\section{Introduction}
Linear colliders, LC, can be operated in several modes among which the $e \gamma$
and $\gamma \gamma$ modes provide a wide field of physics opportunities,
including unique experimental solutions to fundamental problems \cite{r1,r2}.
By means of Compton back-scattering of laser light, almost the entire
energy of electrons/positrons at a linear collider can be transferred to photons
\cite{r3} so
that $e \gamma$ and $\gamma \gamma$ processes can be studied for energies
close to the TeV scale. The luminosities are
expected to be about one third of the $e^+e^-$ luminosity in the high
energy regime \cite{r1,r2}. Since the cross sections for $e \gamma$ and
$\gamma \gamma$ processes are in general significantly larger than the
cross sections for $e^+e^-$ annihilation processes, the event rates will
be of similar size in the three LC modes. Various options of choosing the
photon polarizations, circular and linear, allow unique experimental
analyses of particle properties and interactions. \\

A rich spectrum of interesting physics problems can be studied experimentally
at a photon collider operating at energies up to a TeV: \\ \vspace*{-3mm}

{{-- {\underline{Higgs physics:}}}} The formation of Higgs bosons
in $\gamma \gamma$ collisions can
be used to measure precisely the Higgs-$\gamma\gamma$ coupling \cite{r4,r5}.
Since photons do not couple directly to neutral Higgs bosons, the coupling is
mediated by virtual charged particle loops, being sensitive to scales
potentially far beyond the Higgs mass. In the supersymmetry sector,
$\gamma \gamma$ formation allows us to generate heavy scalar and pseudoscalar
Higgs bosons \cite{r6,r7} in a wedge centered around medium $\tan\beta$
values, in which no other collider, neither the LHC nor the LC $e^+e^-$ mode,
gives access to the spectrum of heavy Higgs bosons. Near
degeneracy of the scalar and pseudoscalar Higgs bosons will give rise to
large asymmetry effects in $\gamma$ polarization experiments \cite{r8,r9}
if CP is broken in the Higgs sector. \\ \vspace*{-3mm}

{{-- {\underline{Supersymmetry:}}}} A photon collider provides unique
opportunities also in the genuine supersymmetric particle sector. In $e
\gamma$ collisions the production of selectrons in association with
light neutralinos can give access to selectron masses in excess to pair
production in the LC $e^\pm e^-$ modes \cite{r10}. Similarly the production of
$e$-sneutrinos in association with charginos may be used to study the
properties of sneutrinos \cite{r11}.\\ \vspace*{-3mm}

{{-- {\underline{Static electromagnetic parameters:}}}} The large
production cross sections for charged particles in $\gamma \gamma$
collisions can be exploited to determine their static electromagnetic
parameters with high precision. The measurement of the electromagnetic
multi-pole moments can be performed in a pure electromagnetic environment
without interference with weak effects. Examples are the magnetic dipole
moment of the top quark \cite{rr11A}, and the magnetic dipole moment and electric
quadrupole moment of the charged $W^\pm$ bosons \cite{r12,r13}. \\ \vspace*{-3mm}

{{-- {\underline{QCD:}}}} Among the QCD problems which can be addressed
at a high energy photon collider, two problems are of particular interest.
The total $\gamma \gamma$ fusion cross section to hadrons is built up by a
mixture of non-perturbative and perturbative interactions including the
difficult transition zone between the two regimes.
Of tantamount importance is the analysis of the
quark and gluon content of the photon {\cite{r14}}-{\cite{r15B}}. These parton
distributions had been predicted to be very different from nucleons, with
characteristic properties governed by asymptotic freedom. \\ \vspace*{-3mm}

{{-- {\underline{Varia:}}}} Experiments at a photon collider open many
other windows for interesting search experiments, extending from heavy
Majorana neutrinos, see e.g. Ref.~\cite{r15C}, to excited electrons. The
key is the character of the photon as an almost pure energy quantum which
is very effective in exciting new degrees of freedom. As a result, new
particles can be
generated with masses very close to the total energy in the system,
most transparent for the electron excitation tower $\{e^\ast\}$
of compositeness models in $e \gamma$ collisions. \\ \vspace*{-3mm}

Some of these elements will be reviewed in the third to fifth section of this
report after the basic principle of photon colliders is briefly described
in the next section.
\section{The basic principle}
In $e^+e^-/e^-e^-$ linear colliders nearly the entire electron/positron
energy can be transferred to photons by Compton back-scattering of laser
light. This method has been proposed
in Refs.~\cite{r3}. The scheme is based on two elements. Kinematically, by
energy-momentum conservation, a low-energy laser photon must carry away
almost the entire energy when scattered backward in a collision with a
high-energy electron/positron. Dynamically, the cross section is maximal,
due to the u-channel exchange of electrons/positrons,
\begin{equation}
A_u \sim {1}/(u-m^2_e) \sim {1}/(1+\beta_e\cos\theta)
\end{equation}
for back-scattering of the photons.
\subsection{Photon Energy}
The spectrum is described in detail by the shape function
\begin{equation}
F(y) = \frac{1}{1-y} + (1-y) - 4r(1-r) + 2 \lambda_e P_c xr(1-2r)(2-y)
\end{equation}
where
\begin{equation}
y = E_\gamma/E_e
\end{equation}
denotes the fraction of energy transferred from the electron/positron to
the photon and $r$ abbreviates the ratio $r = y/(1-y)x$.
The helicities $\lambda_e$ and $P_c$ refer to the incoming electrons/positrons
and laser photons, respectively.
The parameter
\begin{equation}
x = 4 E_e \omega_\gamma / m^2_e
\end{equation}
measures the invariant energy [squared] in units of the electron mass, with
$\omega_\gamma$ denoting the laser $\gamma$ energy.
$x$ determines the upper limit of the final photon energy,
\begin{equation}
E^{max}_\gamma = \frac{1}{1+x^{-1}} E_e
\end{equation}
The larger the value for $x$ is chosen the more energy can be transferred
to the photon. However, to suppress electron-positron pair production in
collisions of the high-energy photons, already generated in the Compton process,
with the numerous
left-over laser photons, an upper bound on $x \leq 2/(\sqrt{2}-1) \simeq 4.8$
must be imposed. \\

For typical values $E_e =$ 250 GeV and $\omega_\gamma =$ 1.17 eV more than
80\% of the electron/positron energy can be transferred to the photon,
\begin{equation}
E^{max}_\gamma = 0.82 E_e
\end{equation}
generating a photon-photon invariant energy of more than 800 GeV at a
1~TeV $e^\pm e^-$ collider. \\

By choosing opposite helicities for the initial electron/positron beams and
the laser photons, i.e. $2\lambda_e P_c = -1$, the spectrum of the final-state
photons can be sharpened dramatically, cf. Fig.\ref{gamspectrum} (left). The
$\gamma$ conversion of
electron beams for which a polarization degree of 95\% may eventually be
achieved, is therefore preferable over positron beams where an upper
limit of about 60\% is expected.
\begin{figure}[h]
\begin{center}
\epsfig{figure=F.spec.eps,bbllx=63,bblly=45,bburx=381,bbury=375,width=5.1cm,clip=}
\epsfig{figure=F.specMC.eps,bbllx=3,bblly=6,bburx=238,bbury=245,width=5.4cm,clip=}
\caption{Photon spectra. Left: The effect of electron/positron and laser
photon polarization on the high-energy photon spectrum. Right: Experimental
simulation of the photon spectrum including non-linear effects; Ref.~\cite{rr17A}.}
\label{gamspectrum}
\end{center}
\end{figure}

Non-linear effects reduce the upper energy slightly, but give rise to
a large number of photons at the lower end of the spectrum. Higher harmonics,
on the other hand, lead to a shallow increase of the upper energy limit,
cf. Fig.\ref{gamspectrum} (right). The main characteristics of the high-energy
photon spectrum, however, remain largely unaltered.
\subsection{Polarization}
-- {{Circular polarization}} is transferred from the initial laser
photons completely to the high energy photons for maximum energy at the peak
of the spectrum, cf. Fig.\ref{circlinspectrum} (left):
\begin{equation}
\lambda_\gamma \to - P_c \;\; \mathrm{[near\,maximum]}
\end{equation}

-- {{Linear polarization}} responds less favorably, cf.
Fig.\ref{circlinspectrum} (right).
The degree of polarization transferred from the laser to the high energy
photons is reduced with rising photon energy, i.e. rising $x$:
\begin{equation}
l_\gamma \to + P_l/[x^2/2(x+1)+1] \;\; \mathrm{[near\,maximum]}
\end{equation}

The variety of polarization states of the high energy photons makes the
photon collider an ideal instrument for investigating the external
spin-parity quantum numbers of particles \cite{r15D} such as Higgs bosons,
and for the study of CP violation effects \cite{r9}.
\begin{figure}[h]
\begin{center}
\epsfig{figure=F.circpol.eps,bbllx=41,bblly=43,bburx=381,bbury=371,width=5.4cm,clip=}
\epsfig{figure=F.linpol.eps,bbllx=41,bblly=43,bburx=381,bbury=371,width=5.4cm,clip=}
\caption{The transfer of circular (left) and linear (right) laser $\gamma$ polarization
to the high-energy photons.}
\label{circlinspectrum}
\end{center}
\end{figure}
\subsection{Luminosities and Cross sections}
The essential characteristics of the $e \gamma$ and $\gamma \gamma$
luminosity distributions can be determined by convoluting the photon
spectrum with the electron spectrum and with itself, respectively. \\ \vspace*{-3mm}

