\magnification 1200
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\centerline {\bf Off-diagonal Long Range Order} 
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\centerline {\bf and Superconductive Electrodynamics}
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\centerline {{by Geoffrey L. Sewell}\footnote{$^{(a)}$}{Electronic mail:
G.L.Sewell@qmw.ac.uk}}
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\centerline {Department of Physics, Queen Mary and Westfield
College}
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\centerline {Mile End Road, London E1 4NS}
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\centerline {\bf Abstract}
\vskip 0.5cm\noindent
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We present a general, model independent, quantum statistical
derivation of superconductive electrodynamics from the
assumptions of off-diagonal long range order (ODLRO), local gauge
covariance and thermodynamic stability. On this basis, we obtain
the Meissner and Josephson effects, the quantisation of trapped
magnetic flux, and the metastability of supercurrents. A key to
these results is that the macroscopic wave function, specified
by the ODLRO condition, enjoys the rigidity property that
London$^{1,2}$ envisaged for the microstate of a superconductor. 
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PACS Numbers 74.20.-z; 64.60.My; 05.30.-d
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\centerline {\bf I.Introduction} 
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The object of this article is to provide a model-independent
derivation of the electromagnetic properties of superconductors
from their order structure, the essential imput being the
assumptions of off-diagonal long range order, gauge covariance
and thermodynamic stability. Thus, our approach to the subject
is centred on very general principles, and is therefore
at the opposite pole from that provided by the computational
techniques of many-body theory.
\vskip 0.2cm
In order to explain the need for such an approach, let us first
recall that, in the case of metallic superconductivity, the
electron pairing hypothesis$^{3,4}$ has been amply substantiated,
at an empirical level, by the the measured values of the magnetic
flux quantisation$^5$ and of the Josephson tunnelling
frequency$^6$. Furthermore, the widely accepted microscopic
theory of superconductivity, devised by Bardeen, Cooper and
Schrieffer (BCS)$^7$ on the basis of this hypothesis, provides
an accurate picture of the thermodynamics$^{8-10}$ of
superconductors, especially of their second order phase
transitions. On the other hand, the electrodynamics of this
theory is seriously flawed in that, firstly, it violates the
principle of local gauge covariance$^{11,12},$ and consequently
does not even admit a precise definition of the local current
density; and, secondly, there is no indication that its ansatz
for current-carrying states have the metastability
properties$^{13,14}$ of supercurrents. In fact, the violation of
the
gauge principle by the BCS theory arises from its {\it
truncation} of a fully gauge invariant model
(Fr\"ohlich's electron-phonon system$^{15}$), that retains only
the interactions that give rise to the electron pairing. 
Attempts$^{16,17}$ to remedy the situation by taking some account 
of the residual interactions have led to derivations of the Meissner
effect that are only {\it approximately} gauge covariant. Since
exact gauge covariance is required for a consistent
electrodynamics, this is no solution of the problem.
As regards ceramic, i.e. high $T_{c},$ superconductors, the
microscopic theory is less developed, and has certainly not led
to an electrodynamics, even though various interesting ideas 
concerning its underlying quantum mechanisms have been proposed, 
e.g. in Refs. 18-23.
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Thus, there is a need for a quantum-based gauge covariant
electrodynamics of the superconducting phase. Since, at an
empirical level, this electrodynamics has such sharply defined
{\it qualitative} characteristics, common to the vast variety of
superconducting materials, a corresponding quantum theory should
surely isolate their origins in general, qualitative terms. We
remark here that the traditional techniques of many-body
theory$^{24,25}$ are unsuited to this purpose, since these are
designed for essentially approximative calculations rather than
precise classifications.
\vskip 0.2cm
In view of these considerations, we have adopted a different
approach$^{26-28}$ to superconductive electrodynamics, based on
the hypothesis of {\it off-diagonal long range order} (ODLRO),
which encapsulates the qualitative features of the electron
pairing in a gauge covariant way. This hypothesis was proposed
by Yang$^{29}$ as a characterisation of the structure of the
superconducting phase, following a similar proposal by
O.Penrose$^{30,31}$ in connection with the theory of superfluid
Helium. Formally, the ODLRO condition may be expressed in terms
of the relevant quantised matter field ${\psi}$ by the formulae
$$\lim_{{\vert}y{\vert}\to{\infty}}{\lbrack}{\langle}
{\psi}(x){\psi}^{\star}(x+y){\rangle}-{\Phi}(x)
{\Phi}^{\star}(x+y){\rbrack}=0 \ for \ bosons, \eqno(1.1a)$$
and
$$\lim_{{\vert}y{\vert}\to{\infty}}\bigl[{\langle}
{\psi}_{\uparrow}(x_{1})
{\psi}_{\downarrow}(x_{2})
{\psi}_{\downarrow}^{\star}(x_{2}^{\prime}+y)
{\psi}_{\uparrow}^{\star}(x_{1}^{\prime}+y){\rangle}
-{\Phi}(x_{1},x_{2}){\Phi}^{\star}(x_{1}+y,x_{2}+y)\bigr]$$
$$=0 \ for \ fermions, \eqno(1.1b)$$
where, in both cases, ${\Phi}$ is a complex-valued function,
often termed the {\it macroscopic wave function}, such that
${\Phi}(x+y),$ or ${\Phi}(x_{1}+y,x_{2}+y),$ does not tend to
zero as ${\vert}y{\vert}{\rightarrow}{\infty}.$ In the
bosonic case, ODLRO is simply a generalisation of Bose-
Einstein condensation$^{30,31}$ to interacting systems.
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We note here that the ODLRO condition is fulfilled not only by
the BCS ansatz, but also some of those for ceramic
superconductivity$^{19,21}$, as well as that of Feynman$^{32}$
for superfluid Helium (cf Ref. 31). It is also satisfied in the
low temperature phases of a number of tractable, gauge
covariant models, namely the ideal Bose gas$^{33}$, the hard
sphere Bose fluid on a lattice$^{34}$, and, at least at zero
temperature, the Hubbard model with {\it attractive} interaction
between electrons on the same site$^{35}$.
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Our project, then, is to derive the principal electrodynamic
properties of superconductors from precisely specified
assumptions of ODLRO, gauge covariance and thermodynamic
stability. In fact, progress towards this objective has already
been achieved in Refs. 26-28 and 36, which have provided
derivations of the Meissner effect$^{26},$ the quantisation of
trapped magnetic flux$^{27,36},$ and the Josephson effect$^{27},$
as well as a sketched approach to the theory of persistent
currents$^{28}$.
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The present article will be devoted to a review and further
development of this work. Our essential aim will be to provide
a coherent account of the way in which the electrodynamics of
superconductors, including the metastabilty of the supercurrents,
stems from their order structure. We shall keep the mathematics
quite simple, formulating the theory within the standard second
quantisation framework of condensed matter physics. We remark
here that one may put the whole theory onto a completely rigorous
basis by recasting it, as in Ref. 27, within the framework of
operator algebraic statistical mechanics.
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We shall organise the presentation of the theory as follows.
We start, in Sec. II, with a general formulation of our 
gauge-covariant model of matter in interaction with a classical
magnetic field. This formulation covers the cases of both
continuous and lattice systems. A key result here is that the
dynamics of the model, in the presence of a uniform magnetic
induction $B,$ is covariant with respect to the {\it regauged
space translations}$^{37,26}$ 
$${\psi}(x){\rightarrow}{\psi}(x+a){\exp}\bigl({-ie(B{\times}a).x
\over 2{\hbar}c}\bigr),\eqno(1.2)$$
where $a$ is an arbitrary displacement and the sinusoidal factor
arises from the corresponding regauging of the magnetic vector
potential. 
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In Sec. III, we employ this result to derive the Meissner
effect, as in Ref. 26. The essential point is that the formula
(1.2) for regauged space translations, together with the 
assumption of ODLRO, leads to the conclusion that {\it either}
${\Phi}$ {\it or} $B$ must vanish. The requirement of
thermodynamic stability of the ordered phase then implies that
the latter condition prevails, for sufficiently weak applied
fields. Thus, we have a Meissner effect, since normally
diamagnetic systems can accommodate non-zero uniform induction.
The key to this result is that the macroscopic wave function
exhibits a rigidity, in the face of the applied field, of the
kind envisaged by London$^1$ at a more microscopic level. It is
worth remarking here that the principle of local gauge
covariance, which was an obstacle to the previous theories, is
an essential ingredient of our argument leading to the Meissner
effect!
\vskip 0.2cm
In Sec. IV, we extend that argument, along the lines of Ref. 36,
to derive the quantisation of trapped magnetic flux in
multiply-connected systems.
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In Sec. V, we formulate the theory of persistent currents in a
multiply-connected body, such as a ring. Here, the supercurrents
are none other than the currents that implement the Meissner
effect by screening the trapped magnetic flux from the interior
of the body$^2,$ and the essential problem is that of the
stabilisation of both the trapped flux and its screening current.
