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\journalname{Few-Body Systems}
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\begin{document}

\title{Spin and dynamics in relativistic quantum theories
\thanks{Research supported by
the US DOE Office of Science, under grant number No. DE-FG02-86ER40286
}}
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%placed here. General acknowledgments should be placed 
%at the end of the article.}
%\subtitle{Do you have a subtitle?\\ If so, write it here}

\titlerunning{Spin and dynamics}        % if too long for running head

\author{W. N. Polyzou %        \and
%        Second Author %etc.
\and W. Gl\"ockle \and H. Wita{\l}a}
\authorrunning{W. N. Polyzou} % if too long for running head

\institute{W. N. Polyzou \at
              The University of Iowa \\
              Tel.: +319-335-1856\\
              Fax: +319-335-1753\\
              \email{polyzou@uiowa.edu}           %  \\
%             \emph{Present address:} of F. Author  %  if needed
%           \and
%           S. Author \at
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}

\date{Received: date / Accepted: date}
% The correct dates will be entered by the editor

\maketitle

\begin{abstract}
The role of relativity and dynamics 
in defining the spin and orbital angular momentum content of 
hadronic systems is discussed.  
\keywords{spin \and orbital angular momentum \and relativity}
\end{abstract}

\section{Introduction}
\label{intro}

There is a great deal of interest in the distribution of spin and
orbital angular momentum in hadronic systems.  In general the
underlying dynamics of partons in hadrons is relativistic.  The
treatment of spin in relativistic systems is different than it is in
non-relativistic systems.  In a relativistic system the spin of a
parton is identified with the angular momentum of the parton in its
rest frame while the spin of the hadron is defined as the angular
momentum of the hadron in its rest frame.  Transforming a parton from
its rest frame to the hadrons rest frame, where the spins can be
coupled, involves boosts which generate dynamical rotations.  These
rotations transform the parton spins and also impact the relative
orbital angular momentum before they can be coupled.  The spin of the
constituents, the internal orbital angular momentum and the spin of
the system are related by Clebsch-Gordan coefficients of the
Poincar\'e group\cite{moussa}\cite{bkwp}\cite{wgwp} .  The
Poincar\'e group Clebsch-Gordan coefficients are labeled by
eigenvalues of mass and spin Casimir operators, which are dynamical
operators.

An additional complication is that boosts to the rest fame are not
unique; a boost to the rest frame followed by a momentum dependent
rotation is a different boost to the rest frame.  There are as many
different kinds of boosts as there are momentum dependent rotations.
Each boost defines a different spin observable.  For example, there
are distinct boosts that are used to define the helicity, light front
spin or canonical spin.  These are three specific choices, that are
distinguished by useful properties, out of an infinite number of
possibilities.  In many-body systems there is another relevant spin,
which I call the constituent spin\cite{wgwp}, which is distinguished by the
property that spins and orbital angular momenta can be combined using
ordinary SU(2) coupling methods to get the hadronic spin.  

While all of the spins satisfy $SU(2)$ commutation relations, the
different spins observables are related by the momentum dependent
(Melosh) rotations\cite{melosh} that relate different boosts.  Because different
spin observables differ by momentum-dependent rotations, partial
derivatives with respect to momentum that hold one spin observable
constant will not commute with a different spin observable.  This
means not only are there an infinite number of possible spin
observables, but each one is associated with a different quantity that
can be identified with an orbital angular momentum.  As a result the
spin and orbital angular momentum content of a hadron is dynamical and
representation dependent.  In what follows we discuss some of the
relevant issues.

The dependence of the spin on the choice of boost is seen in the 
relations between the spin and angular momentum
\begin{equation}
j_x^l := {1 \over 2} \epsilon^{lmn} 
{B_x^{-1} (p)^m{}_{\mu}}
{B_x^{-1} (p)^n{}_{\nu}}J^{\mu \nu}
\label{eq:a1}
\end{equation}
where $B_x^{-1} (p)^m{}_{\mu}$ is a boost that maps $p$ to its rest frame.

This definition can be equivalently expressed in terms of the polarization 
vectors that are three space-like vectors 
$e_x^{m}{}_{\mu}(p)$ that are
orthogonal to the 
four momentum, that can be identified with the 
space-like rows of the boost $B_x^{-1} (p)^m{}_{\mu}$:
\begin{equation}
j_x^l = {1 \over 2} \epsilon^{lmn} 
{e_x^{m}{}_{\mu}(p)
e_x^{n}{}_{\nu}(p)} J^{\mu \nu} .
\label{eq:a2}
\end{equation}
Spins constructed using different boosts are related by momentum-dependent 
Melosh rotations  
\begin{equation}
j_x^l = R_{xy}(p)^l{}_m j_y^m \qquad
\mbox{where} \qquad 
R_{xy}(p)^l{}_m=
B_x^{-1} (p)^l{}_{\mu}
B_y  (p)^{\mu}_{m} .
\label{eq:a3}
\end{equation}
Because the different types of spin observables differ by momentum-dependent
rotations, 
``Position operators'' \cite{newton}\cite{wp}\cite{bkwp}that involve partial derivatives with respect to 
momentum need to specify which kind of spins is being held 
constant during the differentiation,  
\begin{equation}
[\pmb{\nabla}_{P_{\vert_{j_x}}}, \mathbf{j}_x]=0 \,
{\Rightarrow} \, [\pmb{\nabla}_{P_{\vert_{j_x}}}, \mathbf{j}_y] \not= 0 . 
\label{eq:a4}
\end{equation}
These partial derivatives can be written in terms of the Poincar\'e
generators with $\mathbf{V}= \mathbf{P}/M$ by \cite{wp}\cite{bkwp}
\begin{equation}
X^k_x= 
{i \pmb{\nabla}_{\mathbf{P}_{\vert_{\mathbf{j}_x}}}}
=
-{1 \over 2} \{H^{-1},{K}^k\} + i H^{-1} C_{1x}^{kl}(\mathbf{V}) j_x^l . 
\label{eq:xx}
\end{equation}
In terms of these operators the spins and angular momentum are related by
\begin{equation}
J^j = (\mathbf{X}_x \times \mathbf{P})^j + C_{2x}^{jk} (\mathbf{V})j_x^k 
\label{eq:a5}
\end{equation}
where the operators $C_{1x}^{jk}(\mathbf{V})$ and 
$ C_{2x}^{jk} (\mathbf{\mathbf{V}})$ are the following functions 
of the Poincar\'e generators:
\begin{equation}
C_{1x}^{jk}(\mathbf{V}) =
{1 \over 2} \mathbf{Tr} [B_x(\mathbf{V})^{-1}\sigma_j B_x(\mathbf{V}) \sigma_k]
- V^0 \mathbf{Tr} [B_x(\mathbf{V})^{-1}{\partial \over \partial V_l}  
B_x(\mathbf{V}) \sigma_m]
\label{eq:a6}
\end{equation}
\begin{equation}
C_{2x}^{jk} ({\mathbf{V}}) = 
{1 \over 2} \mathbf{Tr} [B_x(\mathbf{V})^{-1}\sigma_j B_x(\mathbf{V}) \sigma_k]
+
i \epsilon_{jlm} 
\mathbf{Tr} [B_x(\mathbf{V})^{-1}{\partial \over \partial V_l}  
B_x(\mathbf{V}) \sigma_m] 
\label{eq:a7}
\end{equation}
and $B_x(\mathbf{V})$ is the $SL(2,\mathbb{C})$ representation of the 
$x$-boost.  The quantity $\mathbf{X}_x \times \mathbf{P}$ is the
associated orbital angular momentum \cite{wp}\cite{bkwp} 

