![3
that the question of whether quantum objects can be differentiated by
properties could merit thorough philosophical scrutiny.
Characteristically, the majority of working physicists remain rather
unimpressed by the metaphysical ramifications of the symmetrization
postulate.1
True, they admit that the postulate is very important from an
empirical and practical point of view, especially when we are interested in
describing the behavior of large collections of particles, for the statistical
predictions regarding indiscernible quantum particles differ significantly
from analogous predictions concerning classical, discernible particles.
Generally speaking, quantum particles are divided into two categories—
bosons and fermions—depending on the mathematical transformation
their joint states undergo under the permutations of objects. Statistical
behavior of bosons differs from that of classical particles in that the
probability of finding a group of bosons occupying same states is higher
than in the classical case (as if bosons “attracted” each other slightly). On
the other hand, same-type fermions never occupy the same state (they
“repel” each other strongly).2
However, when dealing with small numbers of same-type fermions or
bosons, physicists often ignore the symmetrization postulate and write
their states in a non-permutation-invariant form, as if they characterized
distinguishable particles. Here is an interesting quote on that issue from
a well-known textbook on quantum mechanics (Cohen-Tannoudji et al.
1978, p. 1406):
If application of the symmetrization postulate were always indispensable, it
would be impossible to study the properties of a system containing a
restricted number of particles, because it would be necessary to take into
account all the particles in the universe which are identical to those in the
system. We shall see […] that this is not the case. In fact, under certain
special conditions, identical particles behave as if they were actually
1
Of course, there are some notable exceptions—famously including Erwin Schrödinger and Henry
Margenau—but these are physicists who are already influenced by a philosophical way of thinking.
See, for example, Schrödinger (1952); Margenau (1944, 1950).
2
A careful reader may notice a strange inconsistency between this general characteristic of fermi-
onic behavior (whose more precise expression bears the name of Pauli’s exclusion principle) and the
Indiscernibility Thesis mentioned earlier. Much of the subsequent discussions in this book will try
to explain away this inconsistency.
1 Introduction](https://image.slidesharecdn.com/61220-250412055918-bd97da60/85/Identity-and-Indiscernibility-in-Quantum-Mechanics-Tomasz-Bigaj-21-320.jpg)
![16
hardly use the identity between the expectation value of A ⊗ I calculated
for the state of the entire system and the expectation value of A for the
reduced mixed state, as an argument that A ⊗ I and A are physically
equivalent in all scenarios. The only argument we can rely on is an “induc-
tive” one: since the expectation values of A ⊗ I and A coincide in the
special cases when the components possess their own pure states (alterna-
tively, since A ⊗ I possesses a given definite value iff A possesses the very
same value), we stipulate that the expectation values should also coincide
in the remaining cases, and therefore we assume that operators A ⊗ I and
A represent the same physical quantity. We will see later in the book
(Chap. 5) that analogous inductive arguments may be questioned in
some other contexts.
In the previous section we have considered an important type of per-
mutation operation, which was defined as a map on the total Hilbert
space, transforming vectors into vectors. It turns out that permutations
can also be applied to operators acting on Hilbert spaces rather than to
vectors. The general method of how to turn a transformation on a given
space ℋ into the corresponding transformation on the set of operators
on ℋ can be presented as follows. Let P be any linear transformation on
ℋ which has an inverse (thus P must be one-to-one). We will seek a
transformation corresponding to P that sends any operator A into its
counterpart A′ while satisfying the following requirement:
For any vectors and iff .
| | , | | | |
A AP P (2.7)
Expressing this condition informally: the transformed operator A′ acts in
the original space ℋ the same way as the original operator A acts in the
transformed space P[ℋ]. From this we can easily derive the form of the
operator A′ in terms of A and P. By applying the inverse operation P−1
to
both sides of the eq. AP|φ〉 = P|ψ〉, we get P−1
AP|φ〉 = |ψ〉, from which it
follows (given that |φ〉 and |ψ〉 are selected completely arbitrarily) that A′
= P−1
AP (see Fig. 2.1).
We can now return to the case of permutation transformation. What
is the result of applying permutation P12 to the operator A ⊗ I? In order
to calculate the outcome of the transformation P12
−1
(A ⊗ I ) P12, we have
T. Bigaj](https://image.slidesharecdn.com/61220-250412055918-bd97da60/85/Identity-and-Indiscernibility-in-Quantum-Mechanics-Tomasz-Bigaj-33-320.jpg)
![29
| | | | | | | ( ) |
|
A A I P A I P P A I P
1 12 12 12
1
12
I
I A A
| | | .
2
(2.12)
Thus the expectation values for all observables pertaining to either parti-
cle are identical, and this is what the Indiscernibility Thesis amounts to.
The crucial premise in the above argument is the assumption that the
operators A ⊗ I and I ⊗ A are indeed formally accurate representations
of observable A pertaining, respectively, to the first and second particles.
But here we encounter an immediate stumbling block. Operators A ⊗ I
and I ⊗ A are clearly not symmetric, so they should be disallowed on the
basis of our earlier considerations. Since SP implies that only symmetric
operators can have physical meaning when applied to systems of same-
type bosons or fermions, it seems that we should not use the non-sym-
metric products in formalizing the argument for the Indiscernibility
Thesis. French and Redhead are aware of that difficulty, but they are
strangely dismissive about it. First, they interpret the symmetry postulate
with respect to observables not as delimiting physically meaningful opera-
tors but operators that can be observed.15
Having done this, they announce
that “from the point of view of discussing PII [the Principle of the Identity
of Indiscernibles – TB] it seems clear that we should not restrict the dis-
cussion to attributes which can actually be observed” (ibid. p. 239).
French and Redhead’s response to the problem raises several questions.
Firstly, interpreting the requirement of symmetry as applying to opera-
tors that can be observed misses the point of the permutation-invariance
problem. Suppose that we have a non-symmetric operator Ω which, even
though it cannot be observed, is still admissible as a representation of a
particular objective property of the system. This means that an “actualiza-
tion” of a given value of this operator (French and Redhead speak about
15
Cf. (French and Redhead 1988, p. 239). We hasten to explain to logical purists that the phrase
“operators that can be observed” is not supposed to be taken literally (operators, being mathemati-
cal objects, are never observable), but is a mere shorthand for the longer “operators representing
observable properties”. Another minor linguistic issue that may or may not be necessary to clarify
here is that while the standard physical counterparts of mathematical operators are usually called
observables, in the current context this terminology is not particularly felicitous (vide the term
“unobservable observables”).
2 Indiscernibility of Quantum Particles: A Road to Orthodoxy](https://image.slidesharecdn.com/61220-250412055918-bd97da60/85/Identity-and-Indiscernibility-in-Quantum-Mechanics-Tomasz-Bigaj-46-320.jpg)
Identity and Indiscernibility in Quantum Mechanics Tomasz Bigaj
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- 5. Tomasz Bigaj Identity and Indiscernibility in Quantum Mechanics NEW DIRECTIONS IN THE PHILOSOPHY OF SCIENCE
- 6. New Directions in the Philosophy of Science Series Editor Lydia Patton Department of Philosophy Virginia Tech Blacksburg, VA, USA
- 7. These are exciting times for scholars researching the philosophy of sci- ence. There are new insights into the role of mathematics in science to consider; illuminating comparisons with the philosophy of art to explore; and new links with the history of science to examine. The entire relation- ship between metaphysics and the philosophy of science is being re- examined and reconfigured. Since its launch in 2012, Palgrave’s New Directions in the Philosophy of Science series has, under the energetic editorship of Steven French, explored all these dimensions. Topics covered by books in the series dur- ing its first eight years have included interdisciplinary science, the meta- physics of properties, chaos theory, the social epistemology of research groups, scientific composition, quantum theory, the nature of biological species, naturalism in philosophy, epidemiology, stem cell biology, scien- tific models, natural kinds, and scientific realism. Now entering a new period of growth under the enthusiastic guidance of a new Editor, Professor Lydia Patton, this highly regarded series con- tinues to offer the ideal home for philosophical work by both early career researchers and senior scholars on the nature of science which incorpo- rates novel directions and fresh perspectives. The members of the editorial board of this series are: Holly Andersen, Philosophy, Simon Fraser University (Canada) Otavio Bueno, Philosophy, University of Miami (USA) Anjan Chakravartty, University of Notre Dame (USA) Steven French, Philosophy, University of Leeds (UK) series editor Roman Frigg, Philosophy, LSE (UK) James Ladyman, Philosophy, University of Bristol (UK) Michela Massimi, Science and Technology Studies, UCL (UK) Sandra Mitchell, History and Philosophy of Science, University of Pittsburgh (USA) Stathis Psillos, Philosophy and History of Science, University of Athens (Greece) For further information or to submit a proposal for consideration, please contact Brendan George on brendan.george@palgrave.com More information about this series at http://www.palgrave.com/gp/series/14743
- 8. Tomasz Bigaj Identity and Indiscernibility in Quantum Mechanics
- 9. New Directions in the Philosophy of Science ISBN 978-3-030-74869-2 ISBN 978-3-030-74870-8 (eBook) https://doi.org/10.1007/978-3-030-74870-8 © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Palgrave Macmillan imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Tomasz Bigaj Institute of Philosophy University of Warsaw Warsaw, Poland
- 10. v One of the most important purposes that philosophy of science can serve is bridging the gap between scientific inquiries and philosophical specula- tions. Nowhere is the need for building such bridges more pressing than in the case of metaphysical analysis on the one hand and fundamental physical theories on the other hand. If modern metaphysics aspires to be more than a mere footnote to the Great Old Masters, it must receive a generous influx of new ideas and concepts from the rapidly developing field of fundamental physical sciences. However, such an approach to metaphysics places a heavy burden on the practitioners of this ancient art of theoretical reflection. It requires that they enter the maze of highly abstract mathematical concepts that abound in modern theoretical phys- ics. Even more treacherous territory is the issue of a proper physical (and metaphysical) interpretation of the mathematical formalisms of physical theories. Here even seasoned mathematical physicists admit that the task of actually reading physics off the mathematical equations, let alone the challenge of deriving useful metaphysical lessons, is not at all trivial. This book is an attempt to distill some metaphysical contents from the quantum theory of many particles. The metaphysical problems that we will try to sort out with the help of modern quantum theory are old and venerated. They are questions about the most fundamental ontological concepts of identity, individuality and discernibility. Philosophers of dif- ferent stripes have proposed numerous conceptions of what it is to be an Preface and Acknowledgments
