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\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
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\input{tcilatex}

\begin{document}


\section{The Local Laboratory}

\subsection{Lab coordinates}

Each event $P$ that happens `in the laboratory' is identified by a set of
Minkowski coordinates $x^{0}\left( P\right) ,x^{1}\left( P\right)
,x^{2}\left( P\right) ,x^{3}\left( P\right) $. These coordinates are
assigned relative to some reference event $P_{0}$ by specific procedures
such as the Einstein clock synchronization procedure.%
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\]
We will assume that the reference event corresponds to $x^{\mu }\left(
P_{0}\right) =0$. Because the events are happening in the real world as well
as in the laboratory, the functions $x^{\mu }$ are defined on the spacetime
manifold $M$ and provide a map of $M$ into $\mathfrak{R}^{4}$. Thus, they
provide a particular chart on the manifold.

\subsection{Lab Connections}

The coordinates $x^{\mu }$ provide a simple geometry for describing what
happens in a particular laboratory. At each point in the laboratory there is
a set of basis vectors 
\[
e_{\mu }=\partial _{\mu }=\frac{\partial }{\partial x^{\mu }} 
\]%
that are assumed to be parallel everywhere so that the natural connection to
use is one in which the connection coefficients are all zero. Thus, the
derivative of a (possibly non-spacetime) vector field 
\[
u=u^{A}E_{A} 
\]%
in the direction of a spacetime vector field $w$ is expressed within the
laboratory as a vector field $\partial _{w}u$ whose components are just the
derivatives of the components of $u$ 
\[
\partial _{w}u=w^{\delta }\frac{\partial u^{A}}{\partial x^{\delta }}E_{A}. 
\]

In the simplest sort of physical theory, the laboratory coordinates can
extend everywhere in spacetime and the laboratory connection $\partial $ is
actually the same for all laboratories and can therefore be identified as
the physical connection. Notice that this a very restrictive kind of theory
because it implies that the coordinates of any two laboratories must be
related by linear relations 
\[
x^{\alpha ^{\prime }}=U^{\alpha ^{\prime }}{}_{\beta }x^{\beta }+b^{\alpha
^{\prime }} 
\]
where the coefficients $U^{\alpha ^{\prime }}{}_{\beta }$ and the
translation components $b^{\alpha ^{\prime }}$ are the same everywhere.
Special Relativity is this type of highly restrictive physical theory.

\subsection{Physical Connection}

In order to obtain a less restrictive theory, assume that the lab connection 
$\partial $ agrees with the physical connection $D$ on the manifold $M$ only
at the reference event $P_{0}$. Thus, for a vector-field $u,$%
\[
\left. \partial _{w}u\right| _{P_{0}}=\left. D_{w}u\right| _{P_{0}}. 
\]%
In this case, two laboratories with different reference events will
generally not be using the same laboratory connections. Because the
laboratory basis vector fields $E_{A}$ are assumed to be covariantly
constant at the reference event $P_{0}$, their images on the manifold must
have 
\[
\left. D_{w}E_{A}\right| _{P_{0}}=0 
\]%
so that the corresponding connection coefficients must vanish at the
reference event. 
\[
\left. \Gamma ^{A}{}_{B\delta }\right| _{P_{o}}=\Gamma ^{A}{}_{B\delta
}\left( 0\right) =0\text{.} 
\]

\subsection{The Problem with Torsion}

If the physical connection $D$ has a non-zero torsion tensor, then the
antisymmetric part of the tangent-space connection coefficients 
\[
\Gamma ^{\alpha }{}_{\beta \delta }\left( P_{0}\right) -\Gamma ^{\alpha
}{}_{\delta \beta }\left( P_{0}\right) =T^{\alpha }{}_{\delta \beta } 
\]
cannot be zero and therefore, the connection coefficients themselves cannot
be zero at the reference point no matter how the laboratory coordinates are
defined. The difficulty here is that the torsion is actually a tensor and
cannot be made to vanish by coordinate transformations.

There are theories that permit torsion. One way to overcome the difficulty
with the laboratory connection is to introduce a symmetric connection
defined by 
\[
\tilde{D}_{v}u=D_{v}u-\frac{1}{2}T\left( v,u\right) 
\]%
For this connection, the commutator of covariant derivatives on a scalar
field is 
\begin{eqnarray*}
\left[ \tilde{D}_{u},\tilde{D}_{v}\right] f &=&\left[ u,v\right] f+\left( 
\tilde{\nabla}_{v}u-\tilde{\nabla}_{u}v\right) f \\
&=&\left[ u,v\right] f+\left( \nabla _{v}u-\frac{1}{2}T\left( v,u\right)
-\nabla _{u}v+\frac{1}{2}T\left( u,v\right) \right) f \\
&=&\left( \left[ u,v\right] +\nabla _{v}u-\nabla _{u}v+T\left( u,v\right)
\right) f=0
\end{eqnarray*}%
so that the torsion associated with the connection is zero. Thus it is
possible to assume the link 
\[
\left. \partial _{w}u\right| _{P_{0}}=\left. \tilde{D}_{w}u\right| _{P_{0}} 
\]%
between the laboratory connection and the symmetric physical connection $%
\tilde{D}$. This approach amounts to introducing a third-rank tensor field
as an additional part of the theory and is vulnerable to the Occam's Razor
criticism: \textit{Don't introduce things unless they are needed to explain
something}.

