## Monday, September 16, 2019

### Why do some scientists believe that our universe is a hologram?

Today, I want to tell you why some scientists believe that our universe is really a 3-dimensional projection of a 2-dimensional space. They call it the “holographic principle” and the key idea is this.

Usually, the number of different things you can imagine happening inside a part of space increases with the volume. Think of a bag of particles. The larger the bag, the more particles, and the more details you need to describe what the particles do. These details that you need to describe what happens are what physicists call the “degrees of freedom,” and the number of these degrees of freedom is proportional to the number of particles, which is proportional to the volume.

At least that’s how it normally works. The holographic principle, in contrast, says that you can describe what happens inside the bag by encoding it on the surface of that bag, at the same resolution.

This may not sounds all that remarkable, but it is. Here is why. Take a cube that’s made of smaller cubes, each of which is either black or white. You can think of each small cube as a single bit of information. How much information is in the large cube? Well, that’s the number of the smaller cubes, so 3 cube in this example. Or, if you divide every side of the large cube into N pieces instead of three, that’s N cube. But if you instead count the surface elements of the cube, at the same resolution, you have only 6 x N square. This means that for large N, there are many more volume bits than surface bits at the same resolution.

The holographic principle now says that even though there are so many fewer surface bits, the surface bits are sufficient to describe everything that happens in the volume. This does not mean that the surface bits correspond to certain regions of volume, it’s somewhat more complicated. It means instead that the surface bits describe certain correlations between the pieces of volume. So if you think again of the particles in the bag, these will not move entirely independently.

And that’s what is called the holographic principle, that really you can encode the events inside any volume on the surface of the volume, at the same resolution.

But, you may say, how come we never notice that particles in a bag are somehow constrained in their freedom? Good question. The reason is that the stuff that we deal with in every-day life, say, that bag of particles, doesn’t remotely make use of the theoretically available degrees of freedom. Our present observations only test situations well below the limit that the holographic principle says should exist.

The limit from the holographic principle really only matters if the degrees of freedom are strongly compressed, as is the case, for example, for stuff that collapses to a black hole. Indeed, the physics of black holes is one of the most important clues that physicists have for the holographic principle. That’s because we know that black holes have an entropy that is proportional to the area of the black hole horizon, not to its volume. That’s the important part: black hole entropy is proportional to the area, not to the volume.

Now, in thermodynamics entropy counts the number of different microscopic configurations that have the same macroscopic appearance. So, the entropy basically counts how much information you could stuff into a macroscopic thing if you kept track of the microscopic details. Therefore, the area-scaling of the black hole entropy tells you that the information content of black holes is bounded by a quantity which proportional to the horizon area. This relation is the origin of the holographic principle.

The other important clue for the holographic principle comes from string theory. That’s because string theorists like to apply their mathematical methods in a space-time with a negative cosmological constant, which is called an Anti-de Sitter space. Most of them believe, though it has strictly speaking never been proved, that gravity in an Anti-de Sitter space can be described by a different theory that is entirely located on the boundary of that space. And while this idea came from string theory, one does not actually need the strings for this relation between the volume and the surface to work. More concretely, it uses a limit in which the effects of the strings no longer appear. So the holographic principle seems to be more general than string theory.

I have to add though that we do not live in an Anti-de Sitter space because, for all we currently know, the cosmological constant in our universe is positive. Therefore it’s unclear how much the volume-surface relation in Anti-De Sitter space tells us about the real world. And for what the black hole entropy is concerned, the mathematics we currently have does not actually tell us that it counts the information that one can stuff into a black hole. It may instead only count the information that one loses by disconnecting the inside and outside of the black hole. This is called the “entanglement entropy”. It scales with the surface for many systems other than black holes and there is nothing particularly holographic about it.

Whether or not you buy the motivations for the holographic principle, you may want to know whether we can test it. The answer is definitely maybe. Earlier this year, Erik Verlinde and Kathryn Zurek proposed that we try to test the holographic principle using gravitational wave interferometers. The idea is that if the universe is holographic, then the fluctuations in the two orthogonal directions that the interferometer arms extend into would be more strongly correlated than one normally expects. However, not everyone agrees that the particular realization of holography which Verlinde and Zurek use is the correct one.

Personally I think that the motivations for the holographic principle are not particularly strong and in any case we’ll not be able to test this hypothesis in the coming centuries. Therefore writing papers about it is a waste of time. But it’s an interesting idea and at least you now know what physicists are talking about when they say the universe is a hologram.

