## Sunday, January 20, 2013 ... /////

### Mapping all possible physical theories

Young people should get some clue about all possibilities, avoid dead ends

I am kind of – albeit not obsessively – observing young, emerging physicists who have a chance to bring a new significant conceptual development or "revolution" to physics. It's hard to see whether such a future "revolution" will be an accidental side effect of some "modest" technical work or whether it will arise from some people's attempts to think really deeply and conceptually.

It seems to me that on this planet, the number of people below 30 who have independently understood the structure of the "space of ideas" and possible theories and who "really know" why there isn't any alternative to string theory as a theory of quantum gravity, to pick a major example, is at most "a few dozens". There are a few hundred young people who have worked on string theory as well but they were made to do so and they're not really "leaders" of the collaborations. Also, they may easily "change their mind" with a new boss.

A young person's journey to the cutting edge of theoretical physics is neither easy nor straightforward. One may try to take shortcuts and avoid many streets that look like dead ends. On the other hand, it may always turn out to be useful in the future if you have spent some time (hopefully limited time) in such a dead end. Some of the objects found in such a dead end may reappear in places that are not dead ends and they may be useful.

Another reason why dead ends may be useful is that if you get familiar with the "basic dead ends" and perhaps a larger (but still limited) number of dead ends in the foundations of physics, you may get immune against a much larger number of dead ends that lure conceptual researchers when they get to a more advanced stage of research.

Possible classical theories

When I was 4 or so, I started to think about the world as a theoretical physicist. It means that I became certain that the world is isomorphic to a mathematical model – yes, unfortunately, that paradigm was only good for classical physics at that time. The task was to find out the exact mathematical model that describes the real world.

The first category of models involved equations governing "where the matter is and where it isn't". The basic observables in these models were functions$f: \RR^{3+1}\to\{0,1\}.$ For each point of the space and at each moment of time, you either specify whether "matter is there" or "matter is not there". There are different materials – and different colors of matter etc. – so one possibility to guarantee that the theory describes all of them would be to enhance the set $\{0,1\}$ to a larger set. For example, the set could have five elements, corresponding to classical elements. (Yup, it's the same word!)

Alternatively, different materials could arise from the binary data but the microscopic composition of a material could involve both $0$ as well as $1$ in different patterns. This idea was "modern" and indeed, this idea is recycled in Nature many times. Nature doesn't need too many elementary particle species or too many different indivisible "atoms": it may produce them by various combinations of a smaller number of elementary building blocks. String theory is the most principled theory of this sort because even elementary particles are made out of the same string, one kind of a fundamental object.

If you identify the world with the information about the regions where matter is present and where matter is absent, you should hope that these regions are separated by simple enough boundaries. These boundaries may be parameterized by $2+1$ parameters – the coordinates labeling domain walls – and the functions that determine the shape of these domain walls may be subject to dynamical equations. In some sense, this kindergarten model of mine was a model of open D3-branes.

Needless to say, soon afterwords, I was exposed to classical mechanics with point masses. They approximately describe the motion of celestial bodies and other things. The objects' coordinates are functions of time, $x_i(t)$, that may be required to obey some differential equations. One may spend years with classical mechanics but the previous sentence is enough for my current purposes.

The identity of the differential equations that would govern the coordinates is unknown. In fact, all possible forms of the potential energy etc. seem equally justified. There's no principle that would pick the right functions among the $\infty^\infty$ candidates. From such a broader viewpoint, classical mechanics is utterly unpredictive. However, you may try to guess the right functions – which may be simple enough – and voilà, some of the simplest ones will agree with the experiments.

Classical mechanics may be applied to the case of very many mass points. It may also be generalized to classical field theory whose basic time-dependent variables are fields $\phi_j(x,y,z,t)$ that depend not only on time but also on space. In some sense, the extra independent arguments $x,y,z$ play an analogous role to the index $i$ in $x_i(t)$ except that they're continuous: they increase the number of dynamical variables. These functions $\phi_j(x,y,z,t)$ must obey partial differential equations. While $(x,y,z)$ are analogous to $i$ when it comes to imagining what depends on what, you may also see an analogy between $(x,y,z)$ and $t$. In fact, special relativity postulates a symmetry, the Lorentz symmetry, between these four coordinates. All inertial coordinate systems (reference frames) are equally good.

