Many Worlds versus Decoherent Histories

[Morbert]

TL;DR Summary

Continuing a conversation on Many Worlds and Decoherent histories in this dedicated thread.

Isn't DH the same as Everett? Reading James B. Hartle it's hard not to conclude that

There are important differences. E.g. Given a set of decoherent histories of a closed system, both Both MW and DH would resolve a pure initial state into orthogonal branches corresponding to the histories. But MW says all histories occur, while DH says only one history occurs.

If it doesn't define how one 'real' history occur, then it's just semantics?

The interpretations have very different ontological implications. In one, the branches of a wavefunction are real. In the other, the branches are not. These differences are substantive and not trivial. They lead to very different objections being levied against each interpretation. E.g. Probabilities are intuitive and straightforward in DH (they denote the likelihood that a history occurs). In MW, one of the major challenges is making sense of probabilities when all histories occur ( https://link.springer.com/article/10.1007/s10701-014-9862-5 ).

 

[Quanundrum]

Summary:: Continuing a conversation on Many Worlds and Decoherent histories in this dedicated thread.

The interpretations have very different ontological implications. In one, the branches of a wavefunction are real. In the other, the branches are not. These differences are substantive and not trivial. They lead to very different objections being levied against each interpretation. E.g. Probabilities are intuitive and straightforward in DH (they denote the likelihood that a history occurs). In MW, one of the major challenges is making sense of probabilities when all histories occur ( https://link.springer.com/article/10.1007/s10701-014-9862-5 ).

I am well aware of MWI's longstanding problems with probability, but doesn't "DH" really just skirt around this issue by saying that "There's some indeterminism that randomly selects one real world"?

 

[Morbert]

I am well aware of MWI's longstanding problems with probability, but doesn't "DH" really just skirt around this issue by saying that "There's some indeterminism that randomly selects one real world"?

I guess it depends on what you expect from a physical theory. DH simply says that, given a set of decoherent histories, the system of interest is characterised by one history and not the others. By carrying out some experiment whose possible outcomes are highly correlated with the possible histories, we learn which history characterises the system, and QM informs our expectations in the form of probabilities.

It does not offer an explanatory mechanism for one history obtaining over the others, but should we expect this obligation to be met? What about dynamics? Should we also expect an explanatory mechanism for one Hamiltonian obtaining over an alternative Hamiltonian?

 

[A. Neumaier]

Probabilities are intuitive and straightforward in DH (they denote the likelihood that a history occurs).

This view of probabilities is not experimentally accessible as we can observe only a single history of the universe, irrespective of its ''likelihood''.

Hence the notion of probability in the decoherent histories (DH) framework has nothing to do with the notion of probability used everywhere in physics. The latter features probabilities as approximating the relative frequencies in the single history known to us, not relaive frequencies of alternative histories.

 

[A. Neumaier]

I believe Decoherent Histories also shares this virtue, and is as readily generaliseable to quantum gravity.

https://arxiv.org/abs/1803.04605

This is a paper by Gell-Mann and Hartle written in 1989 (though posted only in 2018). A year later, Hartle wrote this paper (also posted in 2018), where he cites the above paper as ''Gell-Mann and Hartle (1990)'' and puts the DH approach squarely into the context of Everett's MWI, though he calls it Post-Everett. On p.5 he says:

The quantum framework for cosmology I shall describe in Section II has its origins in the work of Everett and has been developed by many. Especially taking into account its recent developments, notably the work of Zeh (1971), Zurek (1981, 1982), Joos and Zeh (1985), Griffiths (1984), Omnes (1988abc, 1989) and others, it is sometimes called the post-Everett interpretation of quantum mechanics. I shall follow the development in Gell-Mann and Hartle (1990).

Everett’s idea was to take quantum mechanics seriously and apply it to the universe as a whole. He showed how an observer could be considered part of this system and how its activities — measuring, recording, calculating probabilities, etc. — could be described within quantum mechanics. Yet the Everett analysis was not complete. It did not adequately describe within quantum mechanics the origin of the “classical domain” of familiar experience or, in an observer independent way, the meaning of the “branching” that replaced the notion of measurement.

