Special Relativity and Quantum Wave Nature of Matter in “Bridge Theory”

Massimo AUCI*

Department of Science

Scuola Internazionale Europea “Altiero Spinelli”,

Via Figlie dei Militari 25, 10131 Torino, Italy

(*) Member of the New York Academy of Science

The phenomenology described in a quantum-classical “Bridge Theory”, relative to the

production of a Dipolar Electromagnetic Source (DEMS) during the interaction of a pair of

elementary charged particles, gives to Quantum Mechanics and Special Relativity the

possibility of finding a common and coherent physical origin. The DEMS is responsible for

a quantum-like exchange of energy and momentum between the interacting particles. This

phenomenology allows us to propose a new interpretation of the wave-matter dualism,

supplying new bases for a new “quantum-relativistic” Theory.

2

1) Introduction

The history of Physics in the last century shows how two apparently incompatible

behaviours, such as those of a wave and of a particle, coexisting both in two slits photon

diffraction and in the Davison and Germer electron diffraction experiments, are responsible

for the birth of one of the conceptual fundaments of modern Quantum Physics (MQP): the

wave-matter dualism.

In ref. [1], Louis de Broglie, writing about his "hypothesis" of the autumn 1923 (DBH),

manifests his initial purpose to arrive to a theoretical synthesis of waves and particles in

which the particles would appear as a peculiar behaviour of a wave structure that controls

their propagation. De Broglie affirms that his attempt would have not succeeded without

the introduction of relativistic concepts essential to connect the high-speed peculiarities of

matter with the wave propagation phenomenology.

Today, in the light of the “Bridge Theory” (BT) proposed by the author and by G.

Dematteis in ref. [2], we introduce a new phenomenology able to explain the dualism in

term of real waves. Doing that, we will prove that the joint use of Einstein’s Special

Relativity (SR) and of Quantum Mechanics (QM) is a conceptual obstacle to accept

unconditionally the dualism as an experimental phenomenon, i.e. as a double instrumental

perception of the same phenomenology, as Louis de Broglie believed. In fact, by accepting

dualism as MQP does, we find a phenomenological incompatibility in the contemporary

description of matter like a local corpuscle, the motion of which is described by SR, and as

a wave function, as the Copenaghen interpretation of QM requires. QM only establishes

3

probabilities for a given observation, but a precise description of what might exist or

happen between two successive observations does not belong to the theory. This entails a

deep incompatibility in the contemporary use of QM and SR. This incompatibility has been

analysed in the past quite pertinently by Sachs (for a bibliography see refs. [3-4]) from the

point of view of SR and by many other authors from the point of view of the non-

covariance problem, particularly evident in the usual QM formalism (for a bibliography see

refs. [5-12]).

The fundamental incoherence originated by the synergic use of QM and SR in the wave-

matter description can be synthesised by two simple aspects coexisting in matter

behaviour:

(i) A particle is a localised amount of energy with specific observable properties

such as mass, charge, spin, etc.

(ii) A wave is assumed to be diffuse, and with a wide range of oscillations in

vacuum.

Following the philosophy of MQP, the points (i) and (ii) may alternatively refer to the

same entity forming a complementary description, without SR and QM, which are the

fundaments of MQP, having a true conceptual and formal compatibility.

In this article, we will give great relevance to the phenomenologies described by de

Broglie’s wavelength in QM and by Einstein’s SR (1905), proving how these two different

ways of appearing of matter in nature, can find in Bridge Theory a common and coherent

physical origin. In this context we will use a simple mathematical approach, with a

formalism only apparently having to do with Classical Mechanics, Special Relativity or

4

Quantum Mechanics. In fact, each concept or formalism used is coherent with what is

phenomenologically described in BT. As a consequence the aim of this work is to prove

that wave-matter behaviour in QM and relativistic description of matter represent just two

complementary aspects of the same phenomenology. In this sense, the DBH ends out to be

a winning hypothesis and SR and QM may acquire a new formal and conceptual derivation

from a self-invariant theory, the electromagnetism, which is implicit in the BT formulation.