-- The {{$e \gamma$ luminosity}} distribution coincides with the
high-energy photon energy spectrum. Denoting the scaled $e \gamma$ invariant
mass by $m_{e\gamma} = M_{e\gamma}/\sqrt{s}$, it is given by
\begin{equation}
{d\mathcal{L}}/{dm^2_{e\gamma}} = F(y=m^2_{e\gamma})
\end{equation}
\\

-- The {{$\gamma \gamma$ luminosity}} distribution, on the other
hand, is determined by the standard self-convolution of the photon
spectrum. Denoting the $\gamma \gamma$ invariant mass by $m_{\gamma\gamma}
= M_{\gamma \gamma}/\sqrt{s}$, the luminosity can be written as
\begin{equation}
{d\mathcal{L}}/{dm^2_{\gamma\gamma}} = \int_{m^2_{\gamma\gamma}}^{1}
               \frac{dy}{y} \,F(y) \,F(m^2_{\gamma\gamma} / y)
\end{equation}
If the helicities of the laser photons and the electrons are chosen
opposite to each other, a large fraction of the luminosity is accumulated
in the high-energy region, cf. Fig.\ref{luminosity}. For a polarization degree
of 85\%
of the initial electrons and maximal polarization of the laser photons,
the values of the $e \gamma$ and $\gamma \gamma$ luminosities are
displayed in Tab.1 in units of $10^{34}$cm$^{-2}$s$^{-1}$.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|cc|}
\hline
\rule{0mm}{4mm}$\sqrt{s}_{ee}$ [GeV]  & 500  & 800        \\[0.1cm]
\hline
\hline
\rule{0mm}{4mm}$m_{e\gamma} \geq 0.8 m_{e\gamma}^{max}$
& 0.9  & 1.3        \\[0.1cm]
\rule{0mm}{4mm}$m_{\gamma\gamma} \geq 0.8 m_{\gamma\gamma}^{max}$
& 1.1  & 1.7      \\[0.15cm]
\hline
\rule{0mm}{4mm}$e^+e^-$               & 3.4  & 5.8        \\
\hline
\end{tabular}
\caption{Luminosities for $e \gamma$ and $\gamma \gamma$ within a margin
of size 0.2 below the maximum invariant energies, compared with the LC
$e^+e^-$ luminosity; Ref.~\cite{r1}.}
\end{center}
\end{table}
About one third of the corresponding $e^+e^-$ luminosity is accumulated
within a margin of 20\% below the maximum invariant energy:
\begin{equation}
{\mathcal{L}}_{\gamma\gamma}(\geq 0.8) \sim \frac{1}{3} {\mathcal{L}}_{ee}
\end{equation}
Since the cross sections for the $e \gamma$ and $\gamma \gamma$ processes
are significantly larger than the $e^+e^-$ annihilation cross sections,
the event rates are predicted of similar size for all three types of
collisions. \\ \vspace*{-3mm}

\begin{figure}[h]
\begin{center}
\epsfig{figure=F.ggMC.eps,bbllx=3,bblly=3,bburx=250,bbury=233,width=5.85cm,clip=}
\caption{Luminosity distribution of the $\gamma \gamma$ collider; Ref.~\cite{r1}.}
\label{luminosity}
\end{center}
\end{figure}
A sample of cross sections for $\gamma \gamma$ processes is collected in
Fig.\ref{cxns}. Since the size of the cross sections is large, extending from
10 to $10^5$ fb, a large number of $10^3$ to $10^7$ events can be observed for
a total integrated luminosity of 100 fb$^{-1}$ in the high-energy margin of
the luminosity distribution.
\begin{figure}[h]
\begin{center}
\epsfig{figure=F.totcx.eps,bbllx=3,bblly=3,bburx=531,bbury=656,width=6.6cm,clip=}
\caption{Cross sections for $e \gamma$ and $\gamma \gamma$ processes as functions of the energy; compared with $e^+e^-$ annihilation channels.}
\label{cxns}
\end{center}
\end{figure}
\section{Electroweak Symmetry Breaking}
The discovery of the mechanism which breaks the electroweak symmetries and
the exploration of its nature are among the central experimental tasks at
the next-generation high-energy colliders. The Higgs mechanism is strongly
supported by high-precision analyses in the electroweak sector. However,
these measurements can still be interpreted within a large variety of models
incorporating different realizations of the Higgs mechanism. They extend from
the Standard Model to embeddings in supersymmetric theories up to theories of
extra space dimensions. A photon collider can contribute to the task of uncovering
the underlying structure. The Higgs coupling to two photons, measured by the
size of the Higgs formation cross section in photon collisions, is sensitive
to high scales in the theory. Moreover, photon collisions provide unique
opportunities for the discovery of the heavy Higgs bosons in supersymmetric
theories. Alternatively, strong electroweak symmetry breaking may be studied
by measuring the static electromagnetic properties of the $W^\pm$ bosons in
$\gamma \gamma$ fusion. These opportunities will be illustrated by a few
typical examples.
\subsection{Light Higgs in $\gamma \gamma$ Collisions}
The coupling of Higgs bosons to photons \cite{r16} is mediated by loops of
charged particles. In the Standard Model the main contributions are generated
by top quark and $W$ boson loops. In scenarios beyond the Standard
Model also heavier particles can contribute but the loops are suppressed with
increasing masses according to the rules of quantum mechanics. [The
suppression can be counter-balanced by rising Higgs couplings to the new
particles if their masses are generated by the Higgs mechanism; however, for
masses beyond $\sim 1/2$ TeV the theory would become strongly interacting and
the perturbative loop argument would cease to be valid.]  \\ \vspace*{-3mm}

The Higgs-$\gamma \gamma$ coupling determines the formation cross section
of Higgs bosons in $\gamma \gamma$ fusion. Apart from normalization, the
cross section can conveniently be expressed by the partial $\gamma \gamma$
decay width of the Higgs boson and the luminosity function; for narrow
Higgs bosons:
\begin{equation}
\sigma_{\gamma\gamma} \sim \Gamma_{\gamma\gamma} \,
                        {d\mathcal{L}}/{dm^2_{\gamma\gamma}} (M^2_H)
\end{equation}
Choosing the helicities of the photons opposite to each other enhances the
signal and helps suppress the background. The Higgs boson can be detected
either as a peak in the invariant mass distribution, e.g. in $b\bar{b}$ or
$ZZ$ decays, or by scanning, taking advantage of the sharp upper edge of the
$\gamma \gamma$ spectrum. $b$ decays are leading below a Higgs mass of
140 GeV; $Z$ decays, for which the background is strongly suppressed compared
to $W$ decays, can be used for heavy Higgs masses, cf. Fig.\ref{widthbbzz}.
In the lower mass region the partial width can be measured with an accuracy
of 2.1\% \cite{r4}. \\ \vspace*{-3mm}
\begin{figure}[h]
\begin{center}
\epsfig{figure=F.widthbb.eps,bbllx=1,bblly=12,bburx=556,bbury=612,width=5cm,clip=}
\epsfig{figure=F.widthZZ.eps,bbllx=1,bblly=-17,bburx=554,bbury=500,width=5cm,clip=}
\caption{Final state invariant $bb$, $ZZ$ masses for light and heavy Higgs boson
production in $\gamma \gamma$ collisions; Refs.~\cite{r4,r5}, respectively.}
\label{widthbbzz}
\end{center}
\end{figure}

If the $H \to \gamma \gamma$ decay branching ratio is measured in
Higgs-strahlung, the ratio $\tau_H = BR_{\gamma\gamma}/\Gamma_{\gamma\gamma}$
determines the lifetime $\tau_H$ of the Higgs boson. Depending on the error
of the branching ratio \cite{rr20A}, an accuracy between 15 and 5\% may be reached,
matching eventually the $WW$ channel.
\subsection{Beyond the Standard Model (I)}
The high precision in the measurement of the Higgs-$\gamma \gamma$ coupling
may be exploited to determine or, at least, constrain high scales in theories beyond
the Standard Model in which the Higgs mechanism may be embedded. \\ \vspace*{-3mm}

-- In {2-Higgs doublet models} a situation could arise in which all the
properties of the lightest Higgs boson observed at LHC and ILC may be in
concordance with SM expectations but the heavy Higgs bosons may not have been
observed yet. In this situation the measurement of the $\gamma \gamma$ width
can discriminate the extended Higgs model from the Standard Model \cite{r19}.
Deviations of the $\gamma\gamma$ width from the Standard Model by more than a
factor two could still be possible and they could easily be detected at a
photon collider. \\ \vspace*{-3mm}

-- In models including {{triplet Higgs}} fields, doubly charged Higgs
bosons will
be generated in addition to the singly charged Higgs bosons of 2-doublet
Higgs models. The double electric charge increases the production cross section
by a factor 16 over the cross section for singly charged Higgs bosons. \\ \vspace*{-3mm}

