If you now put a detector in path A , it will find a photon with probability ((1√2)2=12 ), and same for path B. This means that there is a 50% chance of the configuration |photon in path A only>, and 50% chance of the configuration |photon in path B only>. The arrow direction still has no effect on the probability.
Isn’t this kind of assertion implicitly taking a pretty strong stance on a particular philosophical interpretation?
We have some observations (counts of how many times each detector went off in past experiments), and a theory which explains those observations in terms of complex amplitudes and their magnitudes. A more agnostic stance would be to just say that the photon in the experiment is the amplitudes of the two configuration states, and the relative magnitudes of each state in this particular experiment are equal.
Combining the observations and the underlying theory to assign probabilities to future observations, or even talking about past observations in terms of probabilities, introduces a whole bunch of thorny philosophical questions: Bayesian vs. frequentist interpretations, what a “future” even is, and subjective experience.
But you can ignore all of those issues and treat the observation-counts as measuring a magnitude, and then use the theory to infer the underlying complex amplitudes, without ever having to talk about the ratios of the magnitudes as representing anything in particular, e.g. a probability of something happening (whatever it means for something to happen, anyway).
I personally think some or all of those philosophical questions have relatively straightforward answers, but the point is you don’t actually need to resolve them to understand the parts of QM introduced here, if you’re careful not to talk about the probabilities implied by your experimental observations as anything more than an afterthought.
You do sometimes need to introduce probabilities if you want to predict future observations (rather than just accepting the observation-counts as brute facts) for specific real experiments in our actual world, but once you’re doing experimentation and prediction in the real world (rather than just talking about thought experiments that represent different ways a world logically-possibly could be), you unavoidably have to deal with a bunch of philosophical questions anyway, mostly unrelated to QM itself.
Overall, I think this is a nice presentation of the basic concepts and math, and the diagrams in particular are a lot clearer than in the original.
But your prognostications about the intellectual “sinfulness” and “unforgivability” of mistakes in Eliezer’s original presentation are kind of weird and uncharitable.
For one, Eliezer offered his own disclaimer 16 years ago in a comment:
I think some of my readers may be overestimating the degree to which I intend to explain quantum mechanics, here. I’m not doing a textbook. I’m trying to get (reasonably smart nonphysicist) readers to the point where they’re no longer confused, and the remaining difficulties are mere matters of math.
For two, your specific claims about the likely confusion that Eliezer’s presentation could induce in “laymen” is empirically falsified to some degree by the comments on the original post: in at least one case, a reader noticed the issue and managed to correct for it when they made up their own toy example, and the first comment to explicitly mention the missing unitarity constraint was left over 10 years ago. [1]
Finally, I think the actual QM concepts here are sufficiently basic that I expect both you and Eliezer (and many LW readers) would get the right answers on a test consisting of questions about variations on these toy experiments (i.e. I predict that Eliezer, now or in 2008, would not have failed to account for unitarity when it mattered to the final answer).
So the relevant expertise for deciding whether a particular explanation has a “deep flaw” or which context and details are “important”, isn’t expertise in QM (since a solid grasp of the basics likely suffices for getting the right answers on test questions related to the toy problems presented here), but rather depends mostly on judgement and expertise related to pedagogy and technical explanation in general.
I think your presentation is a better and clearer presentation of the basic math and of our actual physical reality, especially for passive readers. (For active readers, e.g. GreedyAlgorithm, there are probably some minor pedagogical advantages to having to find flaws and work out the details on your own, which is a well-known skill for building up durable and true understanding.)
I think Eliezer’s presentation is clearer as an explanation of how and why QM-like theories are meaningful in the first place, and a good gears-level explanation of the kinds of predictions they make, qualitatively if not always quantitatively. It also lays a foundation for the philosophical points Eliezer makes in future posts, posts which are IMO much clearer and more correct treatments of the deeper philosophical issues than any other physicist or philosopher has ever written up.
Incidentally, that commenter suggests pretty much exactly the correction you make, which is to just replace the rule with a more physically-accurate one, without going into the details of why. But the commenter manages to avoid attacking the omission as some kind of sinful, irresponsible flaw and just notes that a different pedagogical choice might make the presentation more clear.