1. It's nothing more than twist in their sobriety. Like Tanita Tikaram: "Look my eyes are just holograms"

2. "Therefore writing papers about it is a waste of time". Reading about it is a waste of time too. It's pseudoscience garbage from people who a) don't understand black holes and b) are Lost in Maths.

1. And yet, the "respected journals" accept and publish such papers.

Mind boggling.

3. Sabine :
You cannot agree that it is a interesting idea if
(1) you dont't believe our space-time is a AdS one (as you criticize)
(2) If you don't believe (as me) that geometrical black-holes of GR exist. Shadow and gravitational wave does not prove that their horizon exist.

4. Sabine :
You cannot agree that it is a interesting idea if
(1) you dont't believe our space-time is a AdS one (as you criticize)
(2) If you don't believe (as me) that geometrical black-holes of GR exist. Shadow and gravitational waves does not prove that their horizon exist.

5. Indeed the entanglement entropy between two systems generally scales as the dimension of the contact surface between them, so usually co-dimension one, but in the AdS case it's the dimension of the boundary. For black holes in general there is no covariant definition of its volume, since the inside metric of a black hole is neither static nor stationary, so the sentence "black hole entropy is proportional to its volume" would not even be a coherent one.

What do you think of a more general claim is that all entropy is entanglement entropy, mainly because, if you trust quantum mechanics, something that is not entangled with us is not observable by us, and hence cannot contribute to anything we measure, including entropy?

6. Hi Sabine,

If only the "community" could somehow stop people like Sean Carroll from writing books, the world would be a slightly better place.

Greg

1. I thought his NYT article was good though.

2. "people like Sean Carroll" ???
Theoretical physicists? Caltech professors? Men? Atheists? People married to Jennifer Ouellette? I suspect you mean "people you don't agree with."

Like many people, I bought Carroll's book, read it, and enjoyed it without being convinced by the arguments he makes. Same with Prof. Hossenfelder, based on her review, except she got a free copy. In what world (see what I did there?) would it be an improvement for a "community" to dictate the thoughts people were allowed to publish?

3. Hi Greg Feild,

Since you failed to provide a link to Sabine's article in Nautilus, I'll provide one here on your behalf. The article is titled, "Mind the Gap Between Science and Religion"

This will save folks the effort needed to search Nautilus to find where it is. And I whole-heartedly concur, it is an excellent article!

-Nick

4. Jim: in Woit's world.

And judging from his comment, in Jim Baggot's world too. See Woit's blog article Quantum Supremacy II

5. I have no idea who Mr. Carroll is and I don't believe in censorship.

I'm sure he is a fine person!

Let the free market decide what is science and what is not.

6. John -- I read the blog post (not for the first time btw). Woit doesn't have much use for Carroll's views, but I don't see where he suggests that Carroll not be allowed to publish them.... Censorship impoverishes debate, disagreement enriches it. That's my only point, which Mr. Feild now appears to agree with.

7. Jim: he would if he could. See his subsequent post:

https://www.math.columbia.edu/~woit/wordpress/?p=11277

8. Yes, he does go from from disagreement to claiming that Carroll is an idiot doing a disservice to science -- but concludes that he should be ignored. I admit though, that in a different context (say,"The Sopranos") the phrase "think hard about what they can do to deal with this situation" might be a bit ... sinister. Check out xkcd.com/1483/ ;-) I think Woit meant "speak out" ....

9. I did pick up one his Carrolls books on GR and the only thing I remember from it is when he introduces tensors and points out the distinction between covariant and contravariant tensors and says that is old-fashioned and obsolete terminology. Except of course it isn't. Covariance and contravariance is a basic distinction in category theory. It justs goes to show what different backgrounds brings to a subject.

7. How do holographic theory proponents explain the positive cosmological constant in the observed universe ?

(not sure, if prev comment was posted)

1. This is an open question. We do not live in an anti-de Sitter spacetime. If we did there would be galaxies red shifted moving away, which we do see, but there would also be galaxies blue shifted moving towards us. There are these geodesic arcs that particles follow, such as on the Poincare half-plane and disk, and we would see galaxies both blue and red shifted. So we are not in an AdS with a negative cosmological constant, but a de Sitter (dS) or its approximation in an FLRW with a positive cosmological constant.

