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A Matter of Degree: Putting Unitary Inequivalence to Work\textbf{*}
Laura Ruetsche$\dagger \ddagger $
University of\ Pittsburgh
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\noindent $\dagger $Send reprint requests to the author, Department of
Philosophy, 1001 Cathedral of Learning, University of Pittsburgh,
Pittsburgh, PA 15260.
\noindent $\ddagger $For comments on earlier drafts, I am obliged to Gordon
Belot, Jeremy Butterfield, Rob Clifton, and John Earman.\newpage
\begin{center}
Abstract
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\noindent If a classical system has infinitely many degrees of freedom, its
Hamiltonian quantization need not be unique up to unitary equivalence. I
sketch different approaches (Hilbert space and algebraic) to understanding
the content of quantum theories in light of this non-uniqueness, and suggest
that neither approach suffices to support explanatory aspirations
encountered in the thermodynamic limit of quantum statistical mechanics.
\newpage
\noindent \textbf{1. Introduction.} A characteristic, and provocative,
feature of quantum field theory (QFT) is the availability of unitarily
inequivalent Hilbert space representations of its canonical commutation
relations (CCRs). Under the reasonable and historically entrenched
assumption that unitary equivalence is a necessary condition for the
physical equivalence of Hilbert space quantizations, this availability
implies that there are myriad physically inequivalent quantizations of any
classical field theory. I aim here to explore this dramatic non-uniqueness,
and its implications for our understanding of the manner in which theories
delimit physical possibilities. Lending both form and interest to this
investigation is the existence of a level of abstraction at which even
unitarily inequivalent Hilbert space quantizations share a common structure.
They are, each of them, a concrete realization of an abstract algebraic
structure--the structure of a $C^{*}$ algebra called \textit{the Weyl algebra%
}, and based upon the CCRs.
Each Hilbert space quantization is also, of course, a lot else, and it is by
differing in features additional to their realization of the Weyl algebraic
structure that quantizations can fail to be unitarily equivalent. Surveying
the roiling mob of inequivalent quantizations from the lofty heights of
algebraic abstraction, one might suppose, as an early advocate of the
algebraic approach to QFT does, that ``all the physical content of the
theory is contained in the algebra itself; nothing of fundamental
significance is added to a theory by its expression in a particular
representation'' (Robinson 1966, 488). It would dissolve the foundational
questions posed by the availability of unitarily inequivalent
representations to deprive differences between those representations of
physical significance.
Such a dissolution comes at a cost. In the QFT context, only a proper subset
of the bounded operators on a Hilbert space representation of the CCRs
instantiate the Weyl algebraic structure. Locating the physics solely in the
abstract algebra exposes the remaining bounded operators--operators \textit{%
\ parochial} to the representation--as unphysical accretions, clinging to
concrete realizations of that algebra. Among the ``accretions'' in a
particular representation are most of its projection operators, including
those in the spectrum of its total number operator. Locating the physics in,
and only in, the abstract algebra could mean investing with physical
significance fewer observables than either scientific practice or our
favored approaches to interpreting quantum theories can bear.
These reservations should not trigger retreat to a reactionary \textit{%
Hilbert space chauvinism}, which identifies physically relevant observables
with the set of bounded self-adjoint operators on some particular Hilbert
space, and physically possible states with the set of density matrices on
that Hilbert space. For the Hilbert space chauvinist runs the risk of
investing with physical significance fewer \textit{states} than our favored
scientific and interpretive practices can bear. One option foreclosed is
that of using states from unitarily \textit{in}equivalent representations in
our accounts of the phenomenon.
QFT is not the only setting where unitarily inequivalent representations
arise. Quantum Statistical Mechanics (QSM), in its thermodynamic limit, is
replete with unitarily inequivalent representations of its fundamental
systems, infinite collections of microentities whose physics is quantum.
What's more, explanations envisioned in the thermodynamic limit promise to
complete the schematic arguments against chauvinism just offered. Thus the
thermodynamic limit of QSM provides a motivation and a model for tempering
chauvinisms, both Hilbert space and algebraic, about the structure and
physical content of quantum theories.
I aim in what follows to make the foregoing somewhat more precise. Section 2
frames issues raised by unitarily inequivalent representations more ornately
than I have so far. It reviews relevant rudiments of both Hilbert space and
algebraic approaches to quantum theories, and describes Hilbert space and
algebraic chauvinisms in more detail. Sketching the use to which QSM can put
unitarily inequivalent representations, and an algebraic framework which
encompasses them, in its treatments of equilibrium and phase transitions,
Section 3 attempts to discredit both chauvinisms. In their stead, Section 4
offers an understanding of the content of physical theories which allows
physical possibility to be a matter of degree.\bigskip
\noindent \textbf{2.} \textbf{Unitary Equivalence and Its Breakdown. }Von
Neumann discerned in both Schr\"{o}dinger's wave and Heisenberg's matrix
mechanics the structure of a \textit{Hilbert space theory}, that is, a
theory which (i) identifies the state space of a physical system with the
set $\rho (\mathcal{H})$ of all positive normalized trace-class operators on
a separable Hilbert space $\mathcal{H}$; (ii) associates the physical
magnitudes (or observables) pertaining to that system with the set $\frak{B}%
_{sa}(\mathcal{H})$ of all bounded, self-adjoint operators acting on $%
\mathcal{H}$; and (iii) where $\hat{\rho}$ is a state and $\hat{A}$ an
observable, assigns $\hat{A}$ the expectation value $Tr(\hat{\rho}\hat{A})$
in the state $\hat{\rho}$.
