What does quantum mechanics tell us about reality? Part II

This blog post follows on from a previous post, “What does quantum mechanics tell us about reality”, in which I tried to give a balanced and non-technical overview of some of the interpretations of quantum mechanics. In this post I will take a different approach: this will be a slightly biased, critical, and more technical follow-up. I recommend reading the earlier post first, but those already familiar with the interpretations of quantum mechanics should be able to dive straight into this.

In the Schrödinger’s cat thought experiment, a cat is placed in a box with a device that contains a radioactive atom and a vial of poison. If the atom decays, then the device is designed to release the poison, thus killing the cat. It is now well known that such an atom can be put into a state in which it has decayed, and not decayed simultaneously – this is known as a superposition state. Now, if this system is studied using the central equation in quantum mechanics, the Schrödinger equation, then the following result will be found: if the atom is in a superposition state, then this will lead to the cat being in a superposition state. The cat will be dead and alive simultaneously! Now suppose you open the box – what will you find? The Schrödinger equation again predicts that, if the cat was in a superposition of being dead and alive, then when you open the box you will also enter into a superposition. You will be in a superposition of either seeing the dead cat, whilst simultaneously seeing the alive cat.

the-voodoo-cat-skull

Artwork by Joe Hollis

This clearly does not fit with our experience of the real world. We never see objects in superpositions, and indeed we never seem to experience superpositions ourselves. And while the above experiment is far too challenging to perform using a real cat, conceptually similar experiments have been performed in which an object is put into superposition, and then observed. The result of these experiments fits with our experience and intuition: we never see a superposition state. So what has gone wrong here? Have we misapplied the Schrödinger equation? Is the Schrödinger equation incorrect? The standard resolution, which can be found in most quantum mechanics textbooks, is to introduce the “collapse postulate”: On observation a superposition state collapses, meaning that only one outcome of an observation, or a measurement, is ever observed. I.e. we only ever see the cat as being dead or alive. But the collapse postulate raises as many problems as it solves. What exactly constitutes an observation or measurement? If macroscopic objects are made of quantum particles, what is so special about a measuring device or a conscious human observer to cause collapse? (This questions are together often termed the measurement problem.)

Is the Schrödinger equation sufficient to solve the problem?

Despite how quantum mechanics is often discussed, there is now a widely accepted and carefully studied solution to these problems that utilises the Schrödinger equation alone, and does not have to introduce the troublesome collapse postulate. The solution lies in the theory of decoherence. Elsewhere I give a more thorough introduction to decoherence, in particular in relation to Schrödinger’s cat. But in this post I will try to give a simple and minimal introduction that still captures the main ideas.

Imagine you have a single atom and some cutting-edge experimental equipment capable of putting this atom into a superposition of two locations, A and B. The crucial question here, which is at the heart of decoherence, is: how do you know it is in a superposition? If you directly measure the atom then you will either see it at position A, or position B, but not both. To confirm the superposition a more advanced step needs to be taken: we must do an interference experiment. This involves the idea of constructive and destructive interference of waves, which can be seen by throwing two stones into a pond close to one another. The waves coming from one stone interfere with the waves coming from the other stone. If two peaks meet they reinforce one another creating a larger peak, whereas if a peak and trough meet they cancel each other out. Quantum mechanical objects, such as the atom we are trying to interfere, are described by equations known as wave functions. As the name suggests, these particles act like waves, and just like the stones in the pond they can demonstrate interference. I’m unsure myself how the exact experiment would work to interfere the two parts of an atom that has been put into a superposition, but by measuring the interference between the two wavefunctions the superposition can indeed be confirmed, and this is now an extremely well measured phenomenon in experiments.

Now suppose you are given two atoms, and you prepare the atoms in the following superposition state: both atoms are in position A, in superposition with both atoms in position B. Again, how can we confirm the superposition? If we directly measure the position of the atoms, then we either find both of them in position A, or both in position B (this is known as an entangled state – the position of the first atom is “entangled” with the position of the second, because we always find them together). Again we do not see, and cannot confirm, the superposition in this way, and we must perform an interference experiment. Now comes the crucial point: the interference experiment must be done on both atoms simultaneously, otherwise we will never see an interference pattern. If we just take the first atom, and try and interfere it with itself, then this will not work. (I explain this in more detail here.)

