2.2 Quantum probabilities do not represent physical reality
After formulating a minimalist account of truth, Wright ([1992]) presents considerations that may incline one to deploy a richer notion of truth as correspondence to objective reality in some domain. By applying these considerations to ascriptions of quantum probabilities we can further articulate the sense in which these are objective even though they do not describe any natural property of or associated with quantum systems.
The first consideration stems from the Cognitive Command constraint, of which this is an abbreviated version of Wright’s first ‘extremely rough’ formulation:
A discourse exhibits Cognitive Command if and only if it isa priori
that differences of opinion arising within it can be satisfactorily explained only in terms of “divergent input”, “unsuitable conditions”, or “malfunction”. (pp.92-3
)
He gives this later formulation to qualify and simplify his first formulation:
It isa priori
that differences of opinion formulated within the discourse, unless excusable as a result of vagueness in a disputed statement, or in the standards of acceptability, or variation in personal evidence thresholds, so to speak, will involve something which may properly be regarded as a cognitive shortcoming. (p.144)
What is the idea of the Cognitive Command constraint, and what motivates it? Here is what Wright says:
The formulation offered is an attempt tocrystallise
a very basic idea we have about objectivity: that, where we deal in a purely cognitive way with objective matters, the opinions which we form are in no sense optional or variable as a function of permissibleidiosyncracy
, but arecommanded
of us - that there will be a robust sense in which a particular point of viewought
to be held, and a failure to hold a particular point of view can be understood only as a rational/cognitive failure. (p.146)
This nicely captures the sense in which I claim quantum probabilities are objective. Indeed, one can locate a twofold source of the command in this case. Quantum theory itself commands that quantum probabilities conform to theBorn
rule: the community’s collective evidence-based judgment commands use of a particular quantum state in the Born rule. Neither command is arbitrary: the authority in each case rests ultimately on experimental and observationalresults and
collective judgment of their evidential bearing. But Wright continues
It is tempting to say that this just is, primitively, what is involved in thinking of a subject matter as purely objective, and of our mode of interaction with it as purely cognitive; and that the Cognitive Command constraint, as formulated, is merely what results when the basic idea is qualified to accommodate various germane kinds of vagueness. [...] But [...] the truth is that the constraint does not reflect a wholly primitive characteristic of the notions of objectivity and cognitive engagement but derives its appeal, at least in part, from a truism to do with the idea ofrepresentation
. For to think of oneself as functioning in purely cognitive mode, as it were, is, when the products of that function are beliefs, to think ofoneself
as functioning in representational mode (ibid
.)
Reflection on the epistemic function of probability ascriptions should prompt one to question this “truism”. For while one is clearly functioning in cognitive mode when assessing probabilities, the products of that function arepartial
beliefs, and while each of these does indeed have some kind of representational content, only in the case of derivative, higher-order applications does this concernprobabilities
- in the fundamental situation, the content of each partial belief represents a possible state of the world free of any natural “probability properties”.
The constitutive function of quantum probability statements is not to represent certain probabilistic aspects of the physical world, but to guide agents in forming appropriate partial beliefs about non-probabilistic aspects of the physical world. The intuition Wright takes his Cognitive Command constraint to express survives undercutting of any possible justification by appeal to alleged truisms about representation.
Wright introduces a second consideration favoring deployment of a richer notion of truth as correspondence to objective reality in some domain.
Let thewidth of cosmological role
of the subject matter of a discourse be measured by the extent to which citing the kinds of states of affairs with which it deals is potentially contributive to the explanation of thingsother than
, orother than via
, our being in attitudinal states which take such states of affairs as object [...]. The crucial question is [...] what else there is, other than our beliefs, of which the citation of such states of affairs can feature in […] explanations. (pp.196-7
)
Ascriptions of quantum probability have very narrow cosmological role. While they may and do play a role in a huge variety of applications of quantum theory in prediction as well as explanation, in each case the contribution of quantum probabilities is indeedvia
an agent’s being in attitudinal states which take quantum probabilities as object. Someone may object that it is a basic role of quantum probabilities to explain frequencies observed, say, in experimental tests of Bell inequalities. But of course a claim about frequencies follows from a claim about probabilitiesonly with a certain probability
: so a judgment that an observed frequency is explained by a quantum probability itself proceedsvia
an agent’s being in attitudinal states which take quantum probabilities as object.