One way to obtain such a structure is to successfully quantize a classical
theory. A standard scheme for quantizing a theory cast in Hamiltonian form
is to promote its canonical position and momentum observables to symmetric
operators $(\hat{q}_{i},\hat{p}_{i})$ acting on a separable Hilbert space $%
\mathcal{H}$ and satisfying CCRs answering to the classical Poisson bracket.
Call any set of Hilbert space operators that does the trick a \textit{\
representation} of the CCRs. According to a theorem announced in 1930 by
Stone and proven the next year by von Neumann, if $(\mathcal{H},\{\hat{O}%
_{i}\})$ and $(\mathcal{H}^{\prime },\{\hat{O}_{i}^{\prime }\})$ are both
irreducible representations of the CCRs for finitely many degrees of freedom,%
$^{\text{1}}$ then $(\mathcal{H},\{\hat{O}_{i}\})$ and $(\mathcal{H}^{\prime
},\{\hat{O}_{i}^{\prime }\})$ are \textit{unitarily\ equivalent,} that is,
there exists a one-to-one, linear, norm-preserving transformation (``unitary
map'') $U:$ $\mathcal{H}\rightarrow \mathcal{H}^{\prime }$ such that $U^{-1}%
\hat{O}_{i}^{\prime }U=\hat{O}_{i}$ for all values of $i$. It follows not
only that Heisenberg's matrix realization of the CCRs for $n$ degrees of
freedom is unitarily equivalent to Schr\"{o}dinger's representation in terms
of differential operators, but also that \textit{any} Hilbert space
representation of the CCRs for $n$ degrees of freedom is equivalent to the
Schr\"{o}dinger representation.
The Stone-von\ Neumann theorem was widely received as proof that the
physical theory arising from the quantization of an $n$-dimensional
classical theory is essentially unique. A number of non-trivial assumptions
about the nature and content of quantum theories underlies this reception.
To reconstruct these assumptions, start with the idea that \textit{the
content of a physical theory is the set }$\Omega $\textit{\ of worlds
possible according to the theory}. Annex to this an assumption about how a
statistical theory characterizes a physical possibility: \textit{A physical
possibility }$\omega \in \Omega $\textit{\ is an assignment of expectation
values to a set }$\mathcal{A}$ \textit{of physical magnitudes. }Then one may
denote the content of a physical theory by the pair $(\Omega ,\mathcal{A)}$,
where $\Omega $ is its set of possibilities, that is, maps $\omega :\mathcal{%
A}\rightarrow \Bbb{R}$ from its set $\mathcal{A}$ of physical magnitudes to
their expectation values. Theories will be physically equivalent exactly
when a suitable isomorphism obtains between the sets of possibilities they
recognize--more precisely, when they satisfy a content coincidence criterion
for physical equivalence, which Clifton and Halvorson 2001 articulate as
follows:
\begin{quotation}
\noindent $(\Omega ,\mathcal{A)}$ and $(\Omega ^{\prime },\mathcal{A}%
^{\prime }\mathcal{)}$ are physically equivalent if and only if there exist
bijections $i_{s}:$ $\Omega $ $\rightarrow $ $\Omega ^{\prime }$ and $i_{o}:%
\mathcal{A}\rightarrow \mathcal{A}^{\prime }$ such that for all $\omega \in
\Omega $ and for all $A\in \mathcal{A}$, $\omega (A)=[i_{s}(\omega
)](i_{o}(A))$.
\end{quotation}
\smallskip To complete the case that quantum theories are physically
equivalent if and only if they're unitarily equivalent, one now need only
assume that quantum theories are Hilbert space theories, in the sense of
satisfying conditions (i)-(iii) announced in the first paragraph of this
section. Then it follows that quantum theories $(\rho (\mathcal{H}),\frak{B}
_{sa}(\mathcal{H}))$ and $(\rho (\mathcal{H}^{\prime }),\frak{B}_{sa}(%
\mathcal{H}^{\prime }))$ are physically equivalent if and only if they're
unitarily equivalent, in which case the unitary map furnishes both the
bijection of possibilities from $\rho (\mathcal{H})$ to $\rho (\mathcal{H}%
^{\prime })$ and the bijection of magnitudes from $\frak{B}_{sa}(\mathcal{H}
) $ to $\frak{B}_{sa}(\mathcal{H}^{\prime }),$ by mapping an operator $X$ on
$\mathcal{H}\ $to an operator$\ X^{\prime }=U^{-1}XU$ on $\mathcal{H}%
^{\prime }$ (for a proof, see Bratteli and Robinson 1987, Thm. 2.3.16).
Filtered through these presuppositions, the Stone-von Neumann theorem issues
remarkable reassurance that the Hamiltonian quantization of a finite
dimensional classical theory issues a unique quantum theory. Physicists
embracing the assumptions catalogued here can quantize classical theories
with confidence that the upshot will be an unambiguous and coherent quantum
theory.
Unless, that is, the theory they're quantizing falls outside the scope of
the Stone-von Neumann theorem. The uniqueness result holds only for
quantizations of classical theories whose configuration spaces are finite in
dimension. Elsewhere, uniqueness breaks down, and breaks down dramatically.
