In this paper a self-excited Rayleigh-type system models the auto-parametric wave-soliton coupling via phase fluctuations. The parameter of dissipative terms determine not only the most likely quantum coupling between solitons and linear waves but also the most likely mass of the solitons. Phase fluctuations are mediated by virtual photons coupling at light-velocity in a permanent Compton scattering process. With a reference to the SI-units and proper scaling relations in length and velocity, the final result shows a highly interesting sequence: the likely soliton Compton mass is about 1.00138 times the neutron and 1.00276 times the proton mass

Binder, Bernd

English

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Soliton Compton Mass from Auto–Parametric Wave–Soliton Coupling
Bernd Binder
88682 Salem, Germany, binder@quanics.com c©2002
(Dated: 20.11.-2.12.2002, updates: http://www.quanics.com)
In this paper a self-excited Rayleigh-type system models the auto-parametric wave-soliton cou-
pling via phase fluctuations. The parameter of dissipative terms determine not only the most likely
quantum coupling between solitons and linear waves but also the most likely mass of the solitons.
Phase fluctuations are mediated by virtual photons coupling at light-velocity in a permanent Comp-
ton scattering process. With a reference to the SI-units and proper scaling relations in length and
velocity, the final result shows a highly interesting sequence: the likely soliton Compton mass is
about 1.00138 times the neutron and 1.00276 times the proton mass.
Introduction. For nonlinear field theory models in
1+1–dimensional space–time the equations of motion
admit finite energy and finite width solutions called soli-
tons [1]. Solitary waves were discovered in the first half
of the nineteenth century by Rusell, the word soliton was
invented by Kruskal, the sine–Gordon (SG) model by
Skyrme [2]. Solitons retain their identity after collisions,
can annihilate with anti–solitons, many–soliton solutions
obey Pauli’s exclusion principle. In 1+1–dimensional
space–time there are two non–trivial minimal quantum
field theories which describe non–perturbative phenom-
ena: the SG model and the massive Thirring model [3]
(a self-coupled Dirac field, see the Lagrangians [4]), both
are intimately related [5]. It is an interesting question
how stationary solitons (like breather) get their absolute
mass/energy. In this paper a self-excited Rayleigh-type
system [6] will model an auto-parametric wave-soliton
coupling via phase-fluctuations. This system can as a
classical example model simplified music instruments (in
the original work of Rayleigh a clarinet reed). In [7] it
has recently been shown, that Rayleigh-type self–excited
auto-parametric systems [8] can stimulate ”whispering
gallery modes” and model Coulomb interaction between
sine-Gordon solitons. In this paper exactly the same
model will be applied to determine the most likely
Compton mass of the soliton.
Quantum field theory. In quantum field theory, par-
ticles correspond to fluctuations of a space-time function
that minimizes the action for some Lagrangian L which
is itself the space integral of a Lagrangian density L. A
low-dimensional (bosonic) SG field of a single hermitian
scalar field θ in 1+1 dimensions with Lagrangian density
L = µ2 ∂νθ∂νθ −V (θ) is a function of one space dimension
and time. The time independent field equations reads
µ∂2rθ = ∂θV which can also be written as
V (θ) =
µ
2
(∂rθ)2 , (1)
where µ is the soliton mass. In the quantum model,
the expectation of the square of the phase gradient
contributes to the soliton self-energy, where the phase
gradient usually scales proportional to the wave number
that is proportional to the frequency.
Rayleigh–type perturbed SG. The coupling between
linear waves and non–linear solitons could be important
for the understanding for scale-dependent electromag-
netic coupling. The Rayleigh-type perturbed SG equa-
tion is given by
(∂rr − ∂tt) θ − sin θ =
[
b2(∂rθ)2 − a2
]
∂tθ , (2)
where ∂rr − ∂tt is the D’Alembert wave operator. On
the left side are the usual SG terms, on the right side
are with eq.(1) the typical Rayleigh wave dissipation
and regeneration terms. These two terms balance
wave dissipation and regeneration and can be found
in self–excited auto-parametric systems [8]. The term
b2(∂rθ)2− a2 controls θ by weighting the dissipative first
order term ∂tθ. The amplitude of the coupling wave can
be assigned to a phase difference between two solitons.
