What is Entanglement?

Wikipedia:

Quantum entanglement is the phenomenon where the quantum state of each particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical physics and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.

Note that it does not say that the state of the particle depends on the state of the distant particle. We have no proof that it does. We only know that the formalism of quantum mechanics is unable to describe the states independently. Maybe the states are independent, but we do not have the math to describe them independently.

Stanford Encyclopedia of Philosophy:

Quantum entanglement is a physical resource, like energy, associated with the peculiar nonclassical correlations that are possible between separated quantum systems. Entanglement can be measured, transformed, and purified. A pair of quantum systems in an entangled state can be used as a quantum information channel to perform computational and cryptographic tasks that are impossible for classical systems.

Here, entanglement is not just a correlation in our mathematical descriptions, it is a resource like energy that can be used for communication and computation.

Is that really what it is? Quantum mechanics is a century old, and Schroedinger wrote a paper stressing the crucial importance of entanglement in 1935, but he never said it was a resource. Dozens of Nobel Prizes have been given for quantum mechanics, but none for using entanglement as a resource.

Use of energy as a resource is evident everywhere we look. Is there some compelling experiment demonstrating entanglement as a resource? A quantum computer showing quantum supremacy would do, and some people have claimed that, but the evidence is unconvincing.

There are experiments using entanglement for communication, but communication is much easier by other methods. Dr. Bee has a video on the coming quantum internet, but I do not see any practical use for it.

Sometimes people think that quantum decoherence explains entanglement, but that is not quite right. Even after Schroedinger's cat decoheres, the live cat state is still said to be entangled with the dead cat state.

I am skeptical that entanglement is anything real, like being a resource. Maybe it is just a property of the mathematical formalism, and no more.

I am guessing that most physicists will say I am wrong about this. Okay maybe so, but where is the experiment that proves me wrong? And why didn't someone get a prize for this discovery?

Maybe you will say that happened in 2022, but here is the announcement:

Alain Aspect, John Clauser and Anton Zeilinger have each conducted groundbreaking experiments using entangled quantum states, where two particles behave like a single unit even when they are separated. Their results have cleared the way for new technology based upon quantum information. ...

For a long time, the question was whether the correlation was because the particles in an entangled pair contained hidden variables, ...

So they demonstrated the correlations, ruled out hidden variables, and cleared the way for new technology. In other words, they confirmed what everyone thought for a century. But where is that new technology?

“It has become increasingly clear that a new kind of quantum technology is emerging. We can see that the laureates’ work with entangled states is of great importance, even beyond the fundamental questions about the interpretation of quantum mechanics,” says Anders Irbäck, Chair of the Nobel Committee for Physics.

So he is convinced that a new technology is emerging. I expect to see a prize given for that new technology, when it emerges. But I don't think we have any proof that it exists.

Sean M. Carroll is probably the leading popularizer of QM, but he gets entanglement wrong, as explained in a recent paper:

A related example comes from Sean Carroll’s book From Eternity To Here: The Quest for the Ultimate Theory of Time (Dutton, 2010). This book contains a fairly lengthy section on entanglement and, nominally, the EPR thought-experiment. Carroll gives the EPR paper itself a brief mention and credit for introducing the concept of an entangled ψ function, but he does not address their motivation, how they wanted to show that quantum particles have definite values of position and momentum even before those quantities are measured. The bulk of the section is a tale of two animals for whom a joint ψ function is written. When “Miss Kitty” is observed, she is found to be either on the table or on the sofa; when “Mr. Dog” is observed, he is found to be either in the living room or in the yard.

Even though we have no idea where Mr. Dog is going to be before we look, if we first choose to look for Miss Kitty, once that observation is complete we know exactly where Mr. Dog is going to be, even without looking for him! That’s the magic of entanglement.

No, it isn’t. It’s an unremarkable possibility that could occur in everyday life. The entire buildup to this declaration is beside the point.5 None of the conceptual or mathematical apparatus of quantum theory is necessary for Carroll’s scenario, and a big sign of why is that the story considers only one observable, the location, of each character.

To elaborate, correlations occur in classical mechanics and in everyday life all the time. In the simplest example, imagine that two billiard balls collide and bounce apart. Knowing something about the initial conditions and measuring one ball will tell you something about the other. There is nothing mysterious about that.

As the paper explains, to get the EPR paradox, you have to combine these correlations with Heisenberg uncertainty. There have to be two observables such that measuring one forces a quantum uncertainty in the other. The correlation in those uncertainties is what requires quantum rules, and cannot be modeled classically. These lame explanations like Miss Kitty and Mr. Dog miss the point of entanglement.

As the paper explains, the Einsteinian talk about elements of physical reality, as some people say EPR abbreviates, also does not get at how QM deviates from a classical theory. To get that you have to look at numerical values for correlations. The qualitative and philosophical arguments prove nothing.

The EPR paper does not look at those numerical correlations, and hence does not show that QM deviates from a classical theory. So it is not clear that Einstein understood entanglement as non-classical. Maybe he wanted a classical theory to explain quantum predictions. The proper resolution of the EPR paradox is that there is no such classical theory. That is the point of the 2022 Nobel Prize.

In a new video:

Matt O'Dowd picks apart the mystery of quantum entanglement, offering different interpretations for this baffling phenomena.

Some of what he says is correct, but he says:

so this is this spooky action at a 1:20 distance and it I mean I just got 1:22 shivers right now thinking about it it 1:25 it it's weird as hell anyway we we've 1:27 now demonstrated that exactly this 1:31 you know redu absurdum that Einstein 1:33 proposed to to to 1:36 discount the idea of of that standard 1:38 quantum mechanics has proved to be real ...

it it really seems like the 3:27 the choice of measurement affects both 3:30 particles

No, there is no spooky action at a distance, and the choice of measurement only affects the particle being measured. I thought he might get it right when he said:

a hidden 3:03 variable interpretation would say that 3:06 this particle really knew what it was 3:08 all along

You could say that in a classical theory, particles know what they are all along, and in a quantum theory they don't.