Three components of the four momentum and the projection of any of these
spin observables on a given axis are labels for vectors in irreducible 
subspaces.  Products of two such irreducible representations can be 
expressed as direct integrals of composite irreducible representations
using the Clebsch-Gordan coefficients for the Poincar\'e group.
Like any set of Clebsch-Gordan coefficients, the actual coefficients
depend on the choice of irreducible basis.
The Poincar\'e group Clebsch-Gordan coefficients for a basis 
labeled by the $x$-type spin are 
\[
_x\langle  (M_1,j_1) \mathbf{P}_1 ,\mu_1  (M_2,j_2) \mathbf{P}_2 ,\mu_2 
\vert k, j (M_1,j_1,M_2,j_2) \mathbf{P} ,  \mu , l , s_{12} \rangle_x =
\]
\[
\sum_{\mu_1',\mu_2',\mu_1'',\mu_2'',\mu_s,m}
\delta (\mathbf{P} - \mathbf{P}_1-\mathbf{P}_2)
{\delta (k -k(\mathbf{P}_1,\mathbf{P}_2)) \over k^2}
\times
\]
\[
\sqrt{{\omega_{M_1} (\mathbf{k})\omega_{M_2} (\mathbf{k})
\over \omega_{M_1} (\mathbf{P}_1)\omega_{M_2} (\mathbf{P}_2)}}
\sqrt{{
\omega_{M_1} (\mathbf{P}_1)+\omega_{M_2} (\mathbf{P}_1)
\over \omega_{M_1} (\mathbf{k})+\omega_{M_2} (\mathbf{k})}}
\times
\]
\[
{D^{j_1}_{\mu_1 \mu_1'}[R_{wx} (B_x(V) ,k_1)]
D^{j_1}_{\mu_1'\mu_1''} [R_{xc}(k_1)]} \times
\]
\[
{D^{j_2}_{\mu_2 \mu_2'}[R_{wx} (B_x(V) ,k_2)]
D^{j_2}_{\mu_2'\mu_2''} [R_{xc}(k_2)]}
\times 
\]
\begin{equation}
Y^{l}_{m} (\hat{\mathbf{k}}(\mathbf{P}_1,\mathbf{P}_2)) 
\langle j_1, \mu_1'', j_2, \mu_2'' \vert s_{12}, \mu_s \rangle
\langle l, m, s_{12}, \mu_s \vert j, \mu \rangle
\label{eq:a8}
\end{equation}
These involve two types of spin rotations.  There are Wigner rotations
$R_{wx} (B_x(V) ,k_i)$ that arise from the $x$-boosts that relate the
system and parton rest frames and generalized Melosh rotations,
$R_{xc}(k_2)$, that transform the resulting spins to the canonical
spin representation where all of the spins and orbital angular momenta
Wigner rotate together so they can be added using ordinary $SU(2)$
spin addition.

The spins obtained by applying these two rotations to the hadronic spins
are the constituent spins mentioned earlier.  These are 
the spins associated with the magnetic quantum numbers
$\mu_i''$ in (\ref{eq:a8}).  It is apparent from
this equation that when these spins are combined with the orbital 
angular momentum using spherical harmonics and SU(2)
Clebsch-Gordan coefficients the result is the total spin.

The Poincar\'e group Clebsch-Gordan coefficients (\ref{eq:a8}) 
simplify in special bases.  If the
spins are defined using the standard rotationless boosts there are no
Melosh rotations, if the rotationless boost is replaced by a
light-front boost there are no Wigner rotations, and if the
rotationless boost is replaced by a helicity boost the Wigner
rotations become multiplication by a phase.

One result of the momentum-dependence of the rotations is that the
momentum dependence of the hadronic wave function affects the
expectation values of both the spins and orbital angular momentum.
Since the structure of the hadronic wave function is dynamical, the
dynamics enters in spin coupling when the the Poincar\'e 
Clebsch-Gordan coefficients are integrated against the hadronic wave
functions.

When one couples two interacting subsystems, one has to ask whether
the masses in the Poincar\'e Clebsch-Gordan coefficients are the
physical masses of the subsystems or the invariant masses of their
constituents.  For example, the mass of a meson or the invariant mass
of a quark antiquark pair?  So far we have treated them as invariant
masses of the constituents.  Cluster properties suggest that one
should really use the physical mass operators of the subsystems.
Fortunately there is a unitary transformation that removes the
interaction dependence from the hadronic spin\cite{fcwp}\cite{wp2}.
In this representation the spins can be coupled by sequential coupling
using the Clebsch-Gordan coefficients of the Poincar\'e group as if
the particles were not interacting.  This unitary transformation
changes the Hamiltonian, generating many-body interactions.  It also
changes the representation of the wave function in a way that
preserves probabilities, expectation values, as well as scattering
observables.  In this case all of the dynamical spin effects can be
absorbed by changing the representation of the wave function.