In fact, this is a problem of {\it metastability}, since the
current-carrying state has higher free energy than that of true
thermal equilibrium at the same temperature. Thus, adopting our
earlier characterisation$^{14}$ of metastable states by
thermodynamic stability against strictly local, rather than
global, disturbances, we formulate the condition for the
persistence of currents in terms of a variational principle, that
is subject to the constraint of the flux quantisation. In this
way, we characterise the phenomenon of superconductivity itself,
i.e. the persistence of currents, by a relatively simple
thermodynamic assumption, together with that of ODLRO.
Furthermore, we show that this phenomenon is intimately related
to a superselection rule, that forbids locally induced
transitions between states with different flux quantum numbers.
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In Sec. VI, we formulate the Josephson effect and derive its
tunnelling frequency form the properties of the macroscopic
wave function.
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We conclude, in Sec. VII, with a brief discussion of open
problems of the theory.
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\centerline {\bf II. The General Model}
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Our model, ${\Sigma},$ is an infinitely extended system,
consisting of electrons and possibly another species of
particles, e.g. phonons or ions, in a space, $X,$ which may be
either a Euclidean continuum or a lattice. Points in $X$ will
usually be denoted by $x,$ sometimes by $y,a$ or $b.$ We shall
denote the electronic component of ${\Sigma}$ by ${\Sigma}_{el},$
and the other component, if any, by ${\Sigma}^{\prime}.$ We shall
assume that the dynamics of the model is covariant w.r.t. gauge
transformations of both first and second kind, space translations
and time reversals. These assumptions represent general demands
of quantum mechanics and electromagnetism, and are fulfilled by
particular models, such as those of Fr\"ohlich$^{15}$ and
Hubbard$^{38}$, on which theories of metallic and high $T_{c}$
superconductivity, respectively, are based.
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For simplicity, we shall generally employ a notation appropriate
to the case where $X$ is a continuum. This may easily be
translated into the corresponding one for lattice case by
standard procedures employed in gauge theories$^{39}$.
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We shall describe the electronic subsystem ${\Sigma}_{el}$ in
terms of a quantised field, 
$${\psi}=\pmatrix{{\psi}_{1}\cr {\psi}_{-1}\cr}{\equiv}
\pmatrix{{\psi}_{\uparrow}\cr {\psi}_{\downarrow}\cr},
\eqno(2.1)$$
which satisfies the canonical anticommutation relations
$$[{\psi}_{s}(x),{\psi}_{s^{\prime}}^{\star}(x^{\prime})]_{+}=
{\delta}_{ss^{\prime}}{\delta}(x-x^{\prime}); \ 
[{\psi}_{s}(x),{\psi}_{s^{\prime}}(x^{\prime})]_{+}=0\eqno(2.2)$$
We shall sometimes use the symbol ${\psi}^{\#}$ to denote
either ${\psi}$ or ${\psi}^{\star}.$ 
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The observables$^{40}$ of ${\Sigma}_{el}$ are formed from
the polynomials in ${\psi}$ and ${\psi}^{\star}$ that are
invariant under gauge transformations of the first
kind, ${\psi}{\rightarrow}{\psi}e^{i{\alpha}},$ with ${\alpha}$
constant. Thus, they are generated algebraically by operators
of the form 
$${\psi}_{s_{1}}^{\star}(x_{1}).. \
.{\psi}_{s_{n}}^{\star}(x_{n}){\psi}_{s_{n+1}}(x_{n+1}).. \
.{\psi}_{s_{2n}}(x_{2n}).$$ 
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We shall be concerned with the properties of the the system in
the presence of a classical electromagnetic field $(E,B),$
represented by a scalar potential ${\phi}$ and a vector
potential, $A:$ thus, $E=-{\nabla}{\phi}-c^{-
1}{\partial}A/{\partial}t$ and $B=curlA.$ We assume that the
dynamics of the model is covariant w.r.t. gauge transformations
of the second kind, as given by the formula
$$A{\rightarrow}A+{\nabla}{\chi}, \ {\phi}{\rightarrow}
{\phi}-c^{-1}{{\partial}{\chi}\over {\partial}t}, \ 
{\psi}{\rightarrow}{\psi}{\exp}({ie{\chi}\over {\hbar}c})
\eqno(2.3),$$
where ${\chi}$ is an arbitrary function of position and time. We
assume that the observables representing the position-dependent
densities of electronic charge, current and magnetic polarisation
in the presence of this field are given by the standard formulae
$${\rho}=-e{\psi}^{\star}{\psi}; \ j(x)=
{ie\over 2}({\psi}^{\star}{\nabla}{\psi}-
({\nabla}{\psi}^{\star}){\psi})+{e\over c}A{\psi}^{\star}{\psi};
\ and \ m={e{\hbar}\over mc}{\psi}^{\star}{\sigma}{\psi},
\eqno(2.4)$$
where $-e$ is the electronic charge and ${\sigma}$ is the spin
vector, whose components
$({\sigma}_{1},{\sigma}_{2},{\sigma}_{3})$
are the Pauli matrices. We note here that ${\rho}, \ j$ and $m$
are invariant w.r.t. all gauge transformations.
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We assume that the formal Hamiltonian of the model is of the form
$$H={\int}({\hbar}{\nabla}{\psi}^{\star}+
{ie\over c}A{\psi}^{\star}).({\hbar}{\nabla}{\psi}-{ie\over
c}A)dx-
{\int}(B.m+{\rho}{\phi})dx+V_{el}+H_{int}+H^{\prime},\eqno(2.5)$$
where $V_{el}$ is the potential energy of the interelectronic
interactions, $H^{\prime}$ is the Hamiltonian for
${\Sigma}^{\prime}$ and $H_{int}$ is the energy of interaction
between ${\Sigma}_{el}$ and ${\Sigma}^{\prime}.$ We assume that
these three contributions to $H$ are all independent of the
potentials ${\phi}$ and $A,$ and thus that $H$ is gauge
invariant. Further, we stipulate that the magnetic interactions
between the electrons, that stem from the sources $j(x)$ and
$m(x)$ according to Maxwell's equations, are not incorporated
into either $V_{el}$ or $H_{int},$ but are represented by the
dependence of $B$ on these sources (cf. Comment at the end of
this Section). In the theory that follows, we shall confine our
considerations to situations where $B$ is static, $E=0,$ and,
except in Sec. 6, we shall take it that ${\phi}=0$ and ${\chi}$
is time-independent. 
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We shall assume that the dynamics is covariant w.r.t. space
translations and time reversals. The transformations of the
fields ${\psi}, \ A,$ corresponding to space translations
$a({\in}X),$ are given by  
$$A(x){\rightarrow}(x+a), \
{\psi}(x){\rightarrow}{\psi}(x+a).\eqno(2.6)$$
The operation of time reversal serves to transform ${\psi}_{s},
\ {\psi}_{s}^{\star}$ and $A$ to ${\psi}_{-s}^{\star}, \
{\psi}_{-s}$ and $-A,$ respectively, and to invert the order of
the terms in the operator products. Thus, its effective
action on the electronic observables and vector potential is
given by 
$${\psi}_{s_{1}}^{\#}(x_{1}). \
.{\psi}_{s_{n}}^{\#}(x_{n})
{\rightarrow}{\psi}_{-s_{n}}^{{\#}{\star}}(x_{n}). \ .
{\psi}_{-s_{1}}^{{\#}{\star}}(x_{1}); \ A{\rightarrow}
-A.\eqno(2.7)$$
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Specialising now to the case where the magnetic induction $B$ is
uniform and so may be represented by the vector potential
$A(x)={1\over 2}B{\times}x,$ and choosing ${\chi}(x)=-
(B{\times}x).a,$ we have the relation
$A(x)+{\nabla}{\chi}(x)={\chi}(x-a).$ Hence, by (2.3) and (2.6),
the dynamics of ${\Sigma}$ is covariant w.r.t. the
transformation$^{37,26}$ 
$${\psi}(x){\rightarrow}{\psi}_{a}(x){\equiv}
{\psi}(x+a){\exp}({-ie(B{\times}x).a)\over e{\hbar}c}), \
A(x){\rightarrow}A(x),\eqno(2.8)$$
together with the transformation of the ${\Sigma}^{\prime}-$
observables corresponding to space translations. Since (2.6)
consists of a space translation, compensated by a gauge
transformation in such a way as to leave $A$ unchanged, we term
it a {\it regauged space translation}.
\vskip 0.2cm
We denote by $Q_{t}$ the time-translate, in Heisenberg
representation, of an arbitrary observable, $Q,$ of ${\Sigma}.$
A dynamical characterisation of thermal equilibrium states of
the system at inverse temperature ${\beta}$ is given by the Kubo-
Martin-Schwinger (KMS) condition$^{41,42}$, i.e.,
$${\langle}Q_{t}Q^{\prime}{\rangle}_{A}=
{\langle}Q^{\prime}Q_{t+i{\hbar}{\beta}}{\rangle}_{A},
\eqno(2.9)$$
for arbitrary observables $Q$ and $Q^{\prime},$ where the angular
brackets denote expectation value and the suffix $A$ indicates
its dependence on $A$. Most importantly, this condition is valid
even for infinite systems$^{43}$, where the traditional Gibbs
canonical formulation is not directly applicable. In the case
where $B$ is uniform, the covariance of the dynamics w.r.t regauged 
space translations ensures that the model supports 
equilibrium states that are invariant under these translations.