We can now return to Schrödinger’s cat. The cat is in a superposition state of being dead and alive, but how can we confirm the superposition? First imagine that the only thing in the box is the cat –it is in a complete vacuum with no air particles or photons or anything. In this case, it is in principle possible to perform an interference experiment with the cat. The dead part of the superposition interferes with the alive part of the superposition, and an interference pattern would be observed, confirming the superposition. This is not practically possible because we would have to interfere every single particle in the cat, and this involves precisely controlling and manipulating every single particle. But according to the laws of physics this is at least in principle possible.

But it is not realistic that the cat could be in a complete vacuum, and no matter how hard we tried there would always be at least a few particles in the box with the cat. These unwanted particles (and photons etc) are often termed the environment, and we assume that we do not have control nor access to them. Now again put the cat into a superposition. The cat will inevitably interact with the unwanted particles in the box, and as soon as they interact the cat and unwanted particles will become entangled with one another. Then, if we want to do an interference experiment, we would have to not only interfere all the particles in the cat, but also all the extra particles and photons in the box. We would have to precisely control and manipulate all of these particles, but as stated above we are assuming that we cannot control them and cannot access them. Therefore, in this case it is not even in principle possible to do an interference experiment. We cannot ever confirm that the cat was in a superposition.

Now what happens when we open the box? As soon as we look at the cat we become entangled with it, and enter into the superposition. The cat is dead and we see a dead cat, in superposition with the cat being alive whilst we see an alive cat. But again there will be unwanted particles and photons, and very quickly the cat and ourselves will become entangled with these particles and photons. Again, if we want to confirm that we are in a superposition, we would have to be able to manipulate and control all of these particles and photons, which is clearly not possible. Therefore, again, we cannot ever confirm that we are in a superposition. Furthermore, it is likely that some of the photons that have interacted with you will escape from the room through window, flying off to space at the speed of light! In this case, seeing that we can’t travel at the speed of light to collect these photons, it is not even in principle possible to confirm the superposition.

We have now solved the main problems in the Schrödinger’s cat thought experiment. Is the Schrödinger equation wrong? No – we can explain our observations, i.e. that we never see the cat in a superposition, just using the Schrödinger equation. Why do we never see the cat in a superposition? You must do an interference experiment to confirm the superposition, but this is not possible when we factor in the other particles in the box with the cat. The question of “what constitutes a measurement?” has not really been answered yet, but I will address this in a future post in which I defend the many worlds interpretation.

I have not yet fully addressed what happens when you open the box – whether you are really in a superposition, and if so, why you don’t “experience” this superposition. The answer to this really depends on how you interpret quantum mechanics, and this is what I will turn to next.

Many worlds interpretation

The introduction above to Schrödinger’s cat and decoherence has, in a sense, been written in the language of the many worlds interpretation. In the many worlds interpretation we firstly assume that the Schrödinger equation is sufficient in itself to explain paradoxes such as Schrödinger’s cat, and secondly we assume that quantum mechanics is a theory that tells us about real objects in the real world. The first of these points is justified in the above introduction to decoherence, and nowadays this explanation is widely accepted. The second point is the usual way we interpret science – normally we assume that our equations and theorems are telling us something about a real world that exists independent of ourselves.

These two assumptions might seem quite straightforward, but they lead to quite a radical picture of the world in which we live. For example, in the many worlds interpretation we say that the cat is indeed in a superposition of being dead and alive. There is technically just one cat, but it is dead and alive simultaneously. However, we have seen that the two parts of the superposition cannot ever interfere with each other. Interference is the only way of confirming that an object in is in a superposition, so the dead and alive cats cannot ever know of each other’s existence. Furthermore, the equations of quantum mechanics are such that the future life of the cat (at least the alive one) does not depend on whether the cat is in a superposition or not. Therefore, for all intents and purposes we can think of this as two cats, one dead and one alive. This is where the idea of “many worlds” comes from. For all intents and purposes there are two worlds, one containing an alive cat and one containing a dead cat.

The same idea holds when you open the box. You split into a superposition of seeing a dead cat and seeing an alive cat. But again the two parts of the superposition cannot ever know of the other’s existence, because they would have to interfere with one another to confirm this, and this isn’t possible. Therefore we can again treat this as being two separate worlds, one in which the cat is dead and you are presumably emotionally and morally scarred by the experience, and another in which the cat is alive and you will be relieved.

