A geometrical meaning to the electron mass from breakdown of Lorentz invariance
ABSTRACT We discuss the problem of the electron mass in the framework of Deformed Special Relativity (DSR), a generalization of Special Relativity based on a deformed Minkowski space (i.e. a four-dimensional space-time with metric coefficients depending on the energy). We show that, by such a formalism, it is possible to derive the value of the electron mass from the space-time geometry via the experimental knowledge of the parameter of local Lorentz invariance breakdown, and of the Minkowskian threshold energy $E_{0,em}$ for the electromagnetic interaction. We put forward the suggestion that mass generation can be related, in DSR, to the possible dependence of mass on the metric background (relativity of mass).
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ABSTRACT: We illustrate the main features of a new Kaluza-Klein-like scheme (Deformed Relativity in five dimensions). It is based on a five-dimensional Riemannian space in which the four-dimensional space-time metric is deformed (i.e. it depends on the energy) and energy plays the role of the fifth dimension. We review the solutions of the five-dimensional Einstein equations in vacuum and the geodetic equations in some cases of physical relevance. The Killing symmetries of the theory for the energy-dependent metrics corresponding to the four fundamental interactions (electromagnetic, weak, strong and gravitational) are discussed for the first time. Possible developments of the formalism are also briefly outlined.06/2005;
Page 1
1
Fabio Cardone1,2, Alessio Marrani3 and Roberto Mignani2-5
1 Dipartimento di Fisica, Università dell'Aquila,
Via Vetoio
I - 67010 COPPITO,L'Aquila, Italy
2 I.N.D.A.M. - G.N.F.M.
3 Università degli Studi ''Roma Tre''
Via della Vasca Navale, 84
I-00146 ROMA, Italy
4 I.N.F.N. - Sezione di Roma III
5 mignani@fis.uniroma3.it
A geometrical meaning to the electron mass from breakdown of
Lorentz invariance
Abstract
We discuss the problem of the electron mass in the framework of Deformed Special
Relativity (DSR), a generalization of Special Relativity based on a deformed Minkowski
space (i.e. a four-dimensional space-time with metric coefficients depending on the
energy). We show that, by such a formalism, it is possible to derive the value of the
electron mass from the space-time geometry via the experimental knowledge of the
parameter of local Lorentz invariance breakdown, and of the Minkowskian threshold
energy E0,e.m. for the electromagnetic interaction. We put forward the suggestion that mass
generation can be related, in DSR, to the possible dependence of mass on the metric
background (relativity of mass).
Page 2
2
1 - Introduction
The problem of the mass spectrum of the known particles (leptons and hadrons) is
still an open one from the theoretical side. As a matter of fact, the Standard Model of
electromagnetic, weak and strong interactions is unable to say why a given particle does
bear that given (experimental) mass. As to the carriers of the four fundamental forces,
symmetry considerations would require they are all massless. However, it is well known
that things are not so simple: weak quanta are massive. It is therefore necessary, in the
framework of the Glashow-Weinberg-Salam model of electroweak interaction, to
hypothesize the Goldstone mechanism, able to give weak bosons a mass by interaction
with the (till now unobserved!) Higgs boson.
On this respect, even the first, best known and familiar particle, the electron, is still
a mysterious object. In spite of the successes of the Dirac equation, which allows one to
The origin of its mass is far from being understood. The classical electron theory (with the
works by Abraham, Lorentz and Poincaré) attempts at considering the mass of the electron
as of purely electromagnetic origin, and is well known to be deficient on several respects.
The basic flaw of such a picture is due to the Ernshaw theorem, stating that it is
impossible to have a stationary nonneutral charge distribution held together by purely
electric forces. Moreover, a purely electromagnetic model of the electron implies the
occurrence of divergent quantities. Such infinities can be dealt with by means of the
renormalization procedure in Quantum Electrodynamics (QED). However, even in this
framework, the value of the electron mass is not the intrinsic one but only that resulting
from its interaction with the vacuum.