2) A short introduction to Bridge Theory

The BT (see ref. [2]), based on the fundamental works in refs. [13-15], is an approach

to unifying Classical and Quantum Electrodynamics based on the role of the transversal

component of the Poynting vector, which is able to localise energy and momentum in

agreement with quantum predictions for a photon.

We start by assuming that matter is created in charged pairs the mass value of which is

not fundamental for the actual theoretical developments. Pairs interact electromagnetically

producing EM sources. The characteristic total energy and momentum for the interacting

real or virtual pairs produced by each Dipolar Electromagnetic Source (DEMS) set the

source wavelength. Consequently, the energy and momentum of the exchanged photon are

fixed by the energy and momentum of the colliding particles.

The theoretical evaluation of the Planck and Sommerfeld’s constants [16] is totally in

agreement with their experimental values, so when in the theory we use the symbol h we

assume the usual numerical value but not ordinary QM, because in BT the quantum

phenomenology is replaced self-consistently by the photon-like behaviour of the DEMS.

5

Every time a real or a virtual pair interacts, a DEMS is produced. The photon-DEMS

equivalency allows us to consider the DEMS like an exchanged photon carrying energy

and momentum between the interacting particles. This corresponds to introducing the first

quantization in the electromagnetic field.

3) The Dipoles

In order to observe a particle in the lab-frame, we need it to interact with at least another

particle producing a measurable effect. Let us consider a charged particle moving through

matter. Independently on the value of its charge, when it crosses the atomic

electromagnetic fields of matter, it interacts with them producing a distribution of virtual

and real dipoles along its path. The effect is the production of a space-time distribution of

Dipolar Electromagnetic Sources (DEMSs), i.e. of propagating EM waves, one for each

combination charge-anti-charge produced during the passage of the particle. To simplify,

we will examine this same phenomenology in the case of just one DEMS.

In BT, we define a collision like an electromagnetic interaction between a pair charge,

anti-charge qq producing a DEMS. Collision could be broken up into two different space-

time phases. The incoming phase, that we define "alpha", corresponds to the DEMS

building, consisting in the approaching phase of the charges, during which the interaction

localises the energy and momentum of a photon in the source zone of the DEMS [17] (see

also ref. [18]). The outcoming phase "omega" corresponds to the charge scattering, during

which the source is destroyed: the products emerging from the source go away from one

another. Wave emission occurs during both phases, but only during the alpha phase, a part

6

of the energy and momentum of the incoming charged particles is electromagnetically

converted and localised inside the source zone, giving the available amounts of energy and

momentum that characterise the exchanged photon.

4) The underlying idea

The de Broglie's wavelength shows formally a direct dependence on the Planck's action

value. In BT the Planck’s action is not a true constant because its value depends on the

external EM conditions in which the DEMS is produced [19]. In this sense, h is a pseudo-

constant slowly varying with the external conditions. Since in standard QM the Planck’s

constant h is a value assumed a priori as a universal constant experimentally derived from

the energy quantization. In QM the DEMS phenomenology is a hidden physical process

that could have to do directly with non-evident physical phenomena.

Using BT, the presence in QM of a phenomenon depending on Planck’s action must be

considered as an evidence of the presence of an active DEMS in which a "quantum" of

energy and a propagating EM wave, respectively associated to the transversal and radial

components of the Poynting vector, are coexisting [20].

Following this idea, we will prove that wave-matter duality may be considered as

originated by the DEMS phenomenology. If we consider arbitrarily the lab at rest, the total

energy and momentum carried by an impinging charged particle set the dipole momentum

of the DEMS and consequently the wavelength of the photon. In this sense, the DEMS acts

like a causal link between the relative motion of two colliding particles and the associated

EM wave.

7

5) Transition →qq DEMS: general phenomenology

Let S0 be the centre-of-mass (CM) of a DEMS of period T; we assume that when the

omega phase starts, the source begins to be active producing a gradual reduction of the

energy background. If in space just one source exists, the evolution of the source is

observed by frames imbedded in the EM field of the DEMS only. Therefore, the time of

each observer can be considered clocked by the local progressive energy degradation of the

source.