Conceptually, these classical field theory models may also be extended to the framework of general relativity in which an arbitrary redefinition of coordinates $(x,y,z,t)$ is allowed and in each set of coordinates, the physical laws are required to have the same simple form (in principle). Such theories have some redundancies but the stuff isn't really conceptually new. The real world is still identified with a mathematical model involving functions of many coordinates that are required to obey partial differential equations.

In all these models, time $t$ is a continuous variable. Because of the Lorentz symmetry that you ultimately get certain about and whose importance you should appreciate, the continuity of $t$ implies the continuity of the spatial coordinates $(x,y,z)$, too. It's useful to spend some time with candidate classical descriptions of Nature in which the time $t$ is modeled by a discrete, not continuous, quantity. But you should do so critically, not mindlessly. When you do so, you will realize that all these models are deeply contrived and almost certainly wrong. Continuous symmetries become impossible if the spacetime coordinates are discrete in any sense. The absence of continuous symmetries will mean that an important principle to constrain the laws will be missing. None of the "real physical problems" of renormalizability or short-distance divergences (in classical physics or, later, quantum physics) would be solved by the discreteness, either. The discreteness is just a regularized definition of a theory you started with and the real problem signaled by the divergences is the infinite number of undertermined parameters (coefficients in front of non-renormalizable couplings).

Discrete physics is a major dead end where tons of young and not-so-young people get stuck. Once you accumulate enough experience from other streets and avenues, it is totally obvious to you that they won't get anywhere if they keep on spending months or years on this direction. However, they haven't really been "elsewhere" which often prevents them from seeing the essential features of theories with a continuous time that their discrete theories simply can't reproduce. This ignorance, usually combined with excessive stubbornness, is a dangerous mix.

That was the last moment I spent with the idea that the time could refuse to be "fundamentally continuous". All discrete things in Nature are ultimately derived, emergent. The fundamental laws of physics must be continuous, and the more they are continuous, the more fundamental they seem to be.

When we switched from classical mechanics to classical field theory, we made the mathematical objects conjectured to match the world more complicated. Instead of functions of one variable, we had functions of several variables. We may also consider functions of infinitely many variables, e.g. functionals$Z[f(x,y,z)]\in\RR.$ For each function of the coordinates $(x,y,z)$, you choose a real number. That's what a functional means. If such an object fundamentally defined a classical theory, this theory would contradict locality. It's just too complicated. Generic functionals of this sort describe voodoo-like objects that affect several places in space simultaneously. The object $Z$ doesn't seem to describe isolated objects. Let me tell you that once you work with a quantum theory, (complex-valued) functionals may describe the state vector in quantum field theory. The voodoo character goes away in this quantum interpretation (and only in the quantum interpretation) because the functionals' being nonzero at several widely separated places means that one OR the other possibility is realized – but both of these possibilities may still correspond to properties of local objects (the functional is a sort of probability distribution).

You could also consider even more complicated objects, "functional-als", which assign a real or complex number to each functional.$\Upsilon(Z[f(x,y,z)]).$ In some sense, these functional-als are one of the possible descriptions of state vectors in string field theory, a definition of string theory that tries to emulate a quantum field theory with many fields. And you could go on and on and on, functional-al-al-al-als, but Al has already gotten \$100 million in donation from the Qatari oil magnates so I don't want to add too many Al's here.

So far I was considering classical theories of physics. There is an objective model that is equivalent to Nature. It's important for every modern physicist who doesn't want to be "completely stuck" about basic issues to understand that this whole classical framework, regardless of its huge number of possibilities and various diverse and "clever" modifications and generalizations you could propose, is fundamentally wrong. As argued in hundreds of articles on this blog, there isn't any objective mathematical function or functional-al-al-al that would explain all observations we make.

