

The Hopfield model revisited: Covariance and Quantization

F. Belgiorno, S.L. Cacciatori, F. Dalla Piazza Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo 32, IT-20133 Milano, Italy and INdAM-GNFM Department of Science and High Technology, Università dell’Insubria, Via Valleggio 11, IT-22100 Como, Italy INFN sezione di Milano, via Celoria 16, IT-20133 Milano, Italy Università “La Sapienza”, Dipartimento di Matematica, Piazzale A. Moro 2, I-00185, Roma, Italy

There are several possible applications of quantum electrodynamics in dielectric media which require a quantum description for the electromagnetic field interacting with matter fields. The associated quantum models can refer to macroscopic electromagnetic fields or, in alternative, to mesoscopic fields (polarization fields) describing an effective interaction between electromagnetic field and matter fields. We adopt the latter approach, and focus on the Hopfield model for the electromagnetic field in a dielectric dispersive medium in a framework in which space-time dependent mesoscopic parameters occur, like susceptibility, matter resonance frequency, and also coupling between electromagnetic field and polarization field. Our most direct goal is to describe in a phenomenological way a space-time varying dielectric perturbation induced by means of the Kerr effect in nonlinear dielectric media. This extension of the model is implemented by means of a Lorentz-invariant Lagrangian which, for constant microscopic parameters, and in the rest frame, coincides with the standard one. Moreover, we deduce a covariant scalar product and provide a canonical quantization scheme which keeps into account the constraints implicit in the model. Examples of viable applications are indicated.

: Physica Scripta

Keywords: Hopfield model, relativistic covariance, quantum theory.


1 Introduction

A longstanding field of investigation for quantum field theory is represented by pair creation in external fields or by moving boundaries. We limit ourselves to quote a couple of seminal papers which were published at the birth of modern quantum field theory [1, 2]. We are mainly interested in photon pair creation associated with variations of the dielectric constant in a dielectric medium, which has been a subject of important investigations, as e.g. a series of papers by Schwinger concerning a possible relation between dynamical Casimir effect and sonoluminescence [3]. In this paper, instead of quantizing phenomenologically the electromagnetic field in presence of a dielectric medium (see e.g. [4]), we delve into a less phenomenological situation in which dielectric properties are modeled as in the Hopfield model: the electromagnetic field interacts with a set of oscillators reproducing sources for dispersive properties of the electromagnetic field in matter [5, 6, 7, 8]. We refer to a more general situation where electric susceptibility, resonance frequencies of the electromagnetic field and also the coupling between electromagnetic field and oscillators (the latter ones can be identified with the mesoscopic polarization fields) depend on space-time variables. In this respect, we can refer our approach to other models which generalize the Hopfield model, see [9, 10, 11], but we stress that we don’t take into account absorption in our paper. The latter assumption is reasonable as far as emission phenomena we are interested in are not too near the absorption region, and the focus is pair-creation. Any framework including absorption would imply a much more tricky approach (cf. e.g. [9, 10, 11]).

