#  The hydrogen atom:
consideration of the electron self-field

Biguaa L.V 111E-mail<EMAIL_ADDRESS>
Kassandrov V.V 222E-mail<EMAIL_ADDRESS>
∗ Quantum technology center
Moscow State University
† Institute of Gravitation and Cosmology,
Peoples’ Friendship University of Russia

Keywords: Spinor electrodynamics. Dirac-Maxwell system of equations.
Solitonlike solutions. Spectrum of characteristics. “Bohrian” law for binding
energies

PACS: 03.50.-z; 03.65.-w; 03.65.-Pm

##  1 THE HYDROGEN ATOM AND CLASSICAL FIELD MODELS OF ELEMENTARY PARTICLES

It is well known that the successful description of the hydrogen atom, first
in Bohr’s theory, and then via solutions to the Schrödinger equation was one
of the main motivations for the development of quantum theory. Later, Dirac
described almost perfectly the observed hydrogen spectrum using his
relativistic generalization of the Schrödinger equation. A slight deviation
from experiment (the Lamb shift between $2s_{1/2}$ and $2p_{1/2}$ levels in
the hydrogen atom) discovered later was interpreted as the effect of electron
interaction with vacuum fluctuations of electromagnetic field and explained in
the framework of quantum electrodynamics (QED) using second quantization.

Nonetheless, the problem of the description of the hydrogen atom still
attracts attention of researchers, and, in some sense, is a “touchstone” for
many new theoretical developments. In particular, it is possible to obtain the
energy spectrum via purely algebraic methods [1]. Theories of the
”supersymmetric” hydrogen atom are also widespread [2]. In the framework of
the classical ideas, the spectrum can be explained by the absence of radiation
for the electron motion along certain “stable” orbits. This effect can be
related with the balance of radiation and absorption of energy from a
hypothetical random electromagnetic background in the framework of stochastic
electrodynamics [3], or it takes place in soliton models of the hydrogen atom
[4].

The difficulties related to the description of the hydrogen atom in canonical
relativistic quantum mechanics and quantum electrodynamics (QED) include the
“strange” singularity $\sim r^{-\alpha},\leavevmode\nobreak\
\alpha=e^{2}/\hbar c\approx 0.007$ of the Dirac wave function at zero for s-
and p- states which appears in spite of the initial requirement of solution
regularity. Consideration of the magnetic field of the proton via perturbation
theory alone does not seem quite correct since the magnetic potential near
zero is of the order $1/r^{2}$ and dominates over the Coulomb potential. This
problem was discussed, in particular, in [5] (see also [6]), where the
possibility of constructing a magnetic theory of nuclear forces was
considered.

Of special topicality is consideration of the electromagnetic self-field of
the electron. From the point of view of quantum ideas, the electron in the
hydrogen atom represents a spatially distributed system according to the
probability density $\Psi^{+}\Psi$ whose elements interact both with the field
of the nucleus and between each other. Since the absolute value of the
electron charge is equal to the proton charge, the proton field turns out to
be screened at large distances.

It is known that consideration of the self-consistent collective field of
electrons is necessary for multielectron  atoms, and is phenomenologically
implemented, for example, in the Hartree–Fock method. The multielectron wave
function, however, does not differ fundamentally from the single-electron
function. Consideration of the self-action of the spatially distributed
electron charge density does not contradict the ideas of its “point-like”
character (see discussion in [7, 8]).

Note also that a similar effect of gravitational selfaction of a “free”
quantum particle was proposed in [9, 10, 11] and other works in the context of
the hypothesis on the objective character of the process of reduction of the
particle wave function.

It should be noted that the problem of consideration of the electron self-
field (self-energy) and its effect on the observed energy levels is one of the
central problems in the context of QED (see, e.g., [12]). According to the
most widely accepted ideas, the effect is related to interaction of the
electron with the second quantized electromagnetic field of “vacuum
fluctuations” [13].The calculation of its value via perturbation theory (see,
e.g., [14]) results in infinite shifts of energy levels which, after mass
renormalization, are reduced to the values of radiative corrections (Lamb
shift etc.) that agree well with experiment.

There were attempts to reproduce all the basic results of QED without second
quantization. Thus, the approach used by Barut [15, 16, 17] is based on the
analysis of the self-consistent Dirac-Maxwell system of equations studied in
this paper, and, essentially, it is purely classical. Nonetheless, the
perturbation theory and a simple regularization procedure ensure exact
reproduction of all radiative corrections usually calculated in the QED
formalism.

The effect of the self-field of the electron, as noted above, is
“nonperturbative” and is not reduced to radiative corrections. Consideration
of this effect completely alters the mathematical structure of the model,
making it essentially nonlinear. It can be assumed that the divergences in
Barut’s approach (and correspondingly, QED) could arise due to the absence of
an exact solution to the initial classical Dirac-Maxwell system which might
serve as the initial approximation in the applied perturbation theory [18].

Thus, we naturally arrive at examination of regular solutions to the self-
consistent effective nonlinear system of coupled Dirac and Maxwell equations
(in the presence of the additional external Coulomb field of the proton). Note
that such systems of equations were considered earlier in the context of
construction of an interesting class of classical soliton models of elementary
particles.

Such models are based on the well proven linear field equations of
Schrödinger, Klein–Gordon, Dirac, and Maxwell 333Or on field structures with
“natural” nonlinearity, Yang–Mills and/or Einstein equations.. In this case,
the effective nonlinearity appears only as a consequence of interaction
between different fields, and the form of this interaction is well defined by
the requirement of gauge invariance.

The founder of this approach is, perhaps, N. Rosen, who considered stationary
spherically symmetric so-called particle-like (i.e., everywhere regular with
finite Noether’s integrals of motion) solutions to the Klein-Gordon and
Maxwell system of equations with minimal electromagnetic interaction between
them in [19].

Later, this approach was extended to the (mathematically much more
complicated) system of Dirac-Maxwell equations with minimal electromagnetic
interaction between fields [20, 21, 22, 23]. It is especially interesting that
this system is a classical analog of operator equations of quantum
electrodynamics. It was shown, in particular, that charged fermions with any
half-integer spin can be described in a unified way in the framework of this
model [21, 24].

Note now that the real particles’ properties are directly manifested if only
they move in external, in particular, electromagnetic, fields. Examination of
the corresponding nonstationary particle-like solutions is quite complicated.
In this regard, stationary solutions to the Dirac-Maxwell system of equations
in the presence of an external Coulomb potential turn out to be physically
attractive and mathematically feasible for consideration.

For hydrogen-like atoms regular solutions to a nonrelativistic analog of the
considered system, the self-consistent system of Schrödinger–Poisson
equations, were considered in [18]; however, just for the case of “pure”
hydrogen ($Z=1$) a solution had not been found. Preliminary examination and
search for particle-like (or, using modern terminology, soliton-like)
solutions in the relativistic Dirac-Maxwell model of the hydrogen atom can be
found in [25] (see also [26]). In this paper, we specify more precisely and
develop the results of these previous studies.

##  2 FIELD EQUATIONS AND BASIC PARAMETERS OF SOLUTIONS

We consider the self-consistent system of Dirac-Maxwell equations in Minkowski
space $\bf M$ with the metric $\eta_{\mu\nu}=diag\\{+1,-1,-1,-1\\}$ and
coordinates $\\{X^{\mu}\\},\leavevmode\nobreak\ \mu=0,1,2,3$. This system of
equations corresponds to the Lagrangian of the form

$L=-\frac{1}{16\pi}F^{2}+\frac{\hbar
c}{2}\\{i\bar{\Psi}\gamma^{\mu}\partial_{\mu}\Psi-i\partial_{\mu}\bar{\Psi}\gamma^{\mu}\Psi-2k\bar{\Psi}\Psi\\}+eA_{\mu}\bar{\Psi}\gamma^{\mu}\Psi.$
(1)

Here, $\partial_{\mu}:=\partial/\partial X^{\mu}$, $\\{\gamma^{\mu}\\}$ are
the $4\times 4$ Dirac matrices in the canonical representation, $\Psi$ is the
Dirac bispinor, $\bar{\Psi}:=\Psi^{+}\gamma^{0}$, $\\{A_{\mu}\\}$ are the
4-potentials of electromagnetic (EM) field, and $F^{2}=F_{\mu\nu}F^{\mu\nu}$
is the main invariant expressed in terms of EM field strength tensor
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$.

It is assumed that the scalar potential $A_{0}$ includes, along with the
regular part, the fixed singular part $A_{0}^{ex}=e/R$, where $e$ is the
elementary charge. This part corresponds to the external Coulomb field of the
proton.

The quantities $\hbar,k$, generally speaking, represent some scaling
coefficients whose numerical values should be found from the comparison of the
parameters of some standard solution (corresponding, for example, to the
ground state of the hydrogen atom) with experimental values. However, it
should be expected from the agreement with the quantum theory that these
quantities correspond to the canonical constants, the Planck constant
($\hbar=h/2\pi$) and the inverse Compton electron length $k=mc/\hbar$, where
$m$ is the electron mass.

By varying the action $S=\int Ld^{4}x$ over $\bar{\Psi}$ and $A_{\mu}$ and
applying the Lorentz gauge condition $\partial_{\mu}A^{\mu}=0$ to the
potentials, we obtain the field equations of the form

$i\gamma^{\mu}(\partial_{\mu}-i(e/\hbar
c)(A_{\mu}+A_{\mu}^{ex})\Psi-k\Psi=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\Box A^{\mu}=4\pi e\bar{\Psi}\gamma^{\mu}\Psi,$ (2)

where $A_{\mu}^{ex}=\\{e/R,0,0,0\\}$.

Now, we impose on the sought solutions of system (2) the condition of
electroneutrality $|Q|=e$, i.e., the condition of equality of the absolute
value of the electron distribution charge $Q=-e\int\Psi^{+}\Psi d^{3}X$ and
the proton charge $e$. Satisfaction of this condition is equivalent to the
following:

$\int\Psi^{+}\Psi d^{3}X=1,$ (3)

so that it automatically guarantees normalization of the “wave function” of
electron distribution, although originally the field $\Psi$ can be considered
classical 444In the linear problem, normalization is achieved through
multiplication by an appropriate constant, which does not affect energy
levels. In our nonlinear case, for the normalization condition to be
satisfied, a nontrivial procedure is required (see Section 5 for details), and
the parameters of normalized and nonnormalized solutions differ..