A second ambiguity with spin has to do with whether the dynamics
formulated using Poincar\'e covariant or Lorentz covariant bases.  The
unitary representation of the Poincar\'e group on positive-mass
positive-energy irreducible basis states has the form
\begin{equation}
U(\Lambda ,0) \vert (M,j)\mathbf{P},\mu \rangle_x = 
\sum \vert (M,j)\pmb{\Lambda}P,\nu \rangle_x 
D^j_{\nu \mu}(R_{wx}(\Lambda,P)).
\label{eq:a9}
\end{equation}
The Wigner rotation can be decomposed into the composition of a
boost followed by a Lorentz transformation followed by an 
inverse boost with the transformed four momentum
\begin{equation}
R_{wx}(\Lambda,P) = B_x^{-1}(\Lambda P) \Lambda B_x (P).
\label{eq:a10}
\end{equation}
The group representation property can be used to split the Wigner function
apart.  The finite dimensional representations of $SU(2)$ are related
to finite dimensional representation of $SL(2,\mathbb{C})$ by analytic 
continuation\cite{weinberg}\cite{wgwp}, so we can 
still use the group representation property.
Absorbing the Wigner functions of the boosts into the
states gives the Lorentz spinor representation of the states:
\begin{equation}
\vert (m,j) \mathbf{P}, {b} \rangle := \sum_{\mu}
\vert (m,j) \mathbf{P}, \mu \rangle_x 
D^j_{\mu {b}}[B_x^{-1}(P/M)].
\label{eq:a11}
\end{equation}
Here the boosts are represented by $2 \times 2 \, SL(2,\mathbb{C})$
transformations.  These spinor basis states (\ref{eq:a11}) 
have the following 
Lorentz covariant transformation property
\begin{equation}
{U(\Lambda, 0)} \vert (m,j) \mathbf{P}, {b} \rangle =
\sum_{{b}'} \vert (m,j) \pmb{\Lambda}{P}, {b}' \rangle
{D^j_{{b}'{b}}[\Lambda]}.
\label{eq:a12}
\end{equation}
The price paid for using the covariant representation is that
the Hilbert space inner product becomes dynamical
\[
\langle \psi \vert \phi \rangle =
\]
\begin{equation}
\int 
\langle \psi \vert (m,j) \mathbf{P}, {b} \rangle 
d^4P \theta(P^0) \delta (P^2 +
{M^2}) 
D^j_{bb'}[P^{\mu}\sigma_{\mu}/{M}]  
\langle (m,j) \mathbf{P}, {b}' \vert \phi \rangle 
\label{eq:a13}
\end{equation}
where we have used 
the hermiticity of the $SL(2,\mathbb{C})$ representation of the
rotationless boost which gives
\begin{equation}
B_x (V)B_x^{\dagger}(V) =
B_c (V)R_{cx}(V) R^{\dagger}_{cx} B_c^{\dagger}(V) = B_c (V)B_c^{\dagger}(V) = B^2_c (V)
= P^{\mu}\sigma_{\mu}/M
\label{eq:a14}
\end{equation}
independent of the type $(x)$ of boost.  Here the dynamics is
contained in the mass-shell condition, which makes the Wigner 
function into a positive matrix.  The inner product (\ref{eq:a13}) 
identical the original Poincar\'e covariant inner product.

Unlike representations of $SU(2)$, the 
representations of $SL(2,C)$ are not equivalent to the
complex conjugate representations.  This means that 
we could alternatively replace (\ref{eq:a11}) by  
\begin{equation}
\vert (m,j) \mathbf{P}, \dot{b} \rangle := \sum_{\mu}
\vert (m,j) \mathbf{P}, \mu \rangle_x 
D^j_{\mu \dot{b}}[B_x^{\dagger}(P/M)].
\label{eq:a11b}
\end{equation}
and (\ref{eq:a12}) by 
\begin{equation}
{U(\Lambda, 0)} \vert (m,j) \mathbf{P}, \dot{b} \rangle =
\sum_{{b}'} \vert (m,j) \pmb{\Lambda}{P}, \dot{b}' \rangle
{D^j_{\dot{b}'\dot{b}}[((\Lambda)^{\dagger})^-1)]}.
\label{eq:a12b}
\end{equation}
This 
gives a representation of the scalar product that has the same
form as (\ref{eq:a13}) with the replacement
\begin{equation}
D^j_{bb'}[P^{\mu}\sigma_{\mu}/{M}] \to 
D^{j}_{\dot{b} \dot{b}'}[P^{\mu}
\sigma_2\sigma^*_{\mu}\sigma_2/{M} ] . 
\label{eq:a16}
\end{equation}
While the inner products in all three representations are identical,
the Lorentz covariant and its complex 
conjugate representations are related by
space reflection.  Space reflection changes the kernel
of the Hilbert-space scalar product in the covariant representations.
Space reflection can be represented as an operator on states
by replacing the representation (\ref{eq:a11}) and (\ref{eq:a11b}) 
by a direct sum of both representations. In the direct sum representation the 
wave function becomes a $2 \times (2j+1)$ component spinor
\begin{equation}
\psi ({P} , b) \to 
\left (
\begin{array}{c} 
\xi ({P} , b) \\
\chi ({P} , \dot{b})
\end{array}
\right ) 
\label{eq:a17}
\end{equation}
and the kernel of the inner product becomes
\begin{equation}
d^4P \theta(P^0) \delta (P^2 +
{M^2}) 
\left (
\begin{array}{cc}
D^j_{bb'}[P^{\mu}\sigma_{\mu}/{M}] & 0 \\
0 & D^{j}_{\dot{b} \dot{b}'}[P^{\mu}
\sigma_2\sigma^*_{\mu}\sigma_2/{M} ] 
\end{array}
\right ).
\label{eq:a18}
\end{equation}.  

One desirable feature of the covariant representation is that the
basis-dependent features are hidden in the wave functions.  To see
this note that for rotations the upper and lower components have
identical transformations laws
\begin{equation}
U(R, 0) \vert (M,j) \mathbf{P}, {b} \rangle =
\sum_{\dot{b}'} \vert (M,j) R\mathbf{P}, {b}' \rangle
D^j_{{b}'{b}}[R]. 
\label{eq:a19}
\end{equation}
and 
\begin{equation}
U(R, 0) \vert (M,j) \mathbf{P}, \dot{b} \rangle =
\sum_{\dot{b}'} \vert (M,j) R\mathbf{P}, \dot{b}' \rangle
D^j_{\dot{b}'\dot{b}}[R]. 
\label{eq:a20}
\end{equation}
which is the standard rotational transformation law 
that leads to standard relation  
\begin{equation}
\mathbf{J} = \mathbf{X} \times \mathbf{P} + \mathbf{j}
\label{eq:a21}
\end{equation}
in the covariant representation.