Thus Hartle sees DH just as a refinement of MWI.

 

[Morbert]

This view of probabilities is not experimentally accessible as we can observe only a single history of the universe, irrespective of its ''likelihood''.

Hence the notion of probability in the decoherent histories (DH) framework has nothing to do with the notion of probability used everywhere in physics. The latter features probabilities as approximating the relative frequencies in the single history known to us, not relaive frequencies of alternative histories.

Roland Omnes remarks that probabilities in consistent histories are used in 'the construction of a language describing the facts of empirical physics' and 'some probabilities will become empirical, with the meaning of predicting the frequencies of random experimental events' [1]. Gell-Mann et al were motivated to develop decoherent histories precisely because we don't have a set of universes to subject to measurements. If a probability is not empirically accessible, but facilitates a physicist in reasoning about a system or recovering some intuition, it is still warranted imho.

And of course, probabilities assigned to histories don't prevent us from predicting relative frequencies. Given a set of decoherent histories, for every history $\alpha$ (represented by the operator $C_\alpha$) we can construct an effective density matrix [2] $$\frac{C_\alpha\rho C^\dagger_\alpha }{\mathrm{Tr}\left[C_\alpha\rho C^\dagger_\alpha\right]}$$ that can be used to predict relative frequencies of outcomes of some future trials, conditioned on that history occurring (and by extension relative frequencies not conditioned on a specific history).

[1] https://press.princeton.edu/books/paperback/9780691036694/the-interpretation-of-quantum-mechanics
[2] https://arxiv.org/abs/gr-qc/9210010

 

[A. Neumaier]

If a probability is not empirically accessible, but facilitates a physicist in reasoning about a system or recovering some intuition, it is still warranted imho.

And of course, probabilities assigned to histories don't prevent us from predicting relative frequencies. Given a set of decoherent histories, for every history $\alpha$ (represented by the operator $C_\alpha$) we can construct an effective density matrix [2] $$\frac{C_\alpha\rho C^\dagger_\alpha }{\mathrm{Tr}\left[C_\alpha\rho C^\dagger_\alpha\right]}$$ that can be used to predict relative frequencies of outcomes of some future trials, conditioned on that history occurring (and by extension relative frequencies not conditioned on a specific history).

Taking your formula as defining relative frequencies inside the single observed history is an additional axiom breaking general coordinate invariance. It cannot be proved from the probability concept for histories, since knowing the distribution of a random object (here a history) tells nothing at all about details inside a particular individual realization. To claim that it does requires making assumptions equivalent to those made in the thermal interpretation.

 

[Morbert]

Taking your formula as defining relative frequencies inside the single observed history is an additional axiom breaking general coordinate invariance. It cannot be proved from the probability concept for histories, since knowing the distribution of a random object (here a history) tells nothing at all about details inside a particular individual realization. To claim that it does requires making assumptions equivalent to those made in the thermal interpretation.

If a physicist simply used the effective density matrix to directly compute relative frequencies, they would make incorrect predictions because they don't know which member of the set of histories $\{\alpha\}$ occurred. They have to treat the probabilities as conditional. If they wanted to use the effective density matrices to e.g. compute the relative frequency of some future experimental outcome $\epsilon$, not conditioned on a specific history, they would instead use the expression
$$p(\epsilon) = \sum_\alpha p(\epsilon|\alpha)p(\alpha) = \sum_\alpha \frac{\mathrm{Tr}\left[ \Pi_\epsilon C_\alpha \rho C^\dagger_\alpha \Pi^\dagger_\epsilon \right]}{\mathrm{Tr}\left[ C_\alpha \rho C^\dagger_\alpha \right]}\mathrm{Tr}\left[ C_\alpha \rho C^\dagger_\alpha \right] = \mathrm{Tr}\left[\rho\Pi_\epsilon\right]$$
In order for a particular history $\alpha$ to be considered the 'observed history', the physicist would have to carry out some experiment whose outcomes correlate with the possible histories. I.e. Whose outcomes constitue a record of the histories (see the discussion around equation 4.11 in [1], or [2]). If they do this, then they can directly use the effective density matrix of the observed history. But this is just the DH equivalent of state collapse. The physicist, upon observing a record of history $\alpha$, updates their probabilities accordingly.

