The shadow of light: non-Lorentzian behavior of photon systems

Physics Letters A 326 (2004) 1–13 www.elsevier.com/locate/pla

The shadow of light: non-Lorentzian behavior of photon systems F. Cardone a,b,c , R. Mignani c,d,e,∗ , W. Perconti f , R. Scrimaglio f a Istituto per lo Studio dei Materiali Nanostrutturati (ISMN-CNR), via dei Taurini 6, 00185 Roma, Italy b Istituto di Radiologia, Università di Roma “La Sapienza”, Roma, Italy c INOA, L.go E. Fermi 6, 50125 Firenze, Italy d Dipartimento di Fisica “E. Amaldi”, Università degli Studi “Roma Tre”, via della Vasca Navale 84, 00146 Roma, Italy e INFN, Sezione di Roma III, Roma, Italy f Dipartimento di Fisica, Università degli Studi de L’Aquila, via Vetoio 1, 67010 Coppito L’Aquila, Italy

Received 28 January 2004; accepted 11 April 2004 Available online 28 April 2004 Communicated by V.M. Agranovich

Abstract We discuss the theoretical foundations and the results of a double slit diffraction-like experiment in the infrared range, aimed at finding departures from the classical predictions. We found indeed an anomalous behavior of such a photon system. Possible interpretations can be given in terms of either the existence of de Broglie–Bohm hollow waves associated to photons, and/or a breakdown of local Lorentz invariance (LLI). The findings of the present experiment do agree with the threshold behavior in energy and space, recently derived (on an experimental basis) for the LLI breaking effect. This leads us to put forward the intriguing hypothesis that the hollow wave is a deformation of the space–time (Minkowskian) geometry. Our experimental findings have been tested in crossed photon-beam experiments, whose preliminary results evidence an anomalous behavior in the photon–photon cross section and confirm the threshold behavior in energy and space of the effect we observed.  2004 Elsevier B.V. All rights reserved.

1. Introduction It is well known the basic role played by optical experiments in testing the very foundations of physics. It is enough to quote, e.g., the Michelson–Morley interference experiment, as a fundamental test of Special Relativity, and the Davisson–Germer diffraction of electrons by a crystal lattice, which checked the wave nature of particles. * Corresponding author.

E-mail address: [email protected] (R. Mignani). 0375-9601/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.04.027

In this Letter, we want just to show that photon experiments may still be crucial in testing two fundamental issues of contemporary physics: the real nature of the quantum wave in Quantum Mechanics, and the possible breakdown of local Lorentz invariance in relativity. Moreover, we shall also put forward the hypothesis of a possible connection between these two seemingly unrelated questions. 1.1. Wave–particle duality A fundamental issue within Quantum Mechanics (QM) is the wave–corpuscle duality, namely—as

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is well known—the different nature shown by subatomic particles and photons, according to the physical process involved. Actually, in spite of the well-established fact of the dual nature of quantum objects, this is far from being completely understood. Let us clarify in what sense the wave–particle duality is still a largely open issue. The main problem rests with the interpretation of the wave associated to the quantum object, in particular to a particle, on account, for instance, of the absence of any medium able to transmit the wave oscillations. The most widely accepted interpretation is, as is well known, the Copenhagen one: the quantum wave associated to a particle is a probability wave, and, as such, devoid of any physical attribute. Such a point of view (essentially due to Bohr) is in clear contrast with that advocated by Einstein, de Broglie and later by Bohm [1]. They proposed that particles are such at all times (not just when they are observed). However, their behavior is determined by the quantum wave, acting as a field that consists of classical forces (like electromagnetism) and an entirely new force (Bohm’s quantum potential), responsible for the truly non-classical effects.1 In this framework, therefore, the quantum wave of an electron does possess a real nature, on the same footing of the electromagnetic wave associated to a photon. Such a quantum wave, however, does not bear either energy or momentum. In this sense, it is to be regarded as a hollow (or ghost) wave (de Broglie and Bohm [1] used also the term “pilot wave”, since it accompanies the particle which, however, is the only physical carrier of observable quantities). Actually, all the experiments which evidence the wave–corpuscle duality require, in order to be correctly interpreted, the theoretical hypothesis of the objective reality of the quantum wave, but they do not provide any direct observation of the hollow wave. However, a way to detect such quantum waves might be through their affecting the probabilities of events to which they superimpose in space–time (for instance, in interference phenomena). This is essentially due (according to de Broglie and Andrade y Silva [2]) to the interaction of quantum objects with

