FROM GHOST TO REAL WAVES: A PROPOSED SOLUTION TO THE WAVE-PARTICLE DILEMMA

G. Tarozzi

Istituto Matematico dell'Universita - Bologna Istituto di Filosofia dell'Universita - Perugia

The most controversial epistemological problem of the foundations of quantum mechanics arises in connection with the interpretation of the fundamental theoretical term of quantal formalism,i.e. the wave-function or state-vector I ~ > , which is at the origin of the problem of the wave-particle duality.

This essential property of microscopic phenomena finds its most significant manifestation in the well-known double-slit expe rimen t .

In this peculiar physical situation one considers a source of photons (so weak that at a given time only one photon can be found in the experimental device) impinging on a diaphragm, en­dowed with two slits A and B, which are detected on a second screen X.

As it can be easily shown the description of the physical situation given starting from hypotheses which seem all absolutely reasonable leads to a result in open contradiction with experimen­tal evidence. Let us consider the event X lithe photon has arrived at some point on the detecting screen X" Pwhich will be given by

(1) X P

where A and B for thePslit p lity P(X ) that

p

X A (A VB) p p p

indicate respectively the passage of the photon A or B respectively. This means that the probabi­the photon arrives at X is

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S. Diner etaL (eds.), The Wave-Particle DUIllism, 139-148. @ 1984 by D. Reidel Publishing Company.

140 G. TAROZZI

(2) P (X ) = P [X A p p

(A VB )1 p p-

which for the principle of distributivity becomes

(3) P (X ) = P [(X A A ) V (X A B )-1 p p p p p-

Moreover as A and B are disjunctive propositions, that the event (A A B p) is

p never true, one can write (3)

p p

(4) P(X ) P(X A A ) + P(X A B ) P P P P P

so as

The relation (4) is however contradicted by quantum me­chanics. According to this last theory the wave-function ~ (x) which describes the photon in the point x of the detecting-screen X is given by

(5) ~ (x) ~ A (~) + ~ (x) B -

where ~ (x) and ~ (x) are complex quantities whose square A - B -modulus corresponds respectively to

(6 ) P (X A A ) ; 1 ~ B (x) 12 p p -

P(X A B ) P P

In a similar way 1'1' (x) 12 represents the probability P(X ) which will therefore in this case correspond to

p

(7)

where (8)

P(X ) = 1'1' A (~) P

~A(~)=P(X A A )+P(X AB )+ I t P P P P

indicates the interference terms of this sort of strange wave-like probability calculus.

As is well known the experimental evidence confirms (7) and contradicts (4). However when one locates behind the slits two measuring instruments allowing to establish through which slit the photon passes, the interference term (8) are destroyed and the right description is proved by (4). The physical situation appears of an even more difficult comprehensibility when one considers all the cases in which one has only one measuring apparatus located behind one slit which has not revealed anything and the photon has

FROM GHOST TO REAL WA YES 141

been anyhow detected in some point of the second screen. In this physical situation the disappearance of the in­

terference terms (8), which is a well-confirmed experimental re­sult, can be justified only assuming that a measuring apparatus acting only as a source of information which allows to establish through which slit the photon has passed, can modify the empirical result of the experiment. But this appears of course a very para­doxical result as Renninger (1) was the first to stress.

This incapability to provide a reasonable explanation to the behaviour of the photon in the double slit experiment, in­capability which can be extended to the more general problem of the wave-particle duality, conceived in the light of the complemen­tary interpretation of the Copenhagen school as an insuperable dilemma has been emphatized also by Feynam: "One might still like to ask: How does it work? What is the machi­nery behind the law? No one has found any machinery behind the law. No one can explain any more than we have just explained. No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced. It is all quite mysterious. And the more you look at it, the more mysterious it seems ( ••• )." (2)

To this completely renounCing epistemological attitude is opposed the realist interpretation of the wave-particle dualism which was historically put forward by Einstein and de Broglie. According to this interpretation the wave-function ~(x, t) asso­ciated to the particle would represent an objectively-existing wave and not only a complex mathematical expression. In particular the particle would be associated to the wave in such a way as to be found in different points of the wave itself with a probability proportional to its square amplitude.

In 1978 Selleri and myself proposed a realist interpre­tation of the double- slit experiment (3), in conformity with the Einstein-de Broglie conceptions of wave-particle duality (4), according to which the events A and B would have to correspond to the more complex events: p p

(9)

B = B I fly , p

where Al ,BI and y mean respectively, (10) Al (BI ) :the particle passed through slit

A(B)

(11) y A 2 B2: the associated quantum wave passed through both A and B.

