1 Introduction

The conventional view of quantum mechanics is that it is a purely mechanical theory of matter. This view leads to the impossibility of finding a physical explanation for phenomena such as the Heisenberg inequalities, the atomic stability and even the transitions between states within a strictly quantum-mechanical context. To explain such phenomena, quantum mechanics is forced to borrow elements from quantum electrodynamics; but even in this more complete theory, questions such as the emergence of operators remain unanswered.

Stochastic electrodynamics (SED) has been developed to address these shortcomings by considering the existence of the zero-point field (ZPF) as a real, omnipresent, fluctuating Maxwellian field in permanent interaction with matter [1,2,3,4,5,6]. The inclusion of the radiation field changes the nature of the system to an electrodynamic one, and provides the necessary elements for a causal explanation of several quantum phenomena, including those mentioned above. For example, atomic stability is the result of the balance achieved between the power radiated by the accelerated electron and the power absorbed from the field [6].

The SED approach also serves to address questions that have remained largely unanswered in the literature concerning the physical mechanism by which the dynamical variables are transformed into quantum operators, and the associated parallelism between Poisson brackets and commutators.

In recent work [7, 8] we have analyzed the peculiar process that brings a charged particle bound by a conservative force and subject to the zero-point field, from its initial classical behavior into the quantum regime. Most importantly, we have shown that in this process the description in terms of classical phase-space variables is replaced by a description in terms of variables expressing the response to the driving radiation field, where the response coefficients correspond to the quantum operators.

Here we take the work a step further by including the entire radiation field and analyzing how the particle in a given stationary state responds resonantly to certain field modes that can take it to another stationary state and back to the initial one. The response coefficients of x(t), p(t) are again identified with the matrix elements of \(\hat{x},\hat{p}\) linking the stationary states. The Poisson bracket of the dynamical variables with respect to the normal variables of the driving field modes gives precisely the corresponding commutator. This provides the grounds for a discussion of the physics underlying the non-commutativity of quantum operators and the meaning of quantum expectation values.

At the cost of a detailed description of the particle trajectory, the quantum formalism is thus a statistical, coarse-grained description of the stationary states and the dynamics of the transitions between them. Ultimately, the dynamics is still governed by the (classical) Hamiltonian, but the kinematics has been radically transformed by the replacement of c-variables by operators, while preserving the (classical) symplectic structure.

The paper is structured as follows. First, the SED analysis of the harmonic oscillator (HO) is revisited and further developed with two objectives: to recall the crucial distinction between vacuum fluctuations and thermal fluctuations (Sect. 2) and of the role of the former in determining the ground state of the oscillator (Sect. 3); and to analyze how the SED oscillator in a stationary state responds resonantly to the driving field, making transitions to other accessible stationary states (Sect. 4). In the description that ensues, the phase-space variables are replaced by the oscillator’s response variables (Sect. 5), which thus constitute the new kinematical elements. The equation describing the energy change due to transitions leads to the quantum formulas for Einstein’s A and B coefficients (Sect. 6). The analysis is then extended to non-linear force problems (Sect. 7), to make contact with the quantum-mechanical operator formalism. In Sect. 8, the relationship between covariances and (quantum) expectation values points to the importance of the order of the dynamical variables involved, as expressed in the quantum commutator. In Sect. 9 we use a similar approach for the canonical description of the radiation field and disclose the origin of the field operators \(\hat{a},\hat{a}^{\dagger }\). The final section briefly discusses the coarse-grained nature of the quantum description and recalls the connection with stochastic quantum mechanics, which describes quantum mechanics as a Markov process.

2 The Harmonic Oscillator: Vacuum Fluctuations vs Thermal Fluctuations

The harmonic oscillator (HO) has been extensively studied in SED since its inception, leading to quantum results both for the ground state and for the state in equilibrium with the thermal radiation field. Different approaches—either by directly solving the oscillator’s equation of motion or by using the corresponding Fokker–Planck equation, see e. g. Refs. [1, 3, 9, 10]—lead to the (quantum) stationary phase-space distribution ( [5] Section 7.4)

$$\begin{aligned} Q(x,p;\beta )=\frac{1}{\pi \hbar \Pi }e^{-H/\mathcal {E}}, \end{aligned}$$
(1)

where \({\mathcal{E}}={\mathcal{E}}_{0}\Pi (\beta \hbar \omega _{0})\), \(\Pi (x)=\coth (x/2)\), \(\beta =1/kT,\) \(H=(p^{2}/2m)+(m\omega _{0}^{2}x^{2}/2)\), and \(\mathcal {E}_{0}=\hbar \omega _{0}/2\) is the ground-state energy of the oscillator of natural frequency \(\omega _{0}\), which coincides with that of the ZPF of the same frequency.

However, as is well known, if this distribution is decomposed into canonical discrete-energy states,

$$\begin{aligned} Q(H)=\frac{1}{Z}\sum _{n}Q_{n}(H)e^{-\beta E_{n}},\;E_{n}=\hbar \omega _{0}(n+\frac{1}{2}), \end{aligned}$$
(2)

where Z is the partition function, the individual (normalized) components of the distribution, given in terms of Laguerre polynomials,

$$\begin{aligned} Q_{n}(\xi )=\frac{(-1)^{n}}{\pi \hbar }e^{-\xi }L_{n}(2\xi ),\;\xi =H/\mathcal {E}_{0}, \end{aligned}$$
(3)

are positive definite only for \(n=0\), the ground state.

This result is a consequence of the treatment of the thermal field as an extension of the ZPF, as if thermal fluctuations and vacuum fluctuations were of the same nature.Footnote 1 Equilibrium of the oscillator in its ground state is reached with the fluctuating zero-point radiation field. However, to obtain the correct distribution of oscillators in equilibrium with Planck’s field, one must instead consider the thermal fluctuations due to the absorption and emission processes that continuously take place between stationary states, precisely the mechanism that leads to thermal equilibrium, as demonstrated by Einstein in 1916 [20]. We recall that Einstein derived the general relation between the coefficients \(A_{nk}\) and \(B_{nk}\) for the rates of spontaneous and induced transitions respectively, by considering two arbitrary energy levels \(E_{n},E_{k}\) connected by radiative transitions, with \(\left| E_{n}-E_{k}\right| =\hbar \omega _{nk}\), and imposing detailed energy balance at a given temperature. The explicit formulas for \(A_{nk}\) and \(B_{nk}\) could only be derived once the quantum formalism was in place; here we will derive them using the SED approach.

