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\noindent
UGVA-DPT 1993/09-834
\title{The Cartan formalism in field
theory\footnote{$^\ast$}{{\rm Partially supported by the Swiss National
Foundation.}}}
\author{By A. O. Barut\footnote{$^\dagger$}{{\rm Physics
Department, University of Colorado, Boulder CO 80309, USA.}}, D. J. Moore and C.
Piron}
\address{D\'epartement de Physique Th\'eorique, Universit\'e de Gen\`eve
CH-1211 Gen\`eve 4, Switzerland}
\abstract{
We consider a generalisation to field theory of the symplectic geometric
approach to particle mechanics. This involves the definition of spacetime
models; space and time as separate entities being taken as the primitive
elements of the theory. Dynamical covariance and the CPT transformation of
the Maxwell-Dirac and Maxwell-Schr\"odinger fields can then be discussed within
the same formalism. A novel matrix formulation of the Schr\"odinger equation
emerges which is a direct limit from the Dirac field.}
\vskip 16truemm
\section{Introduction}
The usual approach to field theory is to perform a formal variational calculus
on a Lagrangian. Here the calculus is formal as we assume that a function and
its derivatives can be varied independently; further complications arise as the
function space of interest is infinite-dimensional. The equivalent
problem in particle mechanics is resolved by using the Cartan formalism. Here
the equations of motion can be rigorously derived from a variational principle
by the use of a 1-form $\omega$, called the Cartan form, which is defined on the
state space of the system. One can then apply all the techniques of
symplectic geometry to describe the motion in the state space in
terms of the integral curve of a vector field; that is a flow [Arnold 1989,
chapter 9].
Here we apply the same idea to field theory. The 1-form $\omega$, used to lift
the time variable into the state space, is thus replaced by a 4-form used to lift
both time and space variables. This leads to the idea of a spacetime model,
which is a 4-manifold constructed to parametrise the evolving field in a
geometric fashion. We stress that it is space and time separately
which are the primitive notions of the theory, the corresponding spacetimes
being viewed as secondary constructions. As we will see, covariance is
maintained at the dynamical level, so the formalism so derived is consistent
with relativity.
Now time has a natural orientation since it only flows in one direction. In
contrast however, space has no intrinsic orientation, even if it is locally
orientable. To build a local spacetime model we must therefore choose whether
to paste a right- or left-handed coordinate system for space to the coordinate
axis for time. This means that each spacetime model
has a canonical dual model. The Galilean equivalence principle
applied to these two models leads
to the CPT transformation, even in the classical
theory. It is interesting to note that we then find
a natural doubling of the spacetime structure similar to that introduced by A.
Connes in the non-commutative differential geometry approach to the standard
model [Connes 1992, Kastler 1992].
The rest of this work is organised as follows. In section 2 we briefy review the
Cartan principle in particle mechanics, extending it to field theory in section
3. We then consider the Maxwell-Dirac and Maxwell-Schr\"odinger fields in a
unified fashion. This allows a simple algebraic understanding of the
relationship between the two models and their limits into each other. Next, in
section 5 the invariances of the Cartan form are exploited. In particular we
discuss the existence of conserved currents which provide a quantum
interpretation of the theory and the dynamical covariance of
the motions.
\section{Particle mechanics}
In this section we give a brief review of the application of the Cartan
formalism to particle mechanics. This will fix the notation to be used when we
come to field theory as well as explaining the concepts involved in a familiar
setting. In particular we will show how the evolution of a single particle can
be recovered in a simple way from a given 1-form on the state space.
This approach is in the same spirit as the usual formal variational approaches,
however it is much more transparent from the mathematical point of view. First
we deal only with the finite-dimensional state space and not an
infinite-dimensional function space. More importantly, in the usual approach one
must postulate that one can vary both a function and its derivative
independently, a hypothesis that is not obvious mathematically.
As we recall in the appendix, the state space for a single elementary particle
in classical mechanics is the seven-dimensional space $\Sigma=(p_k,q^k,t)$. Let
us write $T$ for the time coordinate. This is of course
diffeomorphic to the usual real line ${\Bbb R}$, however
we wish to keep a distinct
notation as this will prove valuable in the context of field theory. As the
state space can be consider as a product manifold we may write it
as a trivial fibre bundle
$\pi:\Sigma\rightarrow T$. A motion is then naturally defined as a map
$s:T\rightarrow\Sigma$ such that $\pi\circ s=1$; that is a motion is a section
of the fibre bundle. This condition merely states that the number used to
parametrise the motion is just the time coordinate itself.
Cartan's principle then states that the motion of a given system can be
determined from a 1-form
$$\omega&=p_kdq^k-H(p_\cdot,q^\cdot,t)dt\cr$$
on $\Sigma$. In future we will call $\omega$ any such Cartan form.
More precisely, given $\omega$ the motion is determined by the requirement that
$s^\ast(i_Xd\omega)=0$ for all vector fields $X$ on $\Sigma$. Here $i_X$ is the
interior product and $s^\ast$ the pullback. Note that if one has two 1-forms
$\omega_1$ and $\omega_2$ which differ only by a closed form, that is
by a term
$\theta$ such that $d\theta = 0$, then $d\omega_1=d\omega_2$ and we
will find the same motion. Hence the Cartan form is only determined
up to an closed form.