The Mind of God is Strings

I wrote a book on How Einstein Ruined Physics, and this man illustrates it. See this new video:

String Theory is Still The Only Game in Town - Michio Kaku

Michio Kaku – world-renowned theoretical physicist and co-founder of string theory – reveals why the long-sought “Theory of Everything” might be within reach. In this mind-expanding interview, he explains how the universe is a symphony of vibrating strings, resonating through eleven dimensions. Kaku explores black holes, time as an illusion, the multiverse, and the tantalizing idea that our universe might be a hologram. This isn't just about physics. It's about the fundamental nature of reality. ...

the universe is a symphony of 1:07 strings. and then what is the mind of God 1:12 that Albert Einstein wrote about 1:14 eloquently for the last 30 years of his 1:17 life? what would be the mind of God the 1:21 mind of God would be? cosmic music 1:25 resonating through 11-dimensional 1:28 hyperspace. that would be the mind of God. 1:31 many scientists believe that the 1:33 ultimate goal of physics is 1:37 unification to unite the left hand and 1:40 the right hand of the universe to unite 1:42 quantum mechanics with relativity to 1:45 create a relativistic theory of quantum 1:48 mechanics. that is the ultimate goal so 1:52 far the only theory the only theory 1:54 which has survived all challenges is 1:58 string theory.

He is now on the fringe, but leading theoretical physicists tell a similar story. That Einstein spent 30 years looking for the mind of God, and now string theory is the only candidate.

The universe is not a symphony of strings. There is no way to understand particles as vibrating strings. There is not really any theory. It certainly has not survived challenges. It does not unite quantum mechanics with relativity. It is all nonsense.

This Einstein dream is not even a worthwhile objective. It would not explain anything in the physical world. The only thing it predicted was supersymmetry, and that was disproved by LHC experiments.

Many-worlds needs another 50 Years

Sean M. Carroll posted his monthly AMA podcast, and he pushes many-worlds as usual:

3:09:38 jonathan Goodson says "I heard you opine that within 50 years many worlds will emerge as the dominant interpretation of 3:09:44 quantum mechanics yet over the past 50 years the number of different interpretations has increased significantly and no major contender has 3:09:51 yet been ruled out." As NDavid Murman remarked "New interpretations appear every year none ever disappear." What 3:09:57 would need to occur for many worlds to buck that trend and emerge as the clear winner and why did you predict the resolution will occur in this century

3:10:05 you know I might be wrong in my prediction but that is my prediction and I think that these processes are gradual i don't think that there's any threshold 3:10:12 you cross and suddenly everyone says oh yes it must be many worlds. i think that 3:10:17 what will happen is there is now here in 2025 a lot more attention being paid to 3:10:23 quantum foundations than there was 50 years ago. so that will lead to more progress and the way the progress will 3:10:30 happen is people will think deeply about the different models and their implications and they will both look for 3:10:36 experimental tests of them and they will ask how those models fit in with other things we think are true in physics 3:10:43 whether it's quantum gravity or particle physics or atomic physics or whatever and certain things will fit better and 3:10:49 better certain things will fit worse and worse and I I do believe that the progress will be made you know maybe 3:10:55 it's wrong that it'll be 50 years maybe it'll be 200 years i don't know for sure 3:11:00 but at the rate at which I see progress being made I'm optimistic that it'll be 50 years.

No, there has been no progress in quantum foundations for about 90 years. Our best theory is more or less the same as Von Neumann's 1927 Trilogy on the Foundations of Quantum Mechanics.

A common complaint is that von Neumann's theorem against hidden variables had a locality assumption, and did not rule out spooky theories like Bohm's. That's right, but those theories are considered unphysical by most.

Many-worlds started in 1957 with Everett's thesis. That was 68 years ago, and it still makes no sense.

Physicist Dismantles Many Worlds Theory

New 16-minute video clip posted: Harvard Physicist Dismantles the “Many Worlds” Theory.

It is an excerpt from the must longer podcast: Harvard Scientist Rewrites the Rules of Quantum Mechanics | Scott Aaronson Λ Jacob Barandes. Aaronson comments on his blog in March.

This is a good criticism of Many-worlds theory, and agrees with what I have posted. Briefly, there is no value to a theory that says anything can happen. The probabilities do not make any sense. Some many-worlds advocates either add a probability axiom, or they have some way of saying you can believe in the probabilities even though they are not literally correct. But these explanations do not really work.

Research on decoherence does not really help either. It may inform about when a split world becomes invisible, but it does not say anything about whether the split world should be regarded as real.

Aaronson has occasionally said that he believes in many-worlds, so I expected more push-back from him. But no, he seemed to concede these points. His main defense of many-worlds was that he finds it useful when explaining quantum computers to his students!

Apparently it is easier to believe in quantum computers if you also believe in many-worlds. That is what David Deutsch says anyway.

I would have commented on this earlier, but I tried to watch the longer video, but I got bored with their discussions of goofy alternative theories.

Sean M. Carroll Debates Eric Weinstein

Piers Morgan Uncensored is a tv show that usually covers political topics, but hosted a Physics debate:

“Don’t Talk About Physics Fight Club” Eric Weinstein vs Sean Carroll Science SHOWDOWN

For centuries, scientists have grappled with the most fundamental question of them all - what is reality?

Is it a matter of common sense? Or can God or some higher being only know? And what was there before the Big Bang created the world we live in?

Einstein revolutionised our way of thinking - and then came along the wild world of quantum physics, where nothing plays by the rules - even Einstein's - and things seemed to exist in two places at once.

Today, the biggest brains on the planet are at loggerheads over what all this means - and for this special Uncensored debate, Piers Morgan is joined by two scientists with very different answers to the big questions - Dr Eric Weinstein and Professor Sean Carroll.

Weinstein declares:

I think that 6:57 Sean and I probably agree almost completely on everything until 7:03 1972 then you have the first uh really serious situation where a mysterious 7:10 term is not only introduced into our common vocabulary uh called quantum 7:15 gravity um but it takes over as a fictitious history that the uh central 7:22 problem of theoretical physics is the quantization of gravity.

Yes, 1972 was the end of the golden age of theoretical physics. The Standard Model was presented, and we then had good theories for the four fundamental forces. Those theories have held up well.

Then theoretical physicists went into wild goose chases for quantum gravity, string theory, and theories of everything. Very little has come out of it.

Weinstein has his own goofy theories that have gone nowhere.

Carroll is more directly tied to mainstream Physics work, but also promotes nonsense like many-worlds theory.