The modern view to the problem of the electron mass is that pioneered by Wheeler
and Feynman(1), according to which it is not of electromagnetic origin but entirely
mechanical(2).
In this paper, we want to show that the electron mass me can be obtained from
arguments related to the breakdown of local Lorentz invariance, in the framework of a
generalization of Special Relativity ( Deformed Special Relativity, DSR), based on a
“deformation” of the Minkowski space (i.e. with metric coefficients depending on the
energy) . This allows one to attribute to me a geometrical meaning, by expressing it in
terms of the parameter d of LLI breakdown.
The organization of the paper is as follows. In Sect. 2 we briefly introduce the
concept of deformed Minkowski space, and give the explicit forms of the
phenomenological energy-dependent metrics for the four fundamental interactions. The
LLI breaking parameter dint for a given interaction is introduced in Sect. 3. In Sect. 4 we
assume the existence of a stable fundamental particle interacting gravitationally,
electromagnetically and weakly, and show (by imposing some physical requirements) that
its mass value (expressed in terms of de.m. and E0,grav is just the electron mass. In Sect. 5 we
briefly introduce the concept of mass relativity in DSR. Sect. 6 concludes the paper.
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3
2- Deformed Special Relativity in four dimensions (DSR)
2.1 - Deformed Minkowski space-time
Deformed Special Relativity is a generalization of Special Relativity (SR) based
on a ''deformed'' Minkowski space, assumed to be endowed with a metric whose
coefficients depend on the energy of the process considered(3). Such a deformation is
essentially aimed at providing a metric representation of the interaction ruling the process
considered (at least in the related energy range, and locally, i.e. in a suitable space-time
region)(3-6). DSR applies in principle to all four interactions (electromagnetic, weak, strong
and gravitational), at least as far as their non-local behavior and non-potential part are
concerned.
The generalized ("deformed'') Minkowski space
with the same local coordinates x of M4 (the four-vectors of the usual Minkowski space),
but with metric given by the metric tensor 1
(
[
)()()(
221100
EbEbEb
mmmm n
dddd
−−−≡
(
∈∀
0
RE
, where the {bµ2(E)} are dimensionless, real, positive functions of the energy(3).
The generalized interval in
4
M is therefore given by (xµ=(x0,x1,x2,x3)=(ct,x,y,z), with c
being the usual light speed in vacuum) (ESC on)
(
.)(
dxdxdxdxE
∗≡=
mn
h
The last step in (2.2) defines the scalar product ∗ in the deformed Minkowski
space
4
M 2. It follows immediately that it can be regarded as a particular case of a
Riemann space with null curvature.
Let us stress that, in this formalism, the energy E is to be understood as the energy
of a physical process measured by the detectors via their electromagnetic interaction in the
usual Minkowski space. Moreover, E is to be considered as a dynamical variable, because
1 In the following, we shall employ the notation "ESC on" ( "ESC off" ) to mean that the Einstein sum
convention on repeated indices is (is not) used.
2 Notice that our formalism - in spite of the use of the word ''deformation'' - has nothing to do with the
''deformation'' of the Poincaré algebra introduced in the framework of quantum group theory (in particular
the so-called k-deformations)(7). In fact, the quantum group deformation is essentially a modification of the
commutation relations of the Poincaré generators, whereas in the DSR framework the deformation concerns
the metrical structure of the space-time (although the Poincaré algebra is affected, too(8),(26)-(28)).
4
~
M (DMS4) is defined as a space
)
(
] )
)( ),( ),( ),()(
2
33
222
2
3
d
2
2
2
1
2
0
Eb
EbEbEbEbdiagE
ESCoff
≡
m
m n
h
−−−=
(2.1)
)
+
~
)
)()()()(
22
3
22
2
22
1
222
0
2
dzEbdyEbdxEbdtcEbds
=++−=
nm
(2.2)
~
Page 4
4
it specifies the dynamical behavior of the process under consideration, and, via the metric
coefficients, it provides us with a dynamical map - in the energy range of interest - of the
interaction ruling the given process. Let's recall that the use of momentum components as
dynamical variables on the same foot of the space-time ones can be traced back to
Ingraham(9) . Dirac(10), Hoyle and Narlikar(11) and Canuto et al.(12) treated mass as a
dynamical variable in the context of scale-invariant theories of gravity.