3 The Oscillator’s Resonant Response

The analysis of the HO sketched above shows that the excited states do not have an existence of their own, which means that they are not stationary in the absolute sense. They may be considered transitory, insofar as they are “visited” by the HO as the result of the transitions taking place. This suggests analyzing the dynamics of the transitions starting from a truly stationary state, which is the ground state. This analysis can be made easily in the case of the HO because the oscillator responds to radiation of its own frequency \(\omega _{0}\), which defines the energy difference between the ground state and the excited state, \(\Delta E=\hbar \omega _{0}\). Furthermore, we know that to a good approximation, the interaction between the electron and the radiation field is through the electric component, and it is a dipole interaction, linear in the electric charge and the field, as expressed in the Braffort–Marshall equation [5, 6]

$$\begin{aligned} m\varvec{\ddot{x}}=\varvec{f}(\varvec{x})+m\tau \varvec{\dddot{x}}+e\varvec{E}(t), \end{aligned}$$
(4)

where \(\varvec{f}(\varvec{x})\) is a (binding) conservative force, and \(m\tau \varvec{\dddot{x}}\) with \(\tau =2e^{2}/3mc^{3}\) is the radiation reaction force. The electric field is taken in the long-wavelength approximation because the wavelengths of the relevant field modes are more than \(10^{3}\) times larger than a typical atomic size.

We will henceforth restrict ourselves to the one-dimensional problem. The stationary solution of Eq. (4) for the 1D HO,

$$\begin{aligned} m\ddot{x}=-m\omega _{0}^{2}x+m\tau \dddot{x}+eE(t), \end{aligned}$$
(5)

obtained by taking the Fourier transform of this equation,

$$\begin{aligned} x(t)=\frac{e}{m}\int _{-\infty }^{\infty }d\omega \frac{E(\omega )e^{i\omega t}}{\omega _{0}^{2}-\omega ^{2}+i\tau \omega ^{3}} \end{aligned}$$
(6)

with \(E(t)=\int _{-\infty }^{\infty }d\omega E(\omega )e^{i\omega t}\), gives for the square average of x

$$\begin{aligned} \left\langle x^{2}\right\rangle =\frac{2\pi c^{3}\tau }{m}\int _{0}^{\infty }d\omega \frac{\rho (\omega )}{\Delta (\omega ;\omega _{0})}, \end{aligned}$$
(7)

where the average is taken over the realizations of the field, \(\rho (\omega )=3\left| E(\omega )\right| ^{2}/4\pi\) is the spectral energy density of the field, and \(\Delta =(\omega _{0}^{2}-\omega ^{2})^{2}+\tau ^{2}\omega ^{6}\). To carry out the integration we take into account that for the relevant (atomic) frequencies \(\omega _{0}\),

$$\begin{aligned} \tau \omega _{0}\leqslant 1, \end{aligned}$$
(8)

and the denominator \(\Delta\) becomes very small for \(\omega \sim \omega _{0}\). Introducing \(y=\omega -\omega _{0}\), noting that the main contribution to the integral comes from y close to zero, and using

$$\begin{aligned} \lim _{\gamma \rightarrow 0}\frac{\gamma }{\gamma ^{2}+y^{2}}=\pi \delta (y), \end{aligned}$$
(9)

we obtain

$$\begin{aligned} \left\langle x^{2}\right\rangle =\frac{\pi ^{2}c^{3}}{m\omega _{0}^{4}}\rho (\omega _{0}). \end{aligned}$$
(10)

When \(\rho (\omega )\) is the spectral energy density of the ZPF,

$$\begin{aligned} \rho ^{0}(\omega )=\frac{\hbar \omega ^{3}}{2\pi ^{2}c^{3}}, \end{aligned}$$
(11)

we get, as expected,

$$\begin{aligned} \left\langle x^{2}\right\rangle _{0}=\frac{\hbar }{2m\omega _{0}}, \end{aligned}$$
(12)

corresponding to a total average energy \(E_{0}=\left\langle H\right\rangle _{0}=\hbar \omega _{0}/2\). (Here and in the following, the subindex 0 refers to the ground state). What is important to note here is that the oscillator responds resonantly to the frequency \(\omega _{0}\), which means that also when it is subject to an external field, it will respond to the field modes of that frequency.

In the following we will analyze in detail the oscillator’s response to an external field.

4 Equation of Evolution Describing the Energy Exchange

We shall describe how an oscillator in a stationary state responds to an external field containing a component of frequency \(\omega _{0}\) that can be absorbed by it. For QM, this is a textbook problem, but our intention is to explain the physics behind the quantum description using the tools of SED. The approach is consistent with linear response theory, where an external field is applied to a system in equilibrium to study the properties of the system. If the field is not too strong, the response of the system is proportional to the driving force of the field, the proportionality factor being defined as the response function ( [21, 22] Chapter 4).

We know that in the present case, the driving force eE(t) takes the system to a state of different energy; in particular, if the oscillator is initially in its ground state, it will be driven to an excited state. We therefore need an equation that describes the energy exchange during the transition.

The general equation of evolution for an average dynamical quantity is obtained from the stochastic Eq. (4) using a standard statistical procedure to construct an equation for the phase-space distribution \(Q(\varvec{x},\varvec{p},t)\). The equation thus obtained is a generalized Fokker-Planck equation (GFPE), i.e. a differential equation with an infinite number of terms (see e.g. [23, 24]). In the Markofian (memory-less) limit, corresponding to long times, this equation is reduced to the second-order (Fokker–Planck) equation [6]

$$\begin{aligned} \frac{\partial Q}{\partial t}+\frac{1}{m}\frac{\partial }{\partial x_{i}}p_{i}Q+\frac{\partial }{\partial p_{i}}\left( f_{i}+m\tau \dddot{x}_{i}\right) Q=\frac{\partial }{\partial p_{i}}D_{ij}^{pp}\frac{\partial Q}{\partial p_{j}}+\frac{\partial }{\partial p_{i}}D_{ij}^{px}\frac{\partial Q}{\partial x_{j}}, \end{aligned}$$
(13)

where the diffusion coefficients are given by

$$\begin{aligned} D_{ij}^{pp}=e^{2}\intop \nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\frac{\partial p_{j}}{\partial p_{i}^{\prime }},\quad D_{ij}^{px}=e^{2}\intop \nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\frac{\partial x_{j}}{\partial p_{i}^{\prime }}, \end{aligned}$$
(14)

and \(\varphi (t-t^{\prime })\) represents the electric field covariance,

$$\begin{aligned} \overline{E_{i}(t)E_{j}(t^{\prime })}^{(i)}=\delta _{ij}\varphi (t-t^{\prime }), \end{aligned}$$
(15)

related to the spectral energy density through the formula

$$\begin{aligned} \varphi (t-t^{\prime })=\frac{4\pi }{3}\intop \nolimits _{0}^{\infty }\rho (\omega )\cos \omega (t-t^{\prime })d\omega . \end{aligned}$$
(16)