I wonder what the average non-physicist thinks of this debate. These were supposed to be leading scientists. Both have large online followings, and respectable credentials, without significant research accomplishments. But no actual science is discussed. It is all just goofy speculation and complaints about wacky ideas not being taken seriously. Weinstein says, in response to a question about time travel:

Well as as Sean has thoroughly digested 48:50 uh my paper he knows that I believe that there are either five or seven dimensions of time in a 14-dimensional 48:58 world which is split five of time uh 9 of space or seven of time seven of space 49:04 so when you talk about time travel it's times travel and only when time is 49:09 one-dimensional is there an arrow of time.

Carroll answers a question about the Big Bang:

my personal favorite theory is there was a pre-existing universe out of which our cosmos arose 52:20 as a baby universe quantum fluctuation what was there before that no it's infinite in it's infinite in time in 52:25 both directions so what was there before so you mean it's never ending yeah. I 52:32 wrote a book if you want to read it, pierce, from eternity to here. it's a great book.

Morgan is bewildered by all this, and says he believes in God. Normally I would say that theism is an unscientific matter of faith, but he seems more reasonable than these physicists.

Sabine Hossenfelder just released a video, Why Theories of Everything Keep Failing, trashing Weinstein among others. But her complaints are largely philosophical, and her dream theories are just as bad.

The real reason that theories of everything keep failing is that we already have one. The Standard Model plus general relativity explains all known tests of the fundamental forces. The unified theories are not even intended to explain any real-world observations, but to satisfy some philosophical desire for supersymmetry, proton decay, 5-dimensional time, or other such nonsense.

I wonder why there are not legitimate physicists pushing sensible science to the general public. If I were a big-shot Physics professor, I would be annoyed that the profession is being represented by podcasters with such silly and unscientific ideas. Where is R.P. Feynman when we need him? Surely a real physicist with real Physics stories could get to go on Piers Morgan and tell a better story.

My guess is that the real physicists are embarrassed by the lack of accomplishments in the field for the last 50 years, and by the wacky stories they and their colleagues have to tell to get media attention. Lisa Randall is one of the more sensible Physics professors, but to get attention she had to tell a crazy story about how dark matter wiped out the dinosaurs. Even still, the press wanted her to tell stories about how she had to make coffee for everyone in the lab, and she did not have such stories.

Professor Dave has 3.8 million subscribers, so he is also a top science communicator. He is also aggressively hostile towards Pres. Trump and everyone else he dislikes. He does a takedown of the debate:

Sean Carroll Humiliates Eric Weinstein (Piers Morgan is Also Dumb)

Eric Weinstein loves pretending to be a physicist. This is something he is able to do on podcasts because most people can't tell that he isn't one. But real physicists can see right through him. And today we get to watch one of these real physicists, Sean Carroll, tell Eric that he isn't a physicist at all right to his smug face, over and over again. It's extremely satisfying.

Carroll sides strongly with Carroll over Weinstein.

It is true that Carroll is closer to mainstream Physics than Weinstein, but they are both pushing goofy and untestable abstract theories. I find them all very annoying. Weinstein is pushing junk that is not even published, and Carroll is pushing junk that is published. Professor Dave is smart enough to see through Weinstein, but not smart enough to see through Carroll.

Sean he deserves a round of applause for sticking it to 51:20 Eric like he deserves. it's still not quite enough fire for me as the segment wrapped without Eric being held 51:27 accountable for most of what he said but it's still a step in the right direction. and I'll just have to accept that 51:32 academics will likely never be as aggressive as I would be. at any rate I hope that Shawn can serve as an example 51:39 for other scholars such that we see more of them dunking on frauds in precisely this manner.

Dave applauds Carroll for exceoriating Weinstein for failing to show that his theory reproduces scattering experiment calculations that the Standard Model does so well. Good point, but Carroll praises string theory and many-worlds theory, and they also fail to make any verifiable quantitative prediction.

If you want more from Weinstein, Physics is Dead | feat. Eric Weinstein also dropped this week. He is the older brother to Bret Weinstein, who has a strong online following in evolutionary biology, and is no relation to Galina Weinstein, whom I criticized last week. He also worked for Peter Thiel.

Gotten Wrong About Quantum Physics

New interview:

What We've Gotten Wrong About Quantum Physics

Are there unresolved foundational questions in quantum physics? Philosopher Tim Maudlin thinks so, and joins Brian Greene to explore possible answers.

This program is part of the Big Ideas series, supported by the John Templeton Foundation.

I think they are both leading us astray, and here is the crux:

[Greene] if we're thinking about bore and Is the Copenhagen approach even a theory? 11:51 the Copenhagen spirit or school or whatever do you consider what is 11:58 normally described as the Copenhagen approach is it actually a theory?

[Maudlin] no No 12:03 it's certainly not a theory. What I mean by a theory I think what anybody ought to mean by a theory is a 12:11 presentation of fairly high precision. I don't want to say perfectly precise precision because we've never had that 12:17 for reasons we might talk about, but of fairly high precision, about saying 12:23 here's what I think exists in the world. so what is there and here's how I think it behaves so that a dynamics which is 12:30 usually presented now in mathematical physics. you use a mathematical formalism 12:36 to represent what there is, and then in terms of that you write down dynamical equations which could be deterministic 12:43 or not probabilistic. Uh that give you a dynamics I would say a theory has to 12:49 have both of those to a reasonably high precision. And if you don't have that you 12:56 just don't have a theory. You may have a perfectly good kind of recipe. what I 13:02 call in my book a recipe for making some predictions but you don't really have a 13:07 theory right.

Got that? Maudlin says that a theory has to have, at a minimum, a precise mathematical representation of what exists, and dynamical equations for making predictions.

Ordinary Copenhagen quantum mechanics satisfies the second condition but fails the first. As Bohr supposedly said, "there is no quantum world." People disagree about what that is supposed to mean, but I take it to mean that QM fails Maudlin's first criterion for a theory.

This sums up my disagreements with Maudlin. I posted my own definition of a theory, and it emphatically does not include Maudlin's first criterion.

Bohr first got famous with his 1911-18 model of the atom. There is tried to precisely describe what exists in the atom. The model had some successes, and was ultimately supplanted by QM. QM describes properties of atom, but it is not clear that it makes sense to have a precise mathematical model of what exists.

Later on, after 42:00, Maudlin tells about attempts to make QM comply with criterion one. In the jargon of von Neumann, Bohm, and Bell, this meant introducing hidden variables. The problem is that such theories are nonlocal and nonrelativistic. But Maudlin persists, in order to achieve what he thinks a theory should be.