Hence we will have six equations which determine the solution submanifold $sT$
of $\Sigma$. These are the equations of motion of the particle. A quick
calculation gives
$$d\omega&=(dp_k+\frac{\partial H}{\partial q^k}dt)\wedge(dq^k-
\frac{\partial H}{\partial p_k}dt)
.\cr$$
Hence, as $s^\ast dt=dt$, $s^\ast
dp_k=\dot{p}_kdt$ and $s^\ast dq^k=\dot{q}^kdt$, we have
$$0&=s^\ast(i_{\partial_{p_k}}d\omega)\cr
&=s^\ast(dq^k-\frac{\partial H}{\partial p_k}dt)\cr
&=(\dot{q}^k-\frac{\partial H}{\partial p_k})dt,\cr$$
so that $\dot{q}^k=\frac{\partial H}{\partial p_k}$. Similarly
$$0&=s^\ast(i_{\partial_{q^k}}d\omega)\cr
&=s^\ast(-dp_k-\frac{\partial H}{\partial q^k}dt)\cr
&=(-\dot{p}_k-\frac{\partial H}{\partial q^k})dt,\cr$$
giving $\dot{p}_k=-\frac{\partial H}{\partial q^k}$.
Cartan's principle can also be applied, for example, to the motion of
a classical particle with spin with
$$\omega &=p_kdq^k+ \bar zdz - H(p_\cdot, q^\cdot, z, \bar z, t)dt.\cr$$
This principle provides a very simple way of computing Noether invariants
for
a system. Let $G$ be a one-parameter Lie group of transformations acting on
the state space $\Sigma$. This action is generated by a vector (vector field)
$X$ on $\Sigma$.
Now if the Cartan form $\omega$ is invariant under the group action,
then the Lie derivative $d_X:=di_X+i_Xd$ annihilates $\omega$:
$d_X\omega=0$. In this case
$$ds^\ast{i_X\omega}&=s^\ast(di_X\omega)\cr
&=-s^\ast(i_Xd\omega)\cr
&=0.\cr$$
Thus the function $\alpha_X:=i_X\omega$ is closed on the motion; that is
$\frac{d\alpha_X}{dt}=0$. This is just Noether's theorem written in geometrical
language.
For example, if $H$ is independent of $t$, then the Cartan form is
invariant under the one-parameter group of translations $t\mapsto t+\tau$
generated by $X=\partial_t$. In this case $\alpha_X=-H$ so that $H$ can be
interpreted as the energy.
A more interesting example is provided by the magnetic dipole, for which
$$H&={p^2\over2m}+{\overrightarrow{m}\cdot\overrightarrow{q}\over|q|^3}.\cr$$
In this case the Cartan form is invariant under the one parameter group of
transformations $p_k\mapsto\lambda^{-1}p_k$, $q^k\mapsto\lambda q^k$,
$t\mapsto\lambda^2 t$ generated by
$X=q^k\partial_{q^k}-p_k\partial_{p_k}+2t\partial_t$. Thus
$\alpha_X=p_kq^k-2Ht$ is closed, the value taken by $\alpha_X$
is constant and thus $\alpha_X$ is a first integral. Hence the Cartan formalism
allows easy access to conserved quantities that depend explicitly on time,
such quantities being difficult to find in the standard approach.
\section{Field theory}
We now turn to the study of fields. At any given moment of time a field is a
function of the space variables. Hence the motion $\tilde{s}$ of the dynamical
variables is parameterised by both $T$ and $S$; that is a motion is defined as a
map $\tilde{s}:T\times S\rightarrow\Sigma$. The Cartan form $\omega$ which
defines $\tilde{s}$ must then be a 4-form, as compared to the case of particle
mechanics where a 1-form is used.
This means that the closed quantities will be 3-forms, that is
currents. To
relate these currents to physical quantities we must integrate them over
suitable 3-dimensional subspaces in $T\times S$. Thus they must be
odd forms in the sense of de Rham [de Rham 1984, p19]: they must change sign
with the
Jacobian under an arbitrary coordinate transformation. This in turn implies that
the Cartan form must be odd. Of course the same situation occurs in the case of
particle mechanics. However $T$ has an orientation
defined by the direction of the flow of time and so in particle mechanics there
is an intrinsic way of passing from odd to even forms and vice versa.
The case of field theory is a little more complicated as the space $S$ is
just a three dimensional topological set with no given coordinates and hence
no orientation.
We now discuss the dynamical variables of the models we will treat in
the next section. We start with the fermionic field. As discussed
in the appendix, an elementary particle with intrinsic angular momentum
(but no internal degrees of freedom) is described by a two-component complex
wave function. But the corresponding field associated to such object is
most conveniently
labeled by four complex numbers. To treat this we take $\psi$ and
$\psi^{\dagger} $ as variables, forming $\omega$ from bilinear products. Thus
the Cartan form will be invariant under gauge transformations
$\psi\mapsto\e^{\i\alpha}\psi$, $\psi^{\dagger}\mapsto\e^{-\i\alpha}\psi^{\dagger} $.
As we will see in the next section, the associated Noether invariant gives
rise to an inner product as thus to the quantum interpretation of
the theory.