[1] https://arxiv.org/pdf/gr-qc/9210010.pdf
[2] https://arxiv.org/abs/1608.04145

 

[A. Neumaier]

They have to treat the probabilities as conditional. [...]
In order for a particular history $\alpha$ to be considered the 'observed history', the physicist would have to carry out some experiment whose outcomes correlate with the possible histories. I.e. Whose outcomes constitue a record of the histories (see the discussion around equation 4.11 in [1], or [2]). If they do this, then they can directly use the effective density matrix of the observed history. But this is just the DH equivalent of state collapse. The physicist, upon observing a record of history $\alpha$, updates their probabilities accordingly.

You are considering the future, but this does not address my concerns. Why should the probabilities already observed in the past in the single available history (i.e, all statistical information we already know about our particular universe) have anything to do with the probabilities of alternative histories defined by the DH approach?

 

[A. Neumaier]

[2] https://arxiv.org/abs/1608.04145

In this paper Hartle assumes on p.3 Born's rule from the start:

We assume that probabilities for records are given by Born’s rule${}^3$. [Where footnote 3 says:]
Thus we are not attempting here to derive Born’s rule from some other assumption.

But an interpretation of quantum mechanics applicable to the universe must be able to obtain Born's rule from the unitary description, the only dynamics that exists in the quantum universe! This is the essential gap in DH.

 

[Halc]

Both Both MW and DH would resolve a pure initial state into orthogonal branches corresponding to the histories. But MW says all histories occur, while DH says only one history occurs.

When does this occur? A device measures a decay and triggers the reaction of the killing of the cat. If DH says that only one history occurs (the cat dead say), then the cat cannot be in superposition of dead and alive, but the lab guy outside the box measures this state. Is the real dead cat able to interfere with the nonexistent live cat, or, since the lab guy has yet to measure the system, is the cat really in superposition still?

 

[Morbert]

You are considering the future, but this does not address my concerns. Why should the probabilities already observed in the past in the single available history (i.e, all statistical information we already know about our particular universe) have anything to do with the probabilities of alternative histories defined by the DH approach?

I'm not sure I understand the issue you are raising so this might not address it but:

Consider a set of histories, each regarding properties at times from $t_1$ to $t_n$. The operator corresponding to history $\alpha$ is a chain of projectors
$$C_\alpha = \Pi_{\alpha_1}(t_1)\Pi_{\alpha_2}(t_2)\Pi_{\alpha_3}(t_3)\dots\Pi_{\alpha_n}(t_n)$$
Say we are interested in the probability of an experimental outcome $\epsilon$ at time $t_1\lt t \lt t_n$. We could compute
$$p(\epsilon) = \sum_\alpha p(\epsilon|\alpha)p(\alpha) = \sum_\alpha \frac{\mathrm{Tr}\left[ C_{\alpha'} \rho C^\dagger_{\alpha'} \right]}{\mathrm{Tr}\left[ C_\alpha \rho C^\dagger_\alpha \right]}\mathrm{Tr}\left[ C_\alpha \rho C^\dagger_\alpha \right] = \mathrm{Tr}\left[\rho\Pi_{\epsilon}(t)\right]$$
where
$$C_{\alpha'} = \Pi_{\alpha_1}(t_1)\Pi_{\alpha_2}(t_2)\Pi_{\alpha_3}(t_3)\dots\Pi_{\epsilon}(t)\dots\Pi_{\alpha_n}(t_n)$$

[edit]

But an interpretation of quantum mechanics applicable to the universe must be able to obtain Born's rule from the unitary description, the only dynamics that exists in the quantum universe!

I'll address this tomorrow

 

[Morbert]

When does this occur? A device measures a decay and triggers the reaction of the killing of the cat. If DH says that only one history occurs (the cat dead say), then the cat cannot be in superposition of dead and alive, but the lab guy outside the box measures this state. Is the real dead cat able to interfere with the nonexistent live cat, or, since the lab guy has yet to measure the system, is the cat really in superposition still?