I was always baffled by Maudlin's take on Bell's Theorem. Now I see the problem. He takes it as axiomatic that a theory must have hidden variables.

There is more elaboration on his ontology in this new two-hour lecture: Tim Maudlin - The Great Rift in Physics: Tension Between Relativity and Quantum Theory. Even though he has written a lot of very good explanations about relativity, he does not believe in it:

I think I 1:27:44 my personal belief is you can super luminally signal but if you ask me how do I set it up in the lab now I need a 1:27:51 specific theory ... if we could show super luminal 1:28:06 signaling then relativity is dead everybody agrees right nobody would defend it anymore I'll make one more.

Yes, that would kill the conceptual basis for much of XX century physics. This puts him on the outer fringe of Physics thinking. It is like believing in perpetual motion machines or time travel to the past. Keep that in mind when you listen to his explanations.

Another new podcast has more elaboration: Tim Maudlin: Physics and Epistemology.

[Q] the the real question is how did they get away from that um how did this shut up and 14:47 calculate idea take root because it's it's a 14:52 deviation from the history of physics.

[Maudlin] it's a very extreme deviation 14:58 and if you talk to physics students and they give you that I can pretty much 15:03 guarantee it was beaten into them i mean they didn't go into physics nobody you know young person says "Oh I want to be 15:09 a physics major because I want to shut up and calculate." Right i mean it just made sense right 15:15 um typically you're interested in physics because it seems to be a way of finding out about the world and the 15:23 calculation is very much secondhand to that project. ...

16:45 the so-called Copenhagen school really tried to argue that physics had reached 16:51 the end of a certain kind of comprehensibility that 16:56 that if you were to seek really what you feel like understanding of what's going 17:02 on you're going to fail and so the only way forward was just to use 17:07 math and calculate some stuff. um now it turns out that Bohr was wrong. i mean he 17:14 was certainly wrong in the sense that we we do have the the poor few straggling people like 17:20 Einstein who who rejected all this and said "No we really want to do what 17:25 Newton was doing. we want to do what what Boltzman was doing. we want to do what Maxwell was doing."

He is really complaining about the positivism of the Copenhagen school, and how it focused on observables. It is not a deviation. Newton focused on that gravity did, and not what gravity was. Likewise Boltzman and Maxwell.

Einstein was a mixed bag. He is most famous for his work on special relativity, but he very much tried to avoid what was really going on. Lorentz had a theory that moving electromagnetic fields caused the length contraction by distorting the fields that hold atoms together. Einstein refused to endorse or reject that theory. He stuck to saying that the contraction was a logical consequence of his postulates, but never explains how it works. If you just read what Einstein wrote about relativity, you would conclude that he was a positivist of the "shut up and calculate" variety.

Then quantum mechanics came along, and he complained that it was incomplete for not explaining everything. This made Einstein more of a realist than a positivist, and it marked the end of his contributions to Physics.

Maudlin gives another explanation of his two requirements for a theory at about 1:32:00.

now you have to tell me about it that's fine so what 1:33:19 is the fundamental ontology that is what is it your theory postulates to exist 1:33:26 and second how does it behave. what does it do right normally you specify with 1:33:31 some equation some dynamics. okay um now the the the problem is that what is 1:33:39 taught as quantum theory. if you got a book a standard physics text called 1:33:45 quantum mechanics or quantum theory and you read it carefully and you said yeah but what is really being postulated to 1:33:51 exist here, you wouldn't find an answer in that book. they don't just don't address that question. what they give you 1:33:58 is what I call a predictive recipe.

So quantum mechanics is not a theory; it is a predictive recipe.

To illustrate, he cites the supposed scandal of a Lutheran minister writing a positivist forward to the famous Copernicus book, undermining the realism of heliocentrism. What you think of this goes to the heart of what science is all about.

In my view, QM is one of the most successful theories ever created. It is so useful that it has created trillions of dollars of wealth. If it does not meet some philosopher's definition of a theory, then there is something wrong with his definition, not the theory.

I don't want to dump on these guys anymore in this post. I have enough already. Watch the videos and decide for yourself.

Update: A new philosophy paper criticizes Maudlin for saying that laws of nature are ontologically primitive. This relates to whether an interpretation of a scientific theory is really a different theory.

Brian Greene on Infinite Universes

New interview:

Brian Greene on Theory of Everything, Big Bang, UFO, Conciseness, Multiverse [INTERVIEW]

In just under an hour, Brian Greene questions whether the Big Bang actually began time, explores a multiverse filled with copies of us, argues that billion-year-old aliens wouldn’t bother with Earth, confronts the hard problem of consciousness — hinting particles may carry “proto-awareness” — and defends string theory as our best yet unproven bridge between gravity and quantum physics, complete with tantalising talk of hidden dimensions.

He is good at explaining many things, but he is mainly famous for overhyping string theory and the multiverse.

I say this is wrong:

[Q] Quantum mechanics is for you 18:41 local or non-local?

[A]Quantum mechanics as far as we can tell is non-local. And this is one of the deep insights that came 18:49 from the work of John Bell working in the 1960s A name that most people have 18:54 never heard of Everybody knows Einstein They should because it was John Bell 18:59 almost as a hobby He was working at CERN particle accelerator was working on 19:04 other things but in his spare time like to think about the foundations of quantum mechanics and in his work he was 19:11 able to reveal a test to determine whether or not certain ideas that 19:17 Einstein had that the universe was local what you do here only affects things here and that it was real that particles 19:24 always have definite features he came up with a test of Einstein's view nobody 19:29 thought you could test Einstein's view It was just metaphysics ideas Go away Einstein You're just thinking philosophy 19:36 We're physicists Which is basically what people said toward the latter decades of Einstein's life But John Bell said "No 19:43 no no This is not just philosophy This is physics We can actually test whether 19:49 these things Einstein was saying are true." And when the tests were done they to most people's satisfaction revealed 19:57 that Einstein's view that the universe was local That that was not true That the universe is non-local In a sense 20:04 what you do here can in a very specific quantum way be connected to something 20:10 that happens over there

No, that was not a deep insight. Bell only disproved a classical model, and said nothing about quantum nonlocality.

Those nonlocal connections are just correlations. Nothing has a nonlocal causal effect. QM is only nonlocal in the sense that if you model it by a classical theory of hidden variables, than that theory must be nonlocal. But QM itself is local.

Here is more foolishness.