Rayleigh–type auto–parametric system. Following [8]
we characterize auto–parametric systems as a vibrating
system which consists of at least two subsystems: an
oscillator that is generally in a vibrating state and the
excited system which is excited indirectly and coupled to
the oscillator in a nonlinear way. A linear wave will be
defined as a disturbance which transports energy from
one location to another location without transporting
matter evolving at light speed c with c ∂rθ = ∂tθ. In
the auto–parametric system the dissipation and excite-
ment parameter will depend on the phase statistics. The
Rayleigh–type system
∂ttθ + θ = a2∂tθ − b2c−2(∂tθ)3 (3)
is called self–excited auto–parametric, and can be
exactly treated by phase averaging methods providing
for the semi–trivial solution [8]. The excitation is here
also balanced by the (soliton vortex) regeneration term
V (θ) of eq.(1) with parameter b2 and (linear mode)
damping term ∂tθ with parameter a2. So a, b, and c
control the amplitude θ of intermediating linear waves.
The coupling mechanism. The vertex solitons are as-
sumed to be identical and not synchronized in phase,
such that the excitement of coupling modes is based on
the (statistical) phase difference. Let the SG solitons be
2coupled such, that the amplitude of the perturbed lin-
ear wave will be the phase difference θ of the perturbed
nonlinear SG solitons controlled by the potential of the
phase gradient V (θ). Usually the non-trivial equilibrium
fluctuation amplitude θ0 can be found with given pa-
rameter a and b or ratio q = b/a, but in our case the
scattering phase fluctuation amplitude θ0 is given by the
relative phase fluctuation range of solitons. Regarding
the kink-antikink solution there exists a characteristic
time scale or oscillation frequency that can be related
via coupling wave velocity to a characteristic length scale
or wavelength λµ. The fermion–like behavior pointed
out by Skyrme [2] suggests to define an additive phase–
fluctuation range by 2×2pi = 4pi. Therefore, the coupling
amplitude for the mediating wave will be given by
θ0 = 4piλµ . (4)
Later λµ will be reintroduced as the Compton wave-
length of wave–soliton coupling.
Phase averaging. Statistically, there are phase oscil-
lations or fluctuations randomly modulated in a special
phase interval, so we can assume in the most simple case
a maximum entropy phase modulation and fluctuation
between two vertex solitons and find the most likely en-
ergy flow and coupling ratio from phase averaged solu-
tions [9]. The phase–amplitude modulation with random
τ as a solution to eq.(3) can be written as [8]
θ = θ0 cos(τ + ψ0) , (5)
averaging over τ yields [8, 9] with constant ψ0
∂tθ0 ∝ 12
(
a2θ0 − 3b
2
4λ2µ
θ30
)
, ∂tψ0 = 0 . (6)
1-dimensional coupling soliton energy. With constant
fluctuation strength ∂tθ0 = 0, the averaged equations
provide with eq.(4) and eq.(6) for a balanced non-trivial
equilibrium
q =
b
a
=
1
4pi
√
3
4
,
2V
Eµ
= (∂rθ)2 = q−2 . (7)
Consequently, the 1-dimensional coupling strength q will
enter the energy definitions via mean unit energy E1d of
1-dimensional coupling
E1d = Eµ = q2(∂rθ)2 = 1µc2 . (8)
To compare our theoretical soliton coupling model to real
existing couplings, mass/energy has to be quantified. Let
the unit mass be the 1d coupling mass–energy of a soli-
ton with Eµ = 1µc2. The mutual 1-d coupling to a pho-
ton with amplitude/wavelenght fluctuation λµ can be re-
garded as a permanent Compton scattering process with
mass–energy value related to λµ via Compton relation
Eµ = 2q2V =
hc
λµ
. (9)
Energy scaling. According to eq.(7) and eq.(8), both
Eµ and V scale with c2. Eµ can be assigned to the soliton
energy and mass. The absolute energy reference is based
on the human artificial energy unit Eu referring to the
kinetic energy of a unit mass 1µ moving at unit velocity
u (in SI 1J = 1kg m2/s2). We will now find the scaling
relations of Eµ and V with respect to the energy unit
Eu provided by the system of units (SI) on one end and
by Planck units (~ = c = µ = 1) on the other end of
the scale. λu = 1m will be the human artificial reference
wavelength. Planck units demand that
• the light velocity equals the unit velocity c = u = 1,
• 2V (c = 1) = Eu(u = 1),
• the the Compton length equals the unit length λµ =
λu = 1,
• Eµ(λµ = 1) = Eu(λu = 1).
