Causality in quantum field theory with classical sources

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 conserved source automatically lead to causal and properly retarded ℏ-independent electromagnetic field strengths. The classical ℏ-independent and gauge invariant Maxwell’s equations naturally emerge in terms of quantum-mechanical expectation values 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-Čherenkov radiation. The inherent and necessary quantum uncertainty, limiting a precise space-time knowledge of expectation values of the quantum fields considered, is, finally, recalled.


Introduction
The roles of causality and retardation in classical, ÿ-independent, and quantum-mechanical versions of electrodynamics are issues that one encounters in various contexts (for recent discussions see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][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 [8]). Even though issues related to causality in physics have been discussed for many years they are still open for investigations and we are 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, e.g., 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 understanding of causality since they describe phenomena that occur behind the light front of electromagnetic signals (see, e.g., [7,12,14] and references cited therein).
In the present paper, we investigate the problems mentioned above, in a quantum-mechanical framework. Some aspects of this were, in fact, already considered a long time ago by Fermi [15]. Here we consider, in particular, the finite time and exact time-evolution as dictated by quantum mechanics with second-quantized electromagnetic fields in the presence of arbitrary classical conserved currents.
In terms of suitable and well-known optical quadratures (see, e.g., [16]), the corresponding ÿ-dependent dynamical equations can then be reduced to a system of decoupled harmonic oscillators with space-time dependent external forces. No pre-defined global causal order is assumed other than the deterministic time-evolution as prescribed by the Schrödinger equation. The classical ÿ-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]. Thus we demonstrate explicitly that the quantized theory of electro-magnetic fields is an adequate framework for solving a whole set of classical field-theory problems. We also clarify the fundamental role of quantum-mechanical interference in comparison to classical interference, as expressed by the linearity of Maxwell's equations.
In a straightforward manner, we have also extended this kind of presentation to a quantized theory of gravitational fluctuations around a flat Minkowski space-time, in the presence of a classical source in terms of a conserved energy-momentum tensor. From this, the weak-field limit of Einstein's general theory of relativity emerges. Our work on this was presented 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 singlemode electromagnetic field in the presence of a time-dependent classical source in section 3, where emergence of the classical ÿ-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 famous classical Vavilov-Čherenkov 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, 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.

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., [19]): 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 f, i.e., where we have introduced a transverse current j T according to is, of course, the well-known wave-equation for the vector potential A T in the Coulomb gauge. The transversality condition ∇·j T =0 follows from charge conservation and in the Coulomb gauge. Equation (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 f degree of freedom can therefore be eliminated entirely in terms of the physical charge density ρ (in this context see, e.g., [20,21]). In passing we also recall that in the Coulomb gauge, the scalar potential f is, according to equation (2.13), given by Due to the conservation of the current, i.e., equation (2.5), the time derivative of f may be written in the form According to the well-known Helmholtz decomposition theorem F=F L +F T for a vector field (see, e.g., [19]), formally written in the form using equation (2.6), we can identify the corresponding longitudinal current j L , i.e., It is now evident that the right-hand side of the wave-equation equation (2.11) for the vector potential can be expressed in terms of the current j(x, t), i.e., The important point here is that j T 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=A L +A T , it follows that the transverse part A T 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=L 3 and with periodic boundary conditions. With k=2π (n x , n y , n z )/L, where n x , n y , n z are integers, we therefore write with time-dependent Fourier components q kλ (t). The, in general, complex-valued polarization vectors k;  l ( ) obey the transversality condition k k; ) . They are normalized in such a way that where we have defined the unit vector k k k ô | |. In the case of linear polarization the real-valued, orthonormal, and linear polarization unit vectors k;  l ( ), with λ=1, 2, are such that k k ; 1 ; Since A T itself is independent of the actual realization of the polarization degrees of freedom k;  l ( ), it is without any difficulty to express equation (2.19) in terms of, e.g., the complex circular polarization vectors with λ=±, i.e., for A T is, of course, consistent with transversality of the current j T in equation (2.11). Due to the transversality of j T , we can then also write that The time-dependence of q kλ (t) is now determined by the dynamical equation equation 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.