Instead, you have pick the observations you make – the input information – translate it to a state vector or a density matrix, apply the transformations dictated by quantum mechanics, and you obtain probabilistic predictions for any meaningful statement about an observable in the future. This quantum framework is entirely different from the classical one and it's the right one.

Even without experimental tests, a good theorist should be able to see that the quantum framework is more natural, more general, and leads to more constrained theories. It's simply consistent to identify observables with mutually non-commuting operators. For this reason, it is an "extremely special and therefore unlikely assumption" that all observables commute with each other. This situation "everything commutes with everything else" may appear in restricted situations and in the classical limit. And the classical limit is a very special, "infinitesimal" subset of quantum mechanics. The constant $\hbar$ is simply not equal to zero. Its being exactly zero is infinitely unlikely, even a priori. The quantum theory is the primary one and the classical one is at most an approximation valid in a limit.

In classical field theories, the functions encoding the potential energy and other things could have been arbitrary. In quantum field theory, they're tightly constrained. Perturbatively speaking, the Lagrangians have to be at most polynomials of the fourth degree (let me not go into the details here) for the theory to be renormalizable (i.e. consistent and predictive up to arbitrarily high energies, if I use a physical language focusing on the implications of the adjective). The number of coefficients is finite. The number of parameters is finite.

It's very important for a modern physicist to understand that quantum mechanics is fundamentally different from classical physics and it has different answers to many questions. Even questions that look "obvious" in classical physics may become tough to prove in quantum physics and some of these claims become just approximations. Nevertheless, quantum mechanics works. And its postulates can't really be "deformed" in any way. For example, the linearity required from the operators is fundamentally equivalent to (or at least equivalent to something that is similar to) addition formulae for probabilities of "A or B" etc. There isn't any logic or probability calculus where the probabilities would add up non-linearly. Analogously, there isn't any non-linear deformation of quantum mechanics worth considering as a theory of physics, and so on.

Once you spend some time with the research – but not a mindless one – of possible frameworks that are neither classical nor quantum, you will realize that classical physics and quantum physics are the only two truly sensible options and the former is known to be wrong. If you search for a theory of Nature, you're searching for a quantum theory.

It means that you have to have a Hilbert space – well, all infinite-dimensional Hilbert spaces are really unitarily equivalent to each other so at this level, there's no freedom. The diversity only emerges once you equip the Hilbert space with natural or interesting observables – linear operators – and one/several of them, e.g. the Hamiltonian (or the unitary S-matrix/evolution operator(s)) – has/have to encode the dynamical evolution in time.

Most of the often studied quantum theories may be obtained by a "straightforward" procedure of quantizing a classical system with an action. In this subset, the search for a quantum theory is equivalent to the search for a classical theory combined with the procedure of quantization. However, you should never forget that the procedure of quantization changes the predictions and the character of the theory. It changes its "moduli spaces", in some cases. It introduces nonzero commutators, quantization conditions, and monodromies. UV divergences and even anomalies appear that make some classical theories inconsistent at the quantum level. Some classical theories refuse to be unique as "generators" of quantum theories – there may be several quantum theories with the same classical limit, and so on.

As I said, the convoluted mathematics needed to calculate things within the quantum framework constrains the possible theories. In some sense, the space of consistent quantum theories is much more special than the space of consistent classical theories – even though some people think that there is a one-to-one correspondence whose map is kind of "straightforward". It's not that straightforward. If you study the space of possible consistent quantum theories, it's a different problem than the problem to map out possible classical theories. One must be aware of these differences. One must think quantum.

The constrained character of the quantum theories becomes really intense once you demand that the theory also contains gravity, approximately in the sense of the general theory of relativity. In this case, you may classify all possible candidate classical theories that could describe such a thing and quantize them, to see whether gravity really emerges. Most of these theories will be crippled by short-distance divergences, anomalies, and other inconsistencies. All "constructible" candidates that you may find and that will work will turn out to be equivalent to string theory in one of its incarnations. This conclusion can't be obvious from the beginning. People couldn't have assumed it. People hadn't known it 45 years ago. But once you really master the "city" with all the possible streets, avenues, and dead ends, you will see that my claim is right. All roads lead to string theory.