[Q] Infinitive universes Do you believe that it could exist

[A]i think the 6:48 possibility of infinite universes is an idea worth having in your toolkit 6:55 because a it naturally emerges from a variety of ideas from the inflationary 7:00 theory to the many worlds interpretation of quantum mechanics So it may be true and if it is true it completely changes 7:09 the landscape of reality And so for me to sit here and imagine there might be a 7:14 copy of me out there in the cosmos infinite copy of us If the universe is 7:19 infinitely big then you can pretty much prove mathematically that there'd be infinite copies of us out there one 7:26 after another after another. Sometimes having exactly the same conversations sometimes having slightly different 7:32 conversations. That's a crazy idea.

No, he cannot prove multiple copies of himself. That is nonsense. There could be infinitely many universes, with all others not having life. Nobody would know what is going on in those other universes.

At least he admits that string theory has no evidence, even though he is the most famous promoter of the theory.

New Biography: Einstein not a Lone Genius

A new biography was just published:

Albert Einstein (1879-1955) was the most influential scientist of the twentieth century, and his influence shows little sign of abating. His work comprises of much of today's understanding of the structure of the microphysical and cosmic universes. Einstein was a man of the modern world, ... His life is interconnected with so many of the important political and intellectual movements of his era - Zionism, pacifism, Nazism, nuclear weapons, philosophy, civil rights, McCarthyism, the League of Nations, and more- that his views shaped the world he lived in while his persona acquired a formidable patina deposited by generations of apocryphal mythmaking, both during and after his lifetime.

Einstein scholar Galina Weinstein has already reviewed it:

This paper examines "Free Creations of the Human Mind: The Worlds of Albert Einstein" by Diana Kormos Buchwald and Michael D. Gordin. The authors seek to dispel the long-standing myths of Einstein as the "lone genius" of Bern and the "stubborn sage" of Princeton, drawing on newly uncovered archival materials to illuminate his intellectual networks and collaborative engagements. ...

Buchwald and Gordin argue that popular narratives often reduce Einstein to two distinct figures: the young Einstein of Bern, the revolutionary scientist who transformed physics through sheer intellectual brilliance. ...

Buchwald and Gordin strive to demythologize the image of Einstein as a ”lone genius” by highlighting his interactions with friends, colleagues, and assistants during his formative years in Bern and later at Princeton. That is a fair point. However, my concern is that this attempt at demythologization may inadvertently reinforce the myth it aims to dismantle.

She knows a lot about Einstein, and is a big fan, so she has a number of quibbles. She mostly seeks to give him more credit.

She does not like saying that Einstein lost the Bohr quantum debates, and writes:

It is crucial to emphasize that although Bohr’s Copenhagen interpretation, along with other interpretations of quantum mechanics that accept entanglement and its inherently probabilistic nature — such as the Many-Worlds interpretation, Quantum Bayesianism, and Relational Quantum Mechanics — has largely dominated the philosophical discourse on quantum theory, Einstein’s special and general relativity profoundly transformed — and continue to transform — our understanding of space, time, and the fundamental structure of the universe.

Far from being a relic of classical physics, Einstein stands as a foundational architect of modern theoretical physics, his insights permeating even the most advanced frontiers of quantum theory and field dynamics. The birth of Quantum Field Theory (QFT) is a testament to this legacy.

No, this is absurd. Einstein was proved wrong about quantum mechanics. And not just because of a Nobel Prize a couple of years ago. The consensus in the 1930s was that he was wrong. Not only that, but he was wrong about most of what he said after about 1920. And Many-Worlds is not inherently probabilistic.

There is a lot here, but I want to address the idea that he transformed our understanding of space and time. He did not.

She is unhappy about this:

The book’s narrative emphasizes that ”eminent theorists of his day, including Hendrik Lorentz and Henri Poincaré, failed to push as far or as rigorously as the Swiss patent clerk,” signaling that Einstein’s work was deeply scaffolded by their contributions [4]. This perspective aligns with a historiographical interpretation that positions Einstein’s work on special relativity as a synthesis and extension of foundational contributions by Lorentz and Poincaré, rather than as an isolated intellectual leap. His genius resided in reassembling these established principles into a coherent and transformative theoretical architecture, rather than generating entirely novel concepts ex nihilo.

She acknowledges that Einstein used without attribution crucial ideas published by Poincare five years earlier, but argues that there is no proof that anyone complained about it at the time.

The truth is that Einstein's work was neither a synthesis and extension, nor an isolated intellectual leap. His work did not advance what had already been published in any way.

She goes on to some nonsense about Michelson-Morley and the aether. Here are the facts. Almost everyone, including Einstein, says that the Michelson-Morley experiment was crucial for the development of special relativity. Modern textbooks often introduce relativity that way. They also explain how the experiment was interpreted in favor of relativity, as opposed to earlier aether-drift models. Lorentz was quite explicit about this in his seminal 1895 paper, as so were others.

Einstein's famous 1905 paper was not directly based on experiment, as he postulated what Lorentz and Poincare had deduced from Michelson-Morley and Maxwell's equations. Again, everyone agrees to this, and that is how Lorentz described Einstein's paper.

She tries to spin this in Einstein's favor:

In later correspondence, Einstein expressed uncertainty about when he first learned of Michelson’s findings, suggesting they did not consciously shape his thinking [19]. He maintained that his path to special relativity was driven by heuristic principles, not ether-drift experiments. Thus, despite its symbolic association with relativity, Einstein’s testimony suggests his conceptual leap was rooted in theoretical reasoning rather than direct experimental influence.

There was no conceptual leap. Einstein just postulated what Lorentz and Poincare previously did, so he had no need to look at the aether-drift experiments.

She goes on to claim that Einstein only pretended to believe in the aether out of deference to Lorentz! In reality, what Einstein said about the aether in 1905 was nearly identical to what Lorentz said in 1895. They had no disagreement about it.

My biggest disagreement is her argument that the great relativity conceptual leap was "rooted in theoretical reasoning rather than direct experimental influence." I wrote a whole book on how this idea ruined Physics. Theoretical physicists today think that they can also make these leaps without experiments, because that is how Einstein did it. Physics has gotten detached from reality.

Our understanding of space and time was transformed by Lorentz, Poincare, and Minkowski, and not by Einstein. Einstein did not even believe or accept what the others did. I have documented this in my book, and on this blog.