Where X is a classical \textit{field} theory, continuously many unitarily
inequivalent quantizations can vie for the title ``quantization of X.''
Supposing that unitary equivalence is criterial for physical equivalence,
one must also suppose that at most one unitary equivalence class of
quantizations can hold the title. This being supposed, the availability of
unitarily inequivalent representations renders ``the quantization of X''
ambiguous at best, if not incoherent.
One reaction to this state of affairs is to disambiguate--to specify that
unitary equivalence class of representations in which resides the content of
the theory. This reaction entails dismissing representations unitarily
inequivalent to the privileged one--and operators parochial to those
representations--as without physical significance. Pledging allegiance to
the idea that the space of quantum theoretic possibilities is given in terms
of a fixed Hilbert space by $(\rho (\mathcal{H}),\frak{B}_{sa}(\mathcal{H}))$%
, this is the reaction of the Hilbert space chauvinist.
Investigations of the abstract structure of standard realizations of quantum
field theoretic CCRs conducted in the 50s and 60s inspire another reaction.
These investigations revealed that each concrete Hilbert space
representation of the CCRs gives rise to an abstract algebra $C^{*}$
algebra, the \textit{Weyl algebra}, which is representation-independent.
(For more on algebraic notions introduced in this section, see Wald, op.
cit.) To frame the interpretive stance which rests on this
representation-independence, I will sketch the rudiments of an algebraic
approach to quantum theories. The algebraic approach identifies quantum
observables with self-adjoint elements of a $C^{*}$ algebra $\mathcal{A}$
(i.e., elements $A$ such that $A^{*}=A$). This algebra can be abstract, like
the Weyl algebra, or it can be an algebra of bounded operators on a fixed
and concrete Hilbert space. Thus the algebraic approach generalizes the
Hilbert space notion of observable.
It is with respect to observables in the more general sense of elements of
an algebra that the algebraic approach constitutes its notion of state. An
algebraic \textit{state} $\omega $ on $\mathcal{A}$ is a linear functional $%
\omega :\mathcal{A}\rightarrow $ $\Bbb{C}$ that is normed ($\omega (I)=1$))
and positive ($\omega (A^{*}A)\geq 0$ for all $A\in \mathcal{A}$). Hence $%
\omega (A)$ may be understood as the expectation value of an observable $%
A\in \mathcal{A}$. Where the algebra is realized as bounded operators on $%
\mathcal{H}$, the set of countably additive algebraic states stands, via the
trace prescription, in one-to-one correspondence with the set $\rho (%
\mathcal{H})$ of density operators on $\mathcal{H}.$ But the general notion
of an algebraic state does not require a Hilbert space middleman.
The set $\Omega $ of states on a $C^{*}$ algebra $\mathcal{A}$ is convex.
Its extremal elements--that is, states $\omega $ which cannot be expressed
as non-trivial convex combinations of other states--are pure states; all
other states are mixed.
Associations can be drawn between algebraic and Hilbert space frameworks. A
\textit{\ Hilbert space representation} of an abstract algebra $\mathcal{A}$
is a structure-preserving map $\pi :\mathcal{A}\rightarrow \frak{B}(\mathcal{%
H})$ , from elements of $\mathcal{A}$ to the set of bounded linear operators
on a Hilbert space $\mathcal{H}$. That even abstract algebras admit concrete
Hilbert space representations enables us to connect Hilbert space and
algebraic notion of states. A state $\hat{\rho}$ in a Hilbert space$\
\mathcal{H}$ carrying a representation $\pi :\mathcal{A}\rightarrow \frak{B}(%
\mathcal{H})$ of an algebra $\mathcal{A}$ naturally gives rise to the
algebraic state $\omega (A)=Tr(\hat{\rho}\pi (A))$ for all $A\in \mathcal{A}$
.
We can also move in the other direction, from an algebraic state to its
realization on a concrete Hilbert space. A state $\omega $ over a $C^{*}$
algebra $\mathcal{A}$ can be recast as state in a Hilbert space bearing a
faithful$^{\text{2}}$ representation of that algebra. That is, for such a
state, there exists a Hilbert space $\mathcal{H}_{\omega }$, a faithful
representation $\pi _{\omega }:\mathcal{A}\rightarrow \frak{B}(\mathcal{H}%
_{\omega })$, and a cyclic$^{\text{3}}$ vector $|\Psi _{\omega }\rangle \in
\mathcal{H}_{\omega }$ such that $\omega (A)=\langle \Psi _{\omega }|\pi
_{\omega }(A)|\Psi _{\omega }\rangle $ for all $A\in \mathcal{A}$. Called
the GNS representation of the state (for Gel'fand, Naimark, and Segal, who
showed how to construct it), the triple $(\mathcal{H}_{\omega },\pi _{\omega
},|\Psi _{\omega }\rangle )$ is unique up to unitary equivalence. An
algebraic state $\omega $ is pure if and only if its GNS representation $\pi
_{\omega }$ is irreducible. Mixed algebraic states give rise to reducible
GNS representations.
The following facts and locutions will be called into service down the road.
