Could you explain why probability is "inherent within the Hamiltonian? — tom

I told you this already, the energy of the system is expressed as probable locations of particles.

Could you explain how the initial measurement is made in say a two-slit experiment, and what difference the result makes? — tom

There is an amount of energy introduced into the system, as you say, a photon particle or number of particles are "fired". The energy of that system is expressed as particles. This expression is used in the Hamiltonian, and therefore the Schrodinger.

If probability is "the energy of the particles", why are different words used to state the same thing? — tom

Have you ever heard of wave/particle duality? Two different words to express the same thing.

No it is easy enough to visualize three groups of four or four groups of three, and to see that it equals twelve. — John

Say you visualize such groups, how would you know that there was twelve there without counting them or performing the math?

Because twelve is a small enough number to be visualized, either as three groups of four or four groups of three. If you can't visualize twelve then just consider the example of two or three objects; which can be visualized either in terms of addition or multiplication.

Probability is inherent within the Hamiltonian and therefore inherent within the Schrodinger. — Metaphysician Undercover

The Hamiltonian tells you how a quantum state will (deterministically) evolve. When it is applied in the Schrodinger equation, it produces a superposition of states which are expressed as probability amplitudes (complex numbers), not probabilities.

This matters. If states in a quantum superposition were merely probabilistic, with only one state being real as with a coin flip, then they could not constructively or destructively interfere with each other to produce interference patterns. That is why all of those states must be real.

It is only the Born rule, which is not part of the Schrodinger equation, that enables us to extract the probabilities for observing those states when a measurement is made.

It is only the Born rule, which is not part of the Schrodinger equation, that enables us to extract the probabilities for observing those states when a measurement is made. — Andrew M

But as we learned from the video on probability, the exact same results are achieved - i.e. we obtain the same Value, without invoking the Born Rule or probability.

Probabilities are not fundamental to QM, they are simply useful.

I told you this already, the energy of the system is expressed as probable locations of particles. — Metaphysician Undercover

You claim that the Hamiltonian operator expresses the "probable locations of particles".

What is the wavefunction for?

There is an amount of energy introduced into the system, as you say, a photon particle or number of particles are "fired". The energy of that system is expressed as particles. This expression is used in the Hamiltonian, and therefore the Schrodinger. — Metaphysician Undercover

If you "fire" a particle into a twin-slit experiment with an "energy" 3, what difference does that make to the outcome of the twin-slit experiment compared to an energy of 2?

Have you ever heard of wave/particle duality? Two different words to express the same thing. — Metaphysician Undercover

You claim that wave/particle duality is just two words to express the same thing - and that "probability" and "energy of the particles" also express the same thing.

But you also claim that "the energy of the system is expressed as probable locations of particles"

You seem to be going round in circles.

The Hamiltonian tells you how a quantum state will (deterministically) evolve. When it is applied in the Schrodinger equation, it produces a superposition of states which are expressed as probability amplitudes (complex numbers), not probabilities. — Andrew M

Yes, you can use the magic of mathematics to turn possibilities into realities if you like, but I think that if the magician is convinced my the magic, that's a problem And I am not convinced by that magic. Logic, of which mathematics is a type, cannot produce anything contrary to its premises. The Hamiltonian operator describes the system in terms of probabilities due to the reality which the uncertainty principle is supposed to represent. How do you think that the Schrodinger equation converts these probabilities into realities? Even the Wikipedia article you referred me to clearly discusses probability amplitudes in terms of probabilities. It says in the first line of the article: "The modulus squared of this quantity represents a probability or probability density." Then the entire article discusses things like probabilities, probabilistic laws, and probability densities.

This matters. If states in a quantum superposition were merely probabilistic, with only one state being real as with a coin flip, then they could not constructively or destructively interfere with each other to produce interference patterns. That is why all of those states must be real. — Andrew M

I don't see the premises whereby you make this conclusion. If we have a coin toss of 50/50 probability, and add another coin to the toss with a 50/50 probability, the fact that the two coins could interfere with each other in the air does not produce the conclusion that our description of the toss is not probabilistic.

But you seem to be missing something fundamental in your reference to "one state being real". When the system is described in terms of probabilities, this does not imply that one state must be the real state. There is no such thing as a "state" within a system, it is an active system. A "state" and an active "system" are two incompatible descriptions. Consider the coin toss, the 50/50 probability refers to the outcome of that activity. It is describing the activity, the coin in the air, with reference to the outcome. There is no "state" being referred to, except the outcome, but the outcome has not yet occurred, so it is probabilistic. It is not a "real state", it is a future potential state. The activity described, being the coin in the air, is an activity, it has no real states. To represent the activity as a state is to stop the activity, and this negates the essence of "activity".

