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\begin{center}
{\LARGE Reduction and Emergence}
{\LARGE in Bose-Einstein Condensates\bigskip }
{\Large Richard Healey\bigskip }
Philosophy Department, University of Arizona,
213 Social Sciences, Tucson, AZ 85721-0027
rhealey@email.Arizona.edu\bigskip
\textit{Abstract}
\end{center}
A closer look at some proposed \textit{Gedanken}-experiments on BECs
promises to shed light on several aspects of reduction and emergence in
physics. These include the relations between classical descriptions and
different quantum treatments of macroscopic systems, and the emergence of
new properties and even new objects as a result of spontaneous symmetry
breaking.
\section{Introduction}
Not long after the first experimental production of a Bose-Einstein
condensate (BEC) in a dilute gas of rubidium in 1995$^{(1)}$, experiments
demonstrated interference between two such condensates$^{(2)}$. Interference
is a wave phenomenon, and here it was naturally taken to involve a
well-defined phase difference between two coherent matter waves---the BECs
themselves. Experimental phenomena associated with well-defined
phase-differences were already familiar from other condensed matter systems.
The alternating current observed across a Josephson junction between two
similar superconductors was (and is) explained by appeal to their varying
phase-difference induced by a constant voltage difference across the
junction. These two phenomena are now considered manifestations of quantum
behavior at the macroscopic---or at least mesoscopic---level since they
involve very large numbers of atomic or sub-atomic systems acting in
concert, and it is the theory of quantum mechanics that has enabled us to
understand and (at least in the second instance) to predict them, both
qualitatively and in quantitative detail. They are among a variety of
phenomena manifested by condensed matter that have been described as emergent%
$^{(3),(4)}$, in part as a way of contrasting them with phenomena amenable
to a reductive explanation in terms of dynamical laws governing the behavior
of their microscopic constituents.
\qquad While some kind of contrast with reduction is almost always intended
by use of the term `emergent' (or its cognates), the term has been widely
applied to items of many categories on diverse grounds. After briefly
commenting in section 2 on philosophers' attempts to regiment usage, I focus
on a cluster of issues surrounding the emergence of a definite phase in BECs
and related systems.
\qquad It is widely (though not universally) believed that the concept of
broken symmetry is key to understanding not only the Josephson effect and
interference of BECs but also many other phenomena involving condensed
matter.
\qquad When the state of a condensate is represented by a mathematical
object with $U(1)$ symmetry, spontaneous breaking of this symmetry is
associated with a definite phase---the complex argument of an order
parameter such as the expectation-value of a field operator. It may be said
that this phase emerges as a result of such spontaneous symmetry breaking.
Analogies are often drawn between this spontaneously broken phase symmetry
and the breaking of rotational symmetry as the magnetization of a Heisenberg
ferromagnet or the axis of a crystal acquires a definite orientation. But
the attribution of a definite value for the phase of a condensate raises a
thicket of problems that challenge these analogies.
\qquad While the orientation of a crystal or a Heisenberg ferromagnet has
direct operational significance, it is at most the relative phase of two or
more condensates that is manifested in interference experiments: the
absolute phase of a condensate is generally taken to be without physical
significance. A second issue concerns measurements of the relative phase of
condensates. In quantum mechanics, a measurable magnitude (an
\textquotedblleft observable\textquotedblright ) is represented by a
self-adjoint operator, and the possible results of a measurement of this
observable are given by the spectrum of this operator. But there are
powerful reasons for denying that observables generally have values for
measurement to reveal. If the relative phase were represented by such an
operator, then the appearance of a definite (relative) phase on measurement
is no indication of a definite pre-existing phase in the condensate. Rather
than emerging spontaneously, the definite phase would be precipitated by the
measurement itself.
\qquad A number of recent papers have treated the emergence of a definite
relative phase between BECs as a stochastic physical process that occurs as
a result of multiple measurements of quantum observables, each on a
different microscopic constituent of the BECs$^{(5)-(11)}$. The measured
observable is not the phase itself, so there is no need to represent this by
an operator. Indeed, as section 4 explains, the emerging relative phase
plays the role of a kind of \textquotedblleft hidden
variable\textquotedblright\ within a standard quantum mechanical analysis.
