Quantum effects due to extra space multiconnectivity
C. GAUTHIER
Department of Mathematics and Statistics
Université de Moncton - Moncton, NB, Canada E1A 3E9
(ricevuto il 31 Ottobre 1996; approvato il 23 Dicembre 1996)
Summary. — Within a new unified field theory built on a hyperspace-time with a multiconnected extra space, we present pure geometrical interpretations of many quantum aspects of observables as well as the unavoidable random part of any observation. This multiconnectivity also affects Heisenberg’s uncertainty principle by introducing lower and upper bounds for uncertainties of conjugates attributes. It follows that both space and time are then formally quantized.
PACS 12.10 – Unified field theories and models.
1. – Introduction
It is nowadays a common belief that the fundamental concepts of quantum mechanics should be essential ingredients of any attempts to unify theoretical physics. However, this point of view raises difficulties so serious that it is not meaningless to try other methods of unification. Many have already been proposed [1]. In order to justify the relevance of a new unified field theory of the Kaluza-Klein type, we examine in this paper the possibility of geometrizing quantum theories instead of attempting to quantize gravitation. Let us immediately notice that adherence to the latter perspective does not imply a support to the idea that the laws of physics are governed by «hidden variables» through a cryptodeterminism ([2], p. 155). Our goal here is rather to show how one can explain the quantum aspect of nature and the existence of the unavoidable random part of any observation through the structure of the geometrical framework within which the new unified field theory is set.
The basic manifold of the unified field theory we consider supposes a hyper-space-time with a multiconnected extra space called superior space. We have shown [3] that this multiconnectivity generates a discontinuous gauge which provides a geometrical interpretation for the apparent discretization of fields, the wave- corpuscle duality and the wave function collapse. After a short review of some of these ideas in a more general context, we will obtain a combinatorial interpretation for the amplitude of probability. We will also indicate how the mass spectrum of fundamental particles could be related, at least partially, to the distribution of divisors of the order of the group associated with the superior space multiconnectivity. Finally, we will show that this
multiconnectivity affects Heisenberg’s uncertainty principle by introducing lower and upper bounds for uncertainties of conjugates attributes. A consequence of this is that both space and time turn out to be naturally quantized that is, to vary discontinuously in quantum jumps.
2. – The superior gauge
Let us consider a hyperspace-time described by a (4 1 d)-dimensional manifold of the form
V4 1d»4 M4
3
(
Sd(K ) OG
)
, (1)where M4 represents space-time. The second term of the product in (1) is called superior space and is here given by the quotient space Sd
(K ) OG, where Sd(K ) is the
sphere of odd dimension d and small curvature K, while G is one of its discrete and cyclic invariance group with a very large order N([4], p. 39, 224). Notice that even if the superior space scalar curvature is constant, the one of V4 1d may vary because the
scalar curvatures of M4and Sd(K ) are here independent. The latter property does not
hold, for instance, in supergravity, where this leads to the cosmological problem ([5], p. 1151; see also [6]).
Let p : Sd
(K ) KSd
(K ) OG be the natural projection of the fibre bundle Sd(K ) with
group G and pg21 its cross-section corresponding to a given g G . The submanifold
pg21
(
Sd(K ) OG)
has the same local geometrical properties as Sd(K ) and can thereforeserve as the effective superior space needed to describe the physics related to one or another of the Lie algebras associated with the geometrical structure of Sd(K ). This submanifold is also congruent with the fundamental polytope determined by G on
Sd(K ) and homeomorphic to Sd[6]. These characteristics of the superior space are at
the origin of our geometrical interpretation of the discretization of continuous gauge fields.
To present this interpretation, let us first notice that the manifold V4 1dcan be seen as a fibre bundle with group G and base M4. This bundle will be called superior fibre
bundle. Each of its fibres is made up of N polytopes identical to the fundamental one. The freedom associated with the choice of one of these polytopes as effective superior space constitutes a degree of physical freedom similar to those of continuous gauges in standard field theories. Since the transfer from one polytope to another is carried out through G, the latter becomes the group of a new gauge called superior gauge. The relation between superior gauge values at two nearby points in M4is then determined by a connection, the form of which is given by the superior gauge potential, called superior potential and denoted by A(x). Any continuous gauge field with characteristics subjected on one hand to one or another of the local symmetries of Sd(K ) and on the
other hand to its discrete symmetry given by G will then be associated with a well-defined superior potential represented by a distribution. This potential affects any continuous gauge field interacting with another field making the former appear through quanta in M4. The process through which quanta are realized will be sketched in the last part of the next section (see also [3]).