Using scale transformations of coordinates (converting lengths to the Compton
scale) and field functions of the form

$X_{\mu}\mapsto x_{\mu}/k,\leavevmode\nobreak\
A_{\mu}\mapsto\frac{ke}{\alpha}a_{\mu},\leavevmode\nobreak\
\Psi\mapsto\left(\frac{k^{3}}{4\pi\alpha}\right)^{1/2}\psi,$ (4)

we obtain the Lagrangian

$l=-\frac{1}{16\pi}f^{2}+\frac{1}{8\pi}\\{i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-i\partial_{\mu}\bar{\psi}\gamma^{\mu}\psi-2\bar{\psi}\psi+2(a_{\mu}+a_{\mu}^{ex})\bar{\psi}\gamma^{\mu}\psi\\},\leavevmode\nobreak\
\leavevmode\nobreak\ a_{\mu}^{ex}=\\{\frac{\alpha}{r},0,0,0\\},$ (5)

and the corresponding field equations

$i\gamma^{\mu}(\partial_{\mu}-i(a_{\mu}+a_{\mu}^{ex}))\psi-\psi=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \Box a^{\mu}=\bar{\psi}\gamma^{\mu}\psi,$ (6)

which do not contain any dimensional parameters. The only dimensionless
parameter, the fine structure constant $\alpha:=e^{2}/\hbar c\approx 1/137$,
enters only the expression for the Coulomb potential $a_{0}^{ex}=\alpha/r$ and
the condition of electroneutrality/normalization which takes the following
form in terms of dimensionless variables:

$\frac{1}{4\pi}\int\psi^{+}\psi d^{3}x=\alpha.$ (7)

Note that dimensional parameters corresponding to original Lagrangian (1) are
connected with the corresponding dimensionless quantities extracted from (5)
and given in Section 4 as follows:

  * •

For the Lagrangian

$L=\frac{\hbar ck^{4}}{\alpha}l;$ (8)

  * •

for the energy

$W=\frac{\hbar ck}{\alpha}w\equiv\frac{mc^{2}}{\alpha}w;$ (9)

  * •

for the total angular momentum vector

${\bf J}=\frac{\hbar}{\alpha}{\bf j};$ (10)

  * •

for electric charge (if (7) is satisfied)

$Q=-e;$ (11)

  * •

and for the magnetic moment vector

${\bf M}=\frac{e}{k\alpha}{\bf
m}\equiv\frac{1}{\alpha}\left(\frac{e\hbar}{mc}\right){\bf m}.$ (12)

Hereinafter, our basic task is to obtain regular solutions to system of
equations (6) with Lorentz gauge and electroneutrality condition (7); we
consider only stationary solutions of the form

$\psi^{T}=\\{\kappa^{T}({\bf r}),\leavevmode\nobreak\ \chi^{T}({\bf
r})\\}e^{-i\omega t},\leavevmode\nobreak\ \leavevmode\nobreak\
a_{\mu}=\\{\phi({\bf r}),\bf a({\bf r})\\},$ (13)

where $\kappa,\chi$ are the 2-spinors, $\phi,\bf a$ are the scalar and vector
electromagnetic potentials, respectively, and the symbol ‘T’ denotes
transposition.

##  3 $\alpha$-EXPANSION
AND THE NONRELATIVISTIC LIMIT $\alpha\rightarrow 0$

In stationary case (13) the basic system of equations (6) takes the form

$\begin{array}[]{l}{\bf\Sigma}\cdot(i\nabla-{\bf
a})\chi=(\omega+\phi+\alpha/r-1)\kappa,\\\ {\bf\Sigma}\cdot(i\nabla-{\bf
a})\kappa=(\omega+\phi+\alpha/r+1)\chi,\\\
\Delta\phi=\kappa^{+}\kappa+\chi^{+}\chi,\\\ \Delta{\bf
a}=\kappa^{+}{\bf\Sigma}\chi+\chi^{+}{\bf\Sigma}\kappa,\end{array}$ (14)

with the electroneutrality condition
$(4\pi)^{-1}\int(\kappa^{+}\kappa+\chi^{+}\chi)d^{3}x=\alpha$. Here
${\bf\Sigma}=\\{{\bf\Sigma}_{a}\\},a=1,2,3$ is the set of Pauli matrices in
the standard representation.

Let us now make one more transformation of coordinates and field functions
[25], to convert length to the Bohrian scale and expand the solutions over the
small parameter $\alpha$. Namely, let us introduce the new coordinates $\bf
X$, 2-spinors $K,N$ and the potentials $\Phi,\bf A$ (not to be confused with
the original dimensional ones) as follows:

$\begin{array}[]{l}{\bf x}={\bf X}/(\alpha(1+\omega)),\\\
\kappa=\alpha^{2}(1+\omega)^{3/2}K,\leavevmode\nobreak\ \leavevmode\nobreak\
\chi=\alpha^{3}(1+\omega)^{3/2}N,\\\
\phi=\alpha^{2}(1+\omega)\Phi,\leavevmode\nobreak\ \leavevmode\nobreak\ {\bf
a}=\alpha^{3}(1+\omega)\bf A.\end{array}$ (15)

Then system (14) takes the form

$\begin{array}[]{l}{\bf\Sigma}\cdot(i\nabla-\alpha^{2}{\bf
A})N=(-\varepsilon+\Phi+1/R)K,\\\ {\bf\Sigma}\cdot(i\nabla-\alpha^{2}{\bf
A})K=(1+\alpha^{2}(\Phi+1/R))N,\\\ \Delta\Phi=K^{+}K+\alpha^{2}N^{+}N,\\\
\end{array}$ (16)

while the additional equation for the magnetic potential reads

$\Delta{\bf A}=K^{+}{\bf\Sigma}N+N^{+}{\bf\Sigma}K,$ (17)

and the electroneutrality condition takes the form
$(4\pi)^{-1}\int(K^{+}K+\alpha^{2}N^{+}N)d^{3}X=1$. Here, the “reduced
frequency” is

$\varepsilon:=(1-\omega)/(\alpha^{2}(1+\omega)),$ (18)

and it follows from the analysis of the asymptotics (see below) that solutions
(16) decreasing at infinity can exist only for $|\omega|<1$, , so the
parameter $\varepsilon$ is always positive.

The new form of Eqs. (16) is convenient because, due to the smallness of the
parameter $\alpha$, the “zero” approximation $\alpha\rightarrow 0$ can be
considered in a consistent way. Indeed, by discarding terms $\sim\alpha^{2}$,
we obtain for the first three Eqs. (16) and the electroneutrality condition

$\begin{array}[]{l}i{\bf\Sigma}\cdot\nabla K=N,\\\ i{\bf\Sigma}\cdot\nabla
N=(-\varepsilon+\Phi+1/R)K,\\\ \Delta\Phi=K^{+}K,\\\ (4\pi)^{-1}\int
K^{+}Kd^{3}X=1,\end{array}$ (19)

System (19) is closed, it contains the only parameter $\varepsilon$ to be
defined, and it does not include terms with the magnetic potential. The latter
can be defined from the prefound solutions to limiting system (19) by
integration of Eq (17). In this case, system (19) after substituting the
expression for the 2-spinor $N$ from the first equation to the second equation
is reduced to one second order equation for the main 2-spinor $K$. After
simple transformations, we obtain the following final form of the system in
the leading approximation with respect to $\alpha$:

$\begin{array}[]{l}\Delta K=(\varepsilon-\Phi-1/R)K,\\\ \Delta\Phi=K^{+}K,\\\
(4\pi)^{-1}\int K^{+}Kd^{3}X=1.\end{array}$ (20)

The structure of system (20) is close (but not identical) to the self-
consistent system of Schrödinger– Newton equations which attracted attention
earlier thanks to the ideas of Diosi [9], Penrose [10] and others, on the
possible self-gravitating nature of reduction of the particle wave function
mentioned above. In particular, the spectrum of stationary spherically
symmetric regular solutions to the Schrödinger–Newton system was numerically
obtained in [28, 11].

The Schrödinger-Newton system differs from system (20) by the absence of an
external Coulomb (Newtonian) potential and the sign of potential in the first
equation of type (20) (which is connected with the different signs of the
gravitational and electrostatic interactions). It can be easily seen that in
reality (20) represents a self-consistent system of Schrödinger-Poisson
equations (written for the case of stationary solutions), so that the
transition to the limit $\alpha\mapsto 0$ and system (19) or (20) represent,
in essence, the nonrelativistic approximation of the original Dirac-Maxwell
system.

Regular solutions the Schrödinger-Poisson equations are in fact the main
subject of the study below. As for the problem of finding relativistic
corrections, it is quite complicated, since, as will be shown later, in the
general case it results in an infinite chain of equations for radial
functions.

It should be noted that systems of equations similar to (20) were proposed
more than once as a physically justified alternative to the linear Schrödinger
equation. Schrödinger himself discussed the need to consider the “spatial
distribution” of the electron charge, including the case of multielectron
atoms [27, p. 116], and the possibility of introducing a “neutralization
potential” for this purpose [27, p. 175], i.e., the potential of the self-
field of electrons. Detailed examination of these issues can be found in [7].

To the best of our knowledge, the first attempts to obtain the regular
solutions to system (20) and the corresponding binding energies were made in
[25] and [18]. In [18] these solutions were found for lowest s-states (with
$n=1,2,3$) for different nuclear charge $Ze$ of hydrogen-like atoms; the
iteration method was used, and wave functions of the canonical linear quantum
mechanical problem were taken as the first approximation.

The obtained energy level shifts, as expected, turned out large, and in order
to agree with the observed values, it was proposed to use an analog of
renormalization. However, the correctness of the procedure of finding
solutions raises doubts 555It is not clear whether the wave function
normalization was preserved during iterations. Besides, the expression for the
binding energy, along with the term corresponding to the eigenvalues of the
“customary” quantum mechanical energy $\varepsilon$, should include the self-
energy of the electron field, see (35) below.. Anyway, the shift was not
determined for the most important case of hydrogen itself ($Z=1$).

