The Planck DistributionPlanck distribution, a Necessary Consequence of the Fluctuating Zero-Point Field

Abstract

The historical problem of the spectral distributionDistribution of the radiation field in equilibrium at a given temperature is revisited in this chapter. Taking into account the inescapable presence of the fluctuatingFluctuations!zero-point zero-point radiation field, a derivation of Planck’s formula is presented that does not introduce any quantum postulate. The starting point is Wien’sWien!law law, which dictates that the mean energy of a Harmonic oscillator!and spinharmonic oscillator of frequency $\omega$—or rather, the mean energy of a monochromatic mode of the radiation field—must be proportional to the frequency. The description obtained through a standard thermodynamic analysis is shown to be incomplete, as it does not provide for the existence of fluctuations at zero temperature. This limitation is lifted by means of a statistical analysis that allows to determine the variance of the Fluctuationsfluctuations of the energy at $T=0.$ A combination of the outcomes of the two analyses leads to a differential equation, which upon integration gives the Planck distribution Planck distributionat any temperature. The different terms contributing Commutator!and correlationto the energy fluctuations acquire thus a new meaning, different from those assigned to it by Planck and by EinsteinEinstein, A.. The implications of the results regarding the question of continuity or discontinuity of the field energy are discussed, and the chapter concludes with a brief discussion on the reality Realityof the zero-point fluctuations and the origin of quantum fluctuations.

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With this chapter we initiate our analysis of the implications of considering the fluctuating zero-point radiation field (zpf) as a fundamental constituent of an otherwise classical system. As announced in the introductory chapter, our journey starts with a fresh look at a simple though physically (and historically) relevant system, namely the electromagnetic radiation field in equilibrium with matter at temperature $T$ . The blackbodyBlackbody law problem, the one that gave birth to quantum mechanics, is thus revisited, taking into account the zpf Correlations!and zpf. The mere existence of this nonthermal field is shown to have far-reaching consequences. In particular, by performing a thermodynamic and statistical analysis of an ensemble of Harmonic oscillator!and spinharmonic oscillators of frequency $\omega$ representing the modes of the radiation field of the respective frequency, we find that Planck’sBoyer!and Planck law law, as well as irreducible (quantum) fluctuations, arise as necessary consequences of allowing for the presence of the pervarsive zpf, without any assumption of discreteness.

3.1 Thermodynamics of the Harmonic Oscillator

Let us start by considering a one-dimensional Harmonic oscillator!and spinharmonic oscillator of frequency $\omega ,$ with the Hamiltonian given by¹

$$\begin{aligned} H=(p^{2}+\omega ^{2}q^{2})/2. \end{aligned}$$

(3.1)

For a material oscillator of mass $m=1,$ $q$ and $p$ stand for the oscillator’s position and momentumMomentum, respectively. Now, of relevance for our purposes is that a monochromatic mode of frequency $\omega$ of the radiation field is equivalent to a harmonic oscillator of that same frequency. In this case $H$ refers to the energy of such mode, and $q$ and $p$ represent its quadratures.

Several basic properties of the Harmonic oscillator!and spinharmonic oscillator can be derived from the structure Structureof (3.1), and thus hold irrespective of the oscillator’sOscillator nature. In particular, for a given constant energy $U,$ the trajectory in phase space is the ellipse

$$\begin{aligned} p^{2}+\omega ^{2}q^{2}=2U, \end{aligned}$$

(3.2)

and its area gives the action

$$\begin{aligned} J=\frac{1}{2\pi }\oint pdq=\frac{1}{2\pi \omega }\oint \sqrt{2U-\omega ^{2}q^{2}}\,d\left( \omega q\right) =\frac{U}{\omega }. \end{aligned}$$

(3.3)

The action $J$ is an adiabatic invariant of the harmonic oscillator Landau, L.(see e.g.Lifshitz, E. M. Landau and Lifshitz (1976), Sect. 49;Saletan E.J. José and Saletan (1998), Sect. 6.4)José, J. V., which means that it remains constant under a slow change of the frequency. Therefore, the change $dU$ in the energy concomitant with the slow change $d\omega$ is given by

$$\begin{aligned} dU=Jd\omega =\frac{U}{\omega }d\omega , \end{aligned}$$

(3.4)

so that the work $dW$ done by the system on the external device effecting the change of frequency is

$$\begin{aligned} dW=-\frac{U}{\omega }d\omega . \end{aligned}$$

(3.5)

From here it follows that if $S(T,\omega )$ stands for the entropy Entropyof the system when this latter is in thermodynamic equilibrium at temperature $T$, for a reversible process one may write

$$\begin{aligned} TdS(T,\omega )=dU(T,\omega )+dW=dU(T,\omega )-\frac{U}{\omega }d\omega , \end{aligned}$$

(3.6)

consequently

$$\begin{aligned} T\left( \frac{\partial S}{\partial T}\right) _{\omega }dT+T\left( \frac{\partial S}{\partial \omega }\right) _{T}d\omega =\left( \frac{\partial U}{\partial T}\right) _{\omega }dT+\left[ \left( \frac{\partial U}{\partial \omega }\right) _{T}-\frac{U}{\omega }\right] d\omega . \end{aligned}$$

(3.7)

Since the changes in the variables $T$ and $\omega$ are independent, this relation naturally splits into the pair of equations

$$\begin{aligned} T\left( \frac{\partial S}{\partial \omega }\right) _{T}=\left( \frac{\partial U}{\partial \omega }\right) _{T}-\frac{U}{\omega }, \end{aligned}$$

(3.8)

$$\begin{aligned} T\left( \frac{\partial S}{\partial T}\right) _{\omega }=\left( \frac{\partial U}{\partial T}\right) _{\omega }. \end{aligned}$$

(3.9)

We utilize these two relations by taking the partial derivative of the first one with respect to $T$ and of the second one with respect to $\omega ,$ and combine the results to get

$$\begin{aligned} \left( \frac{\partial S}{\partial \omega }\right) _{T}=-\frac{1}{\omega }\left( \frac{\partial U}{\partial T}\right) _{\omega }. \end{aligned}$$

(3.10)

Substitution into Eq. (3.8) gives

$$\begin{aligned} \left( \frac{\partial U}{\partial \omega }\right) _{T}-\frac{U}{\omega }=-\frac{T}{\omega }\left( \frac{\partial U}{\partial T}\right) _{\omega }. \end{aligned}$$

(3.11)

The solution of this equation can be found by writing $U=\omega f(T,\omega )$ to cancel the term $U/\omega ,$ whence

$$\begin{aligned} \frac{\omega }{T}\left( \frac{\partial f}{\partial \omega }\right) _{T}=-\left( \frac{\partial f}{\partial T}\right) _{\omega }. \end{aligned}$$

(3.12)

This equation holds for any function $f$ of the single variable $\omega /T$, as can be easily verified; hence Eq. (3.11) admits the general solution

$$\begin{aligned} U=\omega f(\omega /T). \end{aligned}$$

(3.13)

Equation (3.13) is indeed a very important result: it is Wien’sWien!law law, which establishes the general form of the mean energy $U$ of any harmonic oscillator as a function of its frequency $\omega$ and the temperature $T$. This law will be at the basis of our considerations below.² $^{\text {,}}$ ³

We now present some additional results concerning the thermodynamics of the harmonic oscillator that will be useful below. The Helmholtz Free energyfree energy $F$ takes the form

$$\begin{aligned} F(T,\omega )=-k_{B}T\phi (\omega /T), \end{aligned}$$

(3.14)

where $k_{B}$ is BoltzmannBoltzmann’s constant and $\phi$ is a thermodynamic potential from which the thermodynamic functions of the oscillator can be determined. In particular the mean equilibrium energy becomes

$$\begin{aligned} U(T,\omega )=k_{B}T^{2}\left( \frac{\partial \phi }{\partial T}\right) _{\omega }=-k_{B}\omega \frac{d\phi (z)}{dz}, \end{aligned}$$

(3.15)

with $z=\omega /T.$ Comparison with WienWien’s law gives

$$\begin{aligned} f(z)=-k_{B}\frac{d\phi }{dz}. \end{aligned}$$

(3.16)

Finally, the entropy is also a function of the variable $z,$

$$\begin{aligned} S(z)=k_{B}\phi (z)+zf(z). \end{aligned}$$

(3.17)

These results suffice for our purposes.