3. – The toriis
Let g be an Einstein metric on a compact connected CQ-manifold W. By an Einstein
deformation of g we mean a 1-parameter family g(t), t R , of Einstein metrics on W such that g( 0 ) 4g and the volume of g(t) is constant for all tR. If for each Einstein deformation g(t) of g there exists a 1-parameter family d(t), t R , of diffeomorphisms such that g8(0) 4d[d(t)* g]OdtNt 40, where d(t)* is the formal adjoint of d(t) with
respect to the global inner product for tensor fields on W, then g is said to be rigid. According to this definition, we immediately have that the metric on Sd(K ) is rigid [7]; to shorten we will simply say that Sd(K ) is rigid.
The geometrical rigidity of Sd(K ) is a property that also applies to each polytope of
the multiconnected sphere. However, this multiconnectivity allows a direct passage between any two of these polytopes over any path in M4. This enables one to tie up
V4 1d with an intrinsic flexibility F that goes with its geometrical structure. We can ascribe a physical meaning to the field F within a theory of gravitation on V4 1d.
Following Sakharov [8], who has seen the gravitational constant G as an expression of the flexibility of M4, here F would become an expression of the flexibility of
V4 1d[9].
The field F is defined on V4 1dand takes its values in the space of a representation
of the group G. One can therefore admit that F 4NFNexp[iu] has an energy density potential of the form
V(F , R) »4 (21cos Nu)(F4
2 2 B2R22F2) ,
where the angle u [ 0 , 2 p[ is measured with respect to a given initial value, R is the scalar curvature of V4 1dand B is a constant corresponding to its minimal flexibility. If
R c 0, the potential V has a local maximum at F 4 0 and N absolute minima at Fk»4 BR21 exp [ 2 kpiON] , k 40, 1, R, N21 ,
(2)
which are located in a circle when R is constant. The interior of the polytope serving as effective superior space required to describe any observation in M4 is then associated
with the bottom of one of the N basins of V. The field F will play here the same role as the one of Higgs field in standard field theory.
It would be possible to explain the existence and characteristics of the potential V on arguments related to the early evolution of our universe. The existence of the group
G would then be linked up to a crystallisation of the superior space structure which
would have accompanied the drop in the temperature during inflation. One can find a stratification in the inhomogeneity of the universe similar to the one observed if one admits that the order N of G increases discontinuously a certain number of times during inflation. This hypothesis comes down to suppose that the superior space structure crystallises more and more finely when temperature drops. This implies that the present-day order of G should be a round number, that is a number which is the product of a considerable number of comparatively small factors ([10], p. 358). The simplest example of a round number is a number of the form qr, where q is a small
prime number and r is a large positive integer.
For the unified field theory we are looking for, the unicity of the base structure corresponding to Minkowskian M4
compels us to fix in (2) a unique k 4k0for all x M4,
that is to settle the superior gauge over each point of M4. Let A(x) be the new expression of A(x) associated with this choice. One can interpret the modification of
A(x) into A(x) as the result of a transformation principle in V4 1dwhich acts locally in
M4, inside each quantum of the continuous gauge field quantized by A(x). The outcome of this is a geometrical confining of certain regions of V4 1dwhich appear as points in
M4. The geometrical objects in V4 1ddetermined in this way will be called toriis. The
reason of this appellation is that such a geometrical object in V4 1dcan be seen as a site
of transition between two horizontal sections of the superior fibre bundle, like the torii which serves as site of transition between the common world and the world of spirituality inside Shintoist places and temples.
The topological structure of a torii is given by S1
3 Sd, where the circle S1 is
associated with the time of M4[3]. Unlike what happens in string theory, for example,
here the existence and the topological structure of the geometrical objects representing fundamental particles, that is the toriis, follow from the geometrical structure of V4 1d rather than being postulated.
4. – The gauge of the zero
From now on, our argumentation will be limited to a Minkowskian region of M4.
Notice that this hypothesis does not imply that the value of F is infinite in this region because the scalar curvature of V4 1d is the sum of the scalar curvatures of M4 and
Sd(K ); in fact, we then have
NFN4 B[d(d 2 1 ) K]21.