A more reliable method of the numerical solution for system (20), together
with the variational method (in the class of the same trial functions as in
the linear quantum mechanical problem), was used in ([25, 26]). Moreover, the
solutions found therein can be used to find the magnetic potential and the
electron angular momentum in different states of the hydrogen atom by
integrating Eq. (17).

##  4 AXIALLY SYMMETRIC SOLUTIONS: SEPARATION OF ANGULAR DEPENDENCE AND THE
PROBLEM OF ENTANGLEMENT OF HARMONICS

Strictly speaking, the system of equations (19) under study has no spherically
symmetric solutions due to the spinor nature of the functions $K,N$ and the
presence of magnetic potential of the electron field calculated from
additional Eq. (17). System (20) itself, however, can be reduced to an
equation for one spherical harmonic corresponding to the main 2-spinor $K$ and
the corresponding potential $\Phi$. Indeed, by assuming the potential
spherically symmetric, it can be seen that the angular dependence in the
equations for the spinor functions in (19) can be described by any pair of
corresponding spherical spinors, i.e., similarly to the linear problem of
quantum mechanics. On the other hand, the corresponding electric charge
density $\sim K^{+}K$ in the right-hand part of (19) is not necessarily
spherically symmetric, so the system turns out to be unclosed and requires
additional harmonics of the potential (corresponding to the quadrupole and
higher even moments). The inevitable appearance of these harmonics, in turn,
requires introduction of the corresponding pairs of higher order spherical
spinors, and so on. This situation results in the requirement to consider an
infinite chain of equations for radial functions, and it is difficult to
substantiate the cut in this chain because of the absence of any small
parameter. This problem was considered in detail in [24].

The only possibility to obtain a closed system of equations is the choice of
such initial pairs of spherical spinors for which the corresponding electric
charge density is spherically symmetric (below, we denote the radial variable
$R\mapsto r$ for simplicity)

Thus, the choice of the angular dependence of the field functions is limited
by the following two possibilities corresponding, from the point of view of
canonical quantum mechanics, to s- and p-states with the eigenvalues of the
operator of orbital angular momentum $l=0,1$ respectively:

ansatz А:

$\Phi=\phi(r),\leavevmode\nobreak\ \leavevmode\nobreak\
K=k(r)\left(\begin{array}[]{l}1\\\ 0\end{array}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ N=in(r)\left(\begin{array}[]{l}\cos{\theta}\\\
\sin{\theta}\ e^{i\varphi}\end{array}\right),$ (21)

while, after the replacement $K\leftrightarrow N$, we have ansatz B:

$\Phi=\phi(r),\leavevmode\nobreak\ \leavevmode\nobreak\
K=ik(r)\left(\begin{array}[]{l}\cos{\theta}\\\ \sin{\theta}\
e^{i\varphi}\end{array}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
N=n(r)\left(\begin{array}[]{l}1\\\ 0\end{array}\right).$ (22)

Note that two more substitutions are possible, along with the two angular
dependences given above:

$\Phi=\phi(r),\leavevmode\nobreak\ \leavevmode\nobreak\
K=k(r)\left(\begin{array}[]{l}0\\\ 1\end{array}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ N=in(r)\left(\begin{array}[]{l}-\sin{\theta}\
e^{-i\varphi}\\\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \cos{\theta}\end{array}\right),$ (23)

or

$\Phi=\phi(r),\leavevmode\nobreak\ \leavevmode\nobreak\
K=ik(r)\left(\begin{array}[]{l}-\sin{\theta}\ e^{-i\varphi}\\\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\cos{\theta}\end{array}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
N=n(r)\left(\begin{array}[]{l}0\\\ 1\end{array}\right),$ (24)

conjugate to the ansatzes $\bf A$ and $\bf B$ respectively. It can be easily
verified that they result in the same form of system (20), and the
corresponding solutions differ only by the signs of projections of the
magnetic moment and the total angular momentum. Therefore, we will not
consider them separately. For both substitutions $\bf A$ and $\bf B$ main
system of equations (19) and system (20) following from it turn out consistent
and closed. Namely, for $\bf A$ we obtain:

$\begin{array}[]{l}k^{\prime}=n,\leavevmode\nobreak\ \leavevmode\nobreak\
n^{\prime}+(2/r)n=(\varepsilon-\phi-(1/r))k\\\ \mapsto
k^{\prime\prime}+(2/r)k^{\prime}=(\varepsilon-\phi-(1/r))k,\\\
\phi^{\prime\prime}+(2/r)\phi^{\prime}=k^{2},\\\ \int k^{2}r^{2}dr=1,\\\
\lambda^{\prime\prime}+(2/r)\lambda^{\prime}-(2/r^{2})\lambda=2kk^{\prime},\end{array}$
(25)

and for $\bf B$ we obtain:

$\begin{array}[]{l}k^{\prime}+(2/r)k=-n,\leavevmode\nobreak\
\leavevmode\nobreak\ n^{\prime}=-(\varepsilon-\phi-(1/r))k\\\ \mapsto
k^{\prime\prime}+(2/r)k^{\prime}-(2/r^{2})k=(\varepsilon-\phi-(1/r))k,\\\
\phi^{\prime\prime}+(2/r)\phi^{\prime}=k^{2},\\\ \int k^{2}r^{2}dr=1,\\\
\lambda^{\prime\prime}+(2/r)\lambda^{\prime}-(2/r^{2})\lambda=-2k(k^{\prime}+(2/r)k),\end{array}$
(26)

where $(\prime)$ denotes differentiation over the radial variable $r$, and
integration hereinafter is performed over the semi-infinite interval
$[0,\infty)$.

The additional last equation in (25) and (26) represents the equation for the
$\varphi$–component of the magnetic potential
$A_{\varphi}=\lambda(r)\sin{\theta}$, corresponding to (17). There, the
$r,{\theta}$-components of current densities corresponding to substitutions
(21), (22), are identically equal to zero. Therefore, we can set for the
corresponding magnetic potential components $A_{r}=A_{\theta}=0$.

The expression for the magnetic moment of the sought regular solutions to
system (25) can be easily obtained. Indeed, by integrating the last equation
with the weight $r^{3}$ and considering the electroneutrality condition and
the asymptotic behavior $\lambda\sim\mu/r^{2}$, where $\mu$ is the magnetic
moment of the distribution, we obtain

$\int((\lambda^{\prime}-\lambda/r)r^{3})^{\prime}dr=-3\mu=\int(k^{2})^{\prime}r^{3}dr=-3\int
k^{2}r^{2}dr=-3,$ (27)

which yield $\mu=1$, or in terms of dimensional units (12), with consideration
of transformation (15),

$|{\bf M}_{\bf
A}|=\left(\frac{e\hbar}{mc}\right)\frac{1}{1+\omega}\approx\frac{e\hbar}{2mc}.$
(28)

Indeed, since, due to (18),

$1+\omega=2/(1+\varepsilon\alpha^{2})\approx 2(1-\varepsilon\alpha^{2}),$ (29)

the magnetic moment, to extremely small corrections $\sim\alpha^{2}$, has the
canonical Dirac value 666It will be seen below that the numerical value of the
parameter $\varepsilon$ for all regular solutions is very small
($\varepsilon<<1$), so that the deviation of the magnetic moment from the
Dirac value is smaller by many orders of magnitude than the observed and QED-
predicted value of $(\alpha/2\pi)$.

Performing the same operations of integration and dimensionality restoration
for the magnetic moment of solutions of the second class (26), we obtain
$\mu=1/3$, so instead of (28) we now have

$|{\bf M}_{\bf B}|\approx\frac{1}{3}\left(\frac{e\hbar}{2mc}\right).$ (30)

This magnetic moment value seems quite unexpected and interesting. However, it
is difficult to propose a reasonable physical interpretation for such a value.

Now let us find the representation for (dimensionless) energy and angular
momentum of regular solutions. Using the known expression for the symmetric
energy-momentum tensor $T^{(\mu\nu)}$, corresponding to the system of
equations of type (6),

$\begin{array}[]{l}T^{(\mu\nu)}=(4\pi)^{-1}(-\eta_{\beta\rho}F^{\mu\beta}F^{\nu\rho}+\frac{1}{4}\eta^{\mu\nu}F^{2})+\\\
(16\pi)^{-1}(i\bar{\psi}\gamma^{\mu}\partial^{\nu}\psi-i\partial^{\nu}\bar{\psi}\gamma^{\mu}\psi+i\bar{\psi}\gamma^{\nu}\partial^{\mu}\psi-i\partial^{\mu}\bar{\psi}\gamma^{\nu}\psi+2a^{\mu}\bar{\psi}\gamma^{\nu}\psi+2a^{\nu}\bar{\psi}\gamma^{\mu}\psi)\end{array}$
(31)

we obtain for the energy of regular solutions

$w=\frac{\omega}{4\pi}\int\psi^{+}\psi
d^{3}x-\frac{1}{8\pi}\int(\nabla\phi)^{2}d^{3}x$ (32)

On the other hand, considering the so-called “Laue’s theorem” [29] valid for
stationary regular solutions ($\int T^{(aa)}d^{3}x=0,\leavevmode\nobreak\
\leavevmode\nobreak\ a=1,2,3$) we have $w=\int T^{\mu}_{\mu}d^{3}x$, and for
energy (32) we obtain [21, 24] another representation in terms of the (easily
calculated if the field equations are used) trace of the energy-momentum
tensor $T^{\mu}_{\mu}$,

$w=\frac{1}{4\pi}\int\bar{\psi}\psi d^{3}x.$ (33)

Going over to the Bohrian scale (15), considering the electroneutrality
condition and expression (29) for $\omega$ and restoring then the
dimensionality according to (9), we obtain instead of (32)

$W=mc^{2}\left(\frac{1-\varepsilon\alpha^{2}}{1+\varepsilon\alpha^{2}}-\frac{\alpha^{2}}{1+\varepsilon\alpha^{2}}\int(\phi^{\prime})^{2}r^{2}dr\right)=mc^{2}-W_{B}.$
(34)

Here, the binding energy $W_{B}$ up to corrections of order $\alpha^{2}$ is

$W_{B}=4R\left(\varepsilon+\frac{1}{2}\int(\phi^{\prime})^{2}r^{2}dr)\right),$
(35)

where $R:=mc^{2}\alpha^{2}/2\equiv me^{4}/2\hbar^{2}$ corresponds (for $m$
being equal to the electron mass) to the Rydberg constant. The reduced
frequency $\varepsilon$ and the energy of the electric self-field of the
electron (second term in (35)) should be determined from the solution of the
boundary value problem for the system of equations (25) or (26).

