|  Intuitive understanding of extinction of small particles in absorbing and
active host media within the MLWA†
---|---
|  Anton D. Utyushev,∗a Vadim I. Zakomirnyi,b Alexey A. Shcherbakov,a Ilia L.
Rasskazovc and Alexander Morozd
|  In an absorbing or an active host medium characterized by a complex
refractive index $n_{2}=n_{2}^{\prime}+{\rm i}n_{2}^{\prime\prime}$, our
previously developed modified dipole long-wave approximation (MLWA) is shown
to essentially overly with the exact Mie theory results for spherical
nanoparticle with radius $a\lesssim 25$ nm ($a\lesssim 20$ nm) in the case of
Ag and Au (Al and Mg) nanoparticles. The agreement for Au and Ag (Al and Mg)
nanoparticles, slightly better in the case of Au than Ag, continues to be
acceptable up to $a\sim 50$ nm ($a\sim 40$ nm), and can be used, at least
qualitatively, up to $a\sim 70$ nm ($a\sim 50$ nm) correspondingly. A first
order analytic perturbation theory (PT) in a normalized extinction
coefficient, $\bar{\kappa}=n_{2}^{\prime\prime}/n_{2}^{\prime}$, around a
nonabsorbing host is developed within the dipole MLWA and its properties are
investigated. It is shown that, in a suitable parameter range, the PT can
reliably capture the effect of host absorption or gain on the extinction
efficiency of various plasmonic nanoparticles.

##

††footnotetext: a School of Physics and Engineering, ITMO University, 197101,
Saint-Petersburg, Russia††footnotetext: b Beckman Institute for Advanced
Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL
61801, USA††footnotetext: c SunDensity Inc., Rochester, NY 14604, USA
††footnotetext: d Wave-scattering.com ††footnotetext: ∗ Corresponding author:
<EMAIL_ADDRESS>† Electronic Supplementary Information
(ESI) available: See DOI:

##  1 Introduction

Electromagnetic scattering in an absorbing host characterized by a complex
refractive index $n_{2}=n_{2}^{\prime}+{\rm i}n_{2}^{\prime\prime}$ (where
$n_{2}^{\prime}$ and $n_{2}^{\prime\prime}$ are real) has more than fifty
years old history. The traditional scattering theory neglects the host
dissipation and gain 1, 2, because those cases imply either vanishing or
infinite scattering wave at the spatial infinity. Once wave number
$k=k^{\prime}+{\rm i}k^{\prime\prime}$ is a complex number, conventional
expressions for cross sections cannot be straightforwardly extended for
$k^{\prime\prime}\neq 0$, because the expressions yield cross sections as
complex quantities. Not surprisingly, the history of scattering in an
absorbing host is filled in with a number of controversies 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. This is probably why in classical
textbooks it is only fleetingly mentioned in Section 12.1.3 of ref. 2. Already
the definition of an incident intensity is not straightforward, as the field
incident on a particle is different at different points on the particle as a
result of an absorbing medium 4, 5. In an absorbing or an active unbounded
host, the imaginary part $k^{\prime\prime}$ of the complex wave number
$k=k^{\prime}+ik^{\prime\prime}=2\pi n_{2}/\lambda_{0}$, where $\lambda_{0}$
is the vacuum wavelength, is nonzero, $k^{\prime\prime}\neq 0$. A great deal
of effort was required to arrive at suitable definitions of cross sections for
$k^{\prime\prime}\neq 0$.

In contrast to the conventional case of $k^{\prime\prime}=0$, two sets of
cross sections are commonly used: inherent and apparent. The former is
obtained by performing surface integrals of corresponding Poynting vectors
over the particle surface. The approach was developed a long time ago 4, 5,
but currently accepted expressions for the inherent cross-sections were not
presented until 1999 7, 8, 9, 10. The history of apparent cross sections,
which are operationally defined in the far field, began with the optical
theorem of Bohren and Gilra 6. Here, too, it took a long time, until 2007, to
arrive at definitive expressions for the apparent extinction cross-section
cross section, $C_{\rm ext}$, and the so-called apparent scattering cross
section, $C_{\rm sca}$ 11, 12, 13, 14. A curious feature of the apparent
extinction cross-section in an absorbing host is that it can be negative 20,
which is neither an artifact of numerical simulations nor it violates any
physical law. The apparent extinction cross-section quantifies the difference
in the readings of a forward-scattering detector taken with and without the
particle. If the surrounding medium is absorbing, the presence of the particle
can in fact make the detector signal stronger, thereby implying a negative
extinction cross-section. There is no violation of the energy conservation
law, since in this case the extinction cross-section is not used to quantify
the energy budget of a finite volume encompassing the particle 20. Another
curiosity of apparent cross-sections in an absorbing host is that an intrinsic
definition of an apparent absorption cross-sections, $C_{\rm abs}$, is still
missing. The difficulty lies in that the very presence of a particle
necessarily modifies the near field around the particle. The latter may be the
cause of an additional absorption in the host medium outside the particle
compared to what is happening in the absence of the particle 8, 9. The
critical point is that because the field is disturbed by the particle,
additional absorption may be realized in the medium external to the particle
21.

First-principles far-field computations based on the general Lorenz–Mie theory
showed that increasing absorption in the host medium broadens and suppresses
plasmon resonances in the apparent extinction and effective scattering
efficiency factors and suppresses resonance features in the phase function 14.
At the same time, using inherent cross-sections it was shown that a
surrounding lossy medium can narrow the plasmonic resonances, as well as
increase their amplitude dramatically 15, 17. Given this recent revival of
interest in the role of the host dissipation or gain on extinction cross
sections with conflicting predictions of it either enhancing or reducing the
localized surface plasmonic resonance (LSPR) of small particles 14, 15, 17,
18, 19, the focus of present work is on the apparent extinction cross-section
$C_{\rm ext}$ in the so-called modified long-wave approximation (MLWA) 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 17, 19. The MLWA 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 17, 19, which can be viewed as a next-
order approximation beyond the Rayleigh limit, is known to overcome a number
of severe deficiencies of the Rayleigh limit (see eqn (2) below) and, at least
for nonabsorbing hosts, be surprisingly precise 27, 31, 33, 34, 17, 19.

Fig. 1: Schematic representation of the problem under consideration. A
spherical particle with radius $a$ and refractive index $n_{1}$ is embedded in
an absorbing or gain medium with $n_{2}=n_{2}^{\prime}+{\rm
i}n_{2}^{\prime\prime}$, where $n_{2}^{\prime}$ and $n_{2}^{\prime\prime}$ are
real. Imaginary refractive index of the host $n_{2}^{\prime\prime}>0$ for an
absorbing medium and $n_{2}^{\prime\prime}<0$ for a gain medium, and
$n_{2}^{\prime\prime}=0$ for a transparent medium. Normalized extinction
coefficient $\bar{\kappa}=n_{2}^{\prime\prime}/n_{2}^{\prime}$ is introduced
to quantify the magnitude of the host dissipation or gain.

The outline of our contribution is as follows. Section 2 first recalls in its
Subsection 2.1 the expression for distance-independent apparent extinction
cross section $C_{\rm ext}$ in the framework of the Lorenz–Mie theory 13. In
Subsection 2.2 our earlier developed MLWA 33, 34 is summarized. In Subsection
2.3, still within the framework of MLWA, an analytic first-order perturbation
theory in a normalized extinction coefficient,
$\bar{\kappa}=n_{2}^{\prime\prime}/n_{2}^{\prime}$, representing the magnitude
of the host dissipation, is developed. In Section 3, performance of the above
approximations involving small plasmonic particles in different absorbing and
active hosts is investigated. Discussion of some of the observed features is
provided in Section 4. We then conclude with Section 5.

##  2 Theory

###  2.1 Apparent extinction cross section in the framework of the Lorenz–Mie
theory

The specific expression for distance-independent apparent extinction cross
section $C_{\rm ext}$ in the framework of the Lorenz–Mie theory is 13 is given
as an infinite sum over different multipole orders $\ell\geq 1$,

$C_{\rm ext}=-\frac{2\pi}{k^{\prime}}\,\mbox{Re
}\left\\{\sum_{\ell=1}^{\infty}\frac{1}{k}\,(2\ell+1)(T_{E\ell}+T_{M\ell})\right\\}.$
(1)

In the optical convention, $T_{p\ell}$ correspond to the familiar Mie’s
expansion coefficients $a_{\ell}$ and $b_{\ell}$ (eqns (4.53) of ref. 2), i.e.
$T_{E\ell}=-a_{\ell}$ for electric, or transverse magnetic (TM), polarization,
and $T_{M\ell}=-b_{\ell}$ for magnetic, or transverse electric (TE),
polarization. The physical, or operational, meaning of the distance-
independent apparent $C_{\rm ext}$ is that it determines the reading of a
polarization-sensitive well-collimated radiometer (WCR) at a sufficiently
large distance $r$ from the particle 13,

$\mbox{WCR signal}\propto\exp(-2k^{\prime\prime}r)(\Omega-C_{\rm ext})I_{\rm
inc},$

where $\Omega$ is the area of the objective lens of the WCR, and $I_{\rm inc}$
is the intensity of the incident homogeneous (uniform) plane wave at the
center of the particle. Importantly, the distance-independent extinction
cross-section cannot be introduced in the context of evaluating the energy
budget of an arbitrarily shaped volume containing the scattering particle 13.