To obtain the equation of evolution for the average of a dynamical quantity \(\mathcal {G}(\varvec{x},\varvec{p},t)\) we multiply Eq. (13) by \(\mathcal {G}\) and do an integration by parts; the result is

$$\begin{aligned} \frac{d}{dt}\left\langle \mathcal {G}\right\rangle =\left\langle \frac{d\mathcal {G}}{dt}\right\rangle _{\text {nr}}+m\tau \left\langle \dddot{x}_{i}\frac{\partial \mathcal {G}}{\partial p_{i}}\right\rangle +\left\langle D_{ij}^{pp}\frac{\partial ^{2}\mathcal {G}}{\partial p_{i}\partial p_{j}}\right\rangle +\left\langle D_{ij}^{px}\frac{\partial ^{2}\mathcal {G}}{\partial p_{i}\partial x_{j}}\right\rangle , \end{aligned}$$
(17)

where

$$\begin{aligned} \left\langle \frac{d\mathcal {G}}{dt}\right\rangle _{\text {nr}}=\left\langle \dot{x}_{i}\frac{\partial \mathcal {G}}{\partial x_{i}}+f_{i}\frac{\partial \mathcal {G}}{\partial p_{i}}\right\rangle \end{aligned}$$
(18)

is the Liouvillian contribution to \(d\left\langle \mathcal {G}\right\rangle /dt\), and the remaining terms in Eq. (17) originate in the radiation reaction and the fluctuating field.

For any \(\mathcal {G}=\xi (\varvec{x},\varvec{p})\) that represents an integral of motion of the radiationless problem, (18) gives zero and one is left with

$$\frac{d}{{dt}}\left\langle \xi \right\rangle = m\tau \left\langle {\dddot x_{i} \frac{{\partial \xi }}{{\partial p_{i} }}} \right\rangle + \left\langle {D_{{ij}}^{{pp}} \frac{{\partial ^{2} \xi }}{{\partial p_{i} }}\partial p_{j} } \right\rangle + \left\langle {D_{{ij}}^{{px}} \frac{{\partial ^{2} \xi }}{{\partial p_{i} }}\partial x_{j} } \right\rangle ,$$
(19)

which shows that \(\left\langle \xi \right\rangle\) is generally affected by both radiation and diffusion. In particular, for 1D problems the only conserved quantity is the Hamiltonian \(H=p^{2}/2m+V(x)\), and Eq. (19) becomes

$$\begin{aligned} \frac{d}{dt}\left\langle H\right\rangle =\tau \left\langle p\dddot{x}\right\rangle +\frac{1}{m}\left\langle D^{pp}\right\rangle . \end{aligned}$$
(20)

This equation describes the energy exchange during radiative transitions, as will be shown below.

5 The Oscillator’s Response to an Applied Field

The stationary, ground-state solution x(t) of Eq. (5) for the HO satisfies the energy-balance condition, obtained by putting \(d\left\langle H\right\rangle _{0}/dt=0\) in Eq. (20) [6],

$$\begin{aligned} \tau \left\langle p\dddot{x}\right\rangle _{0}=\frac{-1}{m}\left\langle D^{0}\right\rangle _{0}, \end{aligned}$$
(21)

where \(D^{0}\) is the diffusive term due to the ZPF, obtained from the first equation (14) reduced to one dimension,

$$\begin{aligned} D^{0}=e^{2}\intop \nolimits _{-\infty }^{t}dt^{\prime }\varphi ^{0}(t-t^{\prime })\frac{\partial p}{\partial p^{\prime }}, \end{aligned}$$
(22)

with

$$\begin{aligned} \varphi ^{0}(t-t^{\prime })=\frac{2\hbar }{3\pi c^{3}}\intop \nolimits _{0}^{\infty }\omega ^{3}\cos \omega (t-t^{\prime })d\omega , \end{aligned}$$
(23)

according to Eqs. (11) and (16).

When it responds to an applied field, the oscillator is brought out of equilibrium and visits an excited state (called state 1), with energy

$$\begin{aligned} E_{1}=E_{0}+\hbar \omega _{0}. \end{aligned}$$
(24)

The equation describing the breakdown of the energy balance is obtained by subtracting Eq. (21) from (20) (written for the ground state),

$$\begin{aligned} \frac{d}{dt}\left\langle H\right\rangle _{0}=\frac{1}{m}\left\langle D^{e}\right\rangle _{0}, \end{aligned}$$
(25)

where

$$\begin{aligned} D^{e}=e^{2}\intop \nolimits _{-\infty }^{t}dt^{\prime }\varphi ^{e}(t-t^{\prime })\frac{\partial p}{\partial p^{\prime }} \end{aligned}$$
(26)

is the contribution of the applied field to the diffusion coefficient, with

$$\begin{aligned} \varphi ^{e}(t-t^{\prime })=\frac{4\pi }{3}\intop \nolimits _{0}^{\infty }\rho ^{e}(\omega )\cos \omega (t-t^{\prime })d\omega , \end{aligned}$$
(27)

and \(\rho ^{e}(\omega )=\rho (\omega )-\rho ^{0}(\omega )\).

Equation (25) is an important result: it relates the average power absorbed by the oscillator, with its response to the applied field. Therefore, in Eq. (26) the variables \(p(t),p(t')\) represent the response of the oscillator.

To calculate \(D^{e}\), we express the time-dependent response coefficient \(\partial p(t)/\partial p(t')\) as the Poisson bracket,

$$\begin{aligned} \frac{\partial p(t)}{\partial p(t^{\prime })}=\left[ x(t'),p(t)\right] \end{aligned}$$
(28)

and take into account that the response is driven by the field. This means that in the Poisson bracket the derivatives of \(x(t'),p(t)\) must be taken in general with respect to the canonical variables (the quadratures) \(\text{q},\text{p}\) of the field modes to which the particle responds, i.e. [7, 8]

$$\begin{aligned} \left[ x(t'),p(t)\right] _{\text{qp}}=\sum \left( \frac{\partial x(t')}{\partial \text{q}}\frac{\partial p(t)}{\partial \text{p}}-\frac{\partial p(t)}{\partial \text{q}}\frac{\partial x(t')}{\partial \text{p}}\right) . \end{aligned}$$
(29)

In particular, for equal times we have

$$\begin{aligned} \left[ x(t),p(t)\right] _{\text{qp}}=1. \end{aligned}$$
(30)

The \(\text{q,p}\) are such that the average energy per field mode is given by the Einstein relation

$$\begin{aligned} {\mathcal {E}} = \frac{1}{2}\left\langle \omega ^{2}\text{q}^{2}+\text{p}^{2}\right\rangle =\hbar \omega , \end{aligned}$$
(31)

so that they are related to the normal field variables \(a,a^{*}\) through the transformation

$$\begin{aligned} \omega \text{q}=\sqrt{\frac{\hbar \omega }{2}}(a+a^{*}),\;\text{p}=-i\sqrt{\frac{\hbar \omega }{2}}(a-a^{*}). \end{aligned}$$
(32)

The \(a,a^{*}\) corresponding to different frequencies are statistically independent (uncorrelated), and according to Eqs. (31) and (32) they are normalized to 1, i.e.

$$\begin{aligned} \left\langle a(\omega )a^{*}(\omega ')\right\rangle =\delta (\omega -\omega '), \end{aligned}$$
(33)

so that we can write them as

$$\begin{aligned} a=e^{i\theta },\;a^{*}=e^{-i\theta }, \end{aligned}$$
(34)

where \(\theta\) is a purely random phase.