Similarly, another representation for energy (33) after the transition to
dimensional quantities yields the following expression for the binding energy
$\tilde{W}_{B}$

$\tilde{W}_{B}=4R\int n^{2}r^{2}dr$ (36)

The equality of expressions (35) and (36) represents an integral identity
satisfied for any regular solutions to system (25) or (26); it was used for
estimation of the accuracy of the obtained numerical solutions of the
corresponding boundary value problem (see Section 5).

Finally, let us derive the representation of the total angular momentum of
regular solutions. Due to the symmetric character of energy-momentum tensor
(31) the tensor of total angular momentum $M^{\mu[\nu\lambda]}$ has the form

$M^{\mu[\nu\lambda]}:=x^{\nu}T^{(\mu\lambda)}-x^{\lambda}T^{(\mu\nu)},$ (37)

and the conserved 3-vector of angular momentum $\bf j$ is given by the
expression:

${\bf j}=\int{\bf r}\times{\bf p}\leavevmode\nobreak\ d^{3}x,$ (38)

in which the density of the momentum vector ${\bf p}^{c}$ is determined by the
components $T^{(0c)}$ of the energy-momentum tensor

${\bf p}:=\frac{1}{4\pi}({\bf E}\times{\bf H})-\frac{1}{8\pi}({\bf
a}\psi^{+}\psi+(\omega+\phi)\psi^{+}{\bf
s}\psi+\frac{i}{2}(\psi^{+}\nabla\psi-\nabla\psi^{+}\psi)),$ (39)

where ${\bf s}^{c}:=i\gamma^{0}\gamma^{c}$. Using field equations (6) and
integrating by parts, we obtain

${\bf j}=\frac{1}{8\pi}\int{\bf
r}\times(i\nabla\psi^{+}\psi-i\psi^{+}\nabla\psi)\leavevmode\nobreak\
d^{3}x+\frac{1}{8\pi}\int\psi^{+}{\bf\Lambda}\psi\leavevmode\nobreak\ d^{3}x,$
(40)

where ${\bf\Lambda}^{c}:=(i/2)\varepsilon_{abc}\gamma^{a}\gamma^{b}$. In the
case of axial symmetry, the only nonzero component of the angular momentum
takes the form

$j={\bf
j}_{3}=\frac{1}{4\pi}\int\psi^{+}(-i\frac{\partial}{\partial\varphi}+\frac{1}{2}\Lambda_{3})\psi\leavevmode\nobreak\
d^{3}x.$ (41)

After transition to the nonrelativistic limit we obtain the following
ezpression for solutions of both classes A and B:

$j=\frac{1}{2}\int k^{2}r^{2}dr=\frac{1}{2},$ (42)

by considering that the electric charge is equal to the elementary charge. In
terms of dimensional units, as expected 777The relation between the total
angular momentum and electric charge of regular axially symmetric solutions
has a universal character, and in the case of absence of an external
potential, it was obtained in [24] in the form
$J=(Q/e)m\hbar/2,\leavevmode\nobreak\ \leavevmode\nobreak\ m=1,2,...$. Thus,
the models of the considered class can, in principle, describe charged
fermions with any half-integer spin., we have $J=\hbar/2$. For solutions
determined by ansatzes (23), (24), the projection of the angular momentum $J$
has the opposite sign, $J=-\hbar/2$.

##  5  NUMERICAL STUDIES AND BOHRIAN BINDING ENERGY SPECTRUM

Below, we study everywhere regular solutions to system (25) which, after
introduction of the “reduced” potential

$U:=\varepsilon-\phi,$ (43)

and ignoring the additional equation for the magnetic potential $\lambda$
takes the form

$\begin{array}[]{l}k^{\prime}=n,\leavevmode\nobreak\ \leavevmode\nobreak\
n^{\prime}+(2/r)n=(U-(1/r))k\\\ \mapsto
k^{\prime\prime}+(2/r)k^{\prime}=(U-(1/r))k,\\\
U^{\prime\prime}+(2/r)U^{\prime}=-k^{2}.\end{array}$ (44)

The asymptotics of the solutions of interest have the form

$k\sim\beta(1-r/2+...),\leavevmode\nobreak\ \leavevmode\nobreak\
n\sim-\beta/2+...,\leavevmode\nobreak\ \leavevmode\nobreak\
U\sim\delta-\beta^{2}r^{2}/6+...,$ (45)

for small $r\sim 0$, and

$k\sim Ae^{-\sqrt{\varepsilon}r},\leavevmode\nobreak\ \leavevmode\nobreak\
n\sim-\sqrt{\varepsilon}k,\leavevmode\nobreak\ \leavevmode\nobreak\
U\sim\varepsilon+q/r,...$ (46)

for large $r\rightarrow\infty$.

Assuming the existence of such solutions with an arbitrary number of nodes
$N=0,1,2,...$ with respect to the main function $k(r)$, we arrive at the
nonlinear boundary value problem for the “eigen” values: for given
$\delta=U(0)$ it is necessary to fit such $\beta=k(0)$, for which the function
$k(r)$ decreases, asymptotically approaching zero, at large distances from the
center. In this case, the potential $U(r)$ slowly decreases by itself,
according to asymptotics (46), which is used to determine the eigen value
$\varepsilon$.

We use the known (see, e.g., [11]) “bracketing” method, reducing the interval
of $\beta$ values according to the following criterion for numerical
integration. If the function $k(r)$ “blows up” and begins to increase,
$n(r_{0})=k^{\prime}(r_{0})>0$, for some value of the argument, $r=r_{0}$, the
initial value of $\beta$ should be reduced. If, conversely, (for the nodeless
solution with $N=0$) for some $r_{0}$ the function $k(r)$ becomes negative,
$k(r_{0})<0$, the value of $\beta$ should be increased. As a result, we
determine the approximate eigen value $\beta_{0}$, for which the blowing up of
the solution one way or another takes place only at a sufficiently large
$r_{0}$. By finding the value of $q\approx-r_{0}^{2}U^{\prime}(r_{0})$, at
this point, we determine the normalization integral, the charge $q\approx\int
k^{2}r^{2}dr$.

After that, the procedure is repeated for another initial value $\delta$ of
the potential $U$, resulting in the regular solution with another
normalization integral $q$. Finally, we obtain the approximate values of
$\beta_{0},\delta_{0}$, corresponding to the normalized (and therefore,
electrically neutral) nodeless regular solution to system (44), with the form
shown in Fig. 1 (cut-off radius $r_{0}\sim 90)$).The form of the corresponding
electrostatic potential $U$ is shown in Fig. 2.

Figure 1:  Form of the nodeless regular solution corresponding to the ansatz A
Figure 2:  Form of reduced electric potential for the nodeless solution
corresponding to the ansatz A

Then, for the found $\beta_{0}$ we integrate the equation for magnetic
potential in (25), choosing such value of $\gamma_{0}$ for the “derivative
$\lambda^{\prime}$ at zero”, for which the growing term in the asymptotic
$\lambda\sim Hr+\mu/r^{2}$ approximately vanishes at large distances from the
center; then we determine the magnetic moment of the distribution
$\mu=r_{0}^{2}\lambda(r_{0})$, making sure that with good accuracy $\mu\approx
1$ (see (28)). The sought form of the magnetic potential for the nodeless
solution is shown in Fig. 3.

Figure 3: Form of a magnetic potential for the nodeless solution for the
ansatz A

Eventually, using all found values of $\\{\beta_{0},\delta_{0},\gamma_{0}\\}$,
we find the integral characteristics of the nodeless solution by numerical
integration, and verify the accuracy of their determination by satisfaction of
the integral identities (compare, for example, (35) and (36)). Finally, using
the potential $U$ asymptotic, we determine the eigen value for the “reduced
frequency” $\varepsilon\approx U(r_{0})+r_{0}U^{\prime}(r_{0})$, and then
calculate the binding energy using formula (35).

It can be easily seen that a similar procedure can be applied for finding
regular solutions with an arbitrary number of given nodes $N$ of the main
function $k(r)$. To this end, it is necessary to narrow down the corresponding
interval of $\beta_{N}$ values, allowing the function $k(r)$ to change sign
$N$ times. Figure 4 shows the normalized regular solution with the number of
nodes $N=2$, as an example.

Table 1 gives the numerically calculated parameters and characteristics of
normalized regular solutions to system (44) (and corresponding solutions to
(25)) with the number of nodes $N=0\div 10$. It can be seen from Table 1 and
Fig. 5 that the values of the binding energy $W_{B}$ reproduce the canonical
Bohrian distribution with good accuracy,

$W_{n}=W/n^{2},\leavevmode\nobreak\ \leavevmode\nobreak\ n=N+1=1,2,....$ (47)

The next column of Table 1 shows the corresponding values of the binding
energy $W^{var}$, obtained by the variational method. Variation was performed
in the class of trial functions corresponding to the canonical linear quantum
mechanics problem (when the potential of the self-field $\phi(r)$ is not
included in the first of Eqs. (44) and $U(r)=\varepsilon=const$):

$k_{N}=const\cdot L_{N}^{(1)}(x)e^{-x/2},\leavevmode\nobreak\
\leavevmode\nobreak\ x:=r/N,\leavevmode\nobreak\ \leavevmode\nobreak\
N=0,1,2,...,$ (48)

where $L_{N}^{(1)}(x)$ – are the generalized Laguerre polynomials of power $N$
as a function of $x$, and the eigen values of the corresponding linear problem
are

$\varepsilon_{N}=\frac{1}{4(N+1)^{2}}.$ (49)

The corresponding trial functions $\phi_{n}(r)$ for the eigen field potential
were found by integration of the second equation in (44) with the right-hand
part corresponding to (48), so that

$\phi_{N}^{\prime\prime}+\frac{2}{r}\phi_{N}^{\prime}=k_{N}^{2}.$ (50)

The corresponding action functional was varied over two parameters of trial
functions (48) and (50), namely, the amplitude and scale parameters. Good
agreement of the results obtained using the numerical and variational methods
can be explained by the fact that inclusion of the electron self-field
potential does not qualitatively change the behavior of the functions and
their asymptotics both near zero and for large values of radial coordinate.