In this maintaining the same mathematical description of quantum mechanics both the presence of the interference terms (8) as a consequence of the event and their destruction, provoked by

142 G. TAROZZI

the interaction of the measuring apparata in correspondence with the slits with the Einstein-de Broglie wave described by (11), can be explained. The realist interpretation would be able to account also for the paradox of Renninger, since even the insertion of one apparatus behind one slit, although not revealing the particle, would have destroyed the coherence of the wave passing through the two slits and wiped out the interference pattern. In this case one would be dealing of course with the modified events

A' = Al A (A2A B2 ) p

B' P

where A; (B~ ) indicates the absorption of the wave by the apparatus located behind A(B) •

There is however a very strong objection against the possibility of writing relations (11) which regards the inobserv­ability of the Einstein-de Broglie wave. In fact on which basis can we maintain that the apparatus located behind B(A) has ab­sorbed, or at least interacted with the wave when it has not re­vealed any kind of physical effect? This inobservable reality of the wave associated to the particle has been of course object of violent criticism on the part of the supporters of the orthodox interpretation and Einstein himself used to talk in these cases about "ghost fields" (Gespensterfelder).

Anyhow this impossibility of determining any kind of predictable attributes for these "objects" implied that they could not be considered as sensible carriers of predicates in the lan­guage of quantum mechanical events. In a similar way the result of an interaction of this wave with other physical objects could be paradoxically compared to the one produced by other complex mathematical terms like the square root of a negative number.

One is faced therefore with a situation which seems without a way out since if on one hand the assumption of the reality of the Einstein-de Broglie wave can explain what in the orthodox interpretation appears completely inexplicable, on the other hand one must pay the price of postulating the existence of an object which cannot be characterized through any kind of ob­servable quantity like energy momentum charge mass as in the case of every other proper physical object.

An answer to this paradoxical situation is very likely contained in a proposal of experimental test which has been dis­cussed for the first time by F. Selleri (5), and which is now carried out by Gozzini (6), in Pisa.

According to Selleri's proposal one considers a source of photons S whose intensity is reduced by an absorber in such a way that only one photon is present at any given instant in the experimental device (Fig.1). Each of these individual photons falls on a halfsilvered mirror SRM which reflects half its intens­ity towards the counter PH and transmits the other half to the

I counter PH 2 , which are linked together to the extent of performing coincidence measurements. In the path of the reflected"photon"

FROM GHOST TO REAL WAVES 143

is placed a laser gain tube LGT (i.e. a piece of matter composed of unstable physical entities whose excitation energy corresponds to the one of the photons emitted by s).

()-+ S

lGT

Fig.l - Selleri 's proposal for testing the properties of quantum waves.

Let us consider the number of the coincidences of the results of the measurements performed by PH'i and PH 2. If this number will be found higher than the one of the random coinciden­ces this positive result would mean that "something" different from the particle, which has been detected by the counter P2 • has produced a transition of probability on the lifetime of the "atoms of LGT. We assume this "something" to be nothing but the Einstein­de Broglie wave. In this way a positive result would confiLffi the possibility to characterize the wave via the following predictable property: the property of inducing transition of probability on the lifetime of the excited physical systems whose excitation en­ergy is the same one of the associated particle. One would be dealing however with a property of a particular nature which ap­pears weaker than the ones which can be attributed to the particles.

This impossibility of characterizing by means of intrin­sic or monadic properties, as happens for other ordinary physical objects, with the opportunity of determining a relational or bet­ter a diadic property between waves and some particular classes of particles would lead in turn to weaker notion of physical rea­lity. And such a deepening of the conception of realism will be perhaps able to provide a satisfactory explanation of the nature of the wave particle duality.

144 G. TAROZZI

A positive result from the previous experiment would also involve very important consequences for the logic of empiri­cal theories since it would produce an automatic resolution of all the difficulties connected wi th the "weakening" of classical logic - i.e. through the renouncing of the principle of tertium non datur, or to the one of distributivity which has previously assumed in passing from (2) to (3) in the derivation of (4) - re­quired by several logicians to the extent of avoiding a contra­dictory interpretation of microscopic phenomena (7).

The possibility of revealing observable effects induced by the Einstein-de Broglie wave would imply a radical refutation of the "weak" non-standard logics of quantum mechanics and conse­quently the necessity of maintaining the validity of the principles of classical logic as analytical statements which cannot be denied without violating the inventions presupposed by the very procedure of negation allowing however the introduction, in the language of microscopical events, a new class of logical objects which could be sensible carriers of exclusively relational predicates.

If the standard double-slit and Selleri's one, in which the screen with the two holes A and B has been substituted by the beam splitter M (or alternatively by a ~emi-reflecting mirror SRM), are compared, one can easily realize that the events A to "the photon has been transmitted through SRM" and "the photoR has been reflected by SRM". Such an equivalence can be extended of course to all events described by relations (9), (10) and (11).

On the basis of the previously introduced definitions, Selleri's experiment appears as a test of the event

(or wi th a tri vi.al modification of the geometry of the experiment AI1\B:2) corresponding to the statement "the particle has been re­flected by and the wave has been transmitted through SRM".

We are not dealing, therefore with a complete test of the event y, that.is of the' fact that, in conformity with the Einstein-de Broglie realist interpretation of the double.-slit ex­periment, the quantum wave must have been both transmitted and reflected (passed through both A and B according to the language of the standard double slit).