1 The positions of particles in turn serve as hidden variables,

which determine the nature of the quantum wave.

all the pilot waves present in a given space region, through their quantum potential [1]. 1.2. Local Lorentz invariance The fundamental teaching of Einstein’s relativity theories is that physical phenomena occur in four dimensions (three spatial and one time dimension), space–time possessing a global curved (Riemannian) structure and a local flat (Minkowskian) one. However, it is an old-debated problem whether local Lorentz invariance (LLI) preserves its validity at any length or energy scale (far enough from the Planck scale, when quantum fluctuations are expected to come into play). From the experimental side, the main tests of LLI can be roughly divided in three groups [3]: (a) Michelson–Morley (MM)-type experiments, aimed at testing the isotropy of the round-trip speed of light; (b) Tests of the isotropy of the one-way speed of light (based on atomic spectroscopy and atomic timekeeping); (c) Hughes–Drever-type (HD) experiments, testing the isotropy of nuclear energy levels. All such experiments set upper limits on the degree of violation of LLI. From the theoretical side, a lot of generalizations of Special Relativity and/or LLI breaking mechanisms exist in literature. For instance, in order to take account of the LLI breaking effects, an extension of the Standard Model has been proposed by Kostelecky [4], who puts very stringent limits by analyzing the existing data. A possible signature for a breakdown of LLI in electromagnetic interactions was put forward by two of the present authors (F.C. and R.M.), in the framework of a generalization of SR based on a “deformation” of space–time, assumed to be endowed with a metric whose coefficients depend on the energy of the process considered [5]. Such formalism (Deformed Special Relativity, DSR)2 was in particular applied to 2 DSR allows one to parameterize the breakdown of LLI as a departure from the Minkowski metric, and to provide a (local) metric description of all four fundamental interactions. See Ref. [5].

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Fig. 1. Values of the lower limits of the LLI breakdown parameter δ in the isotropic and the anisotropic case.

analyzing the Cologne experiment on the superluminal photon tunneling in undersized waveguides [6]. According to this analysis, the breakdown of local Lorentz invariance for electromagnetic (e.m.) interactions has an energy threshold E0,e.m. ∼ = 4.5 µV [5,7] (in the sense that LLI is expected to be broken under such an energy value). Such a (phenomenological) result on the energy threshold behavior of the LLI breaking in electromagnetic interactions was then confirmed experimentally. The effect observed is essentially a non-zero force between a charge and a coil carrying a steady current,

both at rest in the laboratory frame [8]. The electromagnetic contribution to this effect is just of the order of E0,e.m. evaluated by the analysis of the Cologne experiment. The value of the (isotropic) LLI breakdown parameter δ (defined as δ ≡ [(u2 /c2 ) − 1], where c is the light speed in vacuum and u the maximal causal velocity, i.e., the limiting speed of test particles) [3], corresponding to the threshold energy of 4.5 µV is δ = 3.9 × 10−11 [8]. The existing lower limits of δ for the electromagnetic interaction both in the isotropic case and in the anisotropic one are shown in Fig. 1.