Moreover, it was shown that the DSR formalism is actually a five-dimensional
one, in the sense that the deformed Minkowski space can be naturally embedded in a
larger Riemannian manifold, with energy as fifth dimension(13) . Curved 5-d. spaces have
been considered by several Authors(14). On this respect, the DSR formalism is a kind of
generalized (non-compactified) Kaluza-Klein theory, and resembles, in some aspects, the
so-called ''Space-Time-Mass'' (STM) theory (in which the fifth dimension is the rest
mass), proposed by Wesson(15) and studied in detail by a number of Authors(16).
By putting ds2=0 , we get the maximal causal velocity in
≡
()(
21
EbEb
(i.e. the analogous of the light speed in SR) for the interaction represented by the
deformed metric considered.
In DSR the relativistic energy, for a particle of mass m subjected to a given
interaction and moving along
4
~
M
(3,21)
)(
)(
,
)
)(
,
)(
)(
3
000
Eb
Eb
c
Eb
c
Eb
cEur
(2.3)
ix ˆ ,has the form(3):
)(
~
g
)(
)(
)(
~
g
)(
2
0
2
22
E
Eb
Eb
cmEEumE
i
i
===
(2.4)
where ui is the i-th component of the maximal velocity (2.3) of the interaction considered,
and
(
1
)
i
i
i
ii
i
u
v
Ecb
Ebv
E
≡
−=−≡
−
−
b
bg
~
;
)(
)(
1
~
)(
~
2
1
2
0
2
1
2
(2.5)
In the non-relativistic (NR) limit of DSR, i.e. at energies such that
vi ÷ ui(E)
(2.6)
Eq. (2.4) yields the following NR expression of the energy corresponding to the given
interaction:
Page 5
5
(2.7)
2.2 - Energy-dependent phenomenological metrics
for the four interactions
As far as phenomenology is concerned, we recall that a local breakdown of
Lorentz invariance may be envisaged for all four fundamental interactions
(electromagnetic, weak, strong and gravitational) whereby one gets evidence for a
departure of the space-time metric from the Minkowskian one (at least in the energy
range examined). The experimental data analyzed were those of the following four
physical processes: the lifetime of the (weakly decaying) K0S meson(17); the Bose-Einstein
correlation in (strong) pion production(18); the superluminal photon tunneling(19); the
comparison of clock rates in the gravitational field of Earth(20). A detailed derivation and
discussion of the energy-dependent phenomenological metrics for all the four interactions
can be found in Ref.s [3-6]. Here, we confine ourselves to recall their following basic
features:
1) Both the electromagnetic and the weak metric show the same functional
behavior, namely
(
( ),( ),( , 1)(
EbEbEbdiagE
−−−=
m n
h
), )
222
(2.8)
()
0,1)(1
, 1
0,/
)(
3
1
0
0
0
0
3/1
0
2
>
−
−+=
=
<
≤<
E
=
E
E
E
EE
E
EEEE
Eb
q
(2.9)
(where ?(x) is the Heaviside theta function) with the only difference between them being
the threshold energy E0, i.e. the energy value at which the metric parameters are constant,
i.e. the metric becomes Minkowskian ( ?µ?(E= E0) = gµ? = diag(1,-1,-1,-1)); the fits to the
experimental data yield
E0,e.m. = (4.5± 0.2) µeV;
(2.10)
E0,weak =(80.4± 0.2) GeV;
)(
)(
)(
2
0
2
22
Eb
Eb
mcEmuE
i
i
NR
==
Page 6
6
Notice that for either interaction the metric is isochronous, spatially isotropic and ''sub-
Minkowskian'', i.e. it approaches the Minkowskian limit from below (for E<E0). Both
metrics are therefore Minkowskian for E>E0,weak > 80 GeV, and then our formalism is
fully consistent with electroweak unification, which occurs at an energy scale ~ 100 GeV.