This picture of the universe is clearly unintuitive, and often people reject many worlds outright and come up with all kinds of criticisms of this interpretation. In my opinion most of the standard criticisms are either ill-founded or result from a lack of understanding of the basic theory, and in a future post I will try to flesh out many worlds theory and provide straightforward responses to many of the criticisms.

QBism – does the wavefunction represent reality?

Before continuing, an important comment is needed. Just before uploading this post I was in contact with Chris Fuchs – one of the founders and main promoters of QBism. To cut a long story short, he said (politely but firmly) that (referring to my previous post) “you capture none of the flavor of QBism at all in what you write. You present QBism as a kind of lifeless prediction machine (a positivism or instrumentalism), rather than as an attempt to make a deep statement about the character of the world”. He recommended reading https://arxiv.org/abs/1601.04360 and https://arxiv.org/abs/1207.2141. I have decided to keep my description of QBism in this current post unedited, but bear in mind Chris’s comment when you read this! And please comment on this post if you have an opinion about whether/how I misrepresent QBism…

As introduced above, we can represent quantum mechanical objects using an equation known as a wavefunction. The wavefunction tells us everything we know about this object. For example, we could write down the wavefunction for a single particle in an (equal) superposition of two locations. This wavefunction can then be used to predict what we will see if we perform certain measurements. For example, using the wavefunction we can calculate that, assuming the superposition is equal, if we measure the position of the particle then it will be in position A with 50% probability, or position B with 50% probability. Furthermore, we can use the wavefunction to predict what will happen if we perform an interference experiment. In particular, it will tell us the properties of certain outcomes: it will say that if we perform interference experiment X, then outcome Y will happen with probability Z.

Numerous experiments over the years have confirmed that quantum mechanics is extremely good at correctly predicting outcomes to experiment. But, in a sense, QBism says that this is all that quantum mechanics is good for. It says that we should not interpret the wavefunction as describing a real object, and therefore it is meaningless to ask if the cat is really dead and alive simultaneously. We simply cannot know – all we know is the probability of what will happen if we open the box. More specifically, the wavefunction represents our state of knowledge. It tells us what we know, not what exists. This is similar to Bayesian probability theory, in which probabilities this represent our knowledge of the world, not the world itself. For this reason QBism can also be called quantum Bayesianism.

I certainly have some sympathy with QBism. It takes quantum mechanics seriously, and in particular the Schrödinger equation, and does not try to modify the formulae. And it certainly has a strong point: how do we ever really know what exists? The answer is that we observe it, and we perform measurements on it, and we devise clever experiments to perform measurements on the extremes of scale and energy. But until we measure anything, we cannot truly know what it is, and whether it exists. So in this sense QBism is right that quantum mechanics is just a toolbox for predicting experiments.

But is this all quantum mechanics is? Throughout most of human history the goal of science has been to learn more about the world. We do astronomy and astrophysics to learn about stars and galaxies; we smash particles into one another in colliders to learn about what matter is made of; and we do quantum experiments to learn about the weird and wonderful properties of the quantum world. QBism therefore is a radical departure from how we normally treat the scientific endeavour. It is not necessarily the wrong way to interpret quantum mechanics, but Qbists should at least acknowledge that it is an extreme philosophical position.

To take this further, imagine the Schrödinger’s cat thought experiment, but with your friend opening the box rather than yourself. QBism is perfectly good at predicting what your friend will see when they open the box. But, presumably, you believe that your friend exists, and you might be interested in what happens to them when they open the box. QBism cannot tell us this – you can write down the wavefunction for your friend, but this is only a tool for calculating what you will see when you interact with your friend. Many worlds, on the other hand, is perfectly well-equipped to ask questions about your friend. The answer may be disturbing – that they in effect split into two versions – but at least it is a consistent and coherent answer. And this idea can be extended: many worlds theory predicts that almost continuously the world – and therefore your friend – splits into almost infinite parts of a vast superposition, which we can think of as parallel universes.

Would my assumption that my friend exists be incorrect? Perhaps. Maybe in the “real” world it is meaningless to ask about the state of things before we interact with them. But my friend certainly does exist in my head – I can imagine them walking towards the box, opening it, and looking inside. We can then call this world the “imaginary” world. Even though it might not exist outside my mind, I am still interested in what my imaginary friend is doing in this imaginary world. Removing yourself from the picture, now imagine a scene familiar to yourself, such as your house, or your pet, or your favourite sports team. I wonder what they are doing right now? Are there near infinite numbers of them, in near infinite parallel universes? Or is it meaningless to ask what they are doing right now, and only meaningful to think about what happens when you interact with them in some way?