3.1.1 Unfolding the Zero-Point EnergyOscillator!zero-point energy

In the low-temperature limit $T\rightarrow 0$, Eqs. (3.13) and (3.15) give for the mean energy

$$\begin{aligned} \mathcal {E}_{0}\equiv U(0,\omega )=\omega f(\infty )=-k_{B}\omega \frac{ d\phi }{dz}(\infty )=A\omega , \end{aligned}$$

(3.18)

so that the zero-point energy Oscillator!zero-point energy $\mathcal {E}_{0}$—the mean energy of the oscillator at absolute temperature $T=0$—is determined by the value that the function $f(z)$ (or $d\phi /dz$) attains at infinity.⁴ In the usual thermodynamic analysis the value of the constant $A=f(\infty )$ is arbitrarily chosen as zero, so there is no athermal energy. However, the more general (and more natural) solution corresponds to a nonnull value of $A.$ In the case of the radiation field oscillatorsOscillator, this represents a physically more reasonable choice than a vacuum that is completely devoid of electromagnetic phenomena. By taking $A$ to be nonzero we attest the existence of a zero-point energy Oscillator!zero-point energythat fills the whole space and is proportional to the frequency of the oscillator,⁵ $^{\text {,}}$ ⁶

$$\begin{aligned} \mathcal {E}_{0}=A\omega =\tfrac{1}{2}\hbar \omega . \end{aligned}$$

(3.19)

The value of $A$ (with dimensions of action) must be universal because it determines the equilibriumspectrum at $T=0,$ which, according to Kirchhoff’s lawKirchhoff’s law, has a universal character. We have put it equal to $\hbar /2$ in order to establish contact with present-day knowledge. However, it must be stressed that the presence of the Planck constant here does not imply any quantum connotation. In addition, it should be noticed that many of the results to be obtained in the present chapter do not depend on the precise value of $\mathcal {E}_{0},$ the only requirement being in such instances that it be different from zero.

A nonnull value of $A$ means a violation of energy equipartition Equipartition!violationamong the oscillatorsOscillator, since the equilibrium energy becomes now a function of the oscillator frequency. Though at this stage such violation can strictly be assured only at $T=0,$ the result suggests that the physics ensuing from the existence of $\mathcal {E}_{0}\ne 0$ necessarily transcends classical physics. This opens up interesting possibilities that will be explored along this chapter.

In concluding this section, let us note that the existence of a zero-point energy Oscillator!zero-point energyprovides a natural energy scale, which, along with $k_{B}T$, suggests to introduce the dimensionless quantity

$$\begin{aligned} \mathring{z}=\frac{2\mathcal {E}_{0}}{k_{B}T}=\frac{\hbar }{k_{B}}z=\frac{\hbar \omega }{k_{B}T}. \end{aligned}$$

(3.20)

This will be the natural dimensionless variable of the thermodynamic functions, since the potential $\phi$ in Eq. (3.14) is a dimensionless function of $z$ and can therefore be expressed as a function of $\mathring{z}.$

3.2 General Thermodynamic Equilibrium Distribution

Our aim is to find the average energy $U$ per oscillator in an ensemble of such systems when equilibrium has been reached at a fixed temperature $T$. For this purpose we first follow the standard description of a canonicalDistribution!canonical ensemble (Pathria, R. K. Pathria 1996). In this case, the probability that a member of the ensemble Probability interpretation!ensembleis in a state with energy between $\mathcal {E}$ and $\mathcal {E}+d\mathcal {E}$ can be written in the general form

$$\begin{aligned} W_{g}(\mathcal {E})d\mathcal {E}&=\frac{1}{Z_{g}(\beta )}g(\mathcal {E} )e^{-\beta \mathcal {E}}d\mathcal {E}, \end{aligned}$$

(3.21a)

$$\begin{aligned} Z_{g}(\beta )&=\!\int \!g(\mathcal {E})e^{-\beta \mathcal {E}}d\mathcal {E}, \end{aligned}$$

(3.21b)

where $\beta =1/(k_{B}T)$, $Z_{g}(\beta )$ is the partition function that normalizes $W_{g}(\mathcal {E})$ to unity, and $g(\mathcal {E})$ is a weight function representing the density of states with energy $\mathcal {E}$. The mean value $\left\langle f(\mathcal {E})\right\rangle$ of any function $f( \mathcal {E})$ is thus

$$\begin{aligned} \left\langle f(\mathcal {E})\right\rangle =\!\int \!W_{g}(\mathcal {E})f( \mathcal {E})d\mathcal {E}. \end{aligned}$$

(3.22)

For $f(\mathcal {E})=\mathcal {E}$, (3.22) gives the mean energy

$$\begin{aligned} U=\left\langle \mathcal {E}\right\rangle =\int \!\mathcal {E}W_{g}(\mathcal {E} )d\mathcal {E}. \end{aligned}$$

(3.23)

Equation (3.21a) constitutes the general form of a Boltzmann distribution.⁷ In particular, the corresponding classical distributionDistribution for the harmonic oscillator is obtained from (3.21a) with $g(\mathcal {E})$ given byPathria, R. K. Pathria (1996)

$$\begin{aligned} g_{\text {classic}}(\mathcal {E})=\frac{1}{s\omega }, \end{aligned}$$

(3.24)

where $s$ is a constant with dimensions of action, so $g$ has the dimension of $\left( \text {energy}\right) ^{-1}$. In this case one gets from the above equations

$$\begin{aligned}&W_{\text {cl}}(\mathcal {E})=W_{g_{\text {cl}}}(\mathcal {E})=\frac{e^{-\beta \mathcal {E}}}{\int e^{-\beta \mathcal {E}}d\mathcal {E}}; \end{aligned}$$

(3.25a)

$$\begin{aligned}&Z_{\text {cl}}(\beta )=\!\int \!g_{\text {cl}}(\mathcal {E)}e^{-\beta \mathcal {E }}d\mathcal {E}=\frac{1}{s\beta \omega }; \end{aligned}$$

(3.25b)

$$\begin{aligned}&\left\langle \mathcal {E}\right\rangle =U =-\frac{1}{Z_{\text {cl}}}\frac{dZ_{ \text {cl}}}{d\beta }=\frac{1}{\beta }=k_{B}T. \end{aligned}$$

(3.25c)

From the last equation it follows that $U(T=0)=0$. This means that to allow for a zero-point energy,Oscillator!zero-point energy a form for $g(\mathcal {E})$ different from that given by Eq. (3.24) must be used. The specific structure of this $g(\mathcal {E})$ consistent with a zero-point energy Oscillator!zero-point energyfor the harmonic oscillator will be determined below.

3.2.1 Thermal Fluctuations of the EnergyFluctuations!thermal

Equations (3.21a, 3.21b) and (3.22) lead to a series of important and general results. With $f(\mathcal {E})=\mathcal {E}^{r}$, $r$ a positive integer, it follows that (the prime indicates derivative with respect to $\beta$)⁸

$$\begin{aligned} \left\langle \mathcal {E}^{r}\right\rangle ^{\prime }=-\frac{Z_{g}^{\prime }}{Z_{g}}\left\langle \mathcal {E}^{r}\right\rangle -\frac{1}{Z_{g}}\!\int \!\mathcal {E}^{r+1}g(\mathcal {E})e^{-\beta \mathcal {E}}d\mathcal {E}=-\frac{Z_{g}^{\prime }}{Z_{g}}\left\langle \mathcal {E}^{r}\right\rangle -\left\langle \mathcal {E}^{r+1}\right\rangle , \end{aligned}$$

(3.26)

and further, from (3.21b),

$$\begin{aligned} \left\langle \mathcal {E}\right\rangle =U=\frac{1}{Z_{g}}\!\int \!\mathcal {E}g(\mathcal {E})e^{-\beta \mathcal {E}}d\mathcal {E}=-\frac{Z_{g}^{\prime }}{Z_{g}}. \end{aligned}$$

(3.27)

These two expressions combined give the recurrence Recurrencerelation

$$\begin{aligned} \left\langle \mathcal {E}^{r+1}\right\rangle =U\left\langle \mathcal {E}^{r}\right\rangle -\left\langle \mathcal {E}^{r}\right\rangle ^{\prime }, \end{aligned}$$

(3.28)

which can be extended to any continuous function $h(\mathcal {E})$ to obtain

$$\begin{aligned} -\left\langle h(\mathcal {E})\right\rangle ^{\prime }=\left\langle \mathcal {E}h(\mathcal {E})\right\rangle -U\left\langle h(\mathcal {E})\right\rangle . \end{aligned}$$

(3.29)

Thus ${-}\left\langle h(\mathcal {E})\right\rangle ^{\prime }$ is given in general by the covariance Covarianceof $h(\mathcal {E})$ and $\mathcal {E}$.