\vskip 0.2cm
Turning now to the thermodynamics of the model, we define
$f(B,T)$ to be its global free energy density at temperature $T,$
under the action of the uniform induction, $B.$ This function can
be formulated by standard statistical mechanical methods$^{44}$
in terms of the microscopic description of ${\Sigma},$ with $B$
taken to be a given control variable.
\vskip 0.2cm
On the other hand, this induction is {\it not} given, a priori,
in the physical situation where ${\Sigma}$ is subjected to a
magnetic field, $H_{ex},$ due to a fixed source of current
density $J_{ex}.$ For, in this situation, the induction, $B,$ and
the equilibrium state of ${\Sigma}$ co-determine one another.
\vskip 0.2cm
To formulate the thermodynamics of ${\Sigma}$ under these
conditions, we have to incorporate the energy of the field, $B,$
and of its interaction with the source, $J_{ex},$ into the
picture. Thus, we have to consider the system under consideration
to comprise the composite, ${\tilde {\Sigma}},$ of ${\Sigma}$ and
the field $B.$ The contribution to the internal energy of
${\tilde {\Sigma}}$ due to the induction, $B,$ and its coupling
to the source is then
$${\int}({1\over 2}B^{2}-c^{-1}A.J_{ex})dx,$$
and, in view of the Maxwell equation $curlH_{ex}=c^{-1}J_{ex},$ 
this reduces to
$${\int}({1\over 2}B^{2}-H_{ex}.B)dx.\eqno(2.10)$$
Thus, in the case where $B$ and $H_{ex}$ are uniform, the free
energy density of ${\tilde {\Sigma}}$ is
$${\phi}(B,T,H_{ex})=f(B,T)+{1\over
2}B^{2}-B.H_{ex}.\eqno(2.11)$$
The equilibrium value of $B,$ corresponding to given $H_{ex}$ and
$T,$ is then obtained by minimising ${\phi},$ and the resultant
Gibbs free energy density is
$${\tilde f}(H_{ex},T)=min_{B}{\phi}(B,T,H_{ex}).\eqno(2.12)$$
\vskip 0.2cm
{\bf Comment.} This formulation of the thermodynamics of ${\tilde
{\Sigma}}$ is, of course, semi-classical, since the field $B$ is
treated as classical. In this picture, the field energy (2.10)
stems 
from the magnetic interactions of the currents and spins in
${\Sigma}$
both with one another and with the external source. For a
discussion  
of the problem of a fully quantum formulation, see Ref. 27,
Sec.4.
\vfill\eject
\centerline {\bf III. ODLRO and the Meissner Effect.}
\vskip 0.2cm
In order to formulate ODLRO, we introduce the {\it pair field}
$${\Psi}(x_{1},x_{2})={\psi}_{\uparrow}(x_{1})
{\psi}_{\downarrow}(x_{2}).\eqno(3.1)$$
We then say that a state of ${\Sigma}$ possesses the property of
ODLRO if there is a {\it classical} two-point field
${\Phi}(x_{1},x_{2})$ such that$^{29}$
$$\lim_{{\vert}y{\vert}\to{\infty}}\bigl[{\langle}
{\Psi}(x_{1},x_{2})
{\Psi}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y){\rangle}_{A}-
{\Phi}(x_{1},x_{2})
{\Phi}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)\bigr]=0,\eqno(3.2)$$
and, further, ${\Phi}(x_{1}+y,x_{2}+y),$ does not tend to zero
as ${\vert}y{\vert}{\rightarrow}{\infty}.$ In this case, ${\Phi}$
is termed the {\it macroscopic wave function} of the state.
\vskip 0.2cm
{\bf Note.} Although ${\Psi}$ is not an an observable, in the
sense specified in Sec. II, the quantity in angular brackets
in the ODLRO condition (3.2) is.
\vskip 0.2cm
In order to relate ODLRO to superconductive electrodynamics, we
note that the essential distinction between normal diamagnetism
and the Meissner effect is that the former can support a uniform,
static, non-zero magnetic induction and the latter cannot. Thus,
we base our derivation of the Meissner effect on considerations
of the response of a state possessing ODLRO to the action of a
uniform magnetic field.
\vskip 0.2cm
The following two Propositions, which we shall prove at the end
of this Section, provide us with the key to the relationship
between ODLRO and the electromagnetic properties of our model. 
\vskip 0.2cm
{\bf Proposition 3.1.} {\it (1) The ODLRO condition (3.2) defines
the macroscopic wave function ${\Phi},$ up to a phase factor,
i.e., if ${\Phi}_{1}$ and ${\Phi}_{2}$ are two such functions
which satisfy this condition for the same state, then
${\Phi}_{1}=e^{i{\alpha}}{\Phi}_{2},$ where ${\alpha}$ is
some real constant.
\vskip 0.2cm
(2) In the case where $A=0$ and where the state is invariant
with respect to time reversals, the function ${\Phi}$ is
real-valued, up to a constant phase factor.}
\vskip 0.2cm
{\bf Proposition 3.2.} {\it If the system is in a translationally
invariant state possessing the property of ODLRO, then
\vskip 0.2cm
(1) in the case where $X$ is a continuum, the magnetic induction
$B$ vanishes, and
\vskip 0.2cm
(2) in the case where $X$ is a lattice, $B$ is restricted to the
discrete set of values ${\lbrace}B^{(n)}{\rbrace},$ given by the
condition that $B^{(n)}.(a{\times}b)$ is an integral multiple of
${\pi}e/{\hbar}c,$ for any $a, \ b$ in $X.$ Thus, the vectors
$B^{(n)}$ are the sites of an associated lattice, ${\cal B},$
defined by this condition.} 
\vskip 0.2cm
{\bf Comments.} (1) Prop. 3.2 demonstrates that, in the
continuum case, translationally invariant (including equilibrium)
states with ODLRO do not admit uniform magnetic fields, i.e.,
they exhibit the Meissner effect. 
\vskip 0.2cm
(2) In the case where $X$ is a lattice, this conclusion must be
modified by the possibility that ODLRO might be compatible with
a non-zero quantised induction $B^{(n)}.$  
\vskip 0.2cm
(3) The question of whether ODLRO, with or without the quantised
magnetic induction $B^{(n)},$ prevails is a thermodynamic one.
\vskip 0.2cm
(4) The removal of ODLRO by even infinitessimal changes in $B$
from the value $0$ or, in the lattice case, $B^{(n)}$ suggests
the
advent of a phase transition.
\vskip 0.2cm
(5) In the case of lattice systems, the non-zero $B^{(n)}$'s are
of the order of ${\hbar}c/el^{2},$ where $l$ is the spacing of
the lattice $X.$ Thus, for typical values of $l,$ e.g.
$10^{-8}$cm., they are of the order of $10^{9}$G, which is not
only many orders of magnitude larger than any known critical
fields for superconductors, but also much larger than the
internal fields in ferromagnets.
\vskip 0.2cm
We base our treatment of the thermodynamics and phase structure
of
the system, which is evidently needed in view of Comment (3),
on the following assumptions.
\vskip 0.2cm
{\bf (III.1)} {\it (a) The equilibrium state of ${\Sigma}$
corresponding to any given $(A,T)$ is unique and therefore
translationally invariant. 
\vskip 0.2cm
(b) Similarly, the equilibrium state of ${\tilde {\Sigma}}$
corresponding to given $(H_{ex},T)$ is unique and translationally
invariant.} 
\vskip 0.2cm
{\bf Note.} This assumption precludes the applicability of our
treatment to the mixed phase of type II superconductors, since
this breaks the space translational symmetry of the system. 
\vskip 0.2cm
The next assumption, prompted by the above Comment (5), excludes
ODLRO states with non-zero quantised induction $B^{(n)}$ from the
theory, and thus puts the results of Prop. 3.2 for continuous and
lattice systems onto the same footing.
\vskip 0.2cm
{\bf (III.2)} {\it Even in the case of lattice systems, the
equilibrium states of ${\tilde {\Sigma}}$
that possess the property of ODLRO carry no magnetic induction.}
\vskip 0.2cm
The next assumption follows the line suggested by the above
Comment (4).
\vskip 0.2cm
{\bf (III.3)} {\it (a) For $B=0,$ the system ${\Sigma}$ undergoes
a phase transition at a temperature $T_{c},$ such that, for
$T<T_{c}$ only, it exhibits the property of ODLRO.