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Therefore, given two e.m. systems of the same kind, if one of them behaves upholding LLI, whereas the other exhibits an anomalous behavior, one can state that the anomalous behavior of the latter is a possible signature of LLI breakdown if the measured energies of the two systems differ by an amount of the order of (or less than) the electromagnetic energy threshold E0,e.m. . The formalism of DSR permitted to give a unified description [9] of the experiments of superluminal propagation, carried out at Cologne in wave-guides and at Florence in air between two horn antennas [10], in terms of an under-barrier tunneling. We shall come back to this topic in more detail later on. At present, we want to stress that such an analysis of the Cologne and Florence experiments evidences a further threshold behavior. Namely, the superluminality effect is confined within a space threshold 0 of about 9 cm. Since superluminality is widely accepted as a manifestation of a departure from standard Special Relativity,3 this result is in favor of a space threshold, too, for LLI breakdown (at least at the energies of the e.m. systems involved in those experiments, namely, in the microwave range). We can therefore conclude that the phenomenon of LLI breaking has threshold behavior both in energy and in space. The threshold values E0,e.m. and 0 represent, respectively, the maximum energy difference giving rise to the breakdown effect and its maximum size. 1.3. The shadow of light: hollow wave as space–time deformation We are now ready to put forward the foreseen hypothesis, which establishes a connection between the quantum wave (in the interpretation by de Broglie and Bohm, namely as a hollow or pilot wave) and the breakdown of local Lorentz invariance, described by the formalism of Deformed Special Relativity (i.e., in terms of a modified Minkowski metric). Namely, we state that the hollow wave is nothing but a deformation of space–time geometry, related

to the motion of a quantum object. In this sense, it does carry neither energy nor momentum, but it can affect the occurrence of other phenomena. Moreover, we recover, in this framework, the view by Einstein on the aether as a geometrical entity. Indeed, in the early years of General Relativity, Einstein abandoned the idea of the aether as a material medium, and rather conceived it as a substratum without mechanical and kinematical properties, but able to codetermine mechanical and electromagnetic events. In a modern view, such an Einsteinian concept of aether is nothing but a static gravitational field (locally the Minkowski space–time) [12]. Our hypothesis establishes therefore a link between the breakdown of LLI, related to the superluminal propagation of the waves (observed, e.g., in the Cologne and Florence experiments), and the pilot wave. Of course, such a connection is suggested, among the others, by the well-known non-local features of any quantum theory (as proved by the famous Einstein–Podolski–Rosen effect, experimentally checked by Aspect [13] and then by Alley [14]). Nonlocality is, in turn, strongly interrelated to superluminality. If our working hypothesis is true, then one must be able to detect the quantum wave by experiments designed according to the (energy and space) threshold behavior of the LLI breakdown. We will show that this is indeed the case. Since we shall check our hypothesis for photon systems, we shall use a metaphoric image. The hollow wave of a photon, conceived as a departure of the (local) space–time geometry from the Minkowskian one, is in a sense the shadow of the photon. Like a shadow, it is immaterial (since it does carry neither energy nor momentum), and can find itself in space regions far from the body that casts the shadow. We want just to put in evidence possible observable effects of such a shadow of light.4 The Letter is organized as follows. The motivations of the experiment are discussed in Section 2. It is shown how a previous analysis of the Cologne

4 Still speaking metaphorically, we can state that our problem is 3 Actually, the LLI breakdown in these experiments is related

more to the nonlocal behavior of the barrier [11] than to the fasterthan-light propagation of the signals.

the dual of that by Peter Pan (in Barrie’s “Peter and Wendy”), when he struggles attempting to re-paste his shadow to his body, namely, we want to detach its shadow from the photon.

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and Florence experiments on superluminal photon tunnelling allows us to size the experimental apparatus, that is described in Section 2.2. Section 3 contains the description of the experiment, the measurement procedure and the results obtained. Section 4 gives the preliminary results of two crossed photon-beam experiments that seem to confirm our findings, and the final conclusions.

2. The whys and wherefores of the experiment Our experimental set-up involves two identical photon sources S1 , S2 , three slits F1 , F2 , F3 , and three detectors A, B, C. The resulting final scheme of our apparatus is shown in Fig. 2. 2.1. Sizing the experimental apparatus: connection with the Cologne and Florence experiments The experiment discussed in the present Letter is aimed at detecting the de Broglie–Bohm waves, in the hypothesis that they are nothing but deformations of the space–time metric, and therefore connected to the breakdown of LLI for electromagnetic systems (see Section 1). Then, the experimental set-up was sized according to the results concerning this latter effect, in particular its threshold behavior both in energy and space. We are therefore making the basic (although heuristic) assumption that such behavior is universal, in the sense that the threshold values in energy and space characterizing the e.m. LLI breaking