In order to corroborate the physical interest of our model and its canonical quantization, we recall that, by means of the Kerr effect, it is possible to induce dielectric perturbations which propagate in the dielectric medium. These perturbations represent inhomogeneities which arise as a consequence of intense laser pulse propagating in a dielectric medium (see e.g. [12]). As a result of microscopic interactions, involved in non-linear electrodynamics, a mean motion of a dielectric perturbation with a different refractive index occurs. Instead of attempting immediately a first principle study on this subject, i.e. instead of working out a microscopic model for this case, we begin considering a semi-phenomenological approach, in which we adopt a Lagrangian model which would be ‘first principle-based’ apart for the appearance in the Lagrangian of a (phenomenological) contribution to the refractive index arising from the Kerr effect. Interesting results can be still deduced from this framework, as well as a well-defined route for quantizing the model. Our aim is to get a model which allows to deal with sufficiently general situations of interest, as e.g. the ones created by means of the Kerr effect: traveling dielectric perturbations moving with different laws of motions should be allowed. For example, a uniformly moving perturbation (which is characteristic of the Kerr effect), but also an accelerating one and even a rotating one, each of which represents a very interesting benchmark for photon pair creation in external fields (or under changing external conditions). For situations covering the uniformly moving case, its relation with analogue gravity framework and for experimental measure of Hawking radiation, see e.g. [13, 14, 15, 16, 17, 18, 19]. Thus our approach allows as well the cases where a different space-time evolution is taken into account. As a consequence of microscopic interactions, modified electric susceptibility, proper frequency and coupling constant between electric field and polarization are postulated to occur, which are meant to reproduce a suitable behaviour of the refractive index perturbation.
In the actual construction of the model, we can refer to the aforementioned traveling perturbations, but, in order to be even much more general, we can allow a generic behavior for the susceptibility , which is a scalar function of the space-time coordinates. Moreover, we allow an analogous nature also for , the latter being the proper frequency of the matter oscillators. As e.g. in [9, 10, 11], we generalize the coupling between electric field and polarization field(s), in our case by introducing a scalar function which plays the role of coupling.

The only reasonable constraint we impose are that asymptotically in time are constant, for obtaining well-defined particle states IN and OUT. Moreover, we can allow couplings with N oscillators, in such a way that we can manage with N different susceptibilities, resonances and couplings.
There are two further features of our model which are very important: relativistic covariance and quantization in a covariant gauge. Covariance is fundamental for gettting rid of any ambiguity in actual calculations, and in order to respect a fundamental requirement for a physical model. We stress that these are per sè interesting contributions to the microscopically-grounded works on the subject of electromagnetic field in dielectric media, because a covariant generalization is presented and its canonical quantization in a covariant gauge is performed. Covariance, as is known, and is confirmed since the original work by Minkowski [20] and e.g. by [21, 22], is not simply a speculative exercise in the picture at hand, but allows to get the correct behavior of physical quantities when changing from a inertial observer to another one. An important example is represented by a uniformly travelling perturbation const moving along the -direction, where we can use covariance for passing to the comoving frame, where the theory is static, and where physical interpretation of the scattering process is much more perspicuous [23].
It is also worth mentioning a longstanding series of studies concerning electrodynamics of moving media, where covariance of the formalism plays a key-role. We limit ourselves to quote some seminal papers [24, 25, 26, 27], where phenomenological electrodynamics is adopted (ref. [27] is explicitly devoted to the dispersive case). In particular, [24] and [26] are also important references for the dielectric models in analogue gravity [28]. See also [29], where a covariant model for moving (homogeneous) dispersive dielectric media is studied and quantized in the Coulomb gauge.
As to quantization, we can provide a scheme of constrained quantization where all subtleties of the theory are taken into account (see e.g. [30, 31]), and still covariance plays an important role for obtaining consistent quantization rules which would not be so clearly stated otherwise. As far as dispersive effects are not involving magnetic properties of the material, we can also consider that our model, for constant dielectric susceptibility, represents an improvement of the phenomenological model studied in [32].
It is worth mentioning that a very general and interesting picture is provided in [11], where the susceptibility is a tensor field depending on space and time. Absorption is also included by means of a bath of oscillators whose interactions with the electromagnetic field originate dissipative effects. Still, as a novelty with respect to the aforementioned picture, we develop a formalism leaving room for covariance and also quantization in a covariant gauge, which are not treated therein.

2 A covariant form for the Hopfield model

In the following, we take into consideration the electromagnetic field Lagrangian which is apt for a dispersive lossless dielectric medium, as in Hopfield model [5, 6, 7]. A further field, representing material polarization, is introduced and coupled to the free electromagnetic field as follows:

As an example, a traveling perturbation is described by introducing .
Only a first-principle introduction of the Kerr effect, like e.g. the one obtained by introducing a fourth power of would require substantial modifications (because of the non-linear term), but we do not pursue this problem herein.
In order to pursue a more standard calculation for inferring particle creation, we introduce a covariant generalization of the Hopfield model in 3+1 dimensions: this would make easier to find out an inner product with respect to which one could calculate the Bogoliubov coefficients in order to check if particle creation occurs in the standard way. The problem consists, of course, in finding a covariant form for the polarization part of the Hopfield Lagrangian. This is not a so trivial task. The main problem is represented by the kinetic part of the polarization field Lagrangian; with this aim, let us introduce as the 4-velocity of the bulk dielectric medium (we mean the velocity of the dielectric sample, not the one of the dielectric perturbation). Then, the covariant lagrangian density is:


Minkowski metric is chosen with the standard signature for quantum field theory: . The latter model appears to be the principal candidate in our consideration, because it provides a field equation for which, for constant , gives rise e.g. in the eikonal approximation to the correct covariant dispersion relation.
We have introduced a 4-vector in order to describe the polarization field in a covariant form. We have introduced, as a consequence, also the component of the field, which is absent in the rest frame. Actually, this new component is not an independent one, it depends on the spatial components, as can be easily ascertained by constructing the 4-vector corresponding to the standard polarization field, see e.g. [22, 33]. Indeed, the following condition has to be implemented:


which is required at the level of the classical theory for the polarization vector. Note that in the rest frame, where , the above condition amounts to . We assume this condition too. It is worth mentioning that this condition is the correct one for our harmonic oscillator field coupled to the electromagnetic field, regardless of its specific nature of polarization field.
As to the field equations, in the case of the electromagnetic field we obtain


and for the polarization field we have


By contracting with the equation for the electromagnetic field and taking into account (3) we get , where . This is just the Gauss law for the electric induction field . This is the right condition to identify as a polarization field and is required by compatibility among the transversality condition and the equations of motion. It is also useful to define the induction tensor , which is such that . Notice that, in absence of free charges and currents, Gauss law and Ampere law are summarized by the equation , which, contracted with , amounts to .
It is also easy to realize that, by introducing the , whose first four components coincide with and the remaining four components coincide with 111 is the direct sum of and . we obtain a field theory which is quadratic in

3 Conserved scalar product for the model

We can now determine the conserved scalar product associated to the covariant Hopfield model. The first step in order to determine the scalar product is to complexify the fields. The complexified lagrangian density becomes:


A symmetry of (7) is , where is a constant phase. The associated conserved quantity can be computed by means of the usual Noether method. The computation is immediate and gives the conserved current:


Indeed, a direct computation shows that on the solutions of the equations of motion satisfies . Thus, the standard argument shows that on any spacelike slice the quantity does not depend from . This defines a conserved (Hermitian) quadratic form on :


This gives the conserved scalar product by means of the usual polarization formula. Then we obtain the conserved scalar product:


This scalar product is very important in relation to the quantization of the model. Indeed, it allows to define positive and negative norm states for the solutions of the field equations, i.e. it allows to define in a proper way particles and antiparticles respectively. A proper quantization for the model is discussed in the following section.
For example, if and we work in the lab frame, the scalar product among plane waves (with on shell momenta)




where in (17) and in (19) stand for constant 4-vectors representing the amplitude of the respective plane waves, is


where and .

4 Quantization of the covariant model

The introduction of the condition (3) amounts to a constraint to be imposed on the system. This affects also the quantization of the covariant Hopfield model, in the sense that the covariant form of the Heisenberg commutation relations has to be consistent with the constraints of the theory.
As to the electromagnetic part of the Lagrangian, the procedure we follow slightly departs from what could be considered as standard, e.g. the usual quantization under covariant gauge conditions like the Lorentz gauge , and where a Gupta-Bleuler formalism (see e.g. [34]) can be adopted in order to get rid of spurious degrees of freedom, leaving only transverse (physical) ones. Instead, we follow the Dirac approach in which all first-class constraints which are associated with the gauge freedom are first reduced to second-class ones by means of suitable explicit gauge-fixing terms in the Lagrange formalism, and then quantized [35, 36]. Constraints are then implemented operatorially, rather than in a weak sense (as in the Gupta-Bleuler approach). See e.g. [30, 31, 37, 38].
In order to perform a complete quantization of the full Hopfield covariant model, in a covariant gauge, we have to take into account both the electromagnetic part and the polarization part of the Lagrangian. Both these parts require a suitable implementation of the constraints. We can add the constraints to the covariant Lagrangian, thus obtaining:


where plays the usual role of auxiliary hermitian scalar field, also known as -field [41, 30], and is a constant which is useful for reproducing various gauge conditions (the so-called -gauges). The equations of motions are