Think of two cones glued at their points. There can then be a singly hyperbolic sheet on the outside of this and two parabolic sheets inside the two openings or cones. These two are analogous to AdS and the outside single sheet is dS. These all meet at infinity or I^+ and I^- on the cones. So these all have in effect the same data. They meet at a boundary of one dimension less, which is a sort of holographic principle.

String theory requires a negative ground state, which is one reason it is convenient to do superstring in AdS with is vacuum energy given by Λ < 0. We may however, think of this as a sort of Dirac momentum-energy cone situation, where in a Haldane chain there can be also positive conditions. It is similar to an Ising spin chain. In fact if we are to think of the universe, by that I mean everything, then if "sum_all = 0" then in some way there have to be these inverted dS spacetime with Λ > 0.

8. > Now, in thermodynamics entropy counts the number
> of different microscopic configurations that have
> the same macroscopic appearance.

Please correct me if I am wrong, but my very tenuous understanding of entropy is that it's somehow related to the amount of useful work a system can do thermally. For a chamber filled with gas, the entropy is related to the number of particles as well as the temperature: more particles with higher temperature will be able to do more work, if we can harness all that energy somehow.

But when people talk about entropy as the measure of information related to the number of micro-states, I get lost. I mean, if I take just one particle with some momentum, that's 3 real numbers to describe the position and one more real number for the velocity magnitude. Each real number consists of infinitely many digits, so the amount of information required to code just one of these coordinates is (countably) infinite. If we were to count all the possible micro-states of just one such particle in a closed 1x1x1 box, we would count uncountably many different micro-states (a particle's X coordinate can be any real number between 0 and 1, and there are uncountably many of them). This is the same exact cardinality as that of any other system of countably many particles in the box. In other words, unless space-time is actually grid-like or foam-like, with only finitely many "places" in a given volume (like your black/white cubes), all of this information talk makes zero sense to me.

I would be quite interested in seeing what you think about this issue; perhaps I just don't understand the entropy very well, but the more I think about this, the more convinced I am that something is very fishy is going on with the traditional interpretation.

A somewhat related question is how does one measure entropy of a given box with gas molecules in a particular configuration. Wikipedia says something about cooling it to absolute zero, and then adding temperature little by little until the box is some set number of degrees, and then "integrating" energy input to get the entropy. It's not quite clear to me what they are integrating there, so I am really curious about the details here, as well as perhaps other ways of measuring it.

1. Ivan,

"people talk about entropy as the measure of information related to the number of micro-states"

Then you are talking to the wrong people. The entropy measures (is the logarithm of) the number of microstates. It's not "a measure of information related to", it's a fairly straight-forward definition. (Though, as I mention in the video, there are different types of entropy.)

2. I am sure I am talking to the wrong people every now and then, but right now I am trying to address what you are saying in this very blog post:

From the point of view of mathematical formalism, the space is treated as continuous by every major physical theory, as far as I can tell. If we describe the XYZ coordinates of one particle in a box with real numbers, then there are infinitely (in fact, aleph 1) different positions inside a box, and so infinitely many microstates, and the logarithm of infinity is not defined. If we add more particles, even infinitely (but countably) many particles, then the cardinality of the set of microstates will remain exactly the same: aleph 1. Even we had somehow extended logarithm into transfinite, it would still be unable to tell apart any two boxes with some particles in them.

If the space is discrete, on the other hand, with only finitely many "places" inside the box of a given volume, then there are finitely many microstates, and logarithm will work.

Another possibility is that I don't understand what exactly people talk about when they say "microstates" in a space-time addressed by real and/or complex numbers. Again, the discrete case is very clear to me and makes perfect sense, logarithm and all. Yet no standing theory treats spacetime as discrete, and so all these logarithm applications to entropy really beg the question, in my mind.

3. Ivan,

In classical statistical mechanics, you choose (arbitrarily) your smallest size interval and stick with it. There will always be an arbitrary constant difference between your and some other guy's entropy, but it does not matter.

In quantum mechanics, the states are discrete as long as you are in a finite box. You do not need to choose an arbitrary smallest length.

Of course, if you are in an infinite box to start with, things get nasty, but you take the infinite limit in the right way and everything is fine.

I am of course dramatically over-simplifying, but people have been thinking about these things in enormous detail for well over a century and seem to have it more or less worked out.

I take it you may be coming from a math background?

Most of it has not been done from the view of rigorous math, more like nineteenth-century math. But, there have been some interesting attempts at rigor.

You should be able to explore all this to your heart's content in any decent university library.