\vskip 0.2cm
(b) For $B{\neq}0,$ on the other hand, the thermodynamic
potential $f(B,T)$ of the model is a smooth function of both its
arguments, which tends to a definite limit as $B{\rightarrow}0.$}
\vskip 0.2cm
In view of (III.2) and (III.3), we define the free energy
densities of ${\Sigma}$ for the ODLRO and normal phases at
temperatures below $T_{c}$ to be
$$f_{\cal O}(T)=f(0,T)\eqno(3.3)$$
and
$$f_{\cal N}(B,T)=f(B,T) \ for \ B{\neq}0,\eqno(3.4)$$
respectively. Correspondingly, in view of equns. (2.11), (2.12)
and (3.5), we define the Gibbs potential ${\tilde f}_{\cal N}$
for the normal phase of ${\tilde {\Sigma}}$ by the formula
$${\tilde f}_{\cal N}(H_{ex},T)=min_{B}\bigl(f_{\cal N}(B,T)
+{1\over 2}B^{2}-H_{ex}.B\bigr).\eqno(3.5)$$
We note that, by (III.3b), the potential $f_{\cal N}$ may be
extended to $B=0$ by continuity, according to the formula 
$$f_{\cal N}(0,T)=
{\lim}_{B{\rightarrow}0}f_{\cal N}(B,T).\eqno(3.6)$$
\vskip 0.2cm
Our next assumption serves to relate the phase structure of the
partial system ${\Sigma}$ to that of ${\tilde {\Sigma}}.$
\vskip 0.2cm
{\bf (III.4)} {\it For $T<T_{c}$ and $H_{ex}=0,$ the equilibrium
state of ${\tilde {\Sigma}}$ possesses ODLRO.} 
\vskip 0.2cm
This is equivalent to the demand that 
$$f_{\cal O}(T)<f_{\cal N}(B,T)+{1\over 2}B^{2} \ for \ T<T_{c}, 
\ B{\neq}0,$$
and implies, by equns. (3.4)-(3.6), that
$$f_{\cal O}(T){\leq}{\tilde f}_{\cal N}(0,T) \ for \
T<T_{c}.\eqno(3.7)$$
\vskip 0.2cm
Our final assumption concerning the system ${\tilde {\Sigma}}$
is the following, whose basis we shall discuss below.
\vskip 0.2cm
{\bf (III.5)} {\it (a) The strict inequality prevails in the
condition (3.7), i.e.,
$$f_{\cal O}(T)<{\tilde f}_{\cal N}(0,T) \ for \
T<T_{c}.\eqno(3.8)$$
\vskip 0.2cm
(b) The potential ${\tilde f}_{\cal N}(H_{ex},T)$ is continuous
in both its arguments.}
\vskip 0.2cm
Here, (III.5a) is based on the idea that the radical difference
in structure between the ODLRO and normal phases at $T<T_{c}, \
H_{ex}=0$ should correspond to a difference between their Gibbs
potentials, as in phase transitions of the first kind$^{45,46}$.
(III.5b), on the other hand, represents the assumption that there
are no sub-phase transitions within the normal phase for
$T<T_{c}.$
\vskip 0.2cm
It follows immediately from assumptions (III.5) that, for
$T<T_{c}$ and ${\vert}H_{ex}{\vert}$ sufficiently small,
$$f_{\cal O}(0,T)<{\tilde f}_{\cal N}(H_{ex},T),$$
and hence, by equn. (3.5),
$$f_{\cal O}(0,T)<f_{\cal N}(B,T)+{1\over 2}B^{2}-H_{ex}.B \
{\forall}T<T_{c}, \ B{\neq}0.\eqno(3.9)$$
This signifies that the equilibrium state enjoys the property of
ODLRO and is induction-free. In other words, we have the Meissner
effect. We summarise this result as follows.
\vskip 0.2cm
{\bf Proposition 3.3.} {\it Under the above assumptions, and for
$T<T_{c}$ and ${\vert}H_{ex}{\vert}$ less than some critical
value, $H_{c}(T),$ the equilibrium state of the system
${\tilde {\Sigma}}$ has the property of ODLRO and carries no
magnetic induction. Thus, it exhibits the Meissner effect.} 
\vskip 0.2cm
{\bf Comments.} (1) This result stems from the stability ODLRO
against applied magnetic fields, and corresponds to a rigidity
of the macroscopic wave function similar to that envisaged by
London$^{1}$ for the microstate of the system.
\vskip 0.2cm
(2) In general, when $B$ and $H_{ex}$ might not be uniform, the
quantum statistics of ${\Sigma}$ still yields a constitutive
equation for the position-dependent current density, $J(x),$ of
the form 
$$J(x){\equiv}{\langle}j(x){\rangle}_{A}=F(A,x),\eqno(3.10)$$
where the functional $F$ satisfies the gauge invariance
condition
$$F(A,x)=F(A+{\nabla}{\chi};x).\eqno(3.11)$$
\vskip 0.2cm
(3) According to the London theory$^2$, the essential difference
between a Meissner effect and a normal
diamagnetism may be simply expressed in terms of the linearised
form, $F_{lin},$ of $F$ as follows. For the Meissner effect,
$F_{lin}$ is proportional to the transversely gauged vector
potential $A_{tr}$ in the large scale limit; while for normal
magnetism it is proportional to ${\nabla}^{2}A_{tr} \ (=-curlB)$ 
in this limit.
\vskip 0.2cm
(4) The relation between the critical field, $H_{c}(T),$ and the
thermodynamic potentials of the model takes a particularly simple
form in the case where the magnetic susceptibility is negligible
in the normal phase, e.g. in type-I metallic superconductors.
For, in this case, $f_{\cal N}(B,T)$ may be equated with $f_{\cal
N}(0,T){\equiv}f_{\cal N}(T).$ Hence, the condition (3.9) for the
Meissner effect reduces to the form
$$f_{\cal O}(T)<f_{\cal N}(T)-{1\over 2}H_{ex}^{2},$$
and, consequently, the critical field, $H_{c}$ is given by the
standard formula$^2$
$$H_{c}(T)=(f_{\cal N}(T)-f_{\cal O}(T))^{1/2}.\eqno(3.12)$$ 
\vskip 0.2cm
We shall conclude this Section with the proofs of Props. 3.1 and
3.2.
\vskip 0.2cm
{\bf Proof of Proposition 3.1.} (1) Assuming that
${\Phi}_{1},{\Phi}_{2}$ both satisfy the ODLRO condition (3.2),
it follows from that equation that 
$${\lim}_{{\vert}y{\vert}\to\infty}\bigl[{\Phi}_{1}(x_{1},x_{2})
{\Phi}_{1}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)-
{\Phi}_{2}(x_{1},x_{2})
{\Phi}_{2}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)\bigr]=0.
\eqno(3.13)$$
Since this is valid for all
$x_{1},x_{2},x_{1}^{\prime},x_{2}^{\prime}{\in}X,$ we may
replace $x_{1},x_{2}$ here by arbitrary points 
$x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}},$
thereby obtaining 
$${\lim}_{{\vert}y{\vert}\to\infty}\bigl[{\Phi}_{1}
(x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}})
{\Phi}_{1}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)-
{\Phi}_{2}(x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}})
{\Phi}_{2}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)\bigr]=0.
\eqno(3.14)$$
On multiplying (3.13) by
${\Phi}_{2}(x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}})$ 
and (3.14) by ${\Phi}_{2}(x_{1},x_{2}),$ and then taking the 
difference, we see that
$${\lim}_{{\vert}y{\vert}\to\infty}
{\Phi}_{1}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)
\bigl[{\Phi}_{1}(x_{1},x_{2}){\Phi}_{2}(x_{1}^{{\prime}{\prime}},
x_{2}^{{\prime}{\prime}})-
{\Phi}_{1}(x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}})
{\Phi}_{2}(x_{1},x_{2})\bigr]=0.$$
Since, by the definition of ODLRO, ${\Phi}$ does not tend
to zero at infinity, it follows that the quantity in the square
brackets of this last equation vanishes. Therefore, as
${\Phi}_{1}$ and ${\Phi}_{2}$ are non-zero, by the same
stipulation, 
$${\Phi}_{2}(x_{1},x_{2})=c{\Phi}_{1}(x_{1},x_{2}) \
{\forall}x_{1},x_{2}{\in}X,$$
where $c$ is a complex-valued constant. Consequently, as
${\Phi}_{1},{\Phi}_{2}$ both satisfy (3.2), it follows
immediately that $c$ is just a constant phase factor
${\exp}(i{\alpha}),$ as required.
\vskip 0.2cm
(2) Assuming now that the state is invariant under time-reversals
and that $A=0,$ it follows from equns. (2.2) and (2.7) that, if 
${\Phi}$ satisfies the ODLRO condition (3.2), then so too does
${\Phi}^{\star}.$ Hence, by part (1) of this Proposition, 
${\Phi}^{\star}={\exp}(i{\alpha}){\Phi},$ where ${\alpha}$ is a
real-valued constant. In other words, the function
${\Phi}{\exp}(i{\alpha}/2),$ is real-valued, and this is the
required result.  
\vskip 0.2cm
{\bf Proof of Proposition 3.2.} Let
${\Psi}_{a}{\equiv}{\gamma}(a){\Psi}$ be the transform of
${\Psi}$ under the regauged space translation (2.8). Then, by
equn. (3.1),
$${\Psi}_{a}(x_{1},x_{2}){\equiv}({\gamma}(a){\Psi})(x_{1},x_{2})
={\Psi}(x_{1}+a,x_{2}+a)
{\exp}\bigl({-ie(B{\times}(x_{1}+x_{2})).a\over
2{\hbar}c}\bigr).\eqno(3.15)$$
It follows immediately from this formula that the transformations
${\lbrace}{\gamma}(a){\vert}a{\in}X{\rbrace}$ do not all
intercommute and that, in fact, 
$${\gamma}(-a-b)[{\gamma}(a){\gamma}(b)-{\gamma}(b){\gamma}(a)]=
2i{\sin}\bigl({e(B{\times}a).b\over {\hbar}c}\bigr).\eqno(3.16)$$
We now note that, by the regauged translational invariance of the
equilibrium state,
$${\langle}{\Psi}_{a}(x_{1},x_{2})
{\Psi}_{a}^{\star}(x_{1}^{\prime},x_{2}^{\prime}){\rangle}_{A}
{\equiv}{\langle}{\Psi}(x_{1},x_{2})
{\Psi}^{\star}(x_{1}^{\prime},x_{2}^{\prime}){\rangle}_{A}.
