A Born Rule in Many Worlds Derived?

[Adrian Lee]

https://mateusaraujo.info/2021/12/08/a-satisfactory-derivation-of-borns-rule/

 

[Adrian Lee]

Many articles these years claim that they have derived it.Wonder what your thoughts on this problem are.

 

[vanhees71]

For me there's no need to derive Born's rule, because it's simply a fundamental postulate of quantum theory.

 

[Adrian Lee]

For me there's no need to derive Born's rule, because it's simply a fundamental postulate of quantum theory.

It's just some people argue that it's almost MW's last problem and if it's derived,MW would be "over the other interpretations".

 

[vanhees71]

I must admit that I never understood what MW is good for nor how it interprets quantum states. For me physical theory must clearly relate the mathematical formalism the theory is expressed into observable facts in Nature. I've not seen any convincing interpretation except the minimal statistical interpretation, i.e., a flavor of the Copenhagen Interpretations, where the state is just providing probabilities via Born's Rule (in the general sense for both pure and mixed states) for the outcome of measurements given a preparation procedure.

 

[WernerQH]

Many articles these years claim that they have derived it.

It surely depends what the derivation is based on. For a long time I too believed that Born"s rule ought to be derivable from Schrödinger's equation. But whenever I studied promising articles, the proof contained an innocent looking assumption that was equivalent to Born's rule (if it wasn't just shrouded in mathematIcs). Now I'm convinced that it is an independent ingredient of QM and even more important than the wave function. (What's observable can always be expressed using operators.)

It's just some people argue that it's almost MW's last problem and if it's derived,MW would be "over the other interpretations".

I don't consider MW an interpretation at all. It claims that Schrödinger's equation and continuous evolution is all there is to quantum theory. I can't believe that discrete events like the clicks of a Geiger counter are tricks played on us by our senses while the underlying reality evolves continuously. MWI glosses over the discrepancy with nothing but hand waving.

 

[vanhees71]

This apparently discrete events are just rapid continuous dynamics, at least according to standard quantum theory.

 

[WernerQH]

You mean there is continuous evolution from 5, 4, 3, ... down to 0 undecayed atoms? I think there is some actual granularity that theoreticians should not conceal just because differential equations are easier to work with.

 

[Fra]

I must admit that I never understood what MW is good for nor how it interprets quantum states.

I agree completely.

Many interacting observers - yes, but many worlds? What is the explanatory value and how it helps us forward?

/Fredrik

 

[Demystifier]

I must admit that I never understood what MW is good for nor how it interprets quantum states.

Is there any non-minimal interpretation of QM for which you do understand what is it good for?

 

[vanhees71]

Well, yes. I've an idea, e.g., what's behind the idea of the collapse hypothesis (as an attempt to explain dynamically, how the preparation in a quantum state comes about) or Bohmian mechanics (as a nonlocal deterministic interpretation of the quantum state) although I don't think that these examples have in any sense any advantages in comparison to the more agnostic minimal interpretation, which just states the meaning of the quantum state for the purpose of describing the objective observations made in the lab without any attempt of further "metaphysical" explanations.

 

[MathematicalPhysicist]

I agree completely.

Many interacting observers - yes, but many worlds? What is the explanatory value and how it helps us forward?

/Fredrik

Every observer with his or her own world. :-)

 

[Fra]

Every observer with his or her own world. :-)

Not own world Every observer with its own subjectively inferred imperfect expectation of the one common world.

Observer equivalence is the special case of observer democracy where all the observers evolved their views to be in tune as analogous to a Nash equilibrium. Once in tune, the views asymptotically exhibits the symmetries the traditional pardigm sees as timeless constraints.

/Fredrik

 

[MathematicalPhysicist]

Not own world Every observer with its own subjectively inferred imperfect expectation of the one common world.

Observer equivalence is the special case of observer democracy where all the observers evolved their views to be in tune as analogous to a Nash equilibrium. Once in tune, the views asymptotically exhibits the symmetries the traditional pardigm sees as timeless constraints.

/Fredrik

The "one common world" is the one which needs proof of its existence, otherwise it's as always assumed that it exists.