For solutions with a large number of nodes $N=50\leavevmode\nobreak\ (n=51)$
and $N=100\leavevmode\nobreak\ (n=101)$ the calculations were performed using
the variational method alone. In this case, the binding energy still
corresponds to Bohr’s law (47) with good accuracy..

It can be seen from Table 1 (see the last column) that the average value of
the constant $W$ in (47) is approximately $W\approx 0.120$. The spectrum of
dimensional binding energy for (35) or, equivalently, (36) takes the form

$W_{B}=4RW/n^{2}\approx 0.48R/n^{2}.$ (51)

Therefore, the effective Rydberg constant $R$ is approximately twice as small
as the experimentally observed one. The spectrum of regular solutions with
angular dependence corresponding to the ansatz $\bf B$, (see (22)) is found
similarly by numerical and variational methods. In this case, system for the
radial functions (44) is replaced by the following system if Eq. (19) is used:

$\begin{array}[]{l}k^{\prime}+(2/r)k=-n,\leavevmode\nobreak\
\leavevmode\nobreak\ n^{\prime}=-(U-(1/r))k\\\ \mapsto
k^{\prime\prime}+(2/r)k^{\prime}-(2/r^{2})k=(U-(1/r))k,\\\
U^{\prime\prime}+(2/r)U^{\prime}=-k^{2},\\\ \end{array}$ (52)

with the asymptotics of the form

$k\sim\beta r-...,\leavevmode\nobreak\ \leavevmode\nobreak\
n\sim-3\beta+...,\leavevmode\nobreak\ \leavevmode\nobreak\
U\sim\delta-\frac{1}{20}\beta r^{4}$ (53)

for $r\sim 0$; for $r\rightarrow\infty$ we have

$k\sim Ae^{-\sqrt{\varepsilon}r},\leavevmode\nobreak\ \leavevmode\nobreak\
n\sim\sqrt{\varepsilon}k,\leavevmode\nobreak\ \leavevmode\nobreak\
U\sim\varepsilon+q/r.$ (54)

Figure 4:  Form of two-node (N = 2) regular solution corresponding to the
ansatz A

Variation was performed in the class of trial functions $k(r)$, corresponding
to the particular value of the “orbital” quantum number $l=1$ of solutions of
the linear canonical quantum mechanical problem, namely, ($2l+1=3$):

$k_{N}=const\cdot L_{N}^{(3)}(x)e^{-x/2},\leavevmode\nobreak\
\leavevmode\nobreak\ x:=r/N,\leavevmode\nobreak\ \leavevmode\nobreak\
N=0,1,2,...,$ (55)

for which the eigen values are

$\varepsilon_{N}=\frac{1}{4n^{2}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ n=N+l+1=N+2,$ (56)

while the eigen potential was determined by trial functions found by
integrating former Eq. (50) with the right-hand part defined now by (55).

Table 2 gives the results of the numerical and variational studies of the
spectrum of ground and excited states of regular solutions to system (52),
which again mutually prove each other.

Figure 5: Binding energy $W_{n}$ of regular solutions for the ansatz A as a
function of the “principal quantum number” $n=N+1$ according to (squares) the
numerical and (stars) variational methods
Figure 6: Binding energy $W_{n}$ of regular solutions for the ansatz B as a
function of the “principal quantum number” $n=N+2$ according to (squares) the
numerical and (stars) variational methods

It follows from this table and Fig. 6 that the Bohrian distribution

$\ W_{n}=W/n^{2},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ n=N+2=2,3,...$ (57)

is again satisfied with a good accuracy, especially for solutions with a large
number of nodes $N$. Moreover, the quantity $W$ again has the value $W\approx
0.120$, so the effective Rydberg constant turns out universal, approximately
equal to its value for the ansatz $\bf A$ (nearly two times smaller than its
experimental value).

Figure 7: Form of the nodeless regular solution for the ansatz B
Figure 8: Form of a magnetic potential for the nodeless solution corresponding
to the ansatz B

The form of nodeless solution to system of equations (52) is given in Fig. 7
(the cut-off radius is $r_{0}\sim 200$). The corresponding magnetic potential
is shown in Fig. 8. Note finally that the values of energies of the states of
classes A and B with the same value of the “principal quantum number”
$n=N+l+1$ are close to each other, especially for large $n$. Nonetheless, the
difference between them, in particular, between the energies of analogs of
states $2s_{1/2}$ and $2p_{1/2}$ (for $n=2$, $l=0,1$ respectively) exceeds the
experimentally observed one (corresponding to the “Lamb shift”) by many orders
of magnitude.

Table 1. Parameters and characteristics of regular solutions for ansatz A.

$\begin{array}[]{lccccccc}n&\beta=k(0)&\delta=U(0)&\varepsilon=U(\infty)&W_{n}&W_{n}^{var}&W_{n}\cdot
n^{2}&W_{n}^{var}\cdot n^{2}\\\\[7.0pt]
1&0.492787&0.365947&0.02311&0.1218&0.1182&0.1218&0.1182\\\
2&0.17379&0.09042&0.004515&0.03030&0.03056&0.1212&0.1222\\\
3&0.09471&0.04005&0.001338&0.01335&0.01366&0.1201&0.1229\\\
4&0.06135&0.02243&0.0006103&0.007443&0.007696&0.1191&0.1231\\\
5&0.04382&0.01431&0.0003339&0.004757&0.004930&0.1189&0.1233\\\
6&0.03333&0.009921&0.0002028&0.003200&0.003425&0.1187&0.1233\\\
7&0.02639&0.007278&0.0001344&0.002418&0.002517&0.1185&0.1233\\\
8&0.02158&0.005366&0.00009089&0.001849&0.001927&0.1183&0.1234\\\
9&0.01808&0.004394&0.00006706&0.001461&0.001523&0.1183&0.1234\\\
10&0.01542&0.003556&0.00004866&0.001182&0.001234&0.1182&0.1234\\\
11&0.01336&0.002937&0.00003744&0.0009755&0.001020&0.1180&0.1234\\\
...&...&...&...&...&...&....&....\\\\[7.0pt]
51&.....&.......&......&......&4.745\cdot 10^{-5}&....&0.1234\\\
101&.....&.......&......&......&1.210\cdot 10^{-5}&....&0.1234\end{array}$

Table 2. Parameters and characteristics of regular solutions for ansatz $\bf
B$.

$\begin{array}[]{lccccccc}n&\beta=k(0)&\delta=U(0)&\varepsilon=U(\infty)&W_{n}&W_{n}^{var}&W_{n}\cdot
n^{2}&W_{n}^{var}\cdot n^{2}\\\\[7.0pt]
2&0.020682&0.07992&0.001025&0.02664&0.02534&0.1066&0.1014\\\
3&0.013741&0.03797&0.000705&0.01266&0.01272&0.1139&0.1145\\\
4&0.009477&0.02178&0.000400&0.007256&0.007407&0.1161&0.1185\\\
5&0.006960&0.001405&0.000253&0.004682&0.004813&0.1174&0.1203\\\
6&0.005366&0.009791&0.000168&0.003264&0.003369&0.1175&0.1213\\\
7&0.007205&0.004292&0.000116&0.002402&0.002487&0.1177&0.1219\\\
8&0.003530&0.005524&0.0000861&0.001842&0.001910&0.1179&0.1222\\\
9&0.002966&0.004367&0.0000618&0.001455&0.001512&0.1179&0.1225\\\
10&0.002540&0.003539&0.0000492&0.001180&0.001227&0.1179&0.1227\\\
11&0.002205&0.002925&0.0000379&0.0009751&0.001015&0.1180&0.1228\\\
12&0.001938&0.002458&0.0000285&0.0008194&0.0008535&0.1180&0.1229\\\
...&...&...&...&...&...&....&....\\\\[7.0pt]
52&.....&.......&......&......&4.563\cdot 10^{-5}&....&0.1234\\\
102&.....&.......&......&......&1.186\cdot 10^{-5}&....&0.1234\end{array}$

##  6 DISCUSSION OF RESULTS

It was already noted that consideration of the electric self-field of the
electron in the hydrogen atom looks quite justified, if not inevitable, both
from the general physical point of view, and according to the QED ideas. The
mathematical structure of the system of Dirac-Maxwell equations becomes quite
different from the canonical one; this system is now coupled and effectively
nonlinear.

The spectrum of regular states of such a system can be rather reliably
determined by numerical and variational methods only for the “nonlinear
analogs” of s- or p-states. Indeed, only in these cases successive terms in
the chain of equations for the radial functions have increasing order of
smallness with respect to the fine structure constant. In the nonrelativistic
approximation $\alpha\rightarrow 0$ only the principal harmonics corresponding
to the orbital quantum number $l=0,1$, contribute to the binding energy. It is
possible to obtain relativistic corrections determined for the same solutions
by harmonics with higher values of $l$ and corresponding harmonics of electric
and magnetic potentials of the electron; however, it requires cumbersome
calculations and is not considered here (estimations were obtained in [25]).

Thus, the solutions and the binding energy spectrum given in this paper should
actually be compared with solutions to the Schrödinger equation of s- and
p-types in the presence of the external Coulomb field. Surprisingly, quite a
different structure of the considered nonlinear model reproduces the canonical
Bohrian distribution with good accuracy, and the better, the larger “principal
quantum number” is considered. The effective Rydberg constant turns out to
have very close values for both classes of solutions, thus, it can be assumed
universal. The spinor “wave function” normalization condition is satisfied as
a direct consequence of the natural electroneutrality condition, and the
(projection) of the total angular momentum turns out exactly equal to
$\pm\hbar/2$.