This analysis involves no appeal to spontaneous symmetry-breaking. While
some have embellished the analysis by explicit appeal to von Neumann's
controversial projection postulate (\textquotedblleft
collapse\textquotedblright\ of the wave-function on measurement), this
proves unnecessary: all that is required is standard Schr\"{o}dinger quantum
mechanics, including the Born rule for joint probabilities. One way to look
at this quantum mechanical analysis is as a reduction of the theoretical
treatment of relative phase in terms of spontaneous symmetry-breaking. But
this reduction would also involve elimination, in so far as it assumes there
is no well-defined relative phase prior to the measurements that prompt its
emergence.
\qquad A striking feature of the quantum mechanical analysis is that
macroscopic values for observables also emerge in the stochastic process
that produces a well-defined relative phase. These include transverse spin
polarization in a region occupied by two BECs, each composed of particles
with aligned spins, where the two alignments are in opposite directions. The
measurements that induce this macroscopic spin polarization are themselves
microscopic, and may occur in a distant region. As section 5 explains, this
\textquotedblleft nonlocal\textquotedblright\ emergence of macroscopic
values violates expectations based on a common understanding of the
Copenhagen interpretation, and has been presented as a strengthening of
EPR's challenge to that interpretation$^{(12)}$. Section 6 considers a
possible Bohrian response to this challenge and explains why this is in
tension with the common view that the classical features of macroscopic
objects may be derived from quantum theory. This may prompt one to question
the reduction of classical to quantum physics.
\qquad For a global $U(1)$ symmetry, Noether's first theorem implies the
existence of a conserved quantity, which may in this case be identified with
the number of bosons present in a condensate. Broken global $U(1)$ symmetry
then apparently implies a condensate composed of an indeterminate number of
bosons. While coherent laser light has long been accepted as an example of a
condensate with an indeterminate number of massless bosons, an indeterminate
number of atoms in a BEC/Cooper pairs in a superconductor threatens
cherished beliefs about conservation of mass, and baryon/lepton number.
Section 7 addresses the question: Do we have here an emergent object---an
object not composed of any definite number of its constituents?
\qquad The present paper attempts no more than a preliminary survey of a
cluster of complex interrelated issues concerning reduction and emergence in
Bose-Einstein condensates, each of which will repay detailed further study.
\section{Emergence and Reduction}
In physics and elsewhere, reduction and emergence are characteristically
taken to label opposing views of a single relation, but lack of clarity
about the nature of the relation and the identities of the relata often
results in debates between \textquotedblleft
reductionists\textquotedblright\ and their opponents that generate more heat
than light. One problem is that while it is typically phenomena, behavior,
properties, objects, etc. that are said (or denied) to be emergent,
reduction is more commonly thought of as a relation between theories,
theoretical descriptions, sciences or laws (strictly, law statements). So
while emergence is a relation that may or may not hold between items in the
world that scientists study, reduction is a relation applicable only to
products of that study. This division is not hard and fast\footnote{%
In his qualified defense of reductionism, Weinberg$^{(13)}$ casts reduction
and even reductive explanation in ontological rather then epistemological or
methodological terms. He freely admits that a scientist's best strategy in
understanding a phenomenon is often not to look to the fundamental laws that
govern the elementary constituents of the systems involved, even while
maintaining that it is those laws that \textquotedblleft ultimately
explain\textquotedblright\ it.}. But it is a division I shall respect in my
usage in this paper.