The narrow cosmological role of quantum probability statements provides further support for the conclusion that these do not represent natural properties. But it does nothing to undermine theobjectivity
of quantum probabilities.
Despite his avowed subjectivism about probability, David Lewis ([1980]) undertook to offer a subjectivist’s guide to a kind of objective probability he called chance.
Along with subjective credence we should believe also in objective chance. The practice and the analysis of science require both concepts. Neither can replace the other. Among the propositions that deserve our credence we find, for instance, the proposition that (as a matter of contingent fact about our world) any tritium atom that now exists has a certain chance of decaying within a year. ([1986], p.83)
I will explain in section 4 why quantum probabilities should not be taken to have all the features Lewis attributed to chance. But Lewis was right to believe that objective probabilities figure in science, and that these include quantum probabilities. This makes it particularly interesting that he took a single principle to capture all we know about chance, namely the Principal Principle, whose initial statement was as follows:
LetC
be any initially reasonable credence function. Lett
be any time. Letx
be any real number in the unit interval. LetX
be the proposition that the chance, at timet
, ofA
’s holding equalsx
. LetE
be any proposition compatible withX
that is admissible at timet
. ThenC
(
A/XE
)=x
. ([1986], p.87)
If the only thing we know about objective probabilities is that they command an agent to adjust itscredences
(partial beliefs) so they equal the corresponding objective probabilities, then it is not surprising that they carry so little explanatory weight. The “thinness” of Lewis’s account of probability as it occurs within physics reinforces the application of Wright’s point - that narrowness of cosmological role is convincing evidence against a representational view of quantum probability as a natural property of (something in) the world. But this in no way undermines Lewis’s claim to be offering an account ofobjective
probability.
3. How quantum theory limits description of physical reality
In their famous EPR ([1935]), Einstein,Podolsky
and Rosen assumed that quantum theory has descriptive resources but argued that these do not permit acomplete
description of physical reality. Specifically, their argument set out to show that the wave-function fails to give a complete description of the state of an individual system. But they took it for granted that the wave-function’s role was indeed to describe physical reality, however incompletely: Einstein ([1949], pp.82-7, p.682) suggested that the wave-function should be taken incompletely to describe a statistical ensemble of similarly prepared systems.
On the pragmatist approach I am presenting here, the quantum state does not itself purport to describe physical reality at all - not even incompletely. But in addition to its role in generating quantum probabilitiesvia
theBorn
rule, it has an important secondary role in licensing limited claims about physical reality by an agent applying quantum theory.
One can appreciate the need for such a role only after one has abandoned the idea that quantum theory itself makes available new descriptive or representational resources, either in the form of quantum state ascriptions or in some other way (perhaps by allowing dynamical variables to take on operator-values -q
-numbers-
instead of, or as well as, real numbers -c
-numbers). Rather than thinking of quantum theory as providing distinctively new ways of describing or representing physical reality, focus instead on its effect onnon-quantum
descriptions and representations.
It is tempting to refer to such non-quantum descriptions and representations asclassical
, following Bohr and others. But there are at least two reasons not to yield to this temptation. First, it encourages the mistaken thought that any use of such a description carries with it the full content of classical physics, including dynamical laws such as those of Newton and Maxwell. More importantly, it tends unduly to limit the scope of non-quantum descriptions to exclude what Bell called ‘the familiar language of everyday affairs, including laboratory procedures’ as well as descriptions made available by further advances in non-quantum physics (for example, possible modifications of classical relativity to secure empirical adequacy in light of new observations attributed to high-energy cosmic rays or so-called dark matter). The scope of non-quantum descriptions is very wide: indeed, if quantum theory itself provides no resources for describing or representing physical reality, thenall
present and future ways of describing or representing it will be non-quantum.