The electromagnetic field is described by pairs of forms
$(A_0,{\bf A})$ and $({\bf H},{\bf D})$ on space indexed by the time. Here $A_0$
is an even 0-form, ${\bf A}$ is an even 1-form, ${\bf H}$ is an odd 1-form and
${\bf D}$ is an odd 2-form [see for example
Ingarden and Jamio\l kowski 1985, pp36-47]. As
discussed in the appendix, we can construct
a 4-manifold $M$ such that the {\it motions} of these fields can be expressed as
differential forms $A\in\Lambda^1(M)$ and $H\in\Lambda^2(M)$, where $M$ is
diffeomorphic to $T\times S$. We call $M$ a spacetime model. Note that this
point of view is the reverse of the usually admitted philosophy,
where one starts with a unique given
spacetime and constructs spacelike hypersurfaces.
Both everyday experience and the fact that, from an axiomatic standpoint, time
must be treated as a superselection variable point to a qualitative
difference between time and space. In our opinion it is then better to treat
$T$ and $S$ as separate from the beginning. As we will see later, covariance is
maintained at the dynamical level and so there is no contradiction with the
relativity principle.
Hence the natural generalisation of the Cartan formulation to
field theory is that
the state space $\Sigma$ is a trivial bundle over a spacetime model
$\pi:\Sigma\rightarrow M$ and the evolution is a map $\tilde{s}:T\times
S\rightarrow\Sigma$ such that $\mu:=\pi\circ\tilde{s}:T\times S\rightarrow M$
is a diffeomorphism. We can then define
$s=\tilde{s}\circ\mu^{-1}:M\rightarrow\Sigma$ to recover the interpretation of
the motion as a section of the state space bundle: $\pi\circ s=1$.
This approach can be compared to the usual formalism, where one starts with a
fibre bundle $\sigma:S\rightarrow M$ and defines a Lagrangian
as a function on the
first jet extension of $S$. See Echeverr\'\ii a Enr\'\ii quez and Mu\~noz Lecanda
[1992] for a
discussion of the relation between the Hamiltonian and Lagrangian approaches in
this formalism and Trautman [1967] for a discussion of the Noether theorem.
However one can not treat the electromagnetic field in this formalism in a
simple way, as the gauge covariance of the system causes the Legendre
transformation to be non-regular. As we will see in the next section, such
problems do not occur in the approach discussed here.
\section{The Maxwell-Dirac and Maxwell-Schr\"odinger fields}
In this section we apply the formalism developed in the last section to the
Maxwell-Schr\"odinger and Maxwell-Dirac fields. We find that the same type of
Cartan 4-form can be used for both theories, allowing a simple algebraic
comparison. In the next section we go on to discuss invariances of the system,
in particular the conserved current $J$, the dynamical covariance of the
theory. As discussed in the last section, the state space
$\Sigma~=~(A_\mu,H^{\mu\nu},\psi,\psi^{\dagger},x^\mu)$
has dimension 22. Here the $x^\mu$ are the
coordinates of the space\-time model $M$ and $A=A_\mu dx^\mu$,
$H=\frac14H^{\mu\nu}\epsilon_{\mu\nu\rho\lambda}dx^\rho\wedge dx^\lambda$ are
the forms representing the electrodynamic degrees of freedom. The pseudo-tensor
$\epsilon_{\mu\nu\rho\lambda}$ is defined to be independent of coordinate system
with $\epsilon_{0123}=1$. Hence $H$ is odd in the sense of de Rham.
The form $H$ is related to the magnetic field 2-form $B=\frac12B_{\mu\nu}dx^\mu
\wedge dx^\nu$ by the phenomenological relations. We postulate the existence of
a function $L(x^\cdot,H^{\cdot\cdot})$ such that $B_{\mu\nu}={\partial
L\over\partial H^{\mu\nu}}$. In the linear case we have that
$L=\frac18\mu_{\mu\nu\rho\lambda}H^{\mu\nu}H^{\rho\lambda}$, the function
$\mu_{\mu\nu\rho\lambda}$ then inducing a Lorentz matrix $\hat{g}_{\mu\nu}$ on
$M$ by the relation $\mu_{\mu\nu\rho\lambda}=\hat{g}_{\mu\rho}
\hat{g}_{\nu\lambda}-\hat{g}_{\mu\lambda}\hat{g}_{\nu\rho}$.
For notational convenience we introduce the following odd forms:
$$\eta&:=\frac1{24}\epsilon_{\mu\nu\rho\lambda}dx^\mu\wedge dx^\nu\wedge
dx^\rho\wedge dx^\lambda,\cr
\eta_\mu&:=\frac16\epsilon_{\mu\nu\rho\lambda}dx^\nu\wedge
dx^\rho\wedge dx^\lambda,\cr
\eta_{\mu\nu}&:=\frac12\epsilon_{\mu\nu\rho\lambda}dx^\rho\wedge
dx^\lambda.\cr$$
Hence, for example, $H=\frac12 H^{\mu\nu}\eta_{\mu\nu}$.