 

[MathematicalPhysicist]

I heard once from someone that the axiom that through two points passes one unique straight line follows from some variational principle/s.
So the axiom of one common world might follow from some other principle/s or we might abandon it.
I guess one can use some sort of possible worlds modal logics for QT.
Tried typing google for Quantum Modal Logic... I don't know if it exists but nice term like all quantum something.

 

[Fra]

The "one common world" is the one which needs proof of its existence, otherwise it's as always assumed that it exists.

The only meaning I assign to the "one common world" is "what you get from all physical observers that are in causal contact with each other". Exactly what this is in detail - the microstructure of observers and their relations - is of course what the whole game of inference is about. It´s also necessarily moving target as I think the inference process itself helps to form and a self-organisation will take place. There will never be a complete answer, because the more complex and observer gets, it´s ability to encode more complex relations increase. The ambition is IMO just to understand the abductive inference mechanisms here.

In the research I see implied from this "interpretation" is to identify the most sensible mathematical model for this, and to investigate - in the limit of low complexity of observers - how many options there is. There is some hope that the options are limited, andn that predictions may come out. The hope is that - for any given complexity bound - there may be some uniqe rules that would represent a relation of worldviews that are "optimally" compressed. Like represented by some unique mathematics. But we do not just want to find the mathematics, we also want to understand in deeper WHY its the right one.

So the vision is that once observers interact, their common reality emerges asymptotically. An idea is that, the communication implies a universal unavoidable negotiation driving evolution. So the emergent patterns here should match the particle zoo and their relations. If it does not, then the interpretation is a failure.

/Fredrik

 

[A. Neumaier]

You mean there is continuous evolution from 5, 4, 3, ... down to 0 undecayed atoms? I think there is some actual granularity that theoreticians should not conceal just because differential equations are easier to work with.

Its not much different from the continuous evolution of a discrete number of corona viruses from 0, to 1,2,3,4, to a huge number.

 

[WernerQH]

Its not much different from the continuous evolution of a discrete number of corona viruses from 0, to 1,2,3,4, to a huge number.

It's strange for a mathematician to be lacking in a precise definition of the term "continuous". :-)

 

[A. Neumaier]

It's strange for a mathematician to be lacking in a precise definition of the term "continuous". :-)

This is a physics forum, so I use the term consistent with physics usage. But I presume that the evolution of viruses is governed primarily by classical mechanics, which is continuous even in the mathematical sense.

 

[Fra]

You mean there is continuous evolution from 5, 4, 3, ... down to 0 undecayed atoms? I think there is some actual granularity that theoreticians should not conceal just because differential equations are easier to work with.

I think what's effectively continuous and what is not is observer-dependent. A binary state always has a discrete transition, but if can consider the probability for the transition of the same state by observing the context as well, it can be almost continous. But the latter description contains MORE information and thus requires a sufficiently complex observer.

Its tempting to think that there is always a suffiently complex observer that can have maximal knowledge, and that this would be the "correct" description, but this will not explain the action of the non-maximal systems! It instead leaves us with a silly situation with the wave function of the whole universe. Something no one can compute or fine tune.

/Fredrik

 

[vanhees71]

In quantum theory the evolution of the state (statistical operator) is described by a partial differential equation and thus is continuous. So are the probabilities and related expectation values concerning discrete variables like "the number of infected people" in a epidemics/pandemics simulation.

 

[WernerQH]

Of course. Averages evolve continuously and deterministically. This may have a soothing effect on some physicists. But it doesn't make discontinuities and randomness go away. We should not be misled by Schrödinger's equation to think that physical evolution is necessarily continuous and deterministic. There's ample evidence that it is fundamentally random and discontinuous. After all, it's called quantum theory, and it describes the behaviour of atoms.

 

[vanhees71]

The dynamics of quantum theory is continuous (even smooth) and it's causal. At the same time it's probabilistic and indeterministic. This is no contradiction to the fact that it describes, among all other known "things", atoms.