My favourite thing about QBism is this: the wavefunction is normally written using the Greek letter psi, which is often pronounced “sigh”. Ontology is the study of the existence of things, whereas epistemology is concerned with knowledge rather than existence. Therefore, a Qbist is a psi-epistemist. Whereas someone like me who believes in many worlds and therefore that the wavefunction is real, can be termed a psi-ontologist. It deeply troubles me that I am a psi-ontologist (say this sentence out loud to yourself if you don’t get the joke!).

Collapse theories

Until reasonably recently it was not fully appreciated that the Schrödinger equation alone can lead to the appearance of collapse. Therefore, to explain why we either see the cat as dead or alive a “collapse postulate” was introduced into quantum mechanics. Initially it was just a postulate, and no explanation was given of how collapse takes place, or what causes it. But this introduces many difficult questions: What causes the collapse? It is usually assumed that a measurement causes collapse: but what is a measurement? Often it is said that a “measuring device”, or even a conscious observer, is what causes the collapse. But if macroscopic objects are made of quantum particles, what is so special about a measuring device or a conscious human observer to cause collapse?

Over the years various theories have been introduced to explain collapse with the hope of answering the above questions. Various mechanisms have been proposed: complexity causes collapse – the more complex a system, the more likely it is to collapse; or consciousness itself causes collapse; or gravity causes collapse – the larger the mass, the more likely collapse will occur. These models therefore can explain why Schrödinger’s cat is never seen, or measured, as being in a superposition state.

But now, with the theory of decoherence that I introduced above, we can explain the appearance of collapse without having to add extra postulates into the theory. Collapse theories are therefore unnecessary to explain our observations. So why do they still exist? I have never met anyone who both understands decoherence, and thinks that it is wrong, so collapse would presumably happen in addition to decoherence. And if you are uncomfortable with the conclusion that the cat is in a superposition (many worlds), or that it is meaningless to ask about the state of the cat (QBism), then you can modify quantum mechanics – specifically, modify the Schrödinger equation – so that the state collapses. But for me this seems like a case of changing the science in order to fit our wishes.

This might not be a problem if quantum mechanics was a young and underdeveloped theory. But this is certainly not the case, and the Schrödinger equation itself is responsible for quantum mechanics often being termed “our most successful theory ever”. Do we really want to modify such an equation? Quantum mechanics also works relativistically (i.e. combining it with Einstein’s special relativity), and it has been extended to quantum field theory, which has successfully predicted the Higgs boson. But collapse theories are far from achieving such extensions.

To be fair to gravity-induced-collapse, at some point quantum mechanics, as with any other theory, will be surpassed by some other theory. Quantum mechanics will still be an excellent approximation in many regimes, but in the extremes it will surely break down. But what are these extremes? Potentially the fact that general relativity and quantum mechanics cannot yet fit together gives a clue to this. In this case, might gravity in fact collapse the wavefunction? In my understanding this is at the heart of Roger Penrose’s suggestions to both explain collapse and unify general relativity and quantum mechanics.

For me the main positive to collapse theories is that they are testable. This is especially true for gravity-induced-collapse. If we put bigger and bigger systems into a superposition, while sufficiently isolating them from the environment so that decoherence doesn’t cause the appearance of collapse, then eventually at a certain mass threshold these systems should spontaneously collapse. These experiments should be possible in the relatively near future, and will serve to either confirm this theory, or give extra weight to non-collapse theories such as many worlds.

Consciousness-induced-collapse is in principle testable, but this is far beyond current experiments. To confirm this we would have to put a conscious entity into a superposition. We would have to isolated sufficiently it from the environment so that there is no decoherence, and we would have to be able to control and manipulate every particle in the conscious entity so that we can do an interference experiment. If the consciousness spontaneously collapses, thereby preventing interference, this will be strong evidence that consciousness does induced collapse. The best route to this could be using quantum computers. If we can simulate consciousness on a computer, then we could upload this program to a quantum computer, and subsequently put the consciousness into a superposition. But we don’t even know what consciousness is and such a test is infeasible for now. In addition, I argue elsewhere that if consciousness did cause collapse then the reality this would lead to would be far more bizarre and absurd than even many worlds theory predicts!