In terms of the new variables \(a,a^{*}\), the Poisson bracket (29) is given according to Eqs. (32) by

$$\begin{aligned} \left[ x(t'),p(t)\right] _{\text{qp}}=\frac{1}{i\hbar }\sum \left( \frac{\partial x(t')}{\partial a}\frac{\partial p(t)}{\partial a^{*}}-\frac{\partial p(t)}{\partial a}\frac{\partial x(t')}{\partial a^{*}}\right) =\frac{1}{i\hbar }\left[ x(t'),p(t)\right] _{aa^{*}}. \end{aligned}$$
(35)

The normal variables are associated to oscillations with positive and negative frequencies respectively, i.e. \(a(t)=ae^{-i\omega t},a^{*}(t)=a^{*}e^{i\omega t}\). Therefore, since the particle responds separately to negative and positive frequencies (corresponding to energy emissions and absorptions), it is appropriate to use the Poisson bracket in the \(a,a^{*}\)-representation, which we write simply as

$$\begin{aligned} \left[ x(t'),p(t)\right] _{aa^{*}}\equiv \left[ x(t'),p(t)\right] , \end{aligned}$$
(36)

to express the particle’s response, instead of the Poisson bracket with respect to the quadratures \(\text{q},\text{p}\). Note from Eqs. (30) and (35) that in this representation, the equal-time Poisson bracket is

$$\begin{aligned} \left[ x(t),p(t)\right] =i\hbar . \end{aligned}$$
(37)

Since the oscillator in the ground state responds only to field modes of (positive) frequency \(\omega _{0}\) that take it to an excited state, we write

$$\begin{aligned} x_{0}(t)=x_{10}a^{*}e^{i\omega _{10}t},\;p_{0}(t)=m\dot{x}_{0}(t), \end{aligned}$$
(38)

where \(\omega _{10}=\omega _{0}\) is the frequency of the mode that brings it from the ground state (state 0) to the excited state (state 1) and \(x_{10}\) is the corresponding response coefficient. The response function that brings the oscillator back to the ground state is

$$\begin{aligned} x_{0}^{*}(t)=x_{01}ae^{-i\omega _{10}t},\;p_{0}^{*}(t)=m\dot{x}_{0}^{*}(t), \end{aligned}$$
(39)

with \(x_{01}=x_{10}^{*}\). This gives, using (35) and (36),

$$\begin{aligned} \left[ x(t'),p(t)\right] _{\text{qp}}=\frac{1}{i\hbar }\left[ x(t'),p(t)\right] =\frac{2m}{\hbar }\omega _{0}\left| x_{10}\right| ^{2}\cos \omega _{0}(t-t'). \end{aligned}$$
(40)

Introducing this result into Eq. (26) and using Eq. (27), we get

$$\begin{aligned} \left\langle D^{e}\right\rangle _{0}=\frac{8\pi me^{2}}{3\hbar }\omega _{0}\left| x_{10}\right| ^{2}\intop \nolimits _{-\infty }^{t}dt^{\prime }\intop \nolimits _{0}^{\infty }d\omega \rho ^{e}(\omega )\cos \omega (t-t^{\prime })\cos \omega _{0}(t-t'). \end{aligned}$$
(41)

With

$$\begin{aligned} \intop \nolimits _{0}^{\infty }ds\cos \omega s\cos \omega _{0}s=\frac{\pi }{2}\left[ \delta (\omega _{0}-\omega )+\delta (\omega _{0}+\omega )\right] , \end{aligned}$$
(42)

equation (41) gives

$$\begin{aligned} \left\langle D^{e}\right\rangle _{0}=\frac{4\pi ^{2}me^{2}}{3\hbar }\omega _{0}\left| x_{10}\right| ^{2}\rho ^{e}(\omega _{0}). \end{aligned}$$
(43)

Inserting this result in Eq. (25), we finally obtain

$$\begin{aligned} \frac{d}{dt}\left\langle H\right\rangle _{0}=\hbar \omega _{0}B_{10}\rho ^{e}(\omega _{0}), \end{aligned}$$
(44)

where

$$\begin{aligned} B_{10}=\frac{4\pi ^{2}e^{2}}{3\hbar ^{2}}\left| x_{10}\right| ^{2} \end{aligned}$$
(45)

is the Einstein coefficient for induced absorption. The response coefficient \(x_{10}\) defined in Eq. (38), is therefore correctly identified as the matrix element connecting states 0 and 1 according to the usual quantum formalism.

6 On the Excited States of the Harmonic Oscillator

To the extent that the excited state 1 can be assumed to be a state of “relative equilibrium”, we can apply the same procedure to study the oscillator’s response to field modes that can take it from there to the next excited state (state 2) by energy absorption, or to the ground state (state 0) by energy emission. We will later be able to confirm that it is legitimate to assume the relative stability of state 1 by comparing the average time spent by the HO in state 1, with the characteristic time of its internal motion.

The response function \(x_{1}(t)\) must now contain the upward and downward terms, i.e.

$$\begin{aligned} x_{1}(t)=x_{21}a^{*}e^{i\omega _{21}t}+x_{01}ae^{-i\omega _{10}t},\;p_{1}(t)=m\dot{x}_{1}(t), \end{aligned}$$
(46)

and the response function that brings the oscillator back to state 1 is therefore

$$\begin{aligned} x_{1}^{*}(t)=x_{21}^{*}ae^{-i\omega _{21}t}+x_{01}^{*}a^{*}e^{i\omega _{10}t},\;p_{1}^{*}(t)=m\dot{x^{*}}_{1}(t) \end{aligned}$$
(47)

where \(a^{*}(\omega _{0})=a(-\omega _{0})\), \(\left\langle a^{*}(\omega _{0})a(-\omega _{0})\right\rangle =1\), \(x_{21}^{*}=x_{12}\), \(x_{01}^{*}=x_{10}\), and

$$\begin{aligned} \omega _{21}=\omega _{10}=\omega _{0}. \end{aligned}$$
(48)

In calculating \(d\left\langle H\right\rangle _{1}/dt\) from Eq. (20),

$$\begin{aligned} \frac{d}{dt}\left\langle H\right\rangle _{1}=\tau \left\langle p\dddot{x}\right\rangle _{1}+\frac{1}{m}\left\langle D^{0}\right\rangle _{1}+\frac{1}{m}\left\langle D^{e}\right\rangle _{1}, \end{aligned}$$
(49)

we note that in addition to the absorptions and emissions induced by the external field, represented by the last term, there are those resulting from the capacity of the oscillator to respond to the internal field composed of the radiation reaction plus the ZPF—precisely the two terms responsible for the detailed energy balance in the ground state, as shown in Eq. (21).