It is critical for the present approach to have available non-quantum descriptions of outcomes of quantum measurements. Call a claim expressed by a sentence of the formS
: ‘The value ofA
ons
lies in Δ’ a non-quantum magnitude claim (NQMC). If one could not express the result of a measurement in a NQMC, then theBorn
rule could acquire no empirical support from measurements and we should have little or no reason to believe quantum theory. Quantum theory itself does not imply sentences of the formS
on the present approach, even in a case in which theBorn
rule assignsS
probability 1. But physicists make claims using such sentences (or their equivalents) all the time, when describing the results of quantum measurements and in many other circumstances (e.g. in describing the operation of particle accelerators and nuclear reactors, as well as the position of ions in a crystal and the velocity and polarization of photons propagating through an optical fiber delay). How can one reconcile this practice with their acceptance of quantum theory, in light of the no-go results mentioned in section 2 (see footnote 6)?
Answering this question will require excursions into pragmatist philosophy as well as the quantum physics ofdecoherence
. Claims of environmentally-induceddecoherence
, to solve the quantum measurement problem and explain the emergence of classical behavior of macroscopic objects, are now widely (and wisely) regarded with suspicion.
But it is hard to dismiss the thought thatdecoherence
hassome
important role to play in resolving interpretational problems of quantum theory. AsBacciagaluppi
([2003/7]) andSchlosshauer
([2007]) explain,decoherence
plays different roles within different attempted interpretations of the theory. So after a sketch of relevant quantum physics ofdecoherence
, my main task here will be carefully to explain how a pragmatist can use this to explain when and how quantum theory itself can license the kind of non-quantum descriptive claims physicists do, and must, make in order successfully to apply quantum theory.
The basic idea of environmentally-induced delocalization of coherence is well known.
It may be illustrated by this toy model. Given an arbitrary superposed pure quantum state of a systems
interacting with a systems′
in an appropriate initial quantum state |b
0, , there are Hamiltonians on the tensor product Hilbert space Hs*Hs′ that will induce the following unitary evolution of the total quantum state ofs
+s
ʹ:
∑i
c
i
|a
i
, |b
0, → ∑i
ci
|a
i
, |b
i, (2)
for
some completeorthonormal
basis |a
i
, of Hs. The resulting quantum state ofs
is then given by partial tracing over Hs′ as
ρs
= ∑i
|c
i|2 |a
i
, +a
i
|, which contains no terms diagonal in the preferred |a
i
, basis defined by the Hamiltonian for this interaction. Thinking ofs′
as the environment ofs
, such an interaction with its environment has delocalized the coherence ofs
’s
initial state into the more inclusive systems
+s
′
(which in this case remains pure): every Born probability for an observable ons
alone equals the weighted average (withweights|
c
i|2 ) of Born probabilities of all states |a
i
, Following such an interaction,s
will display none of the interference characteristic of quantum mechanicalsuperpositions
. To observe any interference it would be necessary to perform an appropriatejoint
measurement involvingboth
ofs
ands′
.
Environmentally induceddecoherence
has only relatively recently become the subject of experimental investigation. One particularly revealing set of experiments studies interference phenomena involving large molecules including fullerenes (C60 and C70 molecules).Hackermüller
et
al
. ([2004]) investigated the effects of increased temperature in matter wave interferometer experiments in which C70 molecules lose their quantum behavior by thermal emission of radiation. They prepared a beam of C70 molecules of well-defined velocity, passed them through two gratings of a Talbot-Laue interferometer in a high vacuum, and detected those that passed through a third movable grating set at the appropriate Talbot distance and used as a scanning mask, by ionizing them and collecting the ions at a detector. Each molecule is sufficiently large and complex to be assigned a temperature as it stores a considerable amount of energy in its internal degrees of freedom. Interaction with the electromagnetic vacuum may result in emission of photons withan intensity
and frequency that increases as the internal temperature is raised. These photons may be considered the environment of the molecule. Entanglement between such photon states and the state of the emitting molecule tends to induce environmentaldecoherence
.