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are essentially independent of the energy of the photon system considered. As already stressed in Section 1, the main information on the threshold behavior of LLI breakdown for electromagnetic systems was provided by the 1992 Cologne [6] and 1993 Florence [10] experiments, namely, the first to show superluminal propagation of photons. Let us discuss them in some detail. The Cologne experiment dealt with microwave propagation inside a rectangular waveguide with a reduced-section part. The undersized guide behaves as a high-pass filter: for wave frequencies lower than the cutoff frequency only evanescent waves propagate inside the waveguide. The penetration length of the wave depends on the guide cutoff and the wave frequency. The group speed of the evanescent wave was found to be superluminal. In the Florence experiment, it was investigated the microwave propagation in air between two horn antennas. The experimental scheme is shown in Fig. 3. Here, s, L and d denote, respectively, the distance between the emission centres of the antennas, the horizontal distance between the antenna surfaces, and the vertical distance between the antenna axes. A superluminal speed of the signal was observed for a given value of L and four values of d. In a recent theoretical work [9], two of us (F.C. and R.M.) have shown that in both (seemingly different) experiments, the photon superluminal propagation can be described as a classical tunnel effect of evanescent waves. This result was achieved by exploiting the DSR formalism [5], which allows one to describe

Fig. 2. Above view of the experimental apparatus used in the present experiment. See the text.

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the behavior of the Florence set-up as a barrier, in analogy to the Cologne experiment. In particular, the penetration length of the evanescent wave propagating between the horn antennas was estimated for the four values of d, and found to vary in the range 8.80– 9.30 cm. Such a length is to be meant along the line between the antenna centres, namely, along s (see Fig. 3). Clearly, it provides the size 0 of the space region where the LLI breakdown is expected to occur. We used a kind of geometric analogy with the Florence experiment, by assuming that the photon source S2 and the detector A of our apparatus (see Fig. 2) do correspond to the centres of the horn antennas (see Fig. 3). Then, they belong to parallel planes, but do not lie on the same line orthogonal to the two planes. This amounts to say that the diagonal distance between S2 and A correspond to the distance s of Fig. 3. This is

Fig. 3. Schematic view of the apparatus used in the 1993 Florence experiment.

why we denoted such a distance and the plane distance by the letters s and L, respectively, in Fig. 2. On the basis of the above discussion, s in our apparatus was put equal to 10 cm. Lastly, the other relevant sizes of the apparatus were determined by the requirement that the slit F2 lies outside the opening cone of the source S2 . From the energy threshold E0,e.m. ∼ = 4.5 µV derived from the Cologne experiment (on the basis of the DSR formalism), we expect that—once the minimal conditions of photon intensity above the working threshold of the detector A are reached—the signal measured by A, related to the photon–shadow interaction, to be less than (at most equal to) such a value. 2.2. Experimental set-up In the present experiment (carried out at L’Aquila University in 2002) we used two sources of photons in the infrared range, with a wavelength λ = 8.5 × 10−5 cm. The slits were circular, with a diameter of 0.5 cm, much larger than λ. We worked therefore in absence of single-slit diffraction. However, the Fraunhofer diffraction is still present, and its effects were taken into account in the background measurement. The photon energies involved in the present experiment are about 104 higher than those of both the Cologne and the Florence experiment. However, according to the hypothesis we made on the universality of the electromagnetic LLI breakdown, we expect

Fig. 4. A more detailed view of the internal plan of the box used in the present experiment. See the text.