In this representation the relation between the spin, angular momentum,
and orbital angular momentum add very much like the corresponding
non-relativistic quantities.  The price paid for this simplification
is that the Hilbert space inner product has a non-trivial kernel.
This kernel contains all of the dynamics effects discussed in the
context of Poincare irreducible spins.  The covariant spin is related
to the Poincar\'e irreducible spin of a particle by a boost.  For spin
1/2 particles the usual $u$ and $v$ spinors are direct sum
representations of a Lorentz boost.  The choice (helicity, canonical,
light front spin) appear in the representation of these spinors.

In the end there are many different kinds of spin observables.
In order to measure the spin we need to know how the various 
spin operators couple to the electroweak current operators. 
This will be different for each type of spin observable.

The conclusion is that for relativistic systems the coupling of spin
is more complicated than it is for non-relativistic systems.  It
requires determining the relation between the irreducible
representation of the Poincar\'e group associated with the hadron and
the irreducible representations of the Poincar\'e group associated
with the partons.  In all cases the relation between the 
hadronic and parton spins depends on the momenta of the 
partons in the hadron, which is determined by the dynamics.

In addition, there are many different spin and orbital angular
momentum observables.  How there different quantities contribute to
the hadronic spin is representation dependent.

\begin{thebibliography}{10}
\bibitem{moussa} P. Moussa and R. Stora (1965),
in Lectures in Theoretical Physics, 
Vol VIIA Lorentz Group, Ed.  W. E. Brittin and A. O. Barut, 
The University of Colorado Press, Boulder, CO pp 66-69.
\bibitem{bkwp} B. D. Keister and W. N. Polyzou (1991), Relativistic
Hamiltonian Dynamics, Advances in Nuclear Physics, 
Volume 20, p. 225, Ed J. Negele and
Erich Vogt, Plenum Press, New York-London .  
\bibitem{wgwp}W. N. Polyzou, W. Gl\"ockle and H. Wita{\l}a (2013),
Spin in relativistic quantum theory, Few-Body Systems, 54, p. 1667.
\bibitem{melosh}H. J. Melosh Phys (1974),
Quarks: Currents and constituents,
Rev. D {\bf 9}, p. 1095.
\bibitem{newton}T. D. Newton and E. P. Wigner (1949),
Localized States for Elementary Systems,
Rev. Mod. Phys. 21, p. 400.
\bibitem{wp}W. N. Polyzou (1989),
Relativistic Two-Body Models,
Annals of Physics, N.Y. 193, p. 367. 
\bibitem{fcwp} F. Coester and W. N. Polyzou (1982),
Relativistic Quantum Mechanics of Particles with Direct Interactions
Phys. Rev. D26, 1348. 
\bibitem{wp2} W. N. Polyzou (2010),
Examining the equivalence of Bakamjian-Thomas mass operators 
in different forms Phys. Rev. C 82,064001 .
\bibitem{weinberg}
Weinberg, S, (1964),
Feynman Rules for any spin, Phys. Rev. {\bf 133}, p. B1318-1332.  


\end{thebibliography}

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xxxxxxxxxxxxxxxxxxxxxxxxxxx


So far we have been discussing free particles; where the 
the kernel of the Hilbert space scalar product is the two-point 
Wightman function.  


xxxxxxxxxxxxxxxxxxxxxxxx 

complex conjugage representations are related by space reflection,
This means that if we want to leave 

Because of this it is customary to use doubled 
representations so the parity can be realized by a 
linear operator on these Lorentz covariant states


The nice thing about the Lorenz covariant representations is that 
the choice of boost and the complications assoicated with the 
orbital angular momentum have disappeared.  The complication is
that the Hilbert space inner product has a non-trivial kernel.

  Based on this 
it follows from the definitions that  
that the Hilbert space scalar product has the
form 
\begin{equation}
d^4P \theta(P^0) \delta (P^2 +
{M^2}) 
\left (
\begin{array}{cc}
D^j_{bb'}[P^{\mu}\sigma_{\mu}/{M}] & 0 \\
0 & D^{j}_{\dot{b} \dot{b}'}[P^{\mu}
\sigma_2\sigma^*_{\mu}\sigma_2/{M} ] 
\end{array}
\right ).
\label{eq:xx}
\end{equation}.  
This is essentially the two point Wightman functions for 
a particle with a given mass.  It has the advantage that choice of 
boost has disappeared however the appearance of the mass in this
expression indicates that this is a dynamical quantity.

conclusions

1.) specify what is measured

2.) dynamics can play a role

3.) things can be learned without dynamics.





\[
\psi ({P} , b) := \int \langle (m,j) \mathbf{P}, {b} \vert \psi \rangle
\]

\[
\langle \psi \vert \phi \rangle = 
\sumint \psi^* ({P} , b)
{d^4P \theta (P^0) \delta (P^2 +
{M}^2) D^j_{bb'}[{P^{\mu}}\sigma_{\mu}/{{M}}]
}
\phi ({P} , b')
\]


\[
\langle \psi \vert \phi \rangle = 
\sumint \psi^* ({P} , b)
\underbrace{d^4P \theta (P^0) \delta (P^2 +
{M}^2) D^j_{bb'}[P^{\mu}\sigma_{\mu}/{{M}}]
}_{\mbox{\bf no free lunch}}
\phi ({P} , b')
\]




\[
(B_x (V)B_x^{\dagger}(V) = B_c (V)B_c^{\dagger}(V) = B^2_c (V)
= P^{\mu}\sigma_{\mu}/M)   
\]

\[
\psi ({P} , b) \to 
\left (
\begin{array}{c} 
\xi ({P} , b) \\
\chi ({P} , \dot{b})
\end{array}
\right ) 
\]

\[
d^4P \theta (P^0) \delta (P^2 +
{M}^2) D^j_{bb'}[P^{\mu}\sigma_{\mu}/{{M}}]
\to 
\]

\[
d^4P \theta(P^0) \delta (P^2 +
{M^2}) 
\left (
\begin{array}{cc}
D^j_{bb'}[P^{\mu}\sigma_{\mu}/{M}] & 0 \\
0 & D^{j}_{\dot{b} \dot{b}'}[P^{\mu}
\sigma_2\sigma^*_{\mu}\sigma_2/{M} ] 
\end{array}
\right ).
\]

\[
\vert (m,j) \mathbf{P}, \dot{b} \rangle := \sum_{\mu}
\vert (m,j) \mathbf{P}, \mu \rangle_x 
D^j_{\mu \dot{b}}[B_x^{\dagger}(P/M)],
\]


\[
\langle \psi \vert I^{\mu} (0) \vert \phi \rangle  =
\langle 
\bar{\psi} \vert {A} I^{\mu} (0) {A^{\dagger}}  \vert \bar{\phi} \rangle  
\]