We have seen that the choice of a base structure for the unified field theory comes down to fix k 4k0in (2). The latitude on the localization of the zero angle for the set (2)
means that any function
F8 (x) »4 Fk0 k0exp [ia(x) ] , (3)
where a(x) R , forms a base structure as good as Fk0. The function (3) corresponds to a rotation of (2) and must be interpreted as the effect of a new local and continuous gauge having U(1) as symmetry group. This gauge will be called gauge of the zero or zero gauge. One can observe that the covariant derivative with respect to the superior gauge is also covariant with respect to the gauge of the zero because fixing the latter comes down to assigning the position of F0 around the centre of V. The field here
associated with the gauge of the zero is thus identically zero.
The gauge of the zero notion is useful to obtain a straightforward interpretation of the wave-corpuscle duality. To see that, let us consider the Lagrangian density L describing an isolated interaction between the field F and the superior potential A defined with respect to x0M4:
L »4 1 2 (D
jF)†(D
jF) 2V(F) ,
(4)
where Dj denotes the covariant derivative with respect to the superior gauge, the
dagger superscript indicates the Hermitian conjugation and j 40, 1, 2, 3. Let us now suppose a test particle sensitive to an exterior field discretized by A with a trajectory in
M4 passing through x
one of F 8k0(x) before it reaches x0. The Lagrangian density is then given by 1 2 (D j F8 )k0 †(D jFk8 ) 4 B0 2( 2 R2)21¯j a¯ja ,
for which the Euler-Lagrange equation gives p a 4 0 . (5)
If we interpret the function a as a new field defined on M4, eq. (5) identifies this field
to a scalar wave without proper energy in M4, because it propagates in it accordingly to the minimal energy structure of V.
The existence of this kind of waves, called toriidal waves, can be explained in the following way. The measurement operation of a characteristic of any physical system in
M4 requires to compare at a given point of M4 the value of this characteristic to a
reference value. Since any observer can always compare the measurements of any of its base structure characteristics at any point of M4 and the measurements of each of
these characteristics must be identical amongst themselves, its base structure must correspond to only one value of the function a(x) for all x M4. This means that the
base structure of any observer must be autocoherent. However, before the observation that brings in Fk0, the base structure Fk0has in V the angular freedom associated with the latitude on the localization of the angle zero for the set (2). This base structure will be said free with respect to the autocoherent structure of the observer. When the observation leading to Fk0takes place, a phase will in general appear between the free and the autocoherent base structures. The existence of this phase explains the introduction of the scalar wave a.
When the test particle reaches x0, where it undergoes the interaction described by
(4), the zero and superior gauge symmetries are spontaneously broken. The kinetic energy associated with the field Fk0 is then given by
1 2 (D jF k0 8 )†(DjFk8 ) 4 g0 2B2( 2 R2)21Aj(x 0) Aj(x0) , (6)
where g represents the coupling intensity of the field A with the base structure F 8k0. The quantity g is therefore a function of the wave a characteristics. The right-hand side of (6) corresponds to an energy associated to the field A , energy that we will tie up to the torii arising at x0. At the time of interaction, the transformation of A into A at x0
therefore leads the toriidal wave to emerge as a point in M4and with a specific energy
given by (6) because of the torii presence at x0.
The way toriidal waves are generated implies that they cannot be decomposed in more elementary waves having a physical meaning. Therefore, we will have a unique class of toriidal waves for each positive integer dividing N. It follows that the functions representing these fundamental toriidal waves generate the algebraic structure of a module over the ring of integers [11]. We will call toriion the geometrical object which travels as a toriidal wave in M4and emerges as a point, when this wave is involved in an
interaction. The energy of a toriion has a physical meaning only when its toriidal wave takes part in an interaction. This energy results from the interaction between physical systems and is not a characteristic of an isolated system. If we interpret the toriidal wave as the de Broglie’s wave of the toriion, the preceding observation means that the spontaneous symmetry breaking of the zero and superior gauges resulting from an
interaction can be used to explain its emergence as a point in M4. The zero gauge
symmetry would then be responsible for the wave attributes of the toriion, while the superior gauge symmetry would explain its corpuscular aspect.
We have seen that the propagation of a toriidal wave in a Minkowskian region of M4
results from the gyration in V of a free base structure with respect to an autocoherent base structure. This gyration is observed from the bottom of V. It follows that the invariance of toriidal waves with respect to translations in M4 is related to the invariance of the zero gauge with respect to rotations in V. According to Noether’s theorem ([12], p. 77), the zero gauge invariance with respect to the group U(1) then implies the conservation in M4of the energy-momentum associated with toriidal waves.
