From the hypotheses of realism and stationarity, the latter related to the Leggett-Garg assumption of noninvasive measurabiLity, we derive inequalities, involving three autocorrelation functions, to be fulfilled by any two-state stochastic process. We seek to test a broad class of realist theories against quantum mechanics for a system consisting of a Rydberg atom interacting with a single quantized mode in a superconducting resonant cavity. Departures from quantum predictions are enhanced when the temperature is decreased to about 0.5 K and the state of the cavity approximates a Fock state. PACS number(s): 03.65.8z Recent advances in optics have renewed interest in demonstrating the strange properties of quantum mechanics; for instance, those exemplified by the old Einstein-Podolsky-Rosen (EPR) [1] and Schrodinger's cat [2] paradoxes. Bell was able to refine the EPR paradox introducing (Bell) inequalities able to discriminate between quantum mechanics and local realism [3 -5]. Following his work, many empirical

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We continue the analysis of our previous articles which were devoted to type-I parametric down conversion, the extension to type-II being straightforward. We show that entanglement, in the Wigner representation, is just a correlation that involves both signals and vacuum fluctuations. An analysis of the detection process opens the way to a complete description of parametric down conversion in terms of pure Maxwell electromagnetic waves.The theory of parametric down conversion (PDC) was treated, in the Wigner formalism, in an earlier series of articles [1,2,3]. There we showed that, provided one considers the zeropoint fluctuations of the vacuum to be real, the description of radiation is fully within Maxwell electromagnetic theory. Effectively, because the Wigner function maintains its positivity, we can say that quantization is just the addition of a zeropoint radiation, and there is no need for any further quantization of the light field. In the present article we show that the same result extends, without any difficulty, from the type-I PDC case to the type-II situation.There seems to be a widespread reluctance to accept the reality of the vacuum fluctuations, in spite of the fact that they appear, quite naturally, in the Wigner function of the vacuum state. We remark that such fluctuations have been taken seriously, within a certain school of thought, throughout the entire history of the quantum theory, following the formulation of Max Planck, originating in 1911 [4]. Of course, it is true that, integrated over all frequencies, they give us a vacuum with infinite energy density; why then are all photographic plates not blackened instantaneously? But all photodetectors, including even our own eyes, are very selective, not only as regards the frequency, but even also the wave vectors, of the light components they analyze. This is especially the case with the detectors commonly used in PDC experiments. So, there is a noise to subtract, but it is not infinite! In our previous articles we indicated how the noise subtraction is made, according to the Wigner formalism, and showed how this subtraction is related to the standard calculating device, of normal ordering, used in the Hilbert-space formalism. Here we extend this analysis, in an informal manner, showing that, if we take into account the fact that all detectors integrate the light intensity over a large time window, the process of light detection, like that of light propagation, may also be described entirely in terms of real waves and positive probabilities. We are then able to see that, in terms of a purely wave description, the highly problematic concept of "entangledphoton" states of the field loses all its mystery. Entangled photons are just correlated waves! The only reason this description has taken so long to mature is that the word "classical", in reference to the light field, is restricted in its application to Glauber-classical states [5]. A discussion of the difference between classical and nonclassical effects has been given in Ref. [6]. The

The random pure radiation field postulated in an earlier paper is set up in a relativistically invariant manner. The requirement that this field be isotropic in three dimensions makes much of the formalism identical with the theory of isotropic turbulence, as has been noted by a previous author. It is found, however, that the more stringent requirement of invariance under the Lorentz group, together with the fact that the field components satisfy Maxwell's equations, mean that, to within a multiplicative constant, the entire random process is uniquely specified.In accordance with results obtained previously, the constant is effectively identified as Planck's constant. There then results a striking resemblance between the formalism of the present theory, and that of the quantized pure radiation field, with random variables taking the place of operators, and correlations between these random variables taking the place of commutator brackets.The infinite field energy, which arises also in quantum field theory, where, however, it is usually considered to be purely formal, remains a central difficulty of the random theory. It can be shown to be a natural consequence of assuming Maxwell's equations, and it therefore becomes necessary to consider non-linear modifications of these.The connexion between the present results and the inconsistencies of classical and quantum electrodynamics is discussed. Also some new information is obtained concerning the behaviour of thermodynamic quantities under a Lorentz transformation.1. Introduction. In a previous paper ( (7)), it was shown that, in the presence of a random electromagnetic field, certain charged classical systems, namely, the free particle and the harmonic oscillator, behave essentially like their quantum-mechanical counterparts. The energy spectrum of the field needed to produce this behaviour was obtained, in the non-relativistic limit, and was found to be the familiar Planck spectrum superposed on a certain fixed spectrum. The latter was therefore identified as a zero-temperature spectrum, and it was suggested that such a random radiation field might be capable of explaining other quantum effects in material systems.In the present paper we investigate the requirements which special relativity places on such a random theory. Attention is confined to the pure radiation field, and the question of how non-linear material systems behave in such a field is left for a later paper.As in quantum electrodynamics, we find that a crucial decision has to be made, namely, 'What are the quantities which are observable?' It is necessary to frame