IS THE SPURIOUS SCATTERING A QUANTUM GRAVITY PHENOMENON?
Ágnes Holba & B. Lukács
Central Research Institute for Physics, H1525 Bp. 114. Pf. 49, Budapest, Hungary
Here some arguments are listed suggesting that spurious scattering oftracks in emulsions may be an effect of Quantum Mechanics beyond the orthodox formalism. The statistical behaviours of such tracks are similar than expected at repeated breakdown of superposition; and the few reliable numerical values are conform with that in the gravitational breakdown of superposition by objects in the same order of magnitude for mass as the emulsion grains. A full explanation of the effect, however, would need a description of photographic development in an unorthodox Quantum Mechanics, beyond present possibilities.
1. INTRODUCTION
Since atomic spectra are conform with predictions of the Schrödinger equation or its relativistic counterparts for many digits, extensions of orthodox Quantum Mechanics must be manifested under rather unfamiliar circumstances, say at the transition between macro and microworld, in very strong gravitation, &c. So suggested measurements often involve high technology just above the present possibilities. Here we take the opposite viewpoint and ask if some new effects have already turned up in measurements; more definitely if the phenomenon called spurious scattering can be an effect of unorthodox QM.
This is not an idle curiosity. As we shall see, spurious scattering (briefly SPS) at least has characteristic data compatible with such an origin. We cannot determine a probability for fundamental nature of this phenomenon. However spurious scattering is ideal in that sense that the measurement is cheap and does not involve any high technology at all.
2. ABOUT SPURIOUS SCATTERING
Some readers may not be familiar with SPS, so we briefly recapitulate something about it. SPS was discovered as a byproduct of balloon cosmic radiation measurements in the 30's and 40's. Only emulsions could be lifted, so the energy of particles could not be determined from the geometry of the track. Instead, it was calculated from the serratedness of the track. In average it was a straight line, so take a coordinate system with its x axis along the "average" track. Then divide the x axis with equal steps s (cell length); the ith sagitta is defined as
D_{i} = y(x_{i}+s)2y(x_{i})+y(x_{i}s) (1)
The serratedness is some average of the sagittae. By construction <D>=0, so take
<D^{2}> = D^{2}(s) (2)
Theoretical considerations predict
D^{2}(s) ~ a+c(E)s^{3} (3)
where a comes from the measurement error of the y coordinate, E is the track energy, and the second term on the rhs comes from repeated Coulomb scattering on the emulsion ions. (For details see Ref. 1.) D^{2}(s) is easy to measure at some different s values on the same track, and then c is known. Now, calibration measurements on the same type of emulsion in laboratory yield c(E), so then c of the track gives the energy of the track. Of course, c(E) does differ for emulsions of different compositions or structures. One can see why this technique is out of use now.
When it was used, it needed as good a fit for c as possible. Now, it turned out that the form (3) was not good. Tendencious deviations were seen, dangerous in a fit. The example is taken from Ref. 1. The emulsion was a NIKFI R stack, and the energy is 9 GeV. D^{2}(s) is
shown by Fig. 1.

The dashed line is the best fit with the form (3). However the curve obviously deviates at middle cell lengths, so the Ansatz needs one more term growing with s but more slowly than the cubic one. Statistical considerations would prefer a linear term, so let us try with
D^{2}(s) ~ a+bs+cs^{3} (4)
Then the best fit gives the solid line, which excellently follows the points. (Note that this is not a prefabricated example. Ref. 1 did not perform the fit with form (4), only with form (3).)
Now the term b*s has been named "spurious scattering" since it somehow mimics other scatterings, but is of spurious origin. In the physics of cosmic radiation SPS was not an object but rather a noise to be removed.
However in the time of balloon experiments the determination of b helped the determination of c. Of course nowadays such measurements are not performed.
3. DOES SPURIOUS SCATTERING EXIST?
As a byproduct, some b can always be obtained. In 1991 the present authors performed a systematic analysis of earlier measured data to if such an effect does exist at its own right. Tracks of 4 energies (0.25, 9, 70 and 200 GeV) were evaluated; the details can be found in Ref. 2. The essence of the results is as follows:
1) There is substantial anticorrelation between a and b.
2) However bš0 is significant (at 2s at 0.25 GeV, at 3s at 9 and 70 GeVs and at 20s at 200 GeV).
3) No strong energy dependence is seen in b (however note that measurements differed pairwise both in energy and in emulsion).
So SPS exists. So far, so good. But: what is behind b? Three ideas have been suggested up to now:
A) Mosaiclike disintegration and random shifts in the emulsion during development [1,3]
B) Spontaneous reduction of the "free" wave function of the travelling particle (henceforth proton) [4]
C) Reduction of some wave function in the emulsion [2].