\section{A Simple Example: The Charged Scalar Field}

\subsection{Gauge Transformations of A Complex-valued Field}

Suppose that a particle is described by a wave-function $\phi $. For each
possible set of laboratory coordinates $x=\left\{
x^{0},x^{1},x^{2},x^{3}\right\} $ there is a complex number $\phi \left(
x\right) $. In the simplest interpretation of this sort of field, this
complex number is the probability amplitude for finding a particle at that
set of coordinates. Actually one needs to go beyond this simple
interepretation by quantizing the field and calculating probability
amplitudes for positive frequency states, but that does not change the basic
idea. If $\phi \left( x\right) $ is thought of as a probability amplitude,
only its complex modulus 
\[
\left| \phi \left( x\right) \right| ^{2}=\phi ^{\ast }\left( x\right) \phi
\left( x\right) 
\]
has physical significance and is interpreted as the probability density for
finding the particle. If $\phi \left( x\right) $ is thought of as a complex
field operator, then it cannot be Hermitean and only Hermitean combinations
of $\phi \left( x\right) $ and $\phi ^{\dagger }\left( x\right) $ are
observable. In either case, multiplying $\phi \left( x\right) $ by a phase
factor $e^{i\lambda }$ where $\lambda $ is constant throughout the
laboratory will not change any observable prediction of the theory. This
sort of inconsequential change in a quantity that is not directly observable
is called a \emph{gauge transformation}.

\subsection{Gauge Basis Vectors}

The scalar field $\phi $ has more than one description with any two
descriptions $\phi $ and $\phi ^{\prime }$ connected by a transformation $%
\phi ^{\prime }=e^{i\lambda }\phi $. We have seen this sort of thing before
with the components of a vector, which transform when the basis vectors are
changed. Thus, we introduce a one-dimensional complex vector space $C_{P}$
at each point of the manifold. Only one basis vector $b$ is needed and a
vector can be expanded in this basis as 
\[
\Phi =\phi b. 
\]%
A natural choice for the metric tensor on $V_{P}$ is defined by the inner
product 
\[
\Phi \cdot \Psi =\phi ^{\ast }\psi , 
\]%
which corresponds to insisting that 
\[
b\cdot b=g\left( b,b\right) =1. 
\]%
Thus, $b$ is a unit vector on $C_{P}$.

Consider a new unit vector $b^{\prime }$ and expand it in terms of $b$. 
\[
b^{\prime }=Ub. 
\]%
The condition that $b^{\prime }$ is also a unit vector is 
\[
b^{\prime }\cdot b^{\prime }=g\left( Ub,Ub\right) =U^{\ast }Ug\left(
b,b\right) =1 
\]%
or 
\[
U^{\ast }U=1, 
\]%
which implies that 
\[
U=e^{-i\lambda }. 
\]%
Similarly, the dual space $\hat{C}_{P}$ has a basis one-form $\beta $
defined by 
\[
\beta \cdot b=1 
\]%
and transforms according to the inverse transformation 
\[
\beta ^{\prime }=U^{-1}\beta =U^{\ast }\beta =e^{i\lambda }\beta . 
\]

The object $U$ is just a complex number but we notice that it can be thought
of as a $1\times 1$ matrix that has determinant one (which makes it part of
the `special group' and whose complex conjugate transpose is its inverse
(which makes it unitary). Thus, the gauge transformation group that we have
here is the special unitary group of dimension one or $SU\left( 1\right) $.
The geometrical object that is invariant under this group is the vector
field 
\[
\Phi =\phi b. 
\]%
The original scalar field is the component of this vector in a particular
basis and can be recovered by dotting the vector with the dual basis form $%
\beta $. \ The resulting object, 
\[
\phi =\beta \cdot \Phi 
\]%
depends on the choice of basis and transforms as we noted before: 
\[
\phi ^{\prime }=\beta ^{^{\prime }}\cdot \Phi =\left( U^{-1}\beta \right)
\cdot \Phi =e^{i\lambda }\phi . 
\]%
When the vector $\Phi $ is put back together in the new basis, the result is
the same as before.%
\[
\Phi ^{\prime }=\phi ^{\prime }b^{\prime }=e^{i\lambda }\phi e^{-i\lambda
}b=\phi b=\Phi 
\]

\subsection{The Gauge Connection}

Now suppose that the relationship between the `gauge-vector' spaces $C_{P}$
and $C_{P^{\prime }}$ at different points $P,P^{\prime }$ is not trivial.
The directional derivative $D_{v}$ of a gauge-invariant scalar field $\Phi $
in the direction of a tangent-vector field $v$ is given by the product rule, 
\[
D_{v}\Phi =D_{v}\left( \phi b\right) =\left( v\phi \right) b+\phi D_{v}b, 
\]%
and, in the nontrivial case, the derivative $D_{v}b$ of the gauge-basis
vector is not zero. 
\[
D_{v}b=D_{v^{d}e_{d}}b=v^{d}D_{e_{d}}b. 
\]%
The connection coefficients are just the derivatives in the tangent-space
basis vector directions or 
\[
D_{e_{d}}b=\Gamma _{d}b. 
\]%
Two of the usual indexes on the connection coefficients are missing because
the gauge-vector space $C_{p}$ is one-dimensional.