Let us recall that the phenomenological electromagnetic metric (2.8)-(2.10) was
derived by analyzing the propagation of evanescent waves in undersized waveguides(17). It
allows one to account for the observed superluminal group speed in terms of a nonlocal
behavior of the waveguide, just described by an effective deformation of space-time in its
reduced part(5). As to the weak metric, it was obtained by fitting the data on the meanlife
of the meson K0S (experimentally known in a wide energy range (30÷350 GeV)(17)), thus
accounting for its apparent departure from a purely Lorentzian behavior(3,21).
2) For the strong interaction, the metric was derived(4) by analyzing the
phenomenon of Bose-Einstein (BE) correlation for p -mesons produced in high-energy
hadronic collisions(18). Such an approach permits to describe the BE effect as the decay of
a ''fireball'' whose lifetime and space sizes are directly related to the metric coefficients
b2µ,strong(E), and to avoid the introduction of ''ad hoc'' parameters in the pion correlation
function(4). The strong metric reads
(
),( ),()(
, 1 , 0
EbEb diagE
strongstrongstrong
−=
h
E
) )(),(
2
, 3
2
, 2
22
EbEb
strongstrong
−−
(2.11)
()
0,1)(1
,/
0, 1
E
)()(
, 0
5
2
)(
5
2
)(
2
, 0
, 0
, 0
2
, 0
, 0
<
2
, 3
2
, 0
2
2
, 2
2
2
, 1
b
>
−
−+=
=
≤<
E
==
>∀
=
=
E
E
E
EE
EE
EE
EbEb
Eb
E
strong
strong
strongstrong
strong
strongstrong
strong
strong
q
(2.12)
with
E0,strong = ( 367.5± 0.4) GeV . (2.13)
a deformation of the time coordinate occurs; moreover, the three-space is anisotropic,
with two spatial parameters constant (but different in value) and the third one variable
with energy like the time one.
Let us stress that, in this case, contrarily to the electromagnetic and the weak ones,
Page 7
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experimental data on the relative rates of clocks in the Earth gravitational field(20). Its
explicit form is3:
(
),(),()(
, 1 , 0
EbEb diagE
gravgrav grav
−−=
h
, 1
3) The gravitational energy-dependent metric was obtained(6) by fitting the
) )( ),(
2
, 3
2
, 2
22
EbEb
grav grav
−
(2.14)
0,11
4
1
)(1
,1
4
1
0
)()(
2
, 0
, 0
, 0
2
, 0
, 0
2
, 3
2
, 0
>
−
+−+=
=
<
+
≤<
==
E
E
E
EE
EE
E
E
EE
EbEb
grav
grav
grav
grav
grav
grav grav
q
(2.15)
with
E0,grav = (20.2± 0.1) µeV . (2.16)
Intriguingly enough, this is approximately of the same order of magnitude of the thermal
energy corresponding to the 2.7°K cosmic background radiation in the Universe4.
Notice that the strong and the gravitational metrics are over-Minkowskian (namely,
they approach the Minkowskian limit from above (E0<E), at least for their coefficients
b02(E) = b32(E)).
3 - LLI breaking factor in DSR
The breakdown of standard local Lorentz invariance (LLI) is expressed by the LLI
breaking factor parameter d (23). We recall that two different kinds of LLI violation
parameters exist: the isotropic (essentially obtained by means of experiments based on the
propagation of e.m. waves, e.g. of the Michelson-Morley type), and the anisotropic ones
(obtained by experiments of the Hughes-Drever type(23), which test the isotropy of the
nuclear levels).
In the former case, the LLI violation parameter reads(23)
3 The coefficients b2
4 It is worth stressing that the energy-dependent gravitational metric (2.14)-(2.16) is to be regarded as a local
representation of gravitation, because the experiments considered took place in a neighborhood of Earth, and
therefore at a small scale with respect to the usual ranges of gravity (although a large one with respect to the
human scale).