- 11. vi Preface and Acknowledgments individual object, how we manage to carve reality into separate entities of various kinds, what numerical identity and distinctness is, and how it relates to possessing differentiating qualitative properties and relations. A particularly popular view insists that numerical diversity and qualitative discernibility are intimately connected in that the former guarantees (as a matter of metaphysical necessity) the latter. Yet there are arguments that quantum mechanics may cast serious doubts on the validity of this view. This has something to do with the way quantum mechanics describes systems of many particles that belong to the same category (so-called indistinguishable particles). Given some mathematical restrictions placed on the available states and measurable properties of such systems, it may be argued that quantum particles of the same type are totally indiscern- ible with respect to their physical attributes. However, this conclusion is by no means unquestionable. It relies on a number of tacit interpretational presuppositions which are open to debate. In this book I will carefully scrutinize the mathematical formal- ism of standard, non-relativistic quantum mechanics, and I will show that there is actually substantial freedom in choosing the right interpreta- tion of some parts of this formalism. Even more importantly, depending on which interpretation to follow, the consequences related to the above- mentioned metaphysical issues may vary dramatically. I will give a broad presentation of two alternative readings of the mathematical apparatus used in the quantum theory of many particles (I refer to these readings as “orthodoxy” and “heterodoxy”) that give rise to two distinct metaphysical conclusions regarding the fundamental characteristics of quantum objects, their identities and individualities. In order to fulfill this task properly, some degree of technicality turns out to be necessary. Thus at places this book may read like a textbook in quantum mechanics, with some mathematical theorems and proofs (all rather elementary, to be sure). However, all these technical issues should not obscure the fact that we are ultimately interested in the general lessons that quantum mechan- ics can teach us regarding the nature of the fundamental building blocks of the universe. Allow me to briefly retrace my personal journey leading to the comple- tion of this book, during which I received invaluable help from many people and incurred numerous debts. I got seriously involved in the topic
- 12. vii Preface and Acknowledgments of the identity and individuality of quantum objects back in 2009 when visiting the University of Bristol as a Marie Curie fellow, thanks to James Ladyman. Out of our discussions grew our joint paper on the Principle of the Identity of Indiscernibles in quantum mechanics in which we criti- cized the role of so-called weak discernibility in restoring the objecthood and individuality of quantum particles. At that time I was leaning towards the orthodoxy in the form developed, among others, by Michael Redhead, Paul Teller, Steven French and Decio Krause, with its insistence on the lack of discernibility and individuality of quantum particles. Then, around the year 2010, Marek Kuś from the Center of Theoretical Physics in Warsaw pointed me towards a series of publications by GianCarlo Ghirardi with collaborators on the notion of entanglement applied to “indistinguishable” particles. These papers were a revelation to me. I real- ized that there is a clear formal sense in which electrons, photons and so on can be said to literally possess distinct and differentiating properties. I presented my take on Ghirardi et al. at a workshop in Bristol organized by James Ladyman in June of 2011, and later that year at the 14th Congress of Logic, Methodology and Philosophy of Science in Nancy, France. In attendance of these events were Simon Saunders, Fred Muller and Adam Caulton, whose helpful comments prevented me from making numerous embarrassing mistakes. As a result I prepared a paper com- menting on the role of symmetric projectors in individuating quantum objects, which came out in print in 2015 as part of the CLMPS proceed- ings issue of Philosophia Scientiae. (I should also mention here the related work on the notion of entanglement with James Ladyman and Øystein Linnebo.) In the meantime, Simon Saunders and Adam Caulton wrote a number of excellent articles which essentially pointed in the same direc- tion, developing an approach which Caulton calls “heterodoxy”. Especially one beautiful paper of his, available on arXive, became a bible for me. I cannot fathom why this comprehensive 50-page-long formal analysis hasn’t been published in any of the leading journals in philoso- phy of physics. In a sense the current book may be seen as a long and slightly verbose commentary to Adam’s phenomenal paper. I spent two extremely productive years from 2013 to 2015 at the University of California, San Diego, where I benefitted enormously from discussions with the Philosophy Department members and visiting
- 13. viii Preface and Acknowledgments guests, including Chris Wüthrich, Craig Callender, John Dougherty, Kerry McKenzie, Nick Huggett, Josh Norton and Holger Lyre. Monthly meetings of the Southern California Philosophy of Physics group at UC Irvine were yet another source of inspiration for me. After that, one more year in Bristol gave me the opportunity to talk further about my develop- ing ideas with James Ladyman, Karim Thébault and other faculty and visitors. In 2018 I was kindly invited by Kian Salimkhani and Tina Wachter to participate in a Bonn workshop on quantum individuality and entanglement, where again I had constructive exchanges with Simon Saunders, Adam Caulton, James Ladyman, Fred Muller as well as Cord Friebe, Dennis Dieks, Andrea Lubberdink and Jeremy Butterfield. In 2019 I was a visiting fellow at the Center for Philosophy of Science in Pittsburgh, where I presented and discussed my work. I am particularly grateful to John Norton, Naftali Weinberger and Chungyoung Lee for their incisive comments. I should also thank audiences in Lausanne, Pasadena, Chicago, Helsinki, Leeds, Barcelona, Warsaw, Cracow, Lublin and Santiago de Chile, where I gave presentations on various topics related to this book. To Steven French I extend my thanks for encourag- ing me to submit my work to the series New Directions in Philosophy of Science. Last but not least I would like to thank Ewa Bigaj for her linguis- tic corrections to the manuscript. The writing of this book was supported by grant No. 2017/25/B/ HS1/00620 from the National Science Centre, Poland. Warsaw, Poland Tomasz Bigaj
- 14. ix 1 Introduction 1 References 7 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy 9 2.1 Composite Systems and Their States 10 2.2 Properties of Composite Systems 13 2.3 Projection Operators 18 2.4 Systems of “Indistinguishable” Particles 20 2.5 The Symmetrization Postulate 24 2.6 The Indiscernibility Thesis 27 2.7 Relational Properties of Individual Particles 33 2.8 Indiscernibility and Individuality 36 References 44 3 The Source of the Symmetrization Postulate 47 3.1 Indistinguishability Postulate and Permutations 49 3.2 The Argument from Exchange Degeneracy 53 3.3 Parastatistics 59 References 69 Contents
- 15. x Contents 4 Logic and Metaphysics of Discernibility 71 4.1 From Absolute to Weak Discernibility 72 4.2 The Meaning of Weak Discernibility 77 4.3 Grades of Discernibility in Extended Languages 80 4.4 Discernibility and Symmetry 84 4.5 Weak Discernibility in Quantum Mechanics 86 4.6 Quantum Weak Discernibility and Identity 93 References101 5 Qualitative Individuation of Same-Type Particles: Beyond Orthodoxy103 5.1 Symmetric Operators and Their Meaning 104 5.2 Symmetric Projectors and Disjunctive Properties 109 5.3 Qualitative Individuation via Symmetric Projectors 117 5.4 Singlet-Spin State and Qualitative Individuation 123 5.5 Measurements for “Indistinguishable” Particles 129 References134 6 The Heterodox Approach to Absolute Discernibility and Entanglement137 6.1 Absolute Discernibility in Symmetric Languages 138 6.2 The Scope of Absolute Discernibility for Fermions and Bosons 145 6.3 Entanglement and Properties 152 6.4 Fermionic Entanglement Versus Bosonic Entanglement157 6.5 Entanglement and Non-local Correlations 164 6.6 Entanglement and Discernibility: Summary 170 References174 7 Two Views on Quantum Individuation: A Comparison177 7.1 The Troubles with Orthodoxy 178 7.2 The Fock Space Formalism 183 7.3 The Ambiguity of Qualitative Individuation: Fermions and Bosons 191
- 16. xi Contents 7.4 Ambiguity for Fermions 198 7.5 Two Solutions of the Problem of Ambiguity 203 References210 8 The Metaphysics of Quantum Objects: Transtemporal and Transworld Identities213 8.1 Synchronic Versus Diachronic Identity 214 8.2 Diachronic Identity of Same-Type Particles 219 8.3 Scattering Experiments 223 8.4 Diachronic Identity in Scatterings: Orthodoxy Versus Heterodoxy 229 8.5 Identity Across Possible Worlds 233 8.6 Towards the Metaphysics of Quantum Objects 239 References245 Appendix: Basic Concepts of the Quantum-Mechanical Formalism249 Index257
- 17. xiii Fig. 2.1 The action of transformation P on operator A yields operator A′. Note that the action of A′ on vector |u〉 is equivalent to the composition of transformations P, A and P-1 . Thus A′ = P-1 AP (operators are always put in reverse order—the last in sequence is the first to be applied) 17 Fig. 2.2 The orthogonal projection onto subspace S18 Fig. 2.3 Four possible arrangements of two particles 1 and 2 among two states S and T 39 Fig. 4.1 Structures whose elements are weakly but not relatively discernible (upper diagram) and relatively but not absolutely discernible (lower diagram) 81 Fig. 6.1 Spatial regions used in an example of a GMWB-non-entangled state. Regions L and R represent the locations of measuring devices 169 Fig. 7.1 One particular partition of the set {|φi〉}. There are exactly 2m-1 such partitions 193 Fig. 7.2 Two ways of grouping vectors from the combination (7.3). Vectors in one column are taken from the same symmetric block 194 Fig. 8.1 Two scenarios involving scatterings of distinguishable particles a and b225 List of Figures
- 18. xv Table 6.1 Comparison of entanglement and discernibility for fermions 170 Table 6.2 Comparison of entanglement and discernibility for bosons 174 Table 8.1 The possibility of counterfactual switching in various approaches to quantum individuation and modality de re 239 List of Tables
- 19. 1 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Bigaj, Identity and Indiscernibility in Quantum Mechanics, New Directions in the Philosophy of Science, https://doi.org/10.1007/978-3-030-74870-8_1 1 Introduction In a nutshell, this book is about whether fundamental objects described by quantum mechanics can be distinguished from one another with the help of their physical properties. But why should we be even remotely interested in answering such a trifle question, let alone devote an entire book to its anatomization? And isn’t it clear as day that we can and do differentiate quantum particles using various physical attributes? Surely, quantum mechanics breaks away from many firm beliefs about reality that we hold as self-evident. In classical physics bodies are assumed to be impenetrable, and thus each particle possesses its unique trajectory that differentiates it from any other particle. Quantum mechanics disposes with the idea of a well-defined spatial location, replacing it with a “smeared” presence encoded in a spatially extended wave function. Consequently, two quantum objects may temporarily share the same extended location; they may pass through each other like ghosts, or seemingly disappear and then reappear at the end of their interaction. But it is still perfectly possible for these quantum ghosts to be separated by huge spatial intervals, with virtually no overlap of their respective wave functions. Doesn’t this prove that two electrons, photons or Higgs bosons canbedistinguishedeveninaworldgovernedbythebizarrequantumlaws?