Furthermore, as far as the conjugate momenta are concerned, we obtain:


As to the classical Hamiltonian density, according to the standard procedure we have:


In our model there are three primary constraints . From the analysis of Poisson brackets of the primary constraints with the Hamiltonian we can find the following complete set of constraints:


We have taken into account that all functions of the constraints giving rise to the same submanifold , are to be considered equivalent, and this allows us to get the former simplified expressions for . See also [31, 30]. represent primary second-class constraints of the theory, and they appear explicitly in the expression of our constrained Hamiltonian defined above. We provide in the following some more details of the calculations.
In order to be more explicit, we get


which represent the Poisson brackets with the primary second-class constraints. The first equation (49) fixes and is the same one obtains in QED [30], and implies the same conditions. The second equation (50) determines . The third equation (51) introduces a second-class second-stage constraint. Its Poisson bracket with the Hamiltonian is


which, on the sub manifold defined by the above constraints amounts to requiring , which is then a second-class third-step constraint. Its Poisson bracket with the Hamiltonian is


which, on the submanifold of the previous constraints implies . At this point we have a complete set of constraints, because


simply fixes . See also below.

We can define the matrix whose entries are , where stay for the Poisson brackets. In particular, restoring the dependence on spacetime variables (time is fixed), with , we get the following non-zero entries: . Let us explicit the Dirac brackets as provided from the theory of constrained systems:


We recall that, in a less synthetic form, in the previous formula one has to take into account that, by introducing collective symbols for phase-space variables , we get:


a summation convention on repeated indices is understood. One can also determine the Lagrange multipliers that appear for primary second-class constraints in the constrained Hamiltonian, as shown in our previous discussion. It is now easy to show that the following Dirac brackets hold true:


which represent the basic ingredients for the quantization. Indeed, according to Dirac quantization scheme, we have to impose on quantum operators:


Moreover, constraints , , are implemented as operators (cf. [30], in particular p. 132):


As to the Hamiltonian, we have


As we are dealing with a model in which only second class primary constraints appear, we can also explicit the Lagrange multipliers as functions of the canonical variables. By taking into account our previous analysis, we can limit our attention only to , which display the only non-trivial behaviour, and replace (cf. (50)), (cf. (49)) in (72). (We recall that ).
It is interesting to note that, according to (70),(71), and its conjugate variable are ‘expelled’ by constrained quantization, in the sense that they are reduced both to the zero operator. We can provide the following interpretation. Condition (3) holds true as a consequence of the antisymmetric character of the induction tensor defined at the end of sec. 2. Indeed, we easily get , which, being , and being , necessarily implies also . This condition is then preserved by quantization. This fact corroborates the previous results concerning the quantization of the model.

4.1 Scalar product and constraints

We notice that we have introduced in our setting constraints even in the Lagrangian approach. As a consequence, we have to take into account how the scalar product (15) is modified because of the constraints. After complexification of the constrained Lagrangian , we find that it appears only a further term in the scalar product: if stays for the direct sum of , we get


Notice that, assuming the same fields as in (21), we still obtain the same result.
It is also important to point out that the former scalar product can be found as follows. We introduce the phase-space vectors , to be intended again as a direct sum, by using the symplectic form:


where stays for the identity matrix . Then we can define the following scalar product:


where stays for the usual Euclidean scalar product. Then, by taking into account the definitions for one gets the same result as in (81). This is the extension to a canonical quantization scheme and to 3+1 dimensions of the results obtained for 1+1 dimensions in [18].
It is also possible to introduce the operator such that the Hamiltonian in (72) can be written as follows (see e.g. [39]):


which is such that, by defining , one obtains the Hamiltonian equations in the form


where It is also evident that, due to the trivialization of and , we can restrict our considerations to a dimensional phase space in a straightforward way.

