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boosts,rotations,clifford algebras, geometric product, bounded symmetric domains, symmetry in physics, symmetric velocity, relativistic dynamic equation
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\documentclass[12pt]{article}
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\begin{document}
\title{Relativistic Dynamic Equation in Invariant Form}
\author{Yaakov Friedman*\\
Yuriy Gofman\\
Jerusalem College of Technology\\
P.O.Box 16031\\
Jerusalem 91160 Israel\\
e-mail:friedman@mail.jct.ac.il gofman@mail.jct.ac.il}
\footnotetext[1]{The first author was partially supported by NSF
grant no. DMS-0101153.}
\maketitle
\bigskip
\bigskip
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\begin{abstract}
In some dynamical systems boosts and
rotations occur synchronically. Usually in analysis of such
systems we separate these two types of motion. Unfortunately, the
result of such analysis depends on the order in which operations
are performed.
To avoid the order dependence of the above operations, we propose
a new dynamic variable called the symmetric velocity. This new
velocity could be calculated directly from the regular velocity
and is its relativistic half. The set of all possible symmetric
velocities is a three dimensional ball of radius \textit{c} and
the Lorentz group acts on this ball via conformal maps. The
generators of these maps (elements of the Lie algebra) are second
order transformations expressed by a triple product. This triple
product is the one corresponding to the Bounded Symmetric Domain
of type 4 in Cartan's classification, also called the spin
factor. The product is connected with the Geometric (Clifford)
product, explaining why use of the geometric product simplifies
formulae in several areas of physics.
\end{abstract}
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\section{Introduction}
This paper provides another indication that bounded symmetric
domains and JB*-triples may provide a model for different areas in
physics. A bounded domain $D$ in a Banach space is called a {\it
bounded symmetric domain} if for every $z\in D$ there exists a
smooth automorphism $s_z \in Aut(D)$ of period two on $D$, having
$z$ as the only fixed point. The smoothness of the automorphism
may mean complex analytic, conformal (preserving angles) or
projective (preserving linear segments). It is known that a domain
$D$ is a bounded symmetric domain if it has a symmetry about one
point and is homogeneous in sense that for any two points
$z,w \in D$%
there is an automorphism
$\varphi \in Aut(D)$ such that
$\varphi (z)=w$.
There is a triple product uniquely associated with any bounded
symmetric domain $D$ in a Banach space $A$, obtained as follows
(see \cite{L77} for details): By fixing any point in $D$ (we may
assume for simplicity that this is the zero point of $A$) we may
decompose $Aut(D)$ into rotations and translations. This implies
that the Lie algebra $aut(D)$ will be a direct sum of linear
terms, as generators of rotations, and generators of translations.
It was shown (see \cite{L77} and \cite{K83}) that the generators
of translations are of the form
\begin{equation}\label{xi1}
\xi _a(z)=a-q_a(z), \;\;\; z\in A
\end{equation}
where $q_a(z)$ is quadratic in $z$ and is conjugate linear in $a$.
The quadratic form $q_a(z)$ could be rewritten as a real trilinear
form $q_a(z)=\{z,a,z\}$ and by linearization of the quadratic
dependence in $z$ (polarization) we define a triple product on $A$
with the following properties:
\begin{itemize}
\item[(i)] $\{a,b,c\}$ is linear in $a,c$ and conjugate linear in
$b$,
\item[(ii)] $\{a,b,c\}=\{c,b,a\}.$
\end{itemize}
The operator $D(a,b)$ defined by
\begin{equation}\label{D}
D(a,b)c=\{a,b,c\}, \;\;\; c\in A
\end{equation}
satisfies:
\begin{itemize}
\item[(iii)] the operator $D(a,a)$ is a hermitian with positive
spectrum,
\item[(iv)]
\begin{equation}\label{mainid}
D(a,a)\{x,y,z\}=\{D(a,a)x,y,z\}-\{x,D(a,a)y,z\}+\{x,y,D(a,a)z\},
\end{equation}
for any $a,x,y,z\in A$ and
\item[(v)] $\|a\|^3=\|\{a,a,a\}\|.$
\end{itemize}
A Banach space $A$ with a triple product satisfying the above
properties is called a {\it JB*-triple}. In \cite{K83} it is shown
that the category of bounded symmetric domains is equivalent to
the category of JB*-triples.
Any bounded symmetric domain could be decomposed into
indecomposable domains, called Cartan factors.