Consequently, when a toriidal wave is emitted, the potential V linked to its emission point in M4 is subject to a recoil gyration, with respect to the autocoherent base structure considered, the size of which being proportional to the energy associated with this wave. Since energy-momentum is conserved during the wave propagation in M4, it will be the same with the backward gyration speed of its potential V. The backward gyration of V causes an increase in the toriidal wave frequency n. The energy E of a toriion being in this way proportional to its toriidal wave frequency, we will agree to link them up by Einstein’s formula: E 4hn, where h is Planck’s constant.
To the backward gyration of V, at the time of a toriidal wave emission, corresponds a direct gyration, with respect to the autocoherent base structure, when the wave takes part in an interaction. This direct gyration explains why the toriion then appears with a momentum in the direction of propagation of the toriidal wave in M4 before the
interaction. Each toriion is thus characterized by an energy-momentum due to the form of V, the conservation of which being explained by the circular symmetry of the latter, that is the zero gauge invariance with respect to the group U(1).
One can resume the preceding arguments without a direct reference to the potential V. Indeed, the choice of an autocoherent base structure amounts to fixing a horizontal base section in the superior fibre bundle. The existence of toriidal waves then results from the freedom of the multiconnected sphere Sd(K ) OG to spin on itself around a given axle, with respect to this horizontal base section, over any trajectory in
M4. The toriion appears as a corpuscle when an observation stops the rotation of
Sd
(K ) OG and forces the superior fibre bundle section, with respect to which the observer identifies this rotation, to coincide with the observer’s section.
5. – The probability wave
When an interaction occurs, the zero and superior gauge symmetries are spontaneously broken and the minimal energy value of V is realized as the bottom of one of its N basins. This interaction will be observable only if this basin coincides with the one of the observer when the observation takes place. To him, the toriidal wave then ceases to exist and appears as a corpuscle. However, knowing that a toriidal wave has manifested itself with respect to the bottom of a specific basin of V does not inform us on its properties such as its length wave. When a wave ends at the bottom of a particular basin of V, one can only say that its length wave divides the circumference of the circle of (2) in an integer number of arcs of the same length. We will now show how the last remark leads to a justification for the relevance of the notion of amplitude of probability in the quantum formalism.
functions representing the different toriidal waves capable of a manifestation relative to the basin bottom of the observer. The functions Cngenerate a vector space S having
a dimension given by the cardinality of D(N). The conservation of energy-momentum, which applies to each horizontal section of the superior fibre bundle, implies that when the toriidal wave Cn is absorbed by the torii associated with a given interaction, a
toriidal wave capable of producing the same energy-momentum as Cnis emitted by this
torii. The new toriidal wave corresponds to a gyration in V opposite to the one that has generated the first wave; it is symmetric with respect to time and can be represented by its complex conjugate C *n. A manifestation of the wave Cnin M4is thus related to
the coincidence of Cn and C *n in the torii of the considered interaction.
Taking into account the large number of divisors of the round number N, through all its distinct factorisations where order is irrelevant, we have that the number of its smaller divisors is larger than the number of its larger divisors. To be more explicit, let us determine the distribution of divisors of a round number of the simplest expression, that is of N 4qr, where q , r N* and q is a prime number. Here we have to consider all
distinct factorizations of N where order is irrelevant. We can immediately notice that such factorizations of N contain only divisors larger than 1. It is then easy to see that the distribution of divisors larger than 1 of N, through all its distinct factorizations where order is irrelevant, follows directly from the set of partitions of the exponent r. For example, if we suppose that N 425, we have that its distinct divisors larger than 1
are: 2, 22, 23, 24 and 25. The numbers of times these divisors appear in the set of
distinct factorizations of 25 are then respectively given by the numbers of times the
numbers 1, 2, 3, 4 and 5 appear in the set of partitions of 5, that is ]5, 411, 312, 3 1111, 21211, 2111111, 111111111(. These numbers are respectively: 12, 4, 2, 1 and 1.
In order to deal with case N 4qr, we designate by p(n) the number of partitions
of n where order is irrelevant and by e(r , k) the number of times that the number
k 41, 2, R, r appears in the set of partitions of r. Adopting the convention p(0) 41,
the relation r 4k1 (r2k) leads to
e(r , k) 4p(r2k)1e(r2k, k) .