To calculate the first term on the rhs of (49) we do an integration by parts over time, so that \(\left\langle p\dddot{x}\right\rangle =-\left\langle \dot{p}^{2}\right\rangle /m\); we thus obtain from Eqs. (46) and (47), with \(\tau =2e^{2}/3mc^{3},\)

$$\begin{aligned} \tau \left\langle p\dddot{x}\right\rangle _{1}=-\frac{2e^{2}}{3c^{3}}\omega _{0}^{4}\left[ \left| x_{21}\right| ^{2}+\left| x_{01}\right| ^{2}\right] . \end{aligned}$$
(50)

For the second term on the rhs of (49), we have (in analogy with Eq. (43))

$$\begin{aligned} \frac{1}{m}\left\langle D^{0}\right\rangle _{1}=\frac{4\pi ^{2}e^{2}}{3\hbar }\rho ^{0}(\omega _{0})\left| \omega _{21}\left| x_{21}\right| ^{2}-\omega _{10}\left| x_{01}\right| ^{2}\right| \end{aligned}$$
(51)

or, using Eqs. (11) and (48),

$$\begin{aligned} \frac{1}{m}\left\langle D^{0}\right\rangle _{1}=\frac{2e^{2}}{3c^{3}}\omega _{0}^{4}\left[ \left| x_{21}\right| ^{2}-\left| x_{01}\right| ^{2}\right] . \end{aligned}$$
(52)

By adding Eqs. (50) and (52),

$$\begin{aligned} \tau \left\langle p\dddot{x}\right\rangle _{1}+\frac{1}{m}\left\langle D^{0}\right\rangle _{1}=-\frac{4e^{2}}{3c^{3}}\omega _{0}^{4}\left| x_{01}\right| ^{2}, \end{aligned}$$
(53)

we notice that the first terms in Eqs. (51) and (52) again cancel each other out (there are no upward transitions in the absence of an external field), while the second terms contribute equally to the downward transitions.

Finally, for the third term in Eq. (49) we get

$$\begin{aligned} \frac{1}{m}\left\langle D^{e}\right\rangle _{1}=\frac{4\pi ^{2}e^{2}}{3\hbar }\rho ^{e}(\omega _{0})\omega _{0}\left| \left| x_{21}\right| ^{2}-\left| x_{01}\right| ^{2}\right| . \end{aligned}$$
(54)

Introducing Eqs. (53) and (54) into (49), we obtain

$$\begin{aligned} \frac{d}{dt}\left\langle H\right\rangle _{1}=\hbar \omega _{0}\left[ -A_{01}+B_{21}\rho ^{e}(\omega _{0})-B_{01}\rho ^{e}(\omega _{0})\right] , \end{aligned}$$
(55)

with the A and B coefficients given by

$$\begin{aligned} A_{01}=\frac{4e^{2}\omega _{0}^{3}}{3\hbar c^{3}}\left| x_{01}\right| ^{2}, \end{aligned}$$
(56)
$$\begin{aligned} B_{01}=\frac{4\pi ^{2}e^{2}}{3\hbar ^{2}}\left| x_{01}\right| ^{2},\;B_{21}=\frac{4\pi ^{2}e^{2}}{3\hbar ^{2}}\left| x_{21}\right| ^{2}. \end{aligned}$$
(57)

The analysis of the higher excited states follows the same procedure, leading to the generalization of Eq. (55) for state n with energy \(E_{n}=E_{0}+n\hbar \omega _{0}\) (see Eq. (24)),

$$\begin{aligned} \frac{d}{dt}\left\langle H\right\rangle _{n}=\hbar \omega _{0}\left[ -A_{n-1,n}+B_{n+1,n}\rho ^{e}(\omega _{0})-B_{n-1,n}\rho ^{e}(\omega _{0})\right] , \end{aligned}$$

where

$$\begin{aligned} A_{n-1,n}=\frac{4e^{2}\omega _{0}^{3}}{3\hbar c^{3}}\left| x_{n-1.n}\right| ^{2},\;B_{n+1,n}=\frac{4\pi ^{2}e^{2}}{3\hbar ^{2}}\left| x_{n+1,n}\right| ^{2},\;B_{n-1,n}=\frac{4\pi ^{2}e^{2}}{3\hbar ^{2}}\left| x_{n-1,n}\right| ^{2}. \end{aligned}$$
(58)

To estimate the lifetime of an excited state (for example, state 1) we take into account that for not too strong applied fields, \(\rho ^{e}(\omega _{0})\) is of the order of or smaller than \(\rho ^{0}(s)\), so that—considering \(\left| x_{01}\right| ^{2}\approx \left\langle x^{2}\right\rangle _{1}\approx \hbar /m\omega _{0}\)—the lifetime is approximately given by

$$\begin{aligned} \left( A_{01}\right) ^{-1}\approx \frac{mc^{2}}{\alpha \hbar \omega _{0}}\omega _{0}^{-1}, \end{aligned}$$
(59)

which for normal atomic frequencies is greater than the period of oscillation by a factor of the order of \(\alpha ^{-3}\approx 10^{6}\). Therefore, to a good approximation the excited state can be considered as a state of equilibrium, as is done in quantum mechanics.

If the response of the oscillator to the field could be completely suppressed by eliminating the modes of frequency close to \(\omega _{0}\), it would seem that the excited state would become absolutely stationary. However, we must remember that the excited states—like the ground state—owe their very existence to the support of the background field.

7 Extension to Non-Linear Force Problems

We now extend the results obtained for the SED harmonic oscillator in the previous sections to the general case of a bound particle, typically an atomic electron. Since we are interested in the evolution of the average energy, we will restrict the analysis to the 1D case, where the energy is the only constant of motion.

As in the case of the harmonic oscillator, in the atomic case the electron in a stationary state n responds resonantly to the field, but in this case the resonance frequency depends on the energy difference \(\Delta E_{kn}\) between stationary states. This means that the electron in state n can respond to any of the different modes that bring it to a state k such that \(\Delta E_{kn}=\hbar \omega _{kn}\). Therefore, Eq. (46) must be generalized to include the different responses, i.e.

$$\begin{aligned} x_{n}(t)-x_{nn}=\sum _{k>n}x_{kn}a^{*}e^{i\omega _{kn}t}+\sum _{k<n}x_{nk}ae^{-i\omega _{nk}t},\;p_{n}(t)=m\dot{x}_{n}(t), \end{aligned}$$
(60)

and similarly for \(x_{n}^{*}(t)\); \(x_{nn}=x_{nn}^{*}\) is the value of x in the equilibrium state and the sums are carried out over all states k that are accessible from n through either an upward transition (\(k>n\)) or a downward one (\(k<n\)), accompanied by the corresponding energy gain or loss, \(\Delta E=\hbar \omega _{kn}\). The equation of evolution for the average energy, (49), generalizes to

$$\begin{aligned} \frac{d}{dt}\left\langle H\right\rangle _{n}=\tau \left\langle p\dddot{x}\right\rangle _{n}+\frac{1}{m}\left\langle D^{0}\right\rangle _{n}+\frac{1}{m}\left\langle D^{e}\right\rangle _{n}. \end{aligned}$$
(61)