\[
_0\langle \mathbf{P}, \cdots \vert {\Omega}_{\pm} 
\vert \mathbf{P}_{as}, \cdots
\rangle = \delta( \mathbf{P}-\mathbf{P}_{as}) _0\langle \cdots \vert 
\hat{\Omega}_{\pm}(\mathbf{P}) \vert \cdots \rangle  ~,
\]

\[
\mathbf{X}_0 \bar{\Omega}^{\dagger}_{\pm} = \bar{\Omega}^{\dagger}_{\pm}
\mathbf{X}_{as}.
\]

\[
\mathbf{X}^{\#} {\Omega}^{\dagger}_{\pm} = {\Omega}^{\dagger}_{\pm}
\mathbf{X}_{as}.
\] 
\begin{center}
{\bf 
Lorentz invariance implies
}
\end{center}

\[
S = \bar{\Omega}^{\dagger}_+ \bar{\Omega}^{\dagger}_- =
{\Omega}^{\dagger}_+ {\Omega}^{\dagger}_- 
\]




 One problem is the since a boost to the rest frame followed
by rotation in the rest frame is still a boost,  there are an 
many spin observan




relations are modified by the dynamics,
because the  
Lorentz boost that relate these frames and generate rotations
that depend on the dynamics.   

generally involves the 
dynamics.    


Your text comes here. Separate text sections with
\section{Section title}
\label{sec:1}
Citations \cite{Ref1} and \cite{Ref2} and \cite{Ref3}.

spin crisis

role of relativity

cluster properties = dynamical constraints on spin

spin / boost / cg coefficients

big issuse with orbital 

mom dep


coupling to em - wigner eckart

cov rep

dynamics - wightman

composite spin

orbital not observable

role of dynamics

\[
W^{\mu} := -{1\over 2} \epsilon^{\mu\nu\alpha\beta}P_{\nu} J_{\alpha \beta}
\]
\vfill
\[
{M^2} = -P^{\mu}P_{\mu} 
\qquad  {W^2} = W^{\mu}W_{\mu} = M^2 \mathbf{j}^2
\qquad V^{\mu} = P^{\mu}/M 
\]
\[
B (V)^{\mu}{}_{\nu} (1,0,0,0)^{\nu} = V^{\mu}
\]
\[
{B_x (V)} = B (V) {R_x (V)}
\qquad   
B (V)^{\mu}{}_{\nu} 
= (V^{\mu},\underbrace{{e_1^{\mu}(V),e_2^{\mu}(V) e_3^{\mu}(V)}}_{
\mbox{polarization vectors}})
\]
\[
R_{xc} (V) = B_x^{-1}(V)B_c (V)
\]

\[
R_{wx} (\Lambda,V) = B_x^{-1} (\Lambda V)\Lambda B_x (V)
\]

\[
%\boxed{
P = P^{\mu} \sigma_{\mu} = 
\left ( 
\begin{array}{cc}
P^+ & P_\perp^*\\
P_\perp & P^-
\end{array}
\right ) 
\qquad
P^{\mu}  = {1 \over 2} \mbox{Tr}(\sigma_{\mu}P)
\]
\[
\mbox{det}(P) = (P^0)^2 -\mathbf{P}^2 = M^2
\]

\[
P' = \Lambda P \Lambda^{\dagger}  \qquad \mbox{det} (\Lambda ) =1 
\qquad 
\Lambda^{\mu}{}_{\nu} =
{1 \over 2} \mathbf{Tr}(\sigma_{\mu} \Lambda \sigma_{\nu} \Lambda^{\dagger})  
\]
\vfill
\[
e^{\mu}{}_{ix}(V)  =
{1 \over 2} \mathbf{Tr}(\sigma_{\mu} B_x(V) \sigma_{i} B_x(V)^{\dagger})  
\]

\[
%\boxed{
B_c (\mathbf{V}) := \sqrt{V^0 +1 \over 2} \sigma_0 + 
\hat{\mathbf{V}}\cdot {\boldsymbol \sigma}  \sqrt{V^0 -1 \over 2}.
%}
\]

\[
B_f (V) := 
\left (
\begin{array}{cc} 
\sqrt{V^+}  &0   \\
V_{\perp} /\sqrt{V^+} & 1/\sqrt{V^+}  
\end{array} 
\right )
\]

\[
B_h (\mathbf{V}) =
\]
\[
\left (\sqrt{{V^0+1\over 2}}\sigma_0 +
\sqrt{{V^0-1\over 2}}\hat{\mathbf{V}}\cdot {\pmb{\sigma}}
\right ) \times
\]
\[
\left (\sqrt{{1+\hat{\pmb{V}}\cdot \hat{\mathbf{z}}\over 2}}\sigma_0 +
\sqrt{{1- \hat{\pmb{V}}\cdot \hat{\mathbf{z}} \over 2}}
{(\hat{\mathbf{z}} \times \hat{\mathbf{V}}) \cdot \pmb{\sigma}
\over \sqrt{1-(\hat{\pmb{V}}\cdot \hat{\mathbf{z}})^2}}
\right ) 
\]

\[
R_{wf}(B_f(V') , V) = I 
\qquad R_{fc} = B_f^{-1}(V) B_c (V)
\]

\[
R_{wh}(\Lambda , V) = R_z (\phi)  
\qquad R_{hc} = R(\hat{\mathbf{z}}\to \hat{\mathbf{V}})
\]



\[
B_x (P \leftarrow P_0 ) B_x (P_0 \leftarrow p_{i0}) 
\]



\[
j_x^l := {1 \over 2} \epsilon^{lmn} 
{B_x^{-1} (V)^m{}_{\mu}}
{B_x^{-1} (V)^n{}_{\nu}}J^{\mu \nu} =
\]
\[
{1 \over 2} \epsilon^{lmn} 
{e_x^{m}{}_{\mu}(V)
e_x^{n}{}_{\nu}(V)} J^{\mu \nu}
\]

\[
(0,\mathbf{j}_x)^{\mu}
= {1 \over M} B_x^{-1} (V)^{\mu}{}_{\nu} W^{\nu} =
{1 \over M} e_x^{\mu}{}_{\nu}(V)  W^{\nu}
\]

\[
(0,\mathbf{j}_f)^{\mu}
= {1 \over M} B_f^{-1} (V)^{\mu}{}_{\nu} W^{\nu}
\]


\[
(0,\mathbf{j}_h)^{\mu}
= {1 \over M} B_h^{-1} (V)^{\mu}{}_{\nu} W^{\nu}
\]