A consistent historian would construct a pair of histories:
i) particle decays and then cat dies
ii) particle does not decay and then cat does not die

And compute the probabilities for each. The histories are decoherent with each other so computing probabilities is possible.

 

[Morbert]

In this paper Hartle assumes on p.3 Born's rule from the start:

But an interpretation of quantum mechanics applicable to the universe must be able to obtain Born's rule from the unitary description, the only dynamics that exists in the quantum universe! This is the essential gap in DH.

In DH/CH, probabilities are introduced as measures defined on subspaces of a Hilbert space, and quantum states are introduced as trace class operators related to probabilities via Gleason's theorem. States are not introduced as ontic states evolving unitarily so I do not see any issue with this approach.

 

[A. Neumaier]

In DH/CH, probabilities are introduced as measures defined on subspaces of a Hilbert space, and quantum states are introduced as trace class operators related to probabilities via Gleason's theorem. States are not introduced as ontic states evolving unitarily so I do not see any issue with this approach.

If they are not ontic, what is ontic? Just the single realized history?

 

[A. Neumaier]

In DH/CH, probabilities are introduced as measures defined on subspaces of a Hilbert space

This does not yet equip them with a physical meaning.

How is an (obviously ontic) detector described in DH? And how does DH describe the (equally ontic) single measurement result obtained in an experiment?

 

[Halc]

When does this occur? A device measures a decay and triggers the reaction of the killing of the cat. If DH says that only one history occurs (the cat dead say), then the cat cannot be in superposition of dead and alive, but the lab guy outside the box measures this state. Is the real dead cat able to interfere with the nonexistent live cat, or, since the lab guy has yet to measure the system, is the cat really in superposition still?

A consistent historian would construct a pair of histories:
i) particle decays and then cat dies
ii) particle does not decay and then cat does not die

Question was not answered. I asked when this occurs. You said:

MW says all histories occur, while DH says only one history occurs.

At what point does the dead cat history become the one that actually occurs? The cat measures the poison bottle and dies. The Geiger device measures the decay and releases the poison. The lab guy opens the box hours later. That's three measurements, among others.
Under MWI, as you say, 'the cat' (shortly after the radioactive decay event) is always in superposition of live and dead. Both histories occur. You assert that under DH, only one history of the two above 'occurs'. At what point does this happen and the alternate live-cat history degenerates into a state of 'does not occur' or 'is no longer a potential measurement outcome'?

 

[Morbert]

This does not yet equip them with a physical meaning.

How is an (obviously ontic) detector described in DH? And how does DH describe the (equally ontic) single measurement result obtained in an experiment?

The properties of the detector that record measurement outcomes would be described in the same manner as properties of the measured system. E.g. If we want to measure some observable $A(t) = \sum_i a_i\Pi_{a_i}(t)$ of a quantum system $Q$, the measurement result $a_i$ has a probability
$$p(a_i) = \mathrm{Tr}\left[\rho_Q \Pi_{a_i}(t)\right]$$
and the measuring apparatus $M$ produces a datum $\epsilon_i$ (e.g. a specific dial position) with probability
$$p(\epsilon_i) = \mathrm{Tr}\left[ \rho_M\otimes\rho_Q \Pi_{\epsilon_i}(t) \right]$$
There is a universal correspondence[1] between the datum $\epsilon_i$ and the result $a_i$, which is why the simpler expression for $p(a_i)$ will predict the frequencies in the data produced by the measurement device, and we do not typically have to worry about the more complicated $\rho_M$.

If they are not ontic, what is ontic? Just the single realized history?

There are different flavours of DH/CH with different answers to that question.

Roland Omnes uses CH to construct a language/consistent logic for quantum systems that doesn't rely on external measuring systems. He does not consider the histories or any of the properties they contain to be ontic a priori. Instead, reality "evolves in such a way that the actualisation of facts originating from identical conditions follow the statistical rules predicated by the theory". The properties $a_i$ and $\epsilon_i$ are in the logic of QM, but only $\epsilon_i$ corresponds to a datum experienced by observers.