When a DEMS is produced, an observer placed in an inertial frame S external to the CM

of the DEMS sees the energy and momentum carried by the interacting particles only in

part stored in the source zone, i.e. transferred to the DEMS. In fact, the residual amounts of

energy and momentum supply respectively the kinetic energy and the momentum of the

CM of the source.

In order to characterise two interacting particles independently on their nature, we

define in the frame S their masses, energies and momenta m, E1 1, p and m*, E2 2, p ,

respectively for the particle 1 and 2. During the EM interaction the total energies of the

particles vary as an effect of the contribution given to the EM energy density localised

inside the DEMS, i.e. the energy and momentum acquired by the DEMS depend causally

from the initial dynamical conditions of the two colliding particles.

The observer placed in S measures the total collision energy

E E E= +1 2 (1)

and the total momentum

8

P p p= +1 2 , (2a)

where

2*

2

11

==

cm

cm

βp

βp (2b)

are the momenta of the colliding particles with respect to an external observer.

Let ES and PS be the energy and momentum of the DEMS, i.e. of the photon exchanged

during the particles collision. According to BT

cE ss P= . (3)

Considering the definitions (1-2a), we write the energy and momentum conservation

laws for the DEMS creation as:

0∆+= sEE (4a)

cs

∆+=r

PP (4b)

where 0∆ and c∆r

are respectively the residual energy and momentum not involved in the

source production with respect to the observer. No other assumption concerning the nature

of the total energy (1) is necessary, because we will prove that by using the qq -DEMS

energy and momentum conservation (4a-b) and the fundamental concepts of the BT, we

will obtain results consistent with both SR and QM in their standard formulation.

With respect to an observer placed in the frame S0, i.e. in the CM of the collision, the

energy and momentum of the particles are completely absorbed by the DEMS creation, so

we write:

.0 , 00 =∆≡∆=∆r

(5)

9

Any other inertial frame S ≠ S0, placed externally to the CM is plunged into the EM

field of the source. With respect to his own position, the observer can break the total

momentum of the collision (2a) into two orthogonal components, a radial one and a

transversal one with respect to the observer. The radial component is associated with the

DEMS creation, i.e. with a wave source with total momentum along the direction of the

observer:

kP h=s ; (6)

this component characterises respect the frame S the energy and momentum involved in the

collision during a finite time interval ∆t. On the contrary, the transversal one c∆r

is

associated with the spatial motion of the CM of the DEMS relatively to the observer.

With respect to the inertial frame S, the momentum and the kinetic energy of the CM of

the DEMS appear supplied by the momentum and energy fractions not participating to the

DEMS production, i.e. by the energy and momentum “not appearing exchanged” between

the colliding charges. Since relatively to a same DEMS the observations of each inertial

observer must be equivalent, this requires the energy and momentum of the CM respect to

the observer be balanced:

∆=∆0 . (7)

To prove eq. (7) we square eq. (3), using eqs. (4a), (4b) and (3) we obtain

( ) 220

222 )cos(2 ∆−∆=−−− θPcEEcPE s ; (8)

10

θ is the angle between the momenta P and Ps. Considering that in the frame S0 the right

term of the eq. (8) is identically equal to zero (see eqs. (5)), in each other frame S eq. (8)

must be zero, i.e. for each observer

0220 =∆−∆ . (8a)

Eq. (8a) is equivalent to eq. (7).

Eqs. (8) and (8a) yield

( )θcos2

222

PcE

cPEEs −

−= . (9)

Equation (9) shows that each observer placed in a different inertial frame sees a DEMS

(photon) with a different energy, i.e. by observing a different projection of the resulting

total momentum, he measures a different exchange of energy between the colliding

particles.

