helmholtz decomposition theorem F = FL + FT for a vector ﬁeld (see, e.g., Ref.[19]), formally written in the form 1 F = ∇(∇ · F) − ∇ × (∇ × F) , (2.16) ∇2

using Eq.(2.6), we can identify the corresponding longitudinal current jL, i.e., 1 Z ∇0 · j(x0, t) j (x, t) ≡ − ∇ d3x0 . (2.17) L 4π |x − x0| It is now evident that the right-hand side of the wave-equation Eq.(2.11) for the vector potential can be expressed in terms of the current j(x, t), i.e., 1 Z j(x0, t) j (x, t) ≡ ∇×(∇× d3x0 ) . (2.18) T 4π |x − x0|

The important point here is that jT is an instantaneous and non-local function in space of the physical current j(x, t). When the Helmholtz decomposition theorem is applied to the

vector potential A = AL +AT , it follows that the transverse part AT is gauge-invariant but, again, a non-local function in space of the vector potential A. At the classical level, we now make a normal-mode Ansatz for the real-valued vector ﬁeld A conﬁned in, e.g., a cubic box with volume V = L3 and with periodic boundary conditions.

With k = 2π(nx, ny, nz)/L, where nx, ny, nz are integers, we therefore write r X 1 ik·x ∗ ∗ −ik·x AT (x, t) = qkλ(t)(k; λ)e + qkλ(t) (k; λ)e , (2.19) V 0 kλ

with time-dependent Fourier components qkλ(t). The, in general, complex-valued polarization vectors (k; λ) obey the transversality condition k · (k; λ) = 0. They are normalized in such a way that

ˆ X ∗ ˆ ˆ Pij ≡ Pij(k) ≡ i (k; λ)j(k; λ) = δij − kikj , (2.20) λ

4 where we have deﬁned the unit vector kˆ ≡ k/|k|. In the case of linear polarization the real-valued, orthonormal, and linear polarization unit vectors (k; λ), with λ = 1, 2, are such λ+1 that i(−k; λ) = (−1) i(k; λ). Since AT itself is independent of the actual realization of the polarization degrees of freedom (k; λ), it is without any diﬃculty to express Eq.(2.19) in terms of, e.g., the complex circular polarization vectors with λ = ±, i.e., 1 (k; ±) = √ (k; 1) ± i(k; 2) , (2.21) 2

such that (−k; ±) = ∗(k; ±).

The Ansatz Eq.(2.19) for AT is, of course, consistent with transversality of the current

jT in Eq.(2.11). Due to the transversality of jT , we can then also write that r X 0 j (x, t) = j (t)(k; λ)eik·x + j∗ (t)∗(k; λ)e−ik·x . (2.22) T V kλ kλ kλ

The time-dependence of qkλ(t) is now determined by the dynamical equation Eq.(2.11) for

AT , i.e., 2 q¨kλ(t) + ωk qkλ(t) = jkλ(t) , (2.23)

with ωk = c|k|. If we deﬁne classical real-valued quadratures

∗ Qkλ(t) ≡ qkλ(t) + qkλ(t) , (2.24)

then ¨ 2 ∗ Qkλ(t) + ωk Qkλ(t) = jkλ(t) + jkλ(t) ≡ fkλ(t) . (2.25) This equation has the same form as the dynamical equation for a time-dependent forced harmonic oscillator. The corresponding quantum dynamics will be treated in the next session.

5 3. Single Mode Considerations

As seen in the previous section, a single-mode of the electromagnetic ﬁeld reduces to a dynamical system equivalent to a forced harmonic oscillator with a time-dependent external force. The quantization of such a system is well-known (see, e.g., Refs.[22]-[27]) and is presented here in a form suitable for illustrating a calculational procedure to be used in later sections for ﬁnite time intervals.

With only one mode present, we write Q ≡ Qkλ(t), ω ≡ ωk, as well as f(t) ≡ fkλ(t). Eq.(2.25) then takes the form Q¨ + ω2Q = f(t) . (3.1) This classical equation of motion can, of course, be obtained from the classical time-dependent

Hamiltonian Hcl(t) for a forced harmonic oscillator with m = 1, i.e., P 2 1 H (t) = + ω2Q2 − f(t)Q. (3.2) cl 2 2 We quantize this classical system by making use of the canonical commutation relation Q, P = i~ . (3.3)

We express Q and P in terms of the quantum-mechanical quadratures r ~ ∗ Q = a + a , (3.4) 2ω as well as r ~ω ∗ P = i a − a , (3.5) 2 ∗ where [a, a ] = 1. The classical Hamiltonian Hcl(t) is then promoted to the explicitly time- dependent quantum-mechanical Hamiltonian H(t) according to 1 H (t) → H(t) = ω a∗a + + g(t) a + a∗ , (3.6) cl ~ 2 where we have defined r ~ g(t) ≡ −f(t) . (3.7) 2ω In general, it is notoriously difficult to solve the Schrödingerequation with an explicitly time-dependent Hamiltonian. Due to the at most quadratic dependence of a and a∗ in Eq.(3.6) it is, however, easy to solve exactly for the unitary quantum dynamics. Indeed, if

one considers the dynamical evolution of the system in the interaction picture with |ψ(t)iI ≡

exp(itH0/~)|ψ(t)i, where we for convenience make the choice t0 = 0 of initial time, then

d|ψ(t)iI i~ = HI (t)|ψ(t)iI . (3.8) dt 6 For observables O in the interaction picture we also have that