The algebraic form of the two scaling relations is given
by
• the mean soliton coupling energy V scales with the
square of the wave velocity
2V
Eu
=
c2
u2
= Ξ2, Ξ = 299792458, (10)
• the 1-dimensional quantum energy of waves cou-
pling to particles Eµ is inversely proportional to
the wavelength, especially to the Compton wave-
length
Eµ
Eu
=
λu
λµ
. (11)
Practical necessity motivates to choose a unit velocity
0 < u¿ c with ΞÀ 1 [10].
Results. The main result could be written as a char-
acteristic soliton wavelength of one-dimension coupling
exactly given with eq.(7) - eq.(11) by
λµ =
λu
q2Ξ2
, (12)
and provides for the soliton mass µ via Compton relation
µ = h/(cλµ) = q2Ξ2h/(cλu). Realized in SI units the
values are
µ =
~
c
Ξ2
6pim
≈ 1.67724... · 10−27kg,
λµ =
12pi2m
Ξ2
≈ 1, 31777... · 10−15m. (13)
The corresponding SI-values for the wave number kµ, an-
gular frequency ωµ, and energy Eµ are given by
kµ = 2pi/λµ = 4.7680443... · 1015rad m−1,
ωµ = kµc = 1.4294237... · 1024rad s−1,
Eµ = µc2 = ~ckµ = 940.86369...MeV. (14)
3It turns out, that µ shows a highly interesting and signif-
icant sequence with the fundamental masses of the two
prominent baryons (Neutron with mass Mn and proton
with mass Mp+) [10]:
µ
Mn
≈ 1.001382..., µ
Mp+
≈ 1.002762.... (15)
The system-invariant soliton mass scale is 1.001382
times the neutron and 1.002762 times the proton
scale and could be interpreted as a stability limit.
The neutron is energetically lower than µ, this could
support semi–stability. In this context it is interesting
to note, that two neutrons provide for approximately
the same mass gain as one proton with respect to µ.
This mass differences could be relevant for approach-
ing (nuclear) binding energies with many–soliton models.
Can fundamental baryons be solutions to the SG?
eq.(1) obtained from the SG–Lagrangian is the central
control term of auto-parametric resonance and has
also a central role in [11, 12, 13], where generalized
fine-structure constants coupling between topological
phase fields have been proposed. Solitons keep their
energy in a permanent scattering process, retain their
identity after collisions, can annihilate with anti–
solitons, many–soliton solutions obey Pauli’s exclusion
principle. As pointed out by Skyrme [2] this can be
interpreted as a fermion–like behavior. The interest in
such low-dimensional SG models arises from their inte-
grability, duality properties, non-perturbative aspects,
the electric-magnetic duality in gauge theories, and as an
universal concept in nonlinear science. Solitons appear
in almost all branches of physics, such as hydrodynamics,
condensed matter phenomena, particle physics, plasma
physics, nonlinear optics, low temperature physics,
nuclear physics, biophysics and astrophysics. Coleman’s
”Quantum SG equation as the massive Thirring model”
[5] establish an identity between the quantized SG model
and the Dirac spin-1/2 equation in 1+1 dimensions.
