Transcript Carsten Held, PPT

Can Quantum Mechanics be Shown to be
Incomplete in Principle?
Carsten Held (Universität Erfurt)
Structure of the argument
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QM (axioms A1–A4, no projection postulate), COMP
P0, motivation: ‘probability is quantified possibility’
P1, motivation: a natural interpretation of expectation values
P2, motivation: QM events are spacetime events consisting in the
possession of properties
P3, motivation: QM events are spacetime events
N, a trivially legitimate state ascription
Main argument:
QM + N + COMP + P1 + P0  
First supp. argument:
 P1   P2
 P1 + M1 + M2   P2
Second supp. argument:
 P2   P3
 P2 + M2 + M3   P3
(Meta-premises M1–M3, allowing a more rigorous argument, are just
mentioned)
‘Loophole of type-identity’?
Axioms of QM:
• A1 Any QM system S is associated with a unique Hilbert space H
and its state is represented by a unique density operator W (t) on H,
a function of time.
• A2 Physical quantities A, B …, (called observables) with values
a1, a2, a3…, b1, b2, b3… possibly pertaining to S, are represented by
Hermitian operators A, B …, with eigenvalues a1, a2, a3…, b1, b2, b3
… on H.
• A3 S evolves in time according to: W (t) = U (t) W (t0) U (t)–1 where
U (t) = exp [–iHt], a unitary operator, is a function of time and H is an
operator representing the total energy of S.
• A4 If S is in state W (t) and A is an observable on S, then the
expectation value <A> (t) is:
<A> (t) = Tr (W (t) A)
Axioms of QM (Remarks):
(1) Notice that A1, A3 and A4 mention one and the same time
parameter t. Thus:
M1 QM (axiomatized by A1–A4) contains one parameter t.
A1–A3 are just mentioned to illustrate this claim and play no further
role. A4 will be used below to generate a contradiction with two
principles P0, P1. One tempting way out of the contradiction will be
to implicitly duplicate t, in contrast with what A1–A4 show and M1
explicitly says.
Axioms of QM (Remarks):
(2) Some may object that the axiomatization is empirically inadequate
and thus incomplete without some version of the
projection postulate:
• A5 If S is found to have value ak of A as a result of an A
measurement, then S’s state is Pak immediately after this
measurement.
This objection is irrelevant. The below arguments explicitly refer to
A4 only and will go through whether QM includes A5 or not. For the
green derivation, there is the proviso that A5 must not tacitly
introduce a second t, which it does not seem to do for any
sufficiently precise sense of ‘immediately after’. In this case, M1
would just become: QM (ax. by A1-A5) contains one parameter t.
Axioms of QM (Remarks):
(3) The arguments in fact make reference only to a special case of A4.
For projector Pak, with < Pak > = p (ak), A4 reduces to:
BR
If S is in state W (t) and A is an observable on S with
eigenvalue ak, then the probability that S has ak is:
p (ak) = Tr (W (t) Pak).
(Born Rule)
Throughout, boldface ‘ak’ abbreviates the proposition ‘S has ak’,
naming the simplest candidate for a QM event.
Note also that BR suffers from the defect that in its equation the left
side carries no time-index. In this sense, BR is vague.
Axioms of QM (Remarks):
(3, continued)
Consider two natural ways to specify
the BR equation p (ak) = Tr (W (t) Pak), i.e.:
(i)
p (ak (t) ) = Tr (W (t) Pak),
(ii)
p (t) (ak) = Tr (W (t) Pak)
or more explicitly:
p (ak given ES (t)) = Tr (W (t) Pak)
(where ‘ES (t)’ is some triggering event,
e.g. the onset of an A-measurement on S)
Axioms of QM (Remarks):
(3, continued)
(i)
p (ak (t) ) = Tr (W (t) Pak),
(ii)
p (ak given ES (t)) = Tr (W (t) Pak)
(Notice that (ii) does not essentially make reference to
measurement, but rather encapsulates the idea that QM
probabilities are dispositional properties of S.)
Since it is hard to imagine other ways to specify the left side of the
equation, we might claim:
M2 (i) and (ii) are the only possibilities to specify parameter t in BR.
Completeness of QM:
Completeness is formalized in the standard way, i.e. by the
eigenstate-eigenvalue link:
EE
ak (t) iff S is in state Pak (t).
The converse of the backward direction of EE is:
COMP If S is not in state Pak (t), then not ak (t).
Note that EE and COMP are read as concerning the same type of
events as BR (type-identity of all QM events).
Principles for QM probabilities:
I now introduce four principles [P0-P3] for interpreting QM probabilities
and transform them, one by one, into more formalized prescriptions
P0-P3:
[P0]
If a theory assigns an event a non-zero probability, then, given
the theory’s truth, this event is possible.
Reading possibility as logical possibility, we can write:
P0
If, for a proposition F (describing an event) a theory T yields
another proposition p (F) > 0, then it is not the case
that T, F |– .