Considering now an observer placed in the frame of one of the two particles 1-2

(symmetrically: impinging and target particle), θ is the effective angle between the

direction of the dipole moment of the DEMS and the trajectory of the impinging particle. If

we assume that the frame coincides with the target particle 2 ( S ≡ S2 ), from the previous

definition (2a)

1pP = , (10)

using (2b) we get

cmP 1 β=≡ p , (11)

where β = β1 is the dimensionless velocity of the impinging particle 1 with respect to the

target 2.

11

Following BT, we describe the interaction between a pair of charges by assuming one of

the two particle as a rest target. If R is the vector connecting the two particles, the one

moving along the trajectory r(t) interacts at a time t at the conditions occurring at a time

tt <' when the charge position ( )'' trr = is delayed by a time 'tttdelay −= necessary for the

signal to propagate along ( )'' tRR = . Let us call '' R=R the "effective" distance of

interaction, and R=R the "actual" distance between the charges. If we arbitrarily assume

that the particles achieve the minimal distance ( ) λ=0R when t=0, the interaction acts with

a delay. Then the "effective" position R' depends both on the speed and angle of incidence

at a previous time t' < 0. When the speed component along the trajectory of the impinging

particle is of the same order of magnitude as the speed of light, we can write the ratio

between the delay time of the electromagnetic interaction and the characteristic period T =

λ/c of the DEMS created during the collision, as

θβ cosˆ

=×=c

nT

tdelay v (12a)

(see ref. [22]), since when the actual distance is minimal

( )

+=≅

T

ttRR delay

delay 1' λ , (12b)

eq. (12a) can be rewritten as

1−= ρT

tdelay (12c)

where ρ is defined as the ratio between the effective distance of interaction of two colliding

particles (12b) and the wavelength of the DEMS: in the interval 1cos0 ≤≤ θβ its value is

12

always inside the interval 1 2≤ ≤ρ , therefore by using eqs. (11), (12a), (12c) and (8a), eq.

(8) can be rewritten as:

( ) ( )( )2S

222 12 mcEEcmE −−=− ρβ (13)

Considering a very high energy head-to-head collision, i.e. occurring with an angle

θ = 0, eqs. (12a) and (12c) give β ρ= − 1 , which for ρ ≅ 2 yields β ≅ 1. In this case,

during the alpha phase, the DEMS acquires the maximum available energy of the

interacting particles: 1E≡= EEs , where 1E is the energy of the particle 1 measured in the

frame of the particle 2; eq. (13) becomes

( ) )(2 21 1

2221 mcmc −≅− EEE (14)

yielding

( ) 022

1 ≅− mcE , (15)

with energy solution for the particle 1

21 mc≅E . (16)

The result (16) is interpreted assuming that for the observer in S2 the maximum energy

ES of the source is achieved when the relative velocity between particles converges to c,

i.e. when β = 1. So the energy

21 mcE =≡ E (17a)

is the total energy carried by the particle 1. Symmetrically, the energy of the particle 2 for

the observer S1 is:

22 cmE ∗=≡ E . (17b)

Using the equations (11) and (17a), eq. (7) yields

13

( )242222 1 β−=− cmcPE , (18a)

whereas using eqs. (11) and (17a) we can describe the relative velocity of the particles as a

function of the total energy and momentum

E

cPβ = . (18b)

If β = 0, P = 0; then the eq. (18a) yields the square of eq. (17a) when the impinging

particle is at rest. In this case, the total energy of the particle is not subject to a local

increase of energy due to the contribution of the DEMS produced during the collision,

consequently according to the standard notations of SR, we define formally m0 and m c02 as

the mass and energy at rest of the particle 1. In any other case, as an effect of the energy

contribution of the DEMS, the total energy E is bigger than the value at rest. In general

when the relative velocity varies in the interval 0 ≤ β ≤ 1, we may interpret the eq. (18a) as

the square of the total energy of the particle deprived of the contribution due to the relative

motion between the two colliding particles. In other words, for the observer in S2 , the eq.

