1. Properties, Statistics and the Identity of Quantum Particles
Matteo Morganti
INFN Training School
19-21 December 2016
Frascati
19/12/2016

2. Introduction What sort of things is quantum theory about?
Starting points:
1) Entrenched intuition that reality is made of individual objects
2) It is a widespread belief that quantum theory refutes the intuition, as it is a theory of non-individuals
‘Received View’, dating back to (some of) the founding fathers of the theory - e.g., Born, Heisenberg
«It is impossible for either of these individuals to retain his identity so that one of them will always be able to say ‘I’m Mike’ and the other ‘I’m Ike.’ Even in principle one cannot demand an alibi of an electron!» (Weyl, 1931)
Properties, statistics and the identity of quantum particles

3. Introduction
Aim here: to assess the widespread belief, via a critical analysis of options
‘Naturalistic’ approach whereby science and philosophy complement each other
Input coming from our best physics is essential, but empirical data underdetermine the interpretation of the theory
Clarification of concepts and arguments rather than final answers
Considerations re. non-relativistic quantum mechanics and (more briefly) quantum field theory
Properties, statistics and the identity of quantum particles

4. Plan
Definitions
(Non-relativistic) Quantum Mechanics: Properties, Statistics and the Received View
Alternatives: A Theory of Individuals after All?
Quantum Field Theory
Conclusions
Properties, statistics and the identity of quantum particles

5. 1. Definitions
(Material/physical) entity x is an individual object iff:
x is a thing/object
Not an event, property, fact or…
x is self-identical
x is numerically distinct from everything else
x may also possess:
Diachronic (in addition to synchronic) identity
Sharp localisability
Individual object ≠ object according to classical mechanics/common sense – Particle?
Not obvious: think, e.g., of the difference between an actual coin and a money unit in a bank account
Properties, statistics and the identity of quantum particles

6. 2.1 Non-Relativistic QM: Statistics
Individuality a problem with respect to statistics:
Assume two systems 1 and 2 and two available states x and y
Available arrangements, classically:
|x>1|x>2
|y>1|y>2
|x>1|y>2
|y>1|x>2
Properties, statistics and the identity of quantum particles

7. 2.1 Non-Relativistic QM: Statistics
Individuality a problem with respect to statistics:
Assume two systems 1 and 2 and two available states x and y
Available arrangements, quantum case:
|x>1|x>2
|y>1|y>2
1/2(|x>1|y>2|y>1|x>2)
Only (anti-)symmetric states available in QM
Permutation Symmetry
Confirmation of the Received View?
‘Particles’ more like money units in a bank than real coins
BOSONS
FERMIONS
Properties, statistics and the identity of quantum particles

8. 2.2 Non-Relativistic QM: Properties
In classical mechanics (although not an axiom of the theory!), impenetrability guarantees difference in spatial location/trajectories
(The same in, e.g., Bohmian Mechanics)
Not in (standard) quantum mechanics!
Properties correspond to probabilities (Born Rule)
Two entities can ‘have the same position’, i.e., share the same possible values for position measurements
Properties, statistics and the identity of quantum particles

9. Properties, statistics and the identity of quantum particles

10. Properties, statistics and the identity of quantum particles

11. xy((xy)P(Px&Py))
Underlying intuition: things are individuated by their properties
An important philosophical tradition:
Leibniz  Quine, Hilbert and Bernays, etc.
Principle of the Identity of Indiscernibles (PII):
xyP(PxPy)(x=y) 
xy((xy)P(Px&Py))
(P not ranging over properties such as ‘is identical to a’…)
QM statistics+violation of PIIReceived View?
Properties, statistics and the identity of quantum particles

12. 1/2(|>1|>2+|>1|>2)
3. Alternatives (1)
Individuation via (qualitative) relations, not properties
Consider, e.g., ‘…is taller than…’, ‘… goes in the opposite direction to…’
But, not reducible to monadic properties of relata and symmetric
(Existence and features of relata not ontologically prior)
Singlet state of two fermions
1/2(|>1|>2+|>1|>2)
No well-defined spin properties separately
Yet, Prob=1 of opposite spins!
Properties, statistics and the identity of quantum particles

13. Mass x
Charge y
Spin probability z …
Opposite spin!
Mass x
Charge y
Spin probability z …
Encoded in states like 1/2(|>1|>2+|>1|>2)
Properties, statistics and the identity of quantum particles