All of them would predict a term ~b*s (for Case B see the socalled "anomalous Brownian motion" [5]). However in Case A a strong dependence on emulsion type and development technique would be expected. Such effect was not seen in the analysis of Ref. 2 (with the possible exception of the 70 GeV case, when, however, the emulsion was Sovietmade, highly inhomogeneous for thickness, then stored for a time in an inappropriate way, and therefore curved). Now for further check we reanalyze a completely independent measurement from 1962 [3]. The experimenters deliberately used various emulsions and development techniques; only the track energy was invariantly 6.2 GeV. It turns out that from the D^{2} sagitta data of Ref. 3 bš0 is not significant, the possible reason is that the data do not exceed to middle cell lengths below 500 micron.
However analyses at the other 4 energies already proved the significance of bš0, so we took the means, and the upper bound at 6.2 GeV. Then on Fig. 2 we have 5 energies, 6 emulsions and various development techniques. Still, b does not show large spread, as would be expected in Case A.
We, of course, cannot exclude the possibility of very constant
disintegrations. This problem belongs to some unknown details of materials science, but until that science tells otherwise, we do not believe such insensitivity on composition probable. Now we turn to quantum explanations.

4. REDUCTION OF THE FREE PROTON WAVE FUNCTION
A number of extensions of QM predicts stochastic reduction of free wave funtions. If so, then stochastic jumps would be seen on the path, in a random walk manner. In Ref. 2 we considered this possibility; the details can be found there, here we repeat only some final consequences. Four different types of unorthodoxies were considered:
a) Simple nonlinearity in QM [6]. It does not help. The expansion may stop, but no jump will appear.
b) Repeated Quantum Stochastic Multiplication [7]. It would produce a serratedness and definitely a bs term; but there are no possible parameter values reproducing the measured b values. In the best case 8 orders of magnitude are missed.
c) Influence of a Stochastic Background, say of gravitational origin [5,8,9,10]. A handmade stochastic background may help. However if it is gravity, then proton superpositions will not break down before 10^{53} years.
d) Divine Intervention. Following Bishop Berkeley's Esse est percipi [11] and QM simultaneously, one may believe that objects just not measured by anybody are measured by God, with His operator O repeatedly. This could result in SPS but only if His operator has proton eigenstates no wider than cca. 0.1 mm, and He observes free protons in each 10^{12} s. And Divine Intervention is too serious an
explanation for something spurious.
So there are problems with both mosaiclike disintegration and reduction of free wave functions. Now we turn to the third possibility, unorthodox QM effects in the emulsion.
5. DIMENSIONAL ANALYSES IN THE EMULSION
Let us make first a rather trivial statement. No SPS was ever seen in an undeveloped emulsion. Furthermore, what is seen after development, is not the track itself, but the chain of some Ag grains. First some AgBr grains were influenced by the passing proton, and then somehow the development process led to tremendous enhancement of free Ag by reduction processes in the neighbourhood of the influenced points. Then an explanation of SPS would need a detailed unorthodox QM (ample variety is available), plus a detailed materials science description for the emulsion (not so available), finally a detailed description of the development process from the first principles of the particular QM. The last step of task seems rather hopeless now.
Therefore we choose a less ambitious goal, and anyway proceed backwards. Namely:
1) We accept
b ~ 10^{8 cm} (5)
from measurements.
2) Then we select a special theory by its fundamental constants, and try to calculate from b and from the theory a characteristic time t and a mass m behind SPS.
3) Finally we try to decide from t and m if such data are possible in the system or not. If not, then the particular theory is probably not behind SPS.
By this way no complete explanation of SPS is possible. However, that was not a goal either. We think that in today's physics nobody is overfrustrated for not having the complete explanation of an effect forgotten and not observed in the last 20 years by anybody. We want to get rather suggestions that, if spurious scattering is an effect of something fundamental at all, then what has a possibility to be investigated through it and what has not at all.
The present analysis could be extended to a wide variety of theories. But here we remain at 4 possibilities, all of them "minimal", e.g. without new fundamental constants beyond h, G and c.