Therefore
e(r , k) 4p(r2k)1p(r22k)1R1p
(
r ( mod k))
, (7)for k 41, 2, R, r. The distribution of parts in the set of all partitions of r can thus be obtained from the values of p(n), for n 41, 2, R, r21. The distribution of divisors larger than 1 of N, through all its distinct factorizations where order is irrelevant, then follows from the preceding one when we replace the values of the independent variable
k 41, 2, R, r by respectively qk
, for k 41, 2, R, r. By way of example, the curve of fig. 1 describes the number of divisors u of q100, through all its distinct factorizations where order is irrelevant, vs. the logarithm in base q of this divisor u. The values of
e( 100 , logqu) have been obtained using (7) and the table of values of p(n) given in [13],
pp. 238-240.
The fact that the number of small divisors of N is larger than the number of its large divisors means that most of the Cn, n D(N), correspond to small values of n.
But the smaller n is, the larger the length wave of Cnwill then be. The value of n is
thus proportional to the frequency of Cnand then, to its associated energy through the
Fig. 1. – The distribution of divisors of q100vs. their logarithms in base q.
be related to the distribution of masses in M4. This characterization of masses capitalizes only on the superior space multiconnectivity, that is on its global symmetry associated with the group G. For a more realistic characterization of masses, in particular for those particles subject to one or more continuous gauge fields such as the electromagnetic field, one should also take into account the superior space local symmetries to which these fields are related.
To see how probabilities enter this framework, let anbe the number of waves Cnin
M4. The conservation of the energy-momentum then calls for the existence of anwaves
C *n so that the coincidence of a wave Cn with a wave C *n in a torii can occur in an2
different ways. Knowing that a wave has been observed, the probability that this wave will in fact be described by Cn is then given by
cn2»4 an2
O
!
j D(N)aj2.
If N 4qr, it is easy to see that a
n4 e(r , n). The probability for the observed wave to be
described by a Cn is then given by
c2
n4
(
e(r , n))
2O
!
j 1 rWith the cn, n D(N), one can form the function
C»4
!
n D(N)
cnCn.
The function C is a pure mathematical construction serving solely to list possible physical states with their corresponding probability of observation: only the functions
Cnhave a physical meaning. If we represent a physical system by the function C, then
we only mean that this system will be in the state Cnwith probability cn2. The states Cn
can thus be observed only one by one: no linear combination of Cn has a physical
meaning. The probability of transition from the state Cm to the state Cn then
corresponds to the coincidence of C *n with Cn4 ACm, where A is a matrix operator
acting on S. The fact that S is a vector space guaranteed the probability amplitude additivity, a property which allows one to take care quantitatively of the wave interference effects.
The quantum states of a physical system are fundamental elements of the statistical formalism that lay behind quantum field theories. We have seen that these states can be interpreted by toriidal waves. One can notice that this interpretation is compatible with the idea of identifying the operation of measuring any observable of a physical system with that of an operator acting on the vector space made up of the states of this system. In our interpretation of the wave function, the Hilbert space of quantum mechanics must be replaced by the finite-dimensional vector space generated by the
Cn, n D(N). The fact that a Hilbert space is well suited for this finite vector space is
explained by the property of N to have a very large number of divisors. 6. – The uncertainty principle
We will now show how Heisenberg’s uncertainty principle gets into the perspective adopted in this paper. Let us first say that this principle follows from pure mathematical arguments and applies to any wave phenomenon which manifests itself locally in M4; it will therefore apply to toriidal waves.
When a toriidal wave takes part in an interaction, we know that the zero and superior gauge symmetries are spontaneously broken and the minimal energy value of
V becomes the bottom of one of its N basins. However, the function describing the
toriidal wave does not necessarily have the value corresponding to a basin bottom when those symmetry breakings occur. This implies that the torii then created is endowed with a potential energy, proportional to its height in the basin, which contributes to its total energy. Its movement toward the bottom of the basin resulting from this potential energy can be put forward to explain the existence of uncertainties on the energy and the time related to the appearance of a toriion in M4. These uncertainties are due to the
fact that we do not know the toriidal wave phase when the symmetry breakings occur. We can characterize these uncertainties in the following way.