Now, as to the entropy of a black hole... jump into the pool and join the fun! Lots of people think they know what is going on, but, as far as I can tell, the really smart people do not all agree among themselves. String people, general-relativity people, old fashioned QFT folks -- there's enough fun for everyone.

By the way, the connection between entropy and heat/work/etc. vs. entropy as the logarithm of the number of microstates was worked out long ago ("to physicists' standards of rigor") and again can be explored in any decent university library.

Dave

4. > I take it you may be coming from a math background?

Hehe yeah I kind of gave my game away when I unloaded the cardinals on you people... I got degrees in CS and math logic, and I am extremely interested in physics lately, but I am also very easily confused by the sheer weirdness of it.

> In classical statistical mechanics, you choose (arbitrarily) your smallest
> size interval and stick with it. There will always be an arbitrary constant
> difference between your and some other guy's entropy, but it does not
> matter.

Thanks! This is precisely the bit of info I needed to dispell my confusion. I suppose, as long as spacetime becomes discrete (in this sense) at some (any) scale, we can all boldly write differential equations involving derivatives of entropy, and they will be golden even now, when we have no idea where that final scale is located.

5. Ivan, you raise a good point when you are saying "there are infinitely (in fact, aleph 1) different positions inside a box, and so infinitely many microstates, and the logarithm of infinity is not defined." In reality, you can't know the particles position with infinite accuracy according to the Heisenberg uncertainty principle. Any microstate is function of a cell of uncertainty or ignorance in the state space, a kind of freedom garden where the quantum objects are completely independent or free each other ,Its value is linked to the Planck constant. It's the reason why the Entropy of BH is the number of planck cells on a sphere, in this case the planck cells could be considered as a limit of accuracy, a surface limit under which causality doesn't apply since inside the surface any quantum object are free each others. IMHO, this is similar to say that state space is quantized.

6. Fred Harmand4:27 AM, September 24, 2019

" you raise a good point when you are saying "there are infinitely (in fact, aleph 1) different positions inside a box, "

Congratulations on solving the Continuum Hypothesis. You are assuming that Aleph_1 is the cardinality of the Reals. This famously depends on the axioms chosen. Also, the Reals aren't real and there isn't known to be Aleph_1 or even Aleph_0 of anything in the observable physical universe. It appears to be finite.

9. It is regrettable this blog is becoming a bit of a sounding board for nay-sayers.

There are reasons to take the holographic principle seriously. It goes back to the Bekenstein equation for the entropy of a black hole S = kA/4ℓ_p^2, for ℓ_p = √(Għ/c^3) the Planck length. The area is given by the Schwarzschild radius r_s = 2GM/c^2 and the mass is the Planck mass m_p = √(hc/G). The area A of the black hole is then just A = 4πr_s^2. I leave it to the reader to find the final answer. The upshot though is the entropy of a black hole is determined by its area, not volume. Further, entropy in general is defined by the von Neumann formula S = -k Tr(ρlog(ρ)) and this entropy is ultimately quantum mechanical. The quantum fields or states that are pinned just above the event horizon have entanglement entropy, and this defines the entropy of the black hole.

There are a number of ways of thinking about this. The volume that matter occupied that imploded inwards or has fallen into the black hole has been Lorentz contracted along the radial direction. This contraction is complete on the horizon so that a 3d region → 2d region. This happens only because the horizon is a causal barrier that can't be observed beyond. Then we have Hawking radiation, which tells us that particles or radiation appears outside the black hole and escapes into 3-d space. So there is this strange relationship between matter-fields in 3-d and the fields that make up the entropy of a black hole on its horizon. From this the idea of the holographic principle emerges.

Maldecena found a holographic relationship between an anti-de Sitter (AdS) spacetime of n dimensions and conformal field theory in n-1 dimensions. This is the AdS_n ~ CFT_{n-1} correspondence. This makes a lot of complicated string theory easier to understand. The Ryu-Tankanagi formula also connects entanglement entropy into this picture. AdS_n spacetimes with Lanczos junctions are really interesting, for how one has these junctions that can act as holographic screens with AdS_{n-1} geometry and which has a fibration given by CFT_{n-1}.

If you have read books in string theory you might have noticed the bosonic string and those developments are not that hard to follow. Then say with Polchinski second volume with superstrings things becomes tough. Supersymmetry by itself is not entirely easy, but superstring theory gets crazy. Yet with the above we have concepts involving entanglement that help. Further spacetime may really be built up from entanglement. Supersymmetry is just another route to understanding how aspects of quantum physics connect with spacetime. So there is a general system waiting in the wings folks.