So in our quantum example, the Hamiltonian operator recognizes that the system has no real states. What is real is the activity. So now we can produce a wave function, and the wave function is assumed to be a description of the activity of the system, the activity of the system being what is real. That description describes something real, but the something real is an active system, not a state. Specific spatial temporal positions of particles is a description which refers to a state. So there is an inherent incompatibility between describing an active system and the possible states of that system.

You could ask, which is real, the active system, or the states of the system, and the answer depends on your perspective. The term "energy" refers to the activity of a thing, so if energy is believed to be real, and the system is described in terms of energy, the activity of the system is what is real. The wave function describe the reality, the activity, the coin in the air, and any states are just possible states, produced if we stop the activity. But if the activity is stopped, then the wave function is no longer real. The problem being that there is no principle whereby we can represent both as real.

However, one could take an alternative perspective, and claim that energy, activity, is not real, it is merely conceptual. Activity and energy are how we describe changing states. From this perspective, we would say that "the system" is just a conceptual representation of what is real, and what is real is particular definitive states, which change from one moment to the next. Then we could say that the reason why these states must be represented as possible, or probable states, is that the reality of the system is being represented as energy, activity, and this representation is incapable of reproducing the states, due to the incompatibility expressed above. We do not have the means to say that both of the descriptions, the activity, and the states, are real, because the incompatibility between these two has not been resolved.

It's difficult to address a criticism which is just a slur.

Anyway, Kent doesn't like MW. His solution is to append some extra mathematical structure to QM in order to make it a single-world theory. He is thus an advocate of hidden variables. The trouble with this is that no hidden variable theory that does not contradict QM exists.

So, Kent advocates changing the physics, because he does not like the implication of currently known physics. — tom

The criticism was not 'just a slur'. It was a chapter of reasoned argument of which I gave you the summary and another paragraph of exposition. You didn't address the criticism at all, you just went on to say 'Anyway...' - and then began an entirely different point. The criticism is, to start with, that the axiomisation is invalid in certain reasonable circumstances. It's hard to discuss if you won't answer reasonable points raised. Your entirely different point may be interesting in its turn, but there remains a substantial argument Kent made which you hadn't addressed.

You went to re-recommend me to watch Deutsch's video on probability, which for some reason you find convincing. I've already told you I disliked it, and thought it was particularly weak in that he never represents any of the positions he disagrees with fairly. It's the polemics of a lecture. I doesn't mean it's wrong, I just mean it's obviously a diatribe not a carefully-reasoned piece. I'm happy to read essays which interact with other essays in an academic fashion, but I've never understood why a superficial video is an adequate substitute.

You do, you know, regard 'just a slur' as a reasonable way for you to speak to other people on this forum. It's not ok. If you don't like slurs, I don't think you should make them at other people's expense either.

I must admit, I don't really understand what you are asking me. I've had that problem with your posts in the past. If I can't somehow relate what you're saying to what I said, then a simple one sentence question without any context doesn't explain to me what you are looking for. I can't find the relevance of your questions.

The criticism is, to start with, that the axiomisation is invalid in certain reasonable circumstances — mcdoodle

What axiomatisation?

This matters. If states in a quantum superposition were merely probabilistic, with only one state being real as with a coin flip, then they could not constructively or destructively interfere with each other to produce interference patterns. That is why all of those states must be real. — Andrew M

That is an interesting paragraph as it encapsulates to me the philosophy-of-science debate going back and forth here. Mathematically states in a quantum superposition are probabilistic. 'Merely' is a matter of taste. A measurement occurs, the outcomes are mathematically understood.

You quote Duhem in arguing that putting things merely mathematically like that is not 'an explanation', but Duhem is long dead and there are new sorts of empiricists about who might happily use the word 'explain' about mathematical models. I think here it's realists who are also being rigid about what an explanation must involve; it can be a circular demand, in that if one isn't some sort of realist then what is one explaining? I do think that's a fruitless side-alley.

The realist in turn only feels ok if like you they can point to what is 'real' (albeit hypothetical in that it's unobservable) in order to apply something like ordinary language to what happens in the maths. It *is* hard to talk about. I've a friend who's doing some physics which assumes entanglement in order to try and model what happens in PET scans, for instance. In explaining it to a layperson like me he juggles between talking as if it's all real and then adding, 'or at least that's what the maths says'. I don't see that in remaining agnostic about 'the real' in such cases I'm somehow being anti-scientific, or failing to understand something. I am just a wary sceptic.