###  2.2 The dipole MLWA

Fig. 2: Demonstration of different physical mechanisms described within MLWA
on the extinction spectra of Ag sphere with $a=40$ nm in the host with
$n_{2}^{\prime}=1.33$. Here $\bar{\kappa}=0$ for a transparent host and
$\bar{\kappa}=0.05$ in case of a dissipative host. The refractive indices of
Ag were taken from ref. 35. (a) Quasi-static approximation; (b) radiative
correction taking the effect of retardation with respect to the incident
field; (c) the MLWA including additionally the dynamic depolarization term.
Obviously all the correction terms of the dipole MLWA given by eqn (3) are
necessary to achieve a fairly accurate approximation to the exact Mie theory
shown in panel (d) for a comparison. All the above approximations, but the
exact Mie theory, assume a constant field inside the sphere.

The MLWA is a rational approximation to the Mie coefficients in terms of a
fraction of simple polynomials in size parameter $x$ that in a concise way
combines three different elementary terms, involving size-independent quasi-
static Fröhlich term,

$F_{p\ell}:=\upsilon+\frac{\ell+1}{\ell},$

the dynamic depolarization ($D_{p\ell}\sim x^{2}$) and the radiative reaction
($R_{p\ell}\sim x^{2\ell+1}$ for $p=E$ and $R_{p\ell}\sim x^{2\ell+3}$ for
$p=M$) in the functional form 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 17, 19

$T_{p\ell}=\frac{iR_{p\ell}(x)}{F_{p\ell}+D_{p\ell}(x)-iR_{p\ell}(x)}\cdot$
(2)

Assuming nonmagnetic media, one has $\upsilon=\mu_{1}/\mu_{2}=1$ for magnetic
(TE) polarization ($p=M$), and
$\upsilon=\varepsilon:=\varepsilon_{1}/\varepsilon_{2}$ for electric (TM)
polarization ($p=E$), where the subscript 1 (2) identifies the relevant
quantities of a sphere (host). The functional form of eqn (2) makes it
transparent that the usual Rayleigh limit, which amounts to setting
$D_{p\ell}(x)=R_{p\ell}(x)\equiv 0$ in the denominator for $\ell=1$ and $p=E$,
is recovered for $x,x_{s}\ll 1$. Here $x$ is in general complex size
parameter, $x=2\pi a/\lambda$, with $\lambda$ being the wavelength in the host
medium, whereas $x_{s}=2\pi n_{2}a/n_{1}\lambda=x/\sqrt{\varepsilon}$. The
vanishing of the size-independent $F$ in the denominator yields the usual
quasi-static Fröhlich LSPR condition, which determines the quasi-static LSPR
frequencies $\omega_{0\ell}$. In what follows, we shall focus on the dipole
MLWA which yields 33, 34

$T_{E1}\approx\frac{2i(\varepsilon-1)x^{3}/3}{\varepsilon+2-3(\varepsilon-2)x^{2}/5-2i(\varepsilon-1)x^{3}/3},$
(3)

where $\varepsilon=\varepsilon_{1}/\varepsilon_{2}$ is the relative dielectric
function. An exceptional feature of the MLWA dipole contribution is that one
can determine analytically an exact position of the complex pole of $T_{E1}$
(the Mie coefficient $-a_{1}$), and hence the dipolar LSPR position, at

$\displaystyle\varepsilon_{E1}=-2\times\frac{1+3x^{2}/5+ix^{3}/3}{1-3x^{2}/5-2ix^{3}/3},$
(4)

which corrects formula eqn (6) of ref. 19. The proof of that
$\varepsilon_{E1}$ yields complex zero of the denominator $D$ of $a_{1}$ is
relegated to Sec. S2 “Complex pole of the dipole contribution within the
dipole MLWA” of ESI†. In the limit of small $x$, one can expand the
denominator of $\varepsilon_{E1}$ as $\sim 1+3x^{2}/5+2ix^{3}/3$, whereby eqn
(4) reduces to the familiar classical Bohren and Huffman result (cf. eqn
(12.13) of ref. 2):

$\varepsilon_{E1}\sim\varepsilon_{\rm
BH}\approx-2-\frac{12x^{2}}{5}\qquad(|x|\ll 1).$ (5)

Fig. 3: Extinction spectra, $Q_{\rm ext}$, for individual spherical Ag
nanoparticles with radii $a=10$, $20$, $30$, $40$, $50$ nm in water-like host
($n_{2}^{\prime}=1.33$) with $\bar{\kappa}=0.001,0.005,0.01,0.05$, $0.1$, as
shown in the legend for each plot. Spectra were calculated by eqn (1) of the
exact Mie theory (solid red line), the dipole MLWA by eqn (3) (blue dot-dashed
line), the Mie theory in nonabsorbing host ($n_{2}^{\prime\prime}=0$; dashed
green line), and the first order $\bar{\kappa}$-PT of our eqn (9) (dotted
orange line). Because the Mie theory in the nonabsorbing host is independent
of $\bar{\kappa}$, its plots are identical at each given column. The
refractive indices of Ag were taken from ref. 35.

###  2.3 Perturbation theory in a normalized extinction coefficient within the
MLWA

A useful parametrization of the relative dielectric function $\varepsilon$
between the sphere and the host, suitable for studying the departure from
nonabsorbing host, is (cf. eqn (9) of ref. 19):

$\varepsilon=\frac{\varepsilon_{1}}{\varepsilon_{2}}=\frac{(n_{1}^{\prime}+in_{1}^{\prime\prime})^{2}}{(n_{2}^{\prime}+in_{2}^{\prime\prime})^{2}}=\frac{(n_{1}^{\prime}/n_{2}^{\prime}+in_{1}^{\prime\prime}/n_{2}^{\prime})^{2}}{(1+in_{2}^{\prime\prime}/n_{2}^{\prime})^{2}}=\frac{\varepsilon_{t}}{(1+i\bar{\kappa})^{2}},$
(6)

where
$\varepsilon_{t}=(n_{1}^{\prime}/n_{2}^{\prime}+in_{1}^{\prime\prime}/n_{2}^{\prime})^{2}=\varepsilon_{1}/(n_{2}^{\prime})^{2}$
is the relative (in general complex number if $\varepsilon_{1}$ is complex)
dielectric function between the sphere and a nonabsorbing host with real
refractive index $n_{2}^{\prime}$, and
$\bar{\kappa}=n_{2}^{\prime\prime}/n_{2}^{\prime}$ is a normalized extinction
coefficient representing the magnitude of the host dissipation. On
substituting the Taylor expansion of the electric dipole term $a_{1}$,

$a_{1}(\varepsilon)=a_{1}(\varepsilon_{t})+\bar{\kappa}\left.\frac{{\rm
d}a_{1}}{{\rm d}\bar{\kappa}}\right|_{\varepsilon=\varepsilon_{t}}+{\cal
O}(\bar{\kappa}^{2}),$

in the expression eqn (1) of the apparent cross section, $C_{\rm ext}$, it is
in principle possible to provide in a systematic way the results for the
extinction efficiency, $Q_{\rm ext}=C_{\rm ext}/\pi a^{2}$, in the first order
of $\bar{\kappa}$ for both gain and absorbing media in the dipole
approximation. The derivative ${\rm d}a_{1}/{\rm d}\bar{\kappa}$ is determined
by eqn (S21) of ESI† (Section S5: “The derivative ${\rm d}a_{1}/{\rm
d}\bar{\kappa}$”),

$\displaystyle\frac{{\rm d}a_{1}}{{\rm
d}\bar{\kappa}}=-12n_{2}^{\prime}x^{\prime}\bar{\kappa}\,\mbox{Re}\left[\frac{2-\varepsilon_{t}(\varepsilon_{t}-1)+\frac{1}{5}(\varepsilon_{t}^{2}-\varepsilon_{t}+2)(x^{\prime})^{2}}{n_{2}D^{2}(\varepsilon_{t},x^{\prime})}\right],$
(7)

where $x^{\prime}=x_{0}n_{2}^{\prime}$, with $x_{0}=2\pi a/\lambda$ being the
size parameter in vacuum host, and