Hackermüller
et
al
. ([2004]) present a theoretical model of thisdecoherence
that fits their observations quite well, as the observed interference dies away when the molecules’ temperature is raised from 1000°K to 3000°K. This model bears an interesting correspondence to more informal discussions of how the possibility of observing through which slit a particle passed will prevent observation of any consequent interference pattern.
Such discussions often focus on particular methods for trying to observe through which slit each particle passes, and proceed to argue that quantum features of the required apparatus necessitate a trade-off between success in this attempt and success in obtaining any resulting interference pattern. Following Heisenberg ([1930]), one often considers shining light on the particles and collecting reflected light in a microscope focused on them as they pass the slits. In order to tell through which slit a particle goes one would need to use a microscope capable of resolving distances at least as small as the slit separation. Now the resolving power of a microscope is limited by the wavelength of light used: better resolving power requires light of shorter wavelength. However, photons of light of short enough wavelength would have such a large momentum as to disturb the particle and effectively to destroy the interference pattern. Even though no observation of the positions of C70 molecules as they pass through the apparatus is contemplated in these experiments, and no light is shone on them, the theoretical model ofdecoherence
shows that the possibility of photonemission
of short enough wavelength to make itpossible
to determine through which slit each molecule goes is enough effectively to destroy the interference pattern. Moreover, the detailed form of the quantum state of the fullerene (expressed in the off-diagonal elements of the fullerene center-of-mass position density operator) describes the diffraction limitation of a hypothetical microscope used to obtain which-path information on the molecules.
The phrase “which-path information” (or „welcher
-Weg
-Information”) that occurs repeatedly inHackermüller
et
al
. ([2004]) and many other experimental as well as theoretical treatments of quantum interferenceis
puzzling but highly suggestive. I shall pursue the suggestion after discussing another related experiment recently conducted in the same laboratory in Vienna.
Juffman
et
al.
([2009]) prepared a beam of C60 molecules with well-defined velocity, passed them through two gratings of a Talbot-Laue interferometer in a high vacuum, and collected them on a carefully prepared silicon surface placed at the Talbot distance. They then moved the silicon about a meter into a second high vacuum chamber and scanned the surface with a scanning tunneling electron microscope (STEM) capable of imaging individual atoms on the surface of the silicon. After running the microscope over a square area of approximately 2μm2 they were able to produce an image of some one to two thousand C60 molecules forming an interference pattern.
They reported that the surface binding of the fullerenes was so strong that they could not observe any clustering, even over two weeks. Clearly they felt no compunction in attributing very well defined, stable, positions to the molecules on the silicon surface, and even recommended developing this experiment into a technique for controlled deposition fornano
-technological applications.
Together, these experiments illustrate three different scenarios in which one may contemplate making a claim about the position of an individual fullerene molecule involved in a quantum interference experiment. By reflecting on the inferential commitments entered into by one who makes such a claim, we shall be able to gain a better appreciation of the significance of judgments expressed in NQMC’s of the formS
: ‘The value ofA
ons
lies in Δ’, beginning with the case in whichs
is an individual fullerene molecule,A
is the horizontal distancex
(in nanometers) of its center of mass from a reference point in the plane of the vertically oriented gratings and Δ is an interval of real numbers. To repeat, while quantum theory itself does not imply sentences of the formS
, they play an essential role in any application of quantum theory.
After C60 molecules
has been deposited on the silicon substrate in the experiment ofJuffman
et
al.
([2009]) and imaged by the STEM, their figures 2 and 3 (together with the surrounding discussion) illustrate that some claim of the formS
x
: ‘The positionx
ofs
isx
s±ε
’ for some value of ε<5nm is warranted. The warrant derives substantially from the reliability of the image-forming process, importantly including the (quantum!) theory and practice underlying the successful operation of the STEM used to produce it. But there is a prior issue: given that a C60 molecule may itself be treated as a quantum system, how and why is one entitled to attribute tos
a definite, stable position in the first place?