\eqno(3.17)$$
Further, it follows from (3.15), (3.17) and the ODLRO condition
(3.2) that
$${\lim}_{y\to\infty}\bigl[{\langle}{\Psi}(x_{1},x_{2})
{\Psi}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y){\rangle}_{A}-
{\Phi}_{a}(x_{1},x_{2})
{\Phi}_{a}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)\bigr]=0,
\eqno(3.18)$$
where
$${\Phi}_{a}{\equiv}{\gamma}(a){\Phi},\eqno(3.19)$$
with ${\gamma}$ defined as in (3.15). Since equations (3.2) and
(3.17) imply that ${\Phi}_{a},$ as well as ${\Phi},$
serves as a macroscopic wave function for the equilibrium state,
it follows from Prop. 3.1(1) that
${\Phi}_{a}{\equiv}{\gamma}(a){\Phi}=c(a){\Phi},$ where
$c(a)$ is a complex number of unit modulus. As this is valid
for all spatial displacements $a,$ it follows that
$${\gamma}(a){\gamma}(b){\Phi}={\gamma}(b){\gamma}(a){\Phi}=
c(a)c(b){\Phi} \ {\forall}a,b{\in}X,$$
and therefore, by (3.16), 
$${\sin}\bigl({eB.(a{\times}b)\over {\hbar}c}\bigr){\Phi}=0 \
{\forall}a,b{\in}X.$$
Consequently, since ${\Phi}{\neq}0,$ by the assumption of ODLRO, 
$${\sin}\bigl({eB.(a{\times}b)\over {\hbar}c}\bigr)=0 \ {\forall}
a,b{\in}X,$$
which is equivalent to the required result.
\vskip 0.5cm
\centerline {\bf IV. Flux Quantisation.} 
\vskip 0.2cm
We now consider now the situation where a cylindrical
region ${\Gamma}({\subset}X)$ is removed from the body of
${\Sigma},$ so that the system occupies the complementary region
$X^{\prime}=X-{\Gamma}.$ We assume that, for $T<T_{c},$ the
system exhibits ODLRO and the Meissner effect, and that the
constitutive equation (3.10) remains unchanged, except possibly
for inessential modifications due to the new boundary conditions.
Thus, a sufficiently weak magnetic field in the tunnel ${\Gamma}$
will be screened from the body of the system by currents in 
a 'surface layer', and magnetic flux will be trapped in
${\Gamma}$ and that layer. 
\vskip 0.2cm
We shall derive the quantisation rule for this trapped flux,
${\cal F},$ by an adaptation of the argument of Ref. 36, from
general properties of the macroscopic wave function ${\Phi}$ that
stem from gauge covariance, symmetry and topology. This argument
will be centred exclusively on the properties of the model in the
asymptotic region, far from the tunnel, where the magnetic
induction can be taken to be zero and where we shall assume that
translational covariance prevails. Thus, although we shall not
keep repeating the point, the properties we invoke here will all
be asymptotic ones, that become exact only in a limit where the
regions in question are infinitely far from the tunnel.
\vskip 0.2cm
Suppose now that $C$ is a closed loop in $X^{\prime},$ in the
deep interior of the body, that encircles the tunnel ${\Gamma}.$
Then it follows immediately from the above specifications and
Stokes's theorem that 
$${\int}_{C}A.ds={\cal F}.\eqno(4.1)$$
Thus, as in the Aharonov-Bohm effect$^{47},$ the magnetic vector
potential survives, with non-trivial circulation, even in the
field-free region. 
\vskip 0.2cm
On the other hand, if $C^{\prime}$ is a loop in the asymptotic
field-free region that does not encircle the tunnel, then, by
Stokes's theorem,
$${\int}_{C^{\prime}}A.ds=0.\eqno(4.2)$$
We now use this formula to obtain an asymptotic covariance w.r.t.
certain regauged space translations, similar to those given by
equn. (2.8). To this end, we choose $C^{\prime}$ to be the
parallelogram whose vertices are $x-a, \ x, \ x+h$ and $x-a+h.$
Thus, denoting by ${\int}_{y}^{z}A.dl$ the integral of $A$ along
the straight line leading from $y$ to $z,$ we see from equn.
(4.2) that
$$\bigl({\int}_{x-a}^{x}+{\int}_{x}^{x+h}+{\int}_{x+h}^{x+a+h}+
{\int}_{x+a+h}^{x-a}\bigr)A.dl=0.$$
Hence, defining
$${\theta}_{a}(x)={\int}_{x-a}^{x}A.dl,\eqno(4.3)$$
we obtain the formula
$${\theta}_{a}(x+h)-{\theta}_{a}(x)=
\bigl({\int}_{x}^{x+h}-{\int}_{x-a}^{x-a+h}\bigr)A.dl,$$
from which it follows, on applying the gradient operator w.r.t.
$h,$ at $h=0,$ that
$${\nabla}{\theta}_{a}(x)=A(x-a)-A(x).\eqno(4.4)$$
We now invoke the covariance of the model w.r.t. the gauge
transformation (2.3), and observe that, in view of equn. (4.4),
this reduces to the following form when ${\chi}$ is chosen to be 
${\theta}_{a}.$  
$$A(x){\rightarrow}A(x-a); \ {\psi}(x){\rightarrow}{\psi}(x)
{\exp}\bigl({ie{\theta}_{a}(x)\over {\hbar}c}\bigr).\eqno(4.5)$$ 
Thus, the dynamics is covariant w.r.t. this transformation in the
asymptotic field-free region. 
\vskip 0.2cm
We also assume that it is covariant there w.r.t. space
translations
$$A(x){\rightarrow}A(x+a); \ {\psi}(x){\rightarrow}
{\psi}(x+a).\eqno(4.6)$$
On combining this with the gauge transformation (4.5), we obtain
the {\it regauged space translation} (cf. equn. (2.8))
$$A(x){\rightarrow}A(x); \ {\psi}(x){\rightarrow}
{\psi}(x+a){\exp}\bigl({ie{\theta}_{a}(x+a)\over
{\hbar}c}\bigr).\eqno(4.7)$$ 
Thus, the dynamics is (asymptotically) covariant w.r.t. these
regauged translations. Correspondingly, assuming uniqueness of
the equilibrium state for given $(A,T),$ as in Sec.III (cf.
assumption (III.1)), the expectation value of
\smallbreak\noindent
${\Psi}(x_{1},x_{2}){\Psi}^{\star}(x_{1}',x_{2}')$ becomes
(asymptotically) invariant under this transformation. Hence,
defining
$${\Phi}_{a}(x_{1},x_{2})={\Phi}(x_{1}+a,x_{2}+a)
{\exp}\bigl({i({\theta}_{a}(x_{1}+a)+{\theta}_{a}(x_{2}))\over
{\hbar}c}\bigr),\eqno(4.8)$$
it follows from equns. (3.1), (4.7) and (4.8) that the ODLRO
condition (3.2) remains valid when ${\Phi}$ is replaced is
${\Phi}_{a}.$ Consequently, by Prop. 3.1(1),
${\Phi}_{a}={\exp}(i{\eta}(a)){\Phi},$ for some $a-$dependent
phase angle, ${\eta}(a),$ i.e., by equns. (4.3) and (4.8),
$${\Phi}(x_{1}+a,x_{2}+a)=e^{i{\eta}(a)}{\Phi}(x_{1},x_{2})
{\exp}\bigl({-ie\over {\hbar}c}
({\int}_{x_{1}}^{x_{1}+a}+{\int}_{x_{2}}^{x_{2}+a})A.dl\bigr).
\eqno(4.9)$$
Now let $a_{0}(=0), \ a_{1}, \ a_{2},. \ .,a_{n}$ be a sequence
of displacements, whose sum is zero, and let $C(x)$ be the closed
contour formed from the lines joining $x+a_{0}+ \ +a_{r-1}$ to
$x+a_{0}+ \ +a_{r},$ for $r=1$ to $n.$ For each pair of points
$x_{1}$ and $x_{2},$ we choose the $a_{r}$'s so that the contours
$C(x_{1})$ and $C(x_{2})$ lie in the asymptotic field-free region
and encircle the tunnel, ${\Gamma}.$ Thus, by equn. (4.1),
$${\int}_{C(x_{j})}A.dl={\cal F}, \ for \ j=1,2.$$
It follows now from these specifications and that the application
of equn. (4.9) to the succession of spatial displacements by
$a_{1}, . \ .,a_{n}$ leads to the formula
$${\Phi}(x_{1},x_{2})={\Phi}(x_{1},x_{2})
{\exp}\bigl(i({2e{\cal F}\over {\hbar}c}+{\eta})\bigr),$$
where ${\eta}={\eta}(a_{1})+. \ .+{\eta}(a_{n}).$ Hence
$${\exp}(i({2e{\cal F}\over {\hbar} c}+{\eta}))=1.\eqno(4.10)$$
Furthermore$^{36}$, on replacing $x_{1},x_{2}$ by points
$y_{1},y_{2},$ that are chosen so that the contours $C(y_{1})$
and $C(y_{2})$ do not encircle ${\Gamma},$ we are led by the same
argument to the formula
$${\exp}(i{\eta})=1.$$ 
Consequently, by equn. (4.10), ${\exp}(2ir{\cal F}/{\hbar}c)=1.$
This immediately implies the following result, which accords
with the experiment of Deaver and Fairbank$^5$.