The Hamiltonian operator describes the system in terms of probabilities due to the reality which the uncertainty principle is supposed to represent. — Metaphysician Undercover

So, the "Hamiltonian operator describes the system in terms of probabilities". How does it do that? Where in the Hamiltonian operator are the probabilities? I'm particularly interested as, having applied the Hamiltonian, typically in systems of interest, one obtains a scalar quantity, not a probability distribution.

How do you think that the Schrodinger equation converts these probabilities into realities? — Metaphysician Undercover

How do YOU think the Schrödinger equation achieves that? More pertinently perhaps, why do you think the Schrödinger equation does that, particularly as no one else does?

Mathematically states in a quantum superposition are probabilistic. — mcdoodle

How do they affect each other?

I'm referring back to my previous post, which was in turn referring to Kent's chapter in a multi-author book (including Wallace) called 'Many Worlds?: Everett, Quantum Theory, and Reality' (2010). The summary I mentioned in my post also includes this para, which refers to the proposition of his own you mention which he embeds in the critique of other people's approaches:

This article reviews some ingenious and interesting recent attempts in this direction by Wallace, Greaves– Myrvold and others, and explains why they don't work. An account of one‐world randomness, which appears scientifically satisfactory, and has no many‐worlds analogue, is proposed. A fundamental obstacle to confirming many‐worlds theories is illustrated by considering some toy many‐worlds models. These models show that branch weights can exist without having any role in either rational decision‐making or theory confirmation, and also that the latter two roles are logically separate. — Kent

I'm afraid the detail of the arguments is impossible to summarise in a short space.

How do they affect each other? — tom

In what way is that relevant to what I was saying to AndrewM? I was making an epistemic/ontological distinction, not arguing about what hypothetical entities do or don't do.

I'm afraid the detail of the arguments is impossible to summarise in a short space. — mcdoodle

Really!

You claim that

Mathematically states in a quantum superposition are probabilistic. — mcdoodle

Do these states affect each other? Seems to me a one word answer.

So, the "Hamiltonian operator describes the system in terms of probabilities". How does it do that? Where in the Hamiltonian operator are the probabilities? I'm particularly interested as, having applied the Hamiltonian, typically in systems of interest, one obtains a scalar quantity, not a probability distribution. — tom

Try Google, it's very helpful, but let me try, maybe I can describe it. The Hamiltonian operator describes a system in terms of the energy of all the particles within the system. There is a time-energy uncertainty, so any derived time evolution is inherently probabilistic.

How do YOU think the Schrödinger equation achieves that? More pertinently perhaps, why do you think the Schrödinger equation does that, particularly as no one else does? — tom

Clearly I don't believe that, that seemed to be Andrew M's position, which I objected to.

Try Google, it's very helpful, but let me try, maybe I can describe it. The Hamiltonian operator describes a system in terms of the energy of all the particles within the system. There is a time-energy uncertainty, so any derived time evolution is inherently probabilistic. — Metaphysician Undercover

Google is a bit too hard for me, so if you don't mind, perhaps you could clarify a couple of questions:

What does the Hamiltonian operator operate on?

When you apply the Hamiltonian operator "H" to a ket in Hilbert space "|psi>" what do you get?

As I said before, I find your questions very vague and irrelevant. You asked me one question, I replied, and now you go off in a completely different direction. I'm not a mathematician, nor am I a physicist, I'm a metaphysician. Perhaps you could explain what you are trying to get at. Or, if you think that I am wrong in what I have said, maybe you could explain why. Then you could help me learn something.

Probabilities are not fundamental to QM, they are simply useful. — tom

Agreed. Probabilities relate to observer predictions, not the world itself.

Even the Wikipedia article you referred me to clearly discusses probability amplitudes in terms of probabilities. It says in the first line of the article: "The modulus squared of this quantity represents a probability or probability density." — Metaphysician Undercover

You're misunderstanding that sentence. The modulus squared is the probability that that particular quantity (the amplitude) will be measured.

The amplitude is a complex number associated with a quantum system. It's about the ontology. Whereas the probability (a real number between 0 and 1) is the predicted likelihood that that quantum system will be observed if a measurement were made. It's about the epistemology.

I don't see the premises whereby you make this conclusion. If we have a coin toss of 50/50 probability, and add another coin to the toss with a 50/50 probability, the fact that the two coins could interfere with each other in the air does not produce the conclusion that our description of the toss is not probabilistic. — Metaphysician Undercover

I'm referring to a coin that has already been flipped, where there is a single state that is unknown (e.g., it's hidden under my hand). Whereas a coin held in a superposition of heads and tails has two superposed states.