Unfortunately, at this point coincidences with quantum mechanics and
experiment cease. The numerical value of the “Rydberg constant” is
approximately twice as small as the observed one, and the “Lamb shift” is
larger than the observed one by many orders of magnitude. Also, the magnetic
moment of distributions corresponding to the “analogs” of p-states, strangely,
is three times smaller than the magnetic moment of the electron. It is
possible, of course, following [18] to transform the discovered spectrum of
levels using the classical analog of the renormalization procedure that would
have nothing to do with subtracting of infinities, like in QED, but would be
reduced to simply equating the observed electron mass to the doubled initial
mass of the bare particle $m$ in the Dirac equation. However, this trick,
obviously, does not solve all problems of agreement between the obtained
results and experiment, including the value of magnetic moment of the ansatz B
and the anomalously large difference between the energies of levels $2s_{1/2}$
and $2p_{1/2}$.

In general, it seems that complete agreement with experiment, including
radiative corrections, is possible in the framework of a purely classical
field model of the hydrogen atom free of any divergences and the use of second
quantization. For this purpose, however, it is necessary to consider a certain
additional factor, for example, self-action with respect to the spinor field
[30] or possible supersymmetric structure of the considered self-consistent
model [2]. We leave more detailed discussion of these issues for later.

##  7 ADDENDUM

In the reviewing process, a paper by Ranãda and Uson [31] was found, which is
very close both with respect to the problem formulation, methods of study, and
results obtained 888We are deeply grateful to the referee for attentiveness,
important remarks and sending us paper [31].. Namely, in essence, a similar
hydrogen atom model accounting for the electric self-field of the electron
represented by the self-consistent system of Dirac-Maxwell equation with
Lagrangian (1) was considered. The same stationary regular solutions related
to the nonlinear analogs of s- and p-states and corresponding to our ansatzes
A and B, respectively, were sought.

However, instead of direct reduction to the nonrelativistic case based on
transformation (15) and subsequent $\alpha$-expansion (Section 3), a
cumbersome procedure of the numerical integration of initial system (14) (in
the unsubstantiated assumption on the smallness of terms with magnetic
potential in the equations for spinor harmonics) was applied in [31].

Nonetheless, for the first three levels 1s, 2s, 3s and two levels 1p, 2p the
results obtained in [31] agree with those given above (Tables 1 and 2) with
good accuracy. For example, for the energy $E$ of the ground state 1s,
according to formula (26) from [31], one finds, up to corrections
$\sim\alpha^{4}$):

$E=mc^{2}(1-0.243894\alpha^{2}),$ (58)

which, similar to our study, corresponds to the binding energy $W_{B}\approx
R/2$ and is approximately twice as small as the experimental value. For the
first three s-levels they have, with good accuracy, Bohrian function for the
binding energy (47) (our result is first 10 levels in the case of the
numerical method, and up to $n=100$ in the case of the variational method).
The results for the first two p-states considered in [31], also agree well
with the above data.

Correspondingly, the conclusion made in [31] is similar to ours. Namely, in
spite of the physically grounded statement of the problem, the results do not
improve but violate the agreement with the linear quantum mechanical problem
and experiment. In this relation, similar to [18], it is proposed in [31] to
apply the classical analog of the renormalization procedure by a corresponding
redefinition of the coupling constant $e$ and the “bare” mass $m$, in order to
formally agree with the observed parameters. In this case, it is possible to
provide correct values for the energy difference and absolute values for the
first three s-levels. According to the authors, for p-states the results of
reconciliation are not quite satisfactory.

Unfortunately, as it was already noted above (Section 6), in the considered
model, the charge/mass renormalization is in principle incapable of solving
the problem of achieving agreement with experiment. Obviously, say, it cannot
change the ratio of magnetic moments of the particle in s- and p-state
$M_{s}/M_{p}=3$, or the anomalously large difference between the binding
energies for $2s_{1/2}$ and $2p_{1/2}$ states. Note that in [31] these effects
were not discovered at all.

Moreover, the absolute values of the mechanical $J$ and magnetic $M$ moments
of the particle change in the case of renormalization proposed in [31]. For
example, for s-states we have

$J=XJ_{0},M=XM_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
J_{0}:=\frac{\hbar}{2},\leavevmode\nobreak\
M_{0}=\frac{e_{0}\hbar}{2m_{0}c},\leavevmode\nobreak\ \leavevmode\nobreak\
X:=\frac{e_{0}}{e}\equiv\frac{m_{0}}{m},$ (59)

where $e_{0},m_{0}$ are the electron charge and mass, respectively, and $e,m$
are the bare charge and mass, i.e., the corresponding scaling factors in
initial Lagrangian (1). For formal agreement of the energy of s- and p-states
with experiment, the normalization $X$ should be taken as follows999According
to our calculations (compare with (51)) a more accurate value is as follows:
$X^{2}\approx 1/0.48\approx 2.078$, and according to the data for the first
three s-levels in [31], $X^{2}\approx 2.0453$: $X^{2}\approx 2$. This scale
transformation results thus in incorrect values of the mechanical and magnetic
moments, which was not noticed in [31]

Finally, charge/mass renormalization violates the normalization condition of
the particle spinor field (3), which prevents us from considering it as the
“wave function”; it also violates the agreement of the model with the linear
quantum theory on the whole. Summing up the above said, it should be admitted
that in the form considered here and in [31] the Dirac-Maxwell model, being
mathematically correct and extremely rich with respect to its inner
properties, is unsatisfactory from the physical point of view and should be
substantially changed. Note in this regard, that in [31] an important
conclusion on a negligibly small effect of self-action with respect to the
spinor field (proposed above, see Section 6) on the hydrogen spectrum was
made. The grounds for this conclusion are both the results of the numerical
integration and qualitative considerations on large spatial extension of the
Dirac field ($>10^{-8}$ cm) in the hydrogen atom, i.e., extension to the
region where the nonlinear term is negligible. Thus, other possibilities for
the model modification should be used. We expect to discuss and consider some
of them in the future.

##  8 ACKNOWLEDGMENTS

We thank A.B. Pestov, Yu.P. Rybakov, and B.N. Frolov for helpful discussions
and bibliographic indications.

##  9 FUNDING

The work was supported by the “University program 5-100” of the Russian
Peoples’ Friendship University.

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# Textile Taxonomy and Classification Using Pulling and Twisting

Alberta Longhini, Michael C. Welle, Ioanna Mitsioni and Danica Kragic 1The
authors are with the Robotics, Perception and Learning Lab, EECS, at KTH Royal
Institute of Technology, Stockholm, Sweden albertal, mwelle, mitsioni,
<EMAIL_ADDRESS>
###### Abstract

Identification of textile properties is an important milestone toward advanced
robotic manipulation tasks that consider interaction with clothing items such
as assisted dressing, laundry folding, automated sewing, textile recycling and
reusing. Despite the abundance of work considering this class of deformable
objects, many open problems remain. These relate to the choice and modelling
of the sensory feedback as well as the control and planning of the interaction
and manipulation strategies. Most importantly, there is no structured approach
for studying and assessing different approaches that may bridge the gap
between the robotics community and textile production industry. To this end,
we outline a textile taxonomy considering fiber types and production methods,
commonly used in textile industry. We devise datasets according to the
taxonomy, and study how robotic actions, such as pulling and twisting of the
textile samples, can be used for the classification. We also provide important
insights from the perspective of visualization and interpretability of the
gathered data.

##  I Introduction

Interaction with deformable objects is an integral part of our everyday life
but still a challenge for robotic systems. Work on robotic handling of textile
or fabric traces back several decades [1] and, despite the clear need in
industry and domestic applications, many of the problems related to
perception, planning and control remain open. From the industrial perspective,
textile production and subsequent processes of garment design in fashion
industry are largely not automated. Fashion industry is also undergoing an
important transformation to address sustainability concerns, given that
textile and clothing overproduction has a significant negative impact on the
environment.

Figure 1: System setup: Two 7 DoF Franka Emika Panda arms with force-toque
sensors on the wrists, twisting and pulling a textile sample. Microscope
images are used to define textile classes prior to training.

From the scientific perspective, robotic interaction with deformable materials
has gained significant attention recently [2, 3]. Important milestones
regarding the modelling, perception, planning, control and simulation of
deformable materials have been identified but not yet reached. It may even be
so that until robots reach the dexterity, flexibility and sensing that to some
extend resembles human capabilities, successful interaction with deformable
objects will remain a challenge. In robotics, textile has been used to study
manipulation tasks like folding [4, 3, 5], robot-assisted dressing [6, 7, 8],
garment recognition and classification [9, 10, 11, 12]. In most of these
works, only a subset of textile properties is commonly considered, and textile
is merely a tool for testing sensors [13] or control strategies [14].

In our work, we aim to study textile materials and their properties using
physical interactions and wrist-mounted force-torque sensing. Similarly to
humans, we aim at using actions such as pulling and twisting, to learn more
about the textile properties, see Fig. 1. The properties are defined using a
textile taxonomy that follows the classification used in the textile
production industry. Textile properties in general, and thus interaction
dynamics, are affected by factors such as fiber material and production method
- the fiber may be raw, coated or it may be a blend of several materials. Once
used to produce garments or bed-clothing, the properties will change overtime
based on washing, wearing, steaming - the textile can become harder or softer,
less or more elastic, thinner. The change in properties will also have an
impact on the planning and control strategies used to interact with it - how
we wash, iron and fold them, how we hold and manipulate garments when dressing
somebody, whether we decide to recycle or reuse old garments.

The focus of this paper is to asses how a dual arm robotic system can be used
to identify textile production methods through pulling and twisting. We
propose to do so by learning a classifier on a dataset of textile samples that
are annotated by their construction type, determined by inspecting their
microscopic structure. We first outline a textile classification taxonomy
related to both fiber type and production method, following notation used in
textile industry. We then make a thorough study using a subset of materials
and production techniques to assess the validity of our approach. We analyse
two manipulation strategies as well as investigate which measurements are most
relevant for classification. We conclude by discussing challenges and open
problems.

Figure 2: Textile taxonomy considering yarn/fabric material and production
method. Classes considered in this work are highlighted in red.

##  II Textile Taxonomy

Textile or fabric, is a deformable and flexible material made out of yarns or
threads, which are put together by a construction or manufacturing process
such as weaving, knitting, crocheting, knotting, tatting, felting, bonding or
braiding. Most of the everyday clothing items we wear are constructed through
weaving or knitting prior to sewing, although in high fashion other processes
are used frequently too. Yarns and threads are produced by spinning raw fibers
that may have different origin: animal, plant, mineral, synthetic or their
blend. We summarize some of these aspects in Fig. 2, showing also some of the
further distinctions in terms of differences in the manufacturing process.

