aiming to complete Everett's derivation of the Born Rule

[George Kahrimanis]

Hello Everything. I have a proposal for a common-sense justification of the Born Rule for QM. The idea was motivated with the Many-World Interpretations in mind, but it also works for QM-with-collapse, if that is ever found to be true.

It would be great if you respond with any comment, objection, contribution, or question. Or you can direct me to another discussion forum.

My current draft of the Introduction is at the following link (to save "bandwidth"):
https://drive.google.com/file/d/1CE_qkit5PnS-rzKKmlOoDReBJVN1T0kA/view?usp=sharing

To give you an idea, I paste here just the Abstract and the first subsection of the Introduction.

~~~~

An argument for workability of QM leads to the Born Rule, for QM without collapse and for QM with collapse

George Kahrimanis [, ...]
6 April 2022, incomplete work

ABSTRACT
Any interpretation of QM without collapse (a.k.a. a MWI) crucially needs to produce (not assume) an Everettian analogue of the Born Rule, indispensable not only in practical decisions but also for testing a theory. Related proposals have been controversial. The proposal introduced here is based on an argument for workability of QM and on the old notion of Moral Certainty (formulated by Jean Gerson, cited by Descartes and many others). There are consequences for the foundations of decision theory because chance is undefined for any single outcome, so that Maximisation of Expected Utility is meaningless as a fundamental rational rule, therefore a different decision theory is needed.

1- INTRODUCTION

1.1- Comparison with other derivations of the Born Rule, either in MWI or with collapse

The present study is based on an assessment (not an assumption, strictly speaking) regarding workability of QM (its usability and testability); that is, an argument for workability is presented and the assessment is up to the reader. It avoids a tacit assumption of certain derivations in MWI, developments of the one by [Deutsch 1999], declaring the utility of a bet as a single value, rather than a pair (corresponding to a buying value and a selling value) or an interval -- however, an Everettian agent may well be unwilling to admit a single value, in view of the diversity of outcomes in branching futures. Despite this disagreement, we share an essential common trait: we address the problem outside of pure epistemology, by studying how QM can be a guide to practical applications. Another difference is that the present study is based solely on the status of QM as a workable theory, but Deutsch's derivation also introduces claims about rational behaviour (with which I agree, except for the one mentioned above).

Other derivations not assuming collapse (for example, Zurek's), nonetheless invoke the concept of probability in the interpretation, on the basis of various arguments [Vaidman 2020]. In contrast, the present study adopts a restriction: probability proper will be considered only for outcomes of a randomising process. (It is not enough to know that a black box contains just ten black and ten white balls, or that there are only four aces in a deck of fifty two cards: the cards must be shuffled and the balls stirred, with specifications tailored to the game.) In a single-world interpretation assuming collapse, randomisation is a required assumption (albeit derided as "God plays dice") so that we may legitimately speak of probability; in a MWI though, randomisation makes no sense. Therefore the present study does not invoke a ready concept of probability; it rather discovers what quantum-mechanical quasi-probability is (and what it is not). The results are relevant also to the interpretation of non-QM probability, regardless if it may be ultimately based on QM.

There are derivations of the Born Rule assuming collapse with randomisation, along with some special assumption. (The first such derivation was Gleason's theorem, assuming "non-contextuality" of measurements; for references, see [Vaidman 2020] and [Masanes, Galley, Müller].) These special assumptions are deemed more plausible than assuming the Born Rule directly, because they are qualitative properties rather than quantitative ones; nonetheless any special assumption needs justification, whether on experimental grounds or by some theoretic argument. The present study shows that we can replace both randomisation and the additional special assumption by workability. So the Born Rule is derived from workability alone, whether we assume collapse or not.

1.2- About Moral Certainty

[...]

[Lawrence Crowell]

This is an appeal to some sort of imperative that demands the Born Rule because the counterfactual lack this certainty. This is a sort of "It must be true" type of argument. 