Using Eqs. (60) and following the same algebraic procedure as before, we get

$$\begin{aligned} \tau \left\langle p\dddot{x}\right\rangle _{n} =-\frac{2e^{2}}{3c^{3}} \sum _{k} \omega _{kn}^{4} \left| x_{kn}\right| ^{2},\end{aligned}$$
(62)
$$\begin{aligned} \frac{1}{m}\left\langle D^{0}\right\rangle _{n} =\frac{2e^{2}}{3c^{3}} \sum _{k}\left| \omega _{kn}^{3}\right| \omega _{kn}\left| x_{kn}\right| ^{2},\end{aligned}$$
(63)

and

$$\begin{aligned} \frac{1}{m}\left\langle D^{e}\right\rangle _{n} =\frac{{4\pi ^{2}}e^{2}}{3{\hbar}} \sum _{k} \rho ^{e}\left( \omega _{kn}\right) \omega _{kn}\left| x_{kn}\right| ^{2},\end{aligned}$$
(64)

so that instead of Eq. (55) we obtain

$$\begin{aligned} \frac{d}{dt}\left\langle H\right\rangle _{n}=\left[ -\sum _{k<n}\hbar \omega _{nk}A_{kn}+\sum _{k}\hbar \omega _{kn}B_{kn}\rho ^{e}(\left| \omega _{kn}\right| )\right] , \end{aligned}$$
(65)

where the second sum contains both positive (\(k>n\)) and negative (\(k<n\)) terms, and

$$\begin{aligned} A_{kn}=\frac{4e^{2}}{3\hbar c^{3}}\left| \omega _{kn}^{3}\right| \left| x_{kn}\right| ^{2},\;B_{kn}=\frac{4\pi ^{2}e^{2}}{3\hbar ^{2}}\left| x_{kn}\right| ^{2}. \end{aligned}$$
(66)

The formulas for the induced transition coefficients \(B_{kn}\) agree with those of quantum mechanics. The formula obtained for the so-called spontaneous emission coefficient \(A_{kn}\)—which fails in quantum mechanics by a factor of 2—is in exact agreement with that of quantum electrodynamics, with the advantage that the present derivation makes it clear that radiation reaction and diffusion, given by the terms in Eqs. (62) and (63) with \(k>n\) respectively, contribute equally to “spontaneous” emissions. From the appearance of the factor \(e^{2}\) in these formulas it is clear that they refer only to electric dipole transitions. (For an earlier derivation in the context of SED see [24].)

8 Relationship Between Covariances and Quantum Expectation Values

In quantum mechanics the expectation values of dynamical quantities are calculated in terms of matrix elements; take for example

$$\begin{aligned} \left\langle x^{2}\right\rangle _{n}=\sum _{k}\left| x_{kn}\right| ^{2}. \end{aligned}$$
(67)

As long as the dynamical quantity is a function of commuting operators, as is the case of Eq. (64), the result is uniquely determined. However, as is well known, when the dynamical quantity involves non-commuting operators, there can be more than one expectation value, depending on the order of the operators, so for example

$$\begin{aligned} \left\langle \hat{x}\hat{p}\right\rangle _{n}-x_{nn}p_{nn}=im\sum _{k}\omega _{kn}\left| x_{kn}\right| ^{2}, \end{aligned}$$
(68)
$$\begin{aligned} \left\langle \hat{p}\hat{x}\right\rangle _{n}-x_{nn}p_{nn}=-im\sum _{k}\omega _{kn}\left| x_{kn}\right| ^{2}. \end{aligned}$$
(69)

To bring physical clarity to the issue, let us look at it from the perspective gained from the discussion of the previous sections. We recall that we are considering a stationary state n, and that the \(x_{kn},p_{kn}\) are the coefficients of the response functions involved in the transitions between states n and k. This allows us to calculate quantities such as

$$\begin{aligned} C_{n}(xp)=\left\langle \left( x_{n}^{*}(t)-x_{nn}\right) \left( p(t)-p_{nn}\right) \right\rangle \end{aligned}$$
$$\begin{aligned} =\left\langle x^{*}(t)p(t)\right\rangle _{nn}-x_{nn}p_{nn}, \end{aligned}$$
(70)

which must be understood as the ordered covariance of the response function \(\left( p(t)-p_{nn}\right)\) that can take the particle from state n to any of the states k, followed (on the left) by the response function \(\left( x_{n}^{*}(t)-x_{nn}\right)\) that takes the particle from state k back to state n. If, instead, the response function \(\left( x_{n}(t)-x_{nn}\right)\) takes the particle from state n to any state k, and \(\left( p_{n}(t)-p_{nn}\right)\) takes it back to the initial state, we get the covariance

$$\begin{aligned} C_{n}(px)=\left\langle p^{*}(t)x(t)\right\rangle _{nn}-x_{nn}p_{nn}. \end{aligned}$$
(71)

In Eqs. (67) and (68) we have considered that the right term within the brackets is followed by the left term, to respect the convention used in quantum mechanics. Furthermore, by conjugating the left term we have taken into account that it compensates the effect of the right term by bringing the system back to its initial state. Since in the discrete case, Eq. (33) becomes

$$\begin{aligned} \left\langle a^{*}(\omega _{kn})a(\omega _{k'n})\right\rangle =\delta _{kk'}, \end{aligned}$$
(72)

equations (67) and (68) give, using Eqs. (60),

$$\begin{aligned}{} & {} C_{n}(xp)=im\sum _{k}\omega _{kn}\left| x_{kn}\right| ^{2}, \end{aligned}$$
(73)
$$\begin{aligned}{} & {} C_{n}(px)=-im\sum _{k}\omega _{kn}\left| x_{kn}\right| ^{2}, \end{aligned}$$
(74)

This allows us to identify the quantum expressions (65) and (66) as the ordered covariances of \(\hat{x}\) and \(\hat{p}\), and of \(\hat{p}\) and \(\hat{x}\), respectively.