- 20. 2 Well, not so fast. Quantum theory has a special way of characterizing states of ensembles consisting of many particles that belong to the same type. Given that electrons (photons, neutrinos, etc.) do not differ from each other with respect to their uniquely identifying features, such as mass or electric charge, the quantum theory of many particles imposes the requirement of permutation invariance on the states these particles can jointly occupy. That is, the joint state of same-type particles should remain unchanged under any arbitrary permutation of these particles (this is the essence of what is typically referred to as the symmetrization postulate). And, on the face of it, it looks like attributing distinct properties to separate particles cannot be reconciled with the permutation symmetry of their joint states. Thus enter the Indiscernibility Thesis which proclaims that particles of the same type are indistinguishable with respect to all their properties. This means specifically that if you take, for example, two electrons, then whatever observable characterizing one of them you wish to consider (whether it is position, momentum, energy or spin), the expectation value of this observable should be the same for both electrons. The venerated Leibnizian Principle of the Identity of Indiscernibles seems to be universally violated in the quantum world. Philosophers get all fired up at the prospect of the complete indis- cernibility of quantum particles. Discernibility seems to be a condition sine qua non for individuating objects, selecting and naming them, mak- ing reference to one and not the other. Indiscernible entities, lacking the important feature of individuality, are objects only in the thinnest, Quinean sense of the word. A group of indiscernibles forms a whole that is often referred to as an aggregate rather than a collection. Some philoso- phers insist that for groups of indiscernible entities it is only possible to count them, but not to order them. Aggregates have a mere cardinality but no ordinality. For that reason perhaps the term “electron” should properly function as a mass term, referring to the whole electron mass of the universe. Alternatively, the individuality of the indiscernible quantum particles may be rescued by introducing non-qualitative prinicipia indi- viduationis, such as haecceities (properties of being a particular object). From these remarks we can see that the issue of quantum (in)discernibil- ity can acquire a strong metaphysical flavor. Now it looks more plausible T. Bigaj
- 21. 3 that the question of whether quantum objects can be differentiated by properties could merit thorough philosophical scrutiny. Characteristically, the majority of working physicists remain rather unimpressed by the metaphysical ramifications of the symmetrization postulate.1 True, they admit that the postulate is very important from an empirical and practical point of view, especially when we are interested in describing the behavior of large collections of particles, for the statistical predictions regarding indiscernible quantum particles differ significantly from analogous predictions concerning classical, discernible particles. Generally speaking, quantum particles are divided into two categories— bosons and fermions—depending on the mathematical transformation their joint states undergo under the permutations of objects. Statistical behavior of bosons differs from that of classical particles in that the probability of finding a group of bosons occupying same states is higher than in the classical case (as if bosons “attracted” each other slightly). On the other hand, same-type fermions never occupy the same state (they “repel” each other strongly).2 However, when dealing with small numbers of same-type fermions or bosons, physicists often ignore the symmetrization postulate and write their states in a non-permutation-invariant form, as if they characterized distinguishable particles. Here is an interesting quote on that issue from a well-known textbook on quantum mechanics (Cohen-Tannoudji et al. 1978, p. 1406): If application of the symmetrization postulate were always indispensable, it would be impossible to study the properties of a system containing a restricted number of particles, because it would be necessary to take into account all the particles in the universe which are identical to those in the system. We shall see […] that this is not the case. In fact, under certain special conditions, identical particles behave as if they were actually 1 Of course, there are some notable exceptions—famously including Erwin Schrödinger and Henry Margenau—but these are physicists who are already influenced by a philosophical way of thinking. See, for example, Schrödinger (1952); Margenau (1944, 1950). 2 A careful reader may notice a strange inconsistency between this general characteristic of fermi- onic behavior (whose more precise expression bears the name of Pauli’s exclusion principle) and the Indiscernibility Thesis mentioned earlier. Much of the subsequent discussions in this book will try to explain away this inconsistency. 1 Introduction
- 22. 4 different, and it is not necessary to take the symmetrization postulate into account in order to obtain correct physical predictions. The quoted fragment is baffling. How can indiscernible objects “behave” as if they were discernible, even under “certain special conditions”? If the symmetrization postulate is a universal, exceptionless law of nature, and if its validity implies that whatever measurable property is possessed by one component of a system of “identical” particles, it is also possessed by any other component, then it becomes utterly mysterious how such particles could ever be treated “as if they were actually different”. It is difficult to make sense of a situation in which entirely indistin- guishable objects behave as if they were distinguishable, unless we make some crucial changes in the way we identify these objects. And it turns out that this may be the key to understanding the above-mentioned quote: perhaps what justifies the suspension of the symmetrization postulate is an alternative method of “carving up” the totality of the composite system into smaller components, so that these new components not only behave “as if” they were distinguishable, but really are. Following this lead, in this book I will argue that there are actually two rival methods of individuating quantum particles that compose larger systems. For lack of a better term, I will refer to one of these methods as “orthodoxy”, and the other as “heterodoxy” (I shamelessly borrow this terminology from Adam Caulton). The “orthodox” approach to individuality treats certain parts of the mathematical formalism, namely indices that are attached to the factors in the tensor products of Hilbert spaces, as referring to the components of the composite system under consideration. For the unorthodox approach, on the other hand, the task of individuating the components of a composite system is performed not by unphysical labels used to identify identical copies of a single-particle Hilbert space, but rather by physically meaningful symmetric operators of a certain kind. When this new individuating procedure is executed, it turns out that fermions occupying antisymmetric states are always discernible from each other by some properties, while bosons are not guaranteed to be discernible in that way. However, it is definitely possible to put two (or more) bosons in a state in which they will differ from each other with respect to their possessed properties. T. Bigaj
- 23. 5 This book is devoted to a systematic analysis (both formal and philo- sophical) and comparison of the two competing methods of individuat- ing particles in quantum mechanics. In Chap. 2 I will lay out the formal and conceptual fundamentals of the orthodox approach to this problem, together with the ensuing Indiscernibility Thesis regarding quantum par- ticles of the same type. Chapter 3 delves deeper into the problem of the justification of the symmetrization postulate. It also contains a brief for- mal description of the types of symmetry other than bosonic and fermi- onic, known collectively as parastatistics. Chapter 4 makes a small detour in order to discuss the logic and metaphysics of distinguishability. We identify and formalize three types of discernibility using the standard model-theoretical framework, and we connect them with the issue of symmetry. This chapter also includes a critical analysis of the weak dis- cernibility program, which hopes to provide some semblance of quantum objecthood in the light of the apparent breakage of absolute discernibility. In Chap. 5 we will make first steps towards a non-standard conception of how to individuate quantum particles. The starting point will be an analysis of the physical meaning of certain symmetric projection opera- tors acting in an appropriate tensor-product Hilbert space. Arguments will be presented to the effect that particular symmetric combinations of projectors should be interpreted as representing situations in which the components of a composite quantum system are discernible by their mea- surable properties. As it turns out, there are some serious objections to this interpretation, involving the concept of quantum measurement. The way to repel these objections will be to properly introduce spatial degrees of freedom into our general description of measurement processes. Chapter 6, which is probably the most technical of all chapters in the book, contains proofs of several facts regarding the absolute discernibility of fermions and bosons under the unorthodox approach, and regarding a new concept of entanglement applicable to the case of same-type particles. One section of this chapter also addresses the more general question of whether it is logically possible to formulate symmetric sentences stating facts of absolute discernibility. The main goal of Chap. 7 is to present various pros and cons with respect to the two approaches to quantum individuation developed earlier in the book. In particular, we will discuss a serious problem affecting the unorthodox approach which is caused by 1 Introduction
- 24. 6 the non-uniqueness (ambiguity) of individuation by symmetric operators. Chapter 8 adds the issue of diachronic and counterfactual identity of quantum objects to the discussion. In the closing section of this chapter, we will stress the non-classical character of the metaphysics emerging from the unorthodox approach to individuation. Even though the heterodoxy restores the validity of the Principle of the Identity of Indiscernibles in the majority of cases involving same-type particles, this does not lead to the rehabilitation of the classical picture of the quantum world. Particles of the same type may be individuated synchronically, but they are not full-blown classical individuals, since they typically fail to keep their identities over time and across possible scenarios, and their synchronic individualities suffer from unavoidable ambiguity. I should mention here some presuppositions and limitations of this book. First off, for the most part I restrict myself to standard, non- relativistic quantum mechanics. I make some inroads into quantum field theory in Chap. 7, where I briefly present the Fock space formalism which enables us to talk about variable numbers of particles. However, I do not venture to discuss the ontological problem of identity and individuality in full-blown interacting QFT. One obvious reason for that reservation is that in spite of the tremendous effort of numerous philosophical commentators, the jury is still out whether QFT should be based on the fundamental ontology of particles or fields.3 Another important limitation of the book is that it tries to stay clear of the notorious measurement problem and the ensuing question of the proper interpretation of quantum mechanics. The idea is to present the problem of identity and individuality within the confines of so-called textbook quantum mechanics, which is hopefully neutral with respect to the fun- damental differences between various interpretations of QM.4 Where ref- erence to measurements becomes unavoidable, such as at the end of Chap. 5, I try to use a “unitary” account of measurement interactions as an alternative to the standard but philosophically controversial “collapse” account. Finally, I should warn the readers that they won’t find a complete 3 For an overview of this problem, see Kuhlmann (2020). 4 Of course I am perfectly aware that adopting some specific interpretive variants of QM, such as Bohm’s theory, may change radically our perspective on the issue of the discernibility of quantum objects. See, for example, Brown et al. (1999). T. Bigaj