With the gauge-connection $\Gamma _{b}$ as a freely specifiable object, the
concept of gauge-invariance can be generalized. The parameter $\lambda $ in
the basis transformation 
\[
b^{\prime }=e^{-i\lambda }b 
\]%
no longer needs to be a constant and can now be any real-valued function.
Under this more general kind of transformation, the gauge connection
transforms as 
\[
\Gamma _{d}^{\prime }b^{\prime }=D_{e_{d}}b^{\prime }=D_{e_{d}}\left(
e^{-i\lambda }b\right) =e^{-i\lambda }\Gamma _{d}b-i\left( e_{d}\lambda
\right) e^{-i\lambda }b 
\]%
or 
\[
\Gamma _{d}^{\prime }=\Gamma _{d}-i\left( e_{d}\lambda \right) . 
\]%
The form of this transformation is familiar. It resembles the
electromagnetic gauge transformation except for the fact that $\lambda $ is
dimensionless. Introduce a constant $q$ with the dimensions of electric
charge and rename the variables. 
\begin{eqnarray*}
\Gamma _{d} &=&-i\frac{q}{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}A_{d} \\
\lambda &=&\frac{q}{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}\Lambda
\end{eqnarray*}%
so that the transformation becomes 
\[
A_{d}^{\prime }=A_{d}+e_{d}\Lambda . 
\]%
In terms of the laboratory coordinate basis, the transformation is 
\[
A_{d}^{\prime }=A_{d}+\frac{\partial \Lambda }{\partial x^{d}} 
\]%
which we recognize as the gauge transformation of the electromagnetic vector
potential.

\subsection{Minimal Coupling}

The simplest sort of equation that a scalar field can obey is the
Klein-Gordon equation which, in the laboratory coordinate basis is 
\[
\square \phi +m^{2}\phi =-g^{ab}\frac{\partial ^{2}\phi }{\partial
x^{a}\partial x^{b}}+m^{2}c^{2}\phi =0. 
\]
In terms of the quantum-mechanical momentum operators 
\[
p_{a}=i\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}\frac{\partial }{\partial x^{b}} 
\]
the Klein-Gordon equation simply expresses the relativistic constraint 
\[
p\cdot p=-m^{2}c^{2} 
\]
In the case where the gauge-connection is the same as the laboratory
connection, this equation is just the same as 
\[
-\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}^{2}g^{ab}D_{e_{a}}D_{e_{b}}\Phi +m^{2}c^{2}\phi =0. 
\]
In this case, replacing ordinary derivatives by covariant derivatives
changes nothing.

In the more general case, where the gauge-connection is non-trivial, the
process of replacing ordinary derivatives by covariant derivatives yields a
new set of equations: 
\[
D_{e_{b}}\Phi =D_{e_{b}}\left( \phi b\right) =\left( e_{b}\phi \right) b-i%
\frac{q}{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}A_{b}\phi b=\left( e_{b}\phi -i\frac{q}{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}A_{b}\phi \right) b=\left( e_{b}-i\frac{q}{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}A_{b}\right) \phi b 
\]%
\[
-\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}^{2}g^{ab}\left( e_{a}-i\frac{q}{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}A_{a}\right) \left( e_{b}-i\frac{q}{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}A_{b}\right) \phi +m^{2}c^{2}\phi =0. 
\]%
These equations correspond to replacing the momentum $p_{a}=i$\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}$\frac{\partial }{\partial x^{a}}=i$\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}$e_{a}$ by the combination 
\[
\tilde{p}_{a}=i\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}\left( e_{a}-i\frac{q}{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}A_{a}\right) =i\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}e_{a}+qA_{a}=p_{a}+qA_{a}. 
\]%
That is exactly the replacement that gives the correct equations of motion
for a particle of charge $q$ moving in an electromagnetic field (See
Homework No.21.2).

The general rule for coupling a connection to a physical field is to start
with the equations of motion that apply when the trivial laboratory
connection is valid and then replace ordinary laboratory coordinate
derivatives by covariant derivatives. This rule is called \textit{Minimal
Coupling}. It is the simplest way to get a theory that will reduce back to
the one that works when the connection is trivial.

\subsection{Geometrical Description of Electric Charge}

Notice that we had to introduce a constant $q$ with the dimensions of
electric charge into the relationship between the electromagnetic vector
potential and the complex phase gauge-connection. 
\[
A_{d}=i\frac{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}{q}\Gamma _{d} 
\]%
Since the gauge connection is supposed to be a geometrical object and
neither it nor the vector potential depend on the kinds of charged particles
that are present, we find that any scalar charged field of the form $\Phi
=\phi b$ will have the same charge, $q$. A pair of scalar charged particles
would be described by their product wave functions $\phi \left( x\right)
\psi \left( x\right) $. The corresponding gauge-invariant object would be a
tensor product $\Phi \otimes \Psi =\phi \psi b\otimes b$ and would show a
total charge of $2q$. Evidently, all of the charges that can be constructed
in this way are integer multiples of $q$. Thus, this model `explains' why
all elementary particle charges are integer multiples of a single
fundamental charge and we identify: 
\[
q=e=\text{charge on an electron.} 
\]

Negative charges can be obtained if we have an antilinear operator $C$ that
defines the conjugate of a wave function:%
\[
C\left( \phi b\right) =\phi ^{\ast }C\left( b\right) . 
\]%
Such a conjugated wave function transforms with a phase function $\lambda $
opposite in sign to other wave functions, which corresponds to changing the
sign of the charge $q$.