1,grav(E) and b2
2,grav(E) are presently undetermined at phenomenological level.
Page 8
8
vcu
c
+
u
=
−
=
, 1
2
d
(3.1)
where c is, as usual, the speed of light in vacuo, v is the LLI breakdown speed (e.g. the
speed of the preferred frame) and u is the new speed of light (i.e. the ''maximal causal
speed'' in Deformed Special Relativity(3)). In the anisotropic case, there are different
contributions dA to the anisotropy parameter from the different interactions. In the HD
experiment, it is A=S, HF, ES, W, meaning strong, hyperfine, electrostatic and weak,
respectively. These correspond to four parameters dS (due to the strong interaction), dES
(related to the nuclear electrostatic energy), dHF (coming from the hyperfine interaction
between the nuclear spins and the applied external magnetic field) and dW (the weak
interaction contribution).
All the above tests put upper limits on the value of d (23).
Moreover, at the end of the past century, a new electromagnetic experiment was
proposed(24), aimed at directly testing LLI. It is based on the possibility of detecting a non-
zero Lorentz force between the magnetic field B generated by a stationary current I
circulating in a closed loop γ, and a charge q, in the hypothesis that both q and γ are at rest
in the same inertial reference frame. Such a force is zero, according to the standard
(relativistic) electrodynamics. The results obtained by such a method in two experimental
runs(25) admit as the most natural interpretation the fact that local Lorentz invariance is
in.fact broken.
The value of the (isotropic) LLI breaking factor determined by this electromagnetic
experiment is(25)
d ≅ 4·10-11
and represents the present lowest limit to d.
In order to establish a connection with the electron mass, we can define the LLI
breakdown parameter for a given interaction, dint , as
mm
−
≡
d
(3.2)
. int,
.,
. int,
.,. int,
int
1
in
gravin
in
gravinin
m
m
m
−=
(3.3)
where min.,int. is the inertial mass of the particle considered with respect to the given
interaction5 . In other words, we assume that the local deformation of space-time
corresponding to the interaction considered, and described by the metric (2.1), gives rise
to a local violation of the Principle of Equivalence for interactions different from the
gravitational one. Such a departure, just expressed by the parameter dint, does constitute
also a measure of the amount of LLI breakdown. In the framework of DSR, dint embodies
5 Throughout the present work, ''int'' denotes a physically detectable fundamental interaction, which can be
operationally defined by means of a phenomenological energy-dependent metric of deformed-Minkowskian
type.
Page 9
9
the geometrical contribution to the inertial mass, thus discriminating between two
different metric structures of space-time.
Of course, if the interaction considered is the gravitational one, the Principle of
Equivalence strictly holds, i.e.
min.,grav.= mg
where mg is the gravitational mass of the physical object considered, i.e. it is its
''gravitational charge'' (namely its coupling constant to the gravitational field).
Then, we can rewrite (3.3) as:
mm
−==
d
(3.4)
. .,int
in
..,int
in
..,int
in
m
. int
1
gg
m
m
−
(3.5)
and therefore, when the particle is subjected only to gravitational interaction, it is
dgrav. = 0
In the case of the gravitational metric (2.14)-(2.15), we have
)(
Eb
grav
Therefore, for i=3 , Eq. (2.4) yields, for the gravitational energy of a particle moving along
the z-axis (v3=v):
(3.6)
+
∈∀=
0
., 3
. , 0
, 1
)(
RE
Eb
grav
(3.7)
,1
2
2
1
2
2
g
cm
c
v
cmE
gggrav
=
−=
−
(3.8)
with non-relativistic limit (cfr. Eq. (2.7))
(3.9)
namely, the gravitational energy takes its standard, special-relativistic values.
This means that the special characterization (corresponding to the choice i=3) of
Eq.s (2.4) and (2.7) within the framework of DSR relates the gravitational interaction with
SR, which is - as well known - based on the electromagnetic interaction in its
Minkowskian form.