For two, your specific claims about the likely confusion that Eliezer’s presentation could induce in “laymen” is empirically falsified to some degree by the comments on the original post: in at least one case, a reader noticed the issue and managed to correct for it when they made up their own toy example, and the first comment to explicitly mention the missing unitarity constraint was left over 10 years ago.
Some readers figuring out what’s going on is consistent with many of them being unnecessarily confused.
Apologies for the late reply, but thank you for your detailed response.
Responding to your objection to my passage, I disagree, but I may edit it slightly to be clearer.
I was simply trying to point out the empirical fact that if you put a detector in path A and a detector in path B, and repeat the experiment a bunch of times, you will find the photon in detector A 50% of the time, and the photon in detector B 50% of the time. If the amplitudes had different values, you would empirically find them in different proportions, as given by the squared amplitudes.
I don’t find these probabilities to be an “afterthought”. This is the whole point of the theory, and the reason we consider quantum physics to be “true”. We never see these amplitudes directly, we infer them from the fact that they give correct probabilities via the Born rule. Or more specifically, this is the formula that works. That this formula works is an empirical fact, all the interpretations and debate are a question of why this formula works.
Regarding the defense of the original sequence, I’m sorry, but incorrect math is incorrect math. The people who figured out the mistake in the comments figured it out from other sources. If anything, it is even more damning that people pointed the mistake out 10 years ago, and it still hasn’t been fixed. For every person who figured out the problem or sifted through hundreds of comments to figure out the issue, there are dozens more who accepted the incorrect framework, or decided they were too dumb to understand the math when it was the author who was wrong.
My problem is that the post is misinforming people. I will make no apology for being harsh about that.
I will restrain my opinion on Eliezers other quantum posts for a future post when I tackle the overstated case for many worlds theories.
We never see these amplitudes directly, we infer them from the fact that they give correct probabilities via the Born rule. Or more specifically, this is the formula that works. That this formula works is an empirical fact, all the interpretations and debate are a question of why this formula works.
Sure, but inferring underlying facts and models from observations is how inference in general works; it’s not specific to quantum mechanics. Probability is in the Mind, even when those probabilities come from applying the Born rule.
Analogously, you could talk about various physical properties of a coin and mechanics of a flip, but synthesizing those properties into a hypothesized Coin Rule involves translating from physical properties inherent in the system itself, to facts which are necessarily entangled with your own map. This is true even if you have no way of measuring the physical properties themselves (even in principle) except by flipping the coin and using the Coin Rule to infer them back.
I’m a little confused by what your objection is. I’m not trying to stake out an interpretation here, I’m describing the calculation process that allows you to make predictions about quantum systems. The ontology of the wavefunction is a matter of heated debate, I am undecided on it myself.
Would you object to the following modification:
If you now put a detector in path A , it will find a photon with probability ((1√2)2=12 ), and same for path B. If you repeated this experiment a very large number of times, the results would converge to finding it 50% of the time in the configuration |photon in path A only>, and 50% of the time in the configuration |photon in path B only>. The arrow direction still has no effect on the probability.
I mildly object to the phrase “it will find a photon”. In my own terms, I would say that you will observe the detector going off 50% of the time (with no need to clarify what that means in terms of the limit of a large # of experiments), but the photon itself is the complex amplitudes of each configuration state, which are the same every time you run the experiment.
Note that I myself am taking a pretty strong stance on the ontology question, which you might object to or be uncertain about.
My larger point is that if you (or other readers of this post) don’t see the distinction between my phrasing and yours, or don’t realize that you are implicitly leaning on a particular interpretation (whether you’re trying to do so or not), I worry that you are possibly confused about something rather than undecided.
I actually don’t think this is a huge deal either way for a presentation that is focused on the basic mechanics and math. But I preregister some skepticism of your forthcoming post about the “overstated case for many worlds theories”.
I am assuming you are referring to the many worlds interpretation of quantum mechanics, where superpositions extend up to the human level, and the alternative configurations correspond to real, physical worlds with different versions of you that see different results on the detector.
Which is puzzling, because then why would you object to “the detector finding a photon”? The whole point of the theory is that detectors and humans are treated the same way. In one world, the detector finds the photon, and then spits out a result, and then one You sees the result, and in a different world, the detector finds the photon, spits out the other result, and a different result is seen. There is no difference between “you” and “it” here.