\subsection{Complex Structure and Charge Conjugation}

As we have just noted, we need a way to take the complex conjugate $C\left(
\Phi \right) $of a gauge-invariant field such as $\Phi =\phi b$. The
conjugation operation is antilinear so that, for any object $j$ in the space 
$C_{P}$, 
\[
C\left( fj\right) =f^{\ast }C\left( j\right) 
\]%
and is idempotent, which means that 
\[
C^{2}=1. 
\]%
The basis vector $b$ is an abstract object, so its complex conjugate is not
defined. So define it as a new kind of object: 
\[
b^{\cdot }=C\left( b\right) 
\]%
and note, from the idempotence of $C$, that 
\[
C\left( b^{\cdot }\right) =b. 
\]%
We now have \emph{two }vector spaces at each point, $P$. 
\[
\FRAME{itbpF}{2.5097in}{1.305in}{0in}{}{}{complex.gif}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 2.5097in;height 1.305in;depth
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"1";cropright "1";cropbottom "0";filename
'graphics/complex.gif';file-properties "XNPEU";}} 
\]%
The space $C_{P}$ contains the basis object $b$ and represents fields of
charge $q$ while the space $C_{P}^{\cdot }$ contains the conjugate basis
object $b^{\cdot }$ and represents fields of charge $-q$. The charge
conjugation operator $C$ performs the mappings 
\begin{eqnarray*}
C &:&C_{P}\rightarrow C_{P}^{\cdot } \\
C &:&C_{P}^{\cdot }\rightarrow C_{P}.
\end{eqnarray*}

To complete the family of local spaces, note that a complex-valued function
or scalar field can be thought of as a zero-rank tensor with basis object $%
\mathbf{1}$ satisfying%
\[
C\left( \mathbf{1}\right) =\mathbf{1.} 
\]%
Thus, we have a trivial one-dimensional local vector space at each point
that we can denote by $T_{P}^{0}$ and a real-valued scalar field can be
represented as $f\mathbf{1}$ where $f$ is real. However, we now have \emph{%
other} ways to represent the \emph{same} real field. For example, both%
\[
\frac{1}{2}f\left( b+b^{\cdot }\right) 
\]%
and 
\[
\frac{1}{2}f\left( b\otimes b^{\cdot }+b^{\cdot }\otimes b\right) 
\]%
are self-conjugate. The second repesentation has the advantage of also being
invariant under gauge transformations, $b\rightarrow e^{i\lambda }b,b^{\cdot
}\rightarrow e^{-i\lambda }b^{\cdot },$ so we will focus on that one.

It makes sense to \emph{identify} these different representations of real
functions and require, for example, a linear mapping $\sigma $ between the
tensor space $C_{P}\otimes C_{P}^{\cdot }\oplus C_{P}^{\cdot }\otimes C_{P}$
and the `tangent scalar' space $T_{P}^{0}$. 
\[
\sigma \left( b\otimes b^{\cdot }+b^{\cdot }\otimes b\right) =\mathbf{2.} 
\]%
Similarly, we require a representation of the imaginary number $i$ in the
form 
\[
\sigma \left( b\otimes b^{\cdot }-b^{\cdot }\otimes b\right) =2\mathbf{i} 
\]%
Adding and subtracting these relations gives 
\begin{eqnarray*}
\sigma \left( b\otimes b^{\cdot }\right) &=&\left( 1+i\right) \mathbf{1} \\
\sigma \left( b^{\cdot }\otimes b\right) &=&\left( 1-i\right) \mathbf{1}.
\end{eqnarray*}%
The map $\sigma $ is sometimes called a `soldering map' because it connects
otherwise unrelated abstract spaces.%
\[
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"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
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"1";cropright "1";cropbottom "0";filename
'graphics/solder1.gif';file-properties "XNPEU";}} 
\]

Because the soldering map and charge conjugation operations are doing
relatively trivial jobs, we expect them to commute with covariant
derivatives. Thus, 
\[
D_{v}\sigma \left( b\otimes b^{\cdot }\right) =\sigma \left( D_{v}\left(
b\otimes b^{\cdot }\right) \right) 
\]%
which requires 
\[
D_{v}\left( b\otimes b^{\cdot }\right) =D_{v}\left( 1+i\right) \mathbf{1+}%
\left( 1+i\right) D_{v}\mathbf{1}=0 
\]%
or 
\[
\left( D_{v}b\right) \otimes b^{\cdot }+b\otimes D_{v}b^{\cdot }=0. 
\]%
But 
\[
D_{v}b=v^{d}\Gamma _{d}b 
\]%
and, by charge conjugation, 
\[
D_{v}b^{\cdot }=v^{d}\Gamma _{d}^{\ast }b^{\cdot } 
\]%
so we find that 
\[
v^{d}\Gamma _{d}b\otimes b^{\cdot }+b\otimes v^{d}\Gamma _{d}^{\ast
}b^{\cdot }=0 
\]%
or 
\[
\Gamma _{d}^{\ast }=-\Gamma _{d}. 
\]%
Thus, preserving the gauge-invariant soldering map requires the gauge
connection coeficients to be pure imaginary. The vector potential components 
$A_{d}$ must then be real.