\[
[P^{\mu} , W^{\nu} ]=0
\qquad
[W^{\mu} , W^{\nu} ] = i \epsilon^{\mu \nu \alpha \beta} W_\alpha P_\beta
\]

\[
j^2 := W^2/M^2 = j_x^2 \qquad [j_{xl},j_{xm}] = i \epsilon^{lmn} j_{xn}
\]
\vfill
\[
[j_{xm} ,P^{\mu}]=0
\]
\[
\{ M, \mathbf{j}^2, \mathbf{P}, \hat{\mathbf{z}}\cdot {\mathbf{j}_x}\}
\]
\vfill
\[
\vert (M,j)\mathbf{P},{\mu} \rangle_x = 
U(B_x(P/M),0)  \vert (M,j)\mathbf{0} ,{\mu} \rangle_x 
\]
\vfill
\[ 
U(R,0)\vert (M,j)\mathbf{0},\mu \rangle_x = 
\sum \vert (M,j)\mathbf{0} ,\nu \rangle_x 
{D^j_{\nu \mu}(R)}
\]
\vfill
\[
U(\Lambda ,0) \vert (M,j)\mathbf{P},\mu \rangle_x = 
\sum \vert (M,j)\pmb{\Lambda}P,\nu \rangle_x 
D^j_{\nu \mu}(R_{wx}(\Lambda,P))
\]

\[
\mathbf{j}_x = B^{-1}_x (V)B_y (V)  \mathbf{j}_y = 
R_{xy} (V={P/M}) \mathbf{j}_y
\]

\[
[\pmb{\nabla}_{P_{\vert_{j_x}}}, \mathbf{j}_x]=0 \,
{\not\Rightarrow} \, [\pmb{\nabla}_{P_{\vert_{j_x}}}, \mathbf{j}_y]=0 
\]

\[
X^k_x= 
{i \pmb{\nabla}_{\mathbf{P}_{\vert_{\mathbf{j}_x}}}}
=
-{1 \over 2} \{H^{-1},{K}^k\} + i H^{-1} C_{1x}^{kl}(\mathbf{V}) j_x^l 
\]

\vfill
\[
J^j = (\mathbf{X}_x \times \mathbf{P})^j + C_{2x}^{jk} (\mathbf{V})j_x^k 
\]

\[
C_{1x}^{jk} =
{1 \over 2} \mathbf{Tr} [B_x(\mathbf{V})^{-1}\sigma_j B_x(\mathbf{V}) \sigma_k]
- q^0 \mathbf{Tr} [B_x(\mathbf{V})^{-1}{\partial \over \partial V_l}  
B_x(\mathbf{V}) \sigma_m]
\]

\vfill
\[
C_{2x}^{jk} (\mathbf{\mathbf{V}}) = 
{1 \over 2} \mathbf{Tr} [B_x(\mathbf{V})^{-1}\sigma_j B_x(\mathbf{V}) \sigma_k]
+
i \epsilon_{jlm} 
\mathbf{Tr} [B_x(\mathbf{V})^{-1}{\partial \over \partial V_l}  
B_x(\mathbf{V}) \sigma_m]
\]

\[
\mathbf{J} = {\mathbf{j}_c} + {\mathbf{X}_c} \times \mathbf{P}  
\]
\vfill
\[
{\mathbf{X}_c} = \frac{1}{2} \{ \frac{1} {{H}} ,{\mathbf{K}} \} -
\frac{\mathbf{P} \times ({H}\mathbf{J} + 
\mathbf{P} \times {\mathbf{K}})}{{MH(M+H)}}.
\]

\[
\mathbf{J}\cdot \hat{\mathbf{z}}  = \mathbf{j}_f \cdot \hat{\mathbf{z}} -
{\hat{\mathbf{z}} \over P^+} \cdot (\mathbf{P} \times \mathbf{E}) 
\]
\vfill
\[
{\mathbf{J}_{\perp}} = 
\]
\vfill
\[
{1 \over P^+}
[ {(P^+-{P^-})\over 2}\hat{\mathbf{z}} \times \mathbf{E} - 
(\hat{\mathbf{z}} \times \mathbf{P}) \mathbf{K}\cdot \hat{\mathbf{z}}
+ \mathbf{P}_{\perp}(\hat{\mathbf{z}}\cdot \mathbf{j}_f) +
{M}\mathbf{j}_{\perp f} 
]   
\]

\[
e \int J^{\mu} (x) A_{\mu}(x) d\mathbf{x} 
\]

\[
_x\langle (M',j')\mathbf{P}',\mu' \vert
I^{\mu} (x)
\vert (M,j)\mathbf{P},\mu \rangle_x
\]


\[
U(\Lambda, 0) \vert (m,j) \mathbf{P}, \mu \rangle_x
 = 
\sum_{\mu'} \vert (m,j) \pmb{\Lambda}P , \mu' \rangle_x  
D^j_{\mu' \mu}[{B_x^{-1} (\Lambda P/M)}\Lambda 
{B_x ({P}/M)}].
\]



\[
\vert (m,j) \mathbf{P}, b \rangle := \sum_{\mu'}
\vert (m,j) \mathbf{P}, \mu' \rangle_x
{D^j_{\mu' b}[B_x^{-1} (P)]} .
\]


\[
{U(\Lambda, 0)} \vert (m,j) \mathbf{P}, {b} \rangle =
\sum_{\dot{b}'} \vert (m,j) \pmb{\Lambda}{P}, {b}' \rangle
{D^j_{{b}'{b}}[\Lambda]}. 
\]


\[
SU(2): \quad R = (R^{\dagger})^{-1}
\qquad SL(2,\mathbb{C}): \quad
\Lambda  \not\sim (\Lambda^{\dagger})^{-1}
\]


\[
\vert (m,j) \mathbf{P}, \dot{b} \rangle := \sum_{\mu}
\vert (m,j) \mathbf{P}, \mu \rangle_x 
D^j_{\mu \dot{b}}[B_x^{\dagger}(P/M)],
\]


\[
\vert (m,j) \mathbf{P}, \dot{b} \rangle := \sum_{\mu}
\vert (m,j) \mathbf{P}, \mu \rangle_x 
D^j_{\mu \dot{b}}[B_x^{\dagger}(P/M)],
\]


\[
{U(\Lambda, 0)} \vert (m,j) \mathbf{P}, \dot{b} \rangle =
\sum_{\dot{b}'} \vert (m,j) \pmb{\Lambda}{P}, \dot{b}' \rangle
{D^j_{\dot{b}'\dot{b}}[(\Lambda^{\dagger})^{-1}]}. 
\]