Woven fabric is usually produced by using two sets of yarn, while knitted
fabric employs a single set. To produce woven fabric, the yarn is interlaced,
as opposed to knitting where it is interloped. Due to its construction, woven
fabric is often hard and nonelastic, allowing it to hold creases well, and can
be only stretched diagonally if the yarn itself is not a blend that includes
elastic material. It is commonly used to produce garments such as shirts and
jeans. On the other hand, knitted fabric is soft and can be stretched in all
directions, making it rather wrinkle-resistant. One example of its frequent
use is for t-shirts. One important aspect is that it usually does not stretch
equally in all directions - for example, a t-shirt will stretch more
horizontally than vertically to more naturally follow body shape. A more
complete account of the properties of fabric can be found in [15].

The above is of importance for various robotics applications that consider
active interaction with the textile. For assisted dressing applications, it is
important for the robotic system to generate relevant control strategies when
pulling up pants, helping with the sleeves or pulling down the t-shirt: hard
textile may require completely different manipulation strategies and safety
considerations than the flexible one. Most of the clothing items will have a
content label attached to them and may offer information about the type of
yarn used. However, the manufacturing process is never described on the label,
neither is the fact that a clothing item may be a combination of woven and
knitted parts and a combination of different type of textiles being put
together in the sewing step. For example, both a pair of jeans and a t-shirt
may be made with 100% cotton textile that, in the first case, is woven and in
the second case, knitted. With the proposed taxonomy and the work in this
paper, we report on some initial insights on how some of the textile
properties can be examined by using force-torque measurements and actions such
as pulling and twisting of the textile samples.

Figure 3: Textile can look the same given regular camera (left), but very
different on the microscopic level (right). The different weaving styles
(plain weave and satin weave) with the same material determine the dynamical
properties that are important when manipulating fabric.

Even for humans, the manufacturing method may not be visible with the naked
eye and the label only provides the yarn material. We use our experience of
previously interacting with clothing items to choose appropriate actions when
dressing ourselves or others, washing, ironing, repairing or sewing. To shed
some light on this, we collected microscope images of our textile samples.
While a regular camera image may not provide enough signal resolution in
actual robot interaction with the textile, the high-resolution microscope
images show different ways of interlacing yarn that has a huge effect on the
elasticity of the textile Fig. 3. Similarly, garments with different
reflection or texture properties may look rather different under a regular
camera but their dynamical properties may be the same if the manufacturing
method, and yarn type, is the same, see Fig. 4.

Figure 4: Textile can look different on the macro level, but very similar on
the micro level. Left) two different materials (Wool and Cotton) using a
regular camera. The microscope image (right) reveals that they have the same
underlying construction - knitted jersey.

In this paper, we therefore set out to investigate how force-torque
measurements together with actions such as pulling and twisting may be used to
classify the textiles according to the proposed taxonomy. We chose pulling and
twisting since these are also the two most common actions humans use for
inspecting textile properties.

##  III Related Work

In robotics, textile materials have been considered from perception, learning,
planning and control perspectives. Most notable applications consider folding,
assisted dressing or material classification [16, 2].

Despite the broad interest in the computer vision community, most works
concentrate on building clothing item taxonomies [9] rather than identifying
material properties. Problems such as garment motion prediction [17],
classification [10], dressing 3D simulated humans [18], have also been
addressed. It has also been shown that wrinkle detection may be helpful for
classification [19, 20]. However, with vision alone it may be difficult to
estimate the physical attributes of textiles [21] although the results in [22]
indicate that vibrations captured in video can be correlated to the stiffness
and density of fabrics.

In robotics, identifying textile properties is important, but there is no
common taxonomy that allows for comparison and benchmarking of the proposed
approaches. Recent work in [23] proposes a taxonomy of 184 materials including
leather, fur and plant fiber but there is no focus on textile in particular or
the production method. Haptic feedback has often been used to label various
types of materials [24, 13, 25]. The authors in [26] study compliance and
texture to classify 32 materials including textile. Non-contact techniques
have been used in [27] to distinguish among five material categories, one of
which was textile. Thus, none of these works focuses specifically on textile
material, or considers fiber and production method in particular. When
considering textile classification, it has been studied from the fiber
material perspective [28, 7] or properties such as thickness, softness and
durability [29]. Material texture identification has been addressed in [30,
31, 32], without considering the difference between fiber material and textile
production method. Given these, we believe that our initial study and outlined
taxonomy provides examples of how textile classification can be studied in a
more structured manner.

##  IV Data collection and dataset design

For this initial study, we rely on 40 textile samples. We cut out pieces
$40\times 17$ cm in size. We have 10 polyester and wool samples, and 20 cotton
samples. Polyester samples are woven and wool samples are knitted. Out of the
20 cotton samples, 10 are woven and 10 are knitted. We cut the pieces so that
the yarn direction is along the axis of pulling, with the more elastic
direction of stretching being orthogonal to the axis of pulling as can be seen
in Fig. 5. Two Franka Emika Panda arms are equipped with wrist-mounted
Optoforce 6-axis Force-Torque (FT) sensors and flat 3D-printed grippers.

Figure 5: Upper) Flat 3D-printed grippers holding wool sample. The yarn
direction coincides with the pulling direction. FT sensors are mounted on the
wrist of the manipulators. Lower) Schematic example of pull and twist.

For data collection, we aimed to capture the samples’ properties by means of
two exploratory procedures [33], _pulling_ and _twisting_ , and investigate if
they are consistently classifiable. To further analyse different data
collection strategies, we decided to let just one arm move (also called active
arm) while the other one is kept still (passive arm), see Fig. 5. A precise
definition of these two manipulation actions is the following:

  * •

Pull The active arm exerts force on the sample by steadily moving 2 cm away
from the static passive arm, maintaining a motion direction parallel to the
grasping plane.

  * •

Twist The active arm’s end-effector rotates 90 degrees, while the arm is
pulling to ensure the sample is stretched adequately to capture its reaction
to torsion.

Each textile sample was held with a grasping force of 20N by both robot arms
and pulled and twisted 20 times. As each sequence of pulling and twisting may
result in a slight offset of the contact point, we re-positioned the sample in
the hand to the original starting points after every five pulls/twists. Force
and torque signals were recorded for a duration of $2$s from each sensor at a
frequency of $1$kHz. Thus, for each textile sample, we have 20 examples, one
for each arm, with $2\times 1000\times 6$ raw FT measurements.

###  IV-A Dataset design

We first sub-sampled the raw measurements for each FT dimension. We performed
average downsampling to $150$ values as a trade-off between noise reduction
and information loss. Given these, we build 6 datasets:

  * •

$\mathcal{D}_{active}^{twist}$, $\mathcal{D}_{active}^{pull}$,
$\mathcal{D}_{passive}^{twist}$, $\mathcal{D}_{passive}^{pull}$: 4 datasets
corresponding to the two actions for each arm individually.

  * •

$\mathcal{D}^{twist}$ and $\mathcal{D}^{pull}$: 2 datasets corresponding to
the two actions and the integrated measurements from the two arms.

In summary, the datasets was labelled to represent the samples of the taxonomy
in Fig. 2 highlighted in red. Therefore, by considering the ”Production
Method” branch we obtained 2 classes: woven and knitted, while on the
”Yarn/Fiber Material” branch, we obtained 3 classes: wool, polyester and
cotton.

##  V Data Visualization and Dataset Insights

We first inspect the generated datasets to assess to what extent the collected
data, labelled according to the proposed taxonomy, are representative for
classification. To this end, we employ t-SNE[34] and project the datasets into
a two-dimensional space. More specifically, we want to answer the following
questions:

  * •

Can the generated data and employed actions show a clear distinction between
woven and knitted textiles?

  * •

Is there a difference between pulling and twisting, in terms of how
informative they are, for the woven vs knitted classification?

  * •

What is the effect of fiber type on the classification performance and can we
distinguish not only the production method, but also the fiber type given our
datasets?

Figure 6: Effect of splitting construction method on Cotton samples. t-SNE
plot of measurements obtained from the active (top row) and passive (bottom
row) arms during pulling Cotton samples.

###  V-A Insight 1: Production Method

First, we visually inspect if the datasets are representative of the
production method. Fig.6 shows the distribution of the data projected in 2D
when only cotton is considered, where the left side of the figures shows how
the data is separated by the production method, woven (also called Cotton-
Twill) and knitted (also called Cotton-Jersey). The top row corresponds to the
active arm and the bottom row to the passive arm during a pulling trial. From
the figure we observe that by considering the actual production method we
obtain clearly structured groups of samples that would have been otherwise
masked by categorizing them as the same material.

###  V-B Insight 2: Pulling vs Twisting

Second, we assess whether there is an advantage in using both pulling and
twisting. As a first step, Fig. 7 shows that measurements for the two actions
exhibit different behavior. For example, the force measurements during pulling
exhibit more variety than they do for twisting, indicating that they may carry
more information for the different classes.

Figure 7: Pull vs twist measurements on the active arm.

To further examine how important the different measurements are and how they
can affect classification, we train a simple SVM [35] classifier to predict
the sample’s class, while trained on individual signals ($F_{x}$, $F_{y}$,
$F_{z}$, $T_{x}$, $T_{y}$, $T_{z}$). The classifier has a linear kernel and it
is implemented with Scikit-learn [36]. Table I shows the test set accuracy
based on the individual signals for the _pull_ (left) and the _twist_ (right)
strategies of the _active_ arm, when learning on a material-based dataset and
a construction-based ones.

Signal | _Pull_ | _Twist_
---|---|---
Material | Construction | Material | Construction
$F_{x}$ |  $30$% |  $82$% |  $45$% |  $57$%
$F_{y}$ |  $32$% |  $63$% |  $38$% |  $52$%
$F_{z}$ |  $70$% |  $100$% |  $53$% |  $73$%
$T_{x}$ |  $42$% |  $75$% |  $38$% |  $63$%
$T_{y}$ |  $44$% |  $83$% |  $38$% |  $63$%
$T_{z}$ |  $40$% |  $53$% |  $45$% |  $59$%
All |  $80$% |  $100$% |  $70$% |  $87$%
TABLE I: SVM test set performance based on the individual signals for a
dataset with 3 classes for material distinction and 2 classes for the
production methods.