Let us consider the height of the point in V corresponding to the one in M4where a
toriion appears. The energy associated with this height causes an uncertainty DE on the observed energy of the toriion. We also have that any variation DE on the energy of a toriion is related to a variation Dn on its toriidal wave frequency by DE 4h Dn. However, the uncertainty Dt on the time of appearance of a toriion in M4 is related to
DE by Heisenberg’s inequality:
DE Dt FhO2p . (8)
Therefore, the lower the uncertainty on the toriion energy is, the higher will be the uncertainty on the time of its appearance in M4. Conversely, if the uncertainty on its energy is high, then the time of its appearance can be more precise. The latter interpretation of the uncertainty principle is the standard one. When taking into account the potential V, one can add the following remarks to this interpretation.
The maximal uncertainty on the observed energy of a toriion is in fact determined by the maximal height, with respect to the basin bottoms, of the curve r(u) describing the radial minima of V. This height equals two times the amplitude of the toriidal wave associated with this toriion. The maximal values in V of the curve r(u) are reached when u 4 (2k11) pON, k40, 1, R, N21, and are given by 2B4
OR4; its minimal values are reached when u 42kpON, k40, 1, R, N21, and are given by 23B4
OR4. The integration over the usual 3-space of the difference between these maxima and minima, multiplied by the Dirac distribution defined with respect to the point in M4
where the toriion is realized, gives
DE G2B4OR4. (9)
The units of the constant B being that of the inverse of a length to the cube, the right-hand side of (9) is of the same units as its left-hand side, when one takes into account the integral involved.
The uncertainty Dt on the time of appearance of a toriion in M4 is for its part
characterized by DlOc, where c is the speed of propagation of toriidal waves in M4,
which according to (5) can be identified to that of light, and Dl denotes the uncertainty on toriidal wave length l. Let C be the circle of radius r 4BOR passing through the N basin bottoms of V. The maximal uncertainty on the measure of l is then given by half of the arc length of C corresponding to the angle at the centre of V which intercepts the edges of the basin where the considered toriion is realized. This angle Du equals 2 pON. Therefore
Dl Gr DuO2 . Consequently
Dt GpBOcNR . (10)
The use of (8), (9) and (10) then leads to
hR4 O4 pB4G Dt G pBOcNR (11) and chNRO2p2 B GDEG2B4 OR4. (12)
Being interdependent through (8), the uncertainties Dt and DE associated with the realization of a toriion are thus bounded from below and from above.
We obtain similar results when considering Heisenberg’s inequality for uncertainties on spatial position Dx and momentum Dp . To this end we use the relations Dx 4c Dt and Dp4DEOc. The inequalities corresponding to (11) and (12) are then given by
and
hNRO2p2
B GDpG2B4
OcR4.
Since the numerical values of B, N and R 4d(d21) K are not yet known, it is not possible for now to attach numbers to these bounds. However, one can observe that the existence of positive lower bounds for Dx and Dt means that both space and time are formally quantized: their quantum nature follows from the superior space multiconnectivity. One consequence of a positive lower bound for Dt ([14], p. 131 and references therein) is that the restriction of the unified field theory considered here to quantum phenomenology should be exempt of infrared divergence ([15], p. 168).
7. – Discussion
The fact that toriidal waves are determined by the geometrical structure of the hyperspace-time implies that the analysis of any toriidal wave phenomena should be done directly in the manifold V4 1dinstead of M4. From this point of view, if one identi-fies the toriidal wave with the wave function of the associated toriion, it is easy to understand why the latter collapses when the toriidal wave takes part in an interaction. This collapse results from the spontaneous symmetry breaking of the zero and superior gauges which stops the gyration of the free base structure associated to the toriidal wave considered, gyration with respect to the observer’s autocoherent base structure. To this observer, the spreading over M4 of the toriidal wave then ceases to exist and
the wave appears as a corpuscle with specific physical properties. These properties have physical meanings only when the toriidal wave takes part in an interaction. The same observation can be made by any other observer having an autocoherent base structure in phase with the one of the first observer.
The preceding results are for toriions. Since the wave function of any particle obtained through aggregate of toriions, no matter how they aggregate, is a combination of their wave functions, these results will also apply to particles made of toriions.
The easy use of the scalar curvature of V4 1d to define potential V can give the
feeling that the perspective adopted in this paper could allow one to study gravitation. We will consider this question in a forthcoming paper [9].
* * *
The author is grateful to P. GRAVEL for his helpful comments.
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