We are now in a position to give the Cartan form $\omega$. Let $\alpha^\mu$ and
$u$ be $4\times4$ matrices, $L(x^\cdot,H^{\cdot\cdot})$ represent the
phenomenological relations as above and $J=J^\mu\eta_\mu$, where
$$J^\mu&:=e\psi^{\dagger}\alpha^\mu\psi.\cr$$
The Cartan 4-form is then defined by
$$w&=dA\wedge H-L\eta-A\wedge J-\i\hbar\psi^{\dagger}\alpha^\mu d\psi\wedge\eta_\mu
+\psi^{\dagger} u\psi\eta.\cr$$
The particular forms of the matrices $\alpha^\mu$ and $u$ express the
dynamical covariance of the theory. Hence the differences between the
theories are contained solely in the algebraic properties of these matrices,
allowing a simple comparison of the Galilei and Lorentz evolutions
and their limits into each other.
We now determine the corresponding equations of motion, using the principle that
$s^\ast(i_Xd\omega)=0$ for the motion described by $s:M\rightarrow\Sigma$, where
$X$ runs over all vector fields on $\Sigma$. As for the case of particle
mechanics this condition is trivially satisfied for the vector fields
$\partial_\mu$ on $M$ and so it is sufficient to consider those vector fields in
the kernel of $\pi_\ast$. We have
$$d\omega&=dA\wedge dH-B\wedge dH-dA\wedge J+A\wedge dJ-\i\hbar
d\psi^{\dagger}\alpha^\mu d\psi\wedge\eta_\mu+d(\psi^{\dagger} u\psi)\wedge\eta,\cr$$
where we have $dL\wedge\eta=\mathop{\Sigma}\limits_{\scriptscriptstyle\mu<\nu}
{\partial L\over\partial H_{\mu\nu}}dH^{\mu\nu}\wedge\eta=\frac12
B_{\mu\nu}dH^{\mu\nu}\wedge\eta=B\wedge dH$.
To obtain the equation of motion for $\psi$ we must insert the vector
$\partial_{\psi^{\dagger}}$ into $d\omega$. We find
$$0&=s^\ast(i_{\partial_{\psi^{\dagger}}}d\omega)\cr
&=s^\ast(-eA\wedge\alpha^\mu\psi\eta_\mu-\i\hbar\alpha^\mu d\psi\wedge\eta_\mu
+u\psi\eta)\cr
&=(-\i\hbar\alpha^\mu\partial_\mu\psi-e\alpha^\mu A_\mu\psi+u\psi)s^\ast\eta.
\cr$$
Now by definition $s^\ast\eta\not=0$, hence on the surface describing the
motion we have
$$\alpha^\mu(-\i\hbar\partial_\mu-eA_\mu)\psi+u\psi&=0.\cr$$
Similarly
$$0&=s^\ast(i_{\partial_\psi}d\omega)\cr
&=s^\ast(-eA\wedge\psi^{\dagger}\alpha^\mu\eta_\mu+
\i\hbar d\psi^{\dagger}\wedge\alpha^\mu
\eta_\mu +\psi^{\dagger} u\eta)\cr
&=(\i\hbar\partial_\mu\psi^{\dagger}\alpha^\mu-eA_\mu\psi^{\dagger}\alpha^\mu
+\psi^{\dagger} u)s^\ast\eta,\cr$$
so that
$$(\i\hbar\partial_\mu-eA_\mu)\psi^{\dagger}\alpha^\mu+\psi^{\dagger} u&=0.\cr$$
For the bosonic degrees of freedom the same reasoning leads to
$$0&=s^\ast(i_{\partial_{A_\mu}}d\omega)\cr
&=s^\ast(dx^\mu\wedge(dH-J)).\cr$$
Thus on the motion we have
$$dH&=J.\cr$$
Similarly
$$0&=s^\ast(i_{\partial_{H^{\mu\nu}}}d\omega)\cr
&=s^\ast((dA-B)\wedge\eta_{\mu\nu}),\cr$$
which implies that
$$dA&=B.\cr$$
It is a simple matter to show that the corresponding equations for the
components are $\partial_\nu H^{\mu\nu}=J^\mu$ and $\partial_\mu
A_\nu-\partial_\nu A_\mu=B_{\mu\nu}$.
The form $\omega$ is invariant under the one-parameter group of transformations
$\psi\mapsto\e^{\i\alpha}\psi$, $\psi^{\dagger}\mapsto\e^{-\i\alpha}\psi^{\dagger}$
generated by $X=\psi\partial_\psi-\psi^{\dagger}\partial_{\psi^{\dagger}}$.
Hence we find the closed form
$$i_X\omega&=\i\hbar\psi^{\dagger}\alpha^\mu\psi\eta_\mu=\frac{\i\hbar}eJ.\cr$$
The conservation of charge expressed by the vanishing of $dJ$
on the motion is then a direct consequence of a gauge invariance of the first
kind.
Further, the existence of such a current allows a quantum
interpretation of the theory, as integrating $J^0$ over $S$ defines an
inner product.
It remains to give specific forms for the matrices $\alpha^\mu$ and $u$. For
the Schr\"odinger case we will consider two sets of matrices
since this will allow a
simple interpretation of the non-relativistic limit of the Dirac equation.