- 25. 7 metaphysical conception of quantum objects in the book. While I con- sider myself a philosopher/metaphysician, the aim of this book is primar- ily to gather together some rather elementary facts, provable in the quantum-mechanical formalism, which appear salient with respect to the problem of identity, and raise the question of their proper physical and philosophical interpretation. Thus it is clearly a preparatory work. I do hope that someone may find this work useful while developing some more comprehensive proposals of quantum ontology. References Brown, H., E. Salqvist, and G. Bacciagaluppi. 1999. Remarks on Identical Particles in de Broglie-Bohm Theory. Physics Letters A 251: 229–235. Cohen-Tannoudji, C., B. Diu, and F. Laloë. 1978. Quantum Mechanics. Vol. 2. New York: Wiley. Kuhlmann, M. 2020. Quantum Field Theory. In The Stanford Encyclopedia of Philosophy, ed. E.N. Zalta (Fall 2020 Edition). https://plato.stanford.edu/ archives/fall2020/entries/quantum-field-theory/ Margenau, H. 1944. The Exclusion Principle and Its Philosophical Importance. Philosophy of Science 11: 187–208. ———. 1950. The Nature of Physical Reality. New York: McGraw-Hill. Schrödinger, E. 1952. Science and Humanism. Cambridge: Cambridge University Press. 1 Introduction
- 26. 9 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Bigaj, Identity and Indiscernibility in Quantum Mechanics, New Directions in the Philosophy of Science, https://doi.org/10.1007/978-3-030-74870-8_2 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy It is impossible to constructively engage in a metaphysical discussion inspired by quantum mechanics without possessing rudimentary knowl- edge of the mathematical formalism of this theory. The first chapter of this book is meant primarily as a brief overview of the basics of the quan- tum formalism necessary to understand the debates on the problem of identity and individuality (further definitions and proofs are provided in the Appendix, albeit they are limited to a bare minimum). Obviously, this hasty presentation cannot be treated as a substitute for a thorough, step-by-step introduction to quantum theory, for which I can only refer the reader to one of many excellent textbooks on the market.1 In what follows we will merely touch upon the standard mathematical representa- tion of states and properties of composite systems, that is, physical sys- tems consisting of many smaller components, with its primary tool of the 1 Cohen-Tannoudji et al. (1978), Peres (2003) and Sakurai and Napolitano (2011) are recom- mended comprehensive physical textbooks on quantum mechanics. An elegant introduction to the quantum-mechanical formalism with an eye on the foundational issues is (Griffiths 2002), while the extensive compendium (Greenberger et al. 2009) offers an encyclopedic collection of short articles covering all fundamental concepts and results of quantum theory. My personal favorite among mathematically rigorous yet accessible introductions to philosophical issues in quantum mechanics is Hughes (1989).
- 27. 10 tensor product of vector spaces. We will see that it is not at all trivial to extend the single-system formalism of states and measurable properties to represent states and properties of composite systems. The key question here is how to express a given property of a particular quantum object in a framework which treats this object as part of a broader system consist- ing of many entities. The standard quantum-mechanical way to do this turns out to contain certain loopholes which will later prove to be crucial in the controversy regarding the proper method of individuating quan- tum particles of the same type. Using the formal notion of a permutation operator, in the next step we will articulate the Indistinguishability Postulate which imposes an impor- tant restriction on joint states and properties of so-called indistinguish- able particles (particles belonging to the same type). From this assumption a metaphysical consequence in the form of the Indiscernibility Thesis is usually derived. We will analyze typical arguments in favor of this claim, noting the indispensable role of the tacit assumption referred to as Factorism in these derivations. Finally, we will briefly discuss the meta- physical role of discernibility by properties in clarifying the notion of individuality, as well as its relation to numerical diversity. 2.1 Composite Systems and Their States Let us start with a sketch of how the standard quantum-mechanical for- malism describes states of composite systems, that is, systems consisting of smaller components, each of which can possess its own states. The mathematical representations of the states of any quantum system form a structure known as a vector space. More specifically, vector spaces used in quantum mechanics are spaces over the field of complex numbers, which bear the name of Hilbert spaces (see Appendix for a complete definition). Suppose that we have two quantum systems (e.g. two particles—an elec- tron and a proton), whose states are represented in their respective indi- vidual Hilbert spaces ℋ1 and ℋ2. In order to describe states of the composite system consisting of the two particles (in our example this complex system can be a hydrogen atom, which is composed of one T. Bigaj
- 28. 11 electron and one proton), we have to build a new Hilbert space. The standard mathematical recipe for doing this employs the concept of a ten- sor product of Hilbert spaces. The tensor product of spaces ℋ1 and ℋ2, symbolized as ℋ1 ⊗ ℋ2, is a vector space consisting of all pairs of vectors from ℋ1 and ℋ2 and their linear combinations. More precisely, the defi- nition of ℋ1 ⊗ ℋ2 can be given as follows. Let the set of vectors | ei 1 〉 be an orthonormal basis for ℋ1, and { } ej 2 be an orthonormal basis for ℋ2 (the definition of a basis is given in the Appendix). The tensor product ℋ1 ⊗ ℋ2 is a vector space spanned by the combinations e e i j 1 2 , that is, the space of all linear combinations of the form. ij ij i j c e e 1 2 , (2.1) where cij are complex numbers.2 The key difference between classical and quantum composite systems lies in the fact that the state spaces of quantum compositions are in a sense much larger than in the classical case. In addition to the factorizable combinations of states of the form |φ〉 ⊗ |ψ〉, where |φ〉 ∈ ℋ1, |ψ〉 ∈ ℋ2, space ℋ1 ⊗ ℋ2 includes vectors that cannot be presented as a product of two vectors. These non-factorizable combinations are also known as entangled. An example of an entangled state can be the combination e e e e 1 1 1 2 2 1 2 2 , where ei 1 , ej 2 are, as before, some basis vec- tors of spaces ℋ1 and ℋ2. It is an elementary fact from linear algebra that the above vector cannot be represented in the factorized form |φ〉 ⊗ |ψ〉.3 Thus if the composite system occupies such a non-factorizable, entangled state, its components cannot be assigned any individual states in the form of vectors in their respective Hilbert spaces (states given in the form of vectors are typically referred to as pure). This does not imply that the 2 To avoid clutter, sometimes the symbol ⊗ representing the operation of tensor product is omitted. That is, instead of writing |φ〉 ⊗ |ψ〉 we can simply write |φ〉|ψ〉. 3 Given that each vector |φ〉, |ψ〉 has a unique decomposition in respective bases { } ei 1 and {| } ei 2 〉 (see Appendix for details), it follows that if these decompositions contain vectors e1 1 , e2 1 and e1 2 , e2 2 , the product |φ〉 ⊗ |ψ〉 has to contain “cross terms” e e 1 1 2 2 and e e 2 1 1 2 in its decomposition and hence cannot be written as e e e e 1 1 1 2 2 1 2 2 . 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 29. 12 components of the system cannot be characterized by any quantum- mechanical states whatsoever, but in the case of entanglement, these states will belong to a different category of mixed states, represented not by vectors but by density operators (see Appendix for further explana- tions). Regardless of these technical details, it is important to know that entangled states cannot be fully reduced to the “conjunction” of the (mixed) states of the individual components. That is, while knowing that one particle is in a pure state |φ〉 while the other particle occupies pure state |ψ〉 is sufficient to deduce that the joint state will be the product |φ〉 ⊗ |ψ〉, in the case of entangled states, determining the (mixed) states of thecomponentsdoesnotuniquelydeterminethestateoftheentiresystem. How do the Hilbert spaces ℋ1 and ℋ2 relate to each other? It is an elementary mathematical fact that all Hilbert spaces over complex fields of the same dimensionality are isomorphic. The dimensionality of a state space in quantum mechanics is determined by the number of distinct values that can be possessed by a quantity (or quantities) taken to define states of systems. Thus spaces of states given in terms of position or momentum will be typically infinitely dimensional. On the other hand, spaces of the components of angular momentum (e.g. spin), which are discrete quantities, may have a finite number of dimensions. If we limit ourselves to either position/momentum spaces (spaces of wave func- tions), or to the cases of particles with the same total spin (e.g. spin 1 2 ), we may assume that ℋ1 and ℋ2 have the same number of dimensions and therefore are isomorphic. Consequently, we may stipulate that both ℋ1 and ℋ2 include “the same” vectors, and thus it is possible to drop the superscripts in the basis vectors written above. Henceforth, we will assume that ℋ1 and ℋ2 are two copies of the same Hilbert space spanned by some orthonormal vectors {|ei〉}. The only difference between them is that they are labeled with different numbers 1 and 2 (this labeling is often omitted for the sake of brevity, in which case the labels are assumed to be determined by the place a given vector occupies in the tensor product). However, this does not mean that the mere formal difference in labels does not reflect a deeper physical distinction. For instance, if we consider a system of two particles belonging to different kinds (such as a proton and an electron), the labels used to differentiate identical copies of one T. Bigaj