She goes on:

Einstein was awarded the Nobel Prize for the photoelectric effect — a recognition that, while significant, fell conspicuously short of honoring his magnum opus, the theory of relativity. Instead, the accolade was granted for what could be considered an opusculum amidst the vast landscape of his scientific achievements. Despite the transformative implications of special and general relativity, the Nobel Committee withheld acknowledgment, citing a lack of conclusive experimental verification and lingering skepticism within the scientific community. From 1905 to 1915, special relativity was met with resistance, as many physicists believed that Lorentz’s ether-based theory could equally well explain experiments like Michelson-Morley and Bucherer’s mass-velocity measurements. The Committee’s 1910 report recommended delaying any award for relativity until further empirical validation could be secured, reflecting the broader scientific conservatism of the time. ...

These attacks, coupled with entrenched biases within the Nobel establishment, stymied Einstein’s path to recognition for relativity [17].

She goes on to blame antisemitism and other biases.

This is absurd. Lorentz did get a Nobel Prize in 1902 for his electromagnetism. The citation did not mention relativity, but relativity was a central part of his theory. After 1905, his relativity was known as Lorentz-Einstein Theory, as no one thought that there was any difference. The Nobel committee was not going to give a prize for a theory that was identical to one that already won a prize.

"Einstein’s path to recognition for relativity" was limited by the fact that Einstein did not create special relativity, and everybody knew it. The terms "Lorentz transformation" and "Minkowski space" became accepted because everyone knew that these were not Einstein's ideas. You can go down the list of special relativity concepts, and Einstein did not originate any of them.

She cites original documents, so she gives the impression that she knows what she is saying, but she gives a completely false picture of Einstein.

She has written a bunch of books and papers on Einstein, such as this:

Unfortunately, there are no surviving notebooks and manuscripts, no notes and papers or other primary sources from this critical period to provide any information about the crucial steps that led Einstein to his great discovery. In May 1905, Henri Poincaré sent three letters to Hendrik Lorentz at the same time that Einstein wrote his famous May 1905 letter to Conrad Habicht, promising him four works, of which the fourth one, Relativity, was a rough draft at that point. In the May 1905 letters to Lorentz, Poincaré presented the basic equations of his 1905 “Dynamics of the Electron”, meaning that, at this point, Poincaré and Einstein both had drafts of papers relating to the principle of relativity. The book discusses Einstein’s and Poincaré’s creativity and the process by which their ideas developed.

She gives the impression that the Poincare and Einstein 1905 papers were similar, except that Einstein had a "great discovery".

No, Einstein's was just a regurgitation of previously published ideas, mainly the Poincare synchronization, Lorentz transformations, and Lorentz's theorem of the corresponding states (that the transformed electromagnetic variables satisfy Maxwell's equations). Poincare's paper was much superior, and had spacetime, Lorentz group, 4-vectors, covariance of Maxwell's equations, and Lorentz invariant gravity. Einstein had none of that.

She also gets into Einstein's Zionism, Communism, and other political views, such as how he wanted to bomb Germany but not Russia. I am less interested in this subject, although it may partially explain why some scholars are so fanatical about defending him, to the point of distorting all the facts.

My concern is people who cite the discovery of relativity as one of the great intellectual achievements of all time, and something that all scientists should emulate, and then tell some crazy story about how Einstein figured it all out from pure thought, with no help from experiments.

I have previously critized her idolization of Einstein here, here, here, here, here, here, here, here, here, here, here, here, here, here, here, and here.

It is funny that she had to rush out a long review to defend Einstein and trash this book. The book is not negative about Einstein. Apparently she got triggered by comments on how he built on the work of his predecessors. That is not a criticism. All scientists build on the work of others.

But that is heresy for Einstein, as an army of scholars will leap to his defense, and tell the craziest stories to prop up the Einstein myth. If she wants to take on some real anti-Einstein works, then she should try these videos from a year ago: Einstein Plagiarized Poincaré Word for Word Part 1 Synchronizing Clocks & Relative Simultaneity, Einstein Ripped Off Poincaré's Principle of Relativity and Light Postulate. These have detailed arguments that Einstein plagiarized earlier works.

Update: Weinstein has another new paper nitpicking another Einstein biography, this time on his secular mysticism. Defending him is a full-time job.

Dr. Bee Quotes 'tHooft on Superdeterminism

Dr. Bee has been criticized:

The Fallacy of Sabine Hossenfelder

Dr. Sabine Hossenfelder is a popular science communicator, with often controversial views. In this video, I have attempted to take a critical look at the rise of Sabine, her involvement in string theory wars, recent controversy, and expert opinions.

This ends with an appeal to the authority of a philosopher to badmouth superdeterminism.

Perhaps in response to this, she dropped this video:

Gerard ‘t Hooft won the Nobel Prize in 1999, and the recent Breakthrough Prize, for his work on the Standard Model of Particle physics. He also thinks that quantum mechanics is nonsense. Indeed, he has an alternative theory for quantum mechanics that he says is how the world really works. This theory has been almost entirely ignored by physicists. Which is unfortunate, because he predicts a limit for what quantum computers can do.

Today I want to tell you about ‘t Hooft’s ideas about quantum physics, to the extent that I understand them.

He indeed has one of the best Nobel Prizes in theoretical Physics in recent decades, and he wrote a kooky book promoting superdeterminism. So it is not just her opinion; it is he opinion of one of the world's most respected theoretical physicists.

They are entitled to their opinions, and she seems to be accurately quoting his opinions. So I am not objecting to her posting these videos. I just want to make two points.

Superdeterminism is part of the existential crisis iceberg, like the idea that the universe is all part of your imagination. It cannot be proved or disproved, but accepting it requires rejecting nearly all of the last three millennia of science.

The argument that superdeterminism saves locality from the Bell test experiments is entirely fallacious. Mainstream textbook physics accepts locality and rejects superdeterminism, and does not suffer any contradictions.

Update: The craziest thing about tHooft's argument is that the whole purpose of his conjectured model is to preserve locality by saying that the experimenter has no free will, and his design decisions are determined by the Big Bang in order to constrain the results.

Yes, locality usually means that causality acts withing the light cone, and the experimenter is within the light cone of the Big Bang. So the theory is local in that sense. But it is not local in the sense of depending on events that are close in space and time. It depends on events 14 billion years in the past.