LC

[George Kahrimanis]

On Friday, April 8, 2022 at 3:19:08 AM UTC+3 Lawrence Crowell wrote:

This is an appeal to some sort of imperative that demands the Born Rule because the counterfactual lack this certainty. This is a sort of "It must be true" type of argument.

Thanks for the comments! I wonder though, do you agree with my criticisms of previous proposals for deriving the Born Rule, or are you undecided? I will challenge you (and you all) on this matter, later in this message.

First, a correction: I have not referred to counterfactuals (I think that you meant "alternatives") but now that you mention them, I may have implied one:
"If QM were not a workable theory, we would have no direct, experimental clue that it is a fundamental theory in physics".
(Not the typical use of a counterfactual, which is in an "if..." clause, as in "If I was a rich man...".)

What I say is not exactly
> "It must be true"
but rather
"Although I cannot be certain, it seems to be in my interests to form this assessment now, when I decide how to act in the present situation".

If you find this argument too loose: I have pointed out that it is the same kind of argument that a judge uses to form a decision based on the evidence, or an engineer uses, to trust the theory of real numbers, for her project.

My aim has been to complete Everett's argument, which I outline next. Imagine that we repeat the same trial N times, and we record the ratio {statistical "frequency") r of one among the possible outcomes (eigenstates). Conventional QM assigns a probability R for this outcome, so we need an explanation why r SEEMS to approach R in the long run (though we know that in very many worlds it will not be so!). Everett noted that, for any positive real ε (however small), the measure of all "outlier" sequences, that is: for which r is outside
[R-ε, R+ε],
is small, with limit zero as N increases to infinity. However, a problem remains: why "small measure" or "vanishing measure" have any significance in the interpretation of QM? My proposal answers this question, finding an argument about "small measure" within the reasoned assessment that QM is a workable theory.

Here is my challenge to you. I ask you if you agree with either of the following two proposals (for deriving the Born Rule in a MWI).

First, Deutsch's (1999) proposal, here in a simplified version. Imagine a simulated tossing of a fair coin, using a qubit instead of a coin, with which you either win or lose one dollar. If this bet has a definite, single value to you (presumably, by some kind of intuitive averaging over possible futures) it will necessarily be zero, for symmetry reasons. Caveat: Deutsch points out that we do not derive probability strictly speaking. I accept the reasoning, but not the premise: I am uncomfortable with averaging my future selves, and there is no direct rationale why I SHOULD do so. So, what do you think?

Second (and last), proposals such as Zurek's are of the following pattern (here I reuse the previous example): I am uncertain about the outcome, and I expect the theory to give me some clue, which will be probability -- what else? For symmetry reasons, the probability here must be 1/2. My objection is that there is no randomisation in MWI (no shuffling, stirring, or God playing dice) so that the use of probability is not rationally justified. Again I ask for your opinion.

Clarification. Instead of probability proper, I derive the following. With regard to any given application, an Everettian agent may expect "with moral certainty" (remember the judge and the engineer!) that statistical frequency in the long run will be as close to the Born probability as one needs it to be (in the particular application). Some people may think "po-tah-toes, pot-eight-os", but at some level of thinking this is the crucial issue. In particular, a serious consequence for decision theory results from failing to find any rationale for probability proper!

George K.

[Lawrence Crowell]

The closest there is to a proof of Born's rule is Gleason's theorem. Born's rule in one sense make a lot of sense, but as yet there is no airtight proof of it. I will try to respond more later.

LC

[Brent Meeker]

On 4/12/2022 1:41 PM, George Kahrimanis wrote:

On Friday, April 8, 2022 at 3:19:08 AM UTC+3 Lawrence Crowell wrote:

This is an appeal to some sort of imperative that demands the Born Rule because the counterfactual lack this certainty. This is a sort of "It must be true" type of argument.

Thanks for the comments! I wonder though, do you agree with my criticisms of previous proposals for deriving the Born Rule, or are you undecided? I will challenge you (and you all) on this matter, later in this message.