- 30. 13 Hilbert space are underlain by different state-independent properties defining appropriate kinds (e.g. rest mass). Once we equate basis vectors spanning spaces ℋ1 and ℋ2, it is possi- ble to formally introduce the operation of permutation. The permutation P12 is an operation that acts on the basis vectors of ℋ1 ⊗ ℋ2 in the fol- lowing way: P e e e e i j j i 12 ( ) . (2.2) By linear extension, this operation applies to any vector in ℋ1 ⊗ ℋ2. Clearly, P12 applied to any product state |φ〉 ⊗ |ψ〉 yields its “reverse” |ψ〉 ⊗ |φ〉. The most natural physical interpretation of the permutation oper- ation is that it results in the situation in which particles swap their states, that is, particle 1 now occupies the state initially occupied by particle 2, and vice versa. However, we will have to look carefully at this interpreta- tion later when we discuss the case of so-called indistinguishable particles (see Sect. 3.1 in Chap. 3). 2.2 Properties of Composite Systems Measurable properties of a quantum system are represented by a particu- lar type of linear operators, known as Hermitian (or self-adjoint), acting in the state space for this system. If the system is in a pure state |φ〉, the expectation value for the observable corresponding to a Hermitian opera- tor A is given by the inner product 〈φ|A|φ〉.4 Suppose now that we know the state |ψ(1,2)〉 of an entire composite system consisting of two subsys- tems. State |ψ(1,2)〉 may or may not be factorizable into the product of the states of the components, so we can’t assume that the pure states of these components taken separately are well defined. How, in that case, can we calculate the expectation value for an observable limited to one particle? The answer is given in terms of the tensor products of linear operators. Generally, if A is a linear operator in space ℋ1, and B is a linear 4 See a rationale behind this definition in Hughes (1989, pp. 70–71). 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 31. 14 operator in ℋ2, we can define a new operator A ⊗ B acting in the tensor product ℋ1 ⊗ ℋ2 as follows: A B e e A e B e i j i j ( ) . (2.3) Any linear combination of the products of the form Ak ⊗ Bl is also a lin- ear operator acting in ℋ1 ⊗ ℋ2. Consider now the product A ⊗ I, where I is the identity operator, that is, such that I |φ〉 = |φ〉 for all vectors |φ〉. We can calculate its expecta- tion value in the state |ei〉 ⊗ |ej〉 as follows5 : e e A I e e e A e e e e A e e e i j i j i i j j i i j j , , , since 1 1. (2.4) Thus the expectation value of A ⊗ I in |ei〉 ⊗ |ej〉 turns out to be identical to the expectation value of A in state |ei〉. This gives us a reason to suspect that A ⊗ I in the product space ℋ1 ⊗ ℋ2 represents the very same prop- erty as A in ℋ1. As it is sometimes put, A ⊗ I represents an observable of the composite system which corresponds to the measurement procedure consisting of measuring A on the first component of the system and leav- ing the second component alone (this “leaving alone” is represented by the identity operator which does nothing to any state). Another possible argument for the identification of the physical inter- pretations of operators A ∈ ℋ1 and A ⊗ I ∈ ℋ1 ⊗ ℋ2 uses the standard notions of eigenstates and eigenvalues. The physical meaning of a given Hermitian operator is typically expressed by reference to mathematical eigenequations: A a a a | | , (2.5) where a is a number. Number a, known as an eigenvalue of A (which for Hermitian operators is always a real number—see Appendix), represents a particular value of observable A, while |λa〉 is a corresponding state, called an eigenstate. According to the interpretational rule called the 5 The inner product 〈φ⊗ψ|λ⊗χ〉 in the tensor product space ℋ1 ⊗ ℋ2 is defined as 〈φ|λ〉〈ψ|χ〉 (see Appendix). T. Bigaj
- 32. 15 eigenstate- eigenvalue link (the e/e link),6 a system objectively possesses a given value a of an observable A, iff the system is in an eigenstate |λa〉 corresponding to this value. Now, it can be easily shown that if a vector |λa〉 ∈ ℋ1 is an eigenstate for A with a value a, any vector of the form |λa〉 ⊗ |φ〉, where |φ〉 ∈ ℋ2, is an eigenvector for A ⊗ I with the same value a. The implication holds in the opposite direction too, which proves the following equivalence: (2.6) Particle 1 possesses an objective value a of the observable represented by operator A iff the system of particles 1 and 2 possesses an objective value a of the observable represented by operator A ⊗ I. However, we should not forget the fact that operator A ⊗ I, since it acts in a broader Hilbert space, is applicable in some cases which are not immediately covered by the one-particle operator A. That is, we can cal- culate the expectation value for A ⊗ I in entangled two-particle states, for which no pure state of individual particles exists, and thus the standard formula 〈φ|A|φ〉 for the expectation value of A cannot be applied. This fact may be seen as undermining the perfect physical equivalence between mathematical operators A ⊗ I and A. True, it may be pointed out that the full equivalence is restored by introducing the concept of a mixed state represented by density operators mentioned earlier, together with a new recipe of how to calculate the expectation values of operators when sys- tems are assigned mixed states (the relevant formula is Tr(Aρ), where Tr is the trace operation, and ρ – a density matrix, see Appendix). However, in response we may observe that the assignment of a reduced mixed state ρ to an individual component of an entangled system is motivated precisely by the desire to keep the identity of the expectation values of A ⊗ I and A (the reduced state of a component is defined as the state which attri- butes to A the same expectation value as the expectation value assigned by the total state to A ⊗ I—see Hughes 1989, pp. 149–150). Hence we can 6 The term “eigenstate-eigenvalue link” is admittedly misleading, as observed, for example, in Muller and Leegwater (2020, ft. 18), since it suggests a purely mathematical relation rather than an interpretative rule that may or may not be accepted. I noticed this terminological problem in my Bigaj (2006, p. 375 ft. 1). Nevertheless, I will continue using this nomenclature which has become standard in the literature. 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 33. 16 hardly use the identity between the expectation value of A ⊗ I calculated for the state of the entire system and the expectation value of A for the reduced mixed state, as an argument that A ⊗ I and A are physically equivalent in all scenarios. The only argument we can rely on is an “induc- tive” one: since the expectation values of A ⊗ I and A coincide in the special cases when the components possess their own pure states (alterna- tively, since A ⊗ I possesses a given definite value iff A possesses the very same value), we stipulate that the expectation values should also coincide in the remaining cases, and therefore we assume that operators A ⊗ I and A represent the same physical quantity. We will see later in the book (Chap. 5) that analogous inductive arguments may be questioned in some other contexts. In the previous section we have considered an important type of per- mutation operation, which was defined as a map on the total Hilbert space, transforming vectors into vectors. It turns out that permutations can also be applied to operators acting on Hilbert spaces rather than to vectors. The general method of how to turn a transformation on a given space ℋ into the corresponding transformation on the set of operators on ℋ can be presented as follows. Let P be any linear transformation on ℋ which has an inverse (thus P must be one-to-one). We will seek a transformation corresponding to P that sends any operator A into its counterpart A′ while satisfying the following requirement: For any vectors and iff . | | , | | | | A AP P (2.7) Expressing this condition informally: the transformed operator A′ acts in the original space ℋ the same way as the original operator A acts in the transformed space P[ℋ]. From this we can easily derive the form of the operator A′ in terms of A and P. By applying the inverse operation P−1 to both sides of the eq. AP|φ〉 = P|ψ〉, we get P−1 AP|φ〉 = |ψ〉, from which it follows (given that |φ〉 and |ψ〉 are selected completely arbitrarily) that A′ = P−1 AP (see Fig. 2.1). We can now return to the case of permutation transformation. What is the result of applying permutation P12 to the operator A ⊗ I? In order to calculate the outcome of the transformation P12 −1 (A ⊗ I ) P12, we have T. Bigaj
- 34. 17 | ñ ¢ ñ ñ ñ v |u P|v P|u A A P P Fig. 2.1 The action of transformation P on operator A yields operator A′. Note that the action of A′ on vector |u〉 is equivalent to the composition of transforma- tions P, A and P-1 . Thus A′ = P-1 AP (operators are always put in reverse order—the last in sequence is the first to be applied) to apply this transformed operator to an arbitrary basis vector |ei〉 ⊗ |ej〉, which yields the following: P A I P e e P A I e e P A e i j j i j 12 1 12 12 1 12 1 ( ) ( ) ( ) ( ) ( e e A e i i j ) . (2.8) Thus, the transformed operator P12 −1 (A ⊗ I) P12 acts in the same way on the basis vectors as I ⊗ A, and hence these operators are identical. It should not come as a surprise that applying permutation P12 to the opera- tor A ⊗ I yields the reversed-order variant I ⊗ A which, as we already know, represents the property A possessed by the second component of the composite system. Of particular importance are operators acting on ℋ1 ⊗ ℋ2 whose permutation doesn’t change anything, that is, such that P12 −1 ΩP12 = Ω. Operators invariant under permutations in this sense are called “sym- metric”. A simple example of such an operator is the product A ⊗ A, whose physical interpretation is obvious: it represents the same observ- able assigned to both components of the system. A slightly more complex example of a symmetric operator is provided by the following sum of appropriate operators: A ⊗ B + B ⊗ A. Here the issue of a proper physical interpretation is a bit trickier than in the previous example—we will return to it later in Chap. 5. For now, we will only mention that it would 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 35. 18 be inaccurate to interpret the operator A ⊗ B + B ⊗ A as a representation of the disjunctive property “either A for particle 1 and B for particle 2 or B for particle 1 and A for particle 2”. 2.3 Projection Operators An important category of Hermitian operators are so-called orthogonal projection operators (or projectors, for short). While projectors can be inter- preted as observables analogous to spin, position, momentum and so on, that is, as quantities capable of receiving different values from the admis- sible range, it is more typical to use them as representations of specific properties of quantum systems that may or may not be possessed in a given state. Formally, projection operators stand in one-to-one corre- spondence to subspaces of a given Hilbert space ℋ (including ℋ itself and the zero-subspace containing only the 0-vector). That is, to every subspace S of ℋ, there corresponds a unique projector ES, and each pro- jector defines a subspace of ℋ onto which it projects. Speaking loosely, the projector onto a subspace S acts on an arbitrary vector |φ〉 in such a way that it decomposes |φ〉 into the component |φ〉S lying in S and the component |φ〉S′ perpendicular (orthogonal) to it, and then it selects |φ〉S as the outcome: ES|φ〉 = |φ〉S (see Fig. 2.2). From this loose characteriza- tion, it follows immediately that ES restricted to S is the identity operator and that applying ES twice to any vector is equivalent to applying it only once (this property is called idempotence: ES 2 = ES). |v |v S |v S ES S ñ ¢ ñ ñ Fig. 2.2 The orthogonal projection onto subspace S T. Bigaj