\qquad In their attempts to clarify the notion of emergence, philosophers
have typically begun by concentrating their efforts on the emergence of
properties. No consensus has been reached, and a number of alternative
analyses have been proposed$^{(14)-(17)}$. Rather than take these as rival
attempts to state necessary and sufficient conditions for the correct
application of the term `emergent property', one should view them as
alternative explications of the same rough idea---that an emergent property
is one that is somehow autonomous from more elementary underlying structures
out of which it arises. Each may prove useful in marking some contrast that
is important in a different application. One common application of the
notion of emergence is to the mind: philosophers and cognitive scientists
have debated the emergence of consciousness and other mental properties from
underlying physical processes involving the brain. But here I am interested
in contrasting specific physical properties (or, in one case, objects) with
others as to their autonomy from or dependence on more elementary physical
structures.
\qquad The phase of a condensate is the first such property, and the
underlying structures are the properties and arrangement of its constituent
particles. The phase of a condensate is actually a real-valued magnitude,
though any qualitative (i.e. non-numerical) property may be so
regarded---it's values may be taken to be 1 (for possessed) and 0 (for not
possessed). Other magnitudes of systems of condensates may also be
considered emergent, including spin polarization, magnetization and electric
current. We shall see that not one but several senses of emergence turn out
to be usefully applied to these properties.
\qquad Broken symmetries associated with phase transitions in condensates
have been taken to give rise to emergent phenomena by both physicists and
philosophers$^{(3),(4),(16),(18)}$\footnote{%
Though Anderson$^{(18)}$ doesn't use the word `emergent'. It is an
unfortunate linguistic accident that in the expression `phase transition'
the word `phase' refers to states of matter themselves (e.g.
superconducting), not to the complex argument of a parameter that may be
used to characterize their degree of order.}. Weinberg$^{(19)}$ even defines
a superconductor as \textquotedblleft simply a material in which
electromagnetic gauge invariance is spontaneously broken\textquotedblright .
This at least suggests that it is spontaneously broken symmetry that marks
properties of matter as emergent in a novel phase. If so, properties of
matter in that phase that can be accounted for without appealing to broken
symmetry would not count as emergent.
\qquad In one sense, emergence is a diachronic process rather than a
synchronic condition. Phase transitions occur as dynamical processes,
whether or not the symmetry of the prior state is physically broken during
this process. So a phase of matter with striking properties may emerge
dynamically even though these properties are not sufficiently autonomous
from the underlying structure in the new phase to count as (synchronically)
emergent.
\qquad I think there is another possible use of `emergent', as applied to
properties of a complex system which is, perhaps, illustrated by the
emergence of a definite (relative) phase in BECs. Consider such
\textquotedblleft sensory\textquotedblright\ predicates as red, malodorous,
bitter, silky or even wet or hard\footnote{%
See Wilson's$^{(20)}$ extended exploration of the sensory concomitants of
the first and last of these terms and their bearing on the character of any
corresponding property.}. In paradigm cases, though certainly not always,
these are applied to a macroscopic object on the basis of the response it
elicits in a human who interacts with that object in a minimally invasive
way---unfortunately, looking at a red traffic light is not an effective way
to turn it green, and nor does sniffing rotten meat improve its smell. But
do such predicates pick out a corresponding property of that object?
\qquad Many and varied answers to that question have been proposed
throughout the history of philosophy and natural science. Some have defended
a positive answer by claiming that a property such as the redness of an
object supervenes on more fundamental properties of the microscopic
constituents of that object that are not themselves red. Others have denied
the existence of any property of redness, flushed with the prospect of a
complete scientific explanation of our ability to perceive, classify and
reliably communicate about those things we call red based only on their
fundamental microphysical properties and ours. Philosophical accounts of
emergence generally presuppose that emergent properties are real, even if
they supervene on an underlying microphysical basis. But if one had a
complete scientific explanation of our ability to perceive, classify and
reliably communicate about those things we call red, that might itself be
offered as an account of the emergence of redness even if there is no such
property! For the account would explain the success of our common practice
of calling things red and so license the continuance of that practice.