Pilot wave theory

Einstein famously stated that “God does not play dice”. He simply couldn’t believe that a fundamental theory of nature such as quantum mechanics could really be probabilistic. For example, generally in quantum mechanics we would say that on opening the box containing Schrödinger’s cat it would be random whether the cat is observed as dead or alive (with a certain probability of each). In many worlds theory both outcomes may exist, but it is random whether you end up in the part of the superposition with the dead cat or with the alive cat, so in this sense it is still random. In contrast, theories such as general relativity and Newtonian mechanics are deterministic. For example, if you know all of the positions and velocities of the planets in the solar system, then you can predict with certainty where the planets will be at any given time in the future.

To prevent the randomness of quantum mechanics a “deterministic hidden variable theory” was devised (named Bohmian/De Broglie/pilot wave theory). Taking again the example of the cat, in this theory there are additional variables beyond those in the Schrödinger equation. If we knew the values of all these variables, then we would know with certainty whether the cat will be dead or alive when we open the box. However, these variables are “hidden”, meaning they are fundamentally beyond our measurements and observations. We cannot, and will not, ever be able to determine these values, and therefore quantum mechanics will always appear to be random.

For me this is an even worse case than collapse theories of changing the science so that it more closely fits with our intuition. For protagonists of this theory it is so important that nature must not be random that they are willing to invent an underlying deterministic world that we cannot ever even in principle see. But why should nature be deterministic? In addition, the Schrödinger equation itself is deterministic, so in fact many worlds theory is a deterministic theory. We know with certainty that the cat will be dead and alive. The randomness just comes in when you ask “which universe will I end up in?”. But it is still, from the outside, deterministic.

There are some further complications/criticisms to this theory. John Bell famously showed that, if these hidden variables exist, then they must communicate with one another faster than the speed of light. Furthermore, in a recent paper Renato Renner showed that hidden variable models cannot be self-consistent (although this might not necessarily mean that they are wrong?!).

Conclusion

There are many other interpretations of quantum mechanics, and many more seem to be invented year-on-year. My personal view is that quantum physicists need to stop inventing new interpretations, and consolidate the old ones. Indeed both many worlds and QBism have some features that are unsatisfactory to some and unintuitive to all. But in my understanding there is nothing fundamentally wrong with either of these. Sure there are small problems that need to be ironed out, but this is the same for any theory. My personal prediction is that in 100 years from now, if we survive existential risks such as nuclear war or artificial intelligence taking over the world, pretty much every quantum physicist will either be a Qbist, or believe that we live in a fantastic quantum multiverse!

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About P A Knott

I currently hold a Research Fellowship from the Royal Commission for the Exhibition of 1851. My research project will tackle a key challenge in the quantum technology revolution by designing computer algorithms that automate the engineering of useful quantum states. These algorithms will enable the design of novel experiments to bring forward the development of new technologies such as quantum computing, communications and metrology. In my previous post I worked at the University of Nottingham on a project entitled "Sentient observers in the quantum regime and the emergence of objective reality", with Gerardo Adesso, Marco Piani, and Tommaso Tufarelli. This project involved using quantum information theory to investigate foundational questions concerning the role of the observer in physical theories. More generally, my research interests include quantum metrology, quantum state engineering, quantum sensing networks, and optical interferometry.
This entry was posted in Entanglement, Philosophy, Quantum foundations, Superposition. Bookmark the permalink.

8 Responses to What does quantum mechanics tell us about reality? Part II

  1. juan m. jones says:

    Qbism says a lot about the world but requires that you can change your mind about what is knowledge, science and reality.

    Like

  2. P A Knott says:

    It’s certainly true that QBism says a lot about the world – I was a bit harsh in my representation of it. Though I think it is accurate to say that QBism doesn’t say anything about the state of Schrödinger’s cat, before you open the box?

    Like

    • juan m. jones says:

      It says that the Hilbert space associated with the cat is something about “reality”, but the cat, or any system, is pure potential, not actuality. Actuality arises, as experience, when systems interact. Hilbert space is objetive.

      Liked by 1 person

  3. Pingback: Why the many worlds interpretation of quantum mechanics is fantastic | quanta rei

  4. Hello Paul,

    I have just discovered your very interesting blog via your last paper about decoherence and quantum Darwinism (http://arxiv.org/pdf/1811.09062.pdf), the one you mention here in this post.