OI (t) ≡ exp(itH0/~)O exp(−itH0/~) . (3.9)

∗ In our case H0 = ~ω(a a + 1/2) and therefore

−iωt ∗ iωt HI (t) = g(t) ae + a e . (3.10)

The explicit solution for |ψ(t)iI is then given by

Z t i i 0 0 |ψ(t)iI = exp φ(t) exp − dt HI (t ) |ψ(0)i , (3.11) ~ ~ 0 for any initial pure state |ψ(0)i. Eq.(3.11) can easily be veriﬁed by, e.g., considering the limit

∆t → 0 of (|ψ(t+∆t)iI −|ψ(t)iI )/∆t using that exp(A+B) = exp(A) exp(B) exp(−[A, B]/2) if [A, B] is a c-number. The c-number phase φ(t) can then be computed according to

Z t Z t Z t0 1 0h 0 0 i 1 0 00h 00 0 i iφ(t) = dt N(t ),HI (t ) = dt dt HI (t ),HI (t ) , (3.12) 2~ 0 2~ 0 0 with Z t 0 0 N(t) ≡ dt HI (t ) , (3.13) 0 0 0 since, in our case, [ N(t ),HI (t ) ] is a c-number. We therefore see that, apart from a phase, the time-evolution in the interaction picture is controlled by a conventional displacement operator as used in various studies of coherent states (see, e.g., Refs.[28, 29] and references cited therein). The expectation value of the quantum-mechanical quadrature Q in Eq.(3.4) at time t, i.e.,

1 1 Z t hQi(t) = hQi cos(ωt) + hP i sin(ωt) + dt0f(t0) sin ω(t − t0) , (3.14) ω ω 0 where hOi ≡ hOi(0) for the initial expectation value of an observable O. In Eq.(3.14) we, of course, recognize the general classical solution of the forced harmonic oscillator equations of motion Eq.(3.1), i.e.,

d2 hQi(t) + ω2hQi(t) = f(t) , (3.15) dt2 in terms of its properly retarded Green’s function (see, e.g., Ref.[30]). The last term in Eq.(3.14) is classical in the sense that it does not depend on ~. Possible quantum-interference

7 effects are hidden in the homogeneous solution of Eq.(3.15). Similarly, we find for the P - quadrature in Eq.(3.5) that hP i(t) = dhQi(t)/dt or more explicitly: Z t hP i(t) = hP i cos(ωt) − hQiω sin(ωt) + dt0f(t0) cos ω(t − t0) . (3.16) 0 Even though the classical equation of motion emerges in terms of quantum-mechanical expectation values, intrinsic quantum uncertainty for any observable O as defined by

(∆O)2(t) ≡ hψ(t)|(O − hOi)2|ψ(t)i , (3.17)

are in general present. For O = Q one ﬁnds that

sin2 ωt cos ωt sin ωt (∆Q)2(t) = (∆Q)2 cos2 ωt + (∆P )2 + hPQ + QP − 2hQihP ii , (3.18) ω2 ω independent of the external force f(t). For minimal dispersion states, i.e., states for which ∆Q∆P = ~/2, the last term is zero. For coherent states one then finds the intrinsic and time-independent quantum-mechanical uncertainty (∆Q)2(t) = ~/2ω and (∆P )2(t) = ~ω/2. The classical equation of motion Eq.(3.15) allows for linear superpositions of solutions. Such linear superposition are, however, not directly related to quantum-mechanical superpositions of the initial quantum states since expectation values are non-linear functions of √ quantum states. For number states |ni ≡ (a∗)n|0i/ n!, with a|0i = 0, which in terms of, e.g., a Wigner function have no classical interpretation except for the vacuum state |0i (see, e.g., Ref.[16]), we have that hQi = hP i = 0 but (∆Q)2(t) = (∆P )2(t)/ω2 = (n + 1/2)~/ω. For an √ p initial state of the form |ψ(0)i = (|0i+|1i)/ 2 we find that hQi = ~/2ω and hP i = 0 with an intrinsic time-dependent quantum uncertainty, e.g., (∆Q)2(t) = ~(2 − cos2 ωt)/2ω. This initial state therefore leads to expectation values that do not correspond to a superposition of the classical solutions obtained from the initial states |0i or |1i. This simple example demonstrates the fundamental difference between the role of the superposition principle in classical and in quantum physics. It is a remarkable achievement of experimental quantum optics that such quantum-mechanical interference effects between the vacuum state and a single-photon state have been observed [31, 32] (for a related discussions also see Ref.[33]- [37]). In the next section we extend this simple single-mode case to the general multi-mode space-time dependent situation.