First, let us take
$$\alpha^0_I&=\mat{I&0\cr0&0},\qquad\alpha^i_I=
\mat{0&\sigma^i\cr\sigma^i&0\cr},\qquad u_I=2m\mat{0&0\cr0&-I}.\cr$$
If we then write $\psi_I=\mat{\phi_I\cr\chi_I\cr}$ we find that
$$\i\hbar\partial_0\phi_I&=\sigma^i(-\i\hbar\partial_i-eA_i)\chi_I-eA_0\phi_I\cr
0&=\sigma^i(-\i\hbar\partial_i-eA_i)\phi_I-2m\chi_I.\cr$$
Substituting for $\chi_I$ and using the fact that
$(\sigma^if_i)(\sigma^jg_j)=g^{ij}f_ig_j+\i\epsilon^{ijk}f_ig_j\sigma_k$ we find
that
$$\i\hbar\partial_0\phi_I&=(\frac1{2m}g^{ij}(-\i\hbar\partial_i-eA_i)(-\i\hbar
\partial_j-eA_j)+\frac{\hbar e}{2m}\sigma_i B^i-eA_0)\phi_I,\cr$$
that is Schr\"odinger's equation for a particle with spin$\frac12$. Such a
linearisation of the Schr\"odinger equation was first discovered by J.-M.
L\'evy-Leblond [L\'evy-Leblond 1967] in an attempt to continue the Wigner
programme to the Galilei group.
The second choice is given by
$$\alpha^0_{II}&=\mat{0&0\cr0&I},\qquad\alpha^i_{II}=
\mat{0&\sigma^i\cr\sigma^i&0\cr},\qquad u_{II}=2m\mat{I&0\cr0&0}.\cr$$
If we then write $\psi_{II}=\mat{\phi_{II}\cr\chi_{II}\cr}$ we find that
$$0&=\sigma^i(-\i\hbar\partial_i-eA_i)\chi_{II}+2m\phi_{II}\cr
\i\hbar\partial_0\chi_{II}&=\sigma^i(-\i\hbar\partial_i-eA_i)\phi_{II}
-eA_0\chi_{II}.\cr$$
Substituting for $\phi_{II}$ gives
$$\i\hbar\partial_0\chi_{II}&=(-\frac1{2m}g^{ij}(-\i\hbar\partial_i-eA_i)
(-\i\hbar\partial_j-eA_j)-\frac{\hbar e}{2m}\sigma_i B^i-eA_0)\chi_{II
}.\cr$$
This is the ``negative-energy equivalent'' of the usual Schr\"odinger
equation.
On the other hand, for the Dirac case let us set
$$\alpha^0&=\mat{I&0\cr0&I},\qquad\alpha^i=c
\mat{0&\sigma^i\cr\sigma^i&0\cr},\qquad u=mc^2\beta,\cr$$
where $\beta=\mat{I&0\cr0&-I\cr}$. In this way we find the Dirac equation
$$\i\hbar\partial_0\psi&=(c\alpha^i(-\i\hbar\partial_i-eA_i)+mc^2\beta-eA_0)
\psi.\cr$$
Now each of the two Schr\"odinger equations above have
only two
linearly independent solutions, whereas the Dirac equation has four. We now show
that in the non-relativistic limit the Dirac motions of positive energy tend
to Schr\"odinger motions of the first type, whereas those of negative energy
tend to Schr\"odinger solutions of the second type.
The non-relativistic limit can be taken when the frequency and potential
energies
of the field is small compared to $mc^2$. If the solution has positive energy
then the second component $\chi$ is small with respect to the first, $\phi$, so
that we can ignore its frequency and potential energies with respect to
$\sigma^i(-\i\hbar\partial_i-eA_i)\phi$. This will lead to a Schr\"odinger like
evolution. Of course the converse will hold for solutions with negative energy.
Explicitly, consider a solution of positive energy. If we write
$\psi=(\phi,c^{-1}\chi)$ the Dirac equation takes the form
$$\i\hbar\partial_0\phi&=\sigma^i(-\i\hbar\partial_i-eA_i)\chi+(-eA_0+mc^2)
\phi,\cr
\i\hbar c^{-2}\partial_0\chi&=\sigma^i(-\i\hbar\partial_i-eA_i)\phi
+(-c^{-2}eA_0-m)\chi.\cr$$
We can now perform the gauge transformation
$\psi\mapsto\e^{-\i mc^2t/\hbar}\psi$ in the Dirac equation, corresponding to a
shift in the zero of energy. This leads to
$$\i\hbar\partial_0\phi&=\sigma^i(-\i\hbar\partial_i-eA_i)\chi-eA_0\phi,\cr
\i\hbar c^{-2}\partial_0\chi&=\sigma^i(-\i\hbar\partial_i-eA_i)\phi
+(-ec^{-2}A_0-2m)\chi.\cr$$
Thus we recover the first Schr\"odinger equation in the limit where
$(\i\hbar\partial_0+eA_0)\chi$ can be ignored with respect to $mc^2\chi$.
As we can see, in the non-relativistic limit the small components do not
approach zero, rather they approach a definite relationship with the large
components: $\chi\rightarrow\frac1{2m}\sigma^i(-\i\hbar\partial_i-eA_i)\phi$.
This is exactly the term one has in our Schr\"odinger equation; such
small components then have nothing to do with antiparticles.