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., [22][23][24][25][26][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≡Q kλ (t), ω≡ω k , as well as f (t)≡f kλ (t). Equation (2.25) then takes the form This classical equation of motion can, of course, be obtained from the classical time-dependent Hamiltonian H cl (t) for a forced harmonic oscillator with unit mass, i.e., We quantize this classical system by making use of the canonical commutation relation We express Q and P in terms of the quantum-mechanical quadratures The classical Hamiltonian H cl (t) is then promoted to the explicitly time-dependent quantum-mechanical Hamiltonian H(t) according to In general, it is notoriously difficult to solve the Schrödinger equation with an explicitly time-dependent Hamiltonian. Due to the at most quadratic dependence of a and a* in equation (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 where we for convenience make the choice t 0 =0 of initial time, then For observables  in the interaction picture we also have that The explicit solution for t I y ñ | ( ) is then given by for any initial pure state 0 y ñ | ( ) . Equation (3.11) can easily be verified by, e.g., considering the limit t 0 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., [28,29] and references cited therein).
The expectation value of the quantum-mechanical quadrature Q in equation can now easily be evaluated for an arbitrary initial pure state 0 where 0   á ñ º á ñ( ) for the initial expectation value of an observable . In equation (3.14) we, of course, recognize the general classical solution of the forced harmonic oscillator equations of motion equation in terms of its properly retarded Green's function (see, e.g., [30]). The last term in equation (3.14) is classical in the sense that it does not depend on ÿ. Possible quantum-interference effects are hidden in the homogeneous solution of equation (3.15). Similarly, we find for the P-quadrature in equation Even though the classical equation of motion emerges in terms of quantum-mechanical expectation values, intrinsic quantum uncertainty for any observable  as defined by independent of the external force f (t). For minimal dispersion states, i.e., states for which ΔQΔP=ÿ/2, the last term is zero. For coherent states one then finds the intrinsic and time-independent quantum-mechanical The classical equation of motion equation (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 n a n 0 , which in terms of, e.g., a Wigner function have no classical interpretation except for the vacuum state 0ñ | (see, e.g., [16]), we have that Q P 0 á ñ = á ñ = but Q t P t n 1 2 . For an initial state of the form 0 0 and P 0 á ñ = with an intrinsic timedependent quantum uncertainty, e.g., Q t t 2 cos 2 . This initial state therefore leads to expectation values that do not correspond to a superposition of the classical solutions obtained from the initial states 0ñ | or 1ñ | . 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 [33][34][35][36][37]). In the next section we extend this simple single-mode case to the general multi-mode space-time dependent situation.

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 figure 1. In the multi-mode case the interaction Hamiltonian H I (t) for a classical current j is now an extension of the single-mode version equation (3.10). In the Coulomb gauge and in the interaction picture, we therefore consider Here we have introduced the Fourier transformed current with the basic canonical commutation relation a a , , 4 . 5 and where we recall that c k k w = | |. The vacuum state 0ñ | is then such that a 0 0 k ñ = l | for all quantum numbers kλ. The quantum field A T is then normalized in such a way that where, for the free field in equation The single photon quantum states a k 0 k * lñ º ñ l | | , with λ=±, will then carry the energy ÿω k , momentum ÿk as well as the intrinsic spin angular momentum ±ÿ along the direction k , 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 R ij (θ), which implies that a a i a exp . In terms of the corresponding rotated polarization vectors R k k ; ; 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., [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., [39][40][41] we conclude that the time-evolution for t I y ñ | ( ) in equation (3.11) is, apart from a phase factor, given by a multimode displacement operator Here t k a l ( ) is, as inferred from equation (4.10), explicitly given by The displacement operator D a ( )has the form of a product of independent singlemode displacement operators. By making use of equation (3.11), and by considering the action on the vacuum state, the quantum-mechanical time-evolution generates a multi-mode coherent state D 0 a ñ ( )| , apart from the ÿ-dependent phase f (t) in equation (3.11). As in the single-mode case, the time-dependent expectation value of the transverse quantum field A T (x, t) will then obey a classical equation of motion similar to equation (3.15), i.e., (see appendix) 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 equations (2.1)-(2.4) in terms of quantum-mechanical expectation values.