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Table 1 Comparing the radiation parameters for the Florence and L’Aquila experiments Experiment

Frequency (GHz)

Wavelength (cm)

Energy (eV)

Florence L’Aquila

9.50 3.53 × 105

3.16 8.50 × 10−5

3.93 × 10−5 1.46

the same threshold values to hold true in the present case. We give in Table 1 the photon frequency, wavelength and energy for the Florence and L’Aquila experiments. A Plexiglas box with rectangular basis constituted the experimental apparatus of the present experiment. Two sources of infrared radiation and three detectors were placed inside the box. After removing the upper lid of the box, its interior looks schematically as pictured in Fig. 2. The sources S1 and S2 were two identical leds emitting in the infrared, the detectors A, B and C three identical photodiodes, and F1 , F2 , F3 three identical circular slits. The lines marked by the numbers 1, 2, 3, 4 and 5 in Fig. 2 are the electrical connections with the outside needed to supply sources and detectors and to pick up the response signal of the detectors. This permitted to separate the measuring apparatus from measurement devices and supply devices. The box basis was wooden. All the Plexiglas pieces of the apparatus had a rectangular basis, were 0.4 cm thick, and were covered with a black film. When the box was closed by jointing the upper lid, the box inside was highly insulated from the outside (with respect to the frequencies involved), due to its special internal design and black, reflecting external surfaces. Detectors A and B were set on a common movable support, that allowed one to change their distance L from the source plane, so to investigate the space behavior of the searched effect. The six different possible positions (labelled with numbers 1–6 in brackets) of the movable support containing the two detectors A and B are shown in Fig. 4, where a more detailed view of the internal plan of the experimental set-up is given. The corresponding distances L varied from 1.0 to 8.5 cm (step 1.5 cm) (cf. Table 2). Detector C was fixed, close to the slit and in front of the source S2 . Fig. 4 also shows the opening cones of the emitted infrared radiation through the slits, in the hypothesis of strict validity of geometrical optics (namely, by dis-

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regarding the Fresnel diffraction by the slit borders: this effect would only increase slightly, in a spatially anisotropic way, the angular openings of the slit radiation cones, so that disregarding it does not invalidate any of the geometrical considerations below). Simple geometrical conclusions can be easily drawn from Fig. 4. First, it is easy to see that slit F1 could affect at most detector B only from position (2) on; moreover, the intensity of its signal is expected to decrease as the inverse of the distance F1 –B. Actually, it was experimentally checked that there is always no contribution to the signal at B from F1 irrespective of the B position. Likewise one expects no contribution to the signal detected by either A or B from S2 photons passing trough F2 , since F2 is outside the maximal radiation cone of S2 . This fact, too, was experimentally tested. It is also easily seen from Fig. 5 that the photons of source S2 should not affect the signal of detector A. In particular, therefore, there should be no changes in the signal at A induced by changes in the state (on/off) of S2 . This is indeed the case if source S1 is off. On the contrary, if one assumes the existence of the de Broglie–Bohm wave associated to the photon (and/or the LLI breakdown: see Section 1.3), and source S1 is on, the detector A can detect two different signals according to whether S2 is on or off. This is due to the fact that the S1 photons do interact with the shadows of the photons emitted by S2 and passing through the slit F2 , thus changing the signal detected by A with respect to the case of source S2 off. Moreover, one expects that this effect can occur only when the movable support is in positions (1)– (3), due to the hypothesis made on the nature of the hollow wave (as space–time deformation) and the space threshold derived for the LLI breakdown (see Section 2.2). This clarifies the role of the slit F2 . If hollow waves exist, and they are (as we assumed) space– time deformations, their propagation is affected by matter distribution and density. In particular, they can pass through space regions where the matter density is especially low (as a slit). In summary, the experimental apparatus works as follows. Measuring the energies of the photons emitted by the source S2 completely destroys the eigenstates associated to such photons. Moreover, there is no Young interference of S2 with the other identical

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source S1 , since by construction the emission cone of S2 cannot light the slit F2 . However, the hollow waves associated to the destroyed photons can pass through the slit F2 . Such waves (according to the de Broglie–Andrade y Silva assumption: see Section 1.1) can interact with the photons emitted by the source S1 and passing through the slit F1 . The role of the three detectors is the following. Detector C destroys the eigenstates of the photons coming from S2 . Detector B checks that no photon passes through slit F2 (only hollow waves can pass through it). Detector A measures the photons emitted by S1 , and possible changes in its signal can only be due to the interaction of the S1 photons with the hollow waves passed through F2 . The possible energy difference ∆A in the signal measured by A when source S2 is off or on (and the signal in B is strictly null) is in any case the signature of the light shadow, and of the LLI breakdown iff the energy gap ∆A
E0,e.m. . This is just the effect we searched for.