But probably because of the low dimensionality and
the wide acceptance of the standard model (including
quark model) the interest was limited. Except the
Skyrmion baryon approach (see i.e. [14, 15, 16]),
predicting that the lowest energy Eµ = B is more than
a factor q−2 = 12pi2 lower than the energy given by the
Lagrangian (Fadeev-Bogomolny bound) and confirming
the result of auto–parametric resonance in eq.(7). The
central role approaching the baryon mass scale is the
shift of human artificial units to Planck units (at the
Fermi scale!), this extrapolates the soliton energy by a
factor Ξ2. The deviations in eq.(15) of course suggest to
play with the self-coupling Dirac equation and massive
Thirring model [3] and include other particles as dual
or partner soliton states. This could eventually fill the
gap and build the bridge to a 4-dimensional integrable
quantum field theory.
Relevance to the previous work. Extremely inter-
esting is the topological charge q obtained from the
auto–parametric model regarding topological phase
fields and electromagnetic coupling. Via Gauss relation
the 1-dimensional coupling parameter q can also be
related to a 3-d coupling with spherical “whispering
Gallery modes” (WGM) coupling with a 3-d potential
φ3d = 1137r . Why integral? When the round-trip path fits
integer numbers of the wavelength (single-valuedness),
WGM are formed. The corresponding quantum number
of orbital degeneracy is given by M =
[
4pi
q
]
= 137 where
[ ] means next higher integral value. In [11, 12, 13]
generalized fine-structure constants based on a SG type
pseudospherical coupling of topological phase fields have
been defined. M = 137 with q = e corresponds to the
Sommerfeld fine structure constant α = e2/(4pi²0~c)
according to the Dirac theory of magnetic monopoles
[17] (a generater of the topological Berry phase [18]).
[1] M. J. Ablowitz and H. Segur,Solitons and the Inverse
Scattering Transform, SIAM, Philadelphia, (1981).
[2] T. H. R. Skyrme, Proc. Roy. Soc. A 247, 260 (1958);
ibid. A 262, 237 (1961).
[3] W. Thirring, Ann. Phys. (N.Y.) 3, 91 (1958).
[4] M. Remoissenet, inWaves Called Solitons, Concepts and
Experiments, Springer–Verlag, Heidelberg, 1994.
[5] S. Coleman, Phys. Rev. D 11, 2088 (1975).
[6] L. Rayleigh, Philos. Mag. 27, 100 (1914).
[7] B. Binder,Soliton Coupling Driven by Phase Fluctuations
in Auto-Parametric Resonance; UniTexas mp-arc 02-473.
[8] A. Tondl, T. Ruijgrok, F. Verhulst, R. Nabergoj, Au-
toparametric Resonance in Mechanical Systems Cam-
bridge University Press, New York, (2000).
[9] J. A. Sanders, F. Verhulst, ”Averaging Methods in
Nonlinear Dynamical Systems”, Appl. Math. Sci. 59,
Springer–Verlag, New York (1985).
[10] Groom et al., ”The Review of Particle Physics”, The Eu-
ropean Physical Journal C15,1 (2000).
[11] B. Binder,Josephson Effect, Ba¨cklund Transformations,
and Fine Structure Coupling; PITT-PHIL-SCI00000861.
[12] B. Binder,Topological Phase Fields, Baecklund Transfor-
mations, and Fine Structure; PITT-PHIL-SCI00000841.
[13] B. Binder,Charge as the Stereographic Projection of Ge-
ometric Precession on Pseudospheres;
PITT-PHIL-SCI00000818.
[14] N.S. Mantony, B.M.A.G. Piette, Understanding
Skyrmions using Rational Maps hep-th/0008110.
[15] D.K. Campbell, M. Peyrard, P. Sodano, Physica D19,
165 (1986).
[16] C.J. Houghton, N.S. Manton, P.M. Sutcliffe, Nucl.Phys.
B510, 507 (1998).
[17] P. A. M. Dirac, Proc. Roy. Soc. London A 133, 60 (1931).
[18] M. V. Berry, Proc. Roy. Soc. Lond. A 392, 45 (1984).


Soliton Compton Mass from Auto-Parametric Wave-Soliton Coupling

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Soliton Compton Mass from Auto-Parametric Wave-Soliton Coupling

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