As for the photon “being” the complex amplitudes… That doesn’t sound right to me. Would you say that “you” are the complex amplitudes assigned to world 1 and world 2? It seems more accurate to say that there are two yous, in two different worlds (or many more).
Assuming you are a many worlder, may I ask which solution to the Born probabilities you favour?
I’m a many-worlder, yes. But my objection to “finding a photon” is actually that it is an insufficiently reductive treatment of wave-particle duality—a photon can sometimes behave like a little billiard ball, and sometimes like a wave. But that doesn’t mean photons themselves are sometimes waves and sometimes particles—the only thing that a photon can be that exhibits those different behaviors in different contexts is the complex amplitudes themselves.
The whole point of the theory is that detectors and humans are treated the same way. In one world, the detector finds the photon, and then spits out a result, and then one You sees the result, and in a different world, the detector finds the photon, spits out the other result, and a different result is seen. There is no difference between “you” and “it” here.
Yep! But I think treating the notion of a “you” at this level of reductiveness would actually be overly reductive and distracting in this context. (Picky, aren’t I?)
Would you say that “you” are the complex amplitudes assigned to world 1 and world 2? It seems more accurate to say that there are two yous, in two different worlds (or many more).
I would say that there are two people in two different worlds, but they’re both (almost entirely) me.
It often makes sense to talk about non-ontologically-basic concepts like a photon-as-a-little-billiard-ball, and a person-in-a-single-Everrett-branch as meaningful things. But the true notion of both a “me” and a “photon” requires drawing the conceptual boundaries around the complex amplitudes assigned to multiple worlds.
What part of “finding a photon” implies that the photon is a billiard ball? Wave-particle duality aside, a photon is a quanta of energy: the detector either finds that packet or it doesn’t (or in many worlds, one branched detector finds it and the other branched detector doesn’t).
I’m interested to hear more about how you interpret the “realness” of different branches. Say there is an electron in one of my pinky fingers that is in a superposition of spin up and spin down. Are there correspondingly two me’s, one with with pinky electron up and one with pinky electron down? Or is there a single me, described by the superposition of pinky electrons?
If the photon were only a quanta of energy which is entirely absorbed by the detector that actually fires, how could it have any causal effects (e.g. destructive interference) on the pathway where it isn’t detected?
OTOH, if your definition of “quanta of energy” includes the complex amplitude in the unmeasured path, then I think it’s more accurate to say that the detector finds or measures a component of the photon, rather than that it detects the photon itself. Why should the unmeasured component be any less real or less part of the photon than the measured part?
Say there is an electron in one of my pinky fingers that is in a superposition of spin up and spin down. Are there correspondingly two me’s, one with with pinky electron up and one with pinky electron down? Or is there a single me, described by the superposition of pinky electrons?
If there were a higher-dimensional being simulating a quantum universe, they could treat the up-electron and down-electron people as distinct and do different things to them (perhaps ones which violate the previous rules of the simulation).
But I think your own concept of yourself (for the purposes of making predictions about future observations, making decisions, reasoning about morality or philosophy, etc.) should be drawn such that it includes both versions (and many other closely-related ones) as a single entity.
Okay, let me break in down in terms of actual states, and this time, let’s add in the actual detection mechanism, say an electron in a potential well. Say the detector is in the ground state energy, E=0, and the absorption of a photon will bump it up to the next highest state, E=1. We will place this detector in path A, but no detector in path B.
At time t = 0, our toy wavefunction is:
1/sqrt2 |photon in path A, detector E=0> + 1/sqrt2 |photon in path B, detector E=0>
If the photon in A collides with the detector at time t =1, then at time t=2, our evolved wavefunction is:
Within the context of world A, a photon was found by the detector. This is a completely normal way to think and talk about this.
I think it’s straight up wrong to say “the photon is in the detector and in path B”. Nature doesn’t label photons, and it doesn’t distinguish between them. And what is actually in world A is an electron in a higher energy state: it would be weird to say it “contains” a photon inside of it.
Quantum mechanics does not keep track of individual objects, it keeps track of configurations of possible worlds, and assigns amplitudes to each possible way of arranging everything.
Here’s a crude Google Drawing of t = 0 to illustrate what I mean:
Both the concept of a photon and the concept of a world are abstractions on top of what is ultimately just a big pile of complex amplitudes; illusory in some sense.