In both cases, the 0.5 heads/0.5 tails probabilities are predictions about what state will be observed, not about what states actually exist prior to observation.

That is an interesting paragraph as it encapsulates to me the philosophy-of-science debate going back and forth here. Mathematically states in a quantum superposition are probabilistic. 'Merely' is a matter of taste. A measurement occurs, the outcomes are mathematically understood. — mcdoodle

Mathematically, the states in a quantum superposition are represented as probability amplitudes (complex numbers), not probabilities (real numbers between 0 and 1).

You quote Duhem in arguing that putting things merely mathematically like that is not 'an explanation', but Duhem is long dead and there are new sorts of empiricists about who might happily use the word 'explain' about mathematical models. I think here it's realists who are also being rigid about what an explanation must involve; it can be a circular demand, in that if one isn't some sort of realist then what is one explaining? I do think that's a fruitless side-alley. — mcdoodle

I don't think it's unreasonable to ask why quantum interference effects occur or what nature is really like. I don't see the point in calling interpretations like Copenhagen "explanatory" if they deny that there can be answers to such questions.

The realist in turn only feels ok if like you they can point to what is 'real' (albeit hypothetical in that it's unobservable) in order to apply something like ordinary language to what happens in the maths. It *is* hard to talk about. — mcdoodle

Sure, it is hard to talk about. We don't knowingly encounter superpositions in everyday life, so QM is counterintuitive. But that doesn't mean that we should abandon ordinary language and realism. That's the crucial philosophical issue. Mathematical models help us to correct our intuitions and find the language we need to better reflect the world we find ourselves in.

Sure, it is hard to talk about. We don't knowingly encounter superpositions in everyday life, so QM is counterintuitive. But that doesn't mean that we should abandon ordinary language and realism. That's the crucial philosophical issue. Mathematical models help us to correct our intuitions and find the language we need to better reflect the world we find ourselves in. — Andrew M

On this we can (partly) agree, and thanks for the correction regarding probability amplitude; we are at the outer limit of my ability to talk maths/physics here, but I still hope that it doesn't require postgrad work in the subjects to debate the philosophical issues. I think my version of 'realism' is different from yours but I understand what you're saying. I'm sceptical that we can know 'what nature is really like', which is why I keep asking for the agnostic option in science: what is the minimum ontological commitment involved in such and such a proposition? It feels as if people of a scientific bent sometimes drift from the minimum to a greater 'metaphysical' realism that is to my mind just a metaphysical claim, not something that's necessary to agree on the proposition in question.

The amplitude is a complex number associated with a quantum system. It's about the ontology. Whereas the probability (a real number between 0 and 1) is the predicted likelihood that that quantum system will be observed if a measurement were made. It's about the epistemology. — Andrew M

I don't understand your distinction between ontology and epistemology. It's all mathematics, therefore it's all epistemology. You have arbitrarily singled out a part of the mathematics to say that this is the ontology. But I think you're going in the wrong direction. Instead of following me towards the fundamental principles, which the numbers refer to, and this is where you will find the ontology, for some reason you believe that you will find ontology in some complex mathematics. But that is backward.

I'm referring to a coin that has already been flipped, where there is a single state that is unknown (e.g., it's hidden under my hand). Whereas a coin held in a superposition of heads and tails has two superposed states. — Andrew M

This is a misrepresentation though. What is being referred to is an active system. So you cannot represent this as a case of the coin having been tossed, and the result is hidden under your hand. The system is an active system, so there are no states, there is no result. The ontology of the system must be represented as if the coin is in the air. That is your mistake, you want to represent states, talk about states, when there are no real states in an active system, it is inherently active. The idea that there are particles moving within the system makes you think that the particles must have definitive spatial-temporal positions, and the mathematics represents this. The uncertainty principle demonstrates that this is not the case, there are no definitive positions of particles. So you'll propose "superposition", and desire to assign some ontological status to this.

However, you're not addressing the true ontology of the active system, that it is active, and this is inherently incompatible with states. Consider that under special relativity, electromagnetism is pure energy. There is energy which moves from A to B, but it is impossible that a physical object moves from A to B, because an object cannot move at the speed of light, and this energy moves at the speed of light. The photoelectric effect demonstrates that energy leaves A and gets to B as particles, objects. The question is, how should you represent that energy which is moving from A to B. If you assume a particle, then you look for intermediate states, the positions of the particle. But there is no need to assume a particle. If the energy leaves A and appears at B as a particle, moving at the speed of light, and special relativity states that it is impossible for an object to move at the speed of light, why would you believe that a particle moves from A to B? This is the ontological question, why is the energy within the system represented as moving particles, that makes no sense.