Full Relativistic Quantum Gravity
The
relevant constants are h, G and c. There is a
unique universal length scale in it, the Planck length L:
L ~ (hG/c^{3})^{1/2}
~ 10^{33} cm ~ 10^{25}b (6)
No problem, since we are supposed to have an agent of mass m in the emulsion. Unfortunately Relativistic Quantum Gravity is not an operative theory at the present state of art. We might try to evaluate the development of the emulsion in Supergravity or via Superstrings instead, but that is timeconsuming and one may run beyond the deadline. Therefore here we use only Dimensional Analysis. It says that the only possible result has the form
b ~ (hG/c^{3})^{1/2}f(xşm/(hc/G)^{1/2}) (7)
where f is a dimensionless function of its dimensionless argument. Now x would be in order of 1 at m ~ 10^{5} g, too high a mass in the emulsion for something not identified. Therefore it seems that x << 1, when one might try with f(x)~x^{F}. Then
m ~ 10^{(5+25/F)} g (8)
We do not know F. But obviously a negative F would be needed and in the range 1łFł3 we can get anything for m from 10^{30} g to 10^{13} g. No more definite statement seems possible in this moment.
General Relativity
We would be very surprised if SPS turned to be an overlooked GR effect, but there is no harm to check. The fundamental constants are G and c. So, independently of theoretical details
m ~ bc^{2}/G ~ 10^{+20} g (9a)
t ~ ~ b/c ~ 10^{18} s (9b)
The mass is that of a major asteroid. Since it cannot be within the emulsion, this case is ruled out.
Relativistic Quantum Physics
Now
the fundamental constants are h and c. Since e ~ 10^{1}(hc)^{1/2},
the results will be more or less valid for the inclusion of chemistry as well.
From dimensionality, for any case
m ~ hc/b ~ 10^{9} g (10a)
t ~ b/c ~ 10^{18} s (10b)
Now, these values
are in strange ranges. It is hard to interpret t. Furthermore m would correspond to something of size 10 mm in the emulsion. Such objects would be
macroscopic. The only possibility would be that this size is just the characteristic
size of the mosaiclike domains of disintegration during development. Then, of
course, b is not an effect of some Quantum Field generated by the domains; only
chemistry enters mimicking hc. The result is only a signal that
mosaiclike disintegration is a selfconsistent picture for SPS. We dealt with
this possibility in Sect. 3, and that will be enough. In addition, in such a
picture the value of t is
rather strange. So we do not think the "explanation" of SPS by
Quantum Field Theories probable and proceed.
Newtonian Quantum Gravity
Here N(ewtonian) can be replaced by N(onrelativistic) as well. The
fundamental
constants are now h and G. For the possibility and
characteristics of such a theory see Refs. 9 and 10. Independently of the
details, now
m ~ (h^{2}/Gb)^{1/3}
~ 10^{13} g (11a)
t ~ (b^{5}/hG)^{1/3}
~ 10^{2} s (11b)
Observe that the characteristic time is quite harmless, but would escape any observation since the development time is much longer. As for mass, it correspond to objects of 10^{5} 10^{4} cm size. And such objects are indeed present in any photographic emulsions. Their grain size is always in this range.
6. A POSSIBLE SCENARIO BEHIND SPURIOUS SCATTERING
Now let us go to the opposite direction. SPS is seen in photoemulsions as paths winding in such a way that a parameter b~10^{8} cm appears.
Now
photoemulsions consist of grains of, say, 3*10^{5} cm size. Therefore
the mass inhomogeneities, at least at the end of development, are in the order
of 10^{14}  10^{13} g. From such agents Newtonian Quantum
Gravity should get stochastic paths and by dimensionality the characteristic
parameter of the serratedness can be either 0 or nothing else than b~10^{8}
cm (remember that in bs the linearity in s comes simply stochasticity via
"random walk", and there are no more fundamental constants now than h
and G); and this is just seen. Is this agreement accidental?
The convincing answer again would be the calculation of the exposition and development processes in Newtonian Quantum Gravity. The nonrelativistic nature of the theory may help, but not so much that the calculation be possible until deadline. However, an analogous process has been evaluated in an analogous theory, and that result can be used with some caution.
First let us see the qualitative picture. When the proton passes, it excites the emulsion in some points, and some Ag ions are deliberated.
This is the latent picture of photography. But these "nuclei" are microscopic in any sense; photographic handbooks give the date that such a point contais 440 Ag ions. So it is reasonable to believe that the latent picture is not a unique pattern of microscopic Ag ions, but rather a quantum mechanical superposition of such patterns, naturally similar to each other. If the superposition is broken via gravity, this superposition could live for 10^{38} years.
However during development the masses of inhomogeneities continuously grow. When they are "substantial", the superposition breaks down. We remain with a unique track. But in the breakdown stochastic processes work, so the track is excepted to be winding. And since the mass inhomogeneities have a natural upper bound in the grain mass, the measure of winding has a lower bound. On the other hand, the lifetime of superpositions decreases strongly with increasing mass. So if the grains are oversized, the superposition will break down even before grain size, and the winding "freezes in". Therefore the winding will be more constant than might be naively expected from the variation of the grain size.