I agree that talking in terms of many worlds (“within the context of world A...”) is normal and natural. But sometimes it makes sense to refer to and name concepts which span across multiple (conceptual) worlds.
I’m not claiming the conceptual boundaries I’ve drawn or terminology I’ve used in the diagram above are standard or objective or the most natural or anything like that. But I still think introducing probabilities and using terminology like “if you now put a detector in path A , it will find a photon with probability 0.5” is blurring these concepts together somewhat, in part by placing too much emphasis on the Born probabilities as fundamental / central.
But as a test, may I ask what you think the x-axis of the graph you drew is? Ie: what are the amplitudes attached to?
I’m not claiming the conceptual boundaries I’ve drawn or terminology I’ve used in the diagram above are standard or objective or the most natural or anything like that. But I still think introducing probabilities and using terminology like “if you now put a detector in path A , it will find a photon with probability 0.5” is blurring these concepts together somewhat, in part by placing too much emphasis on the Born probabilities as fundamental / central.
I think you’ve already agreed (or at least not objected to) saying that the detector “found the photon” is fine within the context of world A. I assume you don’t object to me saying that I will find the detector flashing with probability 0.5. And I assume you don’t think me and the detector should be treated differently. So I don’t think there’s any actual objection left here, you just seem vaguely annoyed that I mentioned the empirical fact that amplitudes can be linked to probabilities of outcomes. I’m not gonna apologise for that.
But as a test, may I ask what you think the x-axis of the graph you drew is? Ie: what are the amplitudes attached to?
Position, but it’s not meant to be an actual graph of a wavefunction pdf; just a way to depict how the concepts can be sliced up in a way I can actually draw in 2 dimensions.
If you do treat it as a pdf over position, a more accurate way to depict the “world” concept might be as a line which connects points on the diagram for each time step. So for a fixed time step, a world is a single point on the diagram, representing a sample from the pdf defined by the wavefunction at that time.
“position” is nearly right. The more correct answer would be “position of one photon”.
If you had two electrons, say, you would have to consider their joint configuration. For example, one possible wavefunction would look like the following, where the blobs represent high amplitude areas:
This is still only one dimensional: the two electrons are at different points along a line. I’ve entangled them, so if electron 1 is at position P, electron 2 can’t be.
Now, try and point me to where electron 1 is on the graph above.
You see, I’m not graphing electrons here, and neither were you. I’m graphing the wavefunction. This is where your phrasing seems a little weird: you say the electron is the collection of amplitudes you circled: but those amplitudes are attached to configurations saying “the electron is at position x1” or “the electron is at position x2″. It seems circular to me. Why not describe that lump as “a collection of worlds where the electron is in a similar place”?
If you have N electrons in a 3d space, the wavefunction is not a vector in 3d space (god I wish, it would make my job a lot easier). It’s a vector in 3N+1 dimensions, like the following:
where r1, r2, etc are pointing to the location of electron 1, 2, 3, etc, and each possible configuration of electron 1 here, electron 2 there, etc, has an amplitude attached, with configurations that are more often encountered experimentally empirically having higher amplitudes.
An important point about detecting the photon is that the detector absorbs all the energy of the photon: it’s not as if it is classically sampling part of a distributed EM field. That’s still true if the photon is never a point particle.
Isn’t this kind of assertion implicitly taking a pretty strong stance on a particular philosophical interpretation?
We have some observations (counts of how many times each detector went off in past experiments), and a theory which explains those observations in terms of complex amplitudes and their magnitudes. A more agnostic stance would be to just say that the photon in the experiment is the amplitudes of the two configuration states, and the relative magnitudes of each state in this particular experiment are equal.
Combining the observations and the underlying theory to assign probabilities to future observations, or even talking about past observations in terms of probabilities, introduces a whole bunch of thorny philosophical questions: Bayesian vs. frequentist interpretations, what a “future” even is, and subjective experience.
But you can ignore all of those issues and treat the observation-counts as measuring a magnitude, and then use the theory to infer the underlying complex amplitudes, without ever having to talk about the ratios of the magnitudes as representing anything in particular, e.g. a probability of something happening (whatever it means for something to happen, anyway).
I personally think some or all of those philosophical questions have relatively straightforward answers, but the point is you don’t actually need to resolve them to understand the parts of QM introduced here, if you’re careful not to talk about the probabilities implied by your experimental observations as anything more than an afterthought.