 

[WernerQH]

"Dynamics" is a classical concept. Schrödinger's equation alone does not fully describe quantum processes. I'd say you are so habituated to quantum theory that you can't perceive the contradiction any more. :-)

 

[Fra]

I identified two facets of the discussion here, not sure which one Werner was after but... on the second facet

In quantum theory the evolution of the state (statistical operator) is described by a partial differential equation and thus is continuous. So are the probabilities and related expectation values concerning discrete variables

I think such infinite and uncountable amounts of information is likely a fiction that works well for describing atomic phenomena from the perspective of a dominant classical environment.

In symbolic math, one can easily imagine anything, but whenever you try to actually solve, and compute something, one is typically constrained to fixed precision and fixed information processing capacity. The problem is one asks not just for a description of a small subsystem, but a betting method for action in an unknown environment. Then the non-dominant agent is I think, intutively, certainly saturated with information, and has to resort to lossy retention, and make the right choices to be sucessful. The latter is the scenariou of inside observers (cosmology) and also to find explanatory improvements of unification of interactions, even from the perspective of classical reality.

/Fredrik

 

[vanhees71]

To the contrary! If you have a small subsystem coupled to a macroscopic "environment", you don't have control and thus "trace out" the environment, i.e., we average over a lot of unknown degrees of freedom to effectively describe the "relevant observables" of the subsystem.

Even in cosmology, all we can do is to observe local observables and then extrapolate to the "universe as a whole" assuming the cosmological principle, which is amazingly successful.

 

[martinbn]

Is the Born rule derived in any interpretation?! And why do physicists (at least some) obsess about it? What is the need to have a minimal number of postulates?

 

[WernerQH]

Right. Schrödinger's equation by itself is not enough; we need the Born rule to obtain measurable quantities from the wave function. Accordingly it is not derivable from Schrödinger's equation. But it seems that MWI supporters disagree and prefer to think that Schrödinger's equation is all that quantum theory is about.

What people have qualms about is how unitary evolution and "measurements" fit together. Schwinger's action principle that led to the closed time-path formalism allows direct calculation of observable quantities. It smoothly joins unitary evolution and the Born rule in one formalism.

 

[martinbn]

Right. Schrödinger's equation by itself is not enough; we need the Born rule to obtain measurable quantities from the wave function. Accordingly it is not derivable from Schrödinger's equation. But it seems that MWI supporters disagree and prefer to think that Schrödinger's equation is all that quantum theory is about.

Isn't that about the time evolution only? MWI says that the evolution of the state is given by the Schrödinger's equation, no collapse. It doesn't say that there are no other accpects of QM. At least that is how I always understood it.

What people have qualms about is how unitary evolution and "measurements" fit together. Schwinger's action principle that led to the closed time-path formalism allows direct calculation of observable quantities. It smoothly joins unitary evolution and "measurements" in one formalism.

 

[WernerQH]

Isn't that about the time evolution only? MWI says that the evolution of the state is given by the Schrödinger's equation, no collapse. It doesn't say that there are no other accpects of QM. At least that is how I always understood it.

The other aspects are the other universes, or branches thereof. The central puzzle is: How does the wave function relate to the real world that we perceive around us?

 

[akvadrako]

But it seems that MWI supporters disagree and prefer to think that Schrödinger's equation is all that quantum theory is about.

I think in the early days of MWI, some supporters thought it was an advantage of the interpretation that the Born rule could be derived in it and not in other interpretations. But it turns out most Born rule derivations seem to work (or not work) just as well in any interpretation – it can' t be done without adding other assumptions, so why not just assume it directly? That's the approach Vaidman takes with his Born-Vaidman rule.

 

[A. Neumaier]

What people have qualms about is how unitary evolution and "measurements" fit together. Schwinger's action principle that led to the closed time-path formalism allows direct calculation of observable quantities. It smoothly joins unitary evolution and the Born rule in one formalism.

How does Schwinger's action principle lead to Born's rule?

 

[Fra]

To the contrary! If you have a small subsystem coupled to a macroscopic "environment", you don't have control

It's the small subsystem seens as an "observer" that has no control, yes. This is my point as well! Yet it has to act without full control. How are its actions caused ? Totally random? By deterministic rules? If so, which ones? Can we get insighgs from pondering about this? (I think yes of course)

This the questions want to answer, to understand interactions from the inside.

i.e., we average over a lot of unknown degrees of freedom to effectively describe the "relevant observables" of the subsystem.