Equation (3.28) with $r=1$ gives a most important expression for the energy variance,

$$\begin{aligned} \sigma _{\mathcal {E}}^{2}\equiv \left\langle \left( \mathcal {E}-U\right) ^{2}\right\rangle =\left\langle \mathcal {E}^{2}\right\rangle -U^{2}=-\frac{dU}{d\beta }, \end{aligned}$$

(3.30)

which can be rewritten as the well-known relation (Mandl 1988)Mandl, F.

$$\begin{aligned} \sigma _{\mathcal {E}}^{2}=-\frac{dU}{d\beta }=k_{B}T^{2}\left( \frac{ \partial U}{\partial T}\right) _{\omega }=k_{B}T^{2}C_{\omega } \end{aligned}$$

(3.31)

in terms of the specific heatSpecific heat (or heat capacity) $C_{\omega }$.⁹ Because $C_{\omega }$ is surely finite at low temperatures, the right-hand side of this expression is zero at $T=0$, whence

$$\begin{aligned} \sigma _{\mathcal {E}}^{2}(T=0)=0, \end{aligned}$$

(3.32)

which shows that the description provided by the distribution $W_{g}$ does not allow for the dispersion Momentum!dispersionof the energy at zero temperature. The fact that $W_{g}$ offers a thermodynamic description that admits thermal fluctuations only, and has no room for temperature-independent fluctuations, is an important shortcoming, as is clear when we consider a collection of harmonic oscillators (such as those of the electromagnetic field in equilibrium inside a cavity)Lifetime!cavity effects which are endowed with a zero-point energy Oscillator!zero-point energygiven by (3.19). Indeed, for such system the distributionDistribution $W_{g}$ leaves out the fluctuationsFluctuations of the nonzero Fluctuations!nonthermalnonthermal component of the energy. We continue to work here with the thermodynamic description, but later on we shall introduce a full-fledged statistical description that overcomes this limitation.

3.2.2 Some Consequences of the Recurrence Relation

The recurrence Recurrencerelation (3.28) and the WienWien!law law can be recast into other interesting forms as follows. First we observe that the equation

$$\begin{aligned} \left\langle \mathcal {E}^{r}\right\rangle =\frac{1}{Z_{g}(\beta )}\!\int \! \mathcal {E}^{r}g(\mathcal {E})e^{-\beta \mathcal {E}}d\mathcal {E}, \end{aligned}$$

(3.33)

with the substitutions $\mathcal {E}=\mathcal {E}_{0}\epsilon \ (\epsilon$ dimensionless), and $\mathring{z}=2\mathcal {E}_{0}\beta$, gives

$$\begin{aligned} \left\langle \mathcal {E}^{r}\right\rangle =\mathcal {E}_{0}^{r}f_{r}(\mathcal {E}_{0},\mathring{z}), \end{aligned}$$

(3.34)

where $f_{r}(\mathcal {E}_{0},\mathring{z})$ is defined as

$$\begin{aligned} f_{r}(\mathcal {E}_{0},\mathring{z})=\frac{\int \epsilon ^{r}g(\mathcal {E} _{0}\epsilon )e^{-\mathring{z}\epsilon /2}d\epsilon }{\int \!g(\mathcal {E} _{0}\epsilon )e^{-\mathring{z}\epsilon /2}d\epsilon }. \end{aligned}$$

As follows from Eq. (3.34), $f_{r}$ is an adimensional function, hence it can be expressed as a function of the adimensional parameter $\mathring{z}$ only. For the harmonic oscillator we use Wien’s law to write $\mathcal {E}_{0}=A\omega ,$ so that Eq. (3.34) reads

$$\begin{aligned} \left\langle \mathcal {E}^{r}\right\rangle =\omega ^{r}A^{r}f_{r}(\mathring{z}), \end{aligned}$$

(3.35)

which is a generalization of Wien’s law for any power $r.$

On the other hand, the general recurrence Recurrencerelation between the moments of the energy, Eq. (3.28), can be rewritten as follows, using Eq. (3.31),

$$\begin{aligned} \left\langle \mathcal {E}^{r+1}\right\rangle =U\left\langle \mathcal {E}^{r}\right\rangle -\frac{d}{d\beta }\left\langle \mathcal {E}^{r}\right\rangle =U\left\langle \mathcal {E}^{r}\right\rangle +\sigma _{\mathcal {E}}^{2}\frac{d}{dU}\left\langle \mathcal {E}^{r}\right\rangle , \end{aligned}$$

(3.36)

$$\begin{aligned} {\text {or}} \quad \left\langle \mathcal {E}^{r+1}\right\rangle =\left( U+\sigma _{\mathcal {E}}^{2}\frac{d}{dU}\right) \left\langle \mathcal {E}^{r}\right\rangle . \end{aligned}$$

(3.37)

Successive iterations of this equation yield

$$\begin{aligned} \left\langle \mathcal {E}^{r}\right\rangle =\left( U+\sigma _{\mathcal {E}}^{2}\frac{d}{dU}\right) ^{r-1}U. \end{aligned}$$

(3.38)

This reveals $U+\sigma _{\mathcal {E}}^{2}\left( d/dU\right)$ as a kind of ‘raising’ operator for the higher moments of the energy, beginning with the first moment $\left\langle \mathcal {E}\right\rangle =U$. It is clear that for $\sigma _{\mathcal {E}}^{2}(U)$ even in $U$, the moments $\left\langle \mathcal {E}^{r}\right\rangle (U)$ have the parity of $r.$

The second centered moment of the energy is $\sigma _{\mathcal {E}}^{2}=\left\langle \left( \mathcal {E-}U\right) ^{2}\right\rangle ;$ for the third one we obtain

$$\begin{aligned} \left\langle \left( \mathcal {E-}U\right) ^{3}\right\rangle&=\left\langle \mathcal {E}^{3}\right\rangle -3U\sigma _{\mathcal {E}}^{2}-U^{3} \nonumber \\&=\sigma _{\mathcal {E}}^{2}\frac{d}{dU}\left\langle \left( \mathcal {E-} U\right) ^{2}\right\rangle , \end{aligned}$$

(3.39)

and by induction it can be seen that this last result generalizes into

$$\begin{aligned} \left\langle \left( \mathcal {E-}U\right) ^{r}\right\rangle =\sigma _{ \mathcal {E}}^{2}\frac{d}{dU}\left\langle \left( \mathcal {E-}U\right) ^{r-1}\right\rangle \end{aligned}$$

(3.40)

for any integer $r\ge 1$. This equation shows that at $T=0$, all centered moments are zero because of (3.32); hence the energy is exactly $\mathcal {E}_{0},$ and its distribution function reduces to $\delta (\mathcal {E-E}_{0})$ in this thermodynamic analysis.

