Abstract
The idea that fermions could be solitons was actually confirmed in theoretical models in 1975 in the case when the space-time is two-dimensional and with the sine-Gordon model. More precisely S. Coleman showed that two different classical models end up describing the same fermions particle, when the quantum theory is constructed. But in one model the fermion is a quantum excitation of the field and in the other model the particle is a soliton. Hence both points of view can be reconciliated.The principal aim in this paper is to exhibit a solutions of topological type for the fermions in the wave zone, where the equations of motion are non-linear field equations, i.e. using a model generalizing sine- Gordon model to four dimensions, and describe the solutions for linear and circular polarized waves. In other words, the paper treat fermions as topological excitations of a bosonic field.
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Notes
As we will see, solitons are self-localized wave packets arising from a robust balance between dispersion and nonlinearity. In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. They are a universal phenomenon, exhibiting properties typically associated with particles. Optical soliton in media with quadratic nonlinearity and frequency dispersion are theoretically analyzed over the years.
Meaning that quantum mechanic particles can exhibit wave-like properties and waves (e.g. light) can exhibit particle like properties.
This is clearly a discrete subgroup of a continuous symmetry which rotates p and x into each other.
Te exact extension of electromagnetic duality in N = 4 super Yang-Mills theory should thus be regarded as the harmonic oscillator of four-dimensional gauge theory.
In fact the collection \(\phantom {\dot {i}\!}|\xi _{1}{\cdots } \xi _{j_{p}}\rangle \) is not exactly a basis: one should replace these vectors by smeared vectors using smooth test functions ϕ j : \(\phantom {\dot {i}\!}{\int }_{\mathbb {R}}{\cdots } {\int }_{\mathbb {R}}\phi _{1}(\xi _{1}){\cdots } \phi _{j_{p}}(\xi _{j_{p}}) |\xi _{1}{\cdots } \xi _{j_{p}}\rangle d\xi _{1}{\cdots } d\xi _{j_{p}}.\) But the notation \(\phantom {\dot {i}\!}|\xi _{1}{\cdots } \xi _{j_{p}}\rangle \), although it is not mathematically correct, is more intuitive.
This process, often called second quantization, is quite analogous to the route that in ordinary quantum mechanics leads from a set of classical coordinates q i to a set of quantum operators \(\hat {q}_{i}\). There is one important technical difference, though, since \(\hat {\psi }(x, t)\) is a field, i.e., an object which depends on the coordinate x. The latter plays the role of a continuous-valued index, in contrast to the discrete index i, which labels the set q i . Field theory therefore is concerned with systems having an infinite number of degrees of freedom.
A way to characterize the Bogomol’nyi solution is the following. A consequence of the supersymmetry involved is that the action has the form \(\mathcal {A}[\phi ]:={\int }_{\mathbb {R}\times \mathbb {R}}\left ( {1\over 2c^{2}} \left |\phi _{t}\right |^{2} - {\frac {1}{2}}\left |\phi _{x}\right |^{2} - 2(W^{\prime }(\phi ))^{2} \right ) dtdx\), where here \(W(s)={\frac {1}{2}}\left ( {\frac {1}{3}}s^{3}-s\right )\). It implies in particular that a function f of x is a solution of (6) if and only if it is a critical point of the functional \(\mathcal {E}[f]:={\int }_{\mathbb {R}}\left ((f^{\prime })^{2} + 4\beta ^{2}\left ( W^{\prime }(f)\right )^{2} \right )dx\). Now we can write down this functional as \(\mathcal {E}[f]= {\int }_{\mathbb {R}}\left ( \left (f^{\prime }+ 2\beta W^{\prime }(f)\right )^{2} - 4\beta f^{\prime }W^{\prime }(f)\right ) dx = {\int }_{\mathbb {R}}\left (\left ( f^{\prime }+ 2\beta W^{\prime }(f)\right )^{2} - 4{\beta d\over dx}\left (W(f)\right )\right ) dx\). If we assume that \(\lim _{x\rightarrow \pm \infty }f(x)=f_{\pm }\), then the last term in the right hand side is just C:=4(W(f −)−W(f +)) So \(\mathcal {E}[f]-C\) is the integral of the square of f ′+2β W ′(f) and a trivial solution is to set f ′+2β W ′(f) = 0: this is exactly (7) (see [22]).