- 36. 19 The physical interpretation of a projector ES is rather straightforward: ES is an observable with two possible values 1 and 0 that “measures” whether the state of a given system lies within the corresponding sub- space S.7 If the state vector |φ〉 belongs to subspace S, the observable represented by ES assumes the eigenvalue 1 (since in that case ES|φ〉 = |φ〉). On the other hand, if the state vector is orthogonal to S, the value assumed by ES is 0 (projecting an orthogonal vector onto S gives the zero vector). Any other vector (neither in S nor perpendicular to it) is a non- eigenvector of ES, and hence there is some probability that the value will be 1 and some probability that it will be 0. Projectors are quantum equiv- alents of the characteristic functions of certain sets of values and therefore can be interpreted as representations of the property of possessing one of the set of values associated with a given subspace. A special case of projec- tion operators is one-dimensional projectors, whose corresponding sub- space is a ray (one-dimensional subspace). If the ray signifies a state with a particular value of some measurable property (e.g. spin-up in a given spatial direction), the corresponding projector onto this ray can be assumed to represent this specific property (in the sense that the eigen- value 1 of this projector corresponds to the system’s possession of this property, for instance, spin-up). A standard way to write the projector onto a ray containing a normalized vector |φ〉 is in the form of the so- called dyad |φ〉〈φ| (see Appendix for an explanation of this notation). Projection operators acting in separate one-particle Hilbert spaces can be used to create new projectors acting in the tensor products of these spaces. Exactly as in the general case of Hermitian operators, we can con- sider projectors of the kind E ⊗ I and I ⊗ E, where E acts, respectively, in ℋ1 or ℋ2. If E projects onto the subspace S of, let’s say, ℋ1, the sub- space corresponding to E ⊗ I will be S ⊗ ℋ2. However, not all combina- tions of tensor products of one-particle projectors are themselves projectors. Consider, for instance, the following symmetric operator: E ⊗ F + F ⊗ E, where E and F are any projectors in ℋ1 (ℋ2). It turns out that this new operator will generally not be idempotent and therefore will not belong to the category of projectors in the tensor product ℋ1 ⊗ ℋ2. 7 Alternatively, projectors are interpreted as representing “yes” or “no” questions regarding whether the state of the system lies in a particular subspace. 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 37. 20 Only when E and F are orthogonal to each other (i.e. they project onto orthogonal subspaces) is the above combination idempotent. This can be verified by direct calculation: E F F E E F F E E F F E EF FE FE EF 2 2 2 2 . (2.9) Thanks to the orthogonality assumption EF = FE = 0, and using the idempotence conditions E2 = E and F2 = F, we arrive at the required out- come E ⊗ F + F ⊗ E. But if E and F are not orthogonal, this result is not guaranteed. 2.4 Systems of “Indistinguishable” Particles Virtually all current discussions regarding the notions of the identity, individuality and discernibility of quantum particles center around the concept of “indistinguishable” particles. As is well known, fundamental particles of modern physics are categorized into kinds depending on their basic physical properties. The classification of particles used in particle physics is rather complex and the details need not concern us (see, e.g. Griffiths 2008 for a complete categorization). Suffice it to say that current physics distinguishes three broad types of truly elementary particles (i.e. particles with no proper components): leptons, quarks and mediators. Among leptons we classify electrons, muons and tau particles (plus their antiparticles) with three corresponding types of neutrinos. There are six types of quarks distinguishable by their flavors (up, down, strange, charm, bottom, top) and six corresponding antiquarks. Mediators (or mediating particles) carry forces: electromagnetic (photons), strong (gluons) and weak (particles W± and Z0 ). In addition to genuinely elementary parti- cles, there is a garden variety of particles composed of smaller elements (quarks), of which the best known and certainly most ubiquitous are protons and neutrons. When physicists talk of indistinguishable, or identical particles, they usually mean particles belonging to the narrowest categories described T. Bigaj
- 38. 21 above: electrons, muons, photons, strange quarks and so on. Their indis- tinguishability, or identity, does not involve all their properties, but rather a special kind of properties, the so-called state-independent ones. These are properties that do not and cannot change over time. For instance, every electron, no matter what state it occupies, is characterized by the same rest mass (0.511 MeV), the same electric charge (−1.6 × 10−19 C) and the same spin (½ℏ). No electron can lose these properties without ceasing to be an electron.8 Thus identical, or indistinguishable, particles are those that share all of their state-independent properties. This of course does not exclude the possibility that, for instance, two electrons may differ with respect to their state-dependent features: position, momentum, energy, the spin component in a given direction and so on. For that rea- son, I will try to avoid using the potentially confusing terms “indistin- guishable particles” and “identical particles” (if, for purely stylistic reasons, I occasionally revert to this terminology, I’ll use scare quotes to indicate the metaphorical character of the terms), replacing them with the slightly more cumbersome phrase “particles of the same type”, where type is meant as described above.9 Suppose that we are considering a system consisting of particles of the same type (e.g. a group of electrons). How does the fact that these parti- cles share their state-independent properties bear on the way we should describe their joint state? The standard way of approaching this problem 8 However, we have to admit that the issue of whether properties such as rest mass are always state independent is a bit tricky. For instance, in the well-known phenomenon of neutrino oscillations, the state of a neutrino is a superposition of states with different rest masses, corresponding to dif- ferent types of neutrinos (electron neutrino and muon neutrino). Hence the term “mass eigen- states” is introduced, which clearly suggests that mass becomes part of the state description that can change over time (for details, see Griffiths 2008, pp. 390–392). In response to that one may observe that mass is treated as a state-dependent property if we describe the process in terms of an “unspe- cific” neutrino that may manifest itself as an electron or muon type (in other words what we have here is a superposition of two types of particles). Once we limit ourselves to the states of a specific type of neutrino, mass can no longer vary over time. 9 The fact that the terms “indistinguishability” and “identity” used in the above-mentioned contexts are most certainly misnomers has been noted by many authors (cf. van Fraassen 1991, p. 376; Butterfield 1993, p. 453). Another potential source of terminological confusion is the practice of referring to state-independent properties as “intrinsic”, which is common in physical literature. This unfortunately interferes with the philosophical sense of the term, which roughly means “non- relational” (see Chap. 4, Sect. 4.1, for a more precise characterization of intrinsic properties). For the rest of the book, I will use the term “intrinsic” in the philosophical sense only. 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 39. 22 is through the concept of permutation invariance. Limiting ourselves to the simplest case of two particles of the same type, we may postulate that the states which differ only by the permutation of these particles should not be empirically distinguishable. Switching one electron for another should not create any observable, or measurable, difference in the total state of the system, since all electrons are “alike” (i.e. all electrons possess the same set of state-independent properties). This stands in contrast to the case of particles belonging to different types, such as an electron and a proton. If the electron initially occupies the state “spin-up” in a given direction, and the proton occupies the state “spin-down”, then swapping them creates a new physical situation that may be experimentally distin- guished from the previous one (we may, for instance, use a mass spec- trometer to first select the electron, and then measure its spin, receiving different outcomes in two different scenarios before and after permutation). The empirical indistinguishability of permuted states can be expressed in the form of the following principle, known as the Indistinguishability Postulate (IP)10 : (IP) Let |φ〉 be any available state of a system of N particles of the same type, and P – a permutation of the set of these particles. Then 〈Pφ|Ω|Pφ〉 = 〈φ|Ω|φ〉 for any physically meaningful Hermitian operator Ω. Condition (IP) stipulates that the expectation values for physically mean- ingful operators be the same for all permuted states. This ensures that the permuted states will be indistinguishable by means of experimental pro- cedures. There are two general ways to satisfy the equation in (IP). We can interpret it as a condition imposed on the states available to systems of particles of the same type, or as a condition on the set of admissible 10 This terminology is used, for example, in Saunders (2009), while Bas van Fraassen calls IP “Permutation Invariance” (van Fraassen 1991, p. 382). On the other hand, Nick Huggett and Thomas Imbo in Huggett and Imbo (2009) use the term “Indistinguishability Postulate” slightly ambiguously—once as equivalent to our IP and once as the postulate limiting the set of observables to the symmetric ones. It may be true that IP, as defined above, is equivalent to the assumption of the symmetry of observables, but still conceptually these are two distinct postulates. T. Bigaj