4.2 Construction of physical states and physical operators

In this subsection, we delve into the problem of constructing physical states and observables in our canonical approach in covariant gauges. As well-known, the price to be paid for covariance is the appearance of negative norm states, and also of zero-norm ones. As in QED, one has to face with the problem of non-physical states, which have to be eliminated from the physical spectrum. Also the Fano diagonalization procedure has to be revisited in view of the particular gauge conditions. In the standard approach to quantum electrodynamics, the calculation of the S-matrix requires several ingredients. Lacking a clear and well-defined way to implement a full non-perturbative quantum field theory in the Heisenberg representation, a perturbative approach is implemented in the interaction representation, where the fields evolve as free fields (i.e. their evolution is prescribed by the free field Hamiltonian ). A delicate interplay of the interacting (renormalizable) theory with asymptotic free fields occurs (cf. e.g. [30] for the case of QED). Asymptotic free fields (IN and OUT fields) satisfy canonical commutation relations and their creation and annichilation operators satisfy free-field commutation rules. In the present case, in order to simplify a bit our picture, we limit ourselves to the homogeneous dielectric medium, without losses and without dielectric perturbations e.g. induced by means of the Kerr effect. This is not a so crude limitation, as in region asymptotically far from the traveling dielectric perturbation, homogeneity can be assumed to be preserved. Within the given approximation, we assume a quantization strategy which goes parallel to the one of [40], i.e. we quantize the fields as free, with the coupling , and then perform a Fano diagonalization for the physical part of the Hamiltonian (with ). We expect that the procedure is consistent with the original commutation relations for the fields, as is in [40] (cf. also [9]).
Furthermore, we wish to avoid any problem with the dipole ghost (see e.g. [41]) so we impose the Feynman gauge in what follows. See also the following section.
We start from the polarization field. It is a vector field with four components, so a priori it would involve four degrees of freedom. Still, the transversality condition (3) reduces to three the degrees of freedom which are actually available. This is also evident from (62), which in the rest frame reduces identically to zero. Indeed, the set of eight fields represents a complete but not independent set of initial conditions, because (3) allows to express as a function of the set which is both complete and independent. We can choose, in particular, a set of polarization vectors , such that


which can be easily implemented. See below. As to , we have


where indicates the free field dispersion relation, and is a suitable normalization factor. Our choice for the basis is , where and


in such a way that, for it holds , and


where is the wave-vector, as usual. The third polarization is orthogonal to , and also to . For the operators the following canonical commutation relations (CCR) hold:


For the electromagnetic field and the auxiliary field quantization requires some more efforts. We can introduce a further set of polarization vectors and of operators such that


where is a suitable normalization. Usually, one imposes


In particular, one may choose the set of polarization vectors , where satisfy (88). Still, purposefully, in order to allow a more direct comparison with common literature and with [30], we use the Gitman-Tyutin basis we costruct in the following. Furthermore, in this section, and only in this section, we choose . We choose as in the case of the polarization field. The third polarization can be chosen e.g. as follows (cf. [30]) in the case of the electromagnetic field:


It is also useful to choose


Our choices are the same as in [30], apart from a slightly different normalization of both the vectors. We shall indicate the basis


as the Gitman-Tyutin basis. We stress that in this basis




We also get, in the case of the auxiliary field ,


where is a suitable normalization. We remark that both and are considered as free fields, and then DR is a free field dispersion relation. Furthermore, the equation of motion for (29) implies that we can reduce to four the overall degrees of freedom for the fields , and then only four operators in the set , with are independent. We choose, in analogy with the discussion in [30] (sec. 4.2 therein),


Commutation relations are the same as in [30] , as we are using the same basis for the electromagnetic sector of our model. We find




and also


Also, we find the following commutation brackets for , :


All other commutation relations are equal to zero. In particular, i