The \textit{folium }of an algebraic state $\omega $ is the set of all
algebraic states which may be expressed as density matrices on $\omega $'s
GNS representation. Suppose that algebraic states $\omega $ and $\omega
^{\prime }$ are pure. Then either they give rise to unitarily equivalent GNS
representations, or they do not. In the first case, their folia coincide. In
the second, their folia are \textit{disjoint}, that is, no algebraic state
expressible as a density matrix on $(\mathcal{H}_{\omega },\pi _{\omega })$
is expressible as a density matrix on $(\mathcal{H}_{\omega ^{\prime }},\pi
_{\omega ^{\prime }}),$ and vice versa. A fact that will come to the fore in
Section Three is that mixed algebraic states can be convex combinations of
disjoint algebraic states: consider $\omega =\lambda \omega _{1}+(1-\lambda
)\omega _{2}$, with $\omega _{1}$ and $\omega _{2}$ disjoint.
Setting this framework for algebraic quantum theory alongside the result
that the Weyl algebra is representation-independent one might think that
where there's a Weyl (algebra), there's a way (to do QFT). The position I'll
call \textit{algebraic chauvinism }denies that quantum theories are
essentially Hilbert space theories. The algebraic chauvinist identifies all
physical magnitudes pertaining to a system with self-adjoint elements of its
Weyl algebra, and takes the complete set of states possible for this system
to be given by normed positive linear functionals $\omega $ over this
abstract algebra. For the chauvinist, ``the important thing here is that the
observables form some algebra, and not the representation Hilbert space on
which they act'' (Segal 1967, 128). Withholding significance from
representation-dependent structures, algebraic chauvinists have the luxury
of greeting unitarily inequivalence with a yawn.
\bigskip
\noindent \textbf{3. Unitarily Inequivalent Representations in QSM. }This
section sketches some uses to which QSM would put unitarily inequivalent
representations, uses which, I suggest, should give both chauvinists pause.
(Quantum) statistical mechanics aims at a (quantum) microphysical
underpinning of bulk properties associated with macrosystems--their
temperature, pressure, entropy, and the like--an underpinning which stands
in some suitable explanatory relationship to thermodynamic laws those
macro-properties obey. Articulating canons of suitability, and assessing
putative statistical mechanical explanations against those canons, has been
a mainstay of work on the foundations of statistical mechanics. Until my
concluding anticlimactic postscript, I will bracket questions raised by such
work, in order to explore certain well-entrenched statistical mechanical
explanatory aspirations, and the theoretical/interpretational structures
which sustain them. I will focus in particular on explanatory aspirations
pursued in the \textit{thermodynamic limit }of QSM, i.e. the limit as the
number $N$ of microsystems and the volume $V$ they occupy goes to infinity,
while their density $\frac{N}{V}$ remains finite. Because the thermodynamic
limit for QSM concerns the quantum physics of infinite collections of
particles, there the spectre of unitarily inequivalent representations rears
its head.
Short of the thermodynamic limit, and in the setting of concrete Hilbert
spaces, the \textit{Gibbs state} equips QSM with a notion of equilibrium.
The Gibbs state of a system with Hamiltonian $\hat{H}$ at inverse
temperature $\beta =\frac{1}{kT}$ is the density matrix
\begin{equation}
\hat{\rho}=\exp (-\beta \hat{H})/Tr[\exp (-\beta \hat{H})]
\end{equation}
\noindent For realistic, finite quantum systems the Gibbs state is
well-defined and unique (Ruelle 1969). If, however, the spectrum of $\hat{H}$
fails to be pure discrete, or if we are working in an abstract algebraic
setting, (3.1) fails to be well defined.
Now suppose that we aspire to construct a quantum statistical account of
\textit{phase transitions. }Then we might have reason to seek a notion of
equilibrium suited to these more general settings. For the apparent
macroscopic explanandum is the existence, at certain temperatures, of
multiple thermodynamic phases. The explanatory aspirations I'll recount here
rest on the idea that a statistical account of phase transitions requires
the existence, at these critical temperatures, of \textit{multiple distinct }%
equilibrium states, answering to different thermodynamic phases. Sewell
explains how, short of the thermodynamic limit, the very uniqueness of the
Gibbs state upsets this explanatory applecart$^{\text{4}}$:
\begin{quote}
The traditional form of statistical thermodynamics for large but finite
systems . . .cannot accommodate different phases of a system (e.g. liquid
and vapor) under the same thermodynamic conditions, since the Gibbs ensemble
representing the equilibrium state of a \textit{finite} system is uniquely
determined by the prevailing macroscopic constraints: thus, if the volume,
temperature, and mass are controlled to take specific values, then the
resultant ensemble is the canonical one. (1986, 47)
\end{quote}
Explanatory hopes are revived in the thermodynamic limit by using the
\textit{KMS condition} to explicate a notion of equilibrium more general
than that afforded by the Gibbs state. A naive introduction to KMS states
follow; for an authoritative treatment, see Bratteli and Robinson 1997, \S
5.3. A $C^{*}$ dynamical system $(\mathcal{A},\alpha _{t})$ consists of a $%
C^{*}$ algebra $\mathcal{A}$ whose self-adjoint elements correspond to
physical magnitudes, and a one (real) parameter group $\alpha _{t}$ of
automorphisms on $\mathcal{A}$--that is, maps from $\mathcal{A}$ to itself
which preserve $\mathcal{A}$'s algebraic structure--which encodes dynamics.