\section{Less Simple Examples: Multicomponent Fields}

\subsection{Adding Dimensions}

The simplest way to generalize a charged field and its corresponding
electromagnetic field is to add dimensions to the internal space $C_{P}$ at
each point. Instead of just one basis vector $b$ we have a set of $N$ basis
vectors $b_{A}$. Thus, a field would be a multicomponent object 
\[
\Phi =\phi ^{A}b_{A} 
\]
and the connection coefficients would have both tangent-space indexes and
internal indexes: 
\[
D_{v}b_{A}=b_{B}\Gamma ^{B}{}_{Ad}v^{d}. 
\]
There would be a corresponding dual space $\hat{C}_{P}$ with dual basis
forms $\beta ^{A}$ so we have 
\[
\phi ^{A}=\beta ^{A}\left( \Phi \right) 
\]
and the connection one-forms can be expressed as 
\[
\Gamma ^{B}{}_{A}\left( v\right) =\Gamma ^{B}{}_{Ad}v^{d}=\beta ^{B}\left(
D_{v}b_{A}\right) . 
\]
Just as in the one-dimensional case, there would be a conjugate internal
space $C_{P}^{\cdot }$ with basis vectors $b_{\dot{A}}$ and conjugate fields 
\[
\Psi =\psi ^{\dot{A}}b_{\dot{A}} 
\]
and the conjugation operation would act according to 
\[
C\left( b_{A}\right) =b_{\dot{A}} 
\]
Since $C$ is antilinear, 
\[
C\left( \phi ^{A}b_{A}\right) =\left( \phi ^{A}\right) ^{\ast }C\left(
b_{A}\right) =\left( \phi ^{A}\right) ^{\ast }b_{\dot{A}} 
\]
so that the components of the conjugate field are given by the complex
conjugates of the field components: 
\[
\beta ^{\dot{A}}\left( C\left( \Phi \right) \right) =\left( \beta ^{A}\left(
\Phi \right) \right) ^{\ast }. 
\]

So far, we have four different vector spaces at each point, $%
C_{P},C_{P}^{\cdot },\hat{C}_{P},\hat{C}_{P}^{\cdot }$. The conjugation
operation connects the dotted spaces with the undotted ones. The next step
is to connect each space to its dual space by means of a metric tensor. Once
this step has been taken, we have the additional option of inspecting the
gauge-invariant tensor spaces formed from these spaces and identifying or
`soldering' them to tangent-space objects just as we soldered the space of
symmetric gauge tensors $C_{P}\otimes C_{P}^{\cdot }\oplus C_{P}^{\cdot
}\otimes C_{P}$ to the space $T_{P}^{0}$ of complex scalars at each point.

\subsection{Yang-Mills Fields}

Suppose that $C_{P}$ is a little Hilbert space and the basis vectors $b_{A}$
are orthonormal in the usual Hilbert space metric. Thus, we have the usual
half-antilinear metric tensor 
\begin{eqnarray*}
g\left( \Phi ,f\Psi \right) &=&fg\left( \Phi ,\Psi \right) \\
g\left( f\Phi ,\Psi \right) &=&f^{\ast }g\left( \Phi ,\Psi \right)
\end{eqnarray*}
with 
\[
g\left( \Phi ,\Psi \right) ^{\ast }=g\left( \Psi ,\Phi \right) 
\]
so that we can define inner products of vectors in $C_{P}$%
\[
\Phi \cdot \Psi =g\left( \Phi ,\Psi \right) 
\]
Now, we can define the inner product of the conjugates of vectors according
to 
\[
g^{\cdot }\left( C\Phi ,C\Psi \right) =g\left( \Psi ,\Phi \right) 
\]
so that $C_{P}^{\cdot }$ is also a little Hilbert space. The condition that
the basis vectors are orthonormal is 
\[
g\left( b_{A},b_{B}\right) =\delta _{AB} 
\]
which implies that the conjugate vectors are also orthonormal since 
\[
g^{\cdot }\left( b_{\dot{A}},b_{\dot{B}}\right) =g^{\cdot }\left(
Cb_{A},Cb_{B}\right) =g\left( b_{B},b_{A}\right) =\delta _{BA}. 
\]

Now consider a new set of basis vectors and expand it in terms of the old
ones. 
\[
b_{A^{^{\prime }}}=b_{A}U^{A}{}_{A^{\prime }}, 
\]
The inner products of the new basis vectors are then 
\[
g\left( b_{A^{\prime }},b_{B^{\prime }}\right) =g\left(
b_{A}U^{A}{}_{A^{\prime }},b_{B}U^{B}{}_{B^{\prime }}\right) =\left(
U^{A}{}_{A^{\prime }}\right) ^{\ast }U^{B}{}_{B^{\prime }}\delta _{AB} 
\]
so that the condition that they be orthonormal is 
\[
\left( U^{A}{}_{A^{\prime }}\right) ^{\ast }U^{B}{}_{B^{\prime }}\delta
_{AB}=\delta _{A^{\prime }B^{\prime }} 
\]
and the coefficients $U^{A}{}_{A^{\prime }}$ are required to be the
components of unitary matrix. \ If we rule out reflections, then the result
is the special unitary group of order $N$ or $SU\left( N\right) .$