3. The experiment 3.1. Measurement conditions The two sources of infrared radiation were two identical emitting diodes of kind High Speed Infrared Emitter AlGaAs (HIRL 5010, Hero Electronics Limited) with peak emission at 850 nm and angular opening of about 20◦ . About 65% of the emitted intensity was in the range 840–870 nm. The three detectors, of kind photodiode and amplifier integrated (OPT301, Burr–Brown Corporation), were photodiodes with operational amplifier mounted on a single chip dielectrically insulated. The maximum response of the detectors was in the range 700– 800 nm; at the emission peak of the emitting leds the detector response was about 88% of the maximum response. Since one needs identical sources and identical detectors, the supplies too were identical (separately for sources and detectors). So, the two emitting leds were supplied in parallel by a common variable generator of steady voltage, and analogously for the three detectors.

The main measurement device used was a HP digital multimeter, whereby both supply voltages and detector responses have been measured and checked. The supply voltage of the detectors was chosen so to get the maximum response stability. In fact all measurements have been carried out only when the greatest fluctuation in the main measurement device was 1 µV for all measurements. A statistical study showed that such stability condition was reached in an average time of 100 s. Due to the smallness of the signal difference (with respect to the photon energy involved), which— according to the discussion of the previous section— does constitute the signature of the effect we are looking for, the ideal measurement condition would be the single-photon operation. Therefore, one needs a supply voltage of the emitting leds so small as to get the lowest photon intensity, but enough so to yield a detector response distinguishable from the dark. Then, the search for the optimal supply voltage of the leds was the preliminary step of the measurement procedure. In practice, we started with both sources off and all detectors on, thus getting the dark voltage. Then, we switched only the source S1 with a supply voltage more and more increasing, until detector A yielded a response different from the dark. Such a voltage was found to be 10 µV higher than the dark voltage. The supply voltage thus determined for S1 and A does guarantee a fortiori (when applied to S2 ) that the response of the detector C, too, is higher than the dark voltage, since its distance from S2 is less than the distance of A from S1 . This fact was experimentally tested. As to the detector B, due to its checking role, it is expected to see only and always the dark voltage. 3.2. Measurement procedure Once fixed, the supply voltages were systematically checked in order to avoid (during a single measurement) fluctuations, which may affect the detector response. These supply fluctuations for the accepted measurements were kept below 1 mV (since they were checked to give rise to signal measurements lower than 1 µV). Anyway, the very measurement procedure was designed so to constantly and systematically check the possible spurious effects due to such fluctuations. The measurement procedure was as follows:

F. Cardone et al. / Physics Letters A 326 (2004) 1–13

• Step 0: Measuring the signals of detectors A, B, C with both sources S1 , S2 off; • Step 1: Measuring the signals of detectors A, B, C with S1 on and S2 off; • Step 2: Measuring the signals of detectors A, B, C with S1 off and S2 on; • Step 3: Measuring the signals of detectors A, B, C with both S1 and S2 on.5

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Table 2 Experimental results Position

Distance L (cm)

Energy gap ∆A (1–3) (µV)

1 2 3 4 5 6

1.0 ± 0.1 2.5 ± 0.1 4.0 ± 0.1 5.5 ± 0.1 7.0 ± 0.1 8.5 ± 0.1

2.3 ±0.5 2.2 ±0.4 2.3 ±0.5 no measurement above threshold no measurement above threshold no measurement carried out

