In Fredrik's recent thread: https://www.physicsforums.com/showthread.php?t=506326 some questions arose about this paper: J. B. Hartle, "Quantum Mechanics of Individual Systems", Am. J. Phys., vol 36, no 8, (1968), pp704-712. [For other readers, you'll need to have read or be able to access a copy of Hartle's paper above for what follows to make most sense.] In that paper, Hartle claims to derive statistical aspects of QM (aka Born rule) from the nonstatistical aspects of QM. Fredrik was skeptical about whether Hartle's paper really does derive the Born rule, and had various other doubts about the content of the paper. Since this is all a bit tangential to the main topic of his thread, I post my attempt at a more detailed explanation here as a separate thread. Let A be an observable with eigenvalues and eigenvectors as follows: [tex] \def\<{\langle} \def\>{\rangle} \def\half{\frac{1}{2}} \def\third{\frac{1}{3}} A |i\> = a_i |i\> [/tex] In QM, one asserts that the value of the observable A measured in state [itex]|i\>[/itex] is [itex]a_i[/itex] and that this is definite (i.e., deterministic). This is called a "nonstatistical" assertion of QM about an individual system modeled by [itex]|i\>[/itex]. Hartle's assertion (p706, lower right) is that the statistical predictions of QM can be recovered from QM's nonstatistical assertions about individual systems. His argument involves constructing an N-fold tensor product space and depends on a particular interpretation of the following operator acting on that space [see Hartle's eq(5)]: [tex] f_N^k ~:= \sum_{i_1,\cdots,i_N} |i_1,1\> \cdots |i_N,N\> \left( \frac{1}{N} \sum_{\alpha=1}^N \delta_{k i_\alpha} \right) \<i_N,N| \cdots \<i_1,1| [/tex] which Hartle calls the frequency operator. The sum over each [itex]i[/itex] parameter means we are summing over the eigenvectors of A in each (mutually independent) component space of the N-fold tensor product space. The component spaces are used to model independent repetitions of the experiment (such that the prepared state of the system may or may not all be the same). To explain why [itex]f_N^k[/itex] deserves the name "frequency operator" we consider some specific cases. For the trivial case N=1, the frequency operator reduces to [tex] f_1^k ~=~ \sum_{i_1} |i_1,1\> \left( \delta_{k i_1} \right) \<i_1,1| [/tex] or, dropping the component space label (which is unnecessary when N=1), [tex] f_1^k ~=~ \sum_{i} |i\> \delta_{ki} \<i| ~=~ |k\>\<k| [/tex] For N=2, the frequency operator is [tex] \begin{align} f_2^k &= \sum_{i_1,i_2} |i_1,1\> |i_2,2\> \Big( \half \sum_{\alpha=1}^2 \delta_{k i_\alpha} \Big) \<i_2,2| \<i_1,1| \\ &= \sum_{i_1,i_2} |i_1,1\> |i_2,2\> \half \Big( \delta_{k i_1} + \delta_{k i_2} \Big) \<i_2,2| \<i_1,1| \\ &= \half\Big( \sum_{i_1,i_2} |i_1,1\> |i_2,2\> \delta_{k i_1} \<i_2,2| \<i_1,1| ~+~ \sum_{i_1,i_2} |i_1,1\> |i_2,2\> \delta_{k i_2} \<i_2,2| \<i_1,1| \Big) \\ &= \half\Big( \sum_{i_2} |k,1\> |i_2,2\> \<i_2,2| \<k,1| ~+~ \sum_{i_1} |i_1,1\> |k,2\> \<k,2| \<i_1,1| \Big) \\ &= \half\Big( |k,1\> \<k,1| ~+~ |k,2\> \<k,2| \Big) \end{align} [/tex] where we have used the mathematical theorem that the eigenvectors of the self-adjoint operator A give a resolution of unity. Note that each term in the last line above includes an implicit identity operator on the other component space of the 2-fold tensor product, e.g., [tex] |k,1\> \<k,1| [/tex] acts as a projector on component space 1, but as the identity on component space 2. Similarly, for N=3 we find [tex] f_3^k ~=~ \third\Big( |k,1\> \<k,1| ~+~ |k,2\> \<k,2| ~+~ |k,3\> \<k,3| \Big) [/tex] and so on. Perhaps it would have been clearer to define the frequency operator as [tex] f_N^k ~=~ \frac{1}{N} \sum_{\alpha=1}^N |k,\alpha\> \<k,\alpha| [/tex] Consider now how [itex]f_3^k[/itex] acts on three independent experiments, with preparations corresponding to definite outcomes, i.e., eigenstates of A. If the sequential preparations are, respectively, [itex]|j,1\>[/itex], [itex]|k,2\>[/itex], [itex]|k,3\>[/itex] (with [itex]j\ne k[/itex]), then [itex]f_3^k[/itex] acts on the tensor product of these states as follows: [tex] \begin{align} f_3^k ~ |j,1\> |k,2\> |k,3\> &= \third\Big( |k,1\> \<k,1| ~+~ |k,2\> \<k,2| ~+~ |k,3\> \<k,3| \Big) |j,1\> |k,2\> |k,3\> \\ &= \frac{2}{3} \; |j,1\> |k,2\> |k,3\> \end{align} [/tex] giving the correct answer that [itex](2/3)[/itex] of the initial preparations were in state [itex]|k\>[/itex]. This is the justification for calling [itex]f_N^k[/itex] a "frequency operator" -- in the following sense. Hartle takes the nonstatistical aspects of QM as given. Initial states prepared in an eigenstate of A fall into this category: the result of measuring A for any of these states is definite (deterministic). [This is analogous to a sequence of trivial classical experiments in which a state is prepared (e.g., a coin laid flat on a table by Ian) and then examined (without tossing the coin) by Fred. Fred always observes precisely whatever state Ian prepared.] Now we investigate the behaviour of [itex]f_N^k[/itex] applied to the product of N copies of an arbitrary state [itex]|s\>[/itex] which is in general a linear combination of the [itex]|i\>[/itex] eigenstates of A. We denote N-fold tensor product state in an abbreviated form: [tex] |(s)^N\> ~:=~ |s,1\> |s,2\> \cdots |s,N\> [/tex] Hartle shows that [tex] \lim_{N\to\infty} \Big( f_N^k - |\<k|s\>|^2 \Big) |(s)^N\> ~=~ 0 [/tex] In other words we may say that, [itex]|(s)^\infty\>[/itex] is an eigenstate of [itex]f_\infty^k[/itex] with eigenvalue [itex]|\<k|s\>|^2[/itex]. Relying again on the nonstatistical aspects of QM, this means that if we measure the observable [itex]f_\infty^k[/itex] for the state [itex]|(s)^\infty\>[/itex] we definitely get [itex]|\<k|s\>|^2[/itex]. All that now remains is the question of whether it's still reasonable to call [itex]f_\infty^k[/itex] a "frequency" operator when applied to states that are not eigenstates of A. First, consider other well-known observables in QM, such as momentum or spin. We construct the associated operators so that they have the physically correct spectrum of eigenvalues. (Operationally, this corresponds to the set of outcomes possible from an apparatus which "measures" that observable, which has itself been designed and calibrated by preparing certain states and verifying that the apparatus always gives the same answer whenever presented with that particular prepared state). In other words, the candidate operator must give the correct answer in every classical (deterministic) case. But that's exactly what Hartle did in defining his frequency operator. He constructed it so that it gave the physically correct answers when applied to its (finite-N) eigenstates. So the answer to the question of why it's reasonable to say that [itex]f_N^k[/itex] is a "frequency" operator is essentially the same as the answer to why it's reasonable to consider [itex]i\partial_x[/itex] a "momentum" operator when acting on a space of functions: it has the physically sensible set of eigenvalues, as verified by its action on states which we expect physically should be its eigenstates, corresponding to deterministic classical cases.
Thanks for starting the thread. I have a bunch of visitors here today, so I don't have time to write a reply right now, but I will as soon as I can. Maybe later tonight.