This is from the perspective of the dominany, classical observer - yes. This is statistical description from a different perspective, not a causation.

Even in cosmology, all we can do is to observe local observables and then extrapolate to the "universe as a whole" assuming the cosmological principle, which is amazingly successful.

This extrapolation rules is something I think we can improve. A similar principle could be perhaps that "all agents" have actions causally chosen by a similar logic.

/Fredrik

 

[Fra]

Right. Schrödinger's equation by itself is not enough; we need the Born rule to obtain measurable quantities from the wave function.

This direction of reasoning always seemed conceptually backwards to me. Set aside historical development, how can you start with an equation, and then ask what the dependent variable means? My take on this is to try to first understand in what way the state encodes the observers predictions of the future and how it's inferred. Then ask - what self evolution is implied, when you account for the dependence of some information. I think it's this depencence, that implies the evolution. This is why I interpret the "hamiltonian" as part of the initial information as well. Although it's not the way of standard formalism. In the standard formalism it's just put in there, or considered empirical requiring no explanation.

/Fredrik

 

[WernerQH]

How does Schwinger's action principle lead to Born's rule?

It doesn't. Born's rule is needed only when you think of the wave function as the linchpin of quantum theory, as an additional step (to obtain numbers that you can compare with experiment). But with a path integral over a complete time-path you can write down expressions for all measurable quantities directly. One obtains probabilities (rather than probability amplitudes) automatically. Born's rule is built-in, and there's no need to introduce "measurement" as a separate process that somehow disrupts unitary evolution. In my view QM/QFT is just a machinery for calculating correlation functions, and the closed time-path formalism doesn't need a derivation from a set of historical axioms. It works nicely.

 

[WernerQH]

My take on this is to try to first understand in what way the state encodes the observers predictions of the future and how it's inferred.

I'm sorry, but I can't see the advantage of the views you propose. (If I understand them at all.) No offence!

 

[Fra]

I'm sorry, but I can't see the advantage of the views you propose. (If I understand them at all.) No offence!

No offence, my views are an extremal version of a sort of evolutionary information algorithmic qbism version. The supposed advantage I see, has todo with the quest of unification of forces. My view is that, if we disregard unification problems, and just wants to understand QM - as it is - then I am close to some minimalist statistical interpretation. Beacuse this interpretation makes perfect sense! but only as a limiting case of my general view. For the general case, modifcation of QM is required (this is my take on this).

/Fredrik

 

[A. Neumaier]

It doesn't. Born's rule is needed only when you think of the wave function as the linchpin of quantum theory, as an additional step (to obtain numbers that you can compare with experiment). But with a path integral over a complete time-path you can write down expressions for all measurable quantities directly. One obtains probabilities (rather than probability amplitudes) automatically. Born's rule is built-in, and there's no need to introduce "measurement" as a separate process that somehow disrupts unitary evolution. In my view QM/QFT is just a machinery for calculating correlation functions, and the closed time-path formalism doesn't need a derivation from a set of historical axioms. It works nicely.

This seems to me wishful thinking.

For example, how does one obtain the probabilities that go into Bell experiments (i.e., the relation between the photon count statistics and the incident two_photon states) from the equations deduced from the Schwinger's action principle? Or those for a Stern-Gerlach experiment?

Please point to a paper with enough details.

 

[WernerQH]

For example, how does one obtain the probabilities that go into Bell experiments (i.e., the relation between the photon count statistics and the incident two_photon states) from the equations deduced from the Schwinger's action principle? Or those for a Stern-Gerlach experiment?

There's more than one version of Schwinger's action principle. I had in mind the variant that he used in the paper in which he invented the closed time path formalism (J.Math.Phys. 2, 407):

[...] a knowledge of the transformation function referring to a closed time path determines the expectation value of any desired physical quantity [...]

It has the Born rule sort of "built in", because probability amplitudes on the forward time branch are accompanied by their complex conjugates on the backward time branch.
The probabilities that "go into Bell experiments" can of course be worked out using QED, as you surely are aware. But perhaps you are viewing it from a wrong angle, if your focus is on how a photon wave function collapses to produce a definite measurement result.