That is, for all $A\in \mathcal{A},$ $\alpha _{t}(A)$ represents its
evolution through a time $t$. In a Hilbert space quantum theory, $\mathcal{A}
$ is given by an algebra of bounded observables on a Hilbert space, and $%
\alpha _{t}$ is implemented by a family $\hat{U}_{t}=e^{-i\hat{H}t}$ of
unitary operators generated by the Hamiltonian $\hat{H}$ of the system: $%
\alpha _{t}(\hat{A})=\hat{U}_{t}\hat{A}\hat{U}_{t}^{*}$.
In terms of such a Hilbert space realization of a $C^{*}$ dynamical system,
the Gibbs state $\hat{\rho},$ where it is well-defined, formally satisfies
\begin{center}
\begin{equation}
\omega [A\alpha _{i\beta }(B)]=\omega (BA)\text{ for all }A,B\in \mathcal{A}
\label{KMS}
\end{equation}
\end{center}
\noindent \noindent (here $\omega (x)=Tr(\hat{\rho}x)$ for all $x\in $ $%
\frak{B}(\mathcal{H})$). But formulated in general $C^{*}$ algebraic terms,
(3.2) can apply as well to states and observables abstractly conceived. To
extrapolate the notion of equilibrium beyond circumstances where the Gibbs
state (3.1) is well-defined, make the KMS\ condition (3.2) criterial for
equilibrium. Hence: $\omega $ is a \textit{\ KMS state} with respect to the
automorphism group $\alpha _{t}$ at inverse temperature $\beta $ (an $%
(\alpha _{t},\beta )$-KMS state, for short) if and only if (3.2) holds for
all $A,B$ in a dense subalgebra of $\mathcal{A}$.
If $(\mathcal{A},\alpha _{t})$ admits a standard Gibbs state at inverse
temperature $\beta ,$ the $(\alpha _{t},\beta )$-KMS state is unique and
coincides with that Gibbs state (Bratteli and Robinson 1997, Ex. 5.3.31).
KMS states moreover exhibit a number of stability features, including
invariance under the action of the dynamical group $\alpha _{t}$, putatively
characteristic of equilibrium states. For such reasons, the KMS condition is
generally regarded to be a suitable criterion for equilibrium. So
explicated, the notion applies to systems admitting no Gibbs
states--including infinite quantum systems at the thermodynamic limit.
Now consider a $C^{*}$ dynamical system $(\mathcal{A},\alpha _{t})$ . For $%
\beta \in \Bbb{\ R}$ , let $K_{\beta \text{ }}$denote the set of $(\alpha
_{t},\beta )$-KMS states$.$ Salient results about the structure of the sets $%
K_{\beta \text{ }}$ include (Bratteli and Robinson 1997, Thm. 5.3.30):
\hspace{0.2in}(1) $K_{\beta }$ is convex;
\hspace{0.2in}(2) $\omega \in K_{\beta }$ is extremal (i.e., $\omega $ can't
be expressed as a convex combination of distinct elements of $K_{\beta }$)
if and only if it's a \textit{factor} state (i.e., one for which the
intersection of $\pi _{\omega }(\mathcal{A})$ and its commutant contains
only multiples of the identity);
\hspace{0.2in}(3) Where $\omega _{1}$ and $\omega _{2}$ are extremal
elements of $K_{\beta },$ either they're equal or disjoint.
It follows from (2) that if the set of $(\alpha _{t},\beta )$-KMS states has
only one element, then that state is a factor state. Factor states, examples
of which include the equilibrium states of ideal fermi gases, can often be
characterized by the absence of long-range correlations, and of large
fluctuations for space-averaged observables. Typical of ``pure''
thermodynamic phases, these absences encourage the identification of factor
states with those phases (for more encouragement, see Sewell 1986, \S 4.4,
or Emch and Knops (1970)).
\medskip Now consider $\omega _{1}\in K_{\beta _{1}}$ and $\omega _{2}\in
K_{\beta _{2}}$. Under a technical assumption that holds generally at the
thermodynamic limit,$^{\text{5}}$ if $\beta _{1}\neq \beta _{2},$ then $%
\omega _{1}$ and $\omega _{2}$ are disjoint (Bratteli and Robinson 1997,
theorem 5.3.35). That is, for an infinite quantum statistical system, there
is no single concrete Hilbert space on which its equilibrium states at
different temperatures can be represented as density matrices.
The position of the Hilbert space chauvinist, viewed in the light of this
result, looks unreasonable. Maintaining that all physical possibilities
reside in a single folium, the chauvinist reckons states outside the favored
folium to be physically impossible. But there are systems for which this
amounts to insisting that at most one equilibrium temperature is physically
possible. The Hilbert space chauvinist cannot allow that it's in some sense
physically possible for such systems to reach equilibrium at different
temperatures. While not inconsistent, this consequence offends modal
intuitions.
Setting our sights on the explanation of phase transitions only makes
Hilbert space chauvinism look worse. Recall that for finite systems
admitting Gibbs states, the equilibrium (KMS) state at temperature $\beta $
with respect to an automorphism group $\alpha _{t}$ is unique. But in the
general setting of the thermodynamic limit of QSM, there can be automorphism
groups $\alpha _{t}$ and inverse temperatures $\beta $ such that there are a
plurality of $(\alpha _{t}$, $\beta )$ KMS states. Every $\omega $ in such a
set $K_{\beta }$ can be represented as a \textit{unique }convex combination
of extremal elements of $K_{\beta }$, which extremal elements are pairwise
disjoint (Bratteli and Robinson 1997, Thm. 5.3.30).