3.3 Planck’s Law from the Thermostatistics of the Harmonic OscillatorBoyer!and Planck law

3.3.1 General Statistical Equilibrium Distribution

It now becomes necessary to extend our description so as to allow for nonthermal fluctuations of the zero-point energy Oscillator!zero-point energyof the field, which are excluded by $W_{g}$. This can be achieved by paying attention to the statistical distribution of the energy $W_{s}(\mathcal {E})$. Since for every frequency the field contains a huge number of modes, the central limit theorem applies Grimmett, G. R.(Stirzaker D. R. Grimmett and Stirzaker 1983; Papoulis 1991)Papoulis, A. and hence the field amplitude of frequency $\omega$ follows a Distribution!normalnormal distribution. This means that the energy distribution follows the simple law

$$\begin{aligned} W_{s}(\mathcal {E})=\frac{1}{U}\,e^{-\mathcal {E}/U}, \end{aligned}$$

(3.41)

with

$$\begin{aligned} \int W_{s}(\mathcal {E})d\mathcal {E}=1,\qquad \int \mathcal {E}W_{s}(\mathcal {E})d\mathcal {E}=U, \end{aligned}$$

(3.42)

and the corresponding energy dispersion Momentum!dispersionis given by (the subscript $s$ denotes averages taken with respect to $W_{s}$, to be distinguished from those calculated with $W_{g}$)¹⁰

$$\begin{aligned} \left( \sigma _{\mathcal {E}}^{2}\right) _{s}=U^{2}. \end{aligned}$$

(3.43)

This (exponential) distribution of the energy [subject to the constraints (3.42)] has the property of maximizing the statistical entropy $S_{s}$, defined as

$$\begin{aligned} S_{s}=-k_{B}\int {W_{s}(\mathcal {E})}\ln c_{s}W_{s}(\mathcal {E})d\mathcal {E}, \end{aligned}$$

(3.44)

where $c_{s}$ is an appropriate constant with dimension of energy. Since the entropyEntropy is usually interpreted as a measure of the disorder present in the system (see e.g. Callen, H. B. Callen 1985; Mandl 1988), the maximal entropyMandl, F. property means maximum disorder, which is the natural demand for a system constituted by a huge number of independent components once equilibrium has been reached.

From Eq. (3.43) we see that $W_{s}$ allows indeed for zero-point fluctuations, since at $T=0$

$$\begin{aligned} \left. (\sigma _{\mathcal {E}}^{2})_{s}\right| _{0}=U^{2}(T=0)=\mathcal {E}_{0}^{2}, \end{aligned}$$

(3.45)

which means that there is a nonthermal contribution to the energy fluctuations, with variance $\mathcal {E}_{0}^{2}.$ The thermal contribution $\sigma _{\mathcal {E}_{T}}^{2}$ to the energy fluctuations at any temperature is obtained by subtracting from the total ones this nonthermal term $\mathcal {E}_{0}^{2}.$ This is true because the thermal and nonthermal fluctuations have an entirely different source, so they are statistically independent, with a null correlation [see the discussion following Eq. (3.90)]. That is,

$$\begin{aligned} \sigma _{\mathcal {E}_{T}}^{2}=(\sigma _{\mathcal {E}}^{2})_{s}-\mathcal {E}_{0}^{2}, \end{aligned}$$

(3.46)

whence

$$\begin{aligned} \sigma _{\mathcal {E}_{T}}^{2}=U^{2}-\mathcal {E}_{0}^{2}. \end{aligned}$$

(3.47)

Recalling that $\sigma _{\mathcal {E}}^{2}$ in Eq. (3.31) stands for the thermal fluctuations of the energy, we can combine this latter with (3.47) and write (omitting the subindex $T)$

$$\begin{aligned} \sigma _{\mathcal {E}}^{2}=U^{2}-\mathcal {E}_{0}^{2}=-\frac{dU}{d\beta }. \end{aligned}$$

(3.48)

Before studying the consequences of this relation we observe that the distribution (3.41) leads to recurrence relations incorporating the nonthermal fluctuations. Indeed (3.41) gives for the moments of the energy

$$\begin{aligned} \left\langle \mathcal {E}^{r}\right\rangle _{s}=r!U^{r}, \end{aligned}$$

(3.49)

and making reiterative use of this equation one obtains

$$\begin{aligned} \left\langle \mathcal {E}^{r}\right\rangle _{s}=U\left\langle \mathcal {E}^{r-1}\right\rangle _{s}+(\sigma _{\mathcal {E}}^{2})_{s}\frac{d}{dU}\left\langle \mathcal {E}^{r-1}\right\rangle _{s}. \end{aligned}$$

(3.50)

Thus a sophisticated form of writing (3.49) in terms of a raising operator is

$$\begin{aligned} \left\langle \mathcal {E}^{r}\right\rangle _{s}=\left( U+(\sigma _{\mathcal {E}}^{2})_{s}\frac{d}{dU}\right) \left\langle \mathcal {E}^{r-1}\right\rangle _{s}, \end{aligned}$$

(3.51)

a result analogous to the previous recurrence Recurrencerelation (3.38), but now including the zero-point fluctuations. A much simpler, alternative form of this relation is

$$\begin{aligned} \left\langle \mathcal {E}^{r}\right\rangle _{s}=rU\left\langle \mathcal {E}^{r-1}\right\rangle _{s}. \end{aligned}$$

(3.52)

3.3.2 Mean Energy as Function of Temperature; Planck’s Formula

We note from Eq. (3.48) that knowledge of the variance $\sigma _{\mathcal {E}}^{2}$ as a function of $U$ is enough to determine $U(\beta )$. Indeed, an integration of this equation—which articulates both thermodynamic and statistical information via Eqs. (3.31) and  (3.46), respectively—

$$\begin{aligned} \frac{dU}{d\beta }=\mathcal {E}_{0}^{2}-U^{2}(\beta ) \end{aligned}$$

(3.53)

gives the function $U(\beta )$. Subject to the condition $U\rightarrow \infty$ as $T\rightarrow \infty$, the result is

$$\begin{aligned} U(\beta )= {\left\{ \begin{array}{ll} \frac{1}{\beta }, &{} \text {for}\ \mathcal {E}_{0}=0; \\ \mathcal {E}_{0}\coth \mathcal {E}_{0}\beta , &{} \text {for}\ \mathcal {E}_{0}\ne 0. \end{array}\right. } \end{aligned}$$

(3.54)

Although the case $\mathcal {E}_{0}=0$ can of course be obtained from the last expression in the limit $\mathcal {E}_{0}\rightarrow 0$, it is more illustrative to treat the two cases separately. As seen from Eq. (3.54), the mean energy as a function of the temperature depends critically on the presence of $\mathcal {E}_{0}$. For $\mathcal {E}_{0}=0$ the classical energy equipartitionEquipartition is recovered,

$$\begin{aligned} U_{\text {cl}}=\beta ^{-1}=k_{B}T, \end{aligned}$$

(3.55)

whereas for $\mathcal {E}_{0}=\hbar \omega /2$ Planck’sBoyer!and Planck law law is obtained,

$$\begin{aligned} U_{\text {Planck}}(\omega ,T)=\tfrac{1}{2}\hbar \omega \coth \tfrac{1}{2} \hbar \omega \beta . \end{aligned}$$

(3.56)

By taking the limit $T\rightarrow 0$, we verify that $U_{\text {Planck}}$ includes the zero-point energy,Oscillator!zero-point energy ¹¹

$$\begin{aligned} U_{\text {Planck}}(\beta \rightarrow \infty )=\tfrac{1}{2}\hbar \omega = \mathcal {E}_{0}. \end{aligned}$$

(3.57)

This establishes Planck’sBoyer!and Planck law law as a physical result whose ultimate meaning—or cause—is the existence of a fluctuating zero-point energy Oscillator!zero-point energyof the field oscillators.

It is important to stress that Planck’sBoyer!and Planck law law has been obtained without the introduction of any explicit quantum or discontinuity requirement. Equation (3.53) results from a thermostatistical analysis of the field modes, based on the properties of $W_{g}$ and $W_{s}$, together with Wien!lawWien’s law, which opens the door to their zero-point energy Oscillator!zero-point energy $A\omega$. This leads us to conclude that Wien’s law with $A\ne 0$ in Eq. (3.18) constitutes an extension of classical physics into the quantum domain—as evidenced by the quantum properties of the harmonic oscillator that ensue from Planck’sBoyer!and Planck law law (see below). Thus, strictly speaking, Wien!lawWien’s law stands as a precursor of Planck’sBoyer!and Planck law, and should be considered historically to contain the first quantum law.