Principles for QM probabilities:
[P0] (and thus P0) is not referring specifically to QM. It can be
motivated from the general idea that probability is quantified
possibility.
In the following, I assume without argument that it is not a
reasonable option to give up P0.
Principles for QM probabilities:
[P1]
All statistical expressions in QM have their usual statistical
meanings.
From a consideration of the statistical notion of expectation value
one can make it plausible that [P1] implies:
P1
In BR, every probability, being of the form p (ai) = Tr (W (t) Pai),
is to be interpreted as p (ai (t)),
i.e. as ‘the probability that S has ai at t’.
(This is possibility (i), above.)
Principles for QM probabilities:
Note that, using P1, we can remove the vagueness found in BR,
rewriting it:
BR′
If S is in state W (t) and A is an observable on S with
eigenvalue ak, then the probability that S has ak at t is:
p (ak (t)) = Tr (W (t) Pak).
Principles for QM probabilities:
[P2]
All events that are assigned probabilities in QM are explicitly
spacetime events consisting in the possession of properties.
We can concretise [P2] directly as:
P2
For any expression of the form ‘ai’ in QM there is a
parameter t in the formalism such that the expression is read
as:
‘ai (t)’, i.e. ‘S has ai at t’.
Principles for QM probabilities:
[P3]
All events that are assigned probabilities in QM are explicitly
spacetime events.
We can concretise [P3] as:
P3
For any expression ‘F’ such that BR yields an expression
‘p (F) = Tr (W (t) Pak)’ there is a parameter t in the formalism
qualifying ‘F’ in some way as:
‘F(t)’, i.e. ‘… at t …’.
The main argument:
Assume now (assumption N) that S is in a pure state
W (t1)  Pak (t1) for some t1, such that, by BR, p (ak) > 0.
I now show that BR′, (i.e., BR interpreted via P1) plus N plus COMP
transform QM into a theory that contradicts P0. Recall that rejecting
P0 is not a reasonable option. Likewise, assumption N is trivially
admissible. Hence, the defender of COMP will reject P1.
Here is the argument:
The main argument:
Lines [2], [4] of the following argument follow from N by BR′ and
COMP, respectively:
N
N, BR′
N
N, COMP
[1]
[2]
[3]
[4]
S is in state W (t1).
p (ak (t1)) > 0.
 (S is in state Pak (t1)).
 (ak (t1)).
N
[1], BR′
N
[3], COMP
Let QM, as containing BR′, N and COMP be integrated into one
theory QM′. Then, QM′ assigns ‘ak (t1)’ a positive probability [2], and
yet (by [4]):
QM′, ‘ak (t1)’ |– . This contradicts P0.
Hence: QM + N + COMP + P1 + P0  
P1  P2:
The obvious response now is to reject P1, i.e. find another reading
for BR. One may deny that in p (ak) = Tr (W (t) Pak) the left-hand
expression must be interpreted as:
p (ak(t)) (‘the prob that S has ak at t)’.
Rather:
p (t) (ak) (‘the prob at t that S has ak))’.
(Possibility (ii), above.)
However, QM (A1–A4) provides no second time-index, hence if, in
order to escape the argument, ‘ak’ does not inherit its time-index
from W (t), it can have none at all …
– in contradiction with P2.
 P1 + M1 + M2   P2:
Using M1, M2, we can transform the previous P1  P2 into a
more rigid argument:
 P1 means that ‘p (ak)’ is not specified as ‘p (ak (t) )’. Then, by M2,
it must be specified as p (t) (ak)’. But, by M1, QM contains one
parameter t only. Hence, for any QM event in BR there is no timeindex.
Recall that  P1 is motivated by ‘QM + COMP + P0   P1’ and
that P0 is sacrosanct. We have thus made it plausible (or derived)
that, given COMP and P0, no QM event is a spacetime event
consisting in S having a property.
Die-hard defenders of COMP will bite the bullet, but…
P2  P3:
To see that denying P2 makes P3 implausible, we must take another
look at the positive proposal behind P2,
i.e. that ‘p (ak)’ in BR must be read as: ‘p (ak given ES (t))’.
This gives us another disambiguation of BR:
BR′′
If S is in state W (t) and A is an observable on S with
eigenvalue ak, then the probability that S has ak given ES (t) is:
p (ak given ES (t)) = Tr (W (t) Pak)
P2  P3:
The literature (e.g., Butterfield 1993) offers three possible
interpretations of ‘p (ak given ES (t)) = … ’:
CondProb
‘p (ak | ES (t)) = …’
ProbCons
‘ES (t) > p (ak ) = …’
ProbCond
‘p (ES (t) > ak) = …’
Notice that in the present context (i.e.P2) the conditioned event
must not be time-indexed.
P2  P3:
CondProb
‘p (ak | ES (t)) = …’
(i.e. the BR′′ probs are conditional probs)
No definition à la Kolmogorov of such probabilities is possible for a
non-denumerable set of events ES (e.g., continuously many
observables to be measured on S) is possible (van Fraassen &
Hooker 1976).