The Least Parsimonious Theory Ever

Here is a podcast interview with the leading advocate of many-worlds theory:

Do We Exist in Multiple Realities? David Deutsch and Sam Harris on Quantum Physics

Sam Harris speaks with David Deutsch about quantum physics and current events. They discuss the “many-worlds” interpretation of QM, Schrödinger’s cat, constructor theory, quantum computing and whether it will ever be practically possible, recent developments in AI, the prospects of artificial super-intelligence, the alignment problem, antisemitism and the historical persecution of Jews, misconceptions about Israel, the future of the Jews in Israel and the West, and other topics.

David Deutsch is a visiting professor of physics at the Centre for Quantum Computation at Oxford University, and an honorary fellow of Wolfson College, Oxford. He works on fundamental issues in physics, particularly the quantum theory of computation and information, and constructor theory.

He has written two books, The Fabric of Reality and The Beginning of Infinity, aimed at the general reader.

Deutsch starts by saying that it is not an "interpretation" of quantum mechanics. I agree with that, although for different reasons. He says that saying many-worlds is an alternate interpretation of quantum mechanics is like saying dinosaur theory is an alternate interpretation of Bible Genesis. We do not see the parallel worlds, but we do not see the dinosaurs either. Funny argument.

He wrote a whole book about his supposedly scientific worldview. I think his views are bizarre.

Harris asks:

I 16:42 must admit 16:43 this at first glance and perhaps even at 16:47 second glance seems like the least 16:50 parsimonious theory uh ever 16:54 proffered and we we we seem to be. at 16:57 least we imagine we're in the parsimony 16:59 business in science how how is it that 17:02 this is acceptable this idea. that I mean 17:06 is does this can this be summarized by 17:09 saying 17:10 that everything that can happen does 17:14 happen.

There is no good answer. Nor is there any good answer to Harris pointing out that the theory says that there are copies of Deutsch is parallel universes arguing the opposite.

Harris assumes, as most people do, that if the theory predicts a lot of bizarre things that are never seen, then those things must be very low probability. But that is incorrect.

[Harris] while there's no such 18:20 thing really as possibility everything 18:21 everything that can happen does happen 18:24 it doesn't happen the same number of 18:25 times so there's kind of like a 18:27 frequency difference across the the 18:29 multiverse.

[Deutsch] yes uh it turns out that 18:32 frequency is not good enough to to 18:34 support the notion of probability that 18:37 we need in physics and in everyday life.

That's right, they cannot say that they are low probability, the way that term is understood in math and physics and everyday life. He goes on to explain that he has developed some other sense in which the bizarre events can be rationalized away. That is, you can think of the bizarre events as low probability, even though they are not as we understand probability.

Sean M. Carroll gives a similar explanation, and I don't think that either one makes any sense. But listen for yourself.

Nothing violates the speed of light

New video:Does quantum theory violate the speed of light? | Claudia de Rham, João Magueijo, and Tim Maudlin.

One guy says:

I don't think 0:32 anything is sacred. it's just it's not 0:35 religion./. right this is science.

Okay, but this is just wrong:

what John Bell proved is that there 7:32 are certain constraints on what any 7:34 theory can predict and quantum theory 7:38 violates those constraints and more 7:40 importantly in the lab you violate those 7:43 constraints that's what the Nobel Prize 7:45 was given for two years ago was 7:47 experiments demonstrating violations of 7:49 Bell's inequality. so I don't even think 7:51 it's an open question whether something 7:53 goes faster than light. we know that 7:56 there's causation that's faster than 7:58 light. how you implement that cleanly and 8:01 mathematically is another question and 8:03 that nobody's worked out quite yet and 8:06 it may be that in working that out we 8:08 will have to abandon or certainly deeply 8:10 modify the picture of space and time 8:13 given by general relativity.

No, the consensus is that nothing goes faster than light. Especially not causation. And they certainly did not give a Nobel Prize for showing anything faster than light in the lab.

Later on, he correctly points out that Michelson-Morley experiment had multiple interpretations, and did not directly measure the speed of light. He needs to apply that same thinking to the Bell experiments, and admit that they have multiple interpretations.

Michelson-Morley must have seemed unremarkable, except to people like Maxwell, who followed aether theories. The Bell experiments were even less remarkable, as they only confirmed what the theory had predicted for 50 years.

Dr. Bee rambles on how the present might be caused by the future, instead of the past:

I find it curious that quantum physics is even compatible with this teleological interpretation. 4:49 It suggests to me that our idea of causes coming before effects might be somewhat of 4:54 a bias based on our personal experience, and maybe not a fundamental property of nature.

That would also be causation going faster than light. It is going so fast that it goes backwards in time!

You can say nothing is sacred, and be open to new ideas. But any causation from the future, or from anything outside the light cone, would be bizarre and contrary to our whole scientific worldview. We have no evidence for it. And if we did, we would not know what to do about it. Anyone who proved it would deserve a Nobel Prize. It has not happened.

Physics Blogs Obsessed with Fascism

I have followed the blogs of Peter Woit (Not Even Wrong) and Scott Aaronson (Shtetl-Optimized), but both of them are increasingly unhinged. Woit just posted:

Scott,
Besides the fact that your goals are delusional, what I’m criticizing is your choice to pursue them by collaborating with a Fascist dictatorship. I’m not going to further waste time trying to deal with your delusions about what is going on here.

As for what universities should do now: they should fight in court, they should tell the truth, and they should not collaborate.

By "not collaborate", he means to allow disruptive anti-Israel demonstrations, and not cooperate with Trump administration demands to obey the law.

It is baffling to me how these men go so nuts over this.

Aaronson has typical liberal Jewish political opinions, and his wife is Israeli, so he is pro-Israel. No surprises there. He largely agrees with Trump administration efforts to support Israel and crack down on antisemitism, DEI, wokism, and disruptions on campus. But he is also a hard-core Trump-hater, and hopes Trump fails in in everything he does, even if he agrees with it.

Woit has been obsessively posting sympathies for the anti-Israel protesters, and for fanatical Trump hatred. Aaronson responds:

Let’s get this straight, Peter. You’re saying that you’re angry and miserable, not for any of the reasons that are immediately apparent, but instead, because a shadowy cabal of unnamed Jewish financial elites is pulling the strings behind the scenes for its own incomprehensibly nefarious purposes?

If that’s really what you think, don’t hold back! Say it loudly, clearly, and often, directly into the microphone.