There are six types of Cartan factors
and the JB*-triples associated
with them (\cite{L77} and \cite{DF87}). The type 1 consists of
bounded operators between two Hilbert spaces. The types 2 and 3 consist
of anti-symmetric and symmetric complex matrices, respectively,
representing bounded operators on a Hilbert space.
In all above types the triple
product is defined by
\begin{equation}\label{jctrip}
\{a,b,c\}=\frac{ab^*c+cb^*a}{2},
\end{equation}where $b^*$ is the adjoint operator to $b$.
The types 5 and 6 are
exceptional of dimensions 16 and 27 respectively, and we will not
be concerned with them here.
The Cartan factor of type 4, sometimes called a {\it spin factor}
will be mainly used here. This factor is defined as the space
$R^n$ or $C^n$ with the triple product
\begin{equation}\label{8}
\{a,b,c\}=c+a-\overline{b}.
\end{equation}
As it was shown in \cite{FR86} any JB*-triple is isomorphic to
the sum of an exceptional part and a subspace of operators on a
Hilbert space with the triple product (\ref{jctrip}).
Why are the bounded symmetric domains and JB*-triples a good model
for physics? Any law in physics must satisfy the symmetry or
invariance principle. This principle states that the law should
not change if we change the point from which we observe the
phenomena. Such a principle imposes homogeneity of the state space
of the system, that could be expressed as a symmetry of the domain
representing the possible states of the system. This suggest that
the state space is a symmetric domain. In order to be able to
obtain numerical results we need an algebraic structure. Currently
there is only one algebraic structure that has an origin only in
geometry - the JB*-triple product.
In \cite{FN92} and \cite{F94} it was shown that bounded symmetric
domains occur in Transmission Line theory and in Special
Relativity. Note that the state space of any quantum system must
contain some geometry coming from the measuring process. In
\cite{FR92} and \cite{FR93} it was shown that this geometry
implies that the state space of the system is a bounded symmetric
domain. G\"{u}naydin noticed \cite{G} that all physically
meaningful quantities in Quantum Mechanics depend only on the
Jordan triple product rather than on the binary one. In
\cite{FR01} it was shown that the Canonical Anticommutation
Relations could be efficiently represented on the spin factor and
the Lorentz group is represented on this factor by spin 1 and spin
1/2 representations.
\section{ The meaning of the Einstein velocity addition formula}
One of the assumptions of Special Relativity is that the
velocities of moving objects are bounded by the speed of light
\textit{c}. Thus all possible velocities in any inertial frame
form a 3D ball of radius \textit{c}. If we consider two inertial
frames, then for each velocity in one inertial frame there is
assigned a unique velocity in the second frame. This assignment
generates a smooth
transformation of the ball of possible velocities. The
transformation is projective, meaning preserving line intervals,
but not preserving angles.
The precise formula for this transformation is given by Einstein's
velocity addition formula. To understand this formula consider an
inertial frame K and another inertial frame K' that moves with
velocity $u$ relatively to frame K. Assume that there is an object
moving with constant velocity $v'$ in frame K'. Consider two
events $A(r'_A,t'_A)$ and $B(r'_B,t'_B)$ expressing that the
object was at time $t'_A$ in position $r'_A$ and similarly for
$B$. The space interval between these events in K' is $\triangle
r'=r'_B-r'_A$. If the clocks at the points $A$ and $B$ were
synchronized in K', the time interval between these events is
$\triangle t'= t'_B-t'_A$ and $v'=\triangle r' /\triangle t'$.