First, a correction: I have not referred to counterfactuals (I think that you meant "alternatives") but now that you mention them, I may have implied one:
"If QM were not a workable theory, we would have no direct, experimental clue that it is a fundamental theory in physics".
(Not the typical use of a counterfactual, which is in an "if..." clause, as in "If I was a rich man...".)

What I say is not exactly
> "It must be true"
but rather
"Although I cannot be certain, it seems to be in my interests to form this assessment now, when I decide how to act in the present situation".

If you find this argument too loose: I have pointed out that it is the same kind of argument that a judge uses to form a decision based on the evidence, or an engineer uses, to trust the theory of real numbers, for her project.

My aim has been to complete Everett's argument, which I outline next. Imagine that we repeat the same trial N times, and we record the ratio {statistical "frequency") r of one among the possible outcomes (eigenstates). Conventional QM assigns a probability R for this outcome, so we need an explanation why r SEEMS to approach R in the long run (though we know that in very many worlds it will not be so!). Everett noted that, for any positive real ε (however small), the measure of all "outlier" sequences, that is: for which r is outside
[R-ε, R+ε],
is small, with limit zero as N increases to infinity. However, a problem remains: why "small measure" or "vanishing measure" have any significance in the interpretation of QM? My proposal answers this question, finding an argument about "small measure" within the reasoned assessment that QM is a workable theory.

Here is my challenge to you. I ask you if you agree with either of the following two proposals (for deriving the Born Rule in a MWI).

First, Deutsch's (1999) proposal, here in a simplified version. Imagine a simulated tossing of a fair coin, using a qubit instead of a coin, with which you either win or lose one dollar. If this bet has a definite, single value to you (presumably, by some kind of intuitive averaging over possible futures) it will necessarily be zero, for symmetry reasons. Caveat: Deutsch points out that we do not derive probability strictly speaking. I accept the reasoning, but not the premise: I am uncomfortable with averaging my future selves, and there is no direct rationale why I SHOULD do so. So, what do you think?

Second (and last), proposals such as Zurek's are of the following pattern (here I reuse the previous example): I am uncertain about the outcome, and I expect the theory to give me some clue, which will be probability -- what else? For symmetry reasons, the probability here must be 1/2. My objection is that there is no randomisation in MWI (no shuffling, stirring, or God playing dice) so that the use of probability is not rationally justified. Again I ask for your opinion.

Physics doesn't care about "rationally justified", only about empirically justified.  Both your examples suffer from choosing the simplest case where symmetry can be invoked.  But how you know, or assume, they are symmetric seems to already rely on the QM of qubits.  Once you've assumed the Hilbert space structure of QM, then Gleason's theorem essentially forces the Born rule (correct me if I'm wrong, but I think the theorem has been extended to the two-dimensional case).

Clarification. Instead of probability proper, I derive the following. With regard to any given application, an Everettian agent may expect "with moral certainty" (remember the judge and the engineer!) that statistical frequency in the long run will be as close to the Born probability as one needs it to be (in the particular application).

I think the problem is that MWI (but not Everett) assume all outcomes are equally realized.  So how does a probability become assigned to them, what does it mean.  We're told it's the probability of finding ourself in a particular world...but that seems very much like "collapse of the wave-function".  It introduces the same problems of exactly when and where does it happen; with only the advantage that consciousness is not understood in detail so the mystery can be push off.  Decoherence has gone part way in solving the when/where/what basis questions, but only part way.

Brent

Some people may think "po-tah-toes, pot-eight-os", but at some level of thinking this is the crucial issue. In particular, a serious consequence for decision theory results from failing to find any rationale for probability proper!

George K.

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[George Kahrimanis]

Thanks for the comments!

On Wednesday, April 13, 2022 at 8:55:48 PM UTC+3 meeke...@gmail.com (Brent) wrote:

Physics doesn't care about "rationally justified", only about empirically justified.