1. “The volume that matter occupied that imploded inwards or has fallen into the black hole has been Lorentz contracted along the radial direction. This contraction is complete on the horizon so that a 3d region → 2d region.”

This explanation makes intuitive sense to me, and I think it’s why the Russians used to call black holes “frozen stars,” since for a distant observer, an infalling clock would stop at the event horizon. I’ve wondered for a long time what happens in this situation: a sufficiently long spaceship is falling toward the horizon of a sufficiently large black hole (large enough that curvature isn’t a problem). Two astronauts (Alice and Bob, who else, maybe we can finally get rid of them…) are on the ship, Alice being at the end closest to the event horizon and Bob at the trailing end. They’re flashing lights at one another. What do they each see a) just before Alice crosses the horizon, b) when Alice has crossed but Bob hasn’t, and c) just after Bob crosses? If part of the answer is that Alice's light when she has already crossed can't reach Bob until he has also crossed, so in their frame there's never a time when they are on different sides of the horizon, how does that square with c being constant in any frame of reference? Thanks to anyone who's willing to help sort it for me.

2. If Alice sends a photon to Bob when she is behind the horizon and Bob is not then Bob will cross the horizon before receiving the photon. Bob will notice nothing peculiar about this situation.

3. Yep, thanks, and I guess no matter how long the spaceship is, it will be 2d by then. Appreciate it.

BTW, about the nay-sayers.... No surprise they'd be here; the owner of this blog is a world-class nay-sayer, and we love her for it. I guess it all depends on what someone's saying nay to.

4. All this supposes that extra dimensions are actually there, as with super-symmetry and super-strings. When Dirac predicted the positron it was found within a decade. Can you remind me just how long we've been waiting for either direct or indirect evidence of any of these phenomena? Perhaps if physicists who want to speculate about such concepts prefaced their musings with the word 'hypothetical' or god forbid, 'speculative', that might just help. Ed Witten, I recall, invented a new word 'postdiction' to describe the situation when a new bit of physics was consistent with what had already been established. Isn't this what was already known as the correspondance principle by Bohr?

5. I want to add I'm not a nay-sayer to string theory, just to over-blown and hyperbolic claims. I think thats not un-natural. Personally, the point when I got some physics of string theory, after having worked a little through Beckers and Schwartz book, is actually going back to the history of the subject and realising that they began with the Veneziano amplitude which actually fitted with *actually existing experimental data* of hadronic particles. That helped. To my mind, in a subject like physics, some physical motivation, in the usual sense, and not the speculative sense, really helps. Of course that data is now explained by a SU(3) Yang-Mills.

10. Hi Sabine, !!!

I hope your day is going well.

- I'll keep this in English.

The Holographic
Principal. ...

I

11. "Personally I think that the motivations for the holographic principle are not particularly strong and in any case we’ll not be able to test this hypothesis in the coming centuries. Therefore writing papers about it is a waste of time. But it’s an interesting idea "

Amen.

12. Pascal,

You seem to understand it just fine. By what means does this turn the inside into a hologram? Well, it doesn't.

13. "for all we currently know, the cosmological constant in our universe is positive"
should be
"from all we currently know."
- for all we know - is a statement of uncertainty
- from all we know - is a statement of fact, although perhaps could be expressed differently.

14. If you can reduce the information content of three dimensions to two dimensions, then you can reduce the information content of two dimensions to one dimension, and from one dimension to zero dimension - without loss of information?

Or why only from 3 to 2, but not further on?

1. It does. The AdS_5 ~ CFT_4 is a major case of this.

15. Dr B, Dr Verlinde has also proposed some ideas on gravity which seem very unorthodox. Unorthodox usually means wrong, but perhaps he is trying to push outward from the confines of present scientific thought. Or perhaps he is merely "lost in the math".

16. Well, I'm not a physicist, my background is cognitive behavioural psychology, but as a SF writer I find the whole holographic principle delicious fun. And again, the whole pluralistic debate in science makes for a good jumping off point.

So this is just a thank you for all the hard work that goes into your blog, and expanding my knowledge about physics.