\vskip 0.2cm
{\bf Proposition 4.1.} {\it Under the above assumptions, the
trapped flux, ${\cal F},$ is quantised in integral multiples of
$nhc/2e.$}
\vskip 0.5cm
\centerline {\bf V. Supercurrents and Superselection Rules.} 
\vskip 0.2cm
The phenomenon of superconductivity itself is that of the
persistence of currents induced in a ring, in the absence of any
applied electric field. According to the London macroscopic
theory$^2,$ this phenomenon is intimately connected to the
Meissner effect. Specifically, the supercurrents are the source
of a magnetic field, which they themselves screen from the
interior of the body. Thus, the currents and magnetic field
stabilise one another. Further, it is clear that the
supercurrent-carrying states must be {\it metastable}, rather
than absolutely stable, since both the currents and the magnetic
field carry positive energy. 
\vskip 0.2cm
Our objective now is to provide a quantum mechanical basis for
the above phenomenological picture. We base our considerations
on the quantum mechanics of a system occupying a
multiply-connected region, e.g. with the topology of a torus. For
simplicity, we employ the model of the Sec. IV and, as in
Secs. II and III, we shall denote by  ${\tilde {\Sigma}}$ the
system comprising the matter, ${\Sigma},$ and the magnetic
induction, $B.$ 
\vskip 0.2cm
So again we consider the state of the system in which magnetic
induction is trapped in the tunnel, ${\Gamma},$ and a 'surface
region' of the matter. The supercurrents are then the surface
currents that implement the Meissner effect by screening the
induction out of the deep interior of the body. Our objective now
is to investigate the stability properties of the trapped
magnetic flux and the associated supercurrents, corresponding to
the flux quantum number $n \ ({\neq}0),$ as given by the formula
$${\int}_{S}B.dS=nhc/2e,\eqno(5.1)$$
where $S$ is a surface orthogonal to the generators of the tunnel
${\Gamma}.$
\vskip 0.2cm
We start by noting that since, by equn. (2.4), the densities of
charge, current and polarisation are fully gauge invariant, their
expectation values in the equilibrium state of ${\Sigma}$
corresponding to $(A,T)$ are functionals of the induction $B,$
as defined throughout the space $X,$ and not merely of $A.$ Thus,
we have constitutive equations of the form
$$P(x){\equiv}{\langle}{\rho}(x){\rangle}_{A}
={\cal P}(B;x)\eqno(5.2)$$
$$J(x){\equiv}{\langle}j(x){\rangle}_{A}
={\cal J}(B;x)\eqno(5.3)$$
and
$$M(x){\equiv}{\langle}m(x){\rangle}_{A}
={\cal M}(B;x).\eqno(5.4)$$
These equations apply, of course, only to points $x$ in the
matter. We extend the definitions of the functionals ${\cal
P}, \ {\cal J}$ and ${\cal M}$ to the whole space $X$ by the
specification that they vanish in the tunnel, ${\Gamma}.$ Thus,
the magnetic field $H(x)=B(x)-{\cal M}(B;x)$ and induction $B$
satisfy the Maxwell equations 
$$curlH(x)=c^{-1}{\cal J}(B;x); \ and \ divB=0,$$ 
i.e.,
$$curlB(x)=curl{\cal M}(B;x)+{\cal J}(B;x); \ and \ divB(x)=0,
\ {\forall}x{\in}X.\eqno(5.5)$$
We now make the following assumptions concerning the solution of
these equations for $B.$ 
\vskip 0.2cm
{\bf (V.1)} {\it There is a unique solution, $B_{n},$ of the
equations (5.5) for $B,$ subject to the constraint (5.1) and the
standard conditions on the boundary of ${\Gamma}$ and infinitely
far from this cylinder, where $B$ vanishes, by Meissner effect.}
\vskip 0.2cm
{\bf (V.2)} {\it For $B=B_{n},$ there is a unique equilibrium
state of ${\Sigma}$, as given by the KMS equilibrium conditions
(2.9),
corresponding to the vector potential $A_{n}$ chosen to represent
this induction.}
\vskip 0.2cm
It follows now from our specifications that both $J$ and $B$ are
non-zero in a surface region of the matter when $B=B_{n},$ for
$n{\neq}0,$ and ${\Sigma}$ is in the corresponding equilibrium
state. Thus, the  question of the persistence of the supercurrent
and trapped magnetic field now reduces to that of the stability
of the state of ${\tilde {\Sigma}}$ defined by these conditions.
Moreover, any such stability must, at most, be a metastability,
since the advent of the current and trapped flux ensure that the
free energy of this state exceeds that of true thermal
equilibrium. To be precise, a characterisation of the highest
grade of metastability corresponds to thermodynamic stability
against strictly localised perturbations$^{14},$ as represented
by the condition that such perturbations cannot lead to a
decrease in the free energy of the system. By contrast, true
equilbrium corresponds to a global thermodynamic stability.
\vskip 0.2cm
Thus, we reduce the problem of the persistence of supercurrents
to that of the thermodynamic stability of the above-described
state of ${\tilde {\Sigma}}$ against local perturbations. To
formulate this problem, we define the local perturbations of the
induction, $B_{n},$ to comprise the set, ${\cal B},$ of
divergence-free vector fields, $b,$ such that 
\vskip 0.2cm
(b.1) $b(x)=0$ unless $x$ lies inside some bounded region,
$D_{b},$
and
\vskip 0.2cm
(b.2) the field $b$ does not change the value of the flux, as
given by the l.h.s. of equn. (5.1).
\vskip 0.2cm
In fact, the second condition is redundant if the dimensionality
of $X$ exceeds $2,$ since then the divergence-free property of
$b$ implies that ${\int}_{S}b.dS,$ the flux due to $b,$ is
independent of the particular choice of $S;$ and therefore, by
choosing this surface to lie in the region where $b$ vanishes,
we see that this integral is zero. On the other hand, if $X$ is
two-dimensional and $D_{b}$ is just a bounded region of that
space, then (b) is equivalent to the condition that
\vskip 0.2cm
(b.2)$^{\prime}$ ${\int}_{D_{b}}b.dS=0.$
\vskip 0.2cm
Further, as we shall prove in the Appendix, the conditions
(b.1,2)
imply that $b$ may be represented by a vector potential
$a$ that vanishes outside some bounded region.
\vskip 0.2cm
We formulate the increment ${\phi}_{n}(b)$ in the free energy
of ${\Sigma},$ corresponding to the perturbation of $B_{n}$ by
$b,$ by the following procedure. We first note that it follows
from our above specifications and equns. (2.4) and (2.5) that the
interaction Hamiltonian of the system due to this perturbation
is the localised observable
$$K={\int}_{X^{\prime}}(-c^{-1}j.a+
{1\over 2}c^{-2}{\rho}a^{2}-m.b)dx,\eqno(5.6)$$
where $j, \ {\rho}$ and $m$ are as defined for $A=A_{n}.$
The corresponding interaction Hamiltonian for the perturbative
field ${\lambda}b,$ with ${\lambda}$ real, is therefore
$$K({\lambda})={\int}_{X^{\prime}}(-{\lambda}c^{-1}j.a+
{1\over 2}{\lambda}^{2}c^{-2}{\rho}a^{2}
-{\lambda}m.b)dx.\eqno(5.7)$$
We now invoke the fact that, by classical thermodynamics, the
increment in free energy of a system, due to an infinitesimal
change ${\Delta}H$ in its Hamiltonian, is given by the
equilibrium expectation value of ${\Delta}H.$ In the present
situation, this implies that the change in free energy of
${\Sigma}$ due to an infinitesimal increment increment
$d{\lambda}$ in the parameter ${\lambda}$ of equn. (5.7), is
$${\langle}K^{\prime}({\lambda})
{\rangle}_{A_{n}+{\lambda}a}d{\lambda},\eqno(5.8)$$
where
$$K^{\prime}({\lambda})={\int}(-c^{-1}j.a+
{\lambda}c^{-2}{\rho}a^{2}-m.b)dx.\eqno(5.9)$$
Thus, by (5.8), the incremental free energy, due to the
perturbation $b$ of the magnetic induction $B_{n},$ is
$${\phi}_{n}(b)={\int}_{0}^{1}{\langle}K^{\prime}({\lambda})
{\rangle}_{A_{n}+{\lambda}a}d{\lambda}.\eqno(5.10)$$
The corresponding increment in the free energy of ${\tilde
{\Sigma}}$ is therefore 
$${\tilde {\phi}}_{n}(b)={\phi}_{n}(b)+{\int}_{X}(b.B_{n}
+{1\over 2}b^{2})dx,\eqno(5.11)$$
the integral being the contribution due to the magnetic field
energy. 