-- In {{Little Higgs models}} deviations from the Standard Model
are predicted across the interesting range of the scale parameter $f$
\cite{r20}. Moreover, the breaking of anomalous U(1) gauge symmetries may
generate new axion-type scalar particles \cite{r21} which can be searched
for, as narrow states, in channels parallel to the Higgs bosons, cf.
Fig.\ref{littlehiggs}.  \\ \vspace*{-3mm}
\begin{figure}[h]
\begin{center}
\epsfig{figure=F.etaLH.eps,bbllx=154,bblly=400,bburx=568,bbury=752,width=6.3cm,clip=}
\caption{The formation of an axion-type particle $\eta$ of Little Higgs models in
addition to the Higgs particle in $\gamma \gamma$ collisions; Ref.~\cite{r21}.}
\label{littlehiggs}
\end{center}
\end{figure}

-- Potentially large modifications of the Higgs-$\gamma \gamma$ coupling
are expected quite generally in models with {{strong electroweak
symmetry breaking}}
\cite{r22}. If the photons couple directly to constituents of the Higgs
boson, the $\gamma \gamma$ branching ratio can rise to values significantly
above the Standard Model value and may even become dominant, depending on the
size of the confinement parameters of the Higgs constituents. \\ \vspace*{-3mm}

-- In theories of {{extra space dimensions}}, the Kaluza-Klein states
may alter the Higgs-$\gamma \gamma$ coupling \cite{r23}. In addition, in
specific models, like Randall-Sundrum type (RS) models, the radion field may mix
with the Higgs field and the properties of the Higgs particle may be modified
significantly; for details see e.g. Ref.~\cite{r24}.
\subsection{Beyond the Standard Model (II): Supersymmetry Higgs Sector}
The contribution of charged supersymmetric particles to the loops in the
Higgs-$\gamma \gamma$ coupling can give rise to noticeable effects,
cf. Ref.~\cite{r25}, if the particles are not too heavy. \\ \vspace*{-3mm}

{{-- {{Heavy Higgs Bosons:}}}} If supersymmetry is realized in
nature, a photon collider could be a unique instrument for exploring the
Higgs sector of the theory. In the wedge centered around medium values of
$\tan\beta$ beyond masses of about 200 GeV heavy Higgs bosons cannot be
discovered at LHC and beyond 300 GeV even not at SLHC. Nor can the Higgs
particles be discovered at the $e^+e^-$ linear collider beyond 250 GeV and
500 GeV in the first phase and the second phase, respectively, as heavy
scalar$+$pseudoscalar particles are produced in pairs. Thus only the lightest
Higgs particle would be detected in standard channels while the other members
of the Higgs spectrum would remain undiscovered. This wedge however can be
covered in the $\gamma \gamma$ mode of the linear collider,
\begin{equation}
\gamma\gamma \to H,A
\end{equation}
which extends the mass reach in formation experiments to about 80\% of the
total $e^+e^-$ energy \cite{r6}, i.e. up to 400 GeV and 800 GeV in the first
and second phase of the operation. A clear Higgs signal can be isolated above
the background \cite{r7} as demonstrated in Fig.\ref{heavyhiggs}. \\ \vspace*{-3mm}
\begin{figure}[h]
\begin{center}
\epsfig{figure=F.gagahiggs.eps,bbllx=2,bblly=3,bburx=237,bbury=250,width=5.1cm,clip=}
\caption{Experimental simulation of the production of heavy Higgs bosons
at a photon collider in supersymmetric theories \cite{r7}.}
\label{heavyhiggs}
\end{center}
\end{figure}

Besides the discovery of heavy Higgs bosons, a photon collider could be
very valuable if not even unique in determining some of the parameters in
the Higgs sector of supersymmetric theories. \\ \vspace*{-3mm}

{{-- {{Higgs Mixing $\tan\beta$:}}}}
The measurement of the
mixing parameter $\tan\beta$ is a difficult task for large values. Noticing
that the Higgs Yukawa couplings to $\tau$ pairs are of the order of
$\tan\beta$, the splitting of high-energy photons to $\tau$'s can be
exploited to measure this parameter in $\tau\tau$ fusion, cf. Ref.~\cite{r26}:
\begin{equation}
\sigma[\gamma\gamma \to \tau\tau + h/H/A] \;\; \sim \;\; \tan^2\beta
\end{equation}
For moderate $A,H$ masses, the $h\tau\tau$ coupling is of the order of $\tan\beta$,
while for large $A,H$ masses the size of the $A\tau\tau$ and $H\tau\tau$ couplings
is determined by this parameter;
thus the entire mass parameter range can be covered. Since the splitting
function of photons to leptons is hard, the energy of the $\tau$ beams is
high so that Higgs particles with large masses can be produced. Introducing
proper cuts to suppress the backgrounds, an accuracy of
\begin{equation}
\Delta\tan\beta = 0.9 \; {\rm{to}} \; 1.3
\end{equation}
can be expected across the entire medium to large $\tan\beta$ range \cite{r26}. \\ \vspace*{-3mm}

{{-- {{CP Violation:}}}} The 2-doublet Higgs sector of the
minimal supersymmetric extension of the Standard Model is automatically CP
conserving at the tree level. However, CP violation \cite{r9,r27} can be
induced by radiative corrections transmitting CP violating phases from the
soft SUSY breaking Lagrangian to the Higgs system, in particular the relative
phases between the higgsino mass parameter $\mu$ and the trilinear
sfermion-Higgs parameter $A_f$. \\ \vspace*{-3mm}

CP asymmetries are naturally enhanced \cite{r9} in the decoupling regime
where $M_A \simeq M_H$. The near mass degeneracy of the scalar and
pseudoscalar states allows for frequent mutual transitions which induce large
CP-odd mixing effects in CP violating theories. This is quantitatively
described by the complex mixing parameter
\begin{equation}
\frac{1}{2} \tan 2\theta = \frac{\Delta^2_{HA}}
         {M^2_H - M^2_A - i [M_H\Gamma_H - M_A\Gamma_A]}
\end{equation}
where the off-diagonal CP violating parameter $\Delta_{HA}$ in the Higgs mass
matrix couples the two states. \\ \vspace*{-3mm}

The asymmetry between left- and right-handedly polarized top quark pairs,
produced in $\gamma\gamma$ fusion on the top and in the Breit-Wigner wings
of the Higgs bosons, signals these CP violating effects \cite{r8}. \\ \vspace*{-3mm}

However, CP violation can be studied in a classical way by measuring
asymmetries of inclusive cross sections between left- and right-polarized
photons \cite{r9}:
\begin{equation}
A_{LR} = \frac{\sigma_{++} - \sigma_{--}} {\sigma_{++} + \sigma_{--}}
\end{equation}
Remarkably large asymmetries are predicted on top of the Higgs bosons in
$\gamma \gamma$ fusion, shown in Fig.\ref{lrasymm} as a function of the CP
violating phase of the trilinear stop-Higgs coupling $A_t$.
\begin{figure}[h]
\begin{center}
\epsfig{figure=F.gagaLR.eps,bbllx=5,bblly=3,bburx=237,bbury=227,width=5.8cm,clip=}
\caption{LR asymmetries predicted in CP non-invariant supersymmetric theories
on top of the two heavy Higgs bosons in $\gamma \gamma$ collisions; Ref.~\cite{r9}.}
\label{lrasymm}
\end{center}
\end{figure}
\subsection{Strongly Interacting $W$ Bosons}
If electroweak symmetry breaking is generated dynamically by new strong
interactions at a scale not far above 1~TeV, the longitudinal degrees of
the $W$ bosons, which are equivalent to the Goldstone bosons associated with
spontaneous symmetry breaking, become strongly interacting particles.
As a result, the properties of the $W$ bosons will be affected by the
nearby new strong interactions. In particular deviations are expected in such a
scenario for the static electroweak parameters from the values which are generated
by the electroweak gauge interactions of the Standard Model:
\begin{eqnarray*}
 {\rm{magnetic}}\, W^\pm \,{\rm{dipole\, moment}} &:& \mu_W = 2 \times e/2M_W\\
 {\rm{electric}}\, W^\pm \,{\rm{quadrupole\, moment}} &:& Q_W = -e/M^2_W
\end{eqnarray*}
The deviations are described by linear combinations of parameters $\Delta
\kappa_\gamma$ and $\lambda_\gamma$ which modify the $WW\gamma$ vertex. \\ \vspace*{-3mm}