\section{BEC Phase as Emerging from Spontaneous Symmetry Breaking?}
In his seminal essay Anderson$^{(18)}$ takes the general theory of broken
symmetry to offer an illuminating formulation of how the shift from
quantitative to qualitative differentiation characteristic of emergence
takes place\footnote{%
\textquotedblleft at each level of complexity entirely new properties
appear\textquotedblright\ ($(18)$, p.393).}. In agreement with Weinberg$%
^{(19)}$ he mentions superconductivity as a spectacular example of broken
symmetry, though he gives several others.
\begin{quote}
\qquad The essential idea is that in the so-called $N\rightarrow \infty $
limit of large systems (on our own, macroscopic scale) it is not only
convenient but essential to realize that matter will undergo sharp, singular
\textquotedblleft phase transitions\textquotedblright\ to states in which
the microscopic symmetries, and even the microscopic equations of motion,
are in a sense violated. (\textit{op}. \textit{cit}. p.395)
\end{quote}
After the 1995 experimental production of BECs in dilute gases, Laughlin and
Pines$^{(3)}$ were able to add \textquotedblleft the newly discovered atomic
condensates\textquotedblright\ as examples that display emergent physical
phenomena regulated by higher organizing principles. Since they cite
Anderson's paper approvingly and take a principle of continuous symmetry
breaking to explain (the exact character of) the Josephson effect, it is
reasonable to conjecture that they would join Anderson in taking the phase
transition from a normal dilute gas to a BEC as well as that from a normal
metal to a superconducting state to involve spontaneous symmetry breaking.
\qquad What symmetry is taken to be broken in the transition to the
condensed phase of a BEC? The transition is from a less to a more ordered
state, whose order may be represented by a so-called order parameter.
According to Leggett$^{(21)}$ (p. 38) the order parameter characterizing a
BEC (especially in the case of dilute gases including rubidium) is often
taken to be a complex-valued function---the expectation value of a Bose
field operator in the given quantum state.%
\begin{equation}
\Psi \left( \mathbf{r,}t\right) =\left\langle \hat{\psi}\left( \mathbf{r,}%
t\right) \right\rangle
\end{equation}%
If this is written as%
\begin{equation}
\Psi \left( \mathbf{r,}t\right) =\left\vert \Psi \left( \mathbf{r,}t\right)
\right\vert e^{i\varphi \left( \mathbf{r,}t\right) }
\end{equation}%
then the phase $\varphi \left( \mathbf{r,}t\right) $ parametrizes an element
of the group $U(1)$. If the equations describing the field of the condensate
are symmetric under global $U(1)$ transformations, then changing the order
parameter by addition of an arbitrary constant to the phase will take one
solution into a distinct solution. Global $U(1)$ symmetry will be broken by
choice of one such value.
\qquad An analogy is often drawn to the broken rotation symmetry of the
Heisenberg ferromagnet as the spins of all its magnetic dipoles align along
some arbitrary direction in the ground state. That fits Anderson's quoted
description well, since the phase transition to one such highly ordered
ground state of the ferromagnet is a good example of the kind of
spontaneously broken symmetry amenable to idealized treatment as a quantum
system with an infinite number of degrees of freedom\footnote{%
See, for example, Ruetsche$^{(22)}$.}. In contrast to the case of a quantum
system with a finite number of degrees of freedom, degenerate ground states
of such a system cannot generally be superposed to give another state since
they appear in distinct, unitarily inequivalent, representations of the
fundamental commutation relations. Spontaneous breaking of the rotational
symmetry of the Heisenberg ferromagnet corresponds to the adoption of one
out of the many states in which the dipoles of the ferromagnet are all
aligned. In two or more dimensions, this means breaking of a continuous
rotational symmetry. By Goldstone's theorem$^{(23)}$, when such a continuous
symmetry is broken in quantum mechanics the Hamiltonian has no energy gap%
\footnote{%
As Streater$^{(24)}$ proved for the Heisenberg ferromagnet: this gives rise
to the possibility of spin waves of arbitarily small energy.}: in a quantum
field theory this implies the existence of massless Goldstone bosons.