On the other hand, if $\psi$ is a solution of negative energy we write
$\psi=(c^{-1}\phi,\chi)$. Performing the same steps as above with the gauge
transformation $\psi\mapsto\e^{\i mc^2t/\hbar}\psi$ leads to the equation
$$\i\hbar c^{-2}\partial_t\phi&=\sigma^i(-\i\hbar\partial_i-eA_i)\chi
+(-ec^{-2}A_0+2m)\phi\cr
\i\hbar\partial_0\chi&=\sigma^i(-\i\hbar\partial_i-eA_i)\phi
-eA_0\chi,\cr$$
which gives the second Schr\"odinger equation as a limit with the same
resulting conclusions.
\section{Invariances}
In this section we discuss the covariances of the Cartan form. As shown in
section 3, gauge invariance of the first kind
leads to a conserved current which permits the
quantum interpretation of the theory. Further we deduce the
dynamical covariance of the theory directly from the
Cartan form $\omega$.
We start with the conserved current. For the first
Schr\"odinger equation we have that
$$J^0_I&=e\phi^{\dagger}_I\phi_I,\cr
J^i_I&=e\phi^{\dagger}_I\sigma^i\chi_I+e\chi^{\dagger}_I\sigma^i\phi_I,\cr$$
whereas for the second we have
$$J^0_{II}&=e\chi^{\dagger}_{II}\chi_{II},\cr
J^i&=e\phi^{\dagger}_{II}\sigma^i\chi_{II}+e\chi^{\dagger}_{II}\sigma^i\phi_{II}.\cr$$
As mentioned above, if we integrate $J^0$ over $x^i$ at a fixed
$x^0$ we then find a scalar product
which allows us to construct the
quantum interpretation of the field $\psi$ existing in space.
For the Dirac equation, with $\psi=(\phi,\chi)$, we have
$$J^0&=e\phi^{\dagger}\phi+e\chi^{\dagger}\chi,\cr
J^i&=ec\phi^{\dagger}\sigma^i\chi+ec\chi^{\dagger}\sigma^i\phi.\cr$$
Here it is only the sum of the currents due to $\phi $ and
$\chi $ which is conserved.
Next we discuss the dynamical covariance of the two theories. Performing a
change of the spacetime model induces a change of coordinates
$x^\mu\mapsto x'{}^\mu$. The new expression obtained by pulling back this change
of coordinates in the Cartan 4-form $\omega$ will express the Galilean
equivalence principle if we can define new variables $A'{}_\mu$,
$H'{}^{\mu\nu}$, $\psi'$ and $\psi^\dagger{}'$ with the same interpretation and
an $\omega$ written in the same way. Restricting first to a change of
coordinates
with determinant unity (a proper Galilei or Lorentz transformation) we see
immediately that $\eta$ transforms as a scalar, $\eta_\mu$ as a covector and
$\eta_{\mu\nu}$ as a bicovector. This imposes that $A$, $H$ and $L$ be scalar
and justifies the usual transformation of the electromagnetic fields $A_\mu$,
$H^{\mu\nu}$ and $B_{\mu\nu}$. Note that the $\mu_{\mu\nu\rho\lambda}$ are only
invariant under Lorentz transformations. The transformations of $\psi$ and
$\psi^\dagger$ are more complicated; as we will see they turn out to be
bispinors.
For a rotation of the space coordinates we have that $\psi'=S\psi$ and
$\psi^\dagger{}'=\psi^\dagger S^\dagger$, where
$$S&=\mat{\e^{-\i n_i\sigma^i\theta/2}&0\cr
0&\e^{-\i n_i\sigma^i\theta/2}\cr}.\cr$$
Here $\overrightarrow{n}$ is the unit rotational axis and $\theta$ is the angle
of rotation. This is the usual bispinor transformation well known for the Dirac
theory; here the formula is valid for both Schr\"odinger cases.
For a boost the situation is more complicated and we must consider the two cases
separately. First we consider the first (usual) Schr\"odinger case, as the
calculations for the second follow the same lines. Here only Galilean
transformations can occur, so that for a boost (a pure transformation) we have
that
$$x'{}^0&=x^0,\cr
x'{}^i&=x^i+v^ix^0.\cr$$
The corresponding changes of $A_\mu$ and $H^{\mu\nu}$ can easily be seen to be
$$A'_0&=A_0-v^iA_i,\cr
A'_i&=A_i,\cr
H'{}^{0i}&=H^{0i},\cr
H'{}^{ij}&=H^{ij}+v^iH^{0j}-v^jH^{0i}.\cr$$
On the other hand
$$B'_{0i}&=B_{0i}-v^kB_{ki},\cr
B'_{ij}&= B_{ij}.\cr$$
As we can see, such a law of transformation is in contradiction with the
equivalence principle which imposes the invariance of $\epsilon_0$ and $\mu_0$
in the vacuum. Nevertheless we obtain the following result:
\theorem
For the first Schr\"odinger case, the form $\omega$ is left invariant by the
above transformation with, in addition, $\psi'=\e^{\i\hbar^{-1}f}S\phi$,
$\psi^{\dag}{}'=\psi^\dagger S^\dagger\e^{-\i\hbar^{-1}f}$,
where $f=\frac12mv_iv^ix^0+mv_ix^i$ and
$$S&=\mat{I&0\cr\frac12\sigma^iv_i&I\cr}.\cr$$
\proof
Let us first remark that $J_I^\mu$ transforms as a quadrivector.