(18a) corresponds to the square of the energy in the CM of the particle 1, and as such, it is

frame-invariant:

E P c m c2 2 202 4− = . (19a)

Following eq. (19a), the energy at rest of a colliding pair with respect to an external

inertial frame can be written as:

420

222 ccPE µ=− , (19b)

where we define µ0 as the reduced rest mass of the DEMS, corresponding to the effective

mass observed at rest by each inertial frame. The eq. (19b) results naturally invariant and is

14

in agreement with the relativistic description of the rest energy of the colliding particles.

Like in equation (19a), the invariance of eq. (19b) depends on the creation of the DEMS. In

fact, during the alpha phase the amount of energy localized in the source zone is a function

of the mean relative velocity between moving charges. Since the relative velocity is an

invariant with respect to each inertial observer, the reduced rest mass, energy and

momentum of the DEMS are invariants too.

Eq. (19a) allows to describe in S2 the total energy E carried by the particle 1. We

observe that in agreement with SR, the total energy measured in the frame S2 is given by

two different contributions. The first one due to the energy of the particle at rest measured

in its own CM; the other one due to the total momentum of the interacting particles with

respect to the observer. Both contributions depend on the EM energy produced in the

source zone of the DEMS. Using eqs. (19a), (11) and (17a), the total energy can be

equivalently written as

202

20

1cm

cmE γ

β=

−= . (20)

A symmetric formula can be written for the particle 2. An analogous result can be obtained

for the eq. (19b).

The use of the equation (20) in this theoretical context needs some further explanations.

In fact, though this equation is formally and phenomenologically in agreement with SR, the

physical context where they origin is quite different from Einstein’s theory. We want to

stress that in this paper SR is not assumed a priori, but it comes out as a consequence of

the phenomenological scenery offered by BT.

15

The observer sees the mass of the impinging charge as a variable depending on the

relative motion between charge and frame. In dynamical terms, each charge in motion with

respect to the observer manifests a "massive" behaviour, i.e. each charge sees the other one

as carrying all the energy and momentum that the active DEMS needs. Following this

principle, in agreement with BT phenomenology, we suggest that the rest masses

associated with the rest charges could be originated by the energy localised by a DEMS

produced during a stationary vibration due to the vacuum properties. In this way, different

modes of vibration could be responsible for the rest energy of the particles appearing with

different masses. The observer must conclude that charges have a matter behaviour, i.e they

are real particles.

6) Collisions in the Lab-Frame

From the DEMS charge symmetry, observers placed in the rest frames S1 - S2 measure

reciprocally the energy of the respective impinging particle 2-1 without being able to

measure their own rest energy. In fact, in order to make the measure in these two frames,

they need to create a local DEMS, but in the frames S1 and S2 just a single charge is placed,

so that the observers cannot perform local measurements of energy and momentum.

We consider now a third special frame S that we define as lab-frame. In the frame S

there is a polarizable neutral medium (ordinary neutral matter), therefore when an

interaction between a pair of charges coinciding with the frames 1 and 2 occurs, the

medium in S reacts becoming polarized and four independent DEMSs1 are produced:

1 We denote each DEMS with the symbol "•(frame1)(frame2)".

16

•SS , •SS1 , •SS2 , •S1S2 .

From the experimental point of view, in order to observe the collision we cannot use the

frames S1 and S2, because the DEMSs •S1S2 and the symmetrical one, do not allow

simultaneous measurements of energy and momentum for the colliding particles 1 and 2.

Instead, using the lab-frame S we can measure energy and momentum of all the subjects

involved in the collision. In fact, the lab-frame, being made of polarizable matter, is also

sensible to its own mass, so applying to the lab at rest the eqs. (20) and (2a-2b) we get for

all the subjects involved in the interaction:

==

•0

20

Lab

Lab cMESS

p. (21a)

==

•cm

cmESS

01 11

2011

1 βp γγ

(21b)

==•

cm

cmESS *

*

02 22

2022

2βp γ

γ (21c)

where, M0 is the total polarizable rest mass associated with the lab-frame S (involved

electromagnetically in the collision). The Lorentz's factors and the velocities of the

particles in eqs. (21b-c) are all referred to the lab-frame S.