However, there is still the possibility that the right ultimate theory is not "constructible" in the sense that its "action" would be some functional of some basic fields which are (quantized) functions or functionals or functional-als, and so on. It may be some theory that determines its own structure. In physics, this possibility – and the research programs attempting to change the mere possibility to an operational theory – is known as "bootstrap" because the theory is apparently capable of "pulling itself out by its bootstraps". Heisenberg was an early proponent of this idea and it became very popular in the late 1960s when the strongly interacting "hadrons" started to resemble a zoo that couldn't possibly be "constructed".

If Nature is fundamentally described by a bootstrap-like theory, you can't really map the possibilities in advance. Instead, you must be lucky, find the mysterious principle behind everything, and verify that it is indeed correct. The theory doesn't have to be based on any particular set of functions or functionals or anything else in our sequence. It may be based on mathematical objects of some brutally generalized, vague type that only acquire their identity with a clear name once you solve some mathematical conditions written in terms of maths we can't really comprehend today. These conditions don't have to be any particular, pre-determined differential equations or anything of the sort.

(Two-dimensional conformal field theories – ones relevant for perturbative string theory, among other things – are a "restricted" example of a success of the bootstrap paradigm but they're still "too constructive" relatively to the hypothetical ultimate "full-fledged bootstrap" theories that should really talk about some "consistency" only.)

However, even if string/M-theory is most universally described in this bootstrap way, it doesn't mean that the "constuctive" approaches to find the theory are useless. Quite on the contrary, even if one has a bootstrap theory whose fundamental rules can't be written down in terms of particular functions and/or evolution equations depending on particular potential-energy-like functions, it's always likely that particular environments/solutions (points in its moduli space or configuration space) allowed by this theory will exist in the "extreme limits" and these extreme limits typically do admit "constructible" definitions. And that's what we seem to see in the case of string/M-theory, too. Its extreme limits, such as the weakly coupled limits, admit various types of perturbative and low-energy and other expansions while the "complete bulk of the configuration space" dealing with "generical compactifications and/or cosmology" remains a bit ill-defined and mysterious.

If a bootstrap-like definition of string/M-theory exists, it would be great to find it but you shouldn't forget that we don't really know whether it exists. No one is going to guarantee for you that if you spend years or decades by this pursuit, the investment has to be repaid.

Quite generally, I want to say that as young physicists are getting mature, they must increasingly rely on assumptions they have verified or even "rediscovered" by theselves. Too many young people – well, a big majority – still works on research programs whose rules and limits were defined by someone else. For example, someone decides that quantum gravity is just the addition of some "hats with a nice shape" to the classical observables and we should find it. Dozens if not hundreds of people then spend time by the futile attempts to describe the real world by something like loop quantum gravity, causal dynamical triangulations, spin foams, and several other hopeless proposals like that. The young people going into these programs forget – or don't want to know – that their leaders and advisers aren't infallible Gods. They rely on lots of assumptions about the hypothetical theory that are unverified or, if you verify them, that are demonstrably wrong.

Of course, other (and sometimes the same) young people are doing the same thing in the case of string theory research, too. A difference is that string theory works so they were more lucky about their leaders or advisers. But at the end, a theoretical physicist who wants to get to the conceptual cutting edge must get rid of the dependence on the unreliable shoulders of others. He or she must map out the space of possible ideas in physics, see how large or small various neighborhoods of this space are, which of them are hopeless ghettos or bomb sites – and such bomb sites may be small as well as vast (a suburb's being "large" doesn't mean that it is not a bomb site) – and which of them deserve to be studied in greater detail or generalized.

It's important for a young scientist not to scream resolute conclusions about things that he or she hasn't independently verified. Claiming "I know something" even though in reality it's just "I want to be a partisan and to defend whatever some other people say" is just a deeply unscientific attitude to questions that one should avoid from the beginning, from the low age. For example, it's really bad if someone gets stuck in the trap of the – using a highly diplomatic language – shitty demagogues, crackpots, and assholes such as Lee Smolin and Peter Woit and the mindless movements that these scumbags created around them just in order to harm the proper research in physics. It's likely that most of the people caught in these cesspools will never make it to the fresh air again.