A field $\Phi $ assigns the element $\Phi \left( P\right) =\phi ^{A}b_{A}$
of $C_{P}$ to each spacetime point. Each component $\phi ^{A}$ may be
interpreted as the probability amplitude for finding a particle with label $%
A $ at the point $P$. If the labels of particles can be re-assigned by
changing the internal basis vectors, then the resulting particle theory has $%
SU\left( N\right) $ as a gauge group. That leads to the possibility of
making this gauge-group independent at each point by introducing the
corresponding gauge connection coefficients $\Gamma ^{B}{}_{Ad}$. These
coefficients behave much like the vector potential in electromagnetism
except that they carry a pair of $SU\left( N\right) $ indexes. The resulting
fields, defined in analogy to the electromagnetic vector potential, by 
\[
A^{B}{}_{Ad}=i\frac{\text{\textit{%
%TCIMACRO{\U{127}}%
%BeginExpansion
h{\hskip-.2em}\llap{\protect\rule[1.1ex]{.325em}{.1ex}}{\hskip.2em}%
%EndExpansion
}}}{q}\Gamma ^{B}{}_{Ad} 
\]%
are called Yang-Mills fields.

The Yang-Mills analog of the electromagnetic Maxwell tensor $f$ is the
gauge-curvature tensor 
\[
\mathcal{R}\left( u,v\right) Y=\left[ \nabla _{u},\nabla _{v}\right]
Y-\nabla _{\left[ u,v\right] }Y 
\]
where 
\[
Y=Y^{A}b_{A} 
\]
is a Yang-Mills field. The components of the gauge-curvature tensor are 
\[
F^{A}{}_{B\alpha \beta }=\beta ^{A}\cdot \mathcal{R}\left( e_{\alpha
},e_{\beta }\right) b_{B} 
\]
and the field equations are otained by varying the action 
\[
S=\int d^{4}xg^{\alpha \rho }g^{\beta \sigma }F^{A}{}_{B\alpha \beta
}F^{B}{}_{A\rho \sigma } 
\]
with respect to the potentials $A^{B}{}_{Ad}$. The $N=1$ case of the
Yang-Mills $SU\left( N\right) $ field is just Maxwell's theory of
electromagnetism.

\subsection{Spinor Fields}

Instead of looking for purely internal fields such as the labels on
elementary particle fields, we can seek a metric structure on $C_{P}$ that
makes it possible to identify gauge-invariant tensor products with tangent
space vectors in the same way that we were able to identify the space $%
C_{P}\otimes C_{P}^{\cdot }\oplus C_{P}^{\cdot }\otimes C_{P}$ with the
space of complex scalars.

The metric structure on spacetime has signature (-+++), which is a bit
clumsy to deal with. An orthonormal basis or `tetrad' consists of vectors $%
e_{\alpha }$ such that 
\begin{eqnarray*}
e_{0}\cdot e_{0} &=&-1,\qquad e_{i}\cdot e_{j}=0\text{ (}i\neq j\text{)} \\
e_{1}\cdot e_{1} &=&e_{2}\cdot e_{2}=e_{3}\cdot e_{3}=1.
\end{eqnarray*}%
In the case of a calculation that involves an isolated source, the unit
vector $e_{1}$ is usually taken to point away from the source so that it is
approximately the propagation direction for any waves from the source while $%
e_{2}$ and $e_{3}$ lie in the plane of the wavefronts. This type of
calculation inspired a different set of basis vectors:%
\[
\FRAME{itbpF}{1.8568in}{0.8371in}{0in}{}{}{nullbase.gif}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 1.8568in;height 0.8371in;depth
0in;original-width 7.3172in;original-height 3.2335in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'graphics/nullbase.gif';file-properties "XNPEU";}}
\]%
\[
e_{\pm }=e_{1}\pm e_{0},\qquad m=e_{2}+ie_{3},\qquad \bar{m}=e_{2}-ie_{3}
\]%
For isolated source problems, this basis set is clearly useful because $%
e_{\pm }$ point along the characteristic surfaces of waves while the complex
vector $m$ renders an obvious symmetry of this type of problem --- rotation
around the radial vector --- as just a complex multiplication: $m^{\prime
}=e^{i\phi }m$. More important for our purpose is that the metric structure
takes a simple and symmetrical form when it is expressed in terms of these
vectors: 
\[
e_{+}\cdot e_{+}=e_{-}\cdot e_{-}=m\cdot m=\bar{m}\cdot \bar{m}=0.
\]%
Thus, all of these basis vectors are null vectors. The inner products
between basis vectors take the forms 
\begin{eqnarray*}
e_{+}\cdot e_{-} &=&m\cdot \bar{m}=2 \\
e_{\pm }\cdot m &=&e_{\pm }\cdot \bar{m}=0.
\end{eqnarray*}%
These four vectors are called a \emph{null tetrad}. Converting the
differential geometry of spacetime into null tetrad components and then
giving names to each component results in the \emph{Newman-Penrose formalism}%
.