2
,
cmE
gNRgrav
=
Page 10
10
4 - The electron as a fundamental particle and its ''geometrical'' mass
Let us consider for E the threshold energy of the gravitational interaction:
E = E0,grav
where E0,grav is the limit value under which the metric ?µ?,grav (E) becomes minkowskian
(at least in its known components). Indeed, from Eq.s (2.14), (2.15) it follows:
(
[
gravgrav
EbEb
, 22 , 110
)()(
−−−=
mmm m n
ddddd
Notice that at the energy E = E0,grav the electromagnetic metric (2.8),(2.9) is
Minkowskian, too (because E0,grav >E0,e.m.).
On the basis of the previous considerations, it seems reasonable to assume that the
physical object (particle) p with a rest energy (i.e. gravitational mass) just equal to the
threshold energy E0,grav , namely
E0,grav = mg,pc2
must play a fundamental role for either e.m. and gravitational interaction. We can e.g.
hypothesize that p corresponds to the lightest mass eigenstate which experiences both
force fields (i.e., from a quantum viewpoint, coupling to the respective interaction carriers,
the photon and the graviton). As a consequence, p must be intrinsically stable, due to the
impossibility of its decay in lighter mass eigenstates, even in the case such a particle is
subject to weak interaction, too (i.e. it couples to all gauge bosons of the Glashow-
Weinberg-Salam group SU(2) × U(1), not only to its electromagnetic charge sector).
Since, as we have seen, for E = E0,grav the electromagnetic metric is minkowskian,
too, it is natural to assume, for p:
min,p,e.m. = min,p
namely its inertial mass is that measured with respect to the electromagnetic metric.
Then, due to the Equivalence Principle (see Eq. (3.4)), the mass of p is
characterized by
=
pin
mm
,p,e.m.in,
(4.1)
)
]
(
, 0
)
grav
ESCoff
grav
, 2
gravgrav
EE
EbEb diagE
, 03
22
22
, 1
,
1 ),(),(, 1)(
∈∀
−−−=
m
h
(4.2)
(4.3)
(4.4)
=
pg
mm
p
,gravp,in,
:
(4.5)
Page 11
11
interaction becomes:
Therefore, for such a fundamental particle the LLI breaking factor (3.3) of the e.m.
()
..,,
.,
.,
.,
,.,
..
1
1
mepinpg
p in
pg
p in
pgp
m
in
me
mm
m
mmm
d
d
−=⇔
−=
−
≡
(4.6)
Replacing (4.3) in (4.6) yields:
()
..
2
, 0
c
,
2
.., , 0
1
1
d
1
me
grav
pinmepingrav
E
mcmE
d
−
=⇔−=
(4.7)
Eq. (4.7) allows us to evaluate the inertial mass of p from the knowledge of the
electromagnetic LLI breaking parameter de.m. and of the threshold energy E0,grav of the
gravitational metric.
On account of Eq. (3.1), we can relate the lowest limit to the LLI breaking factor
of electromagnetic interaction, Eq. (3.3) (determined by the coil-charge experiment), with
de.m as follows:
d =1 - de.m. ≅ 4·10-11
Then, inserting the value (2.16) for E0,grav 6 and (4.8) in (4.7), we get
MeV
cc
..
1041
×−
d
(with min,e being the electron mass) where the ≥ is due to the fact that in general the LLI
breaking factor constitutes an upper limit (i.e. it sets the scale under which a violation of
LLI is expected). If experiment [25] does indeed provide evidence for a LLI breakdown
(as it seems the case, although further confirmation is needed), Eq. (4.9) yields min,p=min,e.
We find therefore the amazing result that the fundamental particle p is nothing but the
electron e- (or its antiparticle e+ 7). The electron is indeed the lightest massive lepton
(pointlike, non-composite particle) with electric charge, and therefore subjected to
gravitational, electromagnetic and weak interactions, but unable to weakly decay due to its
(4.8)
e in
me
grav
2
p in
m
c
eV
E
m
,
2211
−
5
, 0
,
5 . 0
1021
==
×
≥=
−
(4.9)
6 Let us recall that the value of E0,grav was determined by fitting the experimental data on the slowing down
of clocks in the Earth gravitational field (20) . See Ref. [6].