This modification can be studied \cite{r13} in the process
\begin{equation}
   \gamma \gamma \to W^+ W^-
\end{equation}
which receives large contributions from the t- and u-channel $W$ exchanges.
For a total energy of 800 GeV and an integrated luminosity of 1 ab$^{-1}$,
the expected sensitivity for the parameters $\Delta\kappa_\gamma$ and
$\lambda_\gamma$ is shown in Table 2, compared with the corresponding
sensitivities in the $e^+e^-$ mode.
\begin{table}
\begin{center}
\begin{tabular}{|r|cc|c|}
\hline
\rule{0mm}{4mm}& $\gamma\gamma$ : $J_z = 2$      &  $\gamma\gamma$ : $J_z = 0$  &
$e^+e^-$ : $J_z = 1$  \\[0.1cm]
\hline
\hline
\rule{0mm}{4mm}$\Delta \kappa_\gamma / 10^{-4}$  &   5.2  & 13.9 & 2.1   \\
\rule{0mm}{4mm}$\lambda_\gamma / 10^{-4}$        &   1.7  &  2.5 & 3.3
\\[0.1cm]
\hline
\end{tabular}
\caption{Sensitivity of $WW$ pair production at $\gamma \gamma$ and $e^+e^-$
colliders to the $WW\gamma$ vertex parameters in theories of strong electroweak
symmetry breaking; Ref.~\cite{r13}.}
\end{center}
\end{table}
Apparently, the sensitivity on $\lambda_\gamma$ at the $\gamma \gamma$ collider
is superior to the $e^+e^-$ collider, though not dramatically.
\section{Supersymmetric Particles}
The genuine supersymmetric particle sector is the second domain
of a photon collider for potentially
unique discoveries of particles which cannot be observed at
other colliders. In supersymmetric scenarios in which sfermions are heavy but
charginos/neutralinos light, cascade decays, the prime source of non-colored
particles at LHC, do not include sleptons at a significant rate. On the other
hand, the sleptons may be too heavy to be produced in pairs in the $e^+e^-$
mode of the linear collider. In this situation an $e \gamma$ collider could
open the window to selectrons and $e$-sneutrinos \cite{r10,r11}.
\begin{figure}[h]
\begin{center}
\epsfig{figure=comp-1000.eps,bbllx=158,bblly=526,bburx=387,bbury=703,width=6.7cm,clip=}
\caption{Total cross sections for the associated production of first generation
sleptons with charginos and neutralinos in $e^-\gamma$ collisions as functions
of the slepton masses; cf.~Ref.~\cite{r10}.}
\label{comp1000}
\end{center}
\end{figure}

While the energy in  the $e^+e^-$ mode must at least be larger than twice
the mass of the lightest selectron,
\begin{equation}
e^+ e^- \to \tilde{e}^+_i \tilde{e}^-_j  \;\;\; : \;\;\;
{\rm{min}}[m_{\tilde{e}}] \leq \frac{1}{2} \sqrt{s}
\end{equation}
the associated production of selectron and neutralino, or sneutrino and
chargino, in the $e \gamma$ mode,
\begin{equation}
e \gamma \to \tilde{e}_i \tilde{\chi}^0_j \;\;\ \mathrm{or} \;\;\
\tilde{\nu}_{e_i} \tilde{\chi}^\pm_j
\;\;\; : \;\;\;
m_{\tilde{\ell}} \leq \sqrt{s} - m_{\tilde{\chi}}
\end{equation}
can give access to heavier selectrons and sneutrinos if the neutralinos
and charginos are light, Fig.\ref{comp1000}. \\ \vspace*{-3mm}

Due to t-channel slepton exchanges the processes set in sharply
at the thresholds proportional
to the velocities $\beta$. The fast rise of the cross section
can be exploited in threshold scans to
determine the masses quite accurately. In the SPS1a$'$ example shown in
Fig.\ref{snumass}, the sneutrino mass can be measured with an
accuracy of 3~GeV, cf. Ref.~\cite{r11}. The
accuracy is less than expected from chargino decays; however, all these
experiments are quite involved due to numerous supersymmetry backgrounds
to the supersymmetry signal so that cross checks are indispensable.
\begin{figure}[h]
\begin{center}
\epsfig{figure=F.snu.eps,bbllx=25,bblly=380,bburx=490,bbury=681,width=6.6cm,clip=}
\caption{Sneutrino mass measurement by scanning the threshold in the process
$e \gamma \to \tilde{\nu} \tilde{\chi}$; Ref.~\cite{r11}.}
\label{snumass}
\end{center}
\end{figure}
\section{Strong interactions and QCD}
\subsection{$\gamma \gamma$ Fusion to Hadrons}
One of the most demanding areas for the understanding of QCD as the
fundamental theory of the strong interactions is hadron production in
$\gamma \gamma$ collisions. Perturbative mechanisms are superimposed on
non-perturbative mechanisms with a difficult transition zone in between.
Since photons can fluctuate to vector mesons, the standard pomeron exchange
will contribute to the process. On the other hand, fluctuations to
quark-antiquark pairs at short distances will give rise to perturbative
QCD components, e.g. perturbative hard pomeron exchange and production of
mini-jets in $\gamma \gamma$ collisions. While for energies in the former
LEP range, the relative weight of the mechanisms could not be determined
properly, the increase of the collision energy up to about 1~TeV
at a photon collider is
expected to discriminate between the various contributions. The theoretical
picture is still evolving so that definite conclusions cannot be drawn yet at
the present time \cite{r32}.
\subsection{Quark-Gluon Structure of the Photon}
The measurement of the photon structure functions \cite{r33} at an $e
\gamma$ collider is of great theoretical interest. In contrast to the
proton structure function, the main characteristics of the photon structure
functions can be predicted theoretically. They are derived from the pointlike
splitting of photons to quark-antiquark pairs \cite{r14} which gives rise to the
increase of the photon structure function $F_2$ for rising Bjorken-$x$ and
to its uniform increase in the logarithm of the momentum transfer $\log Q^2$,
both features in sharp contrast to the proton. The QCD leading order
corrections \cite{r15A} modify the structure function in a characteristic
way. The perturbative radiation of gluons alters the $x$ dependence at
$\mathcal{O}(1)$, though not overturning the characteristic rise in $x$, but it
leaves the uniform $\log Q^2$ rise unchanged. This is a direct consequence
of asymptotic freedom \cite{r31A}; any other than the logarithmic $Q^2$ fall-off
of the QCD coupling would have led to a power dependence of the structure
function. The non-logarithmic terms of the structure function receive
contributions from next-to-leading order of QCD \cite{r15B} but
remnants from the vector-meson like component of the photon preclude
the perturbative prediction of the absolute normalization for
non-asymptotic $Q^2$ values. Nevertheless, the theoretical prediction of the
exceptional $x$ dependence and the uniform $\log Q^2$ rise render deep-inelastic
electron-photon scattering an exciting experimental task at an $e \gamma$
collider. \\ \vspace*{-3mm}

Neutral and charged-current mechanisms can be used to determine the
quark-parton content of the photon over a wide range of $Q^2$:
\begin{eqnarray}
e \gamma \to e X   \;\;&:&\;\; F_2^n \sim [4(u+c) +(d+s)] \nonumber\\
e \gamma \to \nu X \;\;&:&\;\; F_{2,3}^c \sim [(u+c) \pm (d+s)]
\end{eqnarray}
Range and accuracy with which $F_2$ can be measured are shown in
Fig.\ref{f2measur}, Ref.~\cite{r35}.
The analysis of charged-current deep-inelastic scattering allows the
separation of up- and down-quark components of the photon \cite{r34}.
\\ \vspace*{-3mm}
\begin{figure}[h]
\begin{center}
\epsfig{figure=F.F2_gam.ps,bbllx=58,bblly=82,bburx=530,bbury=761,width=5.7cm,clip=,angle=-90}
\caption{Accuracy with which the photon structure function $F_2$ can be
measured over a wide area in the $(x,Q^2)$ plane; Ref.~\cite{r35}.}
\label{f2measur}
\end{center}
\end{figure}

The vector-meson component of the photon as well as perturbative
gluon radiation suggest a gluon component within the photon. The
$(x,Q^2)$ dependence of the gluon
component can be determined in two ways. First, the $\log Q^2$ evolution of the
photon structure function is affected by the splitting of gluons to
quark-antiquark pairs \cite{r35A}. Second, part of the high-transverse momentum
jets in $e \gamma$ scattering,
\begin{equation}
e \gamma \to e + j + X
\end{equation}
are generated at the microscopic level by the subprocess
\begin{equation}
\gamma^\ast + g \to q + \bar{q}
\end{equation}
Studying the rapidity and transverse-momentum distributions of the jets
$j$ can thus be used to measure the gluon distribution of the photon
after the quark and anti-quark initiated Compton jets are subtracted \cite{r36}.
\\ \vspace*{-3mm}

Finally, the spin structure function of the photon, $g_1(x,Q^2)$, can be
measured in deep-inelastic scattering of longitudinally polarized electrons
on circularly polarized photons. As shown earlier, circular
polarization can be realized to a high degree for the photon beams.
In next-to-leading order
the structure function $g_1$ can be predicted in the asymptotic regime,
the result matched closely by predictions based on standard evolution
procedures \cite{r37}.
\section{Summary}
A photon collider provides us with an experimental instrument with which
a large variety of problems across the entire range of physics beyond the Standard
Model can be addressed. Moreover, within the Higgs sector and
the slepton sector of supersymmetric theories such a collider may give
unique access to heavy Higgs particles and heavy selectrons and
$e$-sneutrinos. Thus, by not only offering a platform for studying a rich
experimental bouquet
of interesting problems but also providing unique physics opportunities, a photon
collider can be considered a valuable component of a future linear
collider program.