\vskip 0.2cm
We now recall that$^{14},$ for fixed $A,$ the KMS condition
(2.9) ensures that the free energy of ${\Sigma}$
cannot be reduced by any localised modification of its
equilibrium state. Hence, by equn. (5.11), the condition for
metastability of the state of ${\tilde {\Sigma}},$ in which
$B=B_{n}$ and ${\Sigma}$ satisfies the corresponding KMS
condition, is that ${\tilde {\phi}}_{n}$ is minimised at $b=0.$
In fact, it is a simple matter to infer from our constructions
that ${\tilde {\phi}}_{n}$ is stationary at $b=0.$ For, by equns.
(5.2)-(5.4) and (5.9)-(5.11), together with the stipulation that
${\cal J}$ and ${\cal M}$ vanish in the tunnel,
$${{\partial}\over {\partial}{\lambda}}{\phi}_{n}
({\lambda}b)_{{\vert}{\lambda}=0}=-{\int}_{X}
\bigl(c^{-1}{\cal J}(B_{n};x).a(x)+
{\cal M}(B_{n};x).b(x)\bigr)dx.$$
Consequently, by equn. (5.11),
$${{\partial}\over {\partial}{\lambda}}
{\tilde {\phi}}_{n}({\lambda}b)_{{\vert}{\lambda}=0}
={\int}_{X}\bigl(B_{n}(x).b(x)-{\cal M}(B_{n},x).b(x)-
c^{-1}{\cal J}(B_{n},x).a(x)\bigr)dx.$$
Since $b=curla$ and ${\int}_{X}a.curl{\cal
J}{\equiv}{\int}_{X}{\cal J}.curla,$
it follows from equn. (5.5) that the r.h.s. of this last equation
vanishes. Hence, we have the following result.
\vskip 0.2cm
{\bf Proposition 5.1.} {\it Under the above assumptions, the
incremental free energy function ${\tilde {\phi}}_{n}$ is
stationary at $b=0,$ i.e.}
$${{\partial}\over {\partial}{\lambda}}
{\tilde {\phi}}_{n}({\lambda}b)
_{{\vert}{\lambda}=0}=0 \ {\forall}b{\in}{\cal B}.$$
\vskip 0.2cm
{\bf Comment.} The problem of the metastability of the
current-carrying state ${\tilde {\rho}}_{n}$ now reduces to the
question of whether the stationary point $b=0$ corresponds to a
minimum of ${\tilde {\phi}}_{n}.$ In fact$^{14}$, this
state will be metastable if and only if the following assumption
is valid.
\vskip 0.2cm
{\bf (V.3)} {\it The stationary point of ${\tilde {\phi}}_{n},$
at $b=0,$ corresponds to either 
\vskip 0.2cm
(a) an absolute minimum, in which case the lifetime of the above-
described current-carrying state is infinite; or 
\vskip 0.2cm
(b) a local minimum, surrounded by maxima in such a way that the
activation energy required for escape is sufficently
large by comparison with $k_{Boltzmann}T$ to ensure that the
state has an enormously long lifetime.}
\vskip 0.2cm
Thus, the basic problem concerning the metastability of
supercurrents, i.e. of {\it superconductivity}, is that of
determining which of these alternatives prevails. We remark here
that a similar problem was addressed in Ref. 14, Sec.5, where
we proved that, in the absence of a magnetic field, persistent
translationally invariant states satisfying the KMS conditions
could not carry a current. In the present different context, the
corresponding result would be be that ${\tilde {\phi}}_{n}$ could
not support an absolute minimum at $b=0,$ which would mean that
only the second alternative of (V.3) could be viable. 
\vskip 0.2cm
We may gain another perspective of this picture by relating the
metastability of the superconductive phase to a {\it
superselection rule}, forbidding transitions between states with
different flux quantum numbers. To this end, we note that, in the
asymptotic field-free region, $A_{n}$ is the gradient of a scalar
potential, ${\chi}_{n}.$ Further, by equn. (4.1) and Prop. 4.1, 
this potential is many-valued, its change over a contour passing
once round the tunnel being
$$[{\chi}_{n}]=nhc/2e.\eqno(5.12)$$ 
Furthermore, since a reversal of the magnetic field in the tunnel
corresponds to a reversal of the quantum number $n,$ we see that
$${\chi}_{-n}=-{\chi}_{n}.\eqno(5.13)$$
We now observe that, by the quantisation rule (5.12),
${\exp}(2ie{\chi}_{n}/{\hbar}c)$ is a single-valued function of
position, even though, for $n$ odd,
${\exp}(ie{\chi}_{n}/{\hbar}c)$ is not. Thus, by gauge
covariance of the second kind, the dynamics of the model is
covariant w.r.t. the transformation
$$A{\rightarrow}A+2{\nabla}{\chi}_{n}; \ {\psi}{\rightarrow}
{\psi}{\exp}(2ie{\chi}_{n}/{\hbar}c).\eqno(5.14)$$
Hence, by equns. (3.1) and (3.2), the macroscopic wave-function,
${\Phi}_{A},$ corresponding to the vector potential $A,$
satisfies the condition
$${\Phi}_{A+2{\nabla}{\chi}_{n}}(x_{1},x_{2})=
{\Phi}_{A}(x_{1},x_{2}){\exp}\bigl({2ie\over
{\hbar}c}({\chi}_{n}(x_{1}+{\chi}_{n}(x_{2}))\bigr).$$  
Consequently, choosing $A=-{\nabla}{\chi}_{n}{\equiv}-A_{n},$ and
denoting ${\Phi}_{A_{n}}$ by ${\Phi}_{n},$ it follows from equn.
(5.13) that
$${\Phi}_{-n}(x_{1},x_{2})={\Phi}_{n}(x_{1},x_{2})
{\exp}\bigl({2ie\over {\hbar}c}
({\chi}_{n}(x_{1}+{\chi}_{n}(x_{2}))\bigr).\eqno(5.15)$$
On the other hand, the assumption of covariance w.r.t. time-
reversals implies, by equns. (2.2), (2.7), (3.1) and (3.2), that 
$${\Phi}_{n}^{\star}={\Phi}_{-n}.\eqno(5.16)$$ 
On combining these last two equations, we obtain the following
result.
\vskip 0.2cm
{\bf Proposition 5.2.} {\it Assuming covariance w.r.t. gauge
transformations and time reversals, the phase of the macroscopic
wave-function ${\Phi}_{n}$ in the asymptotic field-free region is
given by}
$$arg\bigl({\Phi}_{n}(x_{1},x_{2})\bigr)={2e\over
{\hbar}c}\bigl({\chi}_{n}(x_{1})+{\chi}_{n}(x_{2})\bigr).
\eqno(5.17)$$
\vskip 0.2cm
{\bf Comment.} In view of equn. (5.12), this result implies that
the equilibrium states corresponding to the different flux
quantum numbers $n$ are {\it globally, i.e. macroscopically,}
different from one another, and thus that there is a {\it
superselection rule}$^{48,49}$ that forbids transitions between
them
via the agency of localised operations. Consequently, the
allowable states of the model fall into {\it superselection
sectors}, indicated by the quantum number $n,$ and the
metastability property of (V.3) pertains to each of these
sectors.
\vskip 0.5cm
\centerline {\bf VI. The Josephson Effect}
\vskip 0.2cm
Suppose now that two superconductors, whose potential difference
is ${\Delta}V,$ are separated by an insulating film. The
Josephson
effect$^6$ is that, in this situation, a current of frequency
$2e{\Delta}V/h$ flows across the insulating barrier. Our aim
here is to derive this frequency from simple general properties
of the macroscopic wave-functions of the superconductors.
\vskip 0.2cm
Our model, ${\Sigma},$ now consists of the two superconductors,
${\Sigma}_{1}, \ {\Sigma}_{2},$ and an insulating film, ${\cal
I},$ which occupy disjoint spatial regions $X_{1}, \ X_{2}$ and
$W,$ respectively, whose union, $X,$ is either a Euclidean space
or a lattice, as in previous Sections. We assume that $W$ is
sandwiched between $X_{1}$ and $X_{2},$ and that its boundaries
with these regions are parallel planes. We note here that the
generalisation of the model to the form where $X_{1}, \ X_{2}$
and $W$ have different crystallographic structures presents no
serious problems.
\vskip 0.2cm
We describe the electronic part of the {\it whole system}
${\Sigma},$ as before, in terms of the wave operator ${\psi}$ and
the pair field ${\Psi}.$ Again, we assume that the system
exhibits ODLRO at temperatures below a certain critical point,
and we denote the macroscopic wave-function by ${\Phi}.$ We
define ${\Phi}_{1}$ and ${\Phi}_{2},$ to be the restrictions of
${\Phi}$ the regions $X_{1}$ and $X_{2},$ respectively, i.e.,
$${\Phi}_{j}(x,x^{\prime})={\Phi}(x,x^{\prime}) \ for \ 
x,x^{\prime}{\in}X_{j}, \ j=1,2.$$
We now introduce an assumption to the effect that 
\vskip 0.2cm
(a) at large distances form their boundaries with ${\cal I},$ the
properties of the subsystems ${\Sigma}_{1}$ and ${\Sigma}_{2}$ 
reduce asymptotically to the form they would have when isolated; 
and 
\vskip 0.2cm
(b) in this latter sitation, each of them satisfies the
conditions of Prop. 3.1. 
