Born Rule in Many Worlds Derived?

[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.

 

[gentzen]

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.

Maybe I misunderstood something, but my impression was that you claimed Schwinger's closed-time-path formalism would have the Born rule sort of "built in". A. Neumaier and others were just reacting to this unexpected and surprising assertion. Your assertion was surprising, because it seemed to violate "conservation of difficulty". Now you seemed to have somewhat scaled back what you assert, but at the same time try to criticize A. Neumaier for pointing out that the CPT-formalism by itself does not provide the link to experiment in the way the Born rule does for "most other formalisms".

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

It is fine if you want to interpret it in that way. But this interpretation seems to imply an ontological commitment to "microscopic events", and that commitment seems to be something in addition to the pure CPT-formalism.

 

[Fra]

I find it odd that you seek a secure foundation of a microscopic theory in a rigorous description of macroscopic devices.

As I see it, the THEORY of the microscopic world, literally LIVES(=informaiton implies in it is inferred and encoded) in the macroscopic world. I do not see this as a problem. I see it as as RELATION between scales (but there are some subtle issues in this which you can handle in different ways).

A theory of measurement without actual measurements, and without actual measurement devices is what is really odd to me.

/Fredrik

 

[WernerQH]

Maybe I misunderstood something, but my impression was that you claimed Schwinger's closed-time-path formalism would have the Born rule sort of "built in".

The split between unitary evolution and measurement has much to do with the idea that the wave function is the linchpin of quantum theory. I think that the two cannot be separated. The Born rule has been added as an afterthought (certainly for Born!), whereas it clearly belongs to the central core of the formalism (in which form whatever). The ongoing discussions about how the wave function relates to the real world has given the Born rule a peculiar status that I find distracting (making QM harder to understand).

It is fine if you want to interpret it in that way. But this interpretation seems to imply an ontological commitment to "microscopic events", and that commitment seems to be something in addition to the pure CPT-formalism.

That's right. If continuous fields evolve continuously, it remains a deep mystery how photons can be counted. But it is easy to visualize a medium as having graininess, as being composed of atoms. Likewise a quantum field can have structure that is not present in the macroscopic theory from which it has been derived by quantization. There are plenty examples in statistical field theory and condensed matter physics. It is possible to express a photon absorption coefficient (i.e. the expected number of absorptions minus the number of stimulated emissions) as a Fourier integral over the current density fluctuations in the medium (a kind of Kubo formula). That's why I said that QFT is just a machinery for calculating correlation functions.

 

[vanhees71]

We should really get rid of the historical approach to QT, again and again stressing, what's so weird about quantum theory but present it as the most successful theoretical description of everything except the gravitational interaction.

It is true that Schrödinger first thought that the wave function describes "the electron" as a "smeared particle-like object" analogously as electromagnetic waves describe "photons" as a "particle-like object" in the sense of the then much discussed socalled "wave-particle dualism". It was almost immediately clear that this original interpretation is contradicting the observed facts, and that was solved already in 1926 by Born in a footnote to his important paper on scattering theory, by introducing the probabilistic interpretation of $|\psi(t,\vec{x})|^2$, which is still considered valid today and is in accordance with all observations for nearly 100 years, and one should be aware that what we can today really observe are single particles/quanta, entangled two-particle (and multi-particle) states, etc. Even the most strange predictions of inseparability, which was the one point Einstein could never accept, are verified with an amazing accuracy and statistical significance. Thanks to the important work by Bell it's also clear that Einstein's hope for a way out, i.e., a deterministic hidden-variable theory (called "realistic" in the (in)famous EPR paper), is definitely closed, but QT is confirmed.

There is still this obsession about the apparent "unsolved" metaphysical problems. From a physical point of view, however, there are none. To put it positive: This obsession has lead to all the above mentioned stringent tests of QT and the development of amazing experimental techniques, which are now becoming a subject of engineering and the hope for ever better new devices useful for practical purposes as "quantum cryptography" and "quantum computers".

 

[Fra]

That's right. If continuous fields evolve continuously, it remains a deep mystery how photons can be counted. But it is easy to visualize a medium as having graininess, as being composed of atoms.

Even though I think we may have too separated views, it's still interesting to try to let ideas meet.

I can relate to what you write here in the following sense: I share an objection to the continuum. I think the continuum is an idealisation.

In my understanding, it corresponds to an infinitely massive, infinitely dominant observer that has unlimited information processing capacity. That "actual limit" IMO, has IMO no place in a real interation, and any deductions from a formalism relying on the ACTUAL continuum will likely not be right, just approximately exact.

Also the "approximate" continuum, from my perspective does is not encoded in the microscopic system, it's encoded in the environment, in a form of compressed statistics. This is how one can form a continuum out of a historical binary flip. We should also know from other QM effects the significance of the boundary. You can not even observer empty space without a boundary where to put detectors. So no meaningful void without a bondary.

In sense sense, if one asks for some ontological inside picture of what is going on inside the microscopic system, I share the view that it is will have some discreteness. But it's the incomplete description of the discrete phenomena from a different scale, that gives the illustion of a continuuum, or optionallly that it's easier to model with calculus. I share this view, even if we may diverge a lot on other perspectives.

It's the allowance of the ACTUAL limit, that I think has given rise to pathologies that forces us to somewhat ambigous renormalisation methods that gives me a bad stomach feeling that something just isn't right about it, not matter how far we get away with it. IMO this is related to the issue of counting, and when the counter is saturated, then what happens? (I see if from an agent picture, but the idea is that the agent actions is indistinguishable from the interactions of matter actions, it's just a angle to gain causal insight).

I can also connet to some hidden variable traits, which I mentioed before. The inseparability is a statistical fact, but the causal explanation is lacking.

/Fredrik