\qquad Pursuing this analogy, spontaneous breaking of the continuous $U(1)$
phase symmetry of a BEC's order parameter could be represented by an
idealized model in which the number of constituent particles is taken to be
infinite, but the density of the BEC is fixed at some low value $\rho $ by
taking the so-called thermodynamic limit $N\rightarrow \infty ,$ $%
V\rightarrow \infty ,$ $N/V=\rho $ (a constant). Then adoption of a definite
phase by a BEC would be an instance of the same kind of spontaneous symmetry
breaking as adoption of a definite direction of magnetization by a
Heisenberg ferromagnet. But there are problems with this analogy, as Leggett$%
^{(21),(25),(26)}$ has noted.
\qquad When rotation symmetry of a Heisenberg ferromagnet is spontaneously
broken, the spins of its components are all aligned along a particular
direction in space. This direction may be operationally defined in many ways
having nothing to do with spin or magnetization: in particular, it need not
be defined in relation to other Heisenberg ferromagnets, either actual or
hypothetical. On the other hand, if the \textit{U(1)} global phase symmetry
of a BEC were to be spontaneously broken, its overall phase would become
well defined only relative to some other BEC of the same kind (for example,
a similarly condensed dilute gas of rubidium 87). At most, a definite phase
consequent upon spontaneously broken symmetry would seem to be an emergent
relational property (cf. Teller$^{(27)}$) of a BEC. Moreover, difficulties
in implementing multiple pairwise phase comparisons between similar BECs
that have never been in contact threaten at least the operational
significance even of such a relational property. Leggett$^{(26)}$ argues
that, at least in the case of superconducting BECs, operational pairwise
phase comparisons among several such BECs will fail to be transitive (though
compare Leggett$^{(28)}$).
\qquad A second problem arises from the need to take the thermodynamic limit
to treat the emergence of relative phase in BECs as an instance of
spontaneous symmetry breaking. No massive BEC system is composed of an
infinite number of elementary bosons. Moreover, while the number of
elementary dipoles in a macroscopic magnet will typically at least be
extremely large (of the order of $10^{23}$ ), the first dilute gas BECs
contained only a few thousand atoms, and even now experimental realizations
have increased this number by only a few factors of $10$. If it were
essential to assume that an \textit{infinite} number of atoms is present in
each of two interfering BECs to explain their observed interference (as the
quote from Anderson might lead one to believe), then one may legitimately
query the value of the explanation. But in fact one need not treat the
emergence of relative phase here as a case of spontaneous symmetry breaking
in the thermodynamic limit, as analyses by Castin and Dalibard$^{(6)}$ and
several subsequent authors have shown.
\qquad In the context of an idealized model of two trapped condensates of
the same atomic species, Castin and Dalibard$^{(6)}$ showed two things:
(1)\qquad No measurements performed on the condensates can allow one to
distinguish between two different quantum representations of this system: By
a uniform average over the unknown relative phase of two coherent states;
and by a Poissonian statistical mixture of Fock states.
(2)\qquad Two different points of view on a system are available: Assuming
an initial pair of coherent states with a definite relative phase,
successive measurements \textquotedblleft reveal\textquotedblright\ that
pre-existing phase in an interference phenomenon; assuming each condensate
is initially\ represented by a definite Fock state, with no well-defined
relative phase, the same sequence of measurements progressively
\textquotedblleft builds up\textquotedblright\ a relative phase between the
condensates as the interference phenomenon is generated.
They take the results of their analysis to show that the notion of
spontaneously broken phase symmetry is not indispensable in understanding
interference between two condensates. I won't explain how they arrived at
these conclusions, since the next section outlines a closely related
analysis by Lalo\"{e} of a similar \textit{Gedankenexperiment} that will
provide a focus for the subsequent discussion. I will merely comment that
Castin and Dalibard$^{(6)}$ assume that the measurements referred to in (2)
are performed in a well-defined temporal sequence on individual elements of
the system of condensates, and that each leaves the rest of the system in
the quantum state it would be assigned if the effect of that measurement
were represented by von Neumann's projection postulate.