Just want to point out that several suggested derivations of Born's rule have been made in addition to Hartle's, eg for a mini review see: On Zurek's derivation of the Born rule
Thanks for the reference. I had quick look through it and mostly they seem to analyze Zurek "envariance" technique and point out some difficulties. They give a brief nod (first page) to preceding attempts to derive the Born rule "in the context of relative-state interpretations", but they include Hartle in this category (their reference 6) which is misleading because Hartle's derivation has nothing to do relative-state/Everett, afaict. Hartle just uses the concept of relative frequency, familiar from ordinary frequentist probability theory. They do however mention a paper of E.J. Squires which criticizes Hartle's approach: E. J. Squires, "On an alleged 'proof' of the quantum probability law", Phys.Lett. A, Volume 145, Issues 2-3, 2 April 1990, Pages 67-68 Abstract: We endeavour to show that, contrary to many claims in the literature, it is not possible to deduce the quantum probability rule by considering many copies of a system and using the fact that we get a unique result when a system is in an eigenstate of the measured observable. <end> Unfortunately, I don't have access to that Squire's paper here. (Elsevier want USD 31.50 for a short 2-page article.) So I don't yet know the substance and precise context of Squire's criticism. Do you? Curiously, Zurek himself also seems to mix Hartle with MWI in quant-ph/0105127. See section 4 where (top of p39, right column) he lumps Hartle in with "no collapse MWI", which is quite unfair, imho. He also mentions the Squires paper above, adding his objection to the need for an infinite ensemble to prove the point. Personally, I'm happy if one has a Cauchy sequence with the right limit, even if the latter is physically an idealization.
If anyone else wants to join the discussion, Hartle's paper can be downloaded in pdf format from several different places. This is one of the download links. I think I understand what Hartle is trying to do a bit better now. These are some of the things that he treats as axioms, or as theorems he's allowed to use: Objects on which we do measurements are representeted by vectors in a Hilbert space. Measuring devices are represented by self-adjoint operators on that Hilbert space. N-particle states are vectors in the tensor product of N copies of the single-particle Hilbert space. If the state vector is an eigenstate of a self-adjoint operator corresponding to a particular measuring device, the result of a measurement with that device will (with probability 1) be the corresponding eigenvalue. [itex]f_N{}^k[/itex] is the operator on the space of N-particle states that corresponds to the measuring device obtained by joining N copies of the device corresponding to A with a component that calculates the frequency of the result [itex]a_k[/itex] in the results of the A measurements. When N is large, the frequency of [itex]a_k[/itex] results in a series of N measurements of A on the state |s> will be approximately the same as the result of a measurement of [itex]f_N{}^k[/itex] on the N-particle state [itex]|(s)^N\rangle =|s\rangle\otimes\cdots\otimes|s\rangle[/itex]. He uses 1-3 to derive the result [tex]f_N{}^k|(s)^N\rangle\approx |\langle k|s\rangle|^2|(s)^N\rangle\quad\text{(for large $N$)}[/tex] and 4-6 to interpret this result as an approximate derivation of the Born rule. (I'm focusing on this approximate result because it seems pointless to even look at the [itex]N\rightarrow\infty[/itex] limit until I'm satisfied that the main issue can be resolved). This isn't nearly bad as I first thought, but I still don't know what to make of it. I'm going to have to think about it some more. The argument doesn't feel like a mathematical proof. As long as it involves correspondence rules, i.e. non-mathematical statements about how things in the mathematics correspond to things in the real world, it isn't a purely mathematical argument. But maybe there's a way to state the assumptions that more clearly separates the mathematics of the theory from the correspondence rules. I also have issues with the use of tensor products, but I don't want to get into that until the main issue is completely settled.
I'm not familiar with Squires's article. I don't even know if it's one of the articles I've come across before. Zurek's approach seems to be treated more thoroughly in this article by Fedor Herbut. This article by Ulrich Mohrhoff may also be useful. I have only read a small part of Herbuts article. I read the whole Mohrhoff article, but I think it was more than 5 years ago.
Following Google Scholar's citations of Hartle's paper turned up this: C. M. Caves, R. Schack, "Properties of the frequency operator do not imply the quantum probability postulate", (Available as quant-ph/0409144) Abstract: We review the properties of the frequency operator for an infinite number of sys- tems and disprove claims in the literature that the quantum probability postulate can be derived from these properties. <end> I've only read a few pages so far, but the essence of their objection is that one cannot take the infinite-N limit in a sensible and unique way. Thus, the discussion relates to the distinction between the weak and strong laws of large numbers. (The weak law does not imply the strong.) I'll try to post some more after I digest the rest of their paper. [Edit: I finished a first pass of the Caves & Schack paper, and I'm starting to understand what goes wrong with the Finklestein-Hartle approach. Briefly, the Hilbert space becomes nonseparable in the infinite-N limit and bizarre things happen in nonseparable spaces. E.g., probability 1 does not necessarily mean certainty. I begin to acquire deeper insight into the merits of Arnold Neumaier's preference for using expectations as a starting point.]