\documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsmath} \setcounter{MaxMatrixCols}{10} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Friday, September 04, 2009 18:12:27} %TCIDATA{LastRevised=Friday, September 25, 2009 09:21:53} %TCIDATA{} %TCIDATA{} %TCIDATA{Language=American English} %TCIDATA{CSTFile=40 LaTeX article.cst} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \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