%slide 18

\[
U(R, 0) \vert (M,j) \mathbf{P}, {b} \rangle =
\sum_{\dot{b}'} \vert (M,j) R\mathbf{P}, {b}' \rangle
D^j_{{b}'{b}}[R]. 
\]
\[
U(R, 0) \vert (M,j) \mathbf{P}, \dot{b} \rangle =
\sum_{\dot{b}'} \vert (M,j) R\mathbf{P}, \dot{b}' \rangle
D^j_{\dot{b}'\dot{b}}[R]. 
\]

\[
\mathbf{J} = \mathbf{X} \times \mathbf{P} + \mathbf{j}
\]

\[
\psi ({P} , b) := \int \langle (m,j) \mathbf{P}, {b} \vert \psi \rangle
\]


\[
\langle \psi \vert \phi \rangle = 
\sumint \psi^* ({P} , b)
\underbrace{d^4P \theta (P^0) \delta (P^2 +
{M}^2) D^j_{bb'}[P^{\mu}\sigma_{\mu}/{{M}}]
}_{\mbox{\bf no free lunch}}
\phi ({P} , b')
\]

\[
(B_x (V)B_x^{\dagger}(V) = B_c (V)B_c^{\dagger}(V) = B^2_c (V)
= P^{\mu}\sigma_{\mu}/M)   
\]

\[
\psi ({P} , b) \to 
\left (
\begin{array}{c} 
\xi ({P} , b) \\
\chi ({P} , \dot{b})
\end{array}
\right ) 
\]

\[
d^4P \theta (P^0) \delta (P^2 +
{M}^2) D^j_{bb'}[P^{\mu}\sigma_{\mu}/{{M}}]
\to 
\]

\[
d^4P \theta(P^0) \delta (P^2 +
{M^2}) 
\left (
\begin{array}{cc}
D^j_{bb'}[P^{\mu}\sigma_{\mu}/{M}] & 0 \\
0 & D^{j}_{\dot{b} \dot{b}'}[P^{\mu}
\sigma_2\sigma^*_{\mu}\sigma_2/{M} ] 
\end{array}
\right ).
\]

\[
\Lambda B_x (P) = 
B_x (\Lambda P) R_{wx} (\Lambda ,P) ~,
\]

\[
S(\Lambda ) = 
\left (
\begin{array}{cc}
D^j(\Lambda)  & 0 \\
0 & D^j((\Lambda^{\dagger})^{-1})  
\end{array}
\right ).
\] 

\[
U(\Lambda ) \Psi (x) U^{\dagger} (\Lambda)
= S(\Lambda^{-1}) \Psi (\Lambda x) 
\]


\[
U(\Lambda ) b_x^{\dagger} (\mathbf{P},\mu) ) U^{\dagger} (\Lambda)
= b_x^{\dagger} (\pmb{\Lambda} P,\nu)
D^j_{\nu\mu}[R_{wx}(\Lambda ,P)]  
\]


\[
\Psi_\gamma (x) = \int d\mathbf{p} (e^{iP\cdot x}
u_{\gamma x} (P,\mu) a_x(\mathbf{P},\mu) +  
e^{-iP\cdot x}
v_{\gamma x} (P,\mu) b_x^{\dagger}(\mathbf{P},\mu))
\]


\[
u_{\gamma x}(P) ={\omega_M (P) \over M} S(B_x(P/M))_{\gamma\gamma'} 
u_{\gamma'}(0)
\]


\[
S(\Lambda) u_x (P) = u_x (\Lambda p) R_{wx}(\Lambda ,P) 
\]


\[
(j_1,M_1) \otimes (j_2,M_2) = \int_{\oplus} (j,M) 
\]

\[
_x\langle  (M_1,j_1) \mathbf{P}_1 ,\mu_1  (M_2,j_2) \mathbf{P}_2 ,\mu_2 
\vert k, j (M_1,j_1,M_2,j_2) \mathbf{P} ,  \mu , l , s_{12} \rangle_x =
\]

\[
\sum_{\mu_1',\mu_2',\mu_1'',\mu_2'',\mu_s,m}
\delta (\mathbf{P} - \mathbf{P}_1-\mathbf{P}_2)
{\delta (k -k(\mathbf{P}_1,\mathbf{P}_2)) \over k^2}
\times
\]

\[
\sqrt{{\omega_{M_1} (\mathbf{k})\omega_{M_2} (\mathbf{k})
\over \omega_{M_1} (\mathbf{k})+\omega_{M_2} (\mathbf{k})}}
\sqrt{
(\omega_{M_1} (\mathbf{k})+\omega_{M_2} (\mathbf{k}))^2+
\mathbf{P}^2} \times
\]

\[
{D^{j_1}_{\mu_1 \mu_1'}[R_{wx} (B_x(V) ,k_1)]
D^{j_1}_{\mu_1'\mu_1''} [R_{xc}(k_1)]} \times
\]

\[
{D^{j_2}_{\mu_2 \mu_2'}[R_{wx} (B_x(V) ,k_2)]
D^{j_2}_{\mu_2'\mu_2''} [R_{xc}(k_2)]}
\times 
\]

\[
Y^{l}_{m} (\hat{\mathbf{k}}(\mathbf{P}_1,\mathbf{P}_2)) 
\langle j_1, \mu_1'', j_2, \mu_2'' \vert s_{12}, \mu_s \rangle
\langle l, m, s_{12}, \mu_s \vert j, \mu \rangle
\]

\[
\mathbf{j}_{cs} = {\mathbf{l}_c} +  
R_{cx} (k_1)R_{wx}(B_x^{-1}(P),P_1) \mathbf{j}_{1x} +                    
R_{cx} (k_2)R_{wx}(B_x^{-1}(P),P_2) \mathbf{j}_{2x} 
\]

\[
\mathbf{j}_{cs1} := R_{cx} (k_1)R_{wx}(B_x^{-1}(P),P_1) \mathbf{j}_{1x}
\]

\[
\mathbf{j}_{cs2} := R_{cx} (k_2)R_{wx}(B_x^{-1}(P),P_2) \mathbf{j}_{2x}
\]

\[
j_{cs}^l = {1 \over 2} \epsilon^{lmn} 
{B_x^{-1} ({P_1+P_2\over M_{012}})^m{}_{\mu}}
{B_x^{-1} ({P_1+P_2\over M_{012}})^n{}_{\nu}}(J_1+ J_2)^{\mu \nu} =
\]