To calculate in the lab-frame S energy and momentum of the CM in the frame S0, we

must obtain the explicit expressions of the reduced rest mass µ0 and of the factors ββββ and γ.

Substituting the equations (21b-c) in (1-2a) and using the equation (19b) we obtain the

definition of the CM rest energy:

17

( )φββγγ

µ

cos12

)(

21210*

02

0*2

02

22

212

212

0

−++=

+−+==

mmmmc

cEEcECM pp , (22)

where φ is the angle between the momenta p1 and p2 of the two colliding particles with

respect to the lab-frame.

Since with respect to the lab-frame, S0 moves with a velocity ββββ, using eq. (18b) we get

( )

20*

10

220*

110

21

21 γγγγ

mm

mm

EEc

++=

++= ββppβ . (23)

To obtain γ, we use its usual definition, so by eq. (23) and by the reduced rest mass:

µ γ γ β β φ0 02

02

0 0 1 2 1 22 1= + + −m m m m* * ( cos ) (24)

as a consequence of the equation (22), we get:

0

2121

µγ mm

E

EE

CM

+=+= . (25)

Using now the eqs. (23), (24) and (25), energy and momentum of the CM of the DEMS

with respect to the lab-frame are given by

==

•c

cESS

0

20

0

µγµγβP

. (26)

We want to point out that the eqs. (19) to (26), introduce effects and formalisms in

agreement with the SR, but they come from eq. (7), by using BT phenomenology. The

evident coincidences between these two formalisms, force the introduction of a theoretical

and phenomenological parallelism between what physically occurs in the BT context and

what is phenomenologically described in SR.

18

7) Doppler Effect in DEMS Emission

Photons can be observed and their energy and momentum measured only when they are

absorbed by matter. During the absorption occurring at the microscopic level, atoms are

sensible to the electromagnetic field of the colliding particles; therefore, matter behaves as

a neutral polarizable medium as the definition of the lab-frame S requires.

During a collision, each of the two particles 1-2 interacts with matter in the lab-frame

allowing the measurement of the dynamical states of both particles. Each state is defined

by the energies and momenta of the DEMSs •SS1 and •SS2 (see eqs. (21b)-(21c)). Since the

frame S0 associated to the CM of •S1S2, is always in motion with respect to the lab-frame,

using the eqs. (9), (18b), (19b) and (26), the observer in the lab-frame obtains for the

energy of the moving DEMS

Ec

s =−

−µ β

β θ0

2 2

2

1

1

cos , (27)

where, β, is the relative velocity between observer and CM, and θ is the angle between the

total momentum P of the particles and the momentum Ps of the source observed in the lab-

frame along the photoemission direction.

Considering that for the DEMS νhcPE ss == is equivalent to the energy of the photon

exchanged between the two colliding particles (see ref. [23]), let

ν µ0

02

2= c

h (28)

be the characteristic rest frequency of the DEMS created during the collision; eq. (27) gives

the frequency measured by the observer placed in the lab-frame S:

19

ν ν ββ θ

= −−0

21

1

cos . (29)

Eqs. (28) and (29) agree with the relativistic Doppler effect of a moving wave source.

This result proves that the observer in S feels the colliding particles in S0 like a wave

source in motion with velocity β.

8) Wave behaviour of matter: the de Broglie's wavelength

When an observer in the lab-frame S is able to interact with an impinging particle

placed in the frame S1, he is able to measure the energy and momentum of the DEMS •SS1

produced during the interaction between lab and particle. Assuming m0 to be the rest mass,

eq. (21b) gives:

==

•111

2 11

1

vP m

cmESS . (30)

If the lab-frame is placed along the direction of motion of the particle, the collision

occurs with an angle θ ≅ 0. In this condition, using eqs. (9) and (30), the energy and

momentum conservation laws between DEMS and particles yield:

1

11

11

11

11

21

2

21

2

ββ

β

+=+=

+=+=

Pc

cPEP

EcPE

E

s

s

. (31)