Let us define the following quantity: ∆d (i − k) ≡ |Σi − Σk | (i, k = 0, 1, 2, 3, i = k; d = A, B, C), i.e., the absolute value of the difference between the signals measured by detector d at steps i, k. To our aims, the physically relevant energy gaps are ∆A (1–3), ∆B (0–3), ∆C (2–3). In particular, it is easy to see (on the basis of the discussion of Section 2) that ∆A (1–3) is a measure of the searched effect. The other two energy gaps are needed in order to check the validity of the measurement of ∆A (1–3). Indeed, both ∆B (0–3) and ∆C (2–3) must be 0 or at most the error voltage 1 µV in order to accept a measurement of ∆A (1–3). In this case, ∆B (0–3) assures that no photon passed through the central slit F2 (see Fig. 2) (otherwise, they would have been detected by B), whereas ∆C (2–3) assures that no change occurred in the source supply, and the photon flux was constant, thus yielding a constant signal from S2 on C and therefore from S1 on A. 3.3. Results We carried out 20 measurements for each position of the movable support from (1) to (5). We accepted only the measurements satisfying the validity conditions given above (∆B (0–3), ∆C (2–3)
1 µV). All the measurements below the apparatus threshold (namely, below the error corresponding to 1 µV) have been discarded. For each position we checked, by means of a suitable test of statistical inference, that the accepted measurements were normally distributed. 5 Of course, step 3 with F closed amounts to step 1. Light 2 shadow remains behind the door if this latter is closed (which is equivalent to open the door and turn the light off).

The mean value of the energy gap ∆A (1–3) for the 5 positions (with the associated errors) are given in Table 2. On the basis of the above experimental results, we can therefore conclude that: 1. The non-zero energy gap ∆A (1–3) is a clear signature of an anomalous interference effect; 2. The value ∆A (1–3) ∼ = 2.3 µV is less than the threshold energy value E0,e.m. ∼ = 4.5 µV for the electromagnetic breakdown of LLI; 3. This non-zero value of the energy gap was found until the characteristic horizontal length of about 4 cm, less than the space threshold 0 ∼ = 8.80 cm for LLI breakdown.

4. Further evidences and conclusions On the basis of the experimental results, we can state we observed indeed an anomalous behavior in a photon system, which cannot be explained in a classical framework. It is possible to interpret it in terms of an interaction of photons with the de Broglie–Bohm waves, and/or as a manifestation of LLI breakdown. This experiment confirms the threshold behavior of the electromagnetic LLI breakdown, both in energy and in space (see points 2 and 3 of Section 3.3), derived from the Cologne, Florence and coil experiments. Moreover, it supports the assumption we made on its universality character, i.e., its independence on the energy of the photon system. The present results are also in favor of our hypothesis of the hollow wave as a deformation of the Minkowskian geometry of space–time, at least for photons. In the metaphoric language we adopted, light is therefore always accompanied by its shadow, which however, once cast by the photons, can penetrate in

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regions physically forbidden to them and affect other photons by “shading” their light. If the shadow of light admits such a geometrical interpretation, we expect that it implies a change (an increase, with respect to the classical and quantum electrodynamics predictions) of the photon–photon cross section, independently from the energy (at least in the infrared-microwave range). Then, effects similar to that we observed are expected to occur in other systems involving photon– photon interaction, in particular in crossed photon beams. Experiments of this kind have been already carried out in the past. We refer, e.g., to that by Pfleegor and Mandel [15], whose result of anomalous interference behavior was just explained by de Broglie and Andrade y Silva [2] as an effect of the pilot wave– photon interaction. Let us stress that the phenomenon of interference between photon beams is strongly depressed not only in classical, but also in quantum electrodynamics, because the photon–photon cross section goes as α 4 (with α being the fine structure constant).

Fig. 5. Proposed crossed-beam experiment in the microwave range, exploiting two horn antennas.

In this connection, two of us (F.C. and R.M.) proposed on February 2003 to two experimental teams of Florence to carry out experiments of photon interference by using orthogonal crossed beams. They are aimed at confirming the results of L’Aquila experiment, in particular the threshold behavior of the anomalous effect in energy and space, and at checking its independence on the energy of the photon system. The proposed interference experiments are presently being carried out, one with microwaves emitted by horn antennas (see Fig. 5), at IFAC-CNR (Ranfagni and coworkers), and the other with infrared CO2 laser beams (Fig. 6), at INOA (Meucci and coworkers). Let us summarize the preliminary results obtained. The main result of the former experiment is given in Fig. 7,6 which shows the signal intensity received by the detecting antenna (see Fig. 5) as a function of the attenuation of the main beam F1 . The frequency of the two crossed microwave beams was ∼ 9.5 GHz, and the crossing region was of the order of 8–9 cm (beam opening). The secondary beam F2 was modulated with a frequency of ∼ 1500 Hz. Due to the smallness of the signal, a lock-in technique was used, in order to magnify the observed effect. These preliminary results cannot be accounted for by a simple interference effect. They exhibit an anomalous behavior. The signal observed is one-two orders of magnitude higher than that we found, due to the much larger number of photons involved in the emission from a microwave antenna. Moreover, it is clearly seen from Fig. 7 that, as attenuation increases, in the limit of low intensity (single photon condition), the signal at the receiver does exhibit an asymptotic behavior for an attenuation increasing from 30 dB up, independently