Now consider spaces $C_{P}$ with just two dimensions, so we have basis
vectors $b_{1}$ and $b_{2}$ and corresponding basis vectors $b_{\dot{1}}$
and $b_{\dot{2}}$ on the conjugate space $C_{P}^{\cdot }$. In the one
dimensional case, we were able to represent real-valued fields by symmetric
tensors. Thus, we consider the following basis tensors for $C_{P}\otimes
C_{P}^{\cdot }\oplus C_{P}^{\cdot }\otimes C_{P}$: 
\begin{eqnarray*}
b_{1\dot{1}} &=&b_{1}\otimes b_{\dot{1}}+b_{\dot{1}}\otimes b_{1} \\
b_{2\dot{2}} &=&b_{2}\otimes b_{\dot{2}}+b_{\dot{2}}\otimes b_{2} \\
b_{1\dot{2}} &=&b_{1}\otimes b_{\dot{2}}+b_{\dot{2}}\otimes b_{1} \\
b_{2\dot{1}} &=&b_{\dot{1}}\otimes b_{2}+b_{2}\otimes b_{\dot{1}}
\end{eqnarray*}%
Notice that 
\[
C\left( b_{1\dot{1}}\right) =b_{1\dot{1}},\qquad C\left( b_{2\dot{2}}\right)
=b_{2\dot{2}} 
\]%
so that the tensors $b_{1\dot{1}}$ and $b_{2\dot{2}}$ are real. The other
two tensors are conjugates of each other, so we have the same pattern as the
null tetrad and can seek to identify 
\begin{eqnarray*}
\sigma \left( b_{1\dot{1}}\right) &=&e_{+},\qquad \sigma \left( b_{2\dot{2}%
}\right) =e_{-} \\
\sigma \left( b_{1\dot{2}}\right) &=&m,\qquad \sigma \left( b_{2\dot{1}%
}\right) =\bar{m}
\end{eqnarray*}%
\[
\FRAME{itbpF}{1.8922in}{0.8371in}{0in}{}{}{spinors.gif}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 1.8922in;height 0.8371in;depth
0in;original-width 7.4581in;original-height 3.2335in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'graphics/spinors.gif';file-properties "XNPEU";}} 
\]%
This identification tells us what we need for a metric. First of all, we
need to know how to take inner products of the tensors $b_{A\dot{A}}.$ That
means taking inner products between all of the basis vectors $b_{A}$ and $b_{%
\dot{A}}$. Thus, we really need a metric on the space $D_{P}=C_{P}\oplus
C_{P}^{\cdot }$ spanned by these four vectors. The usual metric on this
space is defined by 
\begin{eqnarray*}
b_{A}\cdot b_{B} &=&\varepsilon _{AB} \\
b_{\dot{A}}\cdot b_{\dot{B}} &=&\varepsilon _{\dot{A}\dot{B}} \\
b_{A}\cdot b_{\dot{B}} &=&0
\end{eqnarray*}%
where $\varepsilon _{AB}$ is the Levi-Civita tensor in two dimensions with 
\[
\varepsilon _{12}=\varepsilon _{\dot{1}\dot{2}}=1. 
\]

The space $C_{P}$ in this case is called the space of \emph{two-component
Weyl spinors}. The space $D_{P}$ is the space of \emph{Dirac spinors}. Thus,
a Weyl spinor has the form 
\[
\psi =\psi ^{A}b_{A} 
\]
and a Dirac spinor has the form 
\[
\Psi =\Psi ^{A}b_{A}+\Psi ^{\dot{A}}b_{\dot{A}}=\Psi ^{a}b_{a} 
\]
where $a=1,2,\dot{1},\dot{2}$. The identification map $\sigma $ takes pairs
of dirac spinors as arguments and gives spacetime vectors. In terms of
spacetime basis vectors $e_{\mu }$ with dual basis forms $\omega ^{\mu }$,
the components of $\sigma $ are 
\[
\omega ^{\mu }\cdot \sigma \left( b_{a},b_{b}\right) =\gamma ^{\mu }{}_{ab}. 
\]
With one index raised (using the antisymmetric spin metric) these are
usually called the Dirac gamma matrices. 
\[
\left[ \gamma ^{\mu }\right] ^{a}{}_{b}=\gamma ^{\mu a}{}_{b}=g^{ak}\omega
^{\mu }\cdot \sigma \left( b_{k},b_{b}\right) . 
\]

The spacetime tangent-space connection can be expressed in terms of a
connection on spinors 
\begin{eqnarray*}
D_{v}b_{A} &=&b_{B}\Gamma ^{B}{}_{A}\left( v\right) =b_{B}\Gamma
^{B}{}_{A\delta }v^{\delta } \\
D_{v}b_{\dot{A}} &=&b_{\dot{B}}\Gamma ^{\dot{B}}{}_{\dot{A}}\left( v\right)
=b_{\dot{B}}\Gamma ^{\dot{B}}{}_{\dot{A}\delta }v^{\delta }
\end{eqnarray*}
which implies 
\begin{eqnarray*}
D_{v}e_{+} &=&D_{v}\left( \sigma \left( b_{1\dot{1}}\right) \right) =\sigma
\left( D_{v}b_{1\dot{1}}\right) \\
&=&\sigma \left( D_{v}\left( b_{1}\otimes b_{\dot{1}}+b_{\dot{1}}\otimes
b_{1}\right) \right) \\
&=&\sigma \left( D_{v}b_{1}\otimes b_{\dot{1}}+b_{1}\otimes D_{v}b_{\dot{1}%
}+D_{v}b_{\dot{1}}\otimes b_{1}+b_{\dot{1}}\otimes D_{v}b_{1}\right) \\
&=&v^{\delta }\sigma \left( \Gamma ^{B}{}_{1\delta }b_{B}\otimes b_{\dot{1}%
}+\Gamma ^{\dot{B}}{}_{\dot{1}\delta }b_{1}\otimes b_{\dot{B}}+\Gamma ^{\dot{%
B}}{}_{\dot{1}\delta }b_{\dot{B}}\otimes b_{1}+b_{\dot{1}}\otimes \Gamma
^{B}{}_{1\delta }b_{B}\right) \\
&=&v^{\delta }\left( \Gamma ^{B}{}_{1\delta }\gamma _{B\dot{1}}^{\mu
}+\Gamma ^{\dot{B}}{}_{\dot{1}\delta }\gamma _{1\dot{B}}^{\mu }+\Gamma ^{%
\dot{B}}{}_{\dot{1}\delta }\gamma _{\dot{B}1}^{\mu }+\Gamma ^{B}{}_{1\delta
}\gamma _{\dot{1}B}^{\mu }\right) e_{\mu }
\end{eqnarray*}
or 
\[
\Gamma ^{\mu }{}_{+\delta }=2\Gamma ^{B}{}_{1\delta }\gamma _{B\dot{1}}^{\mu
}+2\Gamma ^{\dot{B}}{}_{\dot{1}\delta }\gamma _{1\dot{B}}^{\mu } 
\]
with similar expressions for the other connection coefficients.