The demonstration that the law that gave rise to quantum theory stems from the existence of a fluctuating zero-point energy, brings to the fore the crucial importance of this nonthermal energy for the understanding of quantum mechanics or, more generally, of quantum theory.

A brief comment on the thermal fluctuations of the energy seems in place before ending this section. We have seen that for $\mathcal {E}_{0}\ne 0$ the thermal energy dispersion is given by

$$\begin{aligned} \sigma _{\mathcal {E}_{T}}^{2}(U)=U^{2}-\mathcal {E}_{0}{}^{2}\quad (U=U_{ \text {Planck}}), \end{aligned}$$

(3.58)

whereas in the classical case ($\mathcal {E}_{0}=0)$,

$$\begin{aligned} \sigma _{\mathcal {E}_{T}}^{2}(U)=U^{2}\quad (U=U_{\text {cl}}). \end{aligned}$$

(3.59)

Whilst in the latter case the thermal fluctuations of theCommutator!and correlation oscillator’s energy depend solely on its (purely) thermal mean energy, $U_{\text {cl}},$ in the former case Eq. (3.58) relates the thermal fluctuations with the total mean energy $U_{\text {Planck}}$, which includes the temperature-independent contribution. The statistical description initiated in Sect. 3.3.1 will be resumed below, in Sect.  3.6.

3.4 Planck, Einstein and the Zero-Point EnergyOscillator!zero-point energy

The previous discussion suggests separating the average energy $U_{\text { Planck}}$ (which as of now will be denoted simply by $U$) into a thermal contribution $U_{T}$ and a temperature-independent part $\mathcal {E}_{0},$

$$\begin{aligned} U=U_{T}+\mathcal {E}_{0}, \end{aligned}$$

(3.60)

so that Eq. (3.58) becomes

$$\begin{aligned} \sigma _{\mathcal {E}}^{2}=U_{T}^{2}+2\mathcal {E}_{0}U_{T}. \end{aligned}$$

(3.61)

The first term in Eq. (3.60)

$$\begin{aligned} U_{T}=\mathcal {E}_{0}\coth \mathcal {E}_{0}\beta -\mathcal {E}_{0}=\frac{2 \mathcal {E}_{0}}{e^{2\mathcal {E}_{0}\beta }-1}, \end{aligned}$$

(3.62)

with $\mathcal {E}_{0}=\hbar \omega /2$, is Planck’sBoyer!and Planck law law without the zero-point energy. At sufficiently low temperatures $U_{T}$ takes the form

$$\begin{aligned} U_{T}(\beta \rightarrow \infty )=2\mathcal {E}_{0}e^{-2\mathcal {E}_{0}\beta }. \end{aligned}$$

(3.63)

This is the (approximate) expression suggested by Wien at the end of the 19th century, and considered for some time to be the exact law for the blackbody spectral distribution. Equations (3.63) and (3.61) represent the germ of quantum theory, since it is precisely on their basis that Planck and EinsteinEinstein, A. advanced the notion of the quantum (for the material oscillatorsOscillator and for the radiation field, respectively). The following pages contain a discussion of their respective points of view and of the relations between these and our present notions based on the reality Realityof the zero-point energyOscillator!zero-point energy. A remarkable relationship will thus be disclosed.

3.4.1 Comments on Planck’s Original Analysis

In his initial studies on the radiation field in equilibrium with matter,Planck, M. Planck (1900a, b) used as point of departure the expression for the derivative of the entropy¹²

$$\begin{aligned} \frac{\partial S}{\partial U}=\frac{1}{T}. \end{aligned}$$

(3.64)

In line with the views and knowledge of his time, Planck recognized only the thermal energy, so $U$ should be replaced here by $U_{T}$. In the high-temperature limit the relation (3.64) led him to write (putting $U_{T}(T\rightarrow \infty )=k_{B}T$)

$$\begin{aligned} \frac{\partial ^{2}S}{\partial U_{T}^{2}}=\frac{\partial }{\partial U_{T}} \left( \frac{k_{B}}{U_{T}}\right) =-\frac{k_{B}}{U_{T}^{2}}. \end{aligned}$$

(3.65)

For low temperatures Planck used Wien’s result (3.63), assuming it to afford an exact description of the properties of the equilibrium field. He thus wrote

$$\begin{aligned} U_{T}=2\mathcal {E}_{0}e^{-2\mathcal {E}_{0}\beta }=2\mathcal {E}_{0}e^{-2\mathcal {E}_{0}/k_{B}T}=2\mathcal {E}_{0}e^{-2\left( \mathcal {E}_{0}/k_{B}\right) \left( \partial S/\partial U_{T}\right) }, \end{aligned}$$

(3.66)

whence

$$\begin{aligned}&\frac{\partial S}{\partial U_{T}}=-\frac{k_{B}}{2\mathcal {E}_{0}}\ln \frac{ U_{T}}{2\mathcal {E}_{0}}, \end{aligned}$$

(3.67a)

$$\begin{aligned}&\quad \frac{\partial ^{2}S}{\partial U_{T}^{2}}=-\frac{k_{B}}{2\mathcal {E}_{0}U_{T}}. \end{aligned}$$

(3.67b)

Not surprisingly, Eqs. (3.65) and (3.67b) give different results, since different temperature regimes were used in each case. As the simplest possibility Planck assumed that the description for arbitrary temperatures could be obtained by interpolating Eqs. (3.65) and (3.67b) and consequently he proposed the relation

$$\begin{aligned} \frac{\partial ^{2}S}{\partial U_{T}^{2}}=-\frac{k_{B}}{U_{T}^{2}+2\mathcal {E}_{0}U_{T}}. \end{aligned}$$

(3.68)

This equation leads directly to Boyer!and Planck lawPlanck’s law without the zero-point term [Eqs. (3.62)], a result that Planck (against his will) interpreted, as is well known, as due to the quantization of the energy exchanged between the material oscillators of the cavity Lifetime!cavity effectsand the equilibrium radiation field.¹³

3.4.2 Einstein’s Revolutionary Step

A few years later, Einstein argued that even though Eq. (3.68) was empirically confirmed (through Planck’s law), its full meaning remained to be clarified. For this purpose EinsteinEinstein, A. chose also to take Eq. (3.64) as a safe point of departure, whence he wrote

$$\begin{aligned} \frac{\partial ^{2}S}{\partial U_{T}^{2}}=\frac{\partial }{\partial U_{T}} \frac{1}{T}=-\frac{1}{T^{2}C_{\omega }}, \end{aligned}$$

(3.69)

or

$$\begin{aligned} k_{B}T^{2}C_{\omega }=-k_{B}\left( \frac{\partial ^{2}S}{\partial U_{T}^{2}} \right) ^{-1}. \end{aligned}$$

(3.70)

Equation (3.70) combined with (3.68) and (3.31) gives

$$\begin{aligned} k_{B}T^{2}C_{\omega }=-\frac{dU_{T}}{d\beta }=\sigma _{\mathcal {E} }^{2}=U_{T}^{2}+2\mathcal {E}_{0}U_{T}, \end{aligned}$$

(3.71)

which is the same as (3.61). Einstein recognized the disagreement between this result and the classical expression $\sigma _{\mathcal {E}}^{2}=U_{T}^{2}$. As is well known, it is here where he made his most—according to him (Rigden J.S. Rigden 2005), his only—revolutionary step in physics. He interpreted the first term on the right-hand side of (3.71) as due to the Fluctuationsfluctuations of the thermal field produced by the Interferenceinterference among its modes of a given frequency. This interpretation follows from considering the limit of (3.71) at high temperatures, at which $U_{T}\gg \mathcal {E}_{0}$ and therefore $\sigma _{\mathcal {E}}^{2}=U_{T}^{2}$, as was predicted by Lorentz on the basis of Maxwell’s equations and is discussed in relation with Eq. (3.43). EinsteinEinstein, A. thus saw in this term a direct manifestation of the wavelike nature of light.