On the opposite end there is Robert Griffiths, who describes quantum properties contained in histories as ontic [2]. Under this interpretation, one history occurs, and hence the properties asserted by that history are real properties possessed by the system being modeled.

[1] Chapter 8 https://press.princeton.edu/books/paperback/9780691036694/the-interpretation-of-quantum-mechanics
[2] https://arxiv.org/abs/1105.3932

 

[Morbert]

At what point does the dead cat history become the one that actually occurs? The cat measures the poison bottle and dies. The Geiger device measures the decay and releases the poison. The lab guy opens the box hours later. That's three measurements, among others.
Under MWI, as you say, 'the cat' (shortly after the radioactive decay event) is always in superposition of live and dead. Both histories occur. You assert that under DH, only one history of the two above 'occurs'. At what point does this happen and the alternate live-cat history degenerates into a state of 'does not occur' or 'is no longer a potential measurement outcome'?

There is no point at which a history comes into existence. The theory reports probabilities for alternative histories, only one of which occurs. There's no point time where two histories occur. If a consistent historian observes a live cat, they can infer the history "particle did not decay and the cat survived" just as they could with a classical theory.

The tricky part is QM, unlike classical mechanics, let's us construct sets of histories that will produce inconsistencies if they contain complementary properties and are combined. E.g. Consider a slightly modified case of Schroedinger's cat: A particle in the box is emitted. If it has spin $\uparrow_x$, the cat is killed. If it has spin $\downarrow_x$ the cat lives. A relevant pair of decoherent histories is

1. Particle has spin $\uparrow_x$ and then the cat dies
2. Particle has spin $\downarrow_x$ and then the cat survives

which is fine. But we could also propose the set of histories

1. Particle has spin $\uparrow_z$ and then the cat dies
2. Particle has spin $\uparrow_z$ and then the cat survives
3. Particle has spin $\downarrow_z$ and then the cat dies
4. Particle has spin $\downarrow_z$ and then the cat survives

these histories are inconsistent, because the health of the cat is complementary to the spin-z of the particle. If we want to talk about the spin-z of the particle, we must necessarily exclude complementary properties like the health of the cat in our logic.

 

[Halc]

There is no point at which a history comes into existence.

Of course. There is but the one history, so it never 'comes into existence'. I'm talking effectively wave function collapse. When does that occur? MWI has no collapse, so the question is moot. If there is no live cat, then at some point the wave function that included that possibility has to collapse into a wave function without that outcome. When does that collapse occur?

If there is no point of collapse, then the live cat history still exists, contradicting your statement that at no point in time two histories occur.
I'm trying to distinguish DH from something simple like Copenhagen, which, as I know it, is an epistemological interpretation, not a metaphysical one like most of the others.

 

[Morbert]

Of course. There is but the one history, so it never 'comes into existence'. I'm talking effectively wave function collapse. When does that occur? MWI has no collapse, so the question is moot. If there is no live cat, then at some point the wave function that included that possibility has to collapse into a wave function without that outcome. When does that collapse occur?

If there is no point of collapse, then the live cat history still exists, contradicting your statement that at no point in time two histories occur.
I'm trying to distinguish DH from something simple like Copenhagen, which, as I know it, is an epistemological interpretation, not a metaphysical one like most of the others.

The wavefunction/quantum state in DH is epistemic*. It is part of the procedure a physicist uses to compute probabilities, based on what they know. A collapse is just a physicist updating their knowledge.

*There's a class of interpretations sometimes called psi-epistemic interpretations that supplement the wavefunction $\psi$ with an ontic state $\lambda$. DH is not one of these. DH never tries to do anything more with QM than return probabilities for possible alternatives posed.

 

[A. Neumaier]

There are different flavours of DH/CH with different answers to that question.

Are you considering DH and CH to be the same for the purposes of this thread? What is their distinguishing feature?

Given a set of decoherent histories of a closed system, both Both MW and DH would resolve a pure initial state into orthogonal branches corresponding to the histories. But MW says all histories occur, while DH says only one history occurs.

According to your statement, one history occurs, hence it must be ontic. What else could ''occurs'' mean?