17. @Lawrence Crowell

1293/5000
"It is regrettable this blog is becoming a bit of a sounding board for nay-sayers". You are absolutely right: to deny without argument something admitted is only verbal pollution. What I wanted to point out about the GR is that all this speculative physics - including the entropy of black holes and the holographic conjecture based on it - is based on an interpretation of the metric effects of GR that is not questioned , although this is not the place to do it here.
I read attentively the French edition of Lost in Maths. In my opinion, we should retain that the hierarchy of criteria to be applied to research is (1) the adequacy to experience; (2) logical coherence and conceptual consistency; (3) simplicity (the Occam razor); (4) aesthetic criteria. In particular, if (1) is missing, (2) must apply first. And if two theories are equivalent for (1), it is (2) who must first decide between them.
As for the mathematical consistency, I do not know where to put it: in (0), or at the same level as (2)? As for the order between (1) and (2), we can also wonder what meaning there would be to test a théorie which we already know is not consistent (logically and / or mathematically).

1. Jean,

As for the mathematical consistency, I do not know where to put it: in (0), or at the same level as (2)?

This sort of confusion about the proper role of math in science is a root cause of the "crisis in physics". Simply put, there is no scientific evidence for the conceit, not uncommon among theoretical physicists, that mathematics is fundamental to physics. Mathematical consistency, in your hierarchy, belongs at level (2).

Mathematics is a human construct. It is a set of abstract logical formalisms based on counting. It provides an absolutely essential modeling tool for science, but math is not, itself, science.

There is no scientific evidence for the widely-held belief that mathematics somehow underlies physical reality and therefore math models have a causal relation to physics - level (0). Unfortunately, that belief constitutes the de facto operating paradigm of modern theoretical physics.

The results are two standard models, of particle physics and cosmology, that bear scant structural resemblance to the physical reality of our empirical observations. Both models are heavily populated with entities/events that have no correlates in observed reality. This is not considered a problem by the theoretical physics community - and that is the problem. That is why there is a "crisis in physics".

The math first approach to physics, in which mathematical models are more heavily weighted than empirical observations, is at a self-generated dead-end. The proper role of theoretical physics is to construct mathematical models that conform, in their structural elements, to observed physical reality. What we get instead are models ginned up in abstract mathematical "spaces" that have no obvious relationship to physical reality (substantival spacetime, AdS/CFT, for instance). Experimentalists are then dispatched to seek confirmation of predictions of the theory that usually invoke entities/events at the margins of observational resolution.

If the predictions fail, the model is adjusted by moving the predictions further beyond the current observational limit and experimentalists are tasked with developing more sensitive instruments. This cycle has been regularly repeated over 50 years for gravitational waves, and 30 plus years for dark matter. This approach effectively renders the models unfalsifiable - negative results are never conclusive. "Heads we win, tails we play again" is the game. Science cannot succeed in this manner it can only produce temporary, confirmation-bias results (see LIGO and Bicep2) or endless goal-shifting.

Mathematicism, the belief that math underlies physical reality, needs to be expunged from theoretical physics. Until that is done, the "crisis in physics" will continue stuck as it is, Schroedinger's dead/alive cat chasing its tail, down at the dead-end of a blind-alley.

2. The idea that mathematics is connected to physics runs deep and goes back to the ancient Greeks. The understanding of physics, or astronomy, required measurements. In the ancient world this was often just theodolite measurements of planets against the sky. The changes in these quantities are some sort of "map" between one set of numbers to another. This was made more firm with Galileo and his kinematics for motion, and of course a pillar of physics with Newton.

We have two ways to think of mathematical consistency. One is the mathematics must be internally consistent, so there is no result that is equivalent to 0 = 1. So the mathematics have a set of axioms that are consistent with each other and do not result in a contradiction. The other aspect of consistency is the mathematics has to reflect what one actually measures. This is where empiricism enters in and this is not strictly mathematical. Newtonian-Lagrangian-Hamiltonian formalism of classical mechanics is mathematically consistent, but it does not match all observations outside of some bounded set of conditions.

The condition that 0 = 1 does happen in a funny way with Gödel's theorem, but this has to do with ω-consistency. One can have Peano's number theory where any number has a successor, but the inductive reasoning this continues infinitely is not provable and to state it is is ω-inconsistent. This does not then generally influence most normal mathematics, but is connected in a way to infinity. The induction leading to infinity is not provable, and David Hume even had some insight into this in the late 18th century. It does though pertain to odd numbers that are the reciprocal of infinite numbers, and these are called surreal numbers or supernatural numbers introduced by Robinson. This does have some subtle issues with calculus. Does this have something to do with physics? Maybe, but one is sticking their neck out by proposing such, and I think this might connect with quantum interpretations. So one is sticking their neck out twice as far, and in physics there are plenty ready with guillotines to chop necks stuck out too far.