\[
(j_1,M_1) \otimes (j_{023},M_{023}) = \int_{\oplus} (j,M_0) 
\]



\[
(j_1,M_1) \otimes (j_{23},{M_{23}}) = \int_{\oplus} 
(j,{M}) 
\]

 
\[
\mathbf{J} = {\mathbf{j}_c} + {\mathbf{X}_c} 
\times \mathbf{P}
\]
 
\[
\mathbf{X}_c
 = \frac{1}{2} \{ \frac{1}{H} ,\mathbf{K} \} -
\frac{\mathbf{P} \times (H\mathbf{J} + \mathbf{P} \times 
\mathbf{K})}{MH(M+H)}.
\]

\[
_0\langle \mathbf{P}, \cdots \vert {\Omega}_{\pm} 
\vert \mathbf{P}_{as}, \cdots
\rangle = \delta( \mathbf{P}-\mathbf{P}_{as}) _0\langle \cdots \vert 
\hat{\Omega}_{\pm}(\mathbf{P}) \vert \cdots \rangle  ~,
\]

\[
\mathbf{X}_0 = i \pmb{\nabla}_{P_{j_{c0}}}
\]

\[
_0\langle \mathbf{P}, \cdots \vert \bar{\Omega}_{\pm} 
\vert \mathbf{P}_{as}, \cdots
\rangle := \delta( \mathbf{P}-\mathbf{P}_{as}) _0\langle \cdots \vert 
\hat{\Omega}_{\pm}(\mathbf{0}) \vert \cdots \rangle  ~,
\]

\[
\langle \mathbf{P}_{as}, \cdots \vert S
\vert \mathbf{P}'_{as}, \cdots \rangle =
\delta( \mathbf{P}-\mathbf{P}_{as})
\langle \cdots \vert \hat{\Omega}_{+}(\mathbf{P})^{\dagger}
\hat{\Omega}_{-}(\mathbf{P}) 
\vert \cdots \rangle =
\]

\[ 
\delta( \mathbf{P}-\mathbf{P}_{as})
\langle \cdots \vert \hat{\Omega}_{+}(0)^{\dagger}
\hat{\Omega}_{-}(0) 
\vert \cdots \rangle
\]


\[
S = \bar{\Omega}^{\dagger}_+ \bar{\Omega}^{\dagger}_- =
{\Omega}^{\dagger}_+ {\Omega}^{\dagger}_- 
\]

\[
S= \bar{\Omega}^{\dagger}_+ \bar{\Omega}^{\dagger}_- =
{\Omega}^{\dagger}_+ {\Omega}^{\dagger}_- 
\]

\[
A := \bar{\Omega}^{\dagger}_- {\Omega}_- = \bar{\Omega}^{\dagger}_+ {\Omega}_+
\]


\[
\bar{U}(\Lambda,a) := A U(\Lambda,a) A^{\dagger} \qquad \bar{S} = S
\]

\[
\mathbf{P}_0 A = A \mathbf{P}_0 \qquad 
\mathbf{J}_0 A = A \mathbf{J}_0 \qquad
\mathbf{X}_0 A = A \mathbf{X} .
\]

\[
\bar{\mathbf{j}}_c = \mathbf{J}_0 - \mathbf{X}_0 \times \mathbf{P}_0 =
\mathbf{j}_{c0} 
\]

 
\[
\langle \psi \vert I^{\mu} (0) \vert \phi \rangle  =
\langle 
\bar{\psi} \vert {A} I^{\mu} (0) {A^{\dagger}}  \vert \bar{\phi} \rangle  
\]

\[
_0\langle \mathbf{P}, \cdots \vert {\Omega}_{\pm} 
\vert \mathbf{P}_{as}, \cdots
\rangle = \delta( \mathbf{P}-\mathbf{P}_{as}) _0\langle \cdots \vert 
\hat{\Omega}_{\pm}(\mathbf{P}) \vert \cdots \rangle  ~,
\]

\[
\mathbf{X}_0 \bar{\Omega}^{\dagger}_{\pm} = \bar{\Omega}^{\dagger}_{\pm}
\mathbf{X}_{as}.
\]

\[
\mathbf{X}^{\#} {\Omega}^{\dagger}_{\pm} = {\Omega}^{\dagger}_{\pm}
\mathbf{X}_{as}.
\] 
\begin{center}
{\bf 
Lorentz invariance implies
}
\end{center}

\[
S = \bar{\Omega}^{\dagger}_+ \bar{\Omega}^{\dagger}_- =
{\Omega}^{\dagger}_+ {\Omega}^{\dagger}_- 
\]







\subsection{Subsection title}
\label{sec:2}
as required. Don't forget to give each section
and subsection a unique label (see Sect.~\ref{sec:1}).
\paragraph{Paragraph headings} Use paragraph headings as needed.
\begin{equation}
a^2+b^2=c^2
\end{equation}

% For one-column wide figures use
\begin{figure}
\centering
% Use the relevant command to insert your figure file.
% For example, with the graphicx package use
  \includegraphics{example.eps}
% figure caption is below the figure
\caption{Please write your figure caption here}
\label{fig:1}       % Give a unique label
\end{figure}
%
% For two-column wide figures use
\begin{figure*}
\centering
% Use the relevant command to insert your figure file.
% For example, with the graphicx package use
  \includegraphics[width=0.75\textwidth]{example.eps}
% figure caption is below the figure
\caption{Please write your figure caption here}
\label{fig:2}       % Give a unique label
\end{figure*}
%
% For tables use
\begin{table}[t]
% table caption is above the table
\caption{Please write your table caption here}
\centering
\label{tab:1}       % Give a unique label
% For LaTeX tables use
\begin{tabular}{lll}
\hline\noalign{\smallskip}
first & second & third  \\[3pt]
\tableheadseprule\noalign{\smallskip}
number & number & number \\
number & number & number \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}


%\begin{acknowledgements}
%If you'd like to thank anyone, place your comments here
%and remove the percent signs.
%\end{acknowledgements}

% BibTeX users please use
%\bibliographystyle{spbasic}
%\bibliography{}   % name your BibTeX data base

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\begin{thebibliography}{3}
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% and use \bibitem to create references. Consult the Instructions
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%
% Format for Journal Reference
\bibitem[Author I(1999)]{Ref1}
Author I (year) Article title. Journal Title-Abbreviated Vol: pp--pp
% Format for books
\bibitem[Author and Smith(2001)]{Ref2}
Author I, Smith J (year) Book title. Publisher, Place, pp numbers
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