\vskip 0.2cm
Thus, in view of the second part of that Proposition, the
assumption is the following.
\vskip 0.2cm
{\bf (VI.1)} {\it The asymptotic forms of ${\Phi}_{1}$ and
${\Phi}_{2}$ are given by 
$${\Phi}_{j}(x,x^{\prime})={\Phi}_{j}^{(0)}(x,x^{\prime})
{\exp}(i{\eta}_{j}) \ for \ j=1,2,\eqno(6.1)$$
where ${\Phi}_{1}^{(0)}, \ {\Phi}_{2}^{(0)}, \ {\eta}_{1}$ and
${\eta}_{2}$ are real, and the latter two quantities are
independent of the positions $x$ and $x^{\prime}.$}
\vskip 0.2cm
{\bf Comments.} (1) The angles ${\eta}_{1}$ and ${\eta}_{2}$ are
not neccessarily equal here, since the translational invariance
condition of Prop. 3.1(2) is no longer applicable. We define
$${\eta}={\eta}_{1}-{\eta}_{2}.\eqno(6.2)$$
\vskip 0.2cm
(2) Since equilibrium cannot be maintained when the
superconductors are held at different potentials, the angles
${\eta}, \ {\eta}_{1}$ and ${\eta}_{2}$ might be time-dependent. 
\vskip 0.2cm
In order to consider the possible significance of any time-
dependence of these phases, we first note that, in the case of
a {\it single} superconductor, as represented by the model of
Sections II and III, it follows from equn. (2.3) that the system
is covariant w.r.t. the gauge transformation
${\phi}{\rightarrow}{\phi}+V, \
{\psi}{\rightarrow}{\phi}{\exp}(-ieVt/{\hbar}), \
A{\rightarrow}A,$ 
with $V$ constant. Hence, putting ${\phi}=0,$ we see that the 
raising of the electric potential by $V$ serves to change the
time
translates, ${\psi}_{t}, \ {\Psi}_{t}$ of ${\psi}, \ {\Psi}$ by
the factors
${\exp}(-ieVt/{\hbar}), \ {\exp}(-2ieVt/{\hbar}),$ respectively.
On combining this observation with the above assumption
that, in the compound system, ${\Sigma},$ the local properties
of the deep interiors of ${\Sigma}_{1}$ and ${\Sigma}_{2}$ reduce
asymptotically to the form they have when these subsystems are
uncoupled, we are led to assume the following.
\vskip 0.2cm
{\bf (VI.2)} {\it For $x_{j},x_{j}^{\prime}{\in}X_{j} \ (j=1,2),$
the asymptotic form of the time-dependent expectation value of
${\Psi}(x_{1},x_{1}^{\prime})
{\Psi}^{\star}(x_{2},x_{2}^{\prime})$ carries a time-dependence,
given by the factor ${\exp}(2ie{\Delta}Vt/{\hbar}),$ where}
$${\Delta}V=V_{1}-V_{2}.\eqno(6.3)$$
\vskip 0.2cm
The following Proposition is now a simple consequence of
assumptions (VI.1) and (VI.2), Prop. 3.2 and equns. (3.2) and
6.2.
\vskip 0.2cm
{\bf Proposition 6.1.} {\it Under the assumptions of (VI.1) and
(VI.2), the time-dependence of phase difference ${\eta}$ is given
by
$${\eta}={\eta}_{0}+{2e{\Delta}V\over {\hbar}}t,\eqno(6.4)$$
where ${\eta}_{0}$ is a constant.}
\vskip 0.2cm
This result tells us that, in the case of the present compound
system, the phases of ${\Phi}$ in the deep interiors of the
regions $X_{1}$ and $X_{2}$ differ by the time-dependent angle
${\eta},$ whereas, by contrast, Prop. 3.1(2) implies that, in the
equilibrium states of the homogeneous system of Secs. II and III,
the phase angle of the macroscopic wave function is a
constant. This suggests that, in the present non-equilibrium
situation, ${\eta}$ represents the thermodynamic driving force
of the current across the insulating film, as proposed originally
by Josephson$^6$. Thus, we make the following further
assumption.
\vskip 0.2cm
{\bf (VI.3)} {\it The phase difference ${\eta}$ generates a
current across the insulating film, whose (single-valued) density
is of the form
$$J=F({\eta}).\eqno(6.5)$$
Further, since equn. (6.1) implies the invariance of ${\Phi}_{j}$
under the transformation
${\eta}_{j}{\rightarrow}{\eta}_{j}+2{\pi},$ the function $F$
satisfies the periodicity condition}
$$F({\eta}){\equiv}F({\eta}+2{\pi}).\eqno(6.6)$$ 
\vskip 0.2cm
The following Proposition, which corresponds to the Josephson
effect, is now an immediate consequence of equations (6.4)-(6.6).
\vskip 0.2cm
{\bf Proposition 6.2.} {\it Under the above assumptions, the
frequency of the current due to the potential difference
${\Delta}V$ is $2e{\Delta}V/h.$}
\vskip 0.2cm
{\bf Comment.} The derivation of this result from ODLRO and gauge
covariance may be contrasted with both the original one of
Josephson$^6,$ based on the Ginzburg-Landau model, and also
with the subsequent one obtained by Rieckers$^{50}$ based on a
fully microscopic treatment of a mean-field-theoretic model,
which, however, violated gauge covariance of the second kind.
\vskip 0.5cm
\centerline {\bf VII. Concluding Remarks}
\vskip 0.2cm
The theory presented here provides a general derivation of the
electrodynamics of superconductors from assumptions of
off-diagonal long range order, gauge covariance and thermodynamic
stability. In our view, the outstanding general problems are the
following.
\vskip 0.2cm
(1) The substantiation of our assumptions for concrete models.
\vskip 0.2cm
(2) The extension of our treatment to type-II superconductors.
\vskip 0.2cm
(3) The precise characterisation of the metastability of
supercurrents, within the terms of Sec. V. This amounts to the
determination of which, if either, of the two alternatives
proposed in assumption (V.3) prevails. 
\vskip 0.2cm
(4) The reformulation of the whole theory on a fully quantum
basis, i.e., with the electromagnetic field as well as the matter
treated quantum mechanically. An indication of how this might be
achieved has been given in Ref. 27, Sec. 4.
\vskip 0.5cm
\centerline {\bf Acknowledgement}
\vskip 0.2cm
This work was partially supported by European Capital and
Mobility Contract No. CHRX-Ct. 92-0007.
\vskip 0.5cm
\centerline {\bf Appendix}
\vskip 0.2cm
Our aim here is to show that, under the conditions (b.1,2) of  
Sec. V, the perturbative induction $b$ may be represented by a 
vector potential, $a,$ that vanishes in the region $D_{b}^{(c)},$
exterior to the bounded domain $D_{b}.$ 
\vskip 0.2cm
Suppose, then, that $a^{\prime},$ is a vector potential that 
represents $b,$ i.e. that satisfies the equation
$b=curla^{\prime},$ and that $x_{0}$ is some chosen reference
point in $D_{b}^{(c)}$. Then, for $x{\in}D_{b}^{(c)},$ we
construct a curve, $C,$ leading from $x_{0}$ to $x$ in
$D_{b}^{(c)}$, and define
$${\chi}(x)={\int}_{C}a^{\prime}.dl.\eqno(A.1)$$
To show that this function is independent of the choice of $C,$
we note that, if $C^{\prime}$ is another curve joining $x_{0}$
to $x$ in $D_{b}^{(c)},$ then
$$({\int}_{C}-{\int}_{C^{\prime}})a.dl={\int}_{S}b.dS,
\eqno(A.2)$$
where $S$ is a closed surface bounded by $C$ and $C^{\prime}.$
In the case where the dimensionality of $X$ is greater than $2,$
we may choose $S$ so that it does not intersect $D_{b},$ thereby
ensuring that the r.h.s. of (A.2) vanishes. In the two-
dimensional case, on the other hand, there is the possibility
that the closed curve formed by $C$ and $-C^{\prime}$ loops round
$D_{b},$ in which case the r.h.s. of (A.2) is
${\int}_{D_{b}}b.ds,$ which vanishes, by (b.2)$^{\prime}$. 
Hence, in all cases, the function ${\chi},$ defined by equn.
(A.2), does not depend on the choice of the curve $C.$ It is
therefore a simple consequence of that formula that, outside the
region $D_{b},$
$$A={\nabla}{\chi}.$$
Hence, on extending ${\chi}$ by continuity to a differentiable
function on the whole of $X$ and defining
$$a=a^{\prime}-{\nabla}{\chi},$$
we see that $curla=b$ and that $a$ vanishes outside the bounded
region $D_{b}.$ Thus, the potential $a$ meets all our
requirements.
\vskip 0.5cm
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