This makes available the following template for a quantum statistical
analysis of phase transitions. Phase transitions occur at those inverse
temperatures $\beta $ for which the set $K_{\beta }$ of $(\alpha _{t}$, $%
\beta )$ KMS states is not a singleton set and in those states $\omega \in
K_{\beta }$ which are not extremal. Such $\omega $ are convex combinations
of extremal states $\omega _{i}$. Each extremal state in this decomposition
corresponds to a pure thermodynamic phase, different states to different
phases. Thus the decomposition corresponds to the separation of a system at
equilibrium into pure thermodynamic phases, and a system in $\omega $ at $%
\beta $ exhibits phase transitions (see Sewell 1986, Ch. 4).
The analysis of phase transitions just sketched takes disjoint algebraic
states $\omega _{i}$ to co-exist in the form of different phases present at
a phase transition. Implying that what's \textit{actual} can on its own
correspond to multiple, distinct folia, this explanation is incompatible
with Hilbert space chauvinism, which limits the space of physical
possibilities to a single folium.
Pursuing explanatory aspirations in the thermodynamic limit requires
extending physical possibility beyond the lone folium to which a Hilbert
space chauvinist would confine it. It does not follow that algebraic
chauvinism is ideally supportive of the account of phase transitions just
sketched. The algebraic chauvinist's catechism is that moving to a concrete
representation adds no \textit{physical} content to a theory couched in
terms of an abstract algebra. The foregoing might tempt one to protest that
concrete representations bear crucial physical content, corresponding as
they do to the phase and the temperature of a system at equilibrium. This on
its own needn't trouble the chauvinist, provided that she can understand the
admittedly physical differences between unitarily inequivalent
representations--differences of phase and of temperature--in purely
algebraic terms. If she could, for instance, summon from her algebra a
self-adjoint element $T$ such that for any $(\alpha _{t}$, $\beta )$ KMS
state $\omega $, $\omega (T)=\frac{1}{\beta k}$ , then purely algebraic
resources would suffice for the temperature discriminations her critic
assigns to concrete representations.
Alas, algebraic resources do not, on their own, supply the chauvinist with a
temperature observable. To indicate why, we must articulate the algebraic
approach to QSM in a bit more detail (Primas 1983, \S 4.3 offers a sketch;
see also Kronz and Lupher 2001). That approach associates with each bounded
region $V$ of $\Bbb{R}^{3}$ (i.e., three dimensional physical space) an
algebra $\mathcal{A}(V)$. Call elements of such algebras \textit{strictly
local observables}. From these algebras is constructed a $C^{*}$ algebra,
the \textit{quasi-local algebra }$\mathcal{A},$ roughly as follows: take $%
\bigcup\limits_{V\in \Bbb{R}^{3}}$ $\mathcal{A}(V),$ then close in the
topology furnished by the $C^{*}$ algebraic norm. According to the algebraic
chauvinist, it is in this abstract quasi-local algebra that all physically
relevant observables reside. Now the rub is that classical thermodynamic
observables, including temperature, are absent from this quasi-local
algebra. That is, they are not observables in terms of which the algebraic
chauvinist can distinguish between states.
The accounts of equilibrium and phase transitions just sketched extend
physical possibility to unitarily inequivalent representations. Therein lies
their incompatibility with Hilbert space chauvinism. Those same accounts
distinguish physically between those representations on the basis of
observables without correlate in the abstract algebra. Therein lies their
incompatibility with algebraic chauvinism.
Now, algebraic approaches unfettered by chauvinism can bring classical
thermodynamic observables on board. Here's (again, roughly) how. For every
region $V$ define the algebra $\mathcal{A}^{\bot }(V)$ of \textit{%
quasi-local observables outside }$V$ by taking the norm closure of $\{A:$ $%
A\in \mathcal{A}(V^{\prime }),$ $V^{\prime }\cap V=\emptyset \}$. Given a
Hilbert space representation $\pi $ of the quasi-local algebra $\mathcal{A}$
, one can construct a von Neumann algebra $\mathcal{V}_{\pi }^{\bot }(V)$ by
taking the closure of $\pi (\mathcal{A}^{\bot }(V))$ in the weak operator
topology of the representation's Hilbert space. The \textit{\ von Neumann
algebra `at }$\infty $' $\mathcal{V}_{\pi }^{\infty }$ is defined by $%
\bigcap\limits_{V\in \Bbb{R}^{3}}\mathcal{V}_{\pi }^{\bot }(V)$ (see
Bratteli and Robinson 1987, 119-122). It is in this construction, obtained
only by way of a representation $\pi $, that one encounters classical
thermodynamic observables.\medskip \bigskip
\noindent \textbf{4. Conclusion Coalescing Content. }How can we construe the
content of quantum theories in a way that accommodates explanatory maneuvers
encountered in the thermodynamic limit of QSM? I've just suggested that
neither chauvinism will do. But perhaps they are not the only options open
to us. Their opposition notwithstanding, algebraic and Hilbert space
chauvinism share an assumption about how to interpret a physical theory. The
shared assumption is that a physical theory's content is to be specified by
simply sorting logical possibilities into one of two disjoint and exhaustive
categories: the physically possible and the physically impossible--as though
a physical theory ran a modal toggle with no intermediate settings. The
Hilbert space chauvinist uses a Hilbert space structure of observables to do
the simple sort; the algebraic chauvinist uses the abstract algebraic
structure.