The causality issue
The expectation value of the transverse second-quantized vector field A T is now given by equation (A.4), i.e., where we have carried out a sum over polarizations according to equation (2.20) as in equation (A.5), and where the sum over k in the large volume V limit is replaced by The Fourier transform of the transverse current vector in equation (5.1) is as above given by The time-derivative of equation (5.1) can now be written in the form 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) This Green's function is a solution to the homogeneous wave-equation This is so since the longitudinal vector current j L (x, t) may be written in the form where we make use of the Helmholtz decomposition equation (2.17) and current conservation. After a partial integration in the time variable t¢ and by making use of equation (5.6), the integral I can therefore be written in the following form where we have used the fact that G t t x x , 3 d ¶ ¶ = ( ) ( )at t=0 as well as the initial condition j L (x, 0)=0 for all x. We now perform two partial integrations over the spatial variable and by using equation (2.13), we finally see that neglecting spatial boundary terms and using the initial condition ρ(x, 0)=0 for all x. The first term in equation (5.10) exactly cancels the instantaneous Coulomb potential contribution in the expectation value of the quantized electric field observable For t t > ¢, we therefore obtain the desired result where t x , r ¢ ¢ ¢ ( ) in equation (5.12) has to be evaluated for a fixed value of t t c x x ¢ = --¢ | | . In a similar manner we also see that In equation (5.13), we remark again that t j x , ¢´¢ ¢ ( ) has to be evaluated for a fixed value of t t c x x ¢ = --¢ | | . 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., [19], section 6.5).
The expectation values as given by equations (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 equations (2.1)-(2.4) are invariant under the discrete At the classical level, the corresponding transverse vector potential transforms according to The anti-unitary time-reversal transformation  is implemented on second-quantized fields in the interaction picture according to the rule (see, e.g., [20,[42][43][44]) 14   I  T  I  I  T  I  I  T  | is invariant under time-reversal. We therefore find that

It then follows that
We therefore obtain ) as it should. Due to the form of the Green's function G(x, t) in equation (5.5) it can be verified that equation (5.1) also leads to expectation values t E x, á ñ ( ) and t B x, á ñ ( ) 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 t E x, á ñ ( ) and t B x, á ñ ( ) 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.

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 equation (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 ( ) in terms of the position x(t) of the charged electron in the interaction picture. For the spontaneous single photon transition  using equation (2.20), where, in general, v k cos q ºˆ·ˆin terms of the unit vectors. In summing over the angular distribution of the radiation emitted in equation (6.6), the large T phase-matching condition is taken into account. We then easily find the well-known ÿ-independent power spectrum P e c v c 4 1 cos . 6.8 Alternatively, but in a less rigorous manner, one may consider H t T 0 á ñ( ) and make use of equation (6.4) in the large T-limit, i.e., By inspection we then observe that t k a l ( ) in equation (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 t k a l ( ) is exact.
We have therefore derived a power spectrum that exactly corresponds to the 1937 Frank-Tamm expression [47] in terms of the Čherenkov angle cos C q as obtained from the δ-function constraint in equation (6.9). In the quantum-mechanical perturbation theory language this constraint corresponds to an energy-conservation δfunction as a well as to conservation of momentum taking the refractive index n  º into account. The corresponding energy of the emitted photon is then given by E γ =ÿω and the Minkowski canonical momentum by p k  = g (see, e.g., [49]), with c n k w = | | . The expression for the Čherenkov angle cos C q is then modified according to [50] c nv n v c mc cos 1 1 1 2 . 6 . 1 0 As was first noted by Ginzburg ([50] and references cited therein), and also presented in various text-books accounts (see, e.g., [51,52]), first-order perturbation theory in quantum mechanics actually leads to the same exact power spectrum for Vavilov-Čherenkov 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 f(t) in equation (3.11).

Quantum uncertainty
The displacement of quantum states as induced by D a ( ), as defined in equation (4.11), acting on an arbitrary pure initial state again leads to Maxwell's equations for the expectation value 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 We observe that the uncertainty in equation (7.1) in general does not depend on the complex parameters a when evaluated for the displaced state D 0 a y ñ ( )| ( ) and is therefore determined by the uncertainty as determined by the initial state 0 y ñ | ( ) . In order to be specific, we will evaluate the uncertainty t E x, D ( )for a displaced Fock state with Physical requirements now demand that the uncertainty t E x, D ( )must be smaller than expectation values of the components of the second-quantized electromagnetic field t E x, ( ). If the sum in equation (7.2) had been convergent, the variance would have vanished in the naive limit 0   . Since the natural constant ÿ is non-zero, the sum in equation (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 welldefined, 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 [20]), i.e., we consider