I admit that I have carried the subject to philosophy of physics, but only because this kind of subject cannot be addressed with just experiments. On the other hand, do you admit that your comment needs qualifications? You cannot possibly mean that we just obtain the Born Rule from experiments, and that this is all we care about. Did Einstein make any sense, complaining about who plays dice during a measurement? What was the point of attempts to introduce hidden variables, even at the cost of non-locality? What is the point of MWI, then? I suppose that you are a careful thinker, and if you object to MWI it would be on some rational grounds.

  Both your examples suffer from choosing the simplest case where symmetry can be invoked.

So that you would not be distracted from the basic issue: do you agree with always averaging over future selves (Deutsch) or do you take it for granted that the theory provides probability (Zurek and others)? Or none of the above? Correct me if I am wrong, but I gather that your responce is "who cares", or "po-tah-toes, pot-eight-os".

Once you've assumed the Hilbert space structure of QM, then Gleason's theorem essentially forces the Born rule (correct me if I'm wrong, but I think the theorem has been extended to the two-dimensional case).

I shall respectfully correct you, but not on the question you ask, because I cannot remember now, and I admit that I do not care to look it up (but I explain why). Gleason's theorem also requires the assumption of randomisation (God plays dice) and the assumption of non-contextuality of measurements. The reason I do not care about it any more is that I do not favour these two assumptions (especially the first one).

I think the problem is that MWI (but not Everett) assume all outcomes are equally realized.  So how does a probability become assigned to them, what does it mean.

I agree, and I am glad you are critical on this point.  I am intrigued, though, by the caveat "but not Everett": can you explain, please?

We're told it's the probability of finding ourself in a particular world...but that seems very much like "collapse of the wave-function".  It introduces the same problems of exactly when and where does it happen; with only the advantage that consciousness is not understood in detail so the mystery can be push off.

I emphasise that I have no comment on the above, because I do not endorse probability strictly speaking but only on an "as-if" basis. (I agree with Deutsch on this narrow point -- Physics doesn't care, really?)

Decoherence has gone part way in solving the when/where/what basis questions, but only part way.

I guess you refer to the theoretical possibility of the environment occasionally failing to "decohere" the state. Here is one of the approximations that are required in the translation from objective, deterministic QM (without collapse) to the application of QM in the world of experience (with apparent collapse). Surely this approximation must be scrutinised, I agree.

George K.

[George Kahrimanis]

On Wednesday, April 13, 2022 at 8:55:48 PM UTC+3 meeke...@gmail.com (Brent) wrote:

Decoherence has gone part way in solving the when/where/what basis questions, but only part way.

As I wrote at the end of my first reply to your message, I share your concern about decoherence but I see the glass as half-full; that is, with a little more subtlety I hope that the matter can be formulated in clear terms.

Surely collapse is easier to handle as a general concept (except, on the other hand, that it requires new dynamics). I forgot to mention that my argument for deriving the Born Rule works with collapse, too -- so it is an alternative to Gleason's theorem.

Here I define colapse as an irreversible process, violating unitarity of course, and I keep it separate from randomisation. The latter means that each outcome is somehow randomised -- an assumption we can do without.

Collapse can also be described in a many-world formulation! It differs from the no-collapse MWI only in being irreversible. My argument in outline is
1. assessment that MWI-with-collapse is workable;
2. therefore, outcomes of small enough measure can be neglected in practice;
3. now Everett's argument can proceed, concluding that the Born Rule is a practically safe assumption (to put it briefly).

So I have replaced two assumptions of Gleason's theorem, randomisation and non-contextuality, by the assessment of workability only.

If you don't feel comfortable yet with formulating collapse in a many-world setting, let us also assume randomisation (God plays dice), for the sake of the argument, in a single-world formulation. That is, we ASSUME the existence of probability; then the previous argument just guarantees that this probability follows the Born Rule.

Of course I favour the first version of the argument, using the many-world formulation of collapse, to avoid the "God plays dice" nightmare.

Thanks for the comments so far, because they stirred my thinking and motivated fresh ideas, some of which I hope will prove helpful and worth discussing, if and when they mature.

George K.