The eqs. (31) allow us to consider the case of a particle moving with respect to the

observer with 11 ≅β . In this case, energy and momentum of the DEMS converge to those

carried by the impinging particle:

E m c P ms ≅ ≅12

1 , vs 1 , (32)

20

which, in terms of frequency and wavelength of the source, give

ν λ≅ ≅m c

h

h

m1

2

1

,

v1

. (33)

Eqs. (33) correspond to the frequency and wavelength of the de Broglie's wave of the

impinging particle. These results could be explained considering that when a particle is

"observed", it interacts with the lab-frame and a DEMS with the energy and momentum of

the particle is produced. The energy and momentum exchanged between particle and

observer represent the information concerning their relative dynamical state. When the

DEMS is active, the observer in S is conditioned by its own position on the wave front of

the DEMS and measures the amounts of electromagnetic energy and momentum as if they

were carried by a wave connected with the impinging particle. In this case, the observation

produces a physical phenomenology and formalism consistent with the particle wave

behaviour.

In general, the process of “measuring” performed on a hypothetical charged particle

occurs when there is a direct interaction between particle and matter in the lab-frame.

During the alpha phases of the active DEMSs, all the energy and momentum of the

impinging particle is electromagnetically converted into quanta, one for each DEMS

produced. Therefore, during the interaction, the particle cannot be viewed as classical,

because its initial energy is shared by all the DEMSs produced during the interaction.

When in the DEMSs the alpha phases end, the omega phases start, with the emission of

real waves. The particle emerges from the interaction as described by the harmonic

composition of monochromatic waves yielding a wave-description of the incoming

particle. It follows that, considering a charged particle crossing the matter layer of a

21

detector assumed to be the lab-frame, the particle interacts with the atoms able to feel its

EM field, by polarising them. The EM interactions produce a space-time distribution of

DEMSs with a corresponding wave distribution in which the particle is described by the

constructive waves interference.

Concerning the behaviours of neutral particles such as neutrons or others, we suggest

that their internal structure polarises as an effect of the nuclear electric fields of the lab

matter. The induced DEMSs produce a wave-field in space-time, one for each polarised

charge pair, analogous to that produced by a particle passing through matter.

Vice versa, a purely matter behaviour is observed during the direct interaction of the

impinging charged particle with just one other anti-charged particle. In the case considered,

the DEMS has the effect of exchanging directly energy and momentum between the two

particles. The result is formally and phenomenologically in agreement with the effects of a

classical collision.

9) Pair collision and the Compton effect

As an application of the previous theory, we consider two very different physical

situations.

When a pair collides, eq. (23) yields

( )φββγγµ cos112 212100 −+= m . (34)

The frequency (28) and the corresponding wavelength converge to the Compton's values

only when β β1 2 0≅ ≅ . In this case we obtain:

22

λν

λ 0

0

0

= ≅ =c h

m c c . (35)

The equivalence (35) allows us to consider the eq. (28) as a generalised Compton's

frequency for two slowly colliding particles. It follows that when two particles collide with

respect to the lab-frame at very low relative velocities 021 ≅≅≅ βββ (see eq. (11b)), the

frequencies observed respectively in the frame of the CM (eq. (14)) and in the lab-frame

(eq. (13)), both converge to Compton's values.

Using these results, we attempt now to describe phenomenologically and formally the

Compton’s effect. We know that during a collision of a photon with a q-tron2, the photon is

scattered away like a matter particle. In terms of BT, when the photon with energy E and

momentum E/c impinges on a q-tron at rest, the local intense electric field existing in the

neighbourhood of the particle polarises the photon producing a virtual emerging low

energy pair of q-tron and anti-q-tron. The virtual pair lives a time that agrees with the

Heisemberg uncertainty principle. Instantaneously, the q-tron target interacts with the

polarised virtual anti-q-tron, producing a stable DEMS observed as a photon with energy

E E'< and momentum P P'< . The symmetric residual q-tron not involved directly in the

DEMS production is scattered away as a real particle with an amount of energy bigger than

the rest energy of the target and an amount of momentum bigger than zero. This

phenomenology describes an exchange process between a real and a virtual particle, in fact

during the EM interaction of a pair of q-trons, the target and one of two virtual polarized

particles switch their physical identities.