As for the second term in (3.71), which in the context Contextof classical thermodynamics is completely unexpected, the fact that it leads to the quantum theory of Planck led Einstein to interpret it in terms of light quanta (Einstein 1905a, b)Arons, A. B., seeing in the expression $2\mathcal {E}_{0}U_{T}$ a manifestation of discrete properties of the radiation field, as follows. According to Planck, the average energy exchanged between $n$ material oscillators (representing the walls of the cavity) of frequency $\omega$ and the radiation field is $\Delta U=\hbar \omega \langle n\rangle$, and contributes with $\sigma _{\Delta U}^{2}=2\mathcal {E}_{0}\Delta U=\hbar ^{2}\omega ^{2}\langle n\rangle$ to the fluctuations of the field, as follows from (3.71). For Einstein, the linearity of the variance in $\langle n\rangle$ suggested a Poisson distributionDistribution!Poisson of $n$ independent events, each corresponding to an exchange of energy equal to $\hbar \omega =2 \mathcal {E}_{0}$.¹⁴ It is the interpretation by EinsteinEinstein, A. of the linear term as representing a discrete or ‘corpuscular’ contribution, with each corpuscle being an independent packet of energy $\hbar \omega ,$ what gave birth to the notion of the photon (see Vedral V. Vedral 2005 for a simple derivation). It is clear from Eq. (3.71) that the discrete structure Structureof the field will manifest itself only at very low temperatures, when the linear term dominates over the quadratic, wavelike one. However, it is important to stress, as Einstein did as of 1909, that the two terms coexist at all temperatures, and thus, both particle and wave manifestations of light coexist at all temperatures (Einstein 1909). This observation is sometimes ignored to argue that they are mutually exclusive, although there exist both theoretical arguments and experiments that demonstrate the possible coexistence of the two aspects of the behaviour of light.¹⁵

3.4.3 Disclosing the Zero-Point Field

It is important to note that no zero-point energy was considered by either Planck or EinsteinEinstein, A. in their analysis of Eqs. (3.68) and (3.71), respectively. Instead, as stated above, Planck interpreted the term $2 \mathcal {E}_{0}U_{T}$ in Eq. (3.68) as a result of the discontinuities in the processess of energy exchange between matter and field (more specifically in the emissions, as of 1912). Einstein in his turn saw in $2\mathcal {E}_{0}U_{T}$ a manifestation of the corpuscular nature of the field, and thus pointed to it as the key to Planck’sBoyer!and Planck law law. Now, from the point of view proposed here the consideration of the zero-point energy Oscillator!zero-point energygives rise to a third understanding of Eq. (3.71) that does not depend on the notion of quanta. The elucidation of $U_{T}^{2}$ as the result of the Interferenceinterference of the modes of frequency $\omega$ of the thermal field suggests to interpret $2\mathcal {E}_{0}U_{T}$ as due to additional interferences, now between the thermal field and a zero-point radiation field of mean energy $\mathcal {E}_{0}$ (per mode of frequency $\omega$) that is present at all temperatures. As is by now clear, Eq. (3.71) lacks the extra term $\mathcal {E}_{0}^{2}$ representing the nonthermal fluctuations, just because the thermodynamic description has no room for them; this shortcoming has been overcome with the introduction of the distributionDistribution $W_{s}$, Eq. (3.41).

From this new perspective the notion of intrinsic discontinuities in the energy exchange or in the field itself is unnecessary to explain either Planck’sBoyer!and Planck law law or the linear term in Eq. (3.71); it is the existence of a (fluctuating) zero-point radiation field (zpf)Planck distribution!zpf-field derivation what accounts for that law. This could of course not be Planck’s or Einstein’s interpretation because the zero-point energy (and more so the zero-point field) was still unknown at that time, even though their results were consistent with its existence.

The concept of a zero-point energy Oscillator!zero-point energyof the radiation field appeared for the first time in 1912, in a work where Planck attempted another derivation of his law, motivated by his well-known uneasiness with the idea of introducing discontinuities in our theoretical descriptions Planck, M.(Planck 1912). Some time thereafter EinsteinEinstein, A. and Stern Stern O. (1913) used the idea of a zero-point energy, although applied to molecules, i.e., to mechanical oscillators. Unfortunately the authors were obliged to use the (incorrect) value $\hbar \omega$ for this energy; this along with other difficulties led Einstein to abandon such line of research.¹⁶ Shortly thereafter the notion of a zero-point field was bornBorn, M. anew, when Nernst Nernst, W.made his visionary proposal (Nernst 1916), as briefly mentioned in the preface.

3.5 Continuous Versus Discrete

We have just seen how three alternative approaches provide three quite different readings of the same quantity, $U_{T}^{2}+2\mathcal {E}_{0}U_{T}$. In these approaches, either the zero-point energy Oscillator!zero-point energy(of a continuous field) or the energy quantization is identified as the notion underlying the Planck spectral energy distribution. Therefore the next logical step is to inquire about the relation between the zero-point energy and quantization. Is quantization inevitably linked to Planck’sBoyer!and Planck law law, or is it merely the result of a point of view, of a voluntary but dispensable choice?

3.5.1 The Partition Function

An answer to the above question is found from an analysis of the partition function obtained from (3.54). As follows from Eq. (3.27), $Z_{g}(\beta )$ can be determined by direct integration of

$$\begin{aligned} U=-\frac{d\ln Z_{g}(\beta )}{d\beta }, \end{aligned}$$

(3.72)

with $U(\beta )$ given by the second of Eq. (3.54). The result is

$$\begin{aligned} Z_{g}=\frac{C}{\sin h \mathcal {E}_{0}\beta }, \end{aligned}$$

(3.73)

where $C$ is a numerical constant whose value is determined by requiring the classical result $Z_{g}=\left( s\beta \omega \right) ^{-1}$ [Eq. (3.25b)] to be recovered in the limit $T\rightarrow \infty$. This leads to $C=\mathcal {E}_{0}/s\omega =\hbar /2s,$ so that

$$\begin{aligned} Z_{g}(\beta )=\frac{\mathcal {E}_{0}}{s\omega \sinh \mathcal {E}_{0}\beta }. \end{aligned}$$

(3.74)

On the other hand, from Eqs. (3.15) and (3.72) the thermodynamic potential $\phi$ can be written in the form

$$\begin{aligned} \phi =\ln Z_{g}. \end{aligned}$$

(3.75)

This along with Eq. (3.17) gives for the entropy (up to an additive constant, and writing $S=S_{g}$)

$$\begin{aligned}&\displaystyle _{g} =k_{B}\ln Z_{g}+\frac{U}{T} \nonumber \\ \qquad&=k_{B}\ln \frac{\hbar }{s}-k_{B}\ln (2\sinh \mathcal {E}_{0}\beta )+k_{B}\beta U, \end{aligned}$$

(3.76)

which in the zero-temperature limit reduces to

$$\begin{aligned} S_{g}(\beta \rightarrow \infty )=k_{B}\ln \frac{\hbar }{s}. \end{aligned}$$

(3.77)

To set the origin of the Entropyentropy at $T=0$ one must take $s=\hbar ,$ ¹⁷ hence the partition function takes the form

$$\begin{aligned} Z_{g}(\beta )=\frac{1}{2\sin h \mathcal {E}_{0}\beta }. \end{aligned}$$

(3.78)

3.5.2 The Origin of Discreteness

Once we have determined the partition function $Z_{g}$ we are in position to discuss the discontinuities characteristic of the quantum theory, which are hidden in the continuous description given by the distribution $W_{g}$. To this end we expand Eq. (3.78) and write (see Santos 1975; Theimer O. Theimer 1976; Landsberg, P. T. Landsberg 1981 for related discussions)

$$\begin{aligned} Z_{g}=\frac{1}{2\sinh \mathcal {E}_{0}\beta }=\frac{e^{-\beta \mathcal {E}_{0}} }{1-e^{-2\beta \mathcal {E}_{0}}}=\sum _{n=0}^{\infty }e^{-\beta \mathcal {E} _{0}(2n+1)}=\sum _{n=0}^{\infty }e^{-\beta \mathcal {E}_{n}}, \end{aligned}$$