\section{Introduction}
\label{sec:intr}

It has been shown in this conference that fundamental
questions related to the nature of the AGN are still open.
The accretion of material onto super--massive black
holes and the triggering of relativistic
jets (including their formation, acceleration and further
collimation) are some of the processes that still lack
a detailed understanding.
Observing with the highest angular resolution instruments
offers a good opportunity to learn more about these
processes through the study of the time evolution of
the jets.
An important effort has been made during the last decades
to bring the technique of millimeter Very Long Baseline
Interferometry (mm--VLBI) to progressively higher
sensitivities and shorter wavelengths, offering a
powerful tool to observe the innermost regions of
the jets and study the physics involved in their
behaviour.

During the last years, 7\,mm--VLBI observations, with angular
resolutions of up to $\sim 0.15$ milliarcseconds (mas), have addressed
the triggering of relativistic jets in AGN and their hydrodynamics.
Some particularly important results from these kind of observations
are the first size estimation of the radio--visible jet collimation
region ($\sim 1$\,pc from the core of the jet for
M\,87; Junor, Biretta \& Livio 1999) and the first measurement of
the distance from the central engine to the core of the jet
(of $\sim 0.3$\,pc for 3C\,120; Marscher et al. 2002).
Monitoring programs with adequate time sampling have also
allowed tests of relativistic hydrodynamic models in the
innermost regions of the jets in AGN (e.g. G\'omez et al. 2001
and Jorstad et al. 2005).

\begin{figure*}[t]
  \centering
  \includegraphics[width=14cm,clip]{fig1Agudo_lowres.eps}
  \caption{World distribution of the stations participating in the
          Global mm--VLBI Array.}
  \label{fig:GMVA}
\end{figure*}

At present, 3\,mm--VLBI offers an even better tool
to image deeper jet regions (i.e. closer than
$\sim 0.3$\,pc from the accretion system).
This is because of the lower jet opacities at this shorter
wavelength and to the larger resolving power at 3\,mm,
which can be up to three times higher than at 7\,mm.

\section{The GMVA: sensitive astronomy at
        40 micro--arcsecond resolution}
\label{sec:GMVA}

The most sensitive 3\,mm--VLBI instrument today is the
Global mm--VLBI Array
(GMVA\footnote{http://www.mpifr-bonn.mpg.de/div/vlbi/globalmm},
see Fig.~\ref{fig:GMVA}), composed of the Pico Veleta,
Plateau de Bure, Effelsberg, Onsala and Mets\"ahovi
stations, in addition to eight of the ten Very Long Baseline
Array (VLBA) antennas.
The GMVA achieves angular resolutions of up to
40\,$\mu$as with typical $7\sigma$ baseline sensitivities of
$80$--$100$\,mJy (adopting 20\,s coherence time, 100\,s
segmentation time and the standard GMVA recording rate
of 512\,Mbps).
This yields $7\sigma$ image sensitivities of $1$--$2$\,mJy/beam
(for 12\,h of observation and a duty cycle of 0.5).
With these characteristics the number of AGN which could
be imaged with high dynamic ranges ($\ge$100:1) is now
larger than 100.

In an attempt to obtain a deeper knowledge of the physics
in the innermost regions of jets in AGN, we have started
3\,mm--VLBI monitoring campaigns of some bright
sources. In this paper, we present recent
results about one of them: NRAO\,150.

\section{NRAO\,150: an unusual AGN  ``hidden" by the Milky Way}
\label{sec:NRAO150}

\begin{figure}[t]
  \includegraphics[width=\columnwidth,clip]{fig2Agudo.eps}
  \caption{1.3\,cm and 8\,mm long term light curve of
          NRAO\,150 from 1987 to 2005. Data from the
	  Mets\"ahovi monitoring of radio sources.}
  \label{fig:metsa}
\end{figure}

\begin{figure*}[t]
  \centering
  \includegraphics[width=15cm,clip]{fig3Agudo.eps}
  \caption{From left to right, 6\,cm, 3.6\,cm and 7\,mm VLBA images
          of the jet in NRAO\,150 obtained in February 1994
	  (for the 6\,cm map) and March 2003 (for those at 3.6\,cm and
	  7\,mm). The contours represent the observed total intensity.
	  For the 7\,mm map, the grey scale symbolizes the
	  polarized intensity and the superposed sticks the orientation
	  of the polarization electric vectors.
	  The higher resolution 7\,mm image shows a strong
	  misalignment between the cm and the mm jet of $\sim 120^{\circ}$.}
  \label{fig:misalign}
\end{figure*}

NRAO\,150 is an intense radio to mm source, which was first
cataloged by  Pauliny--Toth, Wade \& Heeschen
(1966).
The source has been monitored regularly in the radio and
millimeter bands since the beginning of the eighties.
Since then, its total flux density light curve has displayed
a quasi--sinusoidal behavior with a characteristic time--scale of
$\sim 20$--$25$)\,yr (see Ter\"asranta et al. 2004 and
Fig.~\ref{fig:metsa}).
The $1.3$\,cm light curve of the source peaked at
the beginning of 2004, when it displayed $\sim 11$\,Jy
(see Fig.~\ref{fig:metsa}).
NRAO\,150 lacks, up to now, an optical identification.
This is probably due to its low Galactic latitude
(-1.6$^{\circ}$), which causes strong Galactic extinction.
Although its distance is still unknown, we
hope to determine its redshift through an ongoing
spectroscopic project in the infrared band, at which
the source is not strongly absorbed.

On cm-VLBI scales, NRAO\,150 shows a compact core plus a
one--sided jet extending beyond 20 mas with a structural
position angle of $\sim 30^{\circ}$ (see Fig.~\ref{fig:misalign}).
Our new 7\,mm--VLBI observations, the first reported at this
wavelength, have revealed a strong misalignment,
of $\sim 120^{\circ}$, between the inner and outer jet
within its first 0.4\,mas (Fig.~\ref{fig:misalign}).

\subsection{The fastest jet rotation in an AGN at $\sim 7^{\circ}$/yr}
\label{sec:rot}

\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth,clip]{fig4Agudo.eps}
  \caption{3\,mm--VLBI image of NRAO\,150 taken on October 2002.
           The positions of the fitted Gaussian components are
	   indicated by the crosses and the circles (of
	   radius equal to the FWHM of each Gaussian) symbolize
	   their size.}
  \label{fig:3mm}
\end{figure}

Making use of the GMVA (and also of the former Coordinated
Millimeter VLBI Array, CMVA),
we have monitored the jet evolution since 1999 up to
date with observations performed about every six months.
Figure~\ref{fig:3mm} shows one of the resulting images from these
observations, which demonstrate the capability of the 3\,mm array to
probe the innermost jet structures with angular resolutions
of 40\,$\mu$as and dynamic ranges of $\sim$100:1.
The adequate image fidelity of our new 3\,mm observations
is also demonstrated by their ({\it u},{\it v})--coverage,
which is comparable to that of our 7\,mm observations
performed with the VLBA (see Fig.~\ref{fig:uv}).

The results from our new NRAO\,150 images have revealed a
clear angular rotation of the inner 0.4\,mas jet with a speed
of $\sim 7^{\circ}$/year -- projected on the plane of the sky --
(see Fig.~\ref{fig:rot}).
To our knowledge, this is so far the fastest jet rotation
reported for an AGN.

This phenomenon not only represents a likely explanation of
the large jet misalignment found in NRAO\,150, but it also
provides clues about the possible origin of the jet rotation.
It is reasonable to think that the quasi--sinusoidal light
curve of the source, its extreme jet misalignment and the
inner jet rotation are related.
In this case, a possible explanation of the NRAO\,150 evolution
would be a precession--like motion of the inner 0.4\,mas of the
jet.
This, together with projection effects and variable Doppler boosting
through small viewing angles, could explain the
strong jet misalignment, the jet rotation in the plane of the
sky and the $\sim 20$--$25$\,yr variability time--scale of the
radio light curves.
If, in the future, a significant correlation between these light
curves and the position angle of the inner jet is found, the
previous explanation will gain stronger support. In that case,
the possible period of the behavior of NRAO\,150 could be
measured from the light curves.

\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth,clip]{fig5Agudo.eps}
  \caption{a) ({\it u},{\it v})--coverage for our 3\,mm--VLBI
           observation performed on 2002 October 24.
	   b) ({\it u},{\it v})--coverage for our 7\,mm--VLBI
           observation performed on 2003 March 14.}
  \label{fig:uv}
\end{figure}

\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth,clip]{fig6Agudo.eps}
  \caption{Position --with respect to the core of the jet emission--
           of the inner model--fit components
           in NRAO\,150.
	   Only results from observations performed between
	   1999 and 2005 at 3\,mm and 7\,mm are drawn.
	   The plot shows a fast change of the jet initial
	   direction with a mean angular speed of
	   $\sim 7^{\circ}$/year.}
  \label{fig:rot}
\end{figure}

\section{Astrophysics from jet wobbling in AGN}
\label{sec:astro}

Like NRAO\,150, several other jets in AGN present
wobblings triggered in their innermost regions
(e.g., in BL\,Lac; Stirling et al. 2003 and Mutel \& Denn
2005, or in OJ\,287; Jorstad et al. 2005).
These wobblings can be induced by the development of helical
instabilities close to the jet base or by the precession of
the accretion disk.