Adding the above results gives

$$\begin{aligned} \left\langle x^{*}(t)p(t)+p^{*}(t)x(t)\right\rangle _{n}=2x_{nn}p_{nn}, \end{aligned}$$
(75)

and subtrating them gives

$$\begin{aligned} \left\langle x^{*}(t)p(t)-p^{*}(t)x(t)\right\rangle _{n}=2im\sum _{k}\omega _{kn}\left| x_{kn}\right| ^{2}. \end{aligned}$$
(76)

On the other hand, using the formula for the Poisson bracket in the \(a,a^{*}\)-representation, and again bearing in mind that the left factor must reverse the effect of the right factor, we have

$$\begin{aligned}{} & {} \left[ x(t),p(t)\right] _{n}=\sum _{k>n}\left( \frac{\partial x_{n}^{*}(t)}{\partial a}\frac{\partial p_{n}(t)}{\partial a^{*}}-\frac{\partial p_{n}^{*}(t)}{\partial a}\frac{\partial x_{n}(t)}{\partial a^{*}}\right) \nonumber \\{} & {} -\sum _{k<n}\left( \frac{\partial x_{n}(t)}{\partial a^{*}}\frac{\partial p_{n}^{*}(t)}{\partial a}-\frac{\partial p_{n}(t)}{\partial a^{*}}\frac{\partial x_{n}^{*}(t)}{\partial a}\right) \nonumber \\{} & {} =2im\sum _{k}\omega _{kn}\left| x_{kn}\right| ^{2}, \end{aligned}$$
(77)

in agreement with (73), which shows that the Poisson bracket is equal to the difference of the covariances. Note that in the second sum, we took into account that the downward transitions (\(k<n\)) depend on the normal variable a, and that \(\left[ a,a^{*}\right] =-\left[ a^{*},a\right]\).

Combining with the quantum equations (65) and (66), we get

$$\begin{aligned} \left[ x(t),p(t)\right] _{n}=\left\langle \hat{x}\hat{p}-\hat{p}\hat{x}\right\rangle _{n}=\left[ \hat{x},\hat{p}\right] _{n}, \end{aligned}$$
(78)

i.e. the Poisson bracket of the response functions x, p to the field in the \(a,a^{*}\)-representation is equal to the quantum commutator of the corresponding operators.

Further, since according to Eq. (37),

$$\begin{aligned} \left[ x(t),p(t)\right] _{n}=i\hbar \end{aligned}$$
(79)

for any state n, from (74) we obtain the Thomas-Reiche-Kuhn sum rule,

$$\begin{aligned} 2m\sum _{k}\omega _{kn}\left| x_{kn}\right| ^{2}=\hbar , \end{aligned}$$
(80)

which fixes the universal scale of the response coefficients.

Finally, we note that just as the commutator \(\left[ \hat{x},\hat{p}\right] _{n}\) is given by the Poisson bracket \(\left[ x(t),p(t)\right] _{n}\), according to (75), the anticommutator \(\left\{ \hat{x},\hat{p}\right\} _{n}\) is obtained from the symmetrized covariance of x(t) and p(t), according to Eq. (72),

$$\begin{aligned} \left\langle x^{*}(t)p(t)+p^{*}(t)x(t)\right\rangle _{n}=\left\{ \hat{x},\hat{p}\right\} _{n}. \end{aligned}$$
(81)

This observation reminds us of the relationship established in linear-response theory (LRT), between the symmetrized covariance, which is a measure of the fluctuations of the system, and the average Poisson bracket, which contains the response or relaxation function; in the context of LRT, such a relationship can generally be expressed in the form of a fluctuation-dissipation theorem [21, 22, 26]. This suggests the need for a deeper analysis of the results obtained in the present paper with a view to establishing a fluctuation-dissipation theorem in the context of SED and its quantum-mechanical counterpart, a task left for a future paper.

9 Quantum Field Operators

We have shown above how, in the description of the stationary regime, the canonical variables x, p of the mechanical system interacting with the field (both the ZPF and the applied field), are replaced by the dipolar response functions to those field modes to which the system responds resonantly. In addition, we have shown that the ordering of the response coefficients defines the order of the respective quantum operators (see e.g. [27]).

In order to complete the quantum description of the whole (matter+field) system in a consistent way, we will now describe how the field operators arise. This description should serve to express any electric or magnetic component of the interacting field, be it the ZPF alone or in combination with an external field.

The fact that matter interacts resonantly with certain field modes, allows us to focus on one of these modes, say the one that connects the states n and k of the material system [28]. We will therefore consider a single component with wave vector and polarization \(\varvec{k},\mathbf {\hat{\epsilon }}\), in the long-wavelength approximation (as explained in connection with Eq. (5)), so that it can be represented as an oscillator with frequency \(\omega _{nk}\equiv \omega\). The canonical variables describing the field in the state \(\text{n}\), \(\mathrm {q_{n}}(t),\mathrm {p_{n}}(t),\) are expressed in general in terms of the normal field variables \(a,a^{*}\) that can take them to a different state \(\text{n}'\), with respective coefficients \(\text{q}_{\mathrm {n'n}}\), \(\text{p}_{\mathrm {n'n}}\). The field mode has a single well-defined frequency \(\omega ,\) which means that \(\left| \omega _{\mathrm {n'n}}\right| =\omega\), i. e., \(\omega _{\mathrm {n'n}}=\pm \omega\). Consequently, only two of the coefficients \(\text{q}_{\mathrm {n'n}}\) connecting state n with some other state n’ of the field are different from zero. We identify the upper state (corresponding to \(\omega _{\mathrm {n'n}}=\omega\)) with \(\mathrm {n'=n}+1\), and the lower state (corresponding to \(\omega _{\mathrm {n'n}}=-\omega\)) with \(\mathrm {\mathrm {n'}=n}-1\), so that

$$\begin{aligned} \text{q}_{\text{n}}(t)-\text{q}_{\text{nn}}=\text{q}_{\mathrm {n+1,n}}a^{*}e^{i\omega t}\mathrm {+\text{q}_{\mathrm {n-1,n}} ae^{-i\omega t} ,\quad }\text{p}_{\text{n}}(t)=\text{p}_{\mathrm {n+1,n}}a^{*}e^{i\omega t}\mathrm {+\text{p}_{\mathrm {n-1,n}} ae^{-i\omega t} ,\negthinspace \;} \end{aligned}$$
(83)

where \(\text{p}_{\mathrm {n\pm 1,n}}=\pm i\omega _{\mathrm {}}\text{q}_{\mathrm {n\pm 1,n}},\) i.e.

$$\begin{aligned} \omega \text{q}_{\mathrm {n+1n}}+i\text{p}_{\mathrm {n+1n}}=0,\;\omega \text{q}_{\text{nn}-1}-i\text{p}_{\text{nn}-1}=0. \end{aligned}$$
(84)