(3.79)

where

$$\begin{aligned} \mathcal {E}_{n}\equiv (2n+1)\mathcal {E}_{0}=\hbar \omega n+\tfrac{1}{2}\hbar \omega . \end{aligned}$$

(3.80)

Equation (3.79) allows now the determination of the function $g(\mathcal {E})$ by means of (3.21b),

$$\begin{aligned} Z_{g}(\beta )=\!\int \!g(\mathcal {E})e^{-\beta \mathcal {E}}d\mathcal {E} =\sum _{n=0}^{\infty }e^{-\beta \mathcal {E}_{n}}=\int \nolimits _{0}^{\infty }\sum _{n=0}^{\infty }\delta (\mathcal {E-E}_{n})e^{-\beta \mathcal {E}}d \mathcal {E}, \end{aligned}$$

(3.81)

whence

$$\begin{aligned} g(\mathcal {E})=\sum _{n=0}^{\infty }\delta (\mathcal {E}-\mathcal {E}_{n}). \end{aligned}$$

(3.82)

The substitution of (3.82) into Eq. (3.21a) finally determines the probability density $W_{g}(\mathcal {E}),$

$$\begin{aligned} W_{g}(\mathcal {E})=\frac{1}{Z_{g}}\sum _{n=0}^{\infty }\delta (\mathcal {E}- \mathcal {E}_{n})e^{-\beta \mathcal {E}}. \end{aligned}$$

(3.83)

This distribution gives for the mean value of any function $f(\mathcal {E})$

$$\begin{aligned} \langle f(\mathcal {E})\rangle =\!\int \!W_{g}(\mathcal {E})f(\mathcal {E})d \mathcal {E}=\frac{1}{Z_{g}}\sum _{n=0}^{\infty }f(\mathcal {E}_{n})e^{-\beta \mathcal {E}_{n}}=\sum _{n=0}^{\infty }w_{n}f(\mathcal {E}_{n}), \end{aligned}$$

(3.84)

with the weights $w_{n}$ given by

$$\begin{aligned} w_{n}=\frac{e^{-\beta \mathcal {E}_{n}}}{Z_{g}}=\frac{e^{-\beta \mathcal {E} _{n}}}{\sum _{n=0}^{\infty }e^{-\beta \mathcal {E}_{n}}}. \end{aligned}$$

(3.85)

The final form of $W_{g}(\mathcal {E}),$ Eq. (3.83), identifies $\left\{ \mathcal {E}_{n}=\hbar \omega (n+1/2)\right\}$ with the set of discrete energy levels accesible to the oscillatorsOscillator. Such discreteness, seemingly excluding all other values of the energy, is due to the highly pathological distribution $g(\mathcal {E}),$ Eq. (3.82). As a result, (3.84) shows that the mean value of a function of the continuous variable $\mathcal {E}$ calculated with the distribution $W_{g}(\mathcal {E}),$ can be obtained equivalently by averaging over the set of discrete indices (or states) $n,$ with respective weights $w_{n}.$ Thus, although both averages are formally equivalent, their descriptions are essentially different: one refers to the continuous energy $\mathcal {E}$, the other one to discrete states (levels) with energy $\mathcal {E}_{n}$. As this latter is completely characterized by the state $n$, it is natural to interpret the last equality in Eq. (3.84) as a manifestation of the discrete (quantized) nature of the energy. Indeed, the last equality in Eq. (3.84) can be recognized as the description afforded by theDensity matrix density matrix for a canonical Distribution!canonicalensembleEnsemble!canonical of quantum oscillators at temperature $T$, with the weights $w_{n}$ given by (3.85) (see e.g. Cohen-Tannoudji et al. 1977).

The above discussion points to the fundamental role played by the zero-point energy in explaining quantization, by putting it at the root of Eq. (3.79) and hence of Eq. (3.82). From the present point of view, and contrary to the usual credo, the radiation field is not intrinsically quantized, but it becomes so when attaining equilibrium through its interaction with matter. In other words, quantization is here exhibited as an emergent property of matter and field in interaction, an idea that is closely examined from several angles in the following chapters, becoming thus the leitmotiv of the book.

3.6 A Quantum Statistical Distribution

The thermostatistical analysis of a canonical Distribution!canonicalensembleEnsemble!canonical of oscillators has led to the conclusion that although $\mathcal {E}$ is a continuous variable, its equilibrium distribution possesses extremely peaked values. In other words, the energies that conform to the thermal equilibrium state described by the distributionDistribution $W_{g}$ belong, roughly speaking, to a discrete spectrum. This explains why the mean value $\langle f(\mathcal {E})\rangle$, which corresponds to an equilibrium state, involves only the discrete set $\mathcal {E}_{n}$. However, the energy still fluctuates and in doing so tends to fill the interspaces between its discrete values.¹⁸ Thus we find that temperature-independent fluctuations appear as a characteristic trait of quantum systems. A closer study of this property allows to establish contact with one of the most frequently used distributions in quantum statisticsQuantum statistics.

3.6.1 Total Energy Fluctuations

The appropriate statistical distribution that includes all (thermal as well as nonthermal) fluctuations is given by Eq.  (3.41),

$$\begin{aligned} W_{s}(\mathcal {E})=\frac{1}{U}\,e^{-\mathcal {E}/U}, \end{aligned}$$

(3.86)

and the variance of the energy at all temperatures (including $T=0$) is $(\sigma _{\mathcal {E}}^{2})_{s}=U^{2}.$ Using the decomposition (3.60) we may write for the total energy fluctuations

$$\begin{aligned} (\sigma _{\mathcal {E}}^{2})_{s}=U^{2}=(U_{T}+\mathcal {E}_{0})^{2}=U_{T}^{2}+2 \mathcal {E}_{0}U_{T}+\mathcal {E}_{0}^{2}. \end{aligned}$$

(3.87)

This result generalizes Eq. (3.59) to include both thermal and nonthermal energy fluctuations. In conformity with the present discussion, the total energy can be written in terms of its thermal and nonthermal fluctuating parts,

$$\begin{aligned} \mathcal {E}=\mathcal {E}_{T}+\mathcal {E}_{0}. \end{aligned}$$

(3.88)

The total energy fluctuations are then given by

$$\begin{aligned} (\sigma _{\mathcal {E}}^{2})_{s}=\sigma _{\mathcal {E}_{T}}^{2}+\sigma _{\mathcal {E}_{0}}^{2}+2\Gamma (\mathcal {E}_{T},\mathcal {E}_{0}), \end{aligned}$$

(3.89)

where $\Gamma (\mathcal {E}_{T},\mathcal {E}_{0})$ is the covarianceCovariance

$$\begin{aligned} \Gamma (\mathcal {E}_{T},\mathcal {E}_{0})=\left\langle \mathcal {E}_{T} \mathcal {E}_{0}\right\rangle -\langle \mathcal {E}_{T}\rangle \,\langle \mathcal {E}_{0}\rangle . \end{aligned}$$

(3.90)

Comparing Eqs. (3.89) and (3.87), and identifying the temperature-dependent part of the fluctuations of the whole field $U_{T}^{2}+2\mathcal {E}_{0}U_{T}$ with $\sigma _{\mathcal {E}_{T}}^{2}$ and $\mathcal {E}_{0}^{2}$ with $\sigma _{\mathcal {E}_{0}}^{2}$, we verify that $\Gamma (\mathcal {E}_{T},\mathcal {E}_{0})=0,$ as was expected considering that the fluctuations of $\mathcal {E}_{T}$ and $\mathcal {E}_{0}$ are statistically independent, due to the independence of their sources.

The entropy $S_{s}$ follows from Eqs. (3.44) and (3.86),

$$\begin{aligned} S_{s}=-k_{B}\int W_{s}(\mathcal {E})\ln c_{s}W_{s}(\mathcal {E})d\mathcal {E} =k_{B}\ln c_{s}^{-1}U+k_{B}, \end{aligned}$$

(3.91)

whence

$$\begin{aligned} \frac{\partial S_{s}}{\partial U}=\frac{k_{B}}{U}. \end{aligned}$$

(3.92)

A comparison with the thermodynamic entropy, which satisfies

$$\begin{aligned} \frac{\partial S_{g}}{\partial U}=\frac{1}{T}, \end{aligned}$$

(3.93)

shows that these two entropies coincide only when $\mathcal {E}_{0}=0,$ i.e., for $U=k_{B}T$.