4. Multi-Mode Considerations

We will now consider emission as well as absorption processes of photons in the presence of a general space-time dependent classical source as illustrated in Fig.1. In the multi-mode

case the interaction Hamiltonian HI (t) for a classical current j is now an extension of the

8 . . .

j(x, t)

Figure 1: Absorption and emission of photons from a classical current j(x, t). single-mode version Eq.(3.10). In the Coulomb gauge and in the interaction picture, we therefore consider Z 3 HI (t) = − d x j(x, t) · AT (x, t) V r X ~ −iωkt ∗ ∗ ∗ iωkt = − j(k, t) · (k; λ)akλe + j (k, t) · (k; λ)akλe , (4.1) 2V 0ωk kλ with the “free” ﬁeld Hamiltonian X 1 H = ω (a∗ a + ) . (4.2) 0 ~ k kλ kλ 2 kλ Here we have introduced the Fourier transformed current Z j(k, t) ≡ d3x eik·x j(x, t) . (4.3) V ˆ ˆ Since jL(k, t) = k(k·j(k, t)) and jT (k, t) = j(k, t)−jL(k, t), it is clear due to the transversality condition k · (k; λ) = 0 that only the transverse part of the current contributes in Eq.(4.1). The interaction Eq.(4.1) therefore corresponds to a system of independent forced harmonic oscillators of the one-mode form as discussed in the previous section. The second- quantized version of the vector-potential in Eq.(2.19) then has the form of a free quantum ﬁeld, i.e., r X ~ i(k·x−ωkt) ∗ ∗ −i(k·x−ωkt) AT (x, t) = akλ(k; λ)e + akλ (k; λ)e , (4.4) 2V 0ωk kλ

9 with the basic canonical commutation relation

∗ [akλ, ak0λ0 ] = δλλ0 δkk0 , (4.5)

and where we recall that ωk = c|k|. The vacuum state |0i is then such that akλ|0i = 0 for

all quantum numbers kλ. The quantum ﬁeld AT is then normalized in such a way that Z 3 1 2 1 2 H0 = d x ε0ET (x, t) + B (x, t) , (4.6) V 2 µ0

where we, for the free ﬁeld in Eq.(4.4), make use of ET = −∂AT /∂t and B = ∇×A = ∇×AT . If we consider the circular polarization vectors (k; ±) according to Eq.(2.21), we have to replace the annihilation operators akλ with 1 ak± = √ (ak1 ∓ iak2) . (4.7) 2 We then observe that X X (k; λ)akλ = (k; λ)akλ , (4.8) λ=1,2 λ=±

which means that the quantum ﬁeld AT does not depend on the actual realization of the choice of polarization degrees of freedom. ∗ The single photon quantum states |kλi ≡ akλ|0i, with λ = ±, will then carry the energy ~ωk, momentum ~k as well as the intrinsic spin angular momentum ±~ along the direction kˆ, i.e., the helicity quantum number of a massless spin-one particle. In passing, we remark that the latter property can be inferred from a consideration of a rotation with an angle θ around the wave-vector k in terms of a rotation matrix Rij(θ), which implies that ak± → ak±(θ) = exp(±iθ)ak±. In terms of the corresponding rotated polarization vectors

i(k; λ|θ) = Rij(θ)j(k; λ) we then have, in accordance with Eq.(4.8), that X X (k; λ)akλ = (k; λ|θ)akλ(θ) . (4.9) λ=1,2 λ=±

In addition to the intrinsic spin angular momentum, photon states can also carry conventional orbital angular momentum which plays an important role in many current contexts (see, e.g., Ref.[38] and references cited therein) but will not be of concern in the present work. A complete set of physical and well-deﬁned Fock-states can then be generated in a conventional manner. By construction, these states have positive norm avoiding the presence of indeﬁnite norm states in manifestly covariant formulations (for some considerations see, e.g., Refs.[39, 40, 41]).

10 Since, obviously, Z t 0 0 dt HI (t ) = 0 r t Z 0 0 X ~ 0 −iωkt 0 ∗ iωkt ∗ ∗ 0 − dt akλ e (k; λ) · j(k, t ) + akλ e (k; λ) · j (k, t ) , (4.10) 2V 0ωk kλ 0

we conclude that the time-evolution for |ψ(t)iI in Eq.(3.11) is, apart from a phase factor, given by a multi-mode displacement operator Z t i 0 0 Y ∗ ∗ D(α) ≡ exp − dt HI (t ) = exp αkλ(t)akλ − αkλ(t)akλ . (4.11) ~ 0 kλ