A quick calculation shows that $S^\dagger\alpha_I^0S=\alpha_I^0$ and
$S^\dagger\alpha^iS=\alpha^i+v^i\alpha_I^0$, where we use the fact that
$\frac12(\sigma^i\sigma_kv^k+\sigma_k\sigma^iv^k)=v^i$. Here $S$ is by
analogy the $(\frac12,0)$ bispinor projective
representation of the Galilei group,
which comes with a pure phase factor $\frac12mv_iv^ix^0$ and a gauge
transformation corresponding to a momentum translation [Piron 1976, 1990]. Next
$S^\dagger U_IS=U_I-(\frac12mv^iv_i\alpha_I^0+mv_i\alpha^i)$, where we use the
fact that $\sigma^iv_i\sigma_jv^j=v^iv_i$. Finally, using the first remark
$$-\i\hbar\psi^\dagger{}'\alpha_I^\mu d\psi'\wedge\eta_\mu'&=
-\i\hbar\e^{-\i f}\psi^{\dagger}\alpha_I^\mu d(\e^{\i f}\psi)\wedge\eta_\mu\cr
&=-\i\hbar\psi^{\dagger}\alpha_I^\mu
d\psi\wedge\eta_\mu+\psi^\dagger(\frac12mv^iv_idx^0+mv_idx^i)\wedge
\alpha_I^\mu\psi\eta_\mu\cr
&=-\i\hbar\psi^{\dagger}\alpha_I^\mu
d\psi\wedge\eta_\mu+\psi^\dagger(\frac12mv^iv_i
\alpha_I^0+mv_i\alpha^i)\psi\eta.\cr$$
Hence the sum $-\i\hbar\psi^{\dagger}\alpha_I^\mu
d\psi\wedge\eta_\mu+\psi^{\dagger} U_I\psi\eta$ is invariant, completing the
proof.
\stop
For the second case a similar calculation shows that the 4-form $\omega$ is left
invariant for $\psi'=\e^{-\i\hbar^{-1}f}S\psi$, $\psi^\dagger{}'=\psi^\dagger
S^\dagger\e^{\i\hbar^{-1}f}$, where $f$ is the same but
$$S&=\mat{I&\frac12\sigma^iv_i\cr0&I\cr}.\cr$$
We can also consider
the space inversion $x^i\mapsto-x^i$, a transformation with negative
determinant. This coordinate change is induced by the passage from one
spacetime model to its dual.
Using the fact that $A$ is even and $H$ odd, the pullbacks of
$A_\mu$ and $H^{\mu\nu}$ are
$$A_0'&=A_0,\cr
A_i'&=-A_i,\cr
H^{0i}{}'&=-H^{0i},\cr
H^{ij}{}'&=H^{ij}.\cr$$
The corresponding transformation for $\psi$ and $\psi^\dagger$ turn out
to be the
same for the Dirac and Schr\"odinger cases:
\theorem
$\psi'=S\psi$, $\psi^\dagger{}'=\psi^\dagger S^\dagger$, where
$$S&=\mat{I&0\cr0&-I\cr}.\cr$$
\proof
By direct computation we have $\psi^\dagger{}'\alpha^i\psi'
=-\psi^\dagger\alpha^i\psi$.
Similarly for the Dirac and both Schr\"odinger cases
$\psi^\dagger{}'\alpha^0\psi'
=\psi^\dagger\alpha^0\psi$. Hence
Next, since $\eta_0\mapsto-\eta_0$ and $\eta_i\mapsto\eta_i$ one has that
$-\i\hbar\psi^\dagger{}'\alpha^\mu d\psi'\wedge\eta_\mu'=\i\hbar\psi^\dagger
\alpha^\mu d\psi\wedge\eta_\mu$. Finally $S^\dagger uS=u$,
completing the proof.
\stop
We are now in a position to give an interpretation of the negative energy
Schr\"odinger equation. If we perform a space inversion and an inversion
of the time coordinate $x^0$ in the equations of motion for $\phi_{II}$
and $\chi_{II}$ we
obtain the usual first Schr\"odinger case but with $e$ replaced by
the charge $-e$ of the antiparticle. This is an exact analogue
of the interpretation
of the negative energy solutions in the Dirac case.
\section{Conclusion}
We have given a rigorous foundation for a ``variational" treatment
of field theory
in terms of motions given by submanifolds of the state manifold. This leads
naturally to the definition of a spacetime model, that is a 4-manifold
constructed from the physical space and time which allows a
differential geometric approach to the physics of the fields.
In the context of the Maxwell-Schr\"odinger and Maxwell-Dirac fields we find
that the Cartan formalism allows a unified treatment, giving a simple
comparison of the two theories. Further, dynamical covariance and the CPT
transformation become natural consequences of the form $\omega$ used, the
latter arising due to the existence of a canonical dual to any spacetime model.
Finally, the existence of conserved currents allows the construction of a scalar
product permitting the quantum interpretation of the theory. In such an
interpretation the first two components of a positive energy solution
characterise the scalar product.
Nevertheless the two other components are not zero and describe dynamical
variables.
\bigbreak\noindent
{\bf Appendix}
\medskip\noindent
{\sectionfont Propositional systems and spacetime models}
\bigskip
In this appendix we will discuss the ideas behind the axiomatic approach to
physical theories of Aerts and Piron. Further details can be found in Aerts
[1982] and Piron [1990], for a review see Piron [1989].