With the spin connection, it is possible to construct a fascinating new
operator that acts on Dirac spinors: 
\[
D\Psi =\left[ \gamma ^{\mu }\right] D_{e_{\mu }}\Psi 
\]%
The Dirac equation for a massless spinor field 
\[
D\Psi =0 
\]%
implies that 
\[
DD\Psi =0 
\]%
or 
\[
\left[ \gamma ^{\nu }\right] D_{e_{\nu }}\left[ \gamma ^{\mu }\right]
D_{e_{\mu }}\Psi =0. 
\]%
The dirac matrices are covariantly constant and obey the relation 
\[
\left[ \gamma ^{\nu }\right] \left[ \gamma ^{\mu }\right] =g^{\mu \nu } 
\]%
so we find that 
\[
g^{\mu \nu }D_{e_{\nu }}D_{e_{\mu }}\Psi =0 
\]%
which is the Klein-Gordon equation. In terms of the momentum operator, the
Dirac equation, which is linear in the momentum, implies the quadratic
momentum constraint 
\[
p\cdot p=0. 
\]

\subsection{Higher Dimensions}

The identification of gauge invariant spin tensors with tangent-space
vectors works in dimensions higher than four. The general rule is that the
identification must be consistent with a spacetime signature that has only
one minus sign. The reason for the rule is that spacetimes with more than
one time direction permit closed timelike paths and have causality problems
(as in shooting your own grandmother). However, even that rule is under
scrutiny and theories with more than one time dimension are not entirely
excluded. Oddly enough, the simplest case is eleven spacetime dimensions
(ten spacelike and one timelike). The corresponding Dirac matrices are 32 by
32 matrices.

\section{The Field Theory Construction Kit}

Although this treatment has left out many important things (such as
supersymmetry), it should have covered enough to convey the general flavor
of how field theories are constructed. The key idea is that the laws of
nature should be constructed from locally linear objects. The basic steps of
field theory construction are:

\begin{enumerate}
\item Assume a finite dimensional spacetime manifold with a
Lorentz-signature metric (usually just one time direction at each point).

\item Associate a \emph{local vector space} with each point of the manifold.
Fields with values in these local vector spaces are \emph{matter fields }and
represent fundamental particles.

\item Specify a transformation group that acts on the these local vector
spaces and is expected to leave the theory invariant. This group is the 
\emph{gauge group}.

\item Identify the tangent space at each point with a subset of the
available gauge-invariant tensors formed from the local vector space at that
point. This identification provides the \emph{spinor fields}.

\item The connection on local vector spaces provides an additional set of
fields, called \emph{gauge-fields }that represent the fundamental forces
available in the theory.

\item Use the connection to construct a curvature tensor and gauge-invariant
covariant derivatives of the matter fields and build dynamical field
equations from these quantities.
\end{enumerate}

The difficulty with this procedure for building field theories is that it
involves choices at each step. The underlying manifold can have any number
of dimensions with some popular choices being:

\begin{itemize}
\item D=1, for matrix theories;

\item D=2, for string theories;

\item D=3, for membrane theories;

\item D=4, for spacetime theories;

\item D=5, for Kaluza-Klein theories;

\item D=10, for the embedding space of supersymmetric string theory;

\item D=11, for supergravity and the low-energy limit of M-theory;

\item D=26, for the embedding space of bosonic string theory.
\end{itemize}

The local vector space can range from one complex dimension, leading to
electromagnetism, to large spaces with extremely complex gauge groups. There
also choices to be made in identifying some local tensors with tangent
vectors. Once all of these choices have been made, there is usually a
variety of different ways to construct dynamical field equations for a given
set of fields.

Only for electromagnetism are the choices simple and the result unique. As
we will see, a classical theory of gravity is also straightforward to
construct in this fashion. Beyond those theories, we are reduced to the
strategy of exhaustion: Try all of the choices and see which ones produce
useable theories.

\end{document}

%%%%%%%%%%%%%%%%%%%%%%% End /document/fields.tex %%%%%%%%%%%%%%%%%%%%%%

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