 

[stevendaryl]

It doesn't. Born's rule is needed only when you think of the wave function as the linchpin of quantum theory, as an additional step (to obtain numbers that you can compare with experiment). But with a path integral over a complete time-path you can write down expressions for all measurable quantities directly. One obtains probabilities (rather than probability amplitudes) automatically. Born's rule is built-in, and there's no need to introduce "measurement" as a separate process that somehow disrupts unitary evolution. In my view QM/QFT is just a machinery for calculating correlation functions, and the closed time-path formalism doesn't need a derivation from a set of historical axioms. It works nicely.

How in QFT do you get probabilities? I thought that the primary quantity that Feynman diagrams(for example) allow you to calculate is amplitudes.

 

[WernerQH]

How in QFT do you get probabilities? I thought that the primary quantity that Feynman diagrams(for example) allow you to calculate is amplitudes.

Of course it's easiest just to take the squared modulus of a scattering amplitude. Schwinger's closed time-path method is more complicated, because what it amounts to is computation of the product of the S-matrix with its time-reverse. For such cases where a single process is dominant it leads to the same result. But the closed time-path method also produces interference terms when more than one process is involved. It is especially useful for non-equilibrium systems.

John Cramer introduced the "transactional interpretation" using essentially the same mathematics. Probabilities (including interference effects) arise when you multiply the offer (forward) waves with the confirmation (backward) waves.

 

[A. Neumaier]

The probabilities that "go into Bell experiments" can of course be worked out using QED, as you surely are aware.

Of course it's easiest just to take the squared modulus of a scattering amplitude.

This only gives the recipes for calculating the probabilities, but not the statement that these are probabilities for detecting individual photons. Here Born's rule needs to be assumed. See, e.g., the discussion relating S-matrix entries to detection rates in Chapter 3 of Weinberg's QFT book, where he gives explicit details.

 

[WernerQH]

This only gives the recipes for calculating the probabilities, but not the statement that these are probabilities for detecting individual photons. Here Born's rule needs to be assumed.

Of course Born's rule has to enter in one way or another. That's what I've been saying all along. Unitary evolution is not the whole story.

For me Schwinger's closed time-path "recipe" has more logical coherence than the usual prescription of (1) calculating the scattering amplitude (unitary dynamics), and then (2) taking the squared modulus (Born rule, "measurement"). You are not saying that Schwinger didn't know how to apply QED, are you?

 

[A. Neumaier]

It doesn't. Born's rule is needed only when you think of the wave function as the linchpin of quantum theory, as an additional step (to obtain numbers that you can compare with experiment). But with a path integral over a complete time-path you can write down expressions for all measurable quantities directly. One obtains probabilities (rather than probability amplitudes) automatically. Born's rule is built-in,

Of course Born's rule has to enter in one way or another. That's what I've been saying all along. Unitary evolution is not the whole story.

For me Schwinger's closed time-path "recipe" has more logical coherence than the usual prescription of (1) calculating the scattering amplitude (unitary dynamics), and then (2) taking the squared modulus (Born rule, "measurement"). You are not saying that Schwinger didn't know how to apply QED, are you?

You had claimed that Born's rule is built-in in his approach. I was just responding that Schwinger's approach does not eliminate the need for Born's rule (or something implying it) in addition to the formal mathematics. Thus it is not built-in but assumed in addition.

The measurement problem - the quest to derive from first principles why the statistical expectations computed from measurement results (read from a large quantum system called detector) agree with the quantum expectations calculated from Schwinger's machinery (or its modern version given e.g., in the book by Calzetta and Hu) - remains unsolved.

 

[WernerQH]

The measurement problem - the quest to derive from first principles why the statistical expectations computed from measurement results (read from a large quantum system called detector) agree with the quantum expectations calculated from Schwinger's machinery (or its modern version given e.g., in the book by Calzetta and Hu) - remains unsolved.

That depends on what you accept as first principle(s). I'm inclined to accept the closed time-path formalism as "the" principle of quantum theory, and be happy about the marvelous agreement with experiments. Asking the question why that is so is like asking why the speed of light is the same in all reference frames.