Woit seems to think that Israel should be destroyed, or that Columbia is unduly influenced by Jewish donors. I am not sure. Woit is chicken to say. Most of the protesters seem to want to destroy Israel, and to celebrate terrorism against Israel.

We can have different opinions about Israel and the Jews. That does not bother me.

What I find bizarre is how both of these guys have total comtempt for the administrations of Columbia and other universities, and yet they are so vehemently opposed to any government accountability. Their one point of agreement is that Pres. Trump is a fascist dictator.

Trump is not a fascist dictator. He was popularly elected, and he is doing what he was elected to do. He is trying to cut government waste, and most federal academic grants go for wasted research.

These universtities do some good research, but I do not expect them to get much public sympathy. A lot of the research is garbage. They are overwhelmingly slanted to the political Left. They arrogantly oppose any accountability. They allows these stupid Gaza protests to disrupt everyone, when Gaza has nothing to do with the university mission.

The Jewish angle to this is weird, as the universities have lot of Jewish professors, students, and donors. They have a history of being pro-Jewish. Trump is also pro-Jewish in his policies. And yet the universities are overwhelmingly anti-Trump.

The universities do not need to be taking a stand on either Zionism or Trump. They get billions of dollars in aid, and they ought to comply with government demands to obey the law. And they ought to be willing to do sensible things like shutting down the Gaza protests, even if the request comes from Trump and they hate Trump.

Reading these blogs just further convinces me that academia has lost its way. Radical reform is needed, and you can sure of a lot more complaints.

Textbooks Define the Orthodox Interpretation

Some modern popularizers of quantum mechanics like to say that the theory lacks foundations; that it has been corrupted by the Copenhagen interpretation; that Copenhagen is nonsense because the ramblings of Niels Bohr were incoherent; that no one understands QM; that we need a new interpretation; etc.

All of that is false. The theory is spelled out in textbooks that are mostly in agreement. You could call the agreement the Copenhagen Interpretation, but calling it that sometimes get sidetracked into what Bohr meant, and not everyone agrees.

A new paper by Geoff Beck dives into what the textbooks say:

This work sets out to answer a single question: what is the orthodox interpretation of quantum mechanics? However, we adopt a different approach to that normally used. Rather than carefully surveying the precise details of the thoughts of Bohr and Heisenberg, we extract an orthodoxy empirically. To do this we review a collection of 33 textbooks on quantum mechanics, encompassing the most popular and prominent works of this nature. We then gauge their response to 12 propositions to build up a picture of exactly what is believed by an orthodox quantum mechanic. We demonstrate that this orthodoxy is largely unchanged over the past century, with some interesting emerging deviations, and has many aspects of Copenhagen-like viewpoints.

This is correct. We do have an orthodox version of the theory, and general agreement on most points for about a century. And it does not include parallel universes, nonlocal interactions, Bohmian ghosts, Bell beables, or any of that.

Orthodox QM is used all the time in computer chip industries, and many others. It has been spectacularly successful, both theoretically and commercially.

What Einstein Would Tell Trump

SciAm opinion piece:

Einstein offers a lesson for scientists who are protesting an out-of-control nationalist administration attacking U.S. science today ...

Einstein was one of the first public critics of the Nazi regime, which he never ceased to criticize. Today his powerful stance may appear natural and uncontroversial. It was different then. ...

What would Einstein tell Trump today? Einstein would urge Trump to strive for high morality in his actions, as the scientist so eloquently presented in a 1950 letter to a minister in Brooklyn, N.Y.:

The most important human endeavor is the striving for morality in our actions. Our inner balance and even our very existence depend on it. Only morality in our actions can give beauty and dignity to life.

Yes, Einstein was a Jew who opposed the Nazi, but his politics were nothing to admire. He was a Communist fellow traveler. UPI reported

In one of the final entries in the file, the FBI noted, 'Extensive investigation in U.S. reflected Einstein affiliated or his name extensively associated with literally hundreds of pro-Communist groups.

'No evidence of CP membership was developed,' it added.

The FBI file described Einstein as a 'pacifist' and a 'liberal thinker' affiliated in some way with more than 30 'Communist-front' organizations.

'He has opposed militarism and universal military training in the United States and has espoused world government,' the file said.

He frequently criticized the USA government, while refusing to criticize the Soviet Union.

Invention of Projective Geometry

SciAm reports:

This demonstration of his recently discovered laws of perspective is said to have occurred sometime between 1415 and 1420, if his biographers are correct. The use of the laws of perspective amazed bystanders, altered the course of Western art for more than 450 years and, more recently, led to mathematical discoveries that enable elliptic curve cryptography. This is the security scheme that underpins Bitcoin and other cryptocurrencies and has become a fast-growing encryption method on other Internet platforms as well.

The article explains how the above art inspired projective geometry, over the next few centuries.

You would think that Microsoft and Google would be able to quickly adapt to a superior software technology, but that has not happened.

Elliptic curve cryptography is a relative latecomer to the encryption game. The first suite of tools did not appear until 2004, far too late to become a standard for the Web but early enough to adopted by the inventors of Bitcoin, which launched in 2009.

Its status as the de facto standard for cryptocurrencies made people more familiar with it and more comfortable implementing it, although it still lags behind RSA encryption, the standard method in use today, by a wide margin.

Yet elliptic curve cryptography has distinct advantages over RSA cryptography: it provides stronger security per bit and is faster than RSA. An elliptic curve cryptographic key of just 256 bits is roughly as secure as a 3,072-bit RSA key and considerably more secure than the 2,048-bit keys that are commonly used.

Current thinking is to abandon elliptic curve cryptography, and switch to new quantum-resistant protocols.

The trouble is, if the industry could not switch from RSA to ECC, how will it switch to quantum-resistant?

ECC is superior to RSA in every respect, except compatibility with ancient systems. ECC is is faster, more secure, less error-prone, and smaller. The new quantum-resistant methods will be worse in all those things, except that it will supposedly resist some quantum computer that might be built in 50 years.

Most people think that non-euclidean geometry means curved manifolds, such as a sphere or hyperbolic space. But the original non-euclidean geometry was projective geometry. Projective geometry also indirectly led to the discovery of relativity.

Think of it this way. A geometry could be defined by the formula for the distance between two points. On a more elementary level, a geometry can be defined by what the lines are. In special relativity, the light rays are the lines of particular interest, and they are different from Euclidean geometry.