\section{The Appearance of Phase Without Symmetry-Breaking}
In 2005 Lalo\"{e}$^{(7)}$ began to develop an elegant framework for
analyzing the emergence of phase in systems of BECs. One important
application is to a system of two BECs, each composed of non-interacting
bosons, and each initially represented by a Fock state corresponding to a
definite number of particles. This provides a simplified and idealized model
for the kind of experimental situation realized by Andrews \textit{et}.
\textit{al}.$^{(2)}$ that first demonstrated interference between two BECs.
An extension of that model is to measurements on BECs in different internal
states---most simply, each in one of two different one-particle \textit{z}%
-spin states. This enables one to consider the BECs to be initially separate
systems no matter what their spatial overlap: and it naturally suggests the
possibility of a variety of different kinds of measurement capable of
revealing interference between them---of spin-component in any direction in
the \textit{x-y} plane. Such measurements are considered in Mullin, Krotkov
and Lalo\"{e}$^{(8)}$, Lalo\"{e}$^{(12)}$, and Lalo\"{e} and Mullin$^{(10)}$%
: here I follow Lalo\"{e}'s$^{(12)}$ presentation.
\qquad Consider a pair of noninteracting spin-polarized BECs in the
normalized Fock state%
\begin{equation}
\left\vert \Phi \right\rangle =\frac{1}{\sqrt{N_{a}!N_{b}!}}\hat{a}\dag
_{u_{a},\alpha }^{N_{a}}\hat{a}\dag _{v_{b},\beta }^{N_{b}}\left\vert
0\right\rangle \label{double Fock}
\end{equation}%
representing $N_{a}$ particles with internal ($z$-spin) state $\alpha $ and
spatial state $u_{a}$ and $N_{b}$ particles with orthogonal internal ($z$%
-spin) state $\beta $ and spatial state $v_{b}$, where $\left\vert
0\right\rangle $ is the vacuum state.
If $\hat{\Psi}_{\alpha }(\mathbf{r})$ is the field operator for $z$-spin $%
\alpha $, $\hat{\Psi}_{\beta }(\mathbf{r})$ for $z$-spin $\beta $, and $%
^{\dag }$ indicates the adjoint operation, \ then the number density
operator of the BECs is%
\begin{equation}
\hat{n}(\mathbf{r})=\hat{\Psi}_{\alpha }^{\dag }(\mathbf{r})\hat{\Psi}%
_{\alpha }(\mathbf{r})+\hat{\Psi}_{\beta }^{\dag }(\mathbf{r})\hat{\Psi}%
_{\beta }(\mathbf{r}) \label{number density}
\end{equation}%
and the density operator for their spin component in a direction in the $x-y$
plane at an angle $\varphi $ from the $x$-axis is%
\begin{equation}
\hat{\sigma}_{\varphi }(\mathbf{r})=e^{-i\varphi }\hat{\Psi}_{\alpha }^{\dag
}(\mathbf{r})\hat{\Psi}_{\beta }(\mathbf{r})+e^{+i\varphi }\hat{\Psi}_{\beta
}^{\dag }(\mathbf{r})\hat{\Psi}_{\alpha }(\mathbf{r}) \label{spin density}
\end{equation}%
Suppose that one measurement is made of the $\varphi $-component of particle
spin in a small region of space $\Delta r$ centered around point $\mathbf{r}$%
. The corresponding spin operator is%
\begin{equation}
\hat{S}(\mathbf{r},\varphi )=\int_{\Delta r}d^{3}\mathbf{r}^{\prime }\hat{%
\sigma}_{\varphi }(\mathbf{r}^{\prime })
\end{equation}%
For sufficiently small $\Delta r$, this has only three eigenvalues $\eta
=0,\pm 1$ since no more than one particle would be found in $\Delta r$. The
single-particle eigenstates for finding a particle there with $\eta =\pm 1$
are%
\begin{equation}