It is in answeringthis question
that it is appropriate to appeal to environmentaldecoherence
. While it may be difficult to formulate and solve the Schrödinger equation for a realistic many-body quantum interaction that bindss
to the silicon surface, it is clear that this will rapidly and strongly couples
to an environment of an exponentially increasing number of degrees of freedom, involving the entire silicon crystal and light reflection from its surface, thermal radiation interacting with phonons in the crystal, vibrations and thermal motion of the supporting structure of the crystal, and eventually the entire laboratory and beyond. Examination of the properties of analytically and computationally solvable models ofdecoherence
in simpler systems justifies one in concluding with a high degree of confidence that the center-of-mass state ofs
alone will extremely quickly become, and remain indefinitely in the absence of external disturbances, very close to diagonal in a preferred “position basis” of states, each close to a delta function of position.
It does not follow that some statement of the formS
x
is true. On the present approach, no analysis of adecoherence
interaction to show the (approximately) diagonal form of the quantum state of adecohering
system ever itself thereby implies any such non-quantum statement. The import of the quantum analysis is more subtle. Whatdecoherence
shows in this example is that what an agent may legitimately infer from a claim about the position ofs
of the formS
x
is, as it relates to any conceivable goal of that agent, exactly what would follow from the simple truth ofS
x
.
This is how the quantum theory ofdecoherence
licenses
the experimenters inJuffman
et
al.
([2009]), anyone reading their paper, and indeed any suitably physically situated agent, human, conscious, or neither, to make some such claim. While quantum theory in this waylicenses
many incompatible claims of this form, each ascribing a different valuex
s1,x
s2 ,
x
s3 ,... tox
, by itself the theorywarrants
an agent in claiming none of them: that requires additional, reliable empirical information of a kind acquired by the skillful use of the STEM used byJuffman
et
al.
([2009]) to produce data like that displayed in figures 2 and 3. Quantum licensing takes the following form: a quantum state of a system and its environment may be such as to grant an agent permission to issue a judgment of a certain kind concerning that system. Equipped with the necessary permission, the agent may be warranted by its “experience” to issue one rather than another judgment of that kind.
Feynman ([1963], vol. III, 1.9) said this about the position of an electron as it passes through an analogous 2-hole interference experiment:
if one has a piece of apparatus which is capable of determining whether the electrons go through hole 1 or hole 2, then onecan
say it goes through either hole 1 or hole 2. [otherwise
] one maynot
say that an electron goes through either hole 1 or hole 2. If onedoes
say that, and starts to make any deductions from the statement, he will make errors in the analysis. This is the logical tightrope on which we must walk if we wish to describe nature successfully.
Consider instead the status of claims of the formS
x
about a fullerenes
as it passes through the interferometer gratings in either of the two experiments just described. In each diffraction grating in these experiments the slits were regularly spaced at a distance of some hundreds of nanometers. So if one could sayS
or
:Sx
s1 orSx
s2 orSx
s3or ...(wherex
si
marks the center of thei
th
slit and ε now corresponds to the width of each slit) then one could say the fullerene goes through slit 1 or slit 2 or slit 3 or . Butcan
one sayS
or
?
In the experiment ofJuffman
et
al.
([2009]) there was no piece of apparatus capable of determining which slit each fullerene goes through. What goes wrong if one saysS
or
? Feynman’s discussion makes clear the nature of the error he thinks would follow from this claim. Suppose one assumes thej
th
fullerene goes through sliti
. It would have made no difference to its subsequent behavior if all the other slits had been closed, so it would have contributed to a single slit interference pattern centered on sliti
. It follows that the total interference pattern will be just a sum of single slit patterns for alli
, weighted by the number of fullerenes going through each slit. Since the actual interference pattern is quite different,S
or
has been empirically falsified.
The form of the argument isreductio
ad absurdum
, but as is typical for such arguments it rests on additional premises, any of whose rejection prevents one from drawing the intended conclusion.Bohmians
, among others, have principled reasons for denying that the behavior of a fullerene passing through one slit is independent of whether other slits are open or closed: roughly, they take its behavior to be governed by a physically real wave-function that passes throughall
the open slits.Bohmian
mechanics shows how to draw many more interesting conclusions ofS
or
consistent with quantum-theoretic predictions, though at the cost of accepting action-at-a-distance.