$D(\varepsilon_{t},x^{\prime})=\varepsilon_{t}+2-3(\varepsilon_{t}-2)(x^{\prime})^{2}/5-2i(\varepsilon_{t}-1)(x^{\prime})^{3}/3$
(8)

is the denominator $D$ in eqn (3) in the limit $\bar{\kappa}\to 0$. One finds
eventually

$\displaystyle Q_{\rm ext}$ $\displaystyle=$ $\displaystyle\frac{C_{\rm
ext}}{\pi
a^{2}}\approx\frac{2}{x^{\prime}}\,\mbox{Re}\left\\{\frac{3}{x}\,\left[a_{1}(\varepsilon_{t})+\bar{\kappa}\left.\frac{da_{1}}{d\bar{\kappa}}\right|_{\varepsilon=\varepsilon_{t}}\right]\right\\}$
(9) $\displaystyle=$
$\displaystyle\frac{2}{x^{\prime}}\mbox{Re}\left\\{\frac{3}{x}a_{1}(\varepsilon_{t})\right\\}-$
$\displaystyle
12n_{2}^{\prime}x^{\prime}\bar{\kappa}\,\mbox{Re}\left[\frac{2-\varepsilon_{t}(\varepsilon_{t}-1)+\frac{1}{5}(\varepsilon_{t}^{2}-\varepsilon_{t}+2)(x^{\prime})^{2}}{n_{2}D^{2}(\varepsilon_{t},x^{\prime})}\right],\hskip
28.45274pt$

which defines the first order expansion of $Q_{\rm ext}$ in the parameter
$\bar{\kappa}$ within the perturbation theory (PT). The first term on the rhs
is the dipole MLWA in a nonabsorbing host characterized by $\varepsilon_{t}$.
The second term on the rhs is the perturbation correction. In what follows, we
will refer to eqn (9) as the $\bar{\kappa}$-PT approximation.

##  3 Results

###  3.1 The dipole MLWA vs perturbation theory

Fig. 4: Extinction spectra, $Q_{\rm ext}$, for Al, Ag, Au and Mg spherical
nanoparticles with $a=30$ nm embedded in a host absorption medium with
$n_{2}^{\prime}=1.33$ (left column) and gain host medium (right column). The
left (right) column is for $\bar{\kappa}=0.01$ ($\bar{\kappa}=-0.01$). $Q_{\rm
ext}$ are shown as calculated by eqn (1) of the Mie theory (solid red line),
within dipole MLWA of eqn (3) (blue dot-dashed line), the Mie theory
approximation with the nonabsorbing host (green dashed line), and by the
$\bar{\kappa}$-PT given by eqn (9) (dotted orange line). The refractive
indices of Ag, Al, Au were taken from ref. 35 and those of Mg from ref. 36.

Fig. 3 compares performance of various approximations relative to the exact
Mie theory involving

  * •

the dipole MLWA (with the sole term $T_{E1}$ in eqn (1) given by eqn (3))

  * •

the Mie theory approximation with nonabsorbing host (determined by eqn (1)
with $n_{2}^{\prime}=0$)

  * •

the first-order $\bar{\kappa}$-PT given by eqn (9)

in water-like host ($n_{2}^{\prime}=1.33$) for different host absorption
characterized by different values of $\bar{\kappa}$ on the example of Ag
spheres with radii between $10$ and $50$ nm. Fig. 4 compares results for
spheres made from most common plasmonic materials Al, Ag, Au, Mg with radius
$a=30$ nm in water-like host ($n_{2}^{\prime}=1.33$) with $\bar{\kappa}=0.01$
and $-0.01$ (similar results for glass-like host ($n_{2}^{\prime}=1.5$) are
presented in Fig. S2I of ESI†). As demonstrated in those figures, the dipole
MLWA can be very precise. In the case of Ag and Au nanoparticles, the dipole
MLWA essentially overlies with the exact Mie theory results for $a\lesssim 25$
nm. The agreement, slightly better in the case of Au than Ag, continues to be
acceptable up to $a\sim 50$ nm (Fig. S1III of ESI†), and can be used, at least
qualitatively, up to $a\sim 70$ nm (Fig. S1IV of ESI†). In the case of Al and
Mg nanoparticles, a slight deviation from Mie theory results becomes visible
by the naked eye already for $a\gtrsim 25$ nm (cf. Figs. S1I and S1II of
ESI†). The agreement continues to be acceptable up to $a\sim 40$ nm and can be
used at least qualitatively up to $a\sim 50$ nm (Fig. S1III of ESI†).

A first general observation is that with increasing sphere radius the
performance of the dipole MLWA worsens. This is not surprising, as the MLWA is
by definition is a long wavelength approximation. The same applies to the
$\bar{\kappa}$-PT. The $\bar{\kappa}$-PT overlies with the dipole MLWA for all
$a\leq 50$ nm, provided that $\bar{\kappa}\leq 0.01$. The latter justifies a
posteriori that our $\bar{\kappa}$-PT is correct. Another interesting tendency
revealed by Fig. 3 (Figs. S1I and S1II of ESI†) is that the maximal radius $a$
for which the dipole MLWA remains accurate increases with increasing
$\bar{\kappa}$.

Fig. 5: Size dependence of $\bar{Q}_{\bar{\kappa}}$, the ratio of the PT
correction to the leading term on the rhs of eqn (9), for Ag sphere at the
(size-dependent) LSPR wavelength for different $\bar{\kappa}$: (a) $0.001$,
(b) $0.005$, (c) $0.01$ and (d) $0.1$. Fig. 6:  Extinction spectra, $Q_{\rm
ext}$, for spherical Ag nanoparticles with (a) $a=50$ nm, (b) $a=60$ nm, (c)
$a=70$ nm, and (d) $a=80$ nm embedded in Poly(3-hexylthiophene) (P3HT) host
medium. Spectra are shown as calculated with the exact Mie theory (solid red
line), the Mie theory for non-absorbing media ($n_{2}^{\prime\prime}=0$)
(green dashed line), the dipole MLWA (blue dot-dashed line), and the
$\bar{\kappa}$-PT (dotted orange line).

Importantly, the results of $\bar{\kappa}$-PT and the dipole MLWA overlie with
the exact Mie theory results for $a\lesssim 25$ nm, which supports earlier
observation on the range of validity of the MLWA in an absorbing host by
Khlebtsov 17. Whereas the dipole MLWA continues to overlie with the exact Mie
theory results for $a\lesssim 25$ nm irrespective of $\bar{\kappa}$, the
results of the $\bar{\kappa}$-PT begin to deviate from those of the dipole
MLWA for $\bar{\kappa}\gtrsim 0.05$. This is to be expected, because the
$\bar{\kappa}$-PT is a first order perturbation theory of the dipole MLWA in
$\bar{\kappa}$.

Surprisingly, the smaller the particle, the greater the deviation of the
$\bar{\kappa}$-PT relative to the dipole MLWA for $\bar{\kappa}\gtrsim 0.05$
(see Fig. S1 and S2 of ESI†). The origin of this behaviour is that
$D(\varepsilon_{t})$ is typically small in a proximity of a LSPR (cf. the
generalized Fröhlich condition eqn (5)). Whereas the first term in eqn (9) is
of the order $1/D(\varepsilon_{t})$, the term proportional to $\bar{\kappa}$
is of the order $1/D^{2}(\varepsilon_{t})$. However, when the size parameter
$x$ increases, $|D(\varepsilon_{t})|$ at a LSPR decreases (due to a larger
separation from the complex zero), and, as illustrated in Fig. 5,
$\bar{Q}_{\bar{\kappa}}$, defined as the ratio of the PT correction to the
leading term on the rhs of eqn (9), gradually decreases, whereby the first-
order $\bar{\kappa}$-PT begins to approximate the MLWA. Note also how the
ratio $\bar{Q}_{\bar{\kappa}}$ increases with $\bar{\kappa}$, which, as
expected, explains why the $\bar{\kappa}$-PT begins to deviate from the MLWA
with increasing $\bar{\kappa}$.