From the SVM results, we observe that for the material-based classification,
the accuracy scores for the two actions are comparable and rather low.
However, when considering the proposed labels, the accuracy increases and in
almost all signal cases, pulling outperforms twisting.

These results reinforce that following construction-based taxonomy is
advantageous as well as the notion that the sensory feedback varies a lot
depending on how textile is manipulated. It is therefore of fundamental
importance to understand how to choose the proper exploration strategy.
Moreover, as mentioned in Section II, the way in which textile threads are
interlocked leads to different elastic properties. Knitted textiles, for
example, can be stretched lengthwise or along the horizontal direction. Woven
textiles instead, are usually not stretchable, apart from a bias direction
that for example, for denim is the diagonal one. All these concepts play an
important role in classification and highly increase the complexity of the
task.

###  V-C Insight 3: Fiber Material vs Production Method

Lastly, we assess to what extent the fiber type can be identified in addition
to the production method. An example of this can be seen in Fig. 8 for dataset
$\mathcal{D}_{passive}^{twist}$ that depicts the difference between samples
categorized using just their production method (knitted or woven) and when
they are labelled based on the material with the further distinction of the
Cotton class, which is split into Cotton-Twill (woven) and Cotton-Jersey
(knitted) to reflect its production methods. We can see that Cotton-Twill
visually belongs to a separate cluster as observed in section V-A. It can be
also noticed that some of the Cotton-Twill samples are closer to Polyester as
to Wool, while Cotton-Jersey is closer to Wool than to Polyester.

Figure 8: Twist on the passive arm: visualization with respect to left)
production method, right) fiber material.

We can also observe the effect of the proposed taxonomy on the individual
signal level, by considering the dataset $\mathcal{D}_{active}^{pull}$ for
example. Fig. 9 depicts how the split of Cotton by construction method
highlights the difference between the mean force used for Cotton-Twill and
Cotton-Jersey at the end of the pulling action, further showing the necessity
of splitting the Cotton class. Moreover, besides the detectable distinction
among Polyester, Wool, Cotton-Twill and Cotton-Jersey signals, woven materials
keep being the ones with higher mean force while knitted ones are in general
less tension-resistant, reflecting the behaviour of the construction methods.

Figure 9: Mean and standard deviation of the measurements sensed by the active
arm while pulling.

##  VI Classification and Interpretability

The next step is to assess the classification performance using a more complex
architecture, like a CNN model. The input of the network are vectors of the
six concatenated FT measurements, the network consists of four 2D
convolutional layers and its activations are rectified linear units (ReLU).
Furthermore, we adopt rectangular kernels of size $5\times 1$ which convolve
across measurements of the same signal as done in [37]. The output sizes of
the convolutional layers are respectively $24,12,8$ and $4$.

The features learned from the last convolutional layer are flattened and fed
to a fully connected layer with 48 hidden neurons. The outputs of this block
are the predicted class probabilities. We also consider the case of the joined
measurements for the _active_ and _passive_ arm, using the same architecture
but adjusting the size of the fully-connected layer to accommodate the
$150\times 12$ signal.

Input Dataset | Materials | Construction
---|---|---
$\mathcal{D}_{active}^{pull}$ | $87.5\%$ | $100\%$
$\mathcal{D}_{passive}^{pull}$ | $85.8\%$ | $96.7\%$
$\mathcal{D}_{active}^{twist}$ | $76.7\%$ | $95.0\%$
$\mathcal{D}_{passive}^{twist}$ | $78.3\%$ | $89.2\%$
$\mathcal{D}^{pull}$ | $95.0\%$ | $100\%$
$\mathcal{D}^{twist}$ | $79.0\%$ | $91.7\%$
TABLE II: Test accuracy with a CNN model for all the different datasets when
following the material-based labelling and the proposed, construction-based
one.

We partitioned each dataset into 90/10 train/test splits. Table II summarizes
the classification results. Firstly, we observe that the construction-based
labeling outperforms the material-based one for any dataset or action. More
specifically, using our taxonomy both actions provide enough information for
accurate predictions. However, twisting is consistently less accurate than
pulling for all labeling and datasets considered.

These observations are validated in the case of the joined datasets with
$\mathcal{D}^{pull}$ achieving excellent performance in distinguishing the
construction method of the textile, leading to the conclusion that pulling is
a better option for classification.

###  VI-A Interpretability and measurement assessment

To further examine the effect of the different measurements for
classification, we interpret the results from the CNN model through GradCAM
[38]. GradCAM is an interpretability technique that produces visual
explanations in the form of heatmaps that portray which parts of the input
contribute the most to the predicted label. We follow the same methodology as
in [39] to produce and inspect the contribution of each feature in samples
from datasets $\mathcal{D}^{pull}$ and $\mathcal{D}^{pull}_{active}$ of Table
II.

An example of the heatmaps can be seen in Fig. 10 for two correctly classified
samples of woven cotton and knitted cotton from the dataset
$\mathcal{D}^{pull}$. Every row corresponds to a different measurement channel
and its color is defined by how important it is for the prediction. The
importance is scaled between 0 and 1 and follows the colormap on the right of
the images. Fig. 10 shows that for both cotton instances, the network is
focusing on the same features between the passive and the active arm.
Concretely, for woven cotton, the forces on both axes $Z$ are the most
important features, followed by the torques $T^{p}_{x},T^{a}_{x}$ and some
parts of $T^{p}_{y},T^{a}_{y}$. However, for knitted cotton, the network
utilized all the force measurements for both arms and the torque on axis $Z$
for the active arm. These results indicate that even when utilizing the
material based-labeling, the CNN network focuses on different patterns when
classifying samples of the same material but different construction methods.

Finally, we inspect two classification results from dataset
$\mathcal{D}^{pull}_{active}$ when it is labelled according to the proposed
taxonomy. The left heatmap corresponds to a correct classification of a woven
sample and the right heatmap on the correct classification of a knitted one.
The important features agree with the intuition gained from Table I as the
decisions are heavily based on the ones highlighted also by the SVM, namely
forces $F_{z}$ and torques $T_{x},T_{y}$.

Figure 10: Heatmaps of feature importance for the classification using dataset
$\mathcal{D}^{pull}$. The intensity of each row (with red being the most
important) denotes what the network focuses on to classify the sample.  Figure
11: Heatmaps for the dataset $\mathcal{D}^{pull}_{active}$: activation for
left) a correctly classified Woven sample, right) correctly classified Knitted
sample.

We note that the visualization shows only where the neurons of the network are
most active for single examples. It is possible for a network to construct
multiple patterns to classify the same class, making generalisation difficult.
We can however, observe that certain measurements are more important for
classification than others, which is a valuable insight when designing future
active exploration strategies.

##  VII Discussion and Conclusion

In this work, we outlined a textile taxonomy and showed our initial results on
textile sample classification using pulling and twisting actions. The focus of
the study was to assess to what extent a taxonomy used in textile industry is
a viable model to structure robotic interaction and provide a basis for a
whole new area of structured studies of this class of deformable materials.

We provided insights into how a combination of different actions and FT
measurements vary with respect to textile production method and fiber
material. Pulling and twisting, as inspired by the human interaction with
textile, are viable choices of actions and these provide relevant information
for classification. One interesting question that arises is what other actions
can be potentially employed and to what extent dexterous hand/finger motion
could be exploited in addition to pulling and twisting. Some previous work
demonstrated the use of specialized fingers and sensors for this purpose and
it is yet to be seen to what extent we can consider such solutions to become
commercial.

Combining multiple actions, as well as passive and active interaction, is also
an interesting aspect to be explored. We may start with pulling/stretching and
based on the first step classification, subsequent routines may be performed
more suitably for identifying categories of interest, such as for example
fiber material, elasticity, whether the textile is wet or dry, etc. Here,
reinforcement learning may be used to learn actions that maximize the utility
of the sensor readings for discriminating various textile properties.

We also performed an initial study using visual feedback under pulling and
twisting. However, for the considered categories, regular cameras do not
provide enough resolution to bring sufficient information on the production
method or the fibre type. One could potentially rely on the reflectance
properties of textile materials, but most of the works in this area that stem
from the computer vision community, are not applicable in uncontrolled
settings that would occur in real-life applications. One idea that arose when
conducting the study was the fact that creases and wrinkles on the textile
fabric may be a useful feature to exploit for certain applications. When
pulling or stretching the fabric in many different directions, creases and
wrinkles will vary dependent on the properties of the textile: dense and hard
textile creases differently from soft and thin textile. In such cases,
integrating vision and FT may be useful. Careful consideration on what visual
features are used needs to be taken into account. For example, using flow-
based methods [40] or specified wrinkle detectors [41] dealing with various
texture properties may be considered.

An additional important aspect to be considered is the ability to assess how
textile properties change over time. Certain textiles are made to be more
durable, fibers are blended, their use and handling in terms of washing,
ironing, folding, will affect how clothing items deteriorate over time. In
other words, the information of the fiber content usually available on the
label sewn on the clothing item, may be helpful but it is not fully relevant.
For example, a T-shirt made out of cotton, may be more elastic and thicker
when new, and rather thin and almost non-elastic after many washings. Thus,
its handling in terms of washing and ironing will be different, as well as one
may decide to keep or reuse a newer one, and recycle a well-used one.

The outlined taxonomy, visualization, CNN classification and measurement
interpretability are important tools that can provide more insight into the
difficulty of the considered problem. The taxonomy provides a structured
approach to study textile materials and has not been previously considered in
the area of robotics. We also need an approach that brings the robotics
community closer to textile production industry and this is one way of
achieving that. We provided several examples of how the generated textile
material classes are a viable approach and how these can be studied together
with actions such as pulling and twisting.

Initial classification results using deep neural networks show a good
potential and we will build on these with a more extensive database of
samples, actions and multimodal sensory feedback. More specifically, we will
study a richer set of pulling actions, with samples of different sizes also
considering standardized textile for the purpose of repeatability,
reproducibility and replicability. We believe that this study is an important
step toward a more robust and versatile textile handling and manipulation for
applications such as various household tasks, assisted dressing and recycling.

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