14. The result is good news for Leibnizians
Some remarks:
The result is good news for Leibnizians
Individuation via qualitative physical features
Due to the Exclusion Principle, ‘identical’ fermions in the same physical system can only be found in states such as the one just described
Basic ontological distinction between bosons and fermions?
Actually a controversial result:
Can relations be prior to their relata?
Are the relevant properties always physically meaningful?
Are weakly discernible entities individuals?
Do we really have a relation between parts?
Properties, statistics and the identity of quantum particles

15. A ? B Spin  Spin correlation Spin  Time t1
Time t2 (after measurement)
Properties, statistics and the identity of quantum particles

16. «Common natures are ‘contracted’ in particular things by haecceitates»
3. Alternatives (2)
An alternative philosophical view:
Not necessary to use PII as a principle
of individuation
Individuality is a primitive, ‘brute’ fact
Duns Scotus:
Individuality based on ‘transcendental’ identities
«Common natures are ‘contracted’ in particular things by haecceitates»
Properties, statistics and the identity of quantum particles

17. What about primitive identities for quantum entities?
Makes sense of violations of PII
As well as of the use of particle names or ‘labels’ in QM
However, two objections:
1) Seems ruled out by permutation symmetry – quantum statistics
2) Methodologically suspect
What is a primitive identity? How can such a thing be empirically relevant?
Replies
Properties, statistics and the identity of quantum particles

18. 1) Primitive identities are compatible with permutation symmetry
For instance, state-dependent properties of many-particle quantum systems (of identical particles) may be holistic
Notice that this accounts for the impossibility of non- symmetric states, which must be explained anyway
Obvious difference: |x>1|y>2 and 1/2(|x>1|y>2|y>1|x>2)
2) Indiscernibles could in fact make a qualitative difference
For example, a system with n indiscernible fermions has n times the unit mass
(Compare with the Leibniz vs. Newton dispute on the nature of space (and time))
Primitive identities need not be ‘mysterious metaphysics’
Properties, statistics and the identity of quantum particles
Identity, Indiscernibility and Naturalised Metaphysics

19. 4. Quantum Field Theory More elements to take into account:
QFT has no ‘labels’ for particles, but rather occupation numbers and annihilation/creation operators
The number operator N can be in a superposition of states!
How many ‘things’??
The relativistic theory makes things even worse:
Not clear that we have a fully consistent theory of both free and interacting fields (Haag’s Theorem)
Expectation values for certain quantities do not vanish for the vacuum state
Properties, statistics and the identity of quantum particles

20. 4. Quantum Field Theory Moreover, in relativistic QFT:
Local measurements do not allow to distinguish a ‘vacuum state’ from any n-particle state (Reeh-Schlieder Theorem)
An accelerated observer in a vacuum detects a ‘thermal bath’ of particles (Unruh Effect)
Particles cannot be localised in finite regions (Malament’s Theorem)
In short, facts about numbers of ‘things’ fail to be completely determinate in QFT
Even the distinction between vacuum and non- vacuum is not clear-cut
Localisation cannot be taken for granted either
Properties, statistics and the identity of quantum particles

21. This may seem to lead us away from individuals and particles
Notice, however, that:
The allegedly problematic results are open to discussion
At any rate, they carry over to an ontology of fields
More importantly, recall that individual object(classical) particle
Properties, statistics and the identity of quantum particles

22. (Material/physical) entity x is an individual object iff:
x is a thing/object
Not an event, property, fact or…
x is self-identical
x is numerically distinct from everything else
x may also possess:
Diachronic (in addition to synchronic) identity
Sharp localisability
Properties, statistics and the identity of quantum particles

23. This may seem to lead us away from individuals and particles
Notice, however, that:
The allegedly problematic results are open to discussion
At any rate, they carry over to an ontology of fields
More importantly, recall that individual object(classical) particle
If we stick to and drop 4.-5., results concerning number and localisability become perhaps less worrisome
(Frame-dependent) individual objects but not (classical) particles?
The ontological question remains open…
Properties, statistics and the identity of quantum particles

24. 5. Conclusions Philosophical issues with identity and individuality
Obvious relevance of science - naturalism
Individuals vs. non-individuals, particles vs. fields vs. …
Non-relativistic quantum mechanics and quantum field theories
Issues not settled: Received View with non-individual entities not an obvious winner
Need for further study of:
Empirical evidence
Physical theory
Philosophical assumptions
Underlying methodology
This case study indicates the importance of the interplay between physics and philosophy: empirical data do not determine a unique interpretation
Properties, statistics and the identity of quantum particles

25. THANK YOU!
Properties, statistics and the identity of quantum particles