6 A. Ranfagni, D. Mugnai and R. Ruggeri: private communica-

tion.

Fig. 6. Proposed crossed-beam experiment in the infrared range, exploiting a CO2 laser.

F. Cardone et al. / Physics Letters A 326 (2004) 1–13

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Fig. 7. Preliminary results of the microwave crossed-beam experiment. The figure shows the signal amplitude received by the detector as function of the attenuation of the primary beam F1 or of the secondary (modulated) beam F2 . The symbols vp and hp refer to vertical and horizontal polarization, respectively (A. Ranfagni et al.: private communication).

of what beam (F1 or F2 ) is attenuated. The asymptotic value of the received signal is ∼ 5 µV, within the experimental errors (the size of the points in the graphs), as expected from the previous experiments on LLI breakdown. The preliminary results of the laser beam experiment are reported in Fig. 8.7 It shows, for the three measurement runs carried out till now, the space distributions of the intensities of the signal received by the detector (a CDC array). The first and second row (from top) depicts, respectively, the signal intensity with only the laser beam F1 on, and in beam-crossing condition; the bottom row shows the difference of the intensity distributions, exhibiting the expected anomalous behavior. We can therefore conclude that: 1. Crossed photon beams exhibit an anomalous behavior, in the sense that a difference in signal ex-

7 R. Meucci: private communication.

ists between the two cases of presence and absence of the crossing beam; 2. The value of this signal difference in the microwave case attains, as expected, the LLI breakdown threshold of 5 µV when the intensity of one of the beam is lowered until the single-photon condition; 3. Such an effect occurs when the crossing beam region is of the order of (or less than) 9 cm and 4 cm, in the microwave and infrared region, respectively; 4. The effect is indeed independent of the energy, at least in the infrared-microwave range. Therefore, although preliminary, the results of the Florence crossed-beam experiments do confirm the findings of the experiment we carried out and discussed in this Letter. Needless to say, our experiment is worth repeating. Possible suggestions for further developments of this kind of experiment are: (i) using, instead of diodes, phototransistors with converging lens (in order to reduce spurious effects), which are more sensitive (at the price of a higher instabil-

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F. Cardone et al. / Physics Letters A 326 (2004) 1–13

ity); (ii) using two different leds as sources with different frequencies (for instance, in the infrared and in the visible).

Acknowledgements We are greatly indebted to Riccardo Meucci (INOA, Florence) and Anedio Ranfagni (IFAC-CNR, Florence) for stimulating discussions and for communicating to us the preliminary results of the crossedbeam experiments they are carrying out. Thanks are also due to Eliano Pessa, of the Centre for Cognitive Sciences of Pavia University, and Fabio Pistella, of the Rome University “Roma Tre”, for kind interest and critical remarks. Preliminary results of our experiment have been presented at the Conference Anomalies and Strange Behavior in Physics: Challenging the Conventional (Napoli, Italy, April 10–12, 2003), and two of us (F.C. and R.M.) are very grateful to the Conference Directors, Daniela Mugnai, Anedio Ranfagni and Lawrence S. Schulman, for their kind invitation and exquisite hospitality.

References

Fig. 8. Preliminary results of the infrared laser crossed-beam experiment. The figure shows, for three measurement runs, the space distribution of the intensity detected. The first and second row (from left) show, respectively, the intensity detected with F1 on, and with both beams on; the third row depicts the difference of the two previous signals (R. Meucci et al.: private communication).

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