    Please excuse my layman question (being at the same time a friendly provocation), but introducing your paper you write:

    > Together these theories explain how our classical reality emerges from an underlying quantum mechanical description

    and in the last sentence of the abstract you announce that finally you are going to

    > demonstrate how decoherence and quantum Darwinism can shed significant light on the measurement problem

    (and indeed you write about the role of the observer and about the measurement problem just near the end, just before comparisions with Everett Interpretation and Conclusion)

    How can you state the problem of emergence of classical reality solved when you can only (!) ,,shed significant light” on the measurement problem?

    (what is more quirky you seem to admit that even having read almost the whole paper ,,it may not be immediately obvious” that decoherence and quantum Darwinism can shed that light)

    Is not the measurement problem (with ubiquitous questions about the role of the observer) the core, the essence of the problem of why and how our classical reality emerges?

    I know your answer:
    > in decoherence and quantum Darwinism this is not the case

    but is it a truly honest answer? or rather, is it the answer of someone from the Decoherence Church? I doubt it as in another paper (https://knottquantum.weebly.com/uploads/9/0/9/4/90944896/does_consciousness_collapse_the_quantum_state.pdf) you write:

    > the so-called measurement problem of quantum mechanics […] lies at the very heart of the theory

    and BTW, I do agree that ,,we cannot ever confirm that we are in a superposition”, but from completely different reasons, I just feel that trying to conduct such measurement would be like trying to bootstrap oneself up, what is more, using bootstrap in a superposition 😉

    Best regards,
    Wojciech

    Like

  5. P A Knott says:

    Hi Wojciech,

    Thanks for your comment and interest!

    In my opinion decoherence does completely solve the measurement problem, but I didn’t fully explain this in my essay, hence the wording I used. But apologies if what I wrote in the abstract was misleading.

    You said: “Is not the measurement problem (with ubiquitous questions about the role of the observer) the core, the essence of the problem of why and how our classical reality emerges?”

    Generally I do agree with this! It depends on exactly how you define the measurement problem, and what aspect of classical reality you are trying to explain, but I agree that generally the measurement problem is a key part.

    So, why do you think that decoherence does not solve the measurement problem? Which specific part of the measurement problem is not solved? If you tell me this, I will try to answer as best I can.

    Best,

    Paul

    Like

  6. Hi Paul,

    Thank you so much for your reply.

    Why do I think that decoherence does not solve the measurement problem?

    Well, thre is so much fuss about decoherence and now quantum Darwinism, so many serious people working in this field, that I can not simply ignore them.

    I have started studying their contributions in earnest with strong feeling that there must be something important about the world that decoherence can reveal, yet the more I read the more I am puzzled and worried.

    I am not very advanced in my study of the formalism and its consequences, but it is not something wrong with the formalism that keeps me awake at nights 🙂

    I have rather serious difficulties to grasp the meaning, the foundations.

    To be honest, I am not a 100% layman, but now I am more interested in philosophy of physics (and were my first name be different, I would have lost my initial enthusiasm for the works of Wojciech H. Zurek long time ago 😉 )

    Reading more about the philosophy of quantum theory I have got infected with doubts that seems fatal and incurable for me.

    The main infector for me was Chris Fields, take a look on his argumentation here: http://arxiv.org/abs/1402.6629v6

    (about decoherence in the last paragraph of chapter 4, I do not understand what he writes in chap 5 and I suspect him of being wrong there but it does not matter)

    or here: https://chrisfieldsresearch.com/quant-freedom.pdf

    As you can see, it is not some specific part of the measurement problem that decoherence have not elucidated yet, the problem is deeper (I feel it must be so, as the observer is the keystone)

    But it is so good to have such problem to think about, what a sad world would it be, were such problems could be solved so easily with decoherence or something like that!

    Best regards,

    Wojciech

    Like

    • P A Knott says:

      Hey,

      It would be great to discuss your problems with decoherence, but you haven’t yet told me what they are! Unfortunately I don’t have time to read this paper you mentioned — I read the paragraph you suggested but it didn’t make sense to me without having read the rest of his argument.

      Perhaps you could start by explaining what is wrong with the examples I gave of decoherence in my paper in the section titled Decoherence, starting on page 2. (Here’s the paper: https://arxiv.org/pdf/1811.09062.pdf)

      Cheers

      Paul

      Like

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