Mathematical physicists discussing algebras and their representations might
be taken to suggest a different take on how physical theories pick out
possibilities. The remainder of this section aims not at a fully satisfying
explication of the suggestion, but at a partial, and admittedly
impressionistic, development of it.
Kadison makes the suggestion this way:
\begin{quotation}
\noindent Mathematically, a representation [of an abstract algebra]
distinguishes a certain ``coherent'' family of states from among [the full
set of algebraic states], and at the same time, in effect, ``coalesces''
some of the algebraic structure. (1965, 186)
\end{quotation}
\noindent The distinguished family is the folium of states expressible as
density matrices on the Hilbert space of the representation; the coalesced
structure includes observables parochial to that representation and
accessible through constructions--e.g. the von Neumann algebra at $\infty $%
--based on that representation.
Kadison takes concrete representations seriously as repositories of physical
content without assuming that \textit{all }physically relevant states must
reside in a single folium. He thereby suggests how to chart a course (a
course I'll call, for reasons which will become apparent, the \textit{Swiss
army approach}) between algebraic and Hilbert space chauvinisms. The Swiss
army approach has as its point of departure a refusal to specify the content
of a physical theory in one fell swoop, cleaving states possible according
to it from states impossible according to it. Rather, the Swiss army
approach takes the specification of content to be (at least) a two-tiered
affair, with a corresponding gradation in the sort of possibilities purveyed
by the theory. The broadest sort of possibility picked out by a quantum
theory is the space $\Omega _{\mathcal{A}}$ of algebraic states on the
appropriate abstract algebra. Self-adjoint elements of the algebra
correspond to the most basic physical magnitudes, those that belong to the
theory automatically. This much of the theory's content can be specified, so
to speak, a priori, before taking physical contingencies into account.
The next tier of physical content specification does take contingencies into
account. From $\Omega _{\mathcal{A}}$ a narrower set of possibilities most
relevant to the contingent empirical situation is distinguished, by appeal
to features of that situation, for instance, equilibrium temperatures. Other
algebraic states aren't impossible; they're simply possibilities more remote
from the present application of the theory than these most relevant states.
This narrowing of possibilities expands the core constituency of relevant
observables from $\mathcal{A}$ to include observables parochial to concrete
GNS representations of states in the narrower set. Again, observables
parochial to the GNS representations of states outside the most relevant set
aren't \textit{unphysical}; they're simply less relevant for the sorts of
discriminations demanded by the application at hand. This is hardly to say
that they couldn't be relevant to other applications.
We can also think of this two-tiered specification of content in terms of
the \textit{universal representation }of an algebra $\mathcal{A}.$ This is
the direct sum, over the set of algebraic states for $\mathcal{A}$, of their
GNS representations. We could construe the theory's broadest set of physical
observables in terms of this universal representation (see Kronz and Lupher
2001 for one version of this proposal, which they attribute to
Muller-Herold; Rob Clifton's also offered a version of this proposal in
conversation). At this stage of content specification, this vast host of
physical observables is just sitting there, like blades folded up in a Swiss
army knife. The next (coalescence) stage appeals to contingent features of
the physical situation to focus on a small set of representations, which are
summands in the universal representation. Observables parochial to those
representations are extracted for application to the situation at hand. Thus
coalescence is something like opening the Swiss army knife to the
appropriate blade or blades, once you've figured out what you're supposed to
do with it.
Though observables thus coalesced are less fundamental than those appearing
in $\mathcal{A}$, there can be call to draft them. We've just seen how the
coalesced von Neumann algebra at $\infty $ sustains quantum statistical
explanatory aspirations. Coalesced observables can be pressed into other
sorts of service. For example, a Hamiltonian parochial to a representation
might serve as the generator of dynamics in the folium of that
representation. (Emch and Knops's (1970) variation on the Ising model of
ferromagnetism develops a dynamics of this sort.) The interplay of abstract
and coalesced structure also figures in accounts which characterize
spontaneous symmetry breaking in terms of symmetries of the algebraic
structure which are not symmetries of coalesced structures. In the case of
phase transitions, a non-extremal KMS state $\omega =\sum \lambda _{i}\omega
_{i}$ might be invariant under symmetries (automorphisms) of the abstract
algebra which fail to be unitarily implementable on representations
coalesced around that states' extremal components $\omega _{i}$.
My suggestions that the Swiss army approach admits and supports the
foregoing applications are sketchy, and the Swiss army approach itself
remains a metaphor. But if these ideas can be developed and defended, they
would make plausible the thesis that the best way to make sense of what
physicists do with quantum theories is to allow physical possibility to be a
matter of degree.
I'll close by acknowledging an objection$^{\text{6}}$ to how I've proceeded.
Steam rises from the surface of my coffee; passing to the thermodynamic
limit to account for this, we attribute my coffee cup infinite volume. But
the volume of my coffee cup is finite! So the objection is that I\ have
rested interpretative conclusions on the consideration of a setting which is
a hotbed of manifest falsehoods and extreme idealizations.