Can physics be done independent of mathematics? I do not think it can in total. With physics we measure things and we correlate these data in some way according to a theory. The only way numbers are correlated is with mathematics of some form. It is not a science that can be done without mathematics as are many areas of biology. It is difficult to foresee a day of math-free physics.

3. The idea that mathematics is connected to physics runs deep and goes back to the ancient Greeks.

That idea is the philosophical stance called mathematicism and it does indeed run deep in modern theoretical physics. Philosophical stances are only beliefs however, and do not constitute a sound basis for doing science. In fact, it is my explicitly stated criticism that mathematicism is the root cause of the so-called "crisis in physics". In several conversations now, you have been at great pains to avoid discussing that criticism.

As usual you spray around several paragraphs of mathematical non sequiturs, and in this instance wrap up with a transparently bogus straw-man argument. It is transparently bogus because I explicitly stated my position re the role of mathematics in science in the comment to which you are responding: [Mathematics] provides an absolutely essential modeling tool for science...

Here is the relevant portion of your final paragraph:

Can physics be done independent of mathematics?...It is difficult to foresee a day of math-free physics.

On the basis of all the evidence so far, I find it difficult to foresee a day in which you frame a sound logical argument in defense of your mathematicist viewpoint.

4. bud rap is right (about mathematicism), as usual.

I explain everything in my book
"Mind Boggling Exploding Swirly Stuff From the Multiverse"

Free upon request ..

18. I take this holography result (AdS/CFT) as a clue that there's some big shift in perspective we're yet to discover. Another clue is entanglement non-locality. These are clues that maybe the degrees of freedom of the universe are not located on particles in 3D+Time as they usually seem to be but maybe they're somewhere else. Maybe particles and spacetime are approximations and if we do clever quantum experiments or examine the limits they break down.

What I don't get is why physicists use intuitive concepts like "hologram" or "string" as ingredients when they try to construct theories of that deeper mechanism. I doubt these are in any way literal. Instead I'd expect some theory that starts with degrees of freedom as abstract quantum numbers and tries to emerge spacetime and particles as the thing that emergent spacetime-particle-people would perceive.

Not a physicist. Trying to understand and follow as much as I can.

19. The solution of a partial differential equation with the right boundary conditions looks to me like an easily understandable example of the holographic principle. The solution of classical field equations in d+1 dimension is determined by d-dimensional initial/boundary data.

Is the gravity-related holographic principle in essence significantly different from this? Just substituting quantum fields for classical fields?

1. The holographic principle is a quantum principle. The stetched horizon fields on the horizon are nonlocally correlated with Hawking radiation. Boundary conditions for PDEs are causal and local.

There may however be a dualism or correspondence between nonlocal quantum entanlements and local causal prinpiles. Quantum gravity might be thought of as some equality or dualism between UV physics of quantum gravity (nonlocal) and local quantum fields at the IR. This is maybe what the Einstein field equation is telling us.

2. Arnold,

The holographic principle is about degrees of freedom. Thus, I guess what is missing in your analogy with partial differential equations (PDEs) in d+1 dim and boundary conditions in d dim is the additional condition of a maximal resolution. Since almost all PDEs only can be solved numerically anyway this translates into the maximal resolution of the dynamical mesh (BTW a nonlocal condition).
Tim Palmer - see e.g. here - found that solutions to PDEs become more accurate when a bit of randomness (stochasticity) is added.
Is there a connection to your thermal interpretation of QM?

3. Our world is probably a solution to PDEs – the physical laws. To numerically solve/integrate PDEs we start from initial conditions and take tiny steps as e.g. shown here with an ODE. The time it takes to determine the next step depends on the required accuracy.
But this (process) time of course has nothing to do with the time t in the PDE or the result of the numerical integration. And each differential step is linear.
This is a bit like the linearity of the unitary evolution in QM between measurements where “...in a superposition nothing dynamical happens...” as Arnold Neumaier said. Only probabilities are (nonlocally) determined for the next “time” step.
When we are dealing with big masses like stars and planets the tiny changes of QM do not matter at all and we are back with Newton or Einstein.
Well, not quite, QM still matters if we want to see these objects or want to explain why they do not collapse or how proteins could develop.