###  3.2 Mie theory approximation with nonabsorbing host

Similar behaviour is observed for the Mie theory approximation with
nonabsorbing host ($n_{2}^{\prime\prime}=0$) that begins to deviate from the
exact Mie theory results, but somewhat earlier, beginning with
$\bar{\kappa}=0.01$. Again the smaller the particle, the greater the
deviation. Nevertheless, the Mie theory approximation with nonabsorbing host
becomes the best approximation with increasing particle radius (e.g. $a\gtrsim
40$ nm and $\bar{\kappa}\leq 0.01$). For example, the Mie theory in
nonabsorbing host begins to overlie with the Mie theory in an absorbing host
for $a\gtrsim 50$ nm and $\bar{\kappa}=0.01$ (see Fig. S1 of ESI†). A
threshold radius for which it happens increases with $\bar{\kappa}$.

Why the Mie theory approximation with nonabsorbing host becomes the best
approximation for sufficiently large $a$ can be explained by increasing
relevance of higher-order multipole contributions with increasing particle
size. This is understandable, because with increasing $a$ higher order
multipoles, which are obviously absent in any dipole approximation, become
more and more relevant. This is demonstrated also in Fig. 6 showing the effect
of increasing the radius of spherical Ag nanoparticles in
Poly(3-hexylthiophene) (P3HT) host medium, which was used in a number of
recent studies 15, 37, 17, 19. Unlike inherent dipole approximations, the Mie
theory approximation with nonabsorbing host captures reasonably well both the
dipole and quadrupole peaks. Obviously, higher-order MLWA of ref. 34 could
have captured the quadrupole peak, but this goes beyond the scope of present
study.

###  3.3 Isolating the host dissipative effects on the extinction cross
sections

The dissipative effects of the host on the extinction cross sections, $Q_{\rm
ext}$, can obviously be isolated by subtracting from the exact value $Q_{\rm
ext}=Q_{{\rm ext};\varepsilon}$ the value of $Q_{{\rm ext};\varepsilon_{t}}$
obtained by the Mie theory for the non-absorbing host characterised by
$n_{2}^{\prime\prime}=0$ (i.e. $\varepsilon\to\varepsilon_{t}$). In what
follows, we denote the difference by $\eta_{\rm eff}:=Q_{{\rm
ext};\varepsilon}-Q_{{\rm ext};\varepsilon_{t}}$. Thus $\eta_{\rm eff}$
quantifies the contribution of the host absorption or gain in the resulting
$Q_{\rm ext}$. To our satisfaction, it turns out that, in a suitable parameter
range, the analytic $\bar{\kappa}$-PT can reliably capture the effect of the
host absorption on the extinction efficiency of a plasmonic nanosphere as
demonstrated in Fig. 7. Not surprisingly, the agreement can also be reached in
the case of gain media, which opens door for analysis of promising
applications involving active media, see Fig. 4 (Fig. S5 of ESI†) such as
spasers 38, 39, 40. Therefore, within the range of its validity, the first
order $\bar{\kappa}$-PT allows one to both intuitively and analytically
understand the mechanisms of host dissipation or gain on the extinction
efficiency of a plasmonic nanosphere.

Fig. 7: The effect $\eta_{\rm eff}$ of the host absorption (left column) and
the gain host medium (right column) on $Q_{\rm ext}$ in the Mie theory (solid
red line) and the $\bar{\kappa}$-PT (dashed dark green line) for Al, Ag, Au
and Mg materials with different radii $a$ embedded in the host medium with
$n_{2}^{\prime}=1.33$ and $\bar{\kappa}=0.001$ (left column) and
$\bar{\kappa}=-0.001$ (right column).

##  4 Discussion

Rather exhaustive MLWA analysis in the absorbing case has been recently
provided by Khlebtsov 17 by employing slightly different MLWA. The study of
ref. 17 arrived at similar conclusions that MLWA is very precise for
$a\lesssim 25$ nm. Our study complements ref. 17 with (i) a first order
analytic perturbation theory around a nonabsorbing host in a normalized
extinction coefficient $\bar{\kappa}$ ($\bar{\kappa}$-PT) and (ii) an
investigation of gain media. In addition, we have examined the behaviour of Al
and Mg nanoparticles.

##  5 Conclusions

Our previously developed modified dipole long-wave approximation (MLWA) was
shown to essentially overly with the exact Mie theory results for $a\lesssim
25$ nm ($a\lesssim 20$ nm) in the case of Ag and Au (Al and Mg) nanoparticles.
The agreement for Au and Ag (Al and Mg) nanoparticles, slightly better in the
case of Au than Ag, continues to be acceptable up to $a\sim 50$ nm ($a\sim 40$
nm), and can be used, at least qualitatively, up to $a\sim 70$ nm ($a\sim 50$
nm). We developed within the dipole MLWA a first order analytic perturbation
theory (PT) around a nonabsorbing host in a normalized extinction coefficient
$\bar{\kappa}$ and investigated its properties. It was shown that, in a
suitable parameter range, the $\bar{\kappa}$-PT can reliably capture the
effect of host absorption or gain on the extinction efficiency of a plasmonic
nanosphere. Considering growing interest in light-matter interactions, we
expect that our results will help in designing optimal systems comprising
plasmonic nanoparticles embedded in suitable dissipative or gain media for
various applications.

## Author Contributions

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

## Electronic Supplementary Information

The supplementary information includes Sec. S1 “Supplementary Figures”, where
additional visual data are provided on apparent extinction efficiencies for
further particle radii, host $n_{2}^{\prime}$, and normalized extinction
coefficient $\bar{\kappa}$. In Sec. S2 “Complex Pole of the Dipole
Contribution within the Dipole MLWA”, the predictive power of the complex pole
knowledge on the LSPR position is investigated. Sec. S3 “Proof of that
$\varepsilon_{E1}$ is the Complex Zero of the Denominator of $a_{1}$” presents
mathematical proof of what is in its title. In Sec. S4 “The Imaginary Part of
$\varepsilon_{E1}$ is Negative” proves the stated statement for any real $x$.
In Sec. S5 “The Derivative $da_{1}/d\bar{\kappa}$” explicit calculations
leading to eqn (7) are presented. Sec. S6 “A Previous Attempt to Formulate a
Perturbation Theory (PT) in Terms of $\bar{\kappa}$” analyses an earlier
failed theoretical attempt. Sec. S7 “Apparent Extinction Cross Section”
reviews the absorption and scattering cross-sections within the MLWA. Finally,
Sec. S8 “Conventional Scattering Theory” reviews some related aspects of
conventional scattering theory.

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# Scattering in algebraic approach to quantum theory. Jordan algebras

Albert Schwarz  A. Schwarz
Department of Mathematics
University of California
Davis, CA 95616, USA,
schwarz @math.ucdavis.edu

###### Abstract

Using geometric approach we formulate quantum theory in terms of Jordan
algebras. We analyze the notion of (quasi)particle (=elementary excitation of
translation-invariant stationary state) and the scattering of (quasi)particles
in this framework.

##  1 Introduction

In algebraic approach physical observables correspond to self-adjoint elements
of $*$-algebra $\cal A.$ The vector space $\cal B$ of self-adjoint elements of
$\cal A$ is not closed with respect to the operation of multiplication, but it
is closed with respect to the operation $a\circ b=\frac{1}{2}(ab+ba).$ It was
suggested long ago [1] that more natural algebraic approach should be based on
axiomatization of the operation $a\circ b.$ This idea led to the notion of
Jordan algebra defined as a commutative algebra over $\mathbb{R}$ with
multiplication $a\circ b$ obeying

$(x\circ y)(x\circ x)=x\circ(y\circ(x\circ x)).$ (1)

(Defining by $R_{a}$ the operator of multiplication by $a$ we can express this
identity saying that operators $R_{a}$ and $R_{a\circ a}$ commute.)

The space $\cal B$ is closed also with respect to the linear operator $Q_{a}$
transforming $x\in\cal B$ into $axa$ (here $a\in\cal B$). Another approach to
Jordan algebras is based on axiomatization of this operator. The operator
$Q_{a}$ is quadratic with respect to $a$, hence it can be extended to operator
$Q_{\tilde{a},a}$ that is symmetric and bilinear with respect to
$\tilde{a},a.$ Imposing some conditions on $Q_{a}$ we obtain the notion of
quadratic Jordan algebra. Starting with original definition of Jordan algebra
we obtain a quadratic Jordan algebra taking $Q_{a}=2R(a)^{2}-R(a\circ a).$
Conversely starting with quadratic Jordan algebra $\cal B$ we obtain a family
of products obeying (1) by the formula

$a\circ_{x}b=Q_{a,b}(x).$

In what follows we work with unital topological Jordan algebra $\cal B$
specified by a product $a\circ b$. It seems, however, that quadratic Jordan
algebras are more convenient to relate Jordan algebras to geometric approach
[4]-[7]. This relation is based on the remark that one can define a cone
${\cal B}_{+}$ in $\cal B$ as the smallest convex closed set , invariant with
respect to operators $Q_{a}$ and containing the unit element. One can consider
also the dual cone ${\cal B}_{+}^{\vee}$ consisting of linear functionals on
$\cal B$ that are non-negative on ${\cal B}_{+}.$ ( See Section 2 for more
details).