We then go on to explain spacetime models within
this framework, and in particular how the differential and metric structure of
spacetime models is induced by the dynamics of the fields considered.
Propositional systems are based on the notion of experimental project. These are
real experiments that one can eventually perform upon the system, where one has
chosen the result or results of interest in advance. We ascribe the response
``yes'' if we obtain one of these results and ``no'' otherwise. Note that we do
not require these experiments to be in any way ideal. The experimental project
is then called true if the response ``yes'' would be certain if we were to
perform the experiment. Two experimental projects are then equivalent if one of
them is true if and only if the other is. The corresponding equivalence classes
are called properties of the system. They represent what one can do with the
system and are elements of reality in the Einstein sense
[Einstein \etal\ 1935]. If
the experimental projects defining a property are true then we call the property
actual, otherwise it is called potential. It is important to emphasise that a
system can very well possess an Einstein element of reality before one
performs the corresponding
experiment and even if we have decided not to perform it.
The state of the system is by definition the complete set of all actual
properties. If one knows the state of the system one knows everything that
can be done to the system; this is the realistic point of view.
If one then imposes some very plausible
physical hypotheses, one can prove that the set of properties is not only a
complete lattice, but is also
atomistic and orthocomplemented. The set of possible states of the system
is then in one to one correspondance
with the atoms of the lattice (the minimal non-trivial properties of
the system).
Here the orthocomplement is defined as usual by the orthogonality
relation. Such a relation is physically constructed by defining
two states to be orthogonal if there exists an experimental
project which is true in one of the states
but impossible for the other. Hence two possible orthogonal
states can be distinguished by performing one experiment only.
A property is called classical if for each possible state
either the property or its
orthocomplement is actual. The set of all such properties is itself a complete
and atomistic sublattice whose atoms will be called macroscopic
states and will define the
superselection variables of the system and in this sense any system appears
at first sight to be classical. If a system
has only the two classical properties 0 and
I we will say that it is purely quantum. The power of these definitions
is that one can
write the property lattice as a family of purely quantum lattices indexed by the
superselection variables.
Some systems, which will be called entities, cannot be divided into
separate parts and satisfy weak
modularity and the covering law. In this case one can show that
a purely quantum
lattices can be described by the lattice of closed
subspaces of a Hilbert space.
Next we define observables as morphisms from a complete Boolean
lattice, linked with the scale of the apparatus, to the given property
lattice. In a
Hilbert space one can prove that each observable can be realised
as the joint spectral family
of some commuting self-adjoint operators.
These concepts now allow us to define
elementary particles in a group theoretical way. Let us consider
as elementary particle, that is an entity whose
only independent observables are just the time, position and momentum.
These observables must
satisfy certain covariance relations allowing us to choose freely the
zeros of the apparatus.
Using group theoretical considerations one finds that there are only two such
models of elementary particles. The first case is the classical
point particle, where the time,
position and momentum are all superselection variables. The second is the
quantum elementary particle, where just the time is a superselection variable,
each of the purely
quantum sublattices is ${\rm L}^2({\Bbb R}^3,d^3x)$ and the position
and momentum satisfy the usual commutation rules. If
we consider a more general system, a particle with intrinsic angular momentum
but no internal degrees of freedom, we find two new models, the spin$\frac12$
particles.
In this paper we are interested in the field created by a quantum particle,
this field being a property (element of reality) of space. We
would like to
reemphasise that the existence of a property does not depend on whether
or not we
have performed the corresponding experimental project or on the
existence of hypothetical test particles. This does not presuppose
the existence
of some kind of substance or ether filling the vacuum and responsible for the
field properties as in the philosophy of Leibnitz and Descartes. It is only
by abandonning this point of view once and for all that we begin to
understand our real physical world.
The bosonic degrees of freedom are described by pairs of differential forms on
space indexed by the time. As shown by Barut \etal\ [1993], we can then describe
the motion of the field in terms of differential forms on a constructed
spacetime model. Note that dynamics is used in the very construction of the
spacetime model; it should not come as a surprise to learn that the rest of the
structure of spacetime models is also dynamically induced. That geometry can be
induced from dynamics is not a concept restricted to general
relativity. For example in
classical mechanics Maupertuis' principle states
that the motion of a classical particle restricted to a fixed energy $E$
and governed by the
Hamiltonian $H=\frac1{2m}p^2+V(q)$ is a geodesic for the metric
$ds^2=2m(E-V(q))dq^2$. Similarly Cartan has shown that for any
such Hamiltonian, even time-dependent, the corresponding motion can be
interpreted as a straight line for a Galilean connection associated to
the geometry of the spacetime.
The geometric properties of the space vacuum are
therefore manifestations of the field dynamics. Note also that the
fact that we treat time and space separately as primitive notions
is not in conflict
with the relativity principles, since its covariance is realised at the
dynamical level.
The spacetime model is constructed by pasting a chosen coordinate system for
space to a time coordinate defined with the orientation
of the flow of time.
The generalised Galilei principle states that the laws of physics
should be equally well formulated for any choice of coordinate system, in
particular for right- or left-handed coordinates. Hence any
spacetime model has a dual, the passage from one to the other
being accomplished by the CPT transformation.
\acknowledgements
The authors would like to thank Professor L. Horwitz for many useful
discussions.
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