In a cosmological context, this history consists of all observations ever made anywhere in the universe, including those made outside of the causal cone of the Earth and those that will be made anytime in the future. The purpose of theoretical physics is to explain the statistical and nonstatistical regularities of this single ontic history on the basis of the state of the universe. All other histories are irrelevant, though they may give food for thought.

The wavefunction/quantum state in DH is epistemic. It is part of the procedure a physicist uses to compute probabilities, based on what they know. A collapse is just a physicist updating their knowledge.

Thus there is no objective state of the universe; all physicist make up their own, depending on what they happen to know. When they get older and forget something, the state of the universe also changes according to their fading knowledge. Thus their alleged state of the universe is a completely subjective item, an object for scientific investigations in mental psychology, not in physics.

An objective covariant quantum theory should be able to predict relative frequencies that are independent of who knows what...

DH never tries to do anything more with QM than return probabilities for possible alternatives posed.

But it should be probabilities that can be objectively and post factum checked in the single history occurring - when all the experiments are done and the checker knows all relevant details about the experiment including their outcomes. Then epistemic collapse cannot be involved as there is no updating of knowledge.

How does DH achieve this?

 

[Morbert]

Are you considering DH and CH to be the same for the purposes of this thread? What is their distinguishing feature?

The difference is primarily historical, with different authors interested in different things. For the purposes of this thread I do not think it's important.

In a cosmological context, this history consists of all observations ever made anywhere in the universe, including those made outside of the causal cone of the Earth and those that will be made anytime in the future. The purpose of theoretical physics is to explain the statistical and nonstatistical regularities of this single ontic history on the basis of the state of the universe. All other histories are irrelevant, though they may give food for thought.

But it should be probabilities that can be objectively and post factum checked in the single history occurring - when all the experiments are done and the checker knows all relevant details about the experiment including their outcomes. Then epistemic collapse cannot be involved as there is no updating of knowledge.

How does DH achieve this?

A DH cosmological model broadly consists of i) A state space, operator algebra etc ii) dynamics iii) Initial conditions. We can make predictions about the frequencies we expect to observe in the universe by casting these frequencies as observables[1][2], embedding these observables in the appropriate histories, and computing their probabilities. So e.g. If we have the set of all frequencies observed from all experiments carried out everywhere and everywhen in the universe $\{f\}$, and our model is reasonable, we could build a history $C_\alpha$ asserting all the observed frequencies, and our model should return the probability $p(C_\alpha) = \mathrm{Tr}\left[C_\alpha \rho\right] \approx 1$.

All other histories are irrelevant, though they may give food for thought.

DH is not normally used to construct a single history with probability 1 (an enormous task for a cosmological model), but to compute conditional probabilities close to 1. To use Hartle's example in [2]: A cosmological theory would not typically predict "the present position of the sun on the sky. It will predict, however, that the conditional probability for the sun to be at the position predicted by classicalcelestial mechanics given a few previous positions is a number very near unity."

Thus there is no objective state of the universe; all physicist make up their own, depending on what they happen to know. When they get older and forget something, the state of the universe also changes according to their fading knowledge. Thus their alleged state of the universe is a completely subjective item, an object for scientific investigations in mental psychology, not in physics.

An objective covariant quantum theory should be able to predict relative frequencies that are independent of who knows what...

According to DH, the quantum state of the universe might not be objective, but it is at least meaningful without needing to embed the universe in some larger system. It would simply serve as an input to our model, generating probabilities for possible histories of the universe and, through these probabilities, predictions all observers in the universe would agree upon. In this sense it would be 'intersubjective' if not wholly objective.

I'll leave the other matter of ontology and 'occurence' for now. For the purposes of this thread we can interpret 'occur' plainly. A history that occurs is a history whose asserted properties are real physical properties.

[1] https://arxiv.org/abs/1907.02953
[2] https://arxiv.org/abs/1805.12246

[edit] - some minor clarifications

 

[Morbert]

and our model should return the probability $p(C_\alpha) = \mathrm{Tr}\left[C_\alpha \rho\right] \approx 1$

Actually this would only hold for very simple universes, since macroscopic events like lab experiments can be contingent on quantum events not guaranteed by initial conditions (like fluctuations in the early universe).