In the case when $\cal B$ consists of self-adjoint elements of $C^{*}$-algebra
$\cal A$ elements of ${\cal B}_{+}^{\vee}$ can be identified with positive
linear functionals on $\cal A$ (states). This means that applying geometric
approach to Jordan algebras we generalize the algebraic approach based on
$*$-algebras. Notice, that Jordan algebras can be regarded as the natural
framework of algebraic approach. This statement is prompted by the following
theorem: Cones of states of two $C^{*}$-algebras are isomorphic iff
corresponding Jordan algebras are isomorphic ( Alfsen - Shultz).

One says that $\cal B$ is a JB-algebra if it is equipped with Banach norm
obeying

$||x\circ
y||\leq||x||\cdot||y||,||x^{2}||=||x||^{2},||x^{2}||\leq||x^{2}+y^{2}||.$

For such an algebra the cone ${\cal B}_{+}$ consists of squares. ( For any
Jordan algebra an element $a\circ a=Q_{a}(1)$ belongs to the cone; for JB-
algebras all elements of the cone have this form.) It follows that for JB-
algebras the cones are homogeneous: automorphism groups of cones act
transitively on the interior of the cone.

It is natural to use homogeneous cones in geometric approach, hence it seems
that JB-algebras can lead to interesting models.

The appearance of cones in the theory of Jordan algebras allows us to apply
general constructions of geometric approach to these algebras. In the Section
3 we consider (quasi)particles in the framework of Jordan algebras. In
Sections 4 and 5 we consider scattering of (quasi)particles. In Section 6 we
define generalized Green functions and show that (inclusive) scattering matrix
can be expressed in terms of these functions. Section 7 is devoted to some
generalizations of Jordan algebras and of above results.

We do not discuss here numerous papers where the theory of Jordan algebras is
related to physics (see, in particular, [9]-[23]). A short review of the
theory of Jordan algebras and Jordan pairs as well as a review of various
relations between Jordan algebras and physics is given in a companion paper
[24].

##  2 Jordan algebras

Let us consider a topological Jordan algebra over $\mathbb{R}$ denoted $\cal
B.$ (Recall that Jordan algebra is defined as a commutative algebra with
multiplication obeying the identity $(x\circ y)(x\circ x)=x\circ(y\circ(x\circ
x)).$ In what follows we consider unital Jordan algebras. If $\cal B$ is a
complete topological vector space and the multiplication is continuous we say
that $\cal B$ is a topological Jordan algebra.) The most important class of
topological Jordan algebras consists of Jordan Banach algebras
($JB$-algebras). The Banach norm in JB-algebra should obey

$||x\circ
y||\leq||x||\cdot||y||,||x^{2}||=||x||^{2},||x^{2}||\leq||x^{2}+y^{2}||.$

(See [3] for a review of the theory of operator Jordan algebras.)

In finite-dimensional case the class of JB-algebras coincides with the class
of Euclidean Jordan algebras classified in the famous paper by Jordan, von
Neumann and Wigner [2]. The most natural simple finite-dimensional JB-algebras
$\textgoth{h}_{n}(\mathbb{R}),\textgoth{h}_{n}(\mathbb{C}),\textgoth{h}_{n}(\mathbb{H})$
consist of Hermitian matrices with real, complex or quaternion entries. One
more series of simple finite-dimensional JB-algebras is the series of spin
factors (or Jordan algebras of Clifford type). These algebras are generated by
elements $1,e_{1},..,e_{n}$ with relations $e_{i}\circ e_{i}=1,e_{i}\circ
e_{j}=0$ for $i\neq j$ . The last simple finite dimensional JB-algebra is
Albert algebra $\textgoth{h}_{3}(\mathbb{O})$ that can be realized as an
algebra of $3\times 3$ Hermitian matrices with octonion entries. This
$27$-dimensional algebra is exceptional ( it cannot be embedded into matrix
Jordan algebra with the operation $a\circ b=\frac{1}{2}(ab+ba).$).

The structure semigroup $Str(\cal B)$ is defined as a semigroup generated by
automorphisms of $\cal B$ and operators $Q_{a}$ that can be expressed in terms
of Jordan triple product $\\{a,x,b\\}$ by the formula $Q_{a}(x)=\\{a,x,a\\}.$
(Jordan triple product is defined by the formula $\\{a,x,b\\}=(a\circ x)\circ
b+(x\circ b)\circ a-(a\circ b)\circ x.)$ The operator $Q_{a}$ is quadratic
with repect to $a$; we use the notation $Q_{\tilde{a},a}$ for the
corresponding bilinear operator: $Q_{\tilde{a},a}=\\{\tilde{a},x,a\\}.$

An element $B\in Str(\cal B)$ ( a structural transformation) obeys

$Q_{Ba}=BQ_{a}B^{t}$ (2)

where $B\to B^{t}$ denotes an involution in the structure semigroup
transforming every operator $Q_{a}$ into itself and every automorphism into
inverse automorphism. The structure group $Strg(\cal B)$ is generated by
automorphisms of $\cal B$ and invertible operators $Q_{a}$.

The inner structure semigroup $iStr(\cal B)$ is generated by operators
$Q_{a}$. The inner structure group $iStrg(\cal B)$ consists of invertible
elements of inner structure semigroup.

We define the positive cone in Jordan algebra as the smallest closed convex
subset of $\cal B$ containing the unit element and invariant with respect to
operators $Q_{a}$.111 We define a cone in topological vector space as a closed
convex subset, that is invariant with respect to dilations $x\to\lambda x$
where $\lambda>0.$ We do not impose any further restrictions, hence in our
terminology a vector space is a cone. One can say also that the positive cone
is the smallest closed convex subset of $\cal B$ that contains unit element
and is invariant with respect to inner structure semigroup. Positive cone is
invariant with respect to automorphisms, hence it is invariant also with
respect to the structure semigroup. It is obvious that the cone contains all
squares; this follows from $Q_{a}(1)=a\circ a.$ For JB-algebras all elements
of the cone can be represented as squares and the structure group acts
transitively on the interior of the cone.

The positive cone is denoted by ${\cal B}_{+}$ and the dual cone is denoted by
${\cal B}_{+}^{\vee}.$

Let us suppose that the Jordan algebra $\cal B$ is obtained from associative
algebra $\cal A$ as a set of self-adjoint elements with respect to involution
∗; this set is equipped with the operation $a\circ b=\frac{1}{2}(ab+ba).$ Then
$Q_{a}(x)=axa,Q_{\tilde{a},a}=\frac{1}{2}(\tilde{a}xa+ax\tilde{a}).$ It
follows that the structure semigroup of $\cal B$ contains all maps $x\to
A^{*}xA$ where $A$ is a self-adjoint or unitary element of $\cal A$ (for self-
adjoint element we get a map $Q_{A}$, for unitary element we get an
automorphism).

If $\cal A$ is a $C^{*}$-algebra then every element of the form $A^{*}A$ can
be represented as a square of self-adjoint element, hence the cone in $\cal B$
is a smallest closed convex set containing all elements of the form $A^{*}A$
where $A\in\cal A.$ The dual cone consists of all linear functionals on $\cal
B$ that are non-negative on the elements of the form $A^{*}A$ (in agreement
with the standard definition of positive linear functional on associative
algebra with involution).

We consider also complexifications of Jordan algebras considered as complex
Jordan algebras with involution. ( If we start with $JB$-algebra then the
complexification is called $JB^{*}$-algebra.)The cones associated with these
algebras can be defined as the cones of their real parts. The Jordan triple
product $\\{a,x^{*},b\\}$ is defined as an operation antilinear with respect
to the middle arguments and linear with respect to other arguments. We
introduce the notation $Q_{a}(x)=\\{a,x,a^{*}\\}$ where $x$ is real. It is
easy to check that $Q_{a}$ belongs to the structure semigroup (it is equal to
$Q_{\alpha}+Q_{\beta}$ where $\alpha$ and $\beta$ are real and imaginary parts
of $a$). It follows that in the case when $x$ is real and $b=a^{*}$ the triple
product belongs to the positive cone. Notice that the map $Q_{a}$ is Hermitian
with respect to $a.$

In geometric approach to scattering theory [7] we are starting with a cone of
states $\cal C\subset\cal L$ and a group $\cal U$ consisting of automorphisms
of the cone. ( Here $\cal L$ denotes a complete topological vector space.)