- 40. 23 observables (cf. Messiah and Greenberg 1964). The second method of making (IP) true follows from a simple transformation of the formal con- dition of the permutation-invariance of expectation values (the transfor- mation is based on the fact that permutation operators are unitary—see Appendix for an explanation): P P P P 1 . (2.10) The last two terms are guaranteed to be identical for all states |φ〉, if only the identity P−1 ΩP = Ω holds. We immediately recognize this equality as the condition that observables be symmetric. However, it is much more common to interpret IP as applying not to observables (at least not directly) but to states. When N = 2, there are two simple ways to make IP true by limiting the set of available states of two same-type particles. One way is to assume that permutation P12 does not change the state of the system, that is, for all |φ〉, P12|φ〉 = |φ〉. An alternative option is to change the sign of the permuted state: P12|φ〉 = − |φ〉. In both cases the expectation value 〈P12φ|Ω|P12φ〉 is guaranteed to be identical to 〈φ|Ω|φ〉 for all operators Ω. Vectors that remain the same under the permutation of two particles are called symmetric, while vectors that change their sign are referred to as antisymmetric. The definitions of symmetric and antisymmetric states can be easily extended for the cases when N 2. Symmetric states of N par- ticles are such that any permutation of the set of particles leaves them unchanged. On the other hand, the case of antisymmetric states is a bit more complicated. We start with the assumption that an antisymmetric state will change its sign under the permutation of any two particles (the permutation swapping two objects is also known as a transposition). However, when we apply an even number of transpositions to a particu- lar antisymmetric state, the result will be the same state. Thus, for anti- symmetric states, odd permutations (i.e. permutations decomposable into an odd number of transpositions) change the sign of the state, while even permutations (consisting of an even number of transpositions) do not alter the initial state. 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 41. 24 2.5 The Symmetrization Postulate Now we are ready to formulate the thesis which has become the corner- stone of the modern debates on the metaphysics of quantum objects: the Symmetrization Postulate (SP). (SP) For any system of particles of the same type, its states are either exclusively symmetric, or exclusively antisymmetric. The Symmetrization Postulate effectively divides up all particles into two categories: those that form groups jointly described by symmetric states and those whose systems are described by antisymmetric states (and for that reason it is referred to as Dichotomy by van Fraassen, 1991, p. 383). Famously, particles of the first category are known as bosons, while the “antisymmetric” particles are referred to as fermions. While other types of symmetry are mathematically possible (and we will discuss them briefly in Chap. 3), so far there is no compelling evidence that particles other than bosons or fermions exist in nature. From a formal point of view, SP amounts to the restriction of the ini- tial N-fold tensor product of one-particle Hilbert spaces to appropriate subspaces (sections) containing only symmetric, or only antisymmetric, vectors. In the case when N = 2, the entire space ℋ1 ⊗ ℋ2 can be decom- posed into two disjoint subspaces: the subspace 𝒜(ℋ1 ⊗ ℋ2) of anti- symmetric states and the subspace 𝒮(ℋ1 ⊗ ℋ2) of symmetric states. Subspace 𝒜(ℋ1 ⊗ ℋ2) is spanned by antisymmetric vectors of the form 1 2 ( ) e e e e i j j i , where i ≠ j, whereas the basis for the symmetric subspace 𝒮(ℋ1 ⊗ ℋ2) can be given in the form of vectors 1 2 ( ) e e e e i j j i plus symmetric products |ei⟩|ei⟩. Note that all these vectors are orthogonal to each other due to the orthogonality rela- tions ⟨ei|ej⟩ = δij, where δij – Kronecker’s delta.11 11 It can be easily verified that if the dimensionality of the one-particle Hilbert space ℋ equals n (and thus the tensor product ℋ ⊗ ℋ has n2 dimensions), then the dimensionality of the antisym- metric subspace 𝒜(ℋ ⊗ ℋ) will be i n i 1 1 , while the dimensionality of 𝒮(ℋ ⊗ ℋ) equals i n i 1 . The T. Bigaj
- 42. 25 Adopting SP ensures that the Indistinguishability Postulate will be true regardless of any restrictions on the admissible observables. On the other hand, limiting the set of observables to those represented by sym- metric Hermitian operators has the same desired consequence even with- out accepting SP. Thus it seems that there are two independent ways to satisfy IP. However, these ways are in fact not entirely independent. It turns out that the requirement of symmetry for observables follows from the Symmetrization Postulate. This is so because the symmetric and anti- symmetric subspaces of the tensor product are not invariant under the action of non-symmetric operators. In other words, it is possible to trans- form a symmetric/antisymmetric vector into one that is neither by acting upon it with a non-symmetric operator. This can be proven as follows (as before, we limit ourselves to the case of N = 2). Let |φS〉 be an arbitrary symmetric vector (i.e. such that P12|φS〉 = |φS〉), and let Ω be a non- symmetric operator in ℋ1 ⊗ ℋ2. Given the assumption of the symmetry of state |φS〉, we have that Ω|φS〉 = ΩP12|φS〉. If vector ΩP12|φS〉 was guar- anteed to be symmetric, this would imply that P12ΩP12|φS〉 = ΩP12|φS〉, which entails that Ω is a symmetric operator when limited to the sym- metric subspace (taking into account that P12 −1 = P12). An analogous argument can be produced for the case of antisymmetric states, which shows, given that in the case when N = 2 the entire tensor product space is spanned by the symmetric and antisymmetric sections, that Ω is a sym- metric operator on the whole space ℋ1 ⊗ ℋ2. Since this contradicts our assumption, Ω has to take us outside either the symmetric or antisym- metric subspaces. Why is the formal requirement that physically meaningful operators should not take us outside the space of available states so important? According to the spectral decomposition theorem, every Hermitian oper- ator in a finitely dimensional vector space can be presented as a linear combination of mutually orthogonal projectors Ei (see Hughes 1989, p. 50): sum of these two expressions is n2 . In particular, if ℋ is two-dimensional, the antisymmetric sub- space will be one-dimensional, while the symmetric subspace will have three dimensions. It is important to keep in mind that in the case of three or more particles (N 2), the antisymmetric and symmetric subspaces do not span the entire tensor product space (see Sect. 3.3 for more details on that). 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 43. 26 A a E i k i i 1 . (2.11) The standard interpretation of this formula is that ai represents a possible value of observable A, while Ei projects onto the corresponding eigens- pace, that is, the space consisting of states for which observable A is well defined and possesses value ai. If an operator A, when applied to a vector |φ〉 from a subspace V, produces a vector A|φ〉 lying outside V, this means that there must be a projector in A’s spectral decomposition that projects onto a space which is neither a subspace of V nor orthogonal to V. But this, in turn, means that some eigenstates of A (states with well-defined values of A) are neither in V nor orthogonal to V. Thus if a particular system occupies a state described by a vector lying in V, there is a non- zero probability that a measurement of A will put the system in a state outside of V (by the standard projection postulate). But SP precludes the possibility that a group of same-type fermions (or bosons) could ever occupy a state that is not antisymmetric (or symmetric). Hence no non- symmetric operators should be allowed to represent physically meaning- ful observables. We have a curious situation now. Typically, the Symmetrization Postulate is argued for by reference to the Indistinguishability Postulate: the argument is that SP makes IP true, and this gives us a reason to adopt SP as a way to ensure the permutation-invariance of expectation values.12 But now we know that SP necessitates the symmetry of admissible observ- ables, and we also know that the condition that observables be symmetric is by itself sufficient to make IP true, regardless of whether we impose any additional restrictions on the available states of same-type particles. So, what additional reasons can we have for adopting SP? Surely, it would be much more cost-effective in terms of the number of extra assumptions to simply accept the symmetry postulate with respect to observables and 12 There are some arguments for SP in the literature that seem to be independent from IP and yet under closer scrutiny turn out to be based on some unwarranted premises. For instance, van Fraassen in his van Fraassen (1991, pp. 389–392) analyzes a simple proof of SP due to Blokhintsev (1964, p. 399ff), which is based on the assumption that all operators on the tensor product are admissible. It is no surprise that if we do not place any restriction on available observables, the only way to satisfy IP is via the superselection rule in the form of SP applied to the available states. T. Bigaj
- 44. 27 forgo a similar postulate with respect to states. That is, unless we can give independent reasons for holding on to SP. We will return to the problem of independent justification for SP in Chap. 3. For now, following the standard approach, we will continue to accept SP as an extra rule govern- ing the behavior of systems of same-type particles.13 2.6 The Indiscernibility Thesis As we have pointed out, the “indistinguishability” of particles of the same type is limited to their state-independent properties. That is, two elec- trons possess the same rest mass and electric charge, but in principle may differ wildly with respect to their state-dependent properties, such as energy, position, spin components and so forth. However, this last state- ment has been challenged in what is known as the Indiscernibility Thesis. It has become part of orthodoxy in the philosophical foundations of quantum mechanics to argue that the Symmetrization Postulate implies that quantum particles of the same type possess the exact same physical properties and therefore cannot be discerned by any physical means. In this section we will discuss typical arguments in favor of this claim, and we will find them wanting. The modern standards for an approach to the problem of the indis- cernibility of same-type quantum particles have been set by Steven French and Michael Redhead (French and Redhead 1988). They start their dis- cussion with formulating the Indistinguishability Postulate and then observing, as we did, that there are two ways of satisfying IP. French and Redhead’s main goal is to argue that given IP, particles of the same type 13 It should be pointed out that the symmetrization of admissible observables (operators) actually yields a principle very similar to SP, at least in the case of systems of two particles. If we limit observables (including Hamiltonians) to the symmetric ones, then it follows that no physical pro- cess (whether a Schrödinger evolution governed by an appropriate Hamiltonian or a measurement- induced collapse) can get us from a symmetric (antisymmetric) state to a non-symmetric (non-antisymmetric) one. This is a simple consequence of the fact that the symmetric/antisym- metric sectors are invariant under the action of symmetric observables. However, this does not mean that a certain type of particles must occupy a given type of state; only that once they start out in a state of a given symmetry type, they can never leave the particular section of states of this type. But restricting ourselves to symmetric operators does not exclude the possibility that a group of fermions could from the outset occupy a state that is not antisymmetric. This may be summarized by saying that SP limits the availability of the states of same-type particles, while the symmetriza- tion of observables merely limits their accessibility (see French and Redhead 1988, p. 239). 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 45. 