For the former, jet--cloud interactions (G\'omez et al. 2000)
or dense ejections filling only part of the jet section could
be possible triggering perturbations.
However, they have not been extensively explored from the
theoretical point of view.
This is most likely due to our lack of knowledge of the jet
formation region and the lack of the adequate relativistic
magneto--hydrodynamic tools to study it.
Nonetheless, the subsequent development of Kelvin--Helmholtz
helical instabilities has been well studied (Hardee 2004
and references therein).

Disk precession seems to be nowadays the preferred
mechanism to test and model the quasi-regular jet structural
position and integrated emission variability of AGN.
Up to now, most precession models applied to AGN are
driven by a companion super--massive black hole or another
massive object (see e.g. Valtonen, Lehto \& Pietil\"a 1999,
for OJ\,287; Lister et al. 2003, for 4C~+12.50; Stirling
et al. 2003, for BL\,Lac; Caproni \& Abraham 2004
for 3C\,120; Lobanov \& Roland 2005, for 3C\,345 ).
However, alternative possibilities for accretion disk precession
-- and hence jet precession -- have appeared in the literature
during the last ten years (e.g., Schandl \& Meyer 1994;
Pringle 1996; Quillen 2001; Liu \& Melia 2002;
Lai 2003).
Among them, of special interest are the models from
Liu \& Melia (2002) and Lai (2003) which drive the precession
through intrinsic properties of the accretion system.
For that reason, they allow one to estimate or constrain
the possible black hole spin and accretion disk
density profile (for Sgr~A*, Liu \& Melia 2002; for a
set of eight AGN, Caproni, Mosquera--Cuesta \& Abraham
2004) and disk infall time (Lai 2003) from the observational
properties of the systems.

Although there is still no general paradigm to explain the
accretion disk (and jet) precession and other kinds of wobbling
for AGN, it is rather likely that, as they are triggered in the
innermost regions of the disks (and jets), their mechanisms have
to be tied to fundamental properties of these regions (i.e., close
to the accretion system).
Hence, further development of
models together with the appropriate characterization of the
observational properties of the innermost regions of jets in
AGN would place our understanding of the jet triggering region
and the super--massive accretion systems on firmer ground.

From the observational point of view, high resolution
mm-VLBI observations such as those presented here for NRAO\,150
are of importance, as they allow to probe the innermost
(sub--pc scale) regions of jets in AGN.

\section{The future Global mm-VLBI Array}
\label{sec:futGMVA}

\subsection{Higher sensitivity, higher image fidelity
           and polarimetry}
\label{sec:futimprov}

Even with the good performance of the GMVA, it is desirable
to further improve the sensitivity and the quality of images.
This would increase the number of observable sources and
astrophysical scenarios.
The most direct way to achieve that is to increase the collecting
area of the present interferometric array.
For the near future, ALMA, the GBT, the LMT, CARMA, SRT, Yebes,
Nobeyama and Noto are some of the most sensitive stations
suitable to participate in 3\,mm--VLBI.
Together with them, the present GMVA would be able to achieve $7\sigma$
baseline sensitivities of (5 to 10)\,mJy, and $7\sigma$ image
sensitivities better than 0.1\,mJy/beam.
These estimates predict an \emph{increase, by a factor of 10}
with respect to the present GMVA sensitivity.
At the same time, the development of the VLBI technique
is providing ever faster data recording speeds.
For the next years, recording rates of at least 2\,Gbps are
expected (Garret 2003), which will increase the expected
sensitivities by an extra factor $\ge\sqrt{2}$.
Further improvements in coherence time for mm--VLBI, through
atmospheric phase correction methods (see Roy, Teuber \& Keller
2004 and also http://www.mpifr-bonn.mpg.de/staff/aroy/wvr.html),
are being developed at present.

\begin{figure}[t]
  \centering
  \includegraphics[bb= 104 111 492 732,width=\columnwidth,clip]{fig7Agudo_lowres.eps}
  \caption{Simulations of the ({\it u},{\it v})--coverages
          of the present GMVA (left) and those of the GMVA
	  plus the suitable stations proposed in
	  \S~\ref{sec:futimprov} (right).
	  $0^{\circ}$, $45^{\circ}$ and $70^{\circ}$ of
	  source declinations are presented from top to
	  bottom.}
  \label{fig:uv_imp}
\end{figure}

But the proposed future array will not only produce an increase in
sensitivity. The new stations will also improve the
({\it u},{\it v})--coverage (see Fig.~\ref{fig:uv_imp}),
and so the image fidelity.
In addition, ALMA will improve the ({\it u},{\it v})--coverage
for sources with low declination (less than $20^{\circ}$) and
will facilitate the VLBI imaging of the Galactic Center source
Sgr\,A*.

Finally, high sensitivity 3mm--VLBI polarimetry is nowadays being
tested for the GMVA and it is expected to be offered as a standard
observing mode during the next years.

\subsection{Future science}
\label{sec:futscience}

If the proposed improvements are achieved in the future, images with
dynamic ranges of up to 1000:1 could be easily obtained.
This would place the sensitivity and image fidelity of 3\,mm--VLBI
at comparable levels than those of present cm-VLBI.
These achievements, together with
the possibility to obtain polarimetric images, would help to
(i) to study the triggering mechanisms of relativistic jets,
(ii) to probe their initial magnetic field configurations
(iii) and to better constrain the properties of their accreting
systems, for several hundreds or possibly thousands of jets in
AGN and microquasars.

\acknowledgements

I. Agudo and U. Bach acknowledge funding by the
European Commission through the TMR program HPRN-CT-2002-00321
(ENIGMA network).
We thank A.~L. Roy for helpful comments on this paper.
The VLBA is a facility of the National Radio Astronomy Observatory
of the USA, which is operated by Associated Universities, Inc., under
co-operative agreement with the National Science Foundation.


\section{Introduction}
\label{intro}

Recently one-dimensional (1D) quantum wires (QW) have attracted much attention
due to their potential applications in nanoelectronics \cite{Bowler} and
quantum computing \cite{Bertoni}. The knowledge of the transport properties of
such structures is crucial for the design and fabrication of the nanodevices.
On the other hand the quantum wires, although conceptually simple, are very
interesting from scientific point of view as they display extremely rich
phenomena, very often different from those in two and three dimensions
\cite{Luttinger,Haldane}. The understanding of the properties of such 1D
objects is a major challenge in the field of nanophysics.

The conductance of the quantum wires has been studied both experimentally and
theoretically by number of authors (see \cite{Agrait} for a review). The
experimental studies require advanced techniques of fabrication of such
structures. Those include: growing of QW on metallic surfaces
\cite{Jalochowski,MK_1}, scanning tunneling microscope techniques
\cite{Yazdani} or mechanically controlled break junctions
\cite{Agrait,Muller,Smit}. Those fabrication techniques allowed for revealing
of many phenomena like charge quantization in units of $G_0 = 2e^2/h$
\cite{Wees}, deviations from that ($0.7 (2e^2/h)$ anomaly) \cite{Thomas},
spin-charge separation (Luttinger liquid) \cite{Auslaender}, oscillations of
the conductance as a function of the length of the chain \cite{Muller,Smit} or
spontaneous spin polarization in QW \cite{Thomas,Kane}.

The purpose of the present paper is two fold. The first one is to investigate
the oscillations of the conductance as a function of the wire length in the
case of strong Coulomb interactions. The oscillatory behavior of the
conductance manifests itself as a maximum of the conductance when a number of
the atoms in a wire is odd and minimum when the number is even. This effect is
known as the even-odd conductance oscillations. Most common examples are the
oscillations with a period of two
\cite{Muller,Smit,Sim,Lang,Emberly,Gutierrez} and four atoms \cite{Thygesen}.
However, the conductance can oscillate with different (from two and four)
periods, depending on the average occupation of the wire. Moreover, recently
the analytical formulas for M-atom (M $\geq$ 2) oscillations have been found
\cite{TK_1}. However, those analytical formulas remain valid for noninteracting
wire only. In the presence of strong Coulomb interactions the even-odd
oscillations with a period of two atoms have been also found
\cite{Oguri_1,Tanaka,Oguri_2,Oguri_3,Molina_1,Meden}. The M $\geq$ 2
conductance oscillations have been only reported for the nearest neighbor
Coulomb interactions \cite{Molina_2}. Therefore we shall study the oscillations
of the conductance in the case of strong on-site Coulomb interactions and see
how their period will be modified.

The second purpose is to see if the wire will exhibit any spontaneous spin
polarization in the presence of strong correlations. Such spontaneous
polarization has been observed experimentally \cite{Thomas,Kane}. But it is
well known that the ferromagnetism in strictly 1D objects is forbidden due to
the Lieb-Mattis theorem \cite{Lieb}.

The paper is organized as follows. In Sec. \ref{model} we present theoretical
description of the model wire, in Sec. \ref{results} we show the results of the
calculations of the conductance, charge and spin polarization. Finally, in Sec.
\ref{conclusions} we provide some conclusions.