You do sometimes need to introduce probabilities if you want to predict future observations (rather than just accepting the observation-counts as brute facts) for specific real experiments in our actual world, but once you’re doing experimentation and prediction in the real world (rather than just talking about thought experiments that represent different ways a world logically-possibly could be), you unavoidably have to deal with a bunch of philosophical questions anyway, mostly unrelated to QM itself.
Overall, I think this is a nice presentation of the basic concepts and math, and the diagrams in particular are a lot clearer than in the original.
But your prognostications about the intellectual “sinfulness” and “unforgivability” of mistakes in Eliezer’s original presentation are kind of weird and uncharitable.
For one, Eliezer offered his own disclaimer 16 years ago in a comment:
For two, your specific claims about the likely confusion that Eliezer’s presentation could induce in “laymen” is empirically falsified to some degree by the comments on the original post: in at least one case, a reader noticed the issue and managed to correct for it when they made up their own toy example, and the first comment to explicitly mention the missing unitarity constraint was left over 10 years ago. [1]
Finally, I think the actual QM concepts here are sufficiently basic that I expect both you and Eliezer (and many LW readers) would get the right answers on a test consisting of questions about variations on these toy experiments (i.e. I predict that Eliezer, now or in 2008, would not have failed to account for unitarity when it mattered to the final answer).
So the relevant expertise for deciding whether a particular explanation has a “deep flaw” or which context and details are “important”, isn’t expertise in QM (since a solid grasp of the basics likely suffices for getting the right answers on test questions related to the toy problems presented here), but rather depends mostly on judgement and expertise related to pedagogy and technical explanation in general.
I think your presentation is a better and clearer presentation of the basic math and of our actual physical reality, especially for passive readers. (For active readers, e.g. GreedyAlgorithm, there are probably some minor pedagogical advantages to having to find flaws and work out the details on your own, which is a well-known skill for building up durable and true understanding.)
I think Eliezer’s presentation is clearer as an explanation of how and why QM-like theories are meaningful in the first place, and a good gears-level explanation of the kinds of predictions they make, qualitatively if not always quantitatively. It also lays a foundation for the philosophical points Eliezer makes in future posts, posts which are IMO much clearer and more correct treatments of the deeper philosophical issues than any other physicist or philosopher has ever written up.
Incidentally, that commenter suggests pretty much exactly the correction you make, which is to just replace the rule with a more physically-accurate one, without going into the details of why. But the commenter manages to avoid attacking the omission as some kind of sinful, irresponsible flaw and just notes that a different pedagogical choice might make the presentation more clear.
Some readers figuring out what’s going on is consistent with many of them being unnecessarily confused.
Apologies for the late reply, but thank you for your detailed response.
Responding to your objection to my passage, I disagree, but I may edit it slightly to be clearer.
I was simply trying to point out the empirical fact that if you put a detector in path A and a detector in path B, and repeat the experiment a bunch of times, you will find the photon in detector A 50% of the time, and the photon in detector B 50% of the time. If the amplitudes had different values, you would empirically find them in different proportions, as given by the squared amplitudes.
I don’t find these probabilities to be an “afterthought”. This is the whole point of the theory, and the reason we consider quantum physics to be “true”. We never see these amplitudes directly, we infer them from the fact that they give correct probabilities via the Born rule. Or more specifically, this is the formula that works. That this formula works is an empirical fact, all the interpretations and debate are a question of why this formula works.
Regarding the defense of the original sequence, I’m sorry, but incorrect math is incorrect math. The people who figured out the mistake in the comments figured it out from other sources. If anything, it is even more damning that people pointed the mistake out 10 years ago, and it still hasn’t been fixed. For every person who figured out the problem or sifted through hundreds of comments to figure out the issue, there are dozens more who accepted the incorrect framework, or decided they were too dumb to understand the math when it was the author who was wrong.
My problem is that the post is misinforming people. I will make no apology for being harsh about that.
I will restrain my opinion on Eliezers other quantum posts for a future post when I tackle the overstated case for many worlds theories.
Sure, but inferring underlying facts and models from observations is how inference in general works; it’s not specific to quantum mechanics. Probability is in the Mind, even when those probabilities come from applying the Born rule.