The Poisson bracket of the canonical field variables pertaining to the interacting field mode, \(\left\{ \mathrm {q_{n}}(t),\mathrm {p_{n}}(t)\right\} =1\), written in the \(a,a^{*}\)-representation and calculated with the same procedure used in Sect. 8, gives, using

$$\begin{aligned}{} & {} \left[ \mathrm {q_{n}}(t),\mathrm {p_{n}}(t)\right] =i\hbar , \end{aligned}$$
(85)
$$\begin{aligned}{} & {} {\displaystyle \sum _{\mathrm {n'=n\pm 1}}}\left( \text{q}_{\text{nn}'}\text{p}_{\mathrm {n'n}}-\text{p}_{\text{nn}'}\text{q}_{\mathrm {n'n}}\right) =i\hbar \end{aligned}$$
(86)

for any state \(\text{n}\mathrm {}\) of the field. By identifying \(\text{q}_{\text{nn}'}\) and \(\text{p}_{\mathrm {n'n}}\) as the elements of matrices \(\hat{\text{q}}\) and \(\hat{\text{p}}\), respectively, Eq. (82) becomes

$$\begin{aligned} \left[ \mathrm {\hat{q}},\mathrm {\hat{p}}\right] =i\hbar . \end{aligned}$$
(87)

Therefore, according to Eqs. (80), the normalized matrix \(\hat{a}\) and its adjoint, defined as

$$\begin{aligned} \hat{a}=\frac{1}{\sqrt{2\hbar \omega }}(\omega \mathrm {\hat{q}}+i\hat{\text{p}}),\;\;\hat{a}^{\dagger }=\frac{1}{\sqrt{2\hbar \omega }}(\omega \mathrm {\hat{q}}-i\hat{\text{p}}), \end{aligned}$$
(88)

have off-diagonal elements either immediately above or immediately below the diagonal, meaning that they play the role of annihilation and creation operators, respectively. In terms of these, (83) becomes

$$\begin{aligned} \left[ \hat{a},\hat{a}^{\dagger }\right] =1. \end{aligned}$$
(89)

The rest of the quantum formalism is obtained as usual by introducing the vectors representing the possible states of the field mode on which the operators act.

It should be stressed that \(\hat{a},\hat{a}^{\dagger }\) are not the normal variables \(a,a^{*}\) formally transformed into operators; instead the matrix elements of \(\hat{a},\hat{a}^{\dagger }\) are the coefficients of the normal variables. Therefore, the quantum-mechanical and the quantum-optical operator formalism contain the response coefficients in the common \(a,a^{*}\)-representation. Interestingly, the \(a,a^{*}\) themselves disappear completely from the description.

By writing the Hamiltonian operator associated with a single mode as

$$\begin{aligned} \mathrm {\hat{H}}=\frac{1}{2}\left( \omega ^{2}\mathrm {\hat{q}^{2}+\hat{\text{p}}^{2}}\right) =\frac{\hbar \omega }{2}\left( \hat{a}\hat{a}^{\dagger }+\hat{a}^{\dagger }\hat{a}\right) , \end{aligned}$$
(90)

one gets the well-known expression for the energy expectation value

$$\begin{aligned} \mathrm {\left\langle n\right| \hat{H}\left| \text{n}\right\rangle }=\mathcal {E}_{\text{n}}=\left( \text{n}+\frac{1}{2}\right) \hbar \omega , \end{aligned}$$
(91)

whence indeed \(\left| \mathcal {E}_{\text{n}}-\mathcal {E}_{\mathrm {n\pm 1}}\right| =\hbar \omega\). Note that Eq. (86) is the symmetrized covariance, which, as said above, is a measure of the fluctuations; in particular, it confirms the existence of zero-point fluctuations with energy \(\frac{1}{2}\hbar \omega\). Further, given that the effect of the operators \(\hat{a},\hat{a}^{\dagger }\) is to lower or raise the number \(\text{n}\), respectively, the separate expressions

$$\begin{aligned} \mathrm {\hat{H}^{a}}=\hbar \omega \hat{a}^{\dagger }\hat{a},\;\mathrm {\hat{H}^{e}}=\hbar \omega \hat{a}\hat{a}^{\dagger }, \end{aligned}$$
(92)

are identified with the operators corresponding to the field energy available for processes involving either absorption or emission of radiation by matter, respectively,

$$\begin{aligned} \mathrm {\left\langle n\right| \mathrm {\hat{H}^{a}}\left| \text{n}\right\rangle }=\text{n}\hbar \omega ,\;\,\mathrm {\left\langle n\right| \mathrm {\hat{H}^{e}}\left| \text{n}\right\rangle }=\mathrm {(n+1)}\hbar \omega . \end{aligned}$$
(93)

10 To Conclude: On the Coarse-Grained Nature of the Quantum Description

The analysis carried out in this work has led us to establish a one-to-one relationship between the response functions of the dynamical variables driven linearly by the radiation field, and the matrices representing their quantum counterparts. The relationship between classical Poisson brackets and commutators, so insightfully established by Dirac in the early quantum times, can thus be understood as the result of a kinematic transformation that preserves the symplectic structure but changes the nature and hence the meaning of the quantities involved in the description.

Thus, quantum mechanics combines in a uniquely compact and elegant way the information about the system’s internal dynamics, contained in the Hamiltonian, and about the system’s response to external forces or perturbations, contained in the response coefficients. The response coefficients (the quantum operators) are the new building blocks for the description of the quantum dynamics—still determined by the Hamiltonian. This is an excellent example that illustrates the point made by Pavarini [26] in the context of linear response theory, in the sense that “everything we know about a physical system comes either from its effects on other physical systems or from its response to external forces.”

Since the (normal) variables associated with the driving force of the field disappear in quantum mechanics, the resulting description appears to be purely mechanical. The field variables are hidden, and only Planck’s constant remains as evidence of their action. However, as mentioned above, if the field itself disappeared, the mechanical system would cease to be quantum.

The change in the nature of the variables used for the description has of course profound consequences. We are no longer dealing with trajectories in phase space, i.e. with the detailed motions of the particles in their (relatively) stationary states. What the quantum formalism describes is the stationary states and the dynamics between states. This indicates that the replacement of the phase-space description by the quantum operator description involves a coarse graining in time: as said above, the times involved in the changes between states are of the order of 10\(^{6}\) larger than the times of the detailed motions.

This coarse-graining in time is expressed in the transition from the generalized Fokker-Planck equation mentioned in Sect. 4, to the (second-order) Fokker-Planck equation (13). By eliminating all the higher-order derivatives from the GFPE, one assumes that enough time has elapsed for the vacuum fluctuations—with the highly colored spectrum given by Eq. (11)—to lead to the build-up of the diffusion coefficients (14); at this rough time scale, the dynamics appears Markofian. And indeed the dynamics of quantum processes involving transitions between quantum states is Markofian; see e.g. Ref. [29].

This explains the success of the stochastic theory of quantum mechanics (SQM) pioneered by Nelson [30, 31] and further developed by a number of authors (see e.g. [32, 5] and references therein): by introducing dynamical quantities (velocities and accelerations) which are statistical averages over a time interval \(\Delta t>0\), one obtains a set of dynamical equations describing a Markov process in configuration space, which are equivalent to the Schrödinger equation and its complex conjugate [5, 6]. For further insights into the connections between SQM, SED and quantum mechanics, see Ref. [33].