There is, moreover, another point connected with the testability of the properties of quantum waves, that must be stressed. As a matter of fact, if one found that quantum waves have in addition to the .property of producing interference pattern, as Janossy and Naray showed in a conclusive way, also the one of inducing stimulated emission, then these physical objects would result characterizable through two different but mutually exclu­sive empirical properties, since interference and stimulated emis­sion would manifest themselves with different experimental devices in two incompatible physical situations. One could, therefore, try

FROM GHOST TO REAL WAVES 145

to apply the notion of complementarity and in this case it would be perhaps more correct to speak of "meta-complementarity", since we could not be dealing with a dualism between the ondulatory and corpuscular properties of physical objects but with a dualism between the two ondulatory properties of microobjects - to the properties of quantum waves of producing either interference or alternatively stimulated emission.

With the aim of bypassing the previous problems I should like to propose two generalizations of Selleri's experiment.

My first variant represents a test of the complete event r. corresponding to the statement "the quantum wave has been both transmitted and reflected" while the second experiment is a unique test of both the empirical properties of quantum waves, i.e. the design of a device able to test at the same time either the stim­ulated emission and the interference pattern produced by this physical object.

My first proposal is shown in fig. 2, where S is a very weak source of photons, SRM and SRMI semireflecting mirrors, LGT and LGTI laser gain tubes, PH PH and PH photomultipliers.

123

(14)

UPHZ UPHa

6~ 6,·" 0--+ C C c=

S PH.

Fig.2 - 'I'es t of the event A. A corresponding to "the particle has been transmitted through and the quantum wave has been both transmitted through e reflected by SRM".

If we denote by Pitt) the event,

"a photon has been detected by PHi at time til

146 G.TAROZZI

it appears clear that each coincidence of the detections of the three counters at any given time t, corresponding to

(15) P 1 (t)AP 2 (t)AP 3 (t)

would represent a confirmation of the event

As a matter of fact

that is, if we have a coincidence in the detection of PH and PH this means that, since the particle has been transmitted through SRM (and SRM ) as is confirmed by the arrival of a photon on PH , the second contemporary arrival of another photon on PH is a con­sequence of the stimulated emission produced by the wave reflected by SRM (Selleri's experiment).

More'over,

(18)

that is, if PHI' PH 2 and PH 3 detect at the same time a photon, we have the previous situation expressed by (17) plus the event cor­responding to the reflection of the quantum wave by SRM which means in turn that such a wave has been not only reflected but also transmitted by SRM.

Our second variant for testing the empirical properties of quantum waves is shown in Fig. 3 where SRM and SRM1 are semi­reflecting mirror, LGT is a laser gain tube, PH is a photomulti­plier and IR a detector devised by Fleegor and Mandel (9) which is built with a stack of thin glass plates each of which has a thickness corresponding to a half-fringe width. The plates are cut and arranged in such a way that photon falling on the odd plates are fed to one photomultiplier PHo while photons falling on the even plates are fed to the other one PHe • Each of these photomul­tipliers is connected to PH for the purpose of performing coinci­dence measurements.

FROM GHOST TO REAL WA YES 147

c= PH

LGT

Fig. 3. - Experimental device for testing both the hy­pothesis that quantum waves induce stimulated emission and their property of producing an interference pattern in a unique physical sit­uation.

We can perform in this way a unique test of both the empirical properties of the event y.

As a matter of fact the coincidences between the detect~ tions of PH PH and PH PH , that is with the notations previously

o e introduced

(19) Ip(t)i\ P (t)IVlp(t)i\ P (tTl - 0 - - e-

would put in evidence the property of quantum wave inducing stim­ulated emission while a difference between the previous coincidence rates, that is

(20) N -,I" N 0 7 - e'

when No and Ne indicate the number of the events p(t)i\ poet) and P(t)i\ Pe(t), respectively, would show the other property of quan­tum wave of producing interference.

148 G. TAROZZI

(1) M. Renninger, Zeitschrift fur Physik 158, 417 (1960)

(2) R. Feynman, Lectures on Physics, ~ (Quantum Mechanics) pp.8 and 13, Addison Wesley (New York, 1972)

(3) F. Selleri and G. Tarozzi, Il Nuovo Cimento, 43 B, 31 (1978)

(4) A. Einstein, Annalen der Physik ~, 132 (1905); L. de Broglie, Annals de la Fondation L. de Broglie, ~, s (1977)

(5) F. Selleri, Nuovo Cimento Lettere, ~, 908 (1969)

F. Selleri, Realism and the Wave-Function of Quantum Mechanics in Foundations of Quantum Mechanics, ed. by B. d'Espagnat, Academic Press (New York, 1971)

F. Selleri, Annals de la Fondation L. de Broglie, ~, 45 (1982)

(6) G. Gozzini, Some experiments on the wave-particle duality of light preprint (1982)

(7) M.L. Dalla Chiara, Logica, ISEDI (Milano, 1974)

(8) L. Janossy and Zs. Naray, Supple Nuovo Cimento, ~, 588 (1958)

(9) R.F. Pfleegor and L. Mandel, Phys. Rev., 159, 1084 (1967). The use of such an interference detector in a similar tests of quantum waves has been proposed for the first time by A. Garuccio, V. Rapisarda and J.P. Vigier, Physics Letters, to be published .