Here αkλ(t) is, as inferred from Eq.(4.10), explicitly given by r t Z 0 i ~ 0 iωkt ∗ 0 ∗ αkλ(t) ≡ dt e j (k, t ) · (k; λ) , (4.12) ~ 2V 0ωk 0 with j∗(k, t) = j(−k, t). The displacement operator D(α) has the form of a product of independent single-mode displacement operators. By making use of Eq.(3.11), and by considering the action on the vacuum state, the quantum-mechanical time-evolution generates a multi-mode coherent state D(α)|0i, apart from the ~-dependent phase φ(t) in Eq.(3.11). As in the single-mode case, the time-dependent expectation value of the transverse quantum

field AT (x, t) will then obey a classical equation of motion like Eq.(3.15), i.e., (see Appendix A) 2 ∂ hAT (x, t)i 2 2 jT (x, t) 2 − c ∇ hAT (x, t)i = , (4.13) ∂t 0 to be investigated in more detail in Section 5. In other words, there are particular quantum states of the radiation field, namely multi-mode coherent states, which naturally lead to the classical electromagnetic fields obeying Maxwell’s equations Eqs.(2.1)-(2.4) in terms of quantum-mechancial expectation values.

5. The Causality Issue

The expectation value of the transverse second-quantized vector ﬁeld AT is now given by Eq.(A.4), i.e., Z 3 Z t 0 Z 1 d k ik·x 0 sin ωk(t − t ) 3 0 −ik·x0 0 0 hAT (x, t)i = 3 e dt d x e jT (x , t ) , (5.1) 0 (2π) 0 ωk where we have carried out a sum over polarizations according to Eq.(2.20) as in Eq.(A.5), and where the sum over k in the large volume V limit is replaced by X V Z = d3k . (5.2) (2π)3 k

11 The Fourier transform of the transverse current vector in Eq.(5.1) is as above given by Z 3 ik·x jT (k, t) ≡ d xe jT (x, t) . (5.3)

The time-derivative of Eq.(5.1) can now be written in the form

Z Z t ∂hAT (x, t)i 1 3 0 0 ∂ 0 0 0 0 0 0 = d x dt G(x − x , t − t ) j(x , t ) − jL(x , t ) , (5.4) ∂t 0 0 ∂t by making use of the Helmholtz decomposition of the current vector j(x, t), and where we identiﬁed the Green’s function G(x, t)

Z 3 1 X ik·x sin(ωkt) d k ik·x sin(ωkt) G(x, t) ≡ lim e = 3 e = V →∞ V ωk (2π) ωk k 1 = δ(t − |x|/c) − δ(t + |x|/c) . (5.5) 4πc2|x| This Green’s function is a solution to the homogeneous wave-equation

∂2G(x, t) = c2∇2G(x, t) , (5.6) ∂t2 such that G(x, 0) = 0. For the second term in Eq.(5.4) we need to consider the integral

Z Z t 3 0 0 ∂ 0 0 0 ∂ 0 0 I ≡ d x dt G(x − x , t − t )∇ 0 φ(x , t ) . (5.7) 0 ∂t ∂t

This is so since the longitudinal vector current jL(x, t) may be written in the form

∂ 1 Z ρ(x0, t) ∂ j (x, t) = ∇ d3x0 = ∇φ(x, t) , (5.8) L ∂t 4π |x − x0| 0 ∂t

where we make use of the Helmholtz decomposition Eq.(2.17) and current conservation. After a partial integration in the time variable t0 and by making use of Eq.(5.6), the integral I can therefore be written in the following form

Z Z t I = ∇φ(x, t) + c2 d3x0 dt0∇0 · ∇0G(x − x0, t − t0)∇0φ(x0, t0) , (5.9) 0 where we have used the fact that ∂G(x, t)/∂t = δ3(x) at t = 0 as well as the initial condition

jL(x, 0) = 0 for all x. We now perform two partial integrations over the spatial variable and by using Eq.(2.13) we, ﬁnally, see that

c2 Z Z t I = ∇φ(x, t) − d3x0 dt0G(x − x0, t − t0)∇0ρ(x0, t0) , (5.10) 0 0

12 neglecting spatial boundary terms and using the initial condition ρ(x, 0) = 0 for all x. The ﬁrst term in Eq.(5.10) exactly cancels the instantaneous Coulomb potential contribution in the expectation value of the quantized electric ﬁeld observable

∂hA (x, t)i hE(x, t)i = − T − ∇φ(x, t) . (5.11) ∂t For t > t0, we therefore obtain the desired result

Z 0 0 ∂ 1 3 0 j x , t − |x − x |/c hE(x, t)i = − 2 d x 0 ∂t 4π0c |x − x | Z 0 0 0 1 3 0 ∇ ρ x , t − |x − x |/c − d x 0 , (5.12) 4π0 |x − x | where ∇0ρ(x0, t0) in Eq.(5.12) has to be evaluated for a ﬁxed value of t0 = t − |x − x0|/c. In a similar manner we also see that Z Z t 1 3 0 0 0 0 0 0 0 hB(x, t)i = ∇ × hAT (x, t)i = d x dt G(x − x , t − t )∇ × jT (x , t ) 0 0 µ Z 1 = 0 d3x0 ∇0× j(x0, t − |x − x0|/c) , (5.13) 4π |x − x0|