In 1917 German mathematician Emilie Emmy Noether had shown that the mass, charge and other attributes of elementary particles are generally conserved because of symmetries. For instance, conservation of energy follows if one assumes that the laws of physics remain unchanged with time, or are symmetric as time passes. And conservation of electrical charge follows from a symmetry of a particle’s wave function. Sometimes, as in our case, attributes may be maintained because of deformations in fields. Such conservation laws are called topological, because topology is that branch of mathematics that concerns itself with the shape of things.
ϕ is also a Bogomol’nyi solution of the form ϕ(t, x) = f(x−v t), where f ′+2β W ′(f) = 0 and \(W(s):= -{2\sqrt {\alpha }\over \lambda ^{2}}\cos {\lambda f\over 2}\).
Skyrme’s explanation was that, in the full quantum theory, it is possible to construct a new quantum field whose fluctuations are the solitons. The new field operator is obtained by an exponential expression in the original field ϕ
$\psi _{\pm }(x)=e^{i\lambda (\phi \pm {\int }_{-\infty }^{x}\,dx^{\prime }{\frac {\partial \phi }{\partial t}})} $with two spin components (and a normal ordering understood).
The construction of this new field operator is an example of the vertex operator construction later to be so important in string theory and in the representation theory of infinite dimensional algebras (resembling quantum field theories).
We use the summation convention that any Latin index that is repeated in a product is automatically summed on from 1 to 3. The arrows on variables in the internal space indicate the set of 3 elements \(\vec {q}=(q_{1}, q_{2}, q_{3})\) or \(\vec {\sigma }=(\sigma _{1}, \sigma _{2}, \sigma _{3})\) and \(\vec {q} \vec {\sigma }= q_{k} \sigma _{k}\).
The choice of the basis vectors does not influence the relations between physical objects. Using the left adjoint basis \(\sigma _{k}^{\prime Q} := Q \sigma _{k}\) instead of the right adjoint basis σ k Q would be completely equivalent and would lead to the same coordinate independlent final results. Gauge dependent quantities like the connection \(\vec {\Gamma }_{s}\) would of course look slightly different \( \vec {\Gamma }^{\prime }_{s} = - \dot {\alpha } \vec {n} - \sin \alpha \cos \alpha \; \dot {\vec {n}} - \sin ^{2} \alpha \; \vec {n} \times \dot {\vec {n}} , \)
In gauge theories expression like (∂ s Q)Q † appearing in (18) and (19) are usually associated with trivial connections and zero curvature. In this respect we would like to remind that the concepts of connection fields and curvature go back to differential geometry of curved spaces by Carl Friedrich Gauss. Our fields are defined on the unit sphere S 3, whose Gaussian curvature is obviously one. The curvature tensor in differential geometry is designed to measure areas on curved surfaces. Therefore, it seems rather natural to allow for non-vanishing curvature tensors for fields dwelling on unit spheres. In our notation the trivial connection would read \(-2 \vec {\Gamma }_{s}\). The factor 2 appears due to the fact that the S U(2) generators in the fundamental representation are σ k /2, whereas in (19) there appears only σ k . It is easy to check that for the trivial connection \(\vec {\Gamma }_{s}^{\prime } = -2 \vec {\Gamma }_{s}\) the Maurer-Cartan equation (33) looks like a vanishing curvature tensor (31).
ε together with the propagation direction defines the polarization plane as it was experimentally defined in crystal optics, “Fresnel definition”.
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Kouneiher, J. Conceptual Foundations of Soliton Versus Particle Dualities Toward a Topological Model for Matter. Int J Theor Phys 55, 2949–2968 (2016). https://doi.org/10.1007/s10773-016-2928-8
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DOI: https://doi.org/10.1007/s10773-016-2928-8