Admittedly up to this point this Section was a mere tale. However, let us turn to the analogous case of a Wilson chamber, which is semiquantitatively calculated in Károlyházy's "many spacetime" model [12. Of course, that theory is not nonrelativistic, since the different spacetimes differ in stochastic "gravitational waves" propagating with light velocity, and a Wilson chamber is not an emulsion. However, for the first difference we note that in that theory light velocity c cancels in a number of formulae [5], and in these points the predictions must be near to those in Newtonian Quantum Gravity. Furthermore, in the two theories the borderlines of micro and macrobehaviour coincide; something whither we will return immediately. And for the second difference, both the Wilson chamber and the emulsion are trackdetecting apparatuses with growing mass inhomogeneities: condensing water droplets are analogous to developing silver grain.
The details of the "Quantum Gravity" explanation of the work of a Wilson chamber can be found in Ref. 12, and will not be repeated here. However the idea is as follows. Originally a superposition of pretracks is present in the chamber. Let one of them be denoted by y_{i}; then in the chamber originally y = ĺc^{i}y_{i}. However the chamber is in such a thermodynamic state that the tiny droplets grow, in average according to a function <m(t)>. When m exceeds a critical value, the superposition breaks down, and we remain with a stochastically selected track. The coherence length in the process would be [5]
a_{c}
~ h^{2}/Gm^{3} (12)
and observe that this is just the result of eqs. (11), and for the breakdown time the identical formula is also derived there, i.e. those formulae are no more dimensional philosophies but consequences of a "Quantum Gravity". Now, taking now the mass from one coherence cell with density r ("just touching droplets") one gets
a_{c}
~ (h^{2}/Gr^{3})^{1/10}. (12')
A
further observation deserves some attention. Since for terrestrial matter r ~ 1 g/cm^{3}, one gets that the
theory singles out the sizes 10^{5}  10^{4} cm and masses 10^{14}
 10^{13} g ("colloidal grains"), and this is just the
range of emulsion grains. The same result is obtained if one considers the
crossing point of the dividing line of micro and macrobehaviour M^{3}R=h^{2}/G
with the terrestrial densities. So, returning to the water droplets, they are
growing until they become too massive and superposition breaks down; it breaks
down when its lifetime becomes short enough, but for that t ~ h^{3}/G^{2}m^{5}, so it breaks down rather sharply in mass [5]. At 10^{14}
g the lifetime of the superposition is ~1000 s, longer than the working time of
the Wilson chamber, but at 10^{13} g it is
only ~0.01 s. So the breakdown happens somewhere between these masses.
We stop here. Mutatis mutandis, one's first guess is that in an emulsion the latent picture may or may not be a superposition (substantial interactions are present), but if it is, then during development the superposition breaks down similarly than in the Wilson chamber. The final result is a unique stochastic track, and b~10^{8} cm winding (measured in SPS) seems to need just the masses present in the emulsion.
This is the reason that we presented the rather surprising idea that spurious scattering may have a Quantum Gravity origin.
7. CONCLUSION
If it has, then by SPS measurements one can measure some deviations from orthodox QM. However three problems hinder the settling the question is or is not SPS an effect of unorthodox QM. First, the necessary calculations in emulsion are nearly impossible technically. Second, the analogous but much simpler Wilson or sparkle chambers cannot be used, because for measuring a b~10^{8} cm, coordinate measurements of less than 1 mm error would be needed, possible only in emulsions. And, third, no new tracks for spurious scattering measurements have been produced in the last 20 years, since the energy determination method via Coulomb scattering have become obsolete.
And still, the topic deserves some attention. We think that this paper at least demonstrated that (some?) Quantum Gravity theories predict a winding of a track in emulsion due to the breakdown of superposition at least with parameters not too different from the measured ones. If the theories reflect something of the physical reality, then in the worst case their effects are hidden by some processes (e.g. the mosaiclike disintegration) mimicking them in similar orders of magnitude. SPS measurements seem so cheap that it would be profitable to check at least the emulsionindependence of b in a methodical way, with a serious accuracy. If there is an independence then SPS cannot be a mosaiclike disintegration. Furthermore, the measurement error of b could be substantially reduced by irradiating some dozen emulsion plates with monoenergetic protons at a few energies and then evaluating the tracks in ~1 manyear. After that we would know if the problem is really viable for theoretical investigation.
ACKNOWLEDGEMENT
This work was partly supported by OTKA Fund 1822.
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