 

[vanhees71]

There's more than one version of Schwinger's action principle. I had in mind the variant that he used in the paper in which he invented the closed time path formalism (J.Math.Phys. 2, 407):

It has the Born rule sort of "built in", because probability amplitudes on the forward time branch are accompanied by their complex conjugates on the backward time branch.
The probabilities that "go into Bell experiments" can of course be worked out using QED, as you surely are aware. But perhaps you are viewing it from a wrong angle, if your focus is on how a photon wave function collapses to produce a definite measurement result.

The closed-time-path formulation (aka Schwinger-Keldysh formalism; when Keldysh was present, it has been better to call it only Keldysh formalism though) is just a method to calculate directly expectation values, taken with respect to a given statistical operator rather than transition amplitudes/S-matrix elements. The definition of expectation values in terms of the quantum-mechanical formalism, however, uses Born's rule to interpret the states, represented by statistical operators. So also with the closed-time-path action functional(s) you can't derive Born's rule but you use it to define the expectation values you calculate.

 

[Fra]

Asking the question why that is so is like asking why the speed of light is the same in all reference frames.

Conincidently I ask this question as well, and I think a potential explanation is not unrelated to the probability question.

The qbist stance is that not confuse descriptive and guiding/betting/normative probabilities.

​

Quantum-Bayesian Coherence: The No-Nonsense Version​

"In the Quantum-Bayesian interpretation of quantum theory (or QBism), the Born Rule cannot be
interpreted as a rule for setting measurement-outcome probabilities from an objective quantum
state. But if not, what is the role of the rule? In this paper, we argue that it should be seen as
an empirical addition to Bayesian reasoning itself. Particularly, we show how to view the Born
Rule as a normative rule in addition to usual Dutch-book coherence."
-- https://arxiv.org/abs/1301.3274

The "problem" in this view, translates to, how come and why, does the descriptive probabilities we measure in physics labs, coincide with the "expectations" that are consistent with the normative betting probabilities of the classical dominant observer/agent?

ie. how come the agents GUESS based on incomplete information, are right? This can be translated into the agent-domain, and one can ask. Does there always exists an agent (with some microstructure) that meets this conditon? IF so, why? or why not?

One could take the view that we why just don't know, and we can settle with that it works. But if we have an idea of why this works, and that it's not a conincidence, then this would likely be a great help in the quests of unifiying forces.

/Fredrik

 

[A. Neumaier]

That depends on what you accept as first principle(s). I'm inclined to accept the closed time-path formalism as "the" principle of quantum theory, and be happy about the marvelous agreement with experiments. Asking the question why that is so is like asking why the speed of light is the same in all reference frames.

The CTP formalism does not say anything about measurement, it talks only about computing N-point functions. Why these N-point functions give expectation values of measurement results is not touched at all by this formalism.

It is an unsolved problem in quantum statistical physics to show that an average over readings from a macroscopic device fed with particle input actually agrees with certain 1-point or 2-point functions. This is the step simply assumed by postulating Born's rule without any argument beyond the marvelous agreement with experiments.

 

[vanhees71]

Yes, and then you compare the predictions based on these assumptions (including Born's rule) with observations in nature. So far there's no contradiction between QT and experiments known. That's why QT is considered the most successful physical theory ever.

 

[WernerQH]

The CTP formalism does not say anything about measurement, it talks only about computing N-point functions. Why these N-point functions give expectation values of measurement results is not touched at all by this formalism.

What you see as a defect of the closed-time-path formalism I see as its biggest virtue. It avoids the discussion of "measurement" and its strange interplay with unitary evolution. It handles reversible (microscopic) and irreversible processes ("measurements", "detection events") on the same footing. One needs quantum theory to understand how the detectors in the Bell-type experiments work. I find it odd that you seek a secure foundation of a microscopic theory in a rigorous description of macroscopic devices. As John Bell has argued, there should be no place for the term "measurement" in the foundations of QM. It does sound silly to phrase the discussion of nuclear reactions in the interior of the sun (for example) in terms of "state preparation" and "measurement".

I don't find the N-point functions as mysterious as they appear to you. They can and should be seen as describing the correlations between microscopic events.

Our diametrically opposed views about the "first principles" on which quantum (field) theory should be based seems to make further discussion pointless. It would certainly leave the scope of the current thread.