\section{Theoretical description}
\label{model}

Our system consists of the quantum wire modeled as a chain of $N$ atoms
coupled to the left $L$ and right $R$ lead described by the following
Hamiltonian in the limit of strong on-site Coulomb interaction
($U_i \rightarrow \infty$) in the slave boson representation where the real
wire electron $d_{i\sigma}$ is replaced by the product of the boson $b_i$ and
the fermion $f_{i\sigma}$ operators ($d_{i\sigma} = b^+_i f_{i\sigma}$)
\cite{Coleman,LeGuillou,MK_2}:
\begin{eqnarray}
H = \sum_{\lambda {\bf k} \sigma} \epsilon_{\lambda {\bf k}}
c^+_{\lambda {\bf k} \sigma} c_{\lambda {\bf k} \sigma} +
\sum_{i\sigma} \varepsilon_i f^+_{i\sigma} f_{i\sigma}
\nonumber \\
+ \sum_{\langle i j \rangle, \sigma}
t_{ij} f^+_{i\sigma} b_i b^+_j f_{j \sigma}
\nonumber \\
+ \sum_{{\bf k} \in L(R), \sigma} V_{L(R) {\bf k}}
c^+_{L(R) {\bf k} \sigma} b^+_{1(N)} f_{1(N) \sigma} + h.c. ,
\label{Hamilt}
\end{eqnarray}
where $c_{\lambda {\bf k} \sigma}$ stands for the electron with the single
particle energy $\epsilon_{\lambda {\bf k}}$, the wave vector ${\bf k}$ and the
spin $\sigma$ in the lead $\lambda = L, R$. $\varepsilon_i$ denotes the wire
energy level at site $i$, $t_{ij}$ is the hopping integral of the electrons
between neighboring wire sites $i$ and $j$, and $V_{L(R) {\bf k}}$ is the
hybridization matrix element between electrons at site $1(N)$ and those
in the lead $L(R)$.

In the linear response and at the zero temperature the conductance $G$ is
proportional to the total transmittance $T$, i. e. $G = \frac{2e^2}{h} T$. In
our case the transmittance is given by \cite{TK_1,Datta}:
\begin{eqnarray}
T_N(E) = \sum_{\sigma} \Gamma_L \Gamma_R |G^r_{1N\sigma}(E)|^2 ,
\label{transmit}
\end{eqnarray}
where $G^r_{1N\sigma}$ is the retarded Green function (GF) connecting the ends
of the wire (sites $1$ and $N$) and $\Gamma_{L(R)}$ is the elastic rate
$\Gamma_{L(R)} = 2 \pi \sum_{\bf k}
|V_{L(R) {\bf k}}|^2 \delta(E - \epsilon_{L(R) {\bf k}})$. In calculations we
have assumed constant bare density of states in the leads.

Using the equation of motion technique for the retarded GF with Hamiltonian
(\ref{Hamilt}) one can write the general matrix equation for $G^r_{ij\sigma}$
in the form:
\begin{eqnarray}
\hat A_{\sigma} \hat G^r_{\sigma} = \hat N_{\sigma} .
\label{EOM}
\end{eqnarray}
Due to the strong on site Coulomb interactions in the wire $U_i$, which is
assumed to be infinity in our case, the problem cannot be solved exactly and
one has to make approximations of the higher order GFs emerging in the equation
of motion for the retarded GF
$G^r_{ij\sigma} = \langle\langle b^+_1 f_{i\sigma} |
f^+_{j\sigma} b_j \rangle\rangle_E$. We have used Hubbard I like approximation
\cite{MK_2} according to which the GF
$\langle\langle f^+_{i-\sigma} f_{i-\sigma} b^+_k f_{k\sigma} |
f^+_{j\sigma} b_j \rangle\rangle_E$ is approximated by
$\langle f^+_{i-\sigma} f_{i-\sigma} \rangle \langle\langle b^+_k f_{k\sigma} |
f^+_{j\sigma} b_j \rangle\rangle_E$ and the other higher order GFs are
neglected. This approximation is reasonably good for not very large values of
the hopping $t$ and neglects higher order processes, like for example the Kondo
effect.

Within the present approximation scheme $\hat A_{\sigma}$ in the Eq.(\ref{EOM})
is $N \times N$ tridiagonal symmetric matrix with the elements:
\begin{eqnarray}
\hat A_{\sigma} = (E - \varepsilon_i) \delta_{i,j} +
i \frac{\Gamma}{2} (\delta_{i,1} \delta_{j,1} + \delta_{i,N} \delta_{j,N}) -
\nonumber \\
t [(1 - n_{i-\sigma}) \delta_{i,i+1} + (1 - n_{i+1-\sigma}) \delta_{i+1,i}]
\label{Asig}
\end{eqnarray}
and $\hat N_{\sigma}$ is the diagonal matrix of the form:
\begin{eqnarray}
\hat N_{\sigma} = (1 - n_{i-\sigma}) \delta_{ij} ,
\label{Nsig}
\end{eqnarray}
with $n_{i\sigma} = \langle f^+_{i\sigma} f_{i\sigma} \rangle =
-\frac{1}{\pi} \int dE Im G^r_{ii\sigma}(E)$ being the average occupation of
the electrons with spin $\sigma$ at site $i$.

\section{Results}
\label{results}

In numerical calculations we have assumed all the wire site energies to be
equal ($\varepsilon_i = \varepsilon_0$) and similarly hopping integrals
$t_{ij} = t$. All energies are measured with respect to the leads Fermi energy
$E_F = 0$ in units of $\Gamma = \Gamma_L = \Gamma_R = 1$. Moreover, the
occupation $n_{i \sigma}$ is calculated self-consistently on each wire site.

To find the condition for M-atom conductance oscillations one has to solve the
relation: $T_N = T_{N+M}$, where $T_N$ ($T_{N+M}$) is the transmittance of the
wire consisted of $N$ ($N+M$) atoms, given by Eq. (\ref{transmit}). In general,
for $U_i \rightarrow \infty$ it is not possible to get analytical expression
for the oscillations condition without further assumptions. Note that for
$U_i = 0$ the problem can be solved exactly and such condition can be found
\cite{TK_1}. In this case it reads $cos{\left(\frac{\pi l}{M}\right)} =
\frac{E_F - \varepsilon_0}{2 t}$, where $l = 1$, $2$, ... $M-1$.

In the case of $U_i \rightarrow \infty$ both matrices $\hat A_{\sigma}$ and
$\hat N_{\sigma}$ depend on the occupation on each wire site $n_{i, \sigma}$
thus the problem has to be solved numerically. However, if one assumes that the
occupation is the same on each wire site and does not depend on spin, i. e.
$n_{i\sigma} = n_{\sigma} = n_{-\sigma}$, similar M-atom oscillations condition
as for $U_i = 0$ can be found. In this case it reads:
\begin{eqnarray}
cos{\left(\frac{\pi l}{M}\right)} =
\frac{E_F - \varepsilon_0}{2 t (1 - n_{\sigma})} ,
\label{cond_osc}
\end{eqnarray}
with $l = 1$, $2$, ... $M-1$. Unfortunately, one has to know the average
occupation $n_{\sigma}$. The only case of $M = 2$ can be solved analytically.
As one can see from Eq. (\ref{cond_osc}) the period of two can be obtained for
$E_F - \varepsilon_0 = 0$, i.e. when the wire single particle energy levels all
coincide with the Fermi energy.

In Fig. \ref{Fig1} the total linear conductance $G = \sum_{\sigma} G_{\sigma}$
is plotted as a function of the wire length $N$ and the energy level
$\varepsilon_0$ for $t = 4$.
\begin{figure}[h]
 \resizebox{\linewidth}{!}{
  \includegraphics{fwire_01.eps}}
 \caption{\label{Fig1} The total linear conductance
          $G = \sum_{\sigma} G_{\sigma}$ as a function of the wire length $N$
	  and the wire energy level $\varepsilon_0$ for $U_i = 0$ (top panel),
	  $U_i \rightarrow \infty$ in paramagnetic configuration
	  $n_{i\sigma} = n_{i-\sigma}$ (middle panel) and ferromagnetic one
	  (bottom panel).}
\end{figure}
As one can see all the figures show similar patterns with the regions of large
and small conductances. Moreover, for $\varepsilon_0 = 0$ behavior of the
conductance is the same in all three cases which leads to the conclusion that
correlations are not important in this case. It always shows even-odd ($M = 2$)
oscillations. Away from $\varepsilon_0 = 0$ correlations strongly modify the
conductance, shifting the maxima of $G$ (for fixed $N$) towards lower (higher)
energies for $\varepsilon_0 > 0$ ($\varepsilon_0 < 0$). This is due to the
modification of the wire hopping, which depends now on the occupation
$n_{i\sigma}$ (see Eq. (\ref{Asig})). This effect also leads to the strong
asymmetry for positive and negative energies. For negative (positive) wire
energies the occupation is large (small), therefore effective hopping
$\tilde t = t (1 - n_{i-\sigma})$ is small (large), thus the conductance
decreases (increases). Note that there is no such asymmetry in the case of
$U_i = 0$. Moreover, different periods of the conductance oscillations can be
observed, depending on the position of the wire energy level.

Another important finding is that the wire shows spontaneous spin polarization.
It is well known that in strictly $1D$ wire the spin polarization is prohibited
due to the Lieb-Mattis theorem \cite{Lieb}. However this theorem is valid for
infinite wire only. In experimental situation the wire is always connected to
the electrodes and this is why the spin polarization is observed experimentally
\cite{Thomas,Kane}. Interestingly, it was predicted recently that even infinite
wire, but of the zig-zag shape, can also exhibit the spin polarization
\cite{Klironomos}.