In order to be able to calculate the time interval of the above
events and the velocity of the object in system K, we must
synchronize first the clocks, moving together with K', at the two
places $r'_A$ and $r'_B$ with respect of the frame K. Let us
denote by $\hat{t'}_B$ the time at clock at $r'_B$ that was
synchronized to the clock at $r'_A$ with respect of the K. The
difference between the times of occurrence of event $B$ on the two
clocks positioned at $r'_B$ is given by the formula for amount of
nonsimultaneity at a displacement $\triangle r'$
$$\hat{t'}_B -t'_B =\frac{1}{c^2}<\triangle r'|u>$$
and the time interval between the events measured by clocks in K'
but synchronized with respect to K is
$$\triangle\hat{t'}=\hat{t'}_B -t'_A=\frac{1}{c^2}<\triangle r'|u>+\triangle t'
=(1+/c^2)\triangle t'.$$
Thus, the time interval between
these events in K will be
$$\triangle t=\frac{\triangle\hat{t'}}{\sqrt{1-|u|^2/c^2}}=
\frac{1+/c^2}{\sqrt{1-|u|^2/c^2}}\triangle t'.$$
Note that the space interval between these events in frame K is
$$\triangle r= u\triangle t +((\sqrt{1-|u|^2/c^2})P_u +(I-P_u))\triangle r'$$
where $P_u$ denotes the orthogonal projection onto direction of
$u$. This lead to the formula for velocity of the object in K
$$ v=\frac{\triangle r}{\triangle t}= u+((\sqrt{1-|u|^2/c^2})P_u +(I-P_u))
\frac{\triangle r'}{\triangle
t'}\frac{\sqrt{1-|u|^2/c^2}}{1+/c^2}=$$
$$ u+((1-|u|^2/c^2)P_u +\sqrt{1-|u|^2/c^2}(I-P_u))
\frac{v'}{1+/c^2}.$$
The above formula defines the transformation $\varphi_u$, called
boosts, of the ball of possible velocities that, by assuming
$c=1$, is given by
\begin{equation}\label{velad}
v=\varphi_u (v')=u +((1-|u|^2)P_u +\sqrt{1-|u|^2}
(I-P_u))\frac{v'}{1+}.
\end{equation}
The set of boosts do not form a group but rather a gyrogroup ( see
\cite{U01} for definitions) but together with rotations they form
a group representing the Lorentz group by projective maps on the
velocity ball. How do the boosts and rotations interact? If we
rotate first the ball by a $\vartheta$ and then perform the boost
in direction $\vartheta u$ we get a map $\varphi_{\vartheta
u}(\vartheta v')$. From (\ref{velad}) we can see that
\begin{equation}\label{inisotropy}
\varphi_{\vartheta u}(\vartheta v')=\vartheta\varphi_{ u}(u')
\end{equation}
only if $\vartheta$ commutes with $(1-|u|^2)P_u +\sqrt{1-|u|^2}
(I-P_u)$. This happen only if the rotation $\vartheta$ is about
the $u$ direction. The failure of (\ref{inisotropy}) is a result
of the space anisotropy under the Lorentz transformations.
\section{Symmetric velocity, conformal group and spin triple
product}
In some dynamical systems the boosts and rotation occur
synchronically. Usually in analysis of such systems we separate
these two types of operations. Unfortunately, the result of such
analysis depends on the order the separations were performed. To
avoid the dependence on the order of the above operations we
propose a new dynamic variable, called the symmetric velocity. The
symmetric velocity was introduced in \cite{FN92} in order to
transform the the transformations $\varphi _u$ to a conformal one.
The symmetric velocity $w$ and velocity $v$ are connected by
\begin{equation}\label{3}
w=F(v)=\frac{v}{1+\sqrt{1-|v|^2/c^2}},\;\:\; v=F^{-1}=\frac{2w}{1+|w|^2/c^2}.
\end{equation}
The symmetric velocity is the relativistic half of the regular
velocity. The set of all possible symmetric velocities in any
inertial frame form a 3D ball $S$ of radius \textit{c}. For
simplicity we will assume from now $c=1$. As it was shown the map
$\psi=F\varphi F^{-1}$ is a conformal map of the ball. A similar
result was obtained by A.Ungar in 1996 (see \cite{U01}) in the
study of so called M\"{o}bius gyrovector space.
An explicit form of the conformal map of the ball was proposed by
Ahlfors in1981 and is given by the formula (see \cite{U01})for the
extended M\"{o}bius transformation
\begin{equation}\label{4}
\psi_u(w)=u \oplus w=\frac{(1+2__ +|w|^2)u
+(1-|u|^2)w}{1+2____+|u|^2|w|^2},\;\;\; u,w\in S
\end{equation}
where $\oplus$ will denote the sum of symmetric velocities.
Note that in this
case if we rotate first the ball by a $\vartheta$ and then perform
the boost in direction $\vartheta u$ we get
\begin{equation}\label{5.1}
\psi_{\vartheta u}(\vartheta w)=\vartheta\psi_{ u}(w)\end{equation}
or
\begin{equation}\label{5}
\vartheta(u \oplus w)=\vartheta u\oplus\vartheta w.