\left\vert \Delta r,\eta \right\rangle =\left\vert \Delta r\right\rangle
\otimes \frac{1}{\sqrt{2}}\left[ e^{-i\varphi /2}\left\vert \alpha
\right\rangle +e^{+i\varphi /2}\left\vert \beta \right\rangle \right]
\end{equation}%
where $\left\vert \Delta r\right\rangle $ is a single-particle spatial state
whose wave-function equals $1$ inside $\Delta r$ but $0$ everywhere outside $%
\Delta r$. The corresponding $N$-particle projector is%
\begin{equation}
\hat{P}_{\eta =\pm 1}(\mathbf{r,}\varphi )=\frac{1}{2}\int_{\Delta r}d^{3}%
\mathbf{r}^{\prime }\left[ \hat{n}(\mathbf{r}^{\prime })+\eta \hat{\sigma}%
_{\varphi }(\mathbf{r}^{\prime })\right] \label{projectors}
\end{equation}%
and the projector for finding no particle there is%
\begin{equation}
\hat{P}_{\eta =0}(\mathbf{r})=\left( \mathbf{1}-\int_{\Delta r}d^{3}\mathbf{r%
}^{\prime }\hat{n}(\mathbf{r}^{\prime })\right)
\end{equation}%
As $\Delta r\rightarrow 0$, the corresponding eigenstates (for variable $%
\mathbf{r}$) form a quasi-complete basis for the $N$-particle space.
Now consider a sequence of $m$ measurements of transverse spin-components $%
\varphi _{j}$ in very small non-overlapping regions $\Delta r_{j}$, each of
volume $\Delta $, centered around points $\mathbf{r}_{j\text{ }}(1\leq j\leq
m)$. Since the projectors for non-overlapping regions commute, the joint
probability for detecting $m$ particles with spins $\eta _{j}$ in regions $%
\Delta r_{j}$ is%
\begin{equation}
\left\langle \Phi \left\vert \hat{P}_{\eta _{1}}(\mathbf{r}_{1}\mathbf{,}%
\varphi _{1})\times \hat{P}_{\eta _{2}}(\mathbf{r}_{2}\mathbf{,}\varphi
_{2})\times ...\times \hat{P}_{\eta _{m}}(\mathbf{r}_{m}\mathbf{,}\varphi
_{m})\times \right\vert \Phi \right\rangle \label{joint probabilities}
\end{equation}%
Using (\ref{projectors}) together with (\ref{number density}) and (\ref{spin
density}) this gives a product of several terms, each containing various
products of field operators. Since these commute, we can push all the
creation operators to the left and all the annihilation operators to the
right. Expanding the field operators in terms of a basis $\left\vert
u_{a},\alpha \right\rangle $, $\left\vert v_{b},\beta \right\rangle $ of
single particle states%
\begin{equation}
\hat{\Psi}_{\alpha }(\mathbf{r})=u_{a}(\mathbf{r})\times \hat{a}%
_{u_{a},\alpha }+...\text{ \ \ ; \ \ }\hat{\Psi}_{\beta }(\mathbf{r})=v_{b}(%
\mathbf{r})\times \hat{a}_{v_{b},\beta }+...
\end{equation}%
But none of the "dotted" terms will contribute to (\ref{joint probabilities}%
), since $\left\vert \Phi \right\rangle $ contains no particles in states
other than $\left\vert u_{a},\alpha \right\rangle $, $\left\vert v_{b},\beta
\right\rangle $.
Each term now contains between $\left\langle \Phi \right\vert $ and $%
\left\vert \Phi \right\rangle $ a string of creation operators followed by a
string of annihilation operators. If a state $\left\vert u_{a},\alpha
\right\rangle $ or $\left\vert v_{b},\beta \right\rangle $ does not appear
exactly the same number of times in each of these, it will not contribute to
(\ref{joint probabilities}): if it does appear exactly the same number of
times in each of these, every creation or annihilation operator will
introduce a factor $\sqrt{N_{a,b}-q}$ where $q$ depends on the term but $q