If we take as a starting point a Jordan algebra $\cal B$ we can take as $\cal
C$ either the cone ${\cal B}_{+}$ or the dual cone. The group $\cal U$ can be
identified with the structure group $Strg(\cal B)$. Sometimes it is convenient
to fix a semiring $\cal W$ consisting of endomorphisms of the cone; if we are
starting with Jordan algebra $\cal B$ the semiring $\cal W$ can be defined as
the smallest semiring containing the structure semigroup $Str(\cal B)$.

##  3  (Quasi)particles

To define (quasi)particles and their scattering we should specify time
translations $T_{\tau}$ and spatial translations $T_{\bf x}$ as elements of
the structure group $Strg(\cal B)$. In other words we should fix a
homomorphism of the commutative translation group $\cal T$ to $Strg(\cal B)$.
The translation group acts also on the cones. We are using the same notations
$T_{\tau},T_{\bf x}$ for time and spatial translations of the cones. As usual
we denote $T_{\tau}T_{\bf x}\alpha$ as $\alpha(\tau,\bf x).$ As in [5], [7] we
define (quasi)particles as elementary excitations of translation-invariant
stationary state.

Applying the general definition of geometric approach we can say that an
elementary excitation of stationary translation- invariant state $\omega$ is a
quadratic or Hermitian map $\sigma$ of the ”elementary space” $\textgoth{h}$
into the cone of states $\cal C$. This map should commute with spatial and
time translations. In addition one should fix a map of $\textgoth{h}$ into the
space $End(\cal L)$ of endomorphisms of $\cal L$ such that
$\sigma(f)=L(f)\omega$ [7]. In the framework of Jordan algebras $\omega$ is an
element of the positive cone ${\cal B}_{+}$ or of the dual cone ${\cal
B}_{+}^{\vee}.$ For definiteness we assume that $\omega$ is an element of the
dual cone; then $\cal L$ should be identified with ${\cal B}^{\vee}.$ (Recall
that that the elementary space $\textgoth{h}$ is defined as a pre Hilbert
space of smooth fast decreasing functions depending on ${\bf
x}\in\mathbb{R}^{d}$ and discrete variable $i\in\cal I.$ The spatial
translations act as shifts by ${\bf a}\in\mathbb{R}^{d}$ , the time
translations commute with spatial translations. We can consider real-valued or
complex-valued functions; we should consider quadratic maps in the first case
and Hermitian maps in the second case.)

Let us start with linear map $\rho:\textgoth{h}\to\cal B$ commuting with
translations. Let us fix translation-invariant stationary state
$\omega\in{\cal B}_{+}^{\vee}$ obeying additional condition
$T^{t}\omega=\omega$ for all $T\in\cal T$. ( Here $T\to T^{t}$ stands for the
involution in structure group entering (2).) Then the map $L:\textgoth{h}\to
End({\cal B}^{\vee})$ transforming $x\in\textgoth{h}$ into $Q_{\rho(x)}$
specifies an elementary excitation of $\omega$ as the map $\sigma:x\to
L(x)\omega$. To verify this statement we notice that the map $\sigma$ is
quadratic or Hermitian because $L$ is quadratic (if the $\cal B$ is a Jordan
algebra over $\mathbb{R}$) or Hermitian (if $\cal B$ is a complex Jordan
algebra with involution), the formula (2) implies that $\sigma$ commutes with
translations. In what follows we consider mostly elementary excitations
constructed this way. Notice, however, that we can start with arbitrary linear
map $\rho:\textgoth{h}\to\cal B$ and impose a weaker condition that the map
$\sigma:x\to L(x)\omega$ commutes with translations. Then $\sigma$ can be
regarded as elementary excitation. (Here again $L$ transforms
$x\in\textgoth{h}$ into $Q_{\rho(x)}$.)

Let us consider as an example a Jordan algebra defined as a set of of self-
adjoint elements of Weyl algebra. We define Weyl algebra corresponding to real
pre Hilbert space $\textgoth{h}$ as a unital $*$\- algebra generated by
elements $a(f),a^{+}(g)$ depending linearly of $f,g\in\textgoth{h}$ and
obeying canonical commutation relations

$[a(f),a(g)]=[a^{+}(f),a^{+}(g)]=0,[a(f),a^{+}(g)]=\langle f,g\rangle.$

We define a map $\rho$ of $\textgoth{h}$ into this Jordan algebra as a map
sending $f\in\textgoth{h}$ to $a(f)+a^{+}(f).$

Let us assume that $\textgoth{h}$ is an elementary space. Then the
translations in $\textgoth{h}$ induce translations in Weyl algebra and in the
corresponding Jordan algebra. The map $\rho$ commutes with translations.
Taking as $\omega$ the state corresponding to the Fock vacuum in positive cone
or a dual cone we obtain an example of elementary excitation of this state.

The same construction works for Clifford algebra specified by canonical
anticommutation relations.

Notice that the Jordan algebra corresponding to Clifford algebra can be
regarded as $JB$-algebra, but starting with Weyl algebra we obtain a
topological Jordan algebra, more precisely a Fréchet Jordan algebra. (Fréchet
vector space is a complete topological vector space where the topology is
specified by a countable family of seminorms. In what follows we are talking
about $JB$-algebras, but our results can be generalized to Fréchet analogs of
these algebras.)

Let us discuss the relation of the above constructions to the construction of
elementary excitations in the approach based on consideration of associative
algebra $\cal A$ with involution ($*$-algebra). The set of self-adjoint
elements of such an algebra can be regarded as Jordan algebra $\cal B$ over
$\mathbb{R}$; the complexification $\mathbb{C}\cal B$ of this Jordan algebra
can be identified with $\cal A$ considered as complex Jordan algebra with
involution. The elements of dual cones of these Jordan algebras can be
identified with not necessarily normalized states of $\cal A.$ Let us assume
that spatial and time translations act on $\cal A$ as automorphisms; this
action generates an action of translations on the cones of Jordan algebras.
Let us fix a translation invariant stationary state $\omega$ of algebra $\cal
A$; corresponding elements of dual cones of Jordan algebras are denoted by the
same symbol. Excitations of $\omega$ can be regarded as elements of pre
Hilbert space $\cal H$ obtained by means of GNS construction applied to
$\omega,$ an elementary excitation is an isometric embedding $\Phi(f)$ of
elementary space $\textgoth{h}$ into $\cal H.$ Following [7] we represent
$\Phi(f)$ in the form $B(f)\theta$ where $B$ is a linear map
$\textgoth{h}\to\cal A$ and $\theta\in\cal H$ is a vector corresponding to the
state $\omega$. If elements $B(f)$ are self-adjoint we can apply the above
construction of elementary excitation of $\omega$ considered as an element of
a dual cone of the Jordan algebra $\cal B$ of self-adjoint elements of $\cal
A$ taking $\rho=B$. Then the state $Q_{\rho(f)}\omega$ corresponds to the
vector $B(f)\theta.$ If $B^{*}(f)=0$ a similar statement can be proved for
$\mathbb{C}\cal B$ ( for Jordan algebra with involution obtained by
complexification of $\cal B$).

##  4 Scattering

To analyze scattering in the framework of Jordan algebras we are starting with
linear map $\rho:\textgoth{h}\to\cal B$ (not necessarily commuting with
translations). We fix translation-invariant stationary state $\omega\in{\cal
B}_{+}^{\vee}$. The map $L:\textgoth{h}\to End({\cal B}^{\vee})$ transforming
$x\in\textgoth{h}$ into $Q_{\rho(x)}$ specifies an elementary excitation of
$\omega$ if the map $\sigma:x\to L(x)\omega$ commutes with translations.