Analogously, you could talk about various physical properties of a coin and mechanics of a flip, but synthesizing those properties into a hypothesized Coin Rule involves translating from physical properties inherent in the system itself, to facts which are necessarily entangled with your own map. This is true even if you have no way of measuring the physical properties themselves (even in principle) except by flipping the coin and using the Coin Rule to infer them back.
I’m a little confused by what your objection is. I’m not trying to stake out an interpretation here, I’m describing the calculation process that allows you to make predictions about quantum systems. The ontology of the wavefunction is a matter of heated debate, I am undecided on it myself.
Would you object to the following modification:
I mildly object to the phrase “it will find a photon”. In my own terms, I would say that you will observe the detector going off 50% of the time (with no need to clarify what that means in terms of the limit of a large # of experiments), but the photon itself is the complex amplitudes of each configuration state, which are the same every time you run the experiment.
Note that I myself am taking a pretty strong stance on the ontology question, which you might object to or be uncertain about.
My larger point is that if you (or other readers of this post) don’t see the distinction between my phrasing and yours, or don’t realize that you are implicitly leaning on a particular interpretation (whether you’re trying to do so or not), I worry that you are possibly confused about something rather than undecided.
I actually don’t think this is a huge deal either way for a presentation that is focused on the basic mechanics and math. But I preregister some skepticism of your forthcoming post about the “overstated case for many worlds theories”.
I am assuming you are referring to the many worlds interpretation of quantum mechanics, where superpositions extend up to the human level, and the alternative configurations correspond to real, physical worlds with different versions of you that see different results on the detector.
Which is puzzling, because then why would you object to “the detector finding a photon”? The whole point of the theory is that detectors and humans are treated the same way. In one world, the detector finds the photon, and then spits out a result, and then one You sees the result, and in a different world, the detector finds the photon, spits out the other result, and a different result is seen. There is no difference between “you” and “it” here.
As for the photon “being” the complex amplitudes… That doesn’t sound right to me. Would you say that “you” are the complex amplitudes assigned to world 1 and world 2? It seems more accurate to say that there are two yous, in two different worlds (or many more).
Assuming you are a many worlder, may I ask which solution to the Born probabilities you favour?
I’m a many-worlder, yes. But my objection to “finding a photon” is actually that it is an insufficiently reductive treatment of wave-particle duality—a photon can sometimes behave like a little billiard ball, and sometimes like a wave. But that doesn’t mean photons themselves are sometimes waves and sometimes particles—the only thing that a photon can be that exhibits those different behaviors in different contexts is the complex amplitudes themselves.
Yep! But I think treating the notion of a “you” at this level of reductiveness would actually be overly reductive and distracting in this context. (Picky, aren’t I?)
I would say that there are two people in two different worlds, but they’re both (almost entirely) me.
It often makes sense to talk about non-ontologically-basic concepts like a photon-as-a-little-billiard-ball, and a person-in-a-single-Everrett-branch as meaningful things. But the true notion of both a “me” and a “photon” requires drawing the conceptual boundaries around the complex amplitudes assigned to multiple worlds.
What part of “finding a photon” implies that the photon is a billiard ball? Wave-particle duality aside, a photon is a quanta of energy: the detector either finds that packet or it doesn’t (or in many worlds, one branched detector finds it and the other branched detector doesn’t).
I’m interested to hear more about how you interpret the “realness” of different branches. Say there is an electron in one of my pinky fingers that is in a superposition of spin up and spin down. Are there correspondingly two me’s, one with with pinky electron up and one with pinky electron down? Or is there a single me, described by the superposition of pinky electrons?
If the photon were only a quanta of energy which is entirely absorbed by the detector that actually fires, how could it have any causal effects (e.g. destructive interference) on the pathway where it isn’t detected?
OTOH, if your definition of “quanta of energy” includes the complex amplitude in the unmeasured path, then I think it’s more accurate to say that the detector finds or measures a component of the photon, rather than that it detects the photon itself. Why should the unmeasured component be any less real or less part of the photon than the measured part?
If there were a higher-dimensional being simulating a quantum universe, they could treat the up-electron and down-electron people as distinct and do different things to them (perhaps ones which violate the previous rules of the simulation).
But I think your own concept of yourself (for the purposes of making predictions about future observations, making decisions, reasoning about morality or philosophy, etc.) should be drawn such that it includes both versions (and many other closely-related ones) as a single entity.