A more general point (in the context of P2): QM is a fundamental
theory, thus no construal of BR can get informative probabilities from
elsewhere. BR′′ (CondProb), by construction, does not deliver probs
like ‘p (ak) and ‘p (ES (t))’, so a Kolmogorov definition is impossible.
P2  P3:
CondProb
‘p (ak | ES (t)) = …’
(i.e. the BR′′ probs are conditional probs)
Interpreters propose to introduce conditional probs via Popper
functions. But this requires probs like ‘p (ES (t) | ak)’ to be welldefined
So a similar point as before applies: QM is a fundamental theory,
thus no construal of BR can get informative probabilities from
elsewhere. BR′′ (CondProb), by construction, does not deliver the
required probs, so a Popper definition is impossible.
P2  P3:
ProbCons
‘ES (t) > p (ak) = …’
(i.e. the BR′′ probs are probabilistic consequents of conditionals)
By BR′′ (ProbCons), the QM probabilities no longer concern
spacetime events in any sense, i.e. we have P3.
P2  P3:
ProbCond
‘p (ES (t) > ak) = …’
(i.e. the BR′′ probs are probabilities of conditionals)
In this version, the QM events are described by conditionals.
Assuming, as we did, a type-identity of QM events in BR and EE, we
must revise COMP:
COMP′
If S is not in state Pak (t), then not (ES (t) > ak).
This looks neat.
However, for a state W (t1)  Pak (t1) we now get:
p (ES (t1) > ak) > 0 and not (ES (t1) > ak).
This again contradicts P0.
P2  P3:
Two remarks:
M3
Butterfield’s CondProb, ProbCons, ProbCond exhaust the
possible analyses of ‘p ([A] = ak given ES (t))’.
Assuming M2, M3 the plausibility argument P2  P3 can be
made rigorous.
Dropping the type-identity of QM events in BR′′ and COMP, BR′′ can
be maintained together with P3.
Escaping the argument for P3:
P3: In BR, no QM events at all get a time-index.
P3 implies: No probs for spacetime QM events. QM is not a theory
about spacetime events, at all. The vagueness in BR cannot be
consistently removed.
This is certainly an unacceptable consequence. The defender of
COMP will use the last loophole and drop the requirement of typeidentity of QM events.
Escaping the argument for P3:
Let’s deny that QM events in versions of BR and COMP are typeidentical. Explicitly, assume BR altered to BR′′ (ProbCond), but keep
the original COMP. E.g., for W (t1)  Pak (t1) :
BR′′ (ProbCond)
COMP
p (ES (t1) > ak) = Tr (W (t1) Pak)
If S is not in state Pak (t1), then not ak (t1).
This recovers (I believe) the standard way to interpret BR in the light
of COMP.
This interpretation is coherent and keeps P3, but it still must pay an
exceedingly high price …
Dropping type-identity?
Indeed, denying type-identity produces an interpretation that is very
close to inconsistency.
Consider the QM events ‘(ES (t1) > ak)’ in BR′′ (ProbCond):
Essentially, the triggering ES must, the triggered ak must not, carry a
time-index. We can, however, easily argue that ak must be true at
some time (indeed in the interval [t1, t3]). So, for every ‘(ES (t1) > ak)’
there is a t2 such that ‘ak (t2)’ is true, but this proposition cannot turn
up in our ‘fundamental’ theory QM.
Why? Because we want to express the completeness of QM by COMP,
where a proposition ‘Not ak (t1)’ does turn up.
So, certain types of propositions about real events ak (i.e. properties
possessed at certain times) cannot turn up in our theory, because in
a supplementary requirement to the theory they do turn up.
Keeping type-identity?
If you think that this is a good plausibility argument for keeping typeidentity, then the last loophole is closed. Then the alternative for QM
is between COMP and P1–P3:
Either QM is complete in the sense of COMP, but does not deal with
spacetime events …
Or QM obeys P1–P3, deals with spacetime events and ascribes probs
to them, but is not complete in the sense of COMP.
Some references:
•
Butterfield, J. [1993]: ‘Forms for Probability Ascriptions’, International Journal of
Theoretical Physics 32, pp. 2271-2286.
•
Halpin, J. F. [1991]: ‘What is the Logical Form of Probability Assignment in Quantum
Mechanics?’, Philosophy of Science 58, pp. 36-60.
•
van Fraassen, B.C. and Hooker, C.A. [1976]: ‘A Semantic Analysis of Niels Bohr’s
Philosophy of Quantum Theory’, in W.L. Harper and C.A. Hooker (eds), Foundations
of Probability Theory, Statistical Inference, and Statistical Theories of Science, vol. III:
Foundations and Philosophy of Statistical Theories in the Physical Sciences,
Dordrecht, Reidel, pp. 221-241.