2 With the term q-tron we denote indifferently an electron or a positron.

23

In order to prove that the phenomenology described above has a good chance to be

physically coherent with what really occurs, using BT we must formally obtain the same

result that characterises the Compton's effect.

We divide the proof into two parts.

1°) We observe that the wavelength of the photons collected in the lab frame along the

direction forming an angle θ with the beam axes is different from the wavelength of the

DEMS produced during the photon-particle interaction. In fact, in the CM of the source we

measure the Compton's wavelength of the particle (35), whereas in the target frame (TF)

we measure the de Broglie's wavelength of the photon exchanged between the particles

(33). When the virtual anti-q-tron (V) interacts with the target q-tron (T), the de Broglie’s

conditions (32) are verified and energy and momentum involved are those of the DEMS

produced (•VT). The frequency and wavelength (33) are associated to the target

antiparticle. By combining the Compton and de Broglie's relations for the wavelengths, we

obtain, in agreement with the formal result proposed by de la Peña and Cetto [24] and

Kracklauer [25], the wavelength transformation from the CM frame to the TF:

λ λβγ

= 0 . (36)

2°) We describe the variation of the wavelength of the scattered photon with respect to

the variation of the collection angle θ. For this, we use the momentum of the DEMS

c

EhP s

s ≡=0λ

(37)

the eq. (27) and the definition of the γ factor, obtaining:

24

h c

λµ

γ β θ

0

20

1

1=

− cos . (38)

Since velocity and mass are constant with respect to the variation of the angle, the

differential of eq. (38) gives

( )

θθβ

θβγ

µλλ

dc

dh

20

0 20 cos -1

sin

2

= . (39)

Eliminating the squared wavelength on the left-hand side of eq. (39) by using the eq. (38),

we obtain

θθβµγλ d

c

hd sin

2

0 0 = . (40)

The eq. (40) describes in the CM, i.e. in the frame of the scattered photon, the

differential behaviour of the wavelength of the DEMS. Applying the eq. (36), we write eq.

(40) in the target frame, obtaining the wavelength variation for the collected photon

θθµ

λ dc

hd sin

2

0

= (41)

where µ0 02= m , because inside the scattered photon the two virtual charges are

considered at rest (see eq. (34)).

Integrating the eq. (41) in the intervals [λ, λ'] corresponding to the collection angle

interval [0, θ ], we obtain

( )θλλ cos1

' 0

−=−cm

h . (42)

Eq. (42) agrees with the result of MQP for the Compton effect.

25

10) Conclusions

The complexity of nature has frequently required new ideas in order to reduce the

number of models or theories needed by description of the phenomena. This work is

dedicated to show that the compatibility between quantum and relativistic effects exists at a

deeper level of phenomenological analysis, when both the apparently incompatible physical

effects acquire a common EM origin.

In this article we have used the Bridge Theory approach [2], because only this

description relative to charged particle-anti-particle interactions allows us to describe

relativistically classical and quantum effects, without introducing in the phenomenological

description conceptual or formal discontinuities. In fact, the relativistic behaviour and the

quantum aspects that characterise the description of nature, spring out from the use of

classical electromagnetism in the processes of collision between the lab-frame and the

particles.

In spite of the usual incompatibility that quantum and classical behaviours seem to

manifest when relativistic and quantum approaches are simultaneously used in a same

theoretical context, BT evidences a formal coherence in the matter and wave description of

particles. We suggest that this coherence follows from the common EM origin of both

phenomena. The use of EM theory as BT requires, gives completeness to the physical

description of the interaction between two charged particles with the consequent vanishing

of the wave-matter duality in favour of a new relativistically consistent EM description of

matter. In this sense, STR and QM could be considered two partial physical models inside

a unified universal BT.

26

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27

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