Okay, let me break in down in terms of actual states, and this time, let’s add in the actual detection mechanism, say an electron in a potential well. Say the detector is in the ground state energy, E=0, and the absorption of a photon will bump it up to the next highest state, E=1. We will place this detector in path A, but no detector in path B.
At time t = 0, our toy wavefunction is:
1/sqrt2 |photon in path A, detector E=0> + 1/sqrt2 |photon in path B, detector E=0>
If the photon in A collides with the detector at time t =1, then at time t=2, our evolved wavefunction is:
1/sqrt2 |no free photon, detector E=1> + 1/sqrt2 |photon in path B, detector E=0>
Within the context of world A, a photon was found by the detector. This is a completely normal way to think and talk about this.
I think it’s straight up wrong to say “the photon is in the detector and in path B”. Nature doesn’t label photons, and it doesn’t distinguish between them. And what is actually in world A is an electron in a higher energy state: it would be weird to say it “contains” a photon inside of it.
Quantum mechanics does not keep track of individual objects, it keeps track of configurations of possible worlds, and assigns amplitudes to each possible way of arranging everything.
Here’s a crude Google Drawing of t = 0 to illustrate what I mean:
Both the concept of a photon and the concept of a world are abstractions on top of what is ultimately just a big pile of complex amplitudes; illusory in some sense.
I agree that talking in terms of many worlds (“within the context of world A...”) is normal and natural. But sometimes it makes sense to refer to and name concepts which span across multiple (conceptual) worlds.
I’m not claiming the conceptual boundaries I’ve drawn or terminology I’ve used in the diagram above are standard or objective or the most natural or anything like that. But I still think introducing probabilities and using terminology like “if you now put a detector in path A , it will find a photon with probability 0.5” is blurring these concepts together somewhat, in part by placing too much emphasis on the Born probabilities as fundamental / central.
Nice graph!
But as a test, may I ask what you think the x-axis of the graph you drew is? Ie: what are the amplitudes attached to?
I think you’ve already agreed (or at least not objected to) saying that the detector “found the photon” is fine within the context of world A. I assume you don’t object to me saying that I will find the detector flashing with probability 0.5. And I assume you don’t think me and the detector should be treated differently. So I don’t think there’s any actual objection left here, you just seem vaguely annoyed that I mentioned the empirical fact that amplitudes can be linked to probabilities of outcomes. I’m not gonna apologise for that.
Position, but it’s not meant to be an actual graph of a wavefunction pdf; just a way to depict how the concepts can be sliced up in a way I can actually draw in 2 dimensions.
If you do treat it as a pdf over position, a more accurate way to depict the “world” concept might be as a line which connects points on the diagram for each time step. So for a fixed time step, a world is a single point on the diagram, representing a sample from the pdf defined by the wavefunction at that time.
“position” is nearly right. The more correct answer would be “position of one photon”.
If you had two electrons, say, you would have to consider their joint configuration. For example, one possible wavefunction would look like the following, where the blobs represent high amplitude areas:
This is still only one dimensional: the two electrons are at different points along a line. I’ve entangled them, so if electron 1 is at position P, electron 2 can’t be.
Now, try and point me to where electron 1 is on the graph above.
You see, I’m not graphing electrons here, and neither were you. I’m graphing the wavefunction. This is where your phrasing seems a little weird: you say the electron is the collection of amplitudes you circled: but those amplitudes are attached to configurations saying “the electron is at position x1” or “the electron is at position x2″. It seems circular to me. Why not describe that lump as “a collection of worlds where the electron is in a similar place”?
If you have N electrons in a 3d space, the wavefunction is not a vector in 3d space (god I wish, it would make my job a lot easier). It’s a vector in 3N+1 dimensions, like the following:
where r1, r2, etc are pointing to the location of electron 1, 2, 3, etc, and each possible configuration of electron 1 here, electron 2 there, etc, has an amplitude attached, with configurations that are more often encountered experimentally empirically having higher amplitudes.
An important point about detecting the photon is that the detector absorbs all the energy of the photon: it’s not as if it is classically sampling part of a distributed EM field. That’s still true if the photon is never a point particle.
The essay Probability is in the Mind doesn’t prove that probability is only in the mind.