28 must possess the same state-dependent quantum properties. In spite of the fact that IP can be interpreted as placing a restriction on observables only, they nevertheless use in their argument the assumption that fermi- onic and bosonic states must be antisymmetric/symmetric. The key premise of their argument is the assumption that monadic (i.e. non-rela- tional, or intrinsic) properties of a particular component of a system of same- type particles are exhausted in statements regarding the probabili- ties of obtaining particular outcomes of measurements for each particle. The way they formalize these probabilistic properties is with the help of the tensor products of observables of the form A ⊗ B. They first calculate, using the standard Born rule, the probability that a joint measurement of observable A on particle 1 and observable B on particle 2 will yield par- ticular outcomes (a, b). The appropriate formula for this probability is the square of the inner product |〈λa ⊗ χb|φ〉|2 , where |λa〉 and |χb〉 are eigenvectors of A and B, respectively, corresponding to values a and b, and |φ〉 is the state of the two-particle system. Using the assumption that |φ〉 is antisymmetric or symmetric, they calculate the probabilities of revealing a given value on particle 1 and on particle 2 given that A = B, and they find these probabilities identical, which supports the claim that two fermions (or bosons) of the same type are never discerned by their properties. French and Redhead’s calculations are slightly complicated due to the necessity of going through the procedure of summing the probabilities of one outcome over all possible outcomes on the other particle. However, this can be significantly simplified by resorting to expectation values rather than probabilities (see Huggett 2003; Dieks and Versteegh 2008).14 Let A1 = A ⊗ I and A2 = I ⊗ A represent observables associated, respec- tively, with the first and the second particles. Then, given the assumption that either P12|φ〉 = |φ〉 (for bosons), or P12|φ〉 = − |φ〉 (for fermions), we arrive at the following sequence of equations: 14 In spite of appearances, there is no loss of generality in moving from probabilities to expectation values. This is so, because the probability of obtaining any outcome of any observable can be recov- ered as the expectation value of the projector onto the subspace corresponding to this outcome (cf. Hughes 1989, p. 71). Thus, fixing the expectation values of all projectors in a given Hilbert space automatically fixes all the probabilities of outcomes for any observable. This proves once again how flexible a tool projection operators are. T. Bigaj
- 46. 29 | | | | | | | ( ) | | A A I P A I P P A I P 1 12 12 12 1 12 I I A A | | | . 2 (2.12) Thus the expectation values for all observables pertaining to either parti- cle are identical, and this is what the Indiscernibility Thesis amounts to. The crucial premise in the above argument is the assumption that the operators A ⊗ I and I ⊗ A are indeed formally accurate representations of observable A pertaining, respectively, to the first and second particles. But here we encounter an immediate stumbling block. Operators A ⊗ I and I ⊗ A are clearly not symmetric, so they should be disallowed on the basis of our earlier considerations. Since SP implies that only symmetric operators can have physical meaning when applied to systems of same- type bosons or fermions, it seems that we should not use the non-sym- metric products in formalizing the argument for the Indiscernibility Thesis. French and Redhead are aware of that difficulty, but they are strangely dismissive about it. First, they interpret the symmetry postulate with respect to observables not as delimiting physically meaningful opera- tors but operators that can be observed.15 Having done this, they announce that “from the point of view of discussing PII [the Principle of the Identity of Indiscernibles – TB] it seems clear that we should not restrict the dis- cussion to attributes which can actually be observed” (ibid. p. 239). French and Redhead’s response to the problem raises several questions. Firstly, interpreting the requirement of symmetry as applying to opera- tors that can be observed misses the point of the permutation-invariance problem. Suppose that we have a non-symmetric operator Ω which, even though it cannot be observed, is still admissible as a representation of a particular objective property of the system. This means that an “actualiza- tion” of a given value of this operator (French and Redhead speak about 15 Cf. (French and Redhead 1988, p. 239). We hasten to explain to logical purists that the phrase “operators that can be observed” is not supposed to be taken literally (operators, being mathemati- cal objects, are never observable), but is a mere shorthand for the longer “operators representing observable properties”. Another minor linguistic issue that may or may not be necessary to clarify here is that while the standard physical counterparts of mathematical operators are usually called observables, in the current context this terminology is not particularly felicitous (vide the term “unobservable observables”). 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 47. 30 actualizations rather than obtaining measurement outcomes, since we are dealing here with unobservable properties) can put the system into a state that is neither symmetric nor antisymmetric, thus violating the Symmetrization Postulate. The claim that we can’t observe this transition does not nullify the fact that SP is made false by its existence. Secondly, and more importantly, the contention that the operator representing a measurable and observable property of one particle gets classified as unob- servable when this particle is taken as part of a broader system together with another particle should be viewed with high suspicion. One possible reply in defense of French and Redhead could be that the reason for the unobservability of operators of the form A ⊗ I and I ⊗ A has nothing to do with the A-property per se, but rather comes from the fact that we don’t have any empirical means to distinguish particle 1 from particle 2, so we don’t know on which particle we are supposed to perform an appro- priate measurement. This seems right, but the moral from this example should be to reevaluate the way we can make reference to individual par- ticles, rather than blindly accept that the attributes of these particles rep- resented by non-symmetric operators “can never be observed” (for more on that see Chap. 5). Is there any other way to represent properties of individual particles without infringing upon the symmetrization requirement with respect to observables? Nick Huggett in (Huggett 2003) has suggested how to approach this problem in a more general fashion. His proposal is to for- mulate a set of minimal conditions that should be met by any operators in the tensor product space that can lay claim to representing attributes of individual particles. Let {O1, O2, … ON} be a set of operators acting in the N-fold tensor product of one-particle Hilbert spaces ℋ1 ⊗ … ⊗ ℋN. Each operator Oi is supposed to represent a particular observable O attributed to the i-th particle. In order to be able to do that, the operators should satisfy two postulates, which Huggett dubs the conjugation con- dition (CC) and independence condition (IC). These conditions are as follows: (CC) Pij −1 Oi Pij = Oj (IC) Pij −1 Ok Pij = Ok, when k ≠ i and k ≠ j. T. Bigaj
- 48. 31 The meaning of these requirements should be clear: the conjugation condition ensures that the permutation of two particles will swap their properties, whereas the independence condition guarantees that permut- ing two particles will not affect the properties of a third one distinct from the two. Huggett then shows that conditions (CC) and (IC) are sufficient to obtain French and Redhead’s indiscernibility result. In the case of monadic properties, the proof is immediate and requires only the conju- gation condition (the case of relational properties will be evaluated in the next section). O P O P i ij i ij by the symmetry antisymmetry of / P O P O ij i ij j 1 formal transformation CC . (2.13) Conditions (CC) and (IC) are obviously satisfied by operators of the form Oi = I ⊗ …⊗ A ⊗ …⊗ I, where A occupies the i-th place in the product. However, other operators can also be shown to conform to (CC) and (IC). Is it possible to find operators that would satisfy (CC) and (IC) and at the same time be symmetric? Generally, the answer is “yes”, but the success turns out to be somewhat limited in scope. If the operators Oi were to be symmetric, this would mean that Pij −1 Oi Pij = Oi which, together with the conjugation condition Pij −1 Oi Pij = Oj, implies that Oi = Oj for all i, j. Thus the only symmetric operators that could possibly rep- resent properties of individual particles would be identical with each other. This obviously trivializes the question of whether particles of the same type can be discerned by their properties. It is hardly an exciting result proving that the expectation values of operators that are identical turn out to be identical too (we don’t even need to rely on the Symmetrization Postulate with respect to states to prove that). French and Redhead’s proposal of how to formally represent properties of individual particles, as well as the general approach advocated by Huggett, both rely on the same implicit assumption, which is so basic that up to a certain point in the history of the debate no one even 2 Indiscernibility of Quantum Particles: A Road to Orthodoxy
- 49. 32 bothered to make it explicit. And yet this assumption, which is constitu- tive of the approach to the individuation of particles that may be called “orthodoxy”, deserves to be seen in broad daylight. This claim, which some refer to as “Factorism”,16 may be spelled out as follows: (F) In the N-fold tensor product of Hilbert spaces ℋ1 ⊗ … ⊗ ℋN that is meant to represent states and properties of sys- tems of N particles of the same type, and whose symmetric and antisymmetric sectors are assumed to contain all the admissible states of N bosons and N fermions respectively, each Hilbert space ℋi represents states and properties of one individual particle. Factorism seems to be presupposed by the way we defined tensor prod- ucts of Hilbert spaces as representations of states of composite systems consisting of a number of component systems. Thus it may be claimed that Factorism is an essential part of the tensor product formalism17 and as such cannot be called into question without abandoning the entire formalism. And yet on a certain level of abstraction this conclusion can be resisted. It is at least conceivable that we could treat the mathematical structure of the N-fold tensor product of Hilbert spaces purely formally, without attaching any physical interpretation to the individual factors in the product. Then, after imposing certain additional restrictions on the admissible states in this product space (e.g. in the form of the 16 I follow the interpretation of Factorism as spelled out in Caulton (2014 p. 11). On the other hand, F.A. Muller and Gijs Leegwater in Muller and Leegwater (2020) introduce a broader reading of Factorism. They consider the general problem of how to factorize a given Hilbert space ℋ into a tensor product of N spaces, and they observe that typically there is more than one way to achieve such a factorization. Consequently they distinguish two general versions of Factorism: ∀-Factorism, stating that for all available factorizations the labels associated with the factors refer to the compo- nents of the system, and ∃-Factorism, asserting that some such factorizations play the referential role. Their main point is that ∃-Factorism may be preserved even for “indistinguishable” particles (see Sect. 4.3 for more on that). However, this conclusion does not invalidate the fact that Factorism as stated above is open to refutation. The variant of Factorism defined above involves one specific factorization—namely the factorization with respect to which we impose the requirement of per- mutation invariance, as explained in Sect. 2.4. 17 Redhead and Teller (1991, 1992) use a longer term Labeled Tensor Product Hilbert Space Formalism. T. Bigaj
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