But there are many more dead ends in the world. And many beautiful and important oases that may remain unexploited by tourists if most tourists spend most of their time in deserts and cesspools.

And that's the memo.

#### snail feedback (17) :

I always enjoy when you let us "in on" (so that we get can get a feel for) how you developed, and how you view the current overall state of, your greatest speciality (aside from your familiarity with outside and inside mushrooms ;>)!

This is a very nice "Wort zum Sonntag", reading this is much more useful and fun than going to church at the proper time and hear a random sermon :-P :-D :-).

As Peter F, I always like it too when Lumo tells us what he thought and how he perscieved the world as a sweet, nice, very young cool kid :-)

This bootstrap things were and still are some kind of mysterious to me (for example when people are talking about it on Physics SE). The explanations that such a bootstrap theory could generally describe string- and M-theory, and the parts which can be studied by well defined methods today are certain extreme limits, seems quite intriguing to me :-)

Dear Lubos, I don't completely understand your problem with discrete field theory models. ( I'm below 30. :) )

In statistical physics, if you want to describe a magnetic phase transition, you can start with the 3d Ising-model and derive an effective low energy Phi^4 model from it. The continuous rotation symmetry is broken at small distance scales but it gets restored at large distance scales.

I am confused by your attitude to Lee Smolin and Peter Woit. I respect all of you and I wonder why you, learned people, cannot get to the same denominator in order sing as a one chorus?

I vaguely understand what the bootstraps were trying to do with the old analytic S matrix theory - basically the s, t and u channel feynman diagrams all generated each other in a self consistent way. But I'd certainly be interested in learning how this is applied in string theory - I know Ron Maimon alluded to it a number of times. Presumably it's related to the fact that the Feynman diagrams are replaced (at least in pert. string theory) by the pair of pants worldsheet joinings and this somehow unifies the s, t and u channel views. Maybe worth asking a question on stackexchange?

Yeah, this old analytic S matrix theory is another thing I do not exactly know what it was or is ... :-/

In this lecture:

Lenny Susskind intruduced these (s,t,u) Mandelstem variables. He explained that in the Veneziano amplitude for meson scattering, there is no need to add up the diagrams of all these three channels because there is a symmetry (or duality) between the S and T channel. By rough outline of how to calculate it, he shows that what the Veneziano amplitude actually describes physically the scattering of strings too.

Your question would make a very decent and nice post on Physics SE, and I would immediately +1 it :-)

The only thing I doubt is that it would get a decent answer too, if Lumo ;-) does not write one. Ron would certainly know a lot about it, but he does not write new answers anymore ...

Personally I am still very annoyed about David Zaslavsky randomly deleting correct physics reasoning mostly in comments but regular posts too, just because in his biased opinion, they are according to his weird personal definition not constructive, add no value, etc. (while they get highly upvoted by other members of the community!).

Thanks yes, I'll have a listen to that lecture when I get a bit of time. I've only really read string theory texts at the Zweibach sort of level, but I know Susskind's lectures are very good.

PW is not a physicist, all he does is permanently bullying research he does not like, insult people who work on it, etc... this seems to be the purpose of his life.

He does this not only on his blog in the internet and by writing books, but by being onmipresent in the popular and influential media and selling his rants against fundamental physics as the "opinion of an expert", which he is not at all (he has no clue about theoretical physics and is not honestly interested in it either...) he tries to destroy research directions he dislikes in the real world too.

From a social science point of view, a person like PW is a picture book example of a so-called concern troll:

http://en.wikipedia.org/wiki/Troll_%28Internet%29#Concern_troll

His blog is, from the same social science point of view, nothing but a troll site

http://en.wikipedia.org/wiki/Troll_%28Internet%29#Troll_sites

where his subtrolls, fans, and spambots can freely propagate and gather, before they fan out to other places in the internet, such as physics blogs and comments below online news articles, where people talk about fundamental or theoretical physics, and inhibit or prevent there any serious discussion about certain topics by trolling

http://en.wikipedia.