The total conductance in ferromagnetic case is shown in the bottom panel of
Fig. \ref{Fig1}. Again, unlike for $U_i = 0$, the conductance pattern shows
strong asymmetry for positive and negative energies. Note that for negative
energies there are no differences between $G_{ferro}$ (bottom panel) and
$G_{para}$ (middle panel). For such energies iterations always converge to
paramagnetic solution. This can be again explained by effect of the hopping
modification. For negative energies the wire occupation is large, thus the
effective hopping $\tilde t = t (1 - n_{i-\sigma})$ is small. One can imagine
in this case that the electrons are more localized and it is more convenient
for them to spend more time on the same site than move to another one.
Moreover in the case of the lack of the inter-site interactions any collective
phenomenon is not possible. Situation is different for positive energies. In
this case the electrons are more mobile, as the effective hopping is larger due
to the small values of the wire occupations. Thus they interact with each other
via hopping and it is possible and energetically more favorable to get the
ferromagnetic state. The hopping, which is the correction to the position of
the wire energy levels, leads to the effective splitting of this levels. Thus
the conductance is spin polarized in this case. It can be read off from Fig.
\ref{Fig2}, where the difference between spin up and spin down conductance is
displayed.
\begin{figure}[h]
 \resizebox{\linewidth}{!}{
  \includegraphics{fwire_02.eps}}
 \caption{\label{Fig2} The difference between spin up and spin down conductance
          as a function of the wire length $N$ and the wire energy level
	  $\varepsilon_0$.}
\end{figure}
It turns out that the strongest differences between $G_{\uparrow}$ and
$G_{\downarrow}$ can be found for intermediate values of $\varepsilon_0$. For
energies close to $E_F$ the modifications are weak due to the small values of
the hopping while for higher energies the wire occupation is very small thus
the difference between spin dependent effective hopping can be neglected.
Finally it is worthwhile to note that for $N = 1$ (single atom) there is no
ferromagnetic solution as the ferromagnetism is governed by inter-site hopping
in our case. Whether this spontaneous spin polarization is a true effect or a
drawback of the approximation used remains an open question, as it is known
that the mean field like theories overestimate the role of the magnetism. The
problem will be further studied.

Corresponding spin polarization $n_{\uparrow} - n_{\downarrow}$
($n_{\sigma} = \sum_i n_{i \sigma}/N$) is shown in Fig. \ref{Fig3}.
\begin{figure}[h]
 \resizebox{\linewidth}{!}{
  \includegraphics{fwire_03.eps}}
 \caption{\label{Fig3} The spin polarization as a function of the wire length
          $N$ and the wire energy level $\varepsilon_0$.}
\end{figure}
As one can see the spin polarization pattern is similar to that of the
conductance differences (see Fig. \ref{Fig2}).

In Fig. \ref{Fig4} we show the conductance (left panels) and the occupation
(right panels) as a function of the wire length for a number of the energy
levels $\varepsilon_0$ in the paramagnetic configuration.
\begin{figure}[h]
 \resizebox{\linewidth}{!}{
  \includegraphics{fwire_04.eps}}
 \caption{\label{Fig4} The total conductance (left panels) and the occupation
          (right panels) vs. wire length. The left panels show the conductance
	  with different oscillations periods ($2$ to $4$) from top to bottom.
	  The positions of the wire energy levels are indicated in the figure.}
\end{figure}
The values of $\varepsilon_0$ have been chosen in such a way that they lead to
the maxima of the conductance for $N = 1$, $2$ and $3$ atom wire. For example,
the maxima of the conductance of the $N = 2$ atom wire correspond to
$\varepsilon_0 = -2.298$ and $3.135$ (see the middle panel of Fig. \ref{Fig1}).
These maxima in $U_i =0$ case give the periods of the conductance oscillations.
To find $M$-atom period it is enough to determine the maxima of the conductance
for $N = M - 1$ atom wire \cite{TK_1}. As one can read off from Fig.
\ref{Fig4}, depending on $\varepsilon_0$, one gets different periods of the
conductance oscillations. Moreover, except for the special case of $M = 2$, the
amplitude of the oscillations decreases with the wire length. This is a kind of
damped oscillations. No such effect has been observed for $U_i = 0$
\cite{TK_1}.

At this point we would like to comment on the other results known in the
literature. The even-odd ($M = 2$) oscillations problem was extensively studied
within the second order perturbation theory in $U_i$ (SOPT)
\cite{Oguri_1,Tanaka} and the numerical renormalization group (NRG) approach
\cite{Oguri_2,Oguri_3}. The results show similar behavior of the conductance
for odd number of atoms in a wire: it always reaches the unitary limit
($\frac{2 e^2}{h}$), independently of $U_i$. Such behavior is a consequence of
the Kondo effect. However, in our case the situation is slightly different, as
we get $M = 2$ oscillations in the mixed valence regime only
($\varepsilon_0 = 0$), where the Kondo effect is excluded. Thus our even-odd
oscillations are caused by the resonances associated with the energy level
structure of the chain rather than the Kondo effect. On the other hand, the
conductance for even number of atoms in a wire is strongly suppressed, in
agreement with SOPT \cite{Oguri_1,Tanaka} and NRG approaches
\cite{Oguri_2,Oguri_3}. However, NRG calculations show that the conductance
exponentially depends on $U_i$, and in the limit of $U_i \rightarrow \infty$
vanishes, contrary to our results, as we get non-zero values of $G$ (see Fig.
\ref{Fig4}), depending on the hopping $t$. The conductance vanishes in the
limit of very large or very small values of $t$. Interestingly, when
$t = \Gamma/2$, the conductance reaches the unitary limit and shows no
oscillations, i.e. is equal to $\frac{2 e^2}{h}$ for even and odd $N$.

Figure \ref{Fig4} (right panels) shows the wire length dependent occupation
which also oscillates with the same period as the conductance does, except for
the special case of $\varepsilon_0 = 0$ ($M = 2$), where the occupation remains
constant. Moreover, the Coulomb interactions $U_i$ lead to the reduction of the
occupation oscillation amplitude. Similar effect, albeit for small $U_i$, has
been found within self-consistent Hartree-Fock approximation \cite{Kostyrko}.

In ferromagnetic case the situation is more complex. For positive values of
$\varepsilon_0$ due to the splitting of the conductance maxima (see Fig.
\ref{Fig1}) no regular oscillations have been observed. On the other hand, for
$\varepsilon_0 < 0$ one gets such oscillations but in this case the solutions
remain always paramagnetic. Figure \ref{Fig5} shows the comparison of the
conductance and the occupations in paramagnetic and ferromagnetic
configurations for the wire consisted of five atoms.
\begin{figure}[h]
 \resizebox{\linewidth}{!}{
  \includegraphics{fwire_05.eps}}
 \caption{\label{Fig5} The total conductance in paramagnetic (left top panel)
          and in ferromagnetic (right top panel) configuration and
	  corresponding occupations (bottom panels) vs. $\varepsilon_0$ of five
	  atom wire.}
\end{figure}
For negative energies, as discussed before, there are paramagnetic solutions
only. For $N > 1$ and $\varepsilon_0 > 0$ the ferromagnetic solutions emerge
in certain energy regimes. In this case $n_{\uparrow} \neq n_{\downarrow}$ and
resulting conductance peaks are split. Interestingly, the spin polarizations
occurs only in the regimes where the occupation has a large slope or the total
conductance has a maximum. Moreover, the splitting of the conductance leads to
the fact that $G_{ferro}$ shows more maxima than $G_{para}$. A number of maxima
of $G_{ferro}$ is related to the wire length (number of atoms - $N$) and for
odd $N$ it gives $(3N-1)/2$ maxima, while for even $N$ one observes $3N/2$
maxima. In paramagnetic case it is always equal to a number of atoms $N$ in the
wire (compare top panels of Fig. \ref{Fig5}).

Finally we would like to comment on the validity of our approach. The present
calculations completely neglect the Kondo effect which is important at low
temperatures and $\varepsilon_d < 0$. We expect some modifications in this
regime, as this effect leads to the corrections of the conductance of the order
of $\frac{e^2}{h}$. Thus our results apply for temperatures higher than the
Kondo temperature. On the other hand, we do not expect any qualitative
modifications in the mixed valence and the empty regimes
($\varepsilon_d \geq 0$).

\section{Conclusions}
\label{conclusions}

In summary we have studied the conductance oscillations of the strongly
interacting wire as a function of the wire length. We have found that strong
Coulomb interactions significantly modify the periods of the oscillations
showing strong asymmetry for negative and positive wire energy levels. They
also lead to the suppression of the conductance with increasing wire length.
There are no such effects for noninteracting wire. Moreover, strong
interactions lead to the spontaneous spin polarization for positive wire
energies, observed in experiments.

\noindent {\bf Acknowledgements} \\
This work has been supported by the grant no.1 P03B 004 28 of the Polish
Committee of Scientific Research. T. K. thanks the Foundation for Polish
Science for a Fellowship for Young Scientists.