The amplitude is a complex number associated with a quantum system. It's about the ontology. Whereas the probability (a real number between 0 and 1) is the predicted likelihood that that quantum system will be observed if a measurement were made. It's about the epistemology. — Andrew M

Don't you find it bizarre that you can (supposedly) go from an ontic state in Reality, to an epistemic state in a mind, just by taking the modulus squared?

I think my version of 'realism' is different from yours but I understand what you're saying. I'm sceptical that we can know 'what nature is really like', which is why I keep asking for the agnostic option in science: what is the minimum ontological commitment involved in such and such a proposition? It feels as if people of a scientific bent sometimes drift from the minimum to a greater 'metaphysical' realism that is to my mind just a metaphysical claim, not something that's necessary to agree on the proposition in question. — mcdoodle

As it happens, Many-Worlds does involve the minimum ontological commitment (albeit more of the same ontology we are already familiar with) - it's bare-bones QM without extra postulates. Whereas it requires extra effort to take the branches out of QM. Other approaches either change the philosophy (Copenhagen, instrumentalism) or change the physics (Bohmian Mechanics, Dynamical Collapse theories).

Regarding knowing 'what nature is really like', this is where translating mathematical models into understandable language is useful. QM is generally regarded as being not merely difficult, but inherently confusing and incomprehensible. Which is a philosophical issue.

It's all mathematics, therefore it's all epistemology — Metaphysician Undercover

The mathematics of QM describes the world. That is what is being modeled. It is not a description of people's knowledge.

This is a misrepresentation though. What is being referred to is an active system. So you cannot represent this as a case of the coin having been tossed, and the result is hidden under your hand. The system is an active system, so there are no states, there is no result. The ontology of the system must be represented as if the coin is in the air. That is your mistake, you want to represent states, talk about states, when there are no real states in an active system, it is inherently active. — Metaphysician Undercover

It's difficult to discuss this when you won't accept a simple, familiar example like the outcome of a coin toss. Of course there can be states in an active system, just as there can be frames in a movie.

The uncertainty principle demonstrates that this is not the case, there are no definitive positions of particles — Metaphysician Undercover

The uncertainty principle does not demonstrate this. It shows that an object can't have a precisely-defined position and precisely-defined momentum at the same time.

Don't you find it bizarre that you can (supposedly) go from an ontic state in Reality, to an epistemic state in a mind, just by taking the modulus squared? — tom

I see the Born rule as an empirical result that is useful for making predictions about the state that will be observed. But I don't have an explanation for the rule. Do you?

It's difficult to discuss this when you won't accept a simple, familiar example like the outcome of a coin toss. Of course there can be states in an active system, just as there can be frames in a movie. — Andrew M

For sure, there could be states like frames in a movie. But this is a completely different premise, not consistent with QM, it's a completely different ontology. If this is the case, then we have to account for a completely different type of activity, the activity of exchanging one frame for the next. This implies that the whole world would be in state X at one moment, and in state Y at the next moment. There would be no "movement" as we now conceive of it, "activity" would be a switching of one frame to the next. The mathematics of QM describes activity as motion though, it doesn't describe a switching of one frame to the next.

The uncertainty principle does not demonstrate this. It shows that an object can't have a precisely-defined position and precisely-defined momentum at the same time. — Andrew M

Have you ever considered that according to relativity theory, "at the same time" is relative, and, everything is moving. Therefore it's impossible that anything has a precisely defined spatial-temporal position. The uncertainty principle demonstrates this in practise.

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I see the Born rule as an empirical result that is useful for making predictions about the state that will be observed. But I don't have an explanation for the rule. Do you? — Andrew M

There are three unrelated derivations of the Born Rule that I'm aware of: Deutsch-Wallace, Zurek, Carrol and Sebens. But after checking Wallace's book "The Emergent Multiverse", there seem to be a variety of arguments of various degrees of formality.

Of course, in all of these derivations, only unitary evolution occurs, and in the end you get something ontological - i.e. branch weights.

I found it perplexing that you were willing to defend Many Worlds against a barrage of repetitive, uninformed criticism of extremely low quality, yet are willing to go all Copenhagen when it comes to the Born Rule! Anything as important as BR that is not an axiom, needs to be derived, and it is!

For sure, there could be states like frames in a movie. But this is a completely different premise, not consistent with QM, it's a completely different ontology. — Metaphysician Undercover

Quantum states are fundamental to QM. The Schrodinger equation describes how the quantum state of a quantum system changes with time.