We define ( following the general theory of [7]) the operator

$L(f,\tau)=T_{\tau}(L(T_{-\tau}f))=T_{\tau}L(T_{-\tau}f)T_{-\tau}.$

where $f\in\textgoth{h}.$ This operator is quadratic or Hermitian with respect
to $f$, therefore we can consider also the operator $L(\tilde{f},f,\tau)$ that
is linear with respect to $\tilde{f}$ and linear or antilinear with respect to
$f$; it coincides with $L(f,\tau)$ for $\tilde{f}=f^{*}.$ ( Notice that in the
case of real vector spaces $f^{*}=f.$) Using the notation
$Q_{\tilde{a},a}x=\\{\tilde{a},x,a^{*}\\}$ we can write

$L(\tilde{g},g,\tau)=T_{\tau}Q_{T_{-\tau}\rho(\tilde{g}),T_{-\tau}\rho(g)}T_{-\tau}=L(\tilde{g},g)T^{t}_{-\tau}$

where $L(\tilde{g},g)=Q_{\rho(\tilde{g}),\rho(g)}$ is a bilinear (or
sesquilinear) form corresponding to the quadratic (or Hermitian) form
$L(g)=Q_{\rho(g)}.$

The state

$\Lambda(f_{1},\cdots,f_{n}|-\infty)=\lim_{\tau_{1}\to-\infty,\cdots,\tau_{n}\to-\infty}L(f_{1},\tau_{1}),...L(f_{n},\tau_{n})\omega$
(3)

describes the collision of (quasi)particles with wave functions
$f_{1},...,f_{n}.$ We say that (3) is a scattering state (or, more precisely,
an $in$-state).

The result of the collision can be characterized by the number

$\lim_{\tau^{\prime}\to+\infty,\tau\to-\infty}\langle\alpha|L(g_{1},\tau^{\prime})...L(g_{m},\tau^{\prime})L(f_{1},\tau)...L(f_{n},\tau)|\omega\rangle$
(4)

where $\alpha$ is a stationary translation-invariant point of the cone ${\cal
B}_{+}$ or of the larger cone ${\cal B}_{+}^{\vee\vee}$ (we use bra-ket
notations). Comparing with the formulas of [7] we see that this number can be
interpreted as a generalization of inclusive scattering matrix. More generally
we can consider a functional

$\sigma(\tilde{g}^{\prime}_{1},g^{\prime}_{1},...,\tilde{g}^{\prime}_{n^{\prime}},g^{\prime}_{n^{\prime}},\tilde{g}_{1},g_{1},...,\tilde{g}_{n},g_{n})=$
$\langle\alpha|\lim_{\tau^{\prime}_{i}\to+\infty,\tau_{j}\to-\infty}L(\tilde{g}^{\prime}_{1},g^{\prime}_{1},\tau^{\prime}_{1})...L(\tilde{g}^{\prime}_{n^{\prime}},g^{\prime}_{n^{\prime}},\tau^{\prime}_{n^{\prime}})L(\tilde{g}_{1},g_{1},\tau_{1})...L(\tilde{g}_{n},g_{n},\tau_{n})|\omega\rangle$
(5)

that is linear or antilinear with respect to all of its arguments. It also can
be regarded as inclusive scattering matrix.

It was proven in [7] that the limit (3) exists for $f_{1},\cdots,f_{n}$ in a
dense open subset of $\textgoth{h}\times\cdots\times\textgoth{h}$ if

$||[T_{\alpha}(L(\phi)),L(\psi)]||\leq\int d{\bf x}d{\bf x}^{\prime}D^{ab}(\bf
x-\bf x^{\prime})|\phi_{\it a}(\bf x)|\cdot|\psi_{\it b}(\bf x^{\prime})|$ (6)

where $D^{ab}(\bf x)$ tends to zero faster than any power as $\bf x\to\infty$
and $\alpha$ runs over a finite interval.

Less formally we can formulate this condition as the requirement that the
commutator $[T_{\alpha}(L(\phi)),L(\psi)]$ is small if the essential supports
of functions $\phi$ and $\psi$ in coordinate representation are far away.

Similar conditions can be formulated for the existence of limits (4), (5) (for
the existence of inclusive scattering matrix). Later we will formulate more
concrete conditions for the existence of the above limits.

Notice that the linear map $\rho:\textgoth{h}\to\cal B$ can be regarded as a
multicomponent generalized function $\rho(\bf x)$ in coordinate representation
or $\rho(\bf k)$ in momentum representation. (This means that we formally
represent $\rho(\phi)$ as $\int d{\bf x}\phi({\bf x})\rho(\bf x)$ or as $\int
d{\bf k}\phi({\bf k})\rho(\bf k).$ Discrete indices are omitted in these
formulas and in what follows.) We assume that the generalized functions
$\rho(\bf x),\rho(\bf k)$ correspond to continuous functions denoted by the
same symbols.

The expression (5) is linear with respect to its arguments, therefore it can
be regarded as a generalized function

$\sigma(\tilde{\bf x}^{\prime}_{1},{\bf x}^{\prime}_{1},...,\tilde{\bf
x}^{\prime}_{n^{\prime}},{\bf x}^{\prime}_{n^{\prime}},\tilde{\bf x}_{1},{\bf
x}_{1},...,\tilde{\bf x}_{n},{\bf x}_{n})=$
$\lim_{\tau^{\prime}_{i}\to+\infty,\tau_{j}\to-\infty}\langle\alpha|L(\tilde{\bf
x}^{\prime}_{1},{\bf x}^{\prime}_{1},\tau^{\prime}_{1})...L(\tilde{\bf
x}^{\prime}_{n^{\prime}},{\bf
x}^{\prime}_{n^{\prime}},\tau^{\prime}_{n^{\prime}})L(\tilde{\bf x}_{1},{\bf
x}_{1},\tau_{1})...L(\tilde{\bf x}_{n},{\bf x}_{n},\tau_{n})|\omega\rangle$
(7)

or, in momentum representation,

$\sigma(\tilde{\bf k}^{\prime}_{1},{\bf k}^{\prime}_{1},...,\tilde{\bf
k}^{\prime}_{n^{\prime}},{\bf k}^{\prime}_{n^{\prime}},\tilde{\bf k}_{1},{\bf
k}_{1},...,\tilde{\bf k}_{n},{\bf k}_{n})=$
$\lim_{\tau^{\prime}_{i}\to+\infty,\tau_{j}\to-\infty}\langle\alpha|L(\tilde{\bf
k}^{\prime}_{1},{\bf k}^{\prime}_{1},\tau^{\prime}_{1})...L(\tilde{\bf
k}^{\prime}_{n^{\prime}},{\bf
k}^{\prime}_{n^{\prime}},\tau^{\prime}_{n^{\prime}})L(\tilde{\bf k}_{1},{\bf
k}_{1},\tau_{1})...L(\tilde{\bf k}_{n},{\bf k}_{n},\tau_{n})|\omega\rangle$
(8)

.

We use the notations

$L(\tilde{g},g,\tau)=\int d\tilde{\bf x}d{\bf x}\tilde{g}(\tilde{\bf x})g({\bf
x})L(\tilde{\bf x},{\bf x},\tau)=\int d\tilde{\bf k}d{\bf
k}\tilde{g}(\tilde{\bf k})g({\bf k})L(\tilde{\bf k},{\bf k},\tau)$

If $\rho$ commutes with translations we can say that

$T_{\tau}\rho(\phi)=\rho(T_{\tau}\phi)=\int d{\bf k}e^{-i\tau E({\bf
k})}\rho({\bf k})\phi(\bf k)$

hence

$L(\tilde{\bf k},{\bf k},\tau)=T_{\tau}(e^{-i\tau[E(\tilde{\bf k})+E({\bf
k})]}Q_{\rho(\tilde{\bf k}),\rho({\bf k})})T_{-\tau}=$ $e^{-i\tau[E(\tilde{\bf
k})+E({\bf k})]}T_{\tau}Q_{\rho(\tilde{\bf k}),\rho({\bf k})}T_{-\tau}$

##  5 Existence of inclusive scattering matrix

Let us consider first of all the case when the Jordan algebra $\cal B$ is
obtained as a set of self-adjoint elements of associative Banach algebra $\cal
A$ with respect to the involution ${}^{*}.$ We impose the condition of
asymptotic commutativity or anticommutativity on the multicomponent function
$\rho(\bf x)$:

$||[\rho({\bf x+a}),\rho({\bf a})]_{\mp}||<\frac{C_{n}}{1+||{\bf x}||^{n}}$
(9)

for every natural number $n.$ It follows from this condition that

$||[\rho(\phi),\rho(\psi)]_{\mp}||<\int d{\bf x}d{\bf x}^{\prime}D^{ab}(\bf
x-\bf x^{\prime})|\phi_{\it a}(\bf x)|\cdot|\psi_{\it b}(\bf x^{\prime})|$
(10)

where $D^{ab}(\bf x)$ tends to zero faster than any power as $\bf x\to\infty.$

The operators $L_{\phi}=Q_{\rho(\phi)}$ transform $b\in\cal B$ into
$\rho(\phi)b\rho(\phi).$ It is easy to check that these operators obey (6),
hence the limits we are interested in exist and we can consider the scattering
of particles. In the case of asymptotic commutativity (anticommutativity) we
are dealing with bosons (fermions).