If one supplementsS
or
withno
additional premises, one will never risk error in making deductions. This is a trivial consequence of two facts: (i
)S
or
is logically consistent, and (ii) no valid deductive argument can lead from logically consistent premises to a contradictory conclusion. But this response misses Feynman’s point, since he is clearly concerned not just with formally valid deductive arguments whose sole premise isS
or
, but with inferences of the kind anyone with a normal understanding ofS
or
will naturally make, for example that:
No particle passes through the material (silicon nitride) in which the slits are cut
It is possible reliably to observe through which slit each particle passed without altering the interference pattern, or
If this is not so, then that can only be because any physical mechanism that permitted reliable observation of through which slit each particle passed would inevitably disturb the particle while doing so
Brandom
([2000]), followingSellars
([1953]), calls inferences such as those fromSor
to I, II, IIImaterial
inferences. Here and inBrandom
([1994]) he develops what he callsaninferentialist
pragmatism about conceptual content. It is a consequence of this kind of pragmatism that thecontent
ofS
or
is a function of the material inferences that connect it to other claims and other actions by a claimant or others in the same linguistic community. Accepting quantum theory in no way undermines the inference fromS
or
toI
: this remains a legitimate material inference even though it is not formally valid. But however natural inferences to II or III may seem,
application of quantum theory shows that both II and III lead to conflict with results of experimental (or at leastGedankenexperimental
)findings.
So while one cansay
S
or
(pace
Feynman), the content of that claim must be understood very differently within a community that has accepted a quantum theoretic analysis of the situation to which the claim applies. Given the possibility of confusion provided by so severely weakening the claim, it may be wise to heed Feynman’s cautionary advice not to sayS
or
at all in the context of the experiment ofJuffman
et
al.
([2009]).
The status ofS
or
in the experiment ofHackermüller
et al.
([2004]) is more complex. In the case of low temperature fullerenes, there is relatively littledecoherence
of their center of mass motion through the interferometer, so the analysis goes through as for the experimentofJuffman
et
al.
([2009]). While onecan
sayS
or
, it is probably safest not to do so, since anyone who did so would naturally be understood as committed to inferences and other actions they did not intend. Without careful qualification, the weakened content of the claim would make it likely subject to misinterpretation. As the temperature of the fullerenes is increased, the interference contrast decreases. The authors comment
This is the signature ofdecoherence
due to the enhanced probability for the emission of thermal photons that carry ‘which-path’ information. [...] They transmit (partial) which-path information to the environment, leading to a reducedobservability
of the fullerene wave nature. [...] Around 3,000°K the molecules have a high probability to emit several visible photons yielding sufficient which-path information to effect a complete loss of fringe visibility in our interferometer.
I believe it would misinterpret their use of the phrase ‘which-path information’ here to take them to presuppose that each fullerene follows a determinate, though unknown, path through the slits, which becomes progressively more open to potential observation as its temperature increases. It is better to regard thecontent
of a claim made by a statement likeS
or
as itself a function of temperature, in the following sense: as the temperature is increased from 1000°K to 3000°K, the inferential power of the claim increases accordingly. This is why it becomes more and more appropriate to think and speak of the fullerenes as having a well-defined path through the interferometer as the degree of thermally induced electromagneticdecoherence
into their environment increases. But note that on the presentinferentialist
view of content, this progressive definition of content has no natural limit such that one could say that when this limit is reached a statement likeS
or
is simply true because one has finally succeeded in establishing a kind of natural language-world correspondence relation in virtue of which the statement correctly represents some radically mind- and language-independent state of affairs.
It is important to bear this in mind when reconsidering the role of measurement in quantum theory. I have been careful not to formulate theBorn
rule narrowly so that it explicitly concerns results of measurements on a quantum system. But it is vital that situations to which theBorn
rule applies include those in which scientists are warranted in making claims about the values of dynamical variables as a result of performing operations they take to constitute measurements of them. Recall from the introduction how Bell introduced his notion ofbeables
to apply to things
…which can be described in ‘classical terms’, because they are there.
