Download Properties, Statistics and the Identity of Quantum Particles

Properties, Statistics and the
Identity of Quantum Particles
Matteo Morganti
INFN Training School
19-21 December 2016
Frascati
19/12/2016
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
2
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
3
Plan
1.
Definitions
2.
(Non-relativistic) Quantum Mechanics: Properties,
Statistics and the Received View
3.
Alternatives: A Theory of Individuals after All?
4.
Quantum Field Theory
5.
Conclusions
Properties, statistics and the identity of quantum particles
4
1. Definitions
•
1.
•
(Material/physical) entity x is an individual object iff:
x is a thing/object
Not an event, property, fact or…
Not obvious: think, e.g., of
the difference between an
actual coin and a money
unit in a bank account
3.
x is self-identical
x is numerically distinct from everything else
•
x may also possess:
4.
Diachronic (in addition to synchronic) identity
Sharp localisability
2.
5.
• Individual object ≠ object according to classical
mechanics/common sense – Particle?
Properties, statistics and the identity of quantum particles
5
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
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, quantum case:
•
– |x>1|x>2
– |y>1|y>2
– 1/2(|x>1|y>2|y>1|x>2)
•
BOSONS
FERMIONS
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
Properties, statistics and the identity of quantum particles
7
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
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Introduce a permutation operator Perm1,2 having the properties that
Perm1,2 O1 Perm1,2 = O2 and (Perm1,2)2 = I (with I being the identity operator)
For all known particles the Indistinguishability Postulate holds, according
to which for any n-particle state of particles of the same type and
observable O,
<Perm |O|Perm > = <|O|> (with Perm being associated with an
arbitrary exchange of particles). Therefore, one obtains
| 
Prob ( x ) O1
=
<|P
O1
x|>
=
<|Perm1,2
P
O2
x
Perm1,2|>
=
|
(
x
)
O2
<|P x|> = Prob
O2
That is, for any observable O (now including location!) and any specific
|
|
(
x
)
O1
value x for it possessed as a monadic property, Prob
= Prob ( x )O
2
Properties, statistics and the identity of quantum particles
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For relational properties, one can prove that
Prob (( x)O1 |(y )O 2 )
|
|
= Prob (( x)O 2 |(y )O1 )
(again for any observable and any
value) as follows:
By a fundamental property of probabilities, the above equality is the
same as
|
|
|
|
Prob (( x)O &(y )O ) / Prob ( y )O = Prob (( x)O 2 &(y )O1 ) / Prob ( y )O1
1
2
2
The denominators have just been shown to be equal. As for the
numerators, they are also equal. For,
|
Prob (( x)O1 &(y )O 2 )
<|Perm1,2 P
O2
<|Perm1,2
P
x
= <|P
O1
x
(Perm1,2)2 P
O2
x
P
O1
y
P
O2
O1
y
y|>
=
Perm1,2|> =
Perm1,2|>
=
<|P
O2
x
P
O1
y|>
|
((
x
)
|(
y
)
)
O2
O1
Prob
Properties, statistics and the identity of quantum particles
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=

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
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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
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Mass x
Charge y
Spin probability z
…
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
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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
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Spin 
Spin correlation
A
Spin 
B
Time t1
Time t2 (after measurement)
Properties, statistics and the identity of quantum particles
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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:

«Common natures are
‘contracted’ in
particular things by
haecceitates»
Individuality based on ‘transcendental’ identities
Properties, statistics and the identity of quantum particles
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•
What about primitive identities for quantum
entities?
•
Makes sense of violations of PII
•
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?
•
•
As well as of the use of particle names or ‘labels’ in QM
Replies
Properties, statistics and the identity of quantum particles
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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 nonsymmetric 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
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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
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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 nonvacuum is not clear-cut
•
Localisation cannot be taken for granted either
Properties, statistics and the identity of quantum particles
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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
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•
1.
•
(Material/physical) entity x is an individual object iff:
x is a thing/object
Not an event, property, fact or…
3.
x is self-identical
x is numerically distinct from everything else
•
x may also possess:
4.
Diachronic (in addition to synchronic) identity
Sharp localisability
2.
5.
Properties, statistics and the identity of quantum particles
22
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 1.-3. 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
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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
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THANK YOU!
Properties, statistics and the identity of quantum particles