For Jordan algebra $\cal B$ coming from associative algebra $\cal A$ with
involution it is easy to formulate sufficient conditions for commutativity of
operators $Q_{a}$ and $Q_{b}$. It is obvious that in the case when $a$ and $b$
commute in $\cal A$ or $a$ and $b$ anticommute in $\cal A$ (equivalently
$a\circ b=0$ in $\cal B$) we have $Q_{a}Q_{b}=Q_{b}Q_{a}.$ Similar statements
are correct in any $JB$-algebra $\cal B$:if operators $R_{a}$ and $R_{b}$
commute or $a\circ b=0$ then the operators $Q_{a},Q_{b}$ commute [27], [25],
[26]. (Here $a,b\in\cal B$, $R_{a}$ stands for the operator of multiplication
by $a$ in $\cal B$. If $R_{a}$ and $R_{b}$ commute one says that $a$ and $b$
operator commute.)

It is natural to conjecture that these statements can be generalized in the
following way:

If $a\circ b$ is small , then the operators $Q_{a}$ and $Q_{b}$ almost
commute.

If the operators $R_{a}$ and $R_{b}$ almost commute, then the operators
$Q_{a}$ and $Q_{b}$ almost commute.

The first of these conjectures is proven in [8]. More precisely,

If the norm of the Jordan product $a\circ b$ of two elements of JB-algebra
$\cal B$ is $\leq\epsilon$ then

$||[Q_{a},Q_{b}]||\leq k(||a||,||b||)\sqrt{\epsilon}.$ (11)

Here $\epsilon\geq 0$ and $k$ is a polynomial function

This statement allows us to give conditions for the existence of limits (3),
(4), (5).

We impose the condition

$||\rho({\bf x+a})\circ\rho({\bf a})||<\frac{C_{n}}{1+||{\bf x}||^{n}}$ (12)

for every natural number $n.$ Then

$||\rho(\phi)\circ\rho(\psi)||<\int d{\bf x}d{\bf x}^{\prime}D^{ab}(\bf x-\bf
x^{\prime})|\phi_{\it a}(\bf x)|\cdot|\psi_{\it b}(\bf x^{\prime})|$ (13)

where $D^{ab}(\bf x)$ tends to zero faster than any power as $\bf x\to\infty.$
Applying (11) we obtain (6) that implies the existence of limits (3), (4), (5)
for dense sets of families of functions in the arguments of these expressions.

It is not clear whether the second conjecture is true. However, the identity
$Q_{a}=2R_{a}^{2}-R_{a^{2}}$ immediately implies the following weaker
statement: if $a$ and $a^{2}$ almost operator commute with $b$ and $b^{2}$
then $Q_{a}$ and $Q_{b}$ almost commute:

$||[Q_{a},Q_{b}]||\leq
8||a||\cdot||b||\cdot||[R_{a},R_{b}]||+4||a||\cdot||[R_{a},R_{b^{2}}||+4||b||\cdot||[R_{b},R_{a^{2}}]||+||[R_{a^{2}},R_{b^{2}}]||.$

Using the identity $Q_{e^{a}}=e^{2R_{a}}$ one can conclude that operators
$Q_{e^{a}}$ and $Q_{e^{b}}$ almost commute if $a$ and $b$ almost operator
commute.

One can use these statements to give conditions for the existence of
scattering states and inclusive scattering matrix.

##  6 Green functions

Let us fix translation-invariant elements $\alpha\in{\cal B}_{+}$,
$\omega\in{\cal B}_{+}^{\vee}$, where ${\cal B}_{+}$ is a positive cone in
Jordan algebra $\cal B$ and ${\cal B}_{+}^{\vee}$ denotes the dual cone. The
quadratic operators $Q_{A}$ act in both cones, the bilinear operators
$Q_{\tilde{A},A}$ act in $\cal B$ and in the dual space $\cal B^{\vee}.$ (Here
$A,\tilde{A}$ are elements of $\cal B.$)

Let us fix elements $\tilde{A}_{1},A_{1},...,\tilde{A}_{n},A_{n}\in\cal B.$ We
introduce the notation $Q_{i}(\tilde{\bf x},\tilde{\tau},{\bf x},\tau)$ for
$Q_{\tilde{A}_{i}(\tilde{\bf x},\tilde{\tau}),A_{i}({\bf x},\tau)}.$

We define (generalized) Green functions by the formula

$G_{n}(\tilde{\bf x}_{1},\tilde{\tau}_{1},{\bf x}_{1},\tau_{1},...,\tilde{\bf
x}_{n},\tilde{\tau}_{n},{\bf x}_{n},\tau_{n})=$
$\langle\alpha|T(Q_{1}(\tilde{\bf x}_{1},\tilde{\tau}_{1},{\bf
x}_{1},\tau_{1})...Q_{n}(\tilde{\bf x}_{n},\tilde{\tau}_{n},{\bf
x}_{n},\tau_{n})|\omega\rangle$

where $T$ stands for the chronological ordering with respect to
$\tau^{\prime}_{i}=\frac{1}{2}(\tilde{\tau}_{i}+\tau_{i}).$

Omitting chronological ordering in this formula we obtain a definition of
correlation functions. Notice that in the case when Jordan algebra $\cal B$ is
constructed as a set of self-adjoint elements of $*$-algebra $\cal A$ we have

$Q(\tilde{\bf x},\tilde{\tau},{\bf x},\tau)a=\frac{1}{2}(A({\bf
x},\tau)a\tilde{A}(\tilde{\bf x},\tilde{\tau})+\tilde{A}(\tilde{\bf
x},\tilde{\tau})aA({\bf x},\tau)).$

Using this remark we can express correlations functions for Jordan algebra
$\cal B$ in terms of correlation functions for $*$\- algebra $\cal A.$

We defined GGreen functions in $({\bf x},\tau)$-representation; as always
taking Fourier transforms we obtain GGreen functions in $({\bf k},\tau)$\- and
$({\bf k},\varepsilon)$-representations.

One can show ( under some conditions) that the inclusive scattering matrix can
be calculated in terms of asymptotic behavior of GGreen functions in $({\bf
k},\tau)$-representation or in terms of poles and residues in $({\bf
k},\varepsilon)$-representation. The proof is similar to the proof of
analogous statement in [6]. It is based on formula (8).

##  7 Generalizations

Let us consider a $\mathbb{Z}_{2}$-graded algebra $\cal A.$ We denote by
${\cal A}_{\Lambda}$ the set of even elements of tensor product ${\cal
A}\otimes\Lambda$ where $\Lambda$ is a Grassmann algebra. This set can be
considered as an algebra; if for every $\Lambda$ the algebra ${\cal
A}_{\Lambda}$ is a Jordan algebra we say that $\cal A$ is a Jordan
superalgebra. (See, for example, [28] for more standard definition of Jordan
superalgebra and for main facts of the theory of Jordan superalgebras.)
Similarly, if the algebra ${\cal A}_{\Lambda}$ is a Lie algebra one says that
$\cal A$ is a Lie superalgebra. One can say that a Jordan superalgebra $\cal
A$ specifies a functor defined on the category of Grassmann algebras and
taking values in the category of Jordan algebras. Analogously, a Lie
superalgebra specifies a functor with values in Lie algebras and a supergroup
can be regarded as a functor with values in groups (all functors are defined
on Grassmann algebras.

Starting with a differential algebra (= $\mathbb{Z}_{2}$-graded algebra
equipped with an od derivation $d$ obeying $d^{2}=0$) we can define a
differential Jordan superalgebra. The simplest way to construct a differential
Jordan superalgebra is to take tensor product of Jordan algebra $\cal B$ by a
differential supercommutative algebra $\cal E$ (for example, one can take as
$\cal E$ a free Grassmann algebra with the differential that calculates
cohomology of Lie algebra).

Using these definitions one can generalize the statements above to Jordan
superalgebras and (in the framework of BRST-formalism) to differential Jordan
superalgebras.

These generalizations can be used to analyze interesting examples.

In particular, one can construct Poincaré invariant theories starting with
simple exceptional Jordan algebra ( Albert algebra
$\textgoth{h}_{3}(\mathbb{O})$). Namely, one should notice that the structure
group of this algebra contains a subgroup isomorphic to $SO(1,9)$ (this
subgroup was used in [9], [10]). Assuming that translations act trivially we
obtain an action of ten-dimensional Poincaré group on Albert algebra. Taking a
tensor product of Albert algebra and (diffferental) supecommutative algebra
where Poincaré group acts by automorphisms we obtain a (differential) Jordan
superalgebra with action of Poincaré group (and in general non-trivial action
of translations).

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