Thebeables
must include the settings of switches and knobs on experimental equipment, the current in coils, and the readings of instruments.
emphasizing
that by ‘classical terms’ he refers simply to the familiar language of everyday affairs, including laboratory procedures, in which objective properties -beables
- are assigned to objects.
His thought seems to be that at least when it comes to descriptions of experimental equipment and laboratory procedures language must be taken to function in a straightforwardly representational way - as simply saying how things are.
What “is there” in these fullerene experiments? Because of the massivedecoherence
between large scale features of the macroscopic laboratory apparatus and its environment, quantum theory licenses claims about the settings of switches and knobs on experimental equipment, the macroscopic current in coils, and the readings of instruments. The content of such claims is almost, but not quite, unchanged by acceptance of quantum theory, since the limits the theory places on their inferential power are of no importance for any practical, or even impractical, purpose. Acceptance of quantum theorydoes
significantly modify the content of claims about the microscopic currents produced by electrons tunneling from the fullerenes and silicon surface when scanned by the STEM in the experiment ofJuffman
et
al.
([2009]). These currents cannot be said to “be there” in the same robust sense, in so far as each results from a tunneling process that is characteristically non-classical. And, as the preceding discussion made clear, acceptance of quantum theory so significantly limits the content of claims about the position of fullerenes as they pass through the interferometer that anyone making them at best courts confusion. But the efficiency ofdecohering
interactions between a fullerene molecule and atoms of the silicon surface in the experiment ofJuffman
et
al.
([2009]) is such that even though the molecule and atoms are microscopic and the interaction is quantum, a claim that the molecule is there at a specific location on the surface has almost the status of a claim that the entire apparatus is there in the laboratory.
There is no explicit reference to measurement in the published report of either fullerene experiment. Moreover, in applying quantum theory to account for the features of the interference patterns it is not necessary to interpret theBorn
rule explicitly to concern probabilities ofmeasured
positions of fullerenes in the pattern: one can take it simply to give probabilities for their positions at the detector. But I think it is clear that deposition of a C60 molecule on the silicon substrate in the experiment ofJuffman
et
al.
([2009]) does count as performance of a quantum measurement of the molecule’s position. Certainly there is no temptation to say that the molecule has no definite position on the surface until and unless a subsequent observation is carried out using the STEM.
We can now see both why it is natural to formulate the Born rule so that it concerns probabilitiesof
measurement outcomes
and why the application of that rule is not restricted to measurement contexts. One reason to explicitly mention measurement in a formulation of theBorn
rule is to stress that the evidence justifying acceptance of quantum theory rests to a large extent on the results of experiments in which observables are measured and the statistical distribution of measurement outcomes compared to those expected on the basis of the Born rule. If the Born rule could not be connected to measurement outcomes in this way, quantum theory would be cut off from its evidential base. But, as we have seen, the link can be preserved by simply assuming that the outcome of a quantum measurement can be expressed in a NQMC, with no mention of any measurement of a kind needed to determine the value of the magnitude in question. This leads to the second, more substantial reason for formulating theBorn
rule in terms of measurement: the no-go results mentioned in section 2. Not all dynamical variables on a quantum system can consistently be assigned simultaneous real values distributed in accordance with theBorn
probabilities - not even so as to match just theextremal
Born probabilities 0 and 1. But this presents no problems on the present approach, since Born-rule probabilities are well-defined only over claims licensed by quantum theory. According to the quantum theory, interaction of a system with its environment typically inducesdecoherence
in such a way as (approximately) to select a preferred basis of states in the system’s Hilbert space. Quantum theory will fully license claims about the real value only of a dynamical variable represented by an operator that is diagonal in a preferred basis: it will grant a slightly less complete license to claims about approximately diagonal observables. All these dynamical variables can consistently be assigned simultaneous real values distributed in accordance with theBorn
probabilities. So there is no need to formulate theBorn
rule so that its probabilities concern only measurement outcomes.
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Author: Richard Healey