4. To get some more mileage out of the PDE analogy and to connect to entropy and action let us look at Penrose’s illustration of the boundary conditions – the blister here. S is the action and δS=0 gives the classical equations of motion, projecting out solutions not on mass shell.
In QM, or better QFT just e^iS/ℏ is used in the path integral. Squaring the added/superposed amplitudes (each can be way off mass shell, i.e. virtual particles, nonlocal) gives the probability for a transition from an initial to a final state, again on mass shell, a “time” step once a measurement is triggered.

On to the two different behaviors of entropy - the area dependent one in gravity and the volume dependent one in statistical thermodynamics (ST).
- area dependent – GR/QM:
Sabine said “It may instead only count the information that one loses ...” - absolutely and this simple “derivation” here following Susskind suggests one may not even have to use entanglement entropy (von Neumann).
- volume dependent – QM/ST:
Particles/photons being exchanged tend to establish equilibrium as in ST and makes the very definition of temperature possible in the first place. The squared amplitudes in QM just provide probabilities, the necessary ingredient for Shannon entropy (*).
That’s “... the stuff that we deal with in every-day life ...”, which “... doesn’t remotely make use of the theoretically available degrees of freedom.”. Gravity compresses this stuff a little bit (**), thus ignites stars and makes all complexity here on earth possible.

Let us have a speculative look at how actions could be related to entropy.
In GR the action is ∼c^4/16πG and in QM ∼ℏ. Let us set these two into relation, i.e. divide by ℏ and further by c (***). This gives ∼c^3/16πGℏ=1/4π 1/4L² with the Planck length L. This might hint to a connection with Bekenstein-Hawking entropy, i.e. in GR action/ℏ ~ entropy.
In QM we are already dealing with an action/ℏ via the path integral ∫...e^iS/ℏ resulting in probabilities ~ entropy.
It is interesting that already de Broglie speculated in his hidden thermodynamics that action and entropy are related (****).
Our world is a subtle interplay between GR and QM and measurement might be the missing link.

---------------------
(*) with p=1/N this even reduces to Boltzmann entropy, i.e. counting microstates.
(**) We already know how Fermions and Bosons “move” in a curved spacetime metric. What we do not yet know is how and “when” the backreaction occurs.
(***) c, the conversion factor between time and space in SR. (By the way: mass is the conversion factor between geometry (the length of the world line) and physics (the action).)
(****) see also the CA-conjecture, i.e. “complexity”~action/ℏ proposed in here, which relies on tracing out a density matrix (mixing QM and statistical probabilities) – the modern way to distill entropy from entanglement.

20. I have never been fond of these escapist-theories which seem to substitute on reality over another and further block questions about the need over the other reality.

21. Happy anniversary.

22. @Bud rap
I forgotted saying that in my hierarchie, (1) and (2) and (3) have to be applied necessarily. They are not optionnal. And that mathematical consistency can bee included in logical consistency. So putting this one on level (0) or (2) is not fondamental, but you are right for putting it at level (2)

742/5000
As for the necessity of using mathematics, the history of classical physics suffices to demonstrate it. Galilean holds that "The Book of Nature is written in mathematical language" (although it may be doubted that this is the only way to access the real, the second qualities are not reducible to the primary qualities). But in his works logical arguments, their application to experience, thought experiments, take much more space than calculations. If he did not specify that, even before being formalized in mathematical language, all that this Book says about the Real rationally articulable first respects the canons of logic is that it was self-evident for him and his contemporaries.

23. "Personally I think that the motivations for the holographic principle are not particularly strong and in any case we’ll not be able to test this hypothesis in the coming centuries. Therefore writing papers about it is a waste of time."

I completely agree with you, Sabine. For this statement however, you don't need to be a theoretical physicist. Just "common sense" is telling me the same. It seems, that a large number of experts working on such topics have lost this common sense to a large extent...

24. Given that time is reversible in QFT, and so in theory you can extrapolate any state at t=0 to all times prior and subsequent, in that sense the four-dimensional history and future of the universe can be reduced to the three-dimensional present moment where time is not a variable. At least, for those who believe in the block universe.

Is the above related in any way to the holographic principle (even as an analogy)?

25. The holographic principle is a bit of a misnomer actually. After all, in a real hologram you still need a three dimensional space for the picture to appear in! As a name it is rather like the Dirac sea which involves no actual sea, and here, no actual hologram.

26. @Greg Feild

Maybe letting the free market decide what is science and what is science is where science is going wrong? Time was when it was scientists that decided that through discussion, debate and experiment.

1. I was making a joke
🙂

27. Correction (Fourth paragraph, first sentence.): "sounds" should be "sound"

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