\end{equation}
This can be expressed as the commuting of the following diagram,
see FIG. 1.
\begin{figure}[h]
\begin{displaymath}
\begin{array}{ccc}
S\times S & \longrightarrow ^{\oplus} & S \\
\vartheta\downarrow & & \vartheta\downarrow \\
S \times S& \longrightarrow ^\oplus & S
\end{array}
\end{displaymath}
\caption{Commuting of symmetric addition and rotation}\label{commuting}
\end{figure}
Thus, in case the dynamics of the system involve both boosts and
rotations the use of symmetric velocities may simplify the
description of the evolution and lead to invariant (under the
order of decomposition into boosts and rotations) dynamic
equations.
If the evolution of the system is described by conformal maps of
the symmetric velocities, the dynamic equation will involve the
generators of such maps. To obtain the generators of the boosts we
have to take a one-parameter family $\psi _{u (\tau)}$ depending
on some real parameter $\tau$ with $u (0) =0$. Then the generator
is given by
\begin{equation}\label{6}
\xi_a(w)=\frac{d}{d\tau}\psi_{u(\tau)}(w)|_{\tau=0}=a-2w +|w|^2a
\end{equation}
with $a=\frac{d}{d\tau} u(\tau)|_{\tau=0}$. This is a general
formula for generators of translation in the Lie algebra
(consisting of the generators of the group) of a bounded symmetric
domain \cite{L77}. As mentioned above, the generators of the
translations are of the form
\begin{equation}\label{7}
\xi_a(w)=a-\{w,a,w\},\end{equation}
with $\{w,a,w\}$ as the triple product associated
with the bounded symmetric domain. In our case the domain is a
real domain of type 4 and dimension 3 in Cartan's classification,
called also the Spin factor with the triple product defined by
(\ref{8}). The norm is the usual Euclidean norm. The ball of
radius one is homogeneous under the group of conformal maps
(\ref{4}).
The Lie algebra of the conformal group consists of generators of
boosts described by (\ref{6}) and (\ref{7}) in terms of the triple
product and of generators of rotations. To describe the generators
of rotations in the symmetric velocity ball we chose first an
orthonormal basis $e_1,e_2,e_3$ in $R^3$ and define
$$\bar{D}=(D(e_2,e_3),D(e_3,e_1),D(e_1,e_2)),$$
where the linear operator $D(a,b)$ is defined by (\ref{D}). Then
the generator of rotation for any symmetric velocity $w$ is
expressed in the triple product by
$$\vartheta (w)=(H\cdot \bar{D}) (w)=H\times w:\;\;H\in R^3.$$
Thus, any element $\zeta$ of our Lie algebra is of the form
\begin{equation}\label{liealg}
\zeta=\zeta_{a,H}(w)=a +(H\cdot \bar{D})(w) -\{w,a.w\} \;\; a,H\in R^3
\end{equation}
and is expressed in terms of the triple product.
Note that also (\ref{4}) could be obtained in terms of the triple
product by exponentiating (\ref{7}) and using the explicit form
for this exponent from \cite{K83}.
\section{Relativistic dynamic equations using symmetric
velocity}
We are going to describe now how to obtain the relativistic
dynamic equation for the symmetric velocities. From the definition
of the symmetric velocity
\begin{equation}\label{9}
\gamma=\frac{1}{\sqrt{1-|v|^2}}=\frac{1+|w|^2}{1-|w|^2}.
\end{equation}
and thus
\begin{equation}\label{10}
mv=m_0\gamma v=m_0\frac{2w}{1-|w|^2}\end{equation}
with $m_0$ the rest mass of the object. The relativistic dynamic
equation $$F=\frac{d}{dt}(mv)$$ now becomes
\begin{equation}\label{11}
F=\frac{d}{dt}m_0\frac{2w}{1-|w|^2}=2m_0(\frac{1}{1-|w|^2}
\frac{dw}{dt}+\frac{2w}{(1-|w|^2)^2}<\frac{dw}{dt}|w>).
\end{equation}
By taking the inner product with $w$ we get
\begin{equation}\label{12}
=2m_0<\frac{dw}{dt}|w>\frac{1+|w|^2}{(1-|w|^2)^2}.
\end{equation}
By substituting $<\frac{dw}{dt}|w>$ from (\ref{12}) into
(\ref{11}) we obtain
\begin{equation}\label{13}
\frac{2m_0}{1-|w|^2}\frac{dw}{dt}=F-\frac{2w}{1+|w|^2}.
\end{equation}
Multiplying both sides of (\ref{13}) by $1+|w|^2$ and using that
$dt=\gamma d \tau$ we obtain \textit{the relativistic dynamic
equation for the symmetric velocities }
\begin{equation}\label{14}
2m_0dw/d\tau =F-c^{-2}\{w,F,w\}=\xi_F(w),\end{equation}
where $\tau$ denote the proper time and the triple product is the
spin triple product defined by (\ref{8}) and $\xi$ by (\ref{7}).
