Can Quantum Mechanics be Shown to be Incomplete in Principle? Carsten Held (Universität Erfurt)

1. Can Quantum Mechanics be Shown to be Incomplete in Principle?Carsten Held (Universität Erfurt)

2. Structure of the argument • 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’?

3. 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)

4. 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.

5. 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, M1would just become: QM (ax. by A1-A5) contains one parameter t.

6. 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.

7. 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)

8. 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.

9. Completeness of QM: Completeness is formalized in the standard way, i.e. by the eigenstate-eigenvalue link: EEak (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).

10. 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 |– .

11. 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.

12. 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 aiat t’. (This is possibility (i), above.)

13. Principles for QM probabilities: Note that, using P1, we can remove the vagueness found in BR, rewriting it: BR′ If S is in stateW (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).

14. 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 aiat t’.

15. 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 …’.

16. 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:

17. The main argument: Lines [2], [4] of the following argument follow from N by BR′ and COMP, respectively: N [1] S is in state W (t1). N N, BR′ [2] p (ak (t1)) > 0. [1], BR′ N [3]  (S is in state Pak (t1)). N N, COMP [4]  (ak (t1)). [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 

18. 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 akat 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.

19.  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 time-index. 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…

20. 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)

21. 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.

22. 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.

23. 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 well-defined 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.

24. 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.

25. 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.

26. 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.

27. 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 type-identity of QM events.

28. Escaping the argument for P3: Let’s deny that QM events in versions of BR and COMP are type-identical. Explicitly, assume BR altered to BR′′ (ProbCond), but keep the original COMP. E.g., for W (t1)Pak(t1) : BR′′ (ProbCond) p (ES(t1) > ak) = Tr (W (t1)Pak) COMP 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 …

29. 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.

30. Keeping type-identity? If you think that this is a good plausibility argument for keeping type-identity, 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.

31. 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.