7 Of course, this last statement does strictly holds only if the CPT theorem maintains its validity in the DSR
framework, too. Although this problem has not yet been addressed in general on a formal basis, we can state
that it holds true in the case we considered, since we assumed that the energy value is E = E0,grav
corresponding to the Minkowskian form of both electromagnetic and gravitational metric.
Page 12
12
small mass. Consequently, e- (e+) shares all the properties we required for the particle p,
whereby it plays a fundamental role for gravitational and electromagnetic interactions.
5 – Mass relativity in DSR
the electron mass to the (local) breakdown of Lorentz invariance. Such a mass would then
be a measure of the deviation of metric from the Minkowskian one. The minimum
measured mass of a particle would be related to the minimum possible metric deviation
compatible with its interactions.
Such a point can be reinforced by the following argument.
physical standpoint, as the speed of the quanta of the interaction locally (and
phenomenologically) described in terms of a deformed Minkowski space. Since these
quanta are associated to lightlike world-lines in
(with respect to the interaction considered), in analogy with photons (with respect to the
e.m. interaction) in the usual SR.
Let us clarify the latter statement. The carriers of a given interaction propagating
with the speed ur typical of that interaction are actually expected to be strictly massless
only inside the space whose metric is determined by the interaction considered. A priori,
nothing forbids that such “deformed photons'' may acquire a non-vanishing mass in a
deformed Minkowski space related to a different interaction.
This might be the case of the massive bosons W+ , W- and Z0, carriers of the weak
interaction. They would therefore be massless in the space
weak interaction, but would acquire a mass when considered in the standard Minkowski
space M of SR (that, as already stressed, is strictly connected to the electromagnetic
interaction ruling the operation of the measuring devices). In this framework, therefore, it
is not necessary to postulate a ''symmetry breaking'' mechanism (like the Goldstone one in
gauge theories) to let particles acquire mass. On the contrary, if one could build up
measuring devices based on interactions different from the e.m. one, the photon might
acquire a mass with respect to such a non-electromagnetic background.
Mass itself would therefore assume a relative nature, related not only to the
interaction concerned, but also to the metric background where one measures the energy
of the physical system considered. This can be seen if one takes into account the fact that
in general, for relativistic particles, mass is the invariant norm of 4-momentum, and what
is usually measured is not the value of such an invariant, but of the related energy.
The considerations carried out in the previous Sections allow us therefore to relate
The maximal causal velocity ur defined by Eq. (2.3) can be interpreted, from a
4
~
M , they must be zero-mass particles
4
~
M (hweak(E)) related to the
Page 13
13
5 - Conclusions
The formalism of DSR describes -among the others -, in geometrical terms (via the
energy-dependent deformation of the Minkowski metric) the breakdown of Lorentz
invariance at local level (parametrized by the LLI breaking factor dint). We have shown
that within DSR it is possible - on the basis of simple and plausible assumptions - to
evaluate the inertial mass of the electron e- (and therefore of its antiparticle, the positron
e+) by exploiting the expression of the relativistic energy in the deformed Minkowski
space
+
∈0
RE
gravitational interaction (in particular its threshold energy), and the LLI breaking
parameter for the electromagnetic interaction de.m..
Therefore, the inertial properties of one of the fundamental constituents of matter
and of Universe do find a ''geometrical'' interpretation in the context of DSR, by admitting
for local violations of standard Lorentz invariance.
We have also put forward the idea of a relativity of mass, namely the possible
dependence of the mass of a particle on the metric background where mass measurements
are carried out. This could constitute a possible mechanism of mass generation alternative
to those based on symmetry breakdown in Relativistic Quantum Theory.
)(
~
M
4
E
, the explicit form of the phenomenological metric describing the
Page 14
14
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