3.6.2 Quantum Fluctuations and Zero-Point Fluctuations

Let us now investigate how the nonthermal fluctuations become manifest in the statistical properties of the ensemble of oscillators. The value of the energy of the harmonic oscillator [(cf. Eq. (3.1)]

$$\begin{aligned} \mathcal {E}=(p^{2}+\omega ^{2}q^{2})/2 \end{aligned}$$

(3.94)

can be used as a starting point to perform a transformation from the energy distribution $W_{s}(\mathcal {E})$ to a distribution $w_{s}(p,q)$ defined in the oscillator’sOscillator phase space $(p,q)$. To this end we introduce the pair of variables $(\mathcal {E},\theta )$ related to the couple $(p,q)$ by¹⁹

$$\begin{aligned} p&=\sqrt{2\mathcal {E}}\cos \theta , \end{aligned}$$

(3.95a)

$$\begin{aligned} q&=\sqrt{\frac{2\mathcal {E}}{\omega ^{2}}}\sin \theta , \end{aligned}$$

(3.95b)

so that $w_{s}(p,q)$ is given by (Papoulis, A. Papoulis 1991; Birnbaum 1961)Birnbaum, Z. W. ²⁰

$$\begin{aligned} w_{s}(p,q)=W_{s}(\mathcal {E}(p,q),\theta (p,q))\left| \frac{\partial (\mathcal {E},\theta )}{\partial (p,q)}\right| , \end{aligned}$$

(3.96)

with the Jacobian of the transformation

$$\begin{aligned} \frac{\partial (p,q)}{\partial (\mathcal {E},\theta )}=\left| \frac{ \partial (\mathcal {E},\theta )}{\partial (p,q)}\right| ^{-1}=\frac{1}{ \omega }. \end{aligned}$$

(3.97)

Now, $W_{s}(\mathcal {E})$ is a marginal probability density that can be obtained from $W_{s}(\mathcal {E},\theta )$ by integrating over the variable $\theta$, so that

$$\begin{aligned} W_{s}(\mathcal {E})=\!\int \nolimits _{0}^{2\pi }W_{s}(\mathcal {E},\theta )d\theta . \end{aligned}$$

(3.98)

For a system of harmonic oscillators in equilibrium, the trajectories (in general, the surfaces) of constant energy do not depend on $\theta$, so all values of $\theta$ are equally probable, which means that

$$\begin{aligned} W_{s}(\mathcal {E},\theta )=\frac{1}{2\pi }W_{s}(\mathcal {E}). \end{aligned}$$

(3.99)

Using Eqs. (3.86), (3.94) and (3.96) we thus obtain for the distributionDistribution in phase space:

$$\begin{aligned} w_{s}(p,q)=\frac{\omega }{2\pi }W_{s}(\mathcal {E(}p,q))=\frac{\omega }{2\pi U}\exp \Big (-\frac{p^{2}+\omega ^{2}q^{2}}{2U}\Big ). \end{aligned}$$

(3.100)

This expression, which is known in quantum theory as the Harmonic oscillator!Wigner functionWigner Quantum distributions!Wignerfunction for Oscillator!Wigner function Wigner functionthe harmonic oscillators Hillery, M.(Hillery et al. 1984), canO’Connell, R. F. be factorized as a product of two Distribution!normalnormal distributions,

$$\begin{aligned} w_{s}(p,q)=w(p)w(q)=\frac{1}{\sqrt{2\pi \sigma _{p}^{2}}}e^{-p^{2}/2\sigma _{p}^{2}}\cdot \frac{1}{\sqrt{2\pi \sigma _{q}^{2}}}e^{-q^{2}/2\sigma _{q}^{2}}, \end{aligned}$$

(3.101)

where $\sigma _{p}^{2}=$ $U$ and $\sigma _{q}^{2}=U/\omega ^{2}$. The product of these dispersions gives

$$\begin{aligned} \sigma _{q}^{2}\sigma _{p}^{2}=\frac{U^{2}}{\omega ^{2}}=\frac{\mathcal {E} _{0}^{2}}{\omega ^{2}}+\frac{\sigma _{\mathcal {E}_{T}}^{2}}{\omega ^{2}}\ge \frac{\mathcal {E}_{0}^{2}}{\omega ^{2}}=\frac{\hbar ^{2}}{4}, \end{aligned}$$

(3.102)

where Eq. (3.58) was used to write the second equality and the value $\mathcal {E}_{0}=\hbar \omega /2$ was introduced into the last one.

Equation (3.102) points to the fluctuating zero-point energy Oscillator!zero-point energyas the ultimate (and irreducible) source of the so-called quantum fluctuations. Indeed, the magnitude of $\sigma _{q}^{2}\sigma _{p}^{2}$ is bounded from below because of the nonthermal energy fluctuations; the Fluctuations!minimum ofminimum value $\hbar ^{2}/4$ is reached when all thermal fluctuations have been suppressed, which means $T=0$. Therefore, descriptions afforded by purely thermal distributions such as $W_{g}$ cannot account for the meaning of these inequalities. This result stresses again the fact that once a zero-point energy has been introduced into the theory, new distributions (specifically statistical rather than thermodynamic) are needed to include its fluctuations and toCommutator!and correlation obtain the corresponding quantum statistical properties. Though here we have arrived at the Heisenberg inequality (3.102) by considering a system of harmonic oscillators, later on (particularly in Chap. 5) we will derive it for an arbitrary system, and again the presence of the zpf Correlations!and zpf will turn out to be decisive in reaching the result. Finally, note that the Heisenberg inequalities should be understood as referring to statistical variances, due to the statistical nature of (3.102).

3.6.3 Comments on the Reality of the Zero-Point Fluctuations

As mentioned earlier, the concept of a zero-point energy Oscillator!zero-point energyof the radiation field entered into scene as early as 1912, with Planck’s second derivation of the blackbody spectrum. Yet further to the frustrated attempt by EinsteinEinstein, A. and Stern Stern O. (1913), and despite the suggestive proposal made byNernst, W. Nernst (1916) to consider the zpf Coherence!zpf modes as responsible for atomic stability, little or no attention was paid to its existence as a real physical entity that could have a role in the newly developing quantum mechanics.²¹ Interestingly, it was the crystallographers who, prompted by Debye’s theoretical work, set out to measure the spectroscopic effects of the zero-point energy through X-rayX rays analysis and thereby seemengly verified its existence James, R. W. Waller, T. Wollan E. O. Hartree, D. R.(James et al. 1928; Wollan 1931).

As mentioned in Sect. 1.4.1, today it is well accepted that the Fluctuationsfluctuations of the electromagnetic vacuum are responsible for important observable physical phenomena. Perhaps their best known manifestations, within the atomic domain, are the Lamb shift Cavity effects!on the Lamb shiftof energy levels Milonni, P. W.(see e.g. Milonni 1994) and their contribution to the spontaneous transitions of the excited states to the ground stateOscillator!ground state. They are known to contribute one half of the Einstein $A$-coefficientEinstein!A coefficient for ‘spontaneous’ transitions, the other half being due to radiation reaction Radiation reaction Davydov, A.S.(see e.g. Milonni 1994; Davydov 1965). ²² A-B coefficients By far the most accepted evidence of the reality Realityof the zpf is the CasimirCasimir, H. B. G. effect, that is, the force between two parallel neutral metallic plates resulting from the modification of the field by the boundaries Boyer, T.H. Bordag, M. Mostepanenko, V. M. Mohideen, U.(see e.g. Boyer 1970; Bordag et al. 2009). The existence of the zpf vcan therefore be considered a reasonably well established physical fact.²³ In the following chapters we will have occasion to study in depth the essential role played more broadly by this random field in its interaction with matter at the atomic level.