My shamefully curt reply to this objection is that I am not resting
interpretative conclusions on artifacts of the idealization committed by the
thermodynamic limit. I am not (for instance) claiming that, notwithstanding
the appearances, steaming cups of coffee are infinite in volume. I am
instead resting interpretative conclusions on the very features of the
idealization--in particular, the structure of equilibrium states it
sustains--that enables it to account for the phenomena--the coexistence of
phases at critical temperatures. Thus I am resting interpretive
conclusions--conclusions about the manners in which theories represent--on
those facets of the thermodynamic limit that appear to do representational
work. But on this question, much more needs to be said, both by the
prosecution and the defense.
\newpage
\begin{center}
REFERENCES
\end{center}
\noindent Bratteli, Ola and Robinson, Derek W. (1987), \textit{Operator
Algebras and Quantum Statistical Mechanics 1}, 2$^{nd}$ edition. Berlin:
Springer-Verlag.
\bigskip
\noindent ---. (1997), \textit{Operator Algebras and Quantum Statistical
Mechanics 2}, 2$^{nd}$ edition. Berlin: Springer-Verlag.
\bigskip
\noindent Callendar, Craig (forthcoming), ``Taking Thermodynamics Too
Seriously,'' \textit{Studies in the History and Philosophy of Modern Physics}%
.
\bigskip
\noindent Clifton, Rob and Halvorson, Hans (2001), ``Are Rindler Quanta
Real?'' \textit{British Journal for the Philosophy of Science} 52: 417-470.
\bigskip
\noindent Emch, G. C. (1984), \textit{Mathematical and Conceptual
Foundations of Twentieth Century Physics}. Amsterdam: North-Holland.
\bigskip
\noindent --- and Knops, H.J.F. (1970), ``Pure thermodynamic phases as
extremal KMS states,'' \textit{Journal of Mathematical Physics 11},
3008-3018.
\bigskip
\noindent Kadison, Richard (1965), ``Transformations of States in Operator
Theory and Dynamics,'' \textit{Topology 3}, Suppl. 2, 177-198.
\bigskip
\noindent Haag, Rudolf (1962), ``The Mathematical Structure of the
Bardeen-Cooper-Schrieffer Model,'' \textit{Nuovo Cimento 25}: 287-299.
\bigskip
\noindent Kronz, Fred and Lupher, Tracy (2001), ``Unitarily Inequivalent
Representations in Algebraic Quantum Theory,'' (pre-print).
\bigskip
\noindent Primas, Hans (1983), \textit{Chemistry, Quantum Mechanics, and
Reductionism. }New York: Springer-Verlag.
\bigskip
\noindent Robinson, Derek W. (1966), ``Algebraic Aspects of Relativistic
Quantum Field Theory,'' in M. Chretien and S. Deser (eds.), \textit{%
Axiomatic Field Theory}. New York: Gordon and Breach.
\bigskip
\noindent Ruelle, D. (1969), \textit{Statistical Mechanics}. New York: W.A.
Benjamin.
\bigskip
\noindent Segal, Irving. (1967), ``Representation of Canonical Commutation
Relations,'' in F. Lurcat (ed.), \textit{Cargese Lectures in Theoretical
Physics}. NY: Gordon and Breach.
\bigskip
\noindent Sewell, Geoffrey L. (1984), \textit{Quantum Theory of\ Collective
Phenomena}. Oxford University Press.
\bigskip
\noindent Wald, Robert M. (1994), \textit{Quantum Field Theory in Curved
Spacetime and Black Hole Thermodynamics}. Chicago: University of Chicago
Press.
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\bibitem{}
\newpage
\end{thebibliography}
\begin{center}
FOOTNOTES
\end{center}
\noindent 1. A representation of the CCRs is\textit{\ irreducible} if and
only if the only subspaces of $\mathcal{H}$ invariant under the action of
all operators in the representation are the zero subspace and $\mathcal{H}$
itself. There are further technical assumptions; for an introduction, see
Wald 1994, Ch. 2.2. (To state the obvious: I don't aim here at a
presentation comprehensive in technical detail, and so will refer interested
readers to more rigorous discussions. To avoid expository clutter, some
peripheral technical notions will be defined in footnotes rather than in the
text.)
\noindent 2. A representation $\pi $ of $\mathcal{A}$ is \textit{faithful}
if and only if $\pi (A)=0\Rightarrow A=0$.
\noindent 3. $\left| \Psi \right\rangle $ is cyclic for $\pi _{\omega }(%
\mathcal{A})$ means $\{\pi _{\omega }(\mathcal{A})\left| \Psi \right\rangle
\}$ is dense in $\mathcal{H}$.
\noindent 4. Another reason for going to the thermodynamic limit, which I do
not consider here, is that it is only in the thermodynamic limit that
discontinuities in thermodynamic functions, which discontinuities are
characteristic of phase transitions, occur.
\noindent 5. Where $\mathcal{U}^{\prime }$ denotes the commutant of $%
\mathcal{U}$, the assumption is that the von Neumann algebras $\pi _{\omega
_{1}}(\mathcal{A})^{\prime \prime }$ and $\pi _{\omega _{2}}(\mathcal{A}%
)^{\prime \prime }$ are Type III ; for a sketch of why it holds generally at
the thermodynamic limit, see Emch 1984, 448-450.
\noindent 6. John Earman, brandishing a copy of Callendar (forthcoming), has
urged this objection upon me.
\end{document}