Let us derive now the relativistic dynamic equation for the
electromagnetic field for the symmetric velocities. Let $E$
denotes the electric strength of the field, $H$ denote the
magnetic strength. For simplicity we will assume $q=1$. Then from
the formula of the Lorentz force for the electromagnetic field,
the dynamic equation becomes
$$\frac{d}{dt}(mv)=E+v\times H.$$
Thus, using equations (\ref{9}) and (\ref{10}) we get
\begin{equation}\label{15}
E+\frac{2w}{1+|w|^2}\times H=\frac{d}{dt}m_0\frac{2u}{1-|w|^2}
=2m_0(\frac{1}{1-|w|^2}
\frac{dw}{dt}+\frac{2w}{(1-|w|^2)^2}<\frac{dw}{dt}|w>).
\end{equation}
By taking the inner product with $w$ we get
\begin{equation}\label{16}
=2m_0<\frac{dw}{dt}|w>\frac{1+|w|^2}{(1-|w|^2)^2}.
\end{equation}
By substituting $<\frac{dw}{dt}|w>$ from (\ref{16}) into
(\ref{15}) we obtain
\begin{equation}\label{17}
\frac{2m_0}{1-|w|^2}\frac{dw}{dt}=E+\frac{2w}{1+|w|^2}\times H-\frac{2w}
{1+|w|^2}.\end{equation}
Multiplying both sides of the last equation by
$1+|w|^2$ and using that $dt=\gamma d \tau$ we obtain
\begin{equation}\label{18}
2m_0dw/d\tau =q(E-c^{-2}\{w,E,w\} + 2w\times H)\end{equation} that
could be considered as the relativistic dynamic equation for the
electromagnetic field.
\section{The Physical meaning of the Geometric product}
In the last decades it was found that the use of Clifford algebra
and the Geometric product ( see \cite{HS84}) associated with it
simplify the description of different physical phenomena in
Classical and Modern physics. But what is the reason that this
description is so successful? So far not much is known why this
algebraic structure is connected with the description of real
phenomena. Often, it is proposed that the success in use of this
algebraic structure is connected with the fact that the geometric
product contains the dot product and the outer (vector) product.
As it was mentioned in Section 3, for description of dynamical
systems in which boost and rotations occur synchronically, it is
important to have a good relation between these two operations.
Such description must be independent as much as possible on the
order in which the operations were performed. We have seen that
use of symmetric velocity, associated with the conformal group,
satisfies partial commutation, as it is given by (\ref{5}). A full
commutation could not be achieved since during the rotation the
direction of the boost was changed.
The symmetric velocity is directly connected to spin triple
product described by (\ref{8}). As it was described in Section 3,
the Lie algebra of the conformal group is fully described by this
triple product. By using the definition of operator $D(u,v)$,
defined above we may rewrite (\ref{8}) as
$$D(u,v) =____I +u\wedge v,$$
where $I$ denote the identity operator. This is similar to the
geometric product between $u$ and $v$ defined as
$$uv=____ +u\wedge v,$$
where the sum of a scalar $____$ and bivector $u\wedge v$ makes
sense in the Clifford algebra.
\section{Conclusions}
This paper is an important step in the new mathematical foundation
to description of physical phenomena. The proposed mathematical
description is based on geometry and symmetry of nature that
results in the symmetry of the state space.
Introduction of a new dynamic variable, called symmetric
velocity, defined by (\ref{3}), leads to distributive law
(\ref{5}) for the velocity addition with respect to multiplication
with rotations. The Lie algebra of the group generated by the
boosts and rotations of symmetric velocities is described in terms
of spin triple product (\ref{liealg}) associated with a bounded
symmetric domain of type 4.
The relativistic dynamic equations (\ref{14}) and (\ref{18}) for
the symmetric velocity state that its derivative is equal to an
element of the Lie algebra. Finally, we show the connection of the
spin triple product to the Clifford or Geometric product,
explaining why this product simplifies equations in different
areas of physics.
% ----------------------------------------------------------------
%\bibliographystyle{amsplain}
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Berlin (1994), 61--82.
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\bibitem{HS84} D. Hestenes, G. Sobczyk, Clifford Algebra to
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\end{thebibliography}
\end{document}
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