0 0 0 0 0 0 0 0 0 since ∇ × jT (x , t ) = ∇ × j(x , t ). In Eq.(5.13), we remark again that ∇ × j(x , t ) has to be evaluated for a fixed value of t0 = t − |x − x0|/c. The causal and properly retarded form of the electric and magnetic quantum field expectation values in terms of the physical and local sources given have therefore been obtained (see in this context, e.g., Ref.[19], Section 6.5). The expectation values as given by Eqs.(5.12) and (5.13) obey Maxwell’s equations in terms of the classical charge density ρ and current j. The quantization procedure above of the electromagnetic field explicitly breaks Lorentz covariance. Since, however, Maxwell’s equations transform covariantly under Lorentz transformations we can, nevertheless, now argue that the special theory of relativity emerges in terms of expectation values of gauge- invariant second-quantized electromagnetic fields. Maxwell’s equations of motion according to Eqs.(2.1)-(2.4) are invariant under the dis- T crete time-reversal transformation t → t0 = −t with E(x, t) −→ E0(x, t) = E(x, −t) and T T T B(x, t) −→ B0(x, t) = −B(x, −t) provided j(x, t) −→ j0(x, t) = −j(x, −t) and ρ(x, t) −→ ρ0(x, t) = ρ(x, −t). At the classical level, the corresponding transverse vector potential trans- T 0 forms according to AT (x, t) −→ AT (x, t) = −AT (x, −t). The anti-unitary time-reversal transformation T is implemented on second-quantized fields in the interaction picture ac-

13 cording to the rule (see, e.g., Refs.[20],[42]-[44])

T 0 I hψ(t)|AT (x, t)|ψ(t)iI −→ I hψ(t)|AT (x, t)|ψ(t)iI

−1 = I hψ(−t)|T AT (x, t)T |ψ(−t)iI . (5.14)

0 −1 It then follows that I hψ(t)|AT (x, t)|ψ(t)iI = − I hψ(−t)|AT (x, −t)|ψ(−t)iI if T akλT = λ a−kλ(−1) and provided the vacuum state |0i is invariant under time-reversal. We therefore 0 T find that hAT (x, t)i = −hAT (x, −t)i. We therefore obtain hE(x, t)i −→ hE(x, −t)i and T hB(x, t)i −→ −hB(x, −t)i as it should. Due to the form of the Green’s function G(x, t) in Eq.(5.5) it can be verified that Eq.(5.1) also leads to average values hE(x, t)i and hB(x, t)i that transform correctly under time-reversal. The arrow of time can therefore, as expected, not be explained by our approach but as soon as the direction of time is defined the observable quantities hE(x, t)i and hB(x, t)i are causal and properly retarded. In the presence of external sources we could have an apparent breakdown of time-reversal invariance unless one also time-reverses the external sources.

6. Electromagnetic Radiation Processes The rate for spontaneous emission of a photon from, e.g., an excited hydrogen atom can now be obtained in a straightforward manner in terms of a slight extension of the interaction Eq.(4.10) as to be made use of in ﬁrst-order time-dependent perturbation theory. We then make use of the long wave-length approximation Z ∂ Z d j(k, t) ≈ d3xj(x, t) = d3xxρ(x, t) = q x(t) , (6.1) ∂t dt taking current conservation Eq.(2.5) into account, where ρ(x, t) = qδ(3)(x − x(t)) in terms of the position x(t) of the charged electron in the interaction picture. For the spontaneous

single photon transition |ii → |fi with |ii = |aii ⊗ |0i and |fi = |af i ⊗ |kλi, we then arrive at the standard dipole radiation ﬁrst-order matrix element r ~ hi|HI (0)|fi = iq ωif (k; λ) · haf |x|aii , (6.2) 2V 0ωk using Eq.(4.10) with q = −e in the interaction picture. The relevant matrix element

haf |x(t)|aii is then given by exp(−iωif t)haf |x|aii. For the atomic transition from |aii =

|nlmi = |2pmi to the ﬁnal atomic ground state |af i = |1si, the corresponding rate is then given by 8 2π X 2 2 4 c Γ ≡ 2 δ(ωk − ωif )|hi|HI (0)|fi| = α , (6.3) 3 aB ~ kλ

14 2 in terms of the fine-structure constant α ≡ e /4π0~c and the Bohr radius aB. The rate Γ is independent of the quantum number m. Stimulated emission gives rise to a multiplicative factor (1 + nkλ). Eq.(6.3) is, of course, a well-known text-book result in agreement with the experimental value (see, e.g., Ref.[45]). The considerations above can be extended to graviton quadrupole radiation processes in an analogous manner [18]. The power of electromagnetic emission from a classical conserved electric current in, e.g., a non-dissipative dielectric medium and the fameous Vavilov-Cherenkovˇ [46] radiation can, furthermore, now also be derived in terms of the quantum-mechanical framework above. This form of radiation was first explained by Frank and Tamm [47] using the framework of Maxwell’s classical theory of electromagnetism. The exact classical ~-independent expression for the power of Vavilov-Cherenkovˇ radiation (see, e.g., Sect. 13.4 in Ref.[19]), neglecting possible spin effects to be discussed elsewhere [48], can now be obtained as follows. For a particle with electric charge q, mass m, and an initial velocity v, moving in dielectric medium such that 0 → 0, with > 1, the interaction HI (t) in Eq.(4.10) leads to a displacement operator with αkλ(t) now replaced by

r t Z 0 Z i ~ 0 iωkt 3 0 ∗ −ik·x αkλ(t) = dt e d xj(x, t ) · (k; λ)e ~ 2V 0ωk t0 r Z t i ~ ∗ 0 i(ω −k·v)t0 = q v · (k; λ) dt e k , (6.4) ~ 2V 0ωk t0 where the relativistic current j(x, t) in an inertial frame is given by

j(x, t) = qvδ(3)(x − vt) . (6.5)

The power P (ω)dω of emitted radiation in the range ω to ω + dω is then obtained by

evaluating the exact expression dhH0i(t)/dt, using Eq.(6.4), where

Z ωc dhH0i(t) d X = |α (t)|2 ≡ dωP (ω) , (6.6) dt dt kλ kλ 0

in the large volume V limit, and by considering the large T ≡ t − t0 limit. Here we can,

of course, disregard the additive divergent zero-point ﬂuctuations in hH0i(t). The large T

limit leads to a phase-matching condition ωk = k · v = vk cos θC , using v ≡ |v| and k ≡ |k|, ˇ expressed in terms of the well-known Cherenkov angle cos θC ≡ c/nv, where ωk = ck/n with √ the refractive index n ≡ . The λ-sum over the polarization degrees of freedom in Eq.(6.6) leads to X |(v · (k; λ)|2 = |v|2(1 − cos2 θ) , (6.7) λ=1,2

15 using Eq.(2.20), where, in general, cos θ ≡ vˆ ·kˆ in terms of the unit vectors. In summing over the angular distribution of the radiation emitted in Eq.(6.6), the large T phase-matching

condition is taken into account. The cut-oﬀ angular frequency ωc is to be determined in a standard manner taken the ω-dependence of into account (see, e.g., Ref.[19]). We then easily ﬁnd the well-known ~-independent power spectrum

2 e v 2 P (ω) = ω 1 − cos θC . (6.8) 4π0c c

Alternatively, but in a less rigorous manner, one may consider hH0i(t)/T and making use of Eq.(6.4) in the large T -limit, i.e., r i ~ ∗ αkλ(t) = 2πq δ (ωk − k · v) v · (k; λ) . (6.9) ~ 2V 0ωk

By inspection we then observe that αkλ(t) in Eq.(6.9) exactly corresponds the quantum- mechanical amplitude for the emission of one photon from the source to ﬁrst-order in time-

dependent perturbation theory even though our expression for αkλ(t) is exact. We have therefore derived a power spectrum that exactly corresponds to the 1937 Frank- ˇ Tamm expression [47] in terms of the Cherenkov angle cos θC as obtained from the δ-function constraint in Eq.(6.9). In the quantum-mechanical perturbation theory language this constraint corresponds to an energy-conservation δ-function as a well as to conservation of √ momentum taking the refractive index n ≡ into account. The corresponding energy of

the emitted photon is then given by Eγ = ~ω and the Minkowski canonical momentum by ˇ pγ = ~k (see, e.g., Ref.[49]), with ω = c|k|/n. The expression for the Cherenkov angle cos θC is then modiﬁed according to [50]

p 2 2 ! c 2 1 − v /c cos θC = 1 + ~ωk(n − 1) . (6.10) nv 2mc2

As was ﬁrst noted by Ginzburg ([50] and references cited therein), and also presented in various text-books accounts (see, e.g., Refs.[51, 52]), ﬁrst-order perturbation theory in quantum mechanics actually leads to the same exact power spectrum for Vavilov-Cherenkovˇ radiation. The explanation of this curious circumstance can be traced back to the fact that all higher order corrections are taken into account by the presence of the phase φ(t) in Eq.(3.11).

7. Quantum Uncertainty

The displacement of quantum states as induced by D(α), as deﬁned in Eq.(4.11), acting on an arbitrary pure initial state again leads to Maxwell’s equations for the expectation value

16 of the quantum field changing, at most, the homogeneous solution of the expectation value of the wave-equation (4.13). The corresponding quantum uncertainty of E(x, t), however, depends on the choice of the initial state along the same reasoning as in the single-mode case in Section 3. An essential and additional ingredient with regard to the approach to the classical limit is to consider the variance of, e.g., the second-quantized E(x, t)-field suitably defined. We consider the scalar quantity

(∆E(x, t))2 ≡ hE2(x, t)i − hE(x, t)i2 . (7.1)

We observe that the uncertainty in Eq.(7.1) in general does not depend on the complex parameters α when evaluated for the displaced state D(α)|ψ(0)i and is therefore determined by the uncertainty as determined by the initial state |ψ(0)i. In order to be speciﬁc, we will evaluate the uncertainty ∆E(x, t) for a displaced Fock

state with |ψ(0)i = |..., nkλ, ...., nk0λ0 , ..i. We then obtain

2 ~ X 1 (∆E(x, t)) = ωk(nkλ + ) . (7.2) V ε0 2 kλ Physical requirements now demand that the uncertainty ∆E(x, t) must be smaller than expectation values of the components of the second-quantized electromagnetic field E(x, t). If the sum in Eq.(7.2) had been convergent, the variance would have vanished in the naive limit ~ → 0. Since the natural constant ~ is non-zero, the sum in Eq.(7.2) is, however, divergent. Even though the expectation value of the quantum field at a space-time point (x, t) in our case is well-defined, the corresponding uncertainty is therefore actually divergent. This means that the observable value of the quantum field in a space-time point (x, t) is physically ill-defined. In the early days of quantum field theory, this fact was actually noticed already in 1933 by Bohr and Rosenfeld [53] and later proved in a rigorous manner by Wightman [54]. Bohr and Rosenfeld also provided a solution of this apparent physical contradiction. The basic idea is to introduce quantum field observables averaged over some finite space-time volume. Bohr and Rosenfeld made use of a cube centered at the space-point x at a fixed time t which, however, makes some of the expressions obtained rather complicated. We will follow another approach which makes the expressions more tractable (see, e.g., problem 2.3 in Ref.[20]), i.e., we consider Z ∞ Z 0 3 0 0 0 0 0 Eσ(x, t) ≡ dt d x fs(x − x )ft(t − t )E(x , t ) , (7.3) −∞ V where 1 x2 fs(x) = 2 3/2 exp(− 2 ) , (7.4) (2πσs ) 2σs

17 and 1 t2 ft(t) = 2 1/2 exp(− 2 ) . (7.5) (2πσt ) 2σt

The parameter σs gives a characteristic scale for the space-volume around the point x where

we perform the space average. Correspondingly, the parameter σt gives a characteristic time-scale for the time average procedure. The linear classical Maxwell’s equations can then again be obtained as in the previous

sections in terms of the quantum-mechanical average of ﬁelds like Eσ(x, t) provided that the classical sources are space and/or time averaged in the same manner. It now follows that

r 2 2 2 2 X ~ωk σs k σt ωk Eσ(x, t) = i exp(− − ) 2V ε0 2 2 kλ i(k·x−ωkt) ∗ ∗ −i(k·x−ωkt) × akλ(k; λ)e − akλ (k; λ)e . (7.6)

2 The variance (∆Eσ(x, t)) of the space and time averaged electric quantum field Eσ(x, t), in a sufficiently large quantization volume V , will then be finite and corresponds to an energy

2 3 1 2π~c ε (∆E (x, t)) σ ≥ E ≡ , (7.7) 0 σ σ (2π)3 σ

3 localized in a volume Vσ ≡ σ V , where

2 2 2 2 σ ≡ σs + c σt . (7.8)

It is now clear that Eσ will be ﬁnite in the cases σt = 0, σs 6= 0 as well as σs = 0, σt 6= 0. Physically, the expression in Eq.(7.7) corresponds, apart from an irrelevant numerical factor, to the energy of a photon with a wave-length λ ' σ and, hence, a wave-number

k ' 2π/σ and therefore to an energy Eσ ' ~ck ' 2π~c/σ, in a typical localization volume

Vσ. It is now clear that Eσ will tend to inﬁnity as σ → 0, i.e., we would then obtain an arbitrarily large energy and/or energy density if we try to localize the quantum ﬁeld in

the sense above in an arbitrary small Vσ. A macroscopic ﬁeld, however, corresponds to a

localization volume much larger than Vσ, and therefore these quantum uncertainties can be disregarded in the classical regime. 2 This latter feature can be illustrated by evaluating (∆Eσ(x, t)) for a thermal Planck

distribution of nkλ at a temperature T with a typical coherence length scale σT ≡ ~c/kBT .

For localization scales σ σT , i.e., at suﬃciently small temperatures, one then ﬁnds that 2 3 4 4 ε0(∆Eσ(x, t)) σ = Eσ(1 + 4π (σ/σT ) /15). The thermal induced uncertainty can therefore

be neglected in comparison with Eσ for large thermal coherence lengths σT as compared

to σ. If, on the other hand, σT ≤ σ, i.e., at suﬃciently high temperatures, it follows that

18 2 3 2 3/2 ε0(∆Eσ(x, t)) σ = kBT (1 + (σT /σ) /8)/4π and, as expected, the thermal uncertainty will then be dominating at sufficiently high temperatures. As was predicted a long time ago for single-mode quantum fields [55], it is possible to reduce the uncertainty (∆E(x, t))2 below the vacuum value by making use of initial squeezed quantum states |ψ(0)i. This feature has recently been confirmed experimentally (Ref.[56] references cited therein). For multi-mode considerations, relevant for the framework of the 2 present work, this may also be possible for (∆Eσ(x, t)) but this will not be a topic in the present paper.

8. Final Remarks

We have seen that a quantum-mechanical framework offers a platform to study causality and retardation issues in the classical theory of Maxwell. As has been shown elsewhere, this framework can rather easily be extended to a derivation of the weak-field limit of Ein- stein’s general theory of relativity [18]. From second-quantization of the physical degrees of freedom under the condition of current conservation the well established classical theory for electromagnetism naturally emerges. The overwhelming experimental support for Maxwell’s classical theory does not necessarily imply the existence of photons and doubts on the existence of such quantum states are sometimes put forward (see, e.g., Ref.[57]). However, the quantum-mechanical derivation of the classical theory necessarily implies the existence of single particle quantum states corresponding to a photon. We have also observed that various radiation processes including the classical, i.e., ~- independent, Vavilov-Cherenkovˇ radiation can be obtained in a straightforward manner. It may come as a surprise that a first-order quantum-mechanical perturbation theory calcula- tion can give an exact ~-independent answer. This, as it seems, remarkable fact is explained by the factorization of the time-evolution operator in terms of a displacement operator for quantum states in the interaction picture according to Eq.(3.11) making use of Eq.(4.10). The phase φ(t) then contains the non-perturbative effects of all higher-order corrections to the first-order result. As a matter of fact, similar features are known to occur also in some other situations. As is well-known, the famous differential cross-section for Rutherford scattering can be obtained exactly in terms of the first-order Born approximation. All higher order corrections will then contribute with an overall phase for probability amplitudes which follows from the exact solution (see, e.g., the excellent discussion in Ref.[58]). The classical Thomson cross- section for low-energy light scattering on a charged particle is also exactly obtained from a Born approximation due to the existence of an exact low-energy theorem in quantum electrodynamics (see, e.g., the discussions in Refs.[2, 3, 59]).

19 Appendix

A. Expectation Value of the Quantum Field AT (x, t)

Apart from a phase-factor, the time-evolution in the interaction picture is controlled by the operator Z t i 0 0 Y ∗ ∗ exp − dt HI (t ) = exp αkλ(t)akλ − αkλ(t)akλ , (A.1) ~ 0 kλ where αkλ(t) by is given by Eq.(4.12) in the main text. Since expectation values are independent of the picture used, i.e.,

Shψ(t)|O|ψ(t)iS = I hψ(t)|OI (t)|ψ(t)iI ≡ hOI (t)i , (A.2) we see that r X ~ ik·x−iωkt ∗ ∗ −ik·x+iωkt hAT (x, t)i = (k; λ)αkλ(t) e + (k; λ)αkλ(t) e . (A.3) 2V ωk0 kλ

If we, in particular, consider initial states |ψ(0)i such that hψ(0)|akλ|ψ(0)i = 0 for all kλ, like quantum states with a ﬁxed number of photons, we ﬁnd that t Z 0 X i 0 iωk(t −t)+ik·x ∗ ∗ 0 hAT (x, t)i = (k; λ) dt e (k; λ) · j (k, t ) 2V ωk0 kλ 0 t Z 0 −∗(k; λ) dt0 e−iωk(t −t)−ik·x (k; λ) · j(k, t0) 0

0 X 1 Z t sin ωk(t − t) = − eik·x dt0 (k; λ) ∗(k; λ) · j∗(k, t0) , (A.4) V 0 ωk kλ 0 after a change k → −k in the last term above, using j(−k, t) = j∗(k, t) as well as Eq.(4.12). The second time-derivative of this expression will then contain the following factor: X 1 eik·x (k; λ) ∗(k; λ) · j∗(k, t) V 0 kλ X 1 = eik·x j∗(k, t) − kˆ kˆ · j∗(k, t) , (A.5) V 0 k

∗ ˆ ˆ where jT (k, t) = jT (−k, t) ≡ j(−k, t)−k k·j(−k, t) corresponds to the Fourier-components

of a transverse current jT (x, t) such that ∇ · jT (x, t) = 0, and where use have been made

20 of Eq.(2.20). The transverse term obtained using Eq.(A.5) can therefore be written in the form:

1 X ik·x e jT (−k, t) V 0 k Z 1 X ik·x 3 0 −ik·x0 0 1 = e d x e jT (x , t) = jT (x, t) , (A.6) V 0 0 k where we make use of the fact that

1 X 0 eik·(x−x ) = δ(3)(x − x0) . (A.7) V k We have therefore reproduced the source-term in the wave-equation Eq.(4.13) in the main 2 2 text and all the terms we have left out above in the evaluation of ∂ hAT (x, t)i/∂t satisfy the homogeneous wave-equation.

ACKNOWLEDGMENT

The research by B.-S. S. was supported in part by Molde University College and NTNU. K.-E. E. gratefully acknowledges hospitality at Chalmers University of Technology. The research of P. K. R. was supported by Molde University College. The authors are grateful for the hospitality provided for at Department of Space, Earth and Environment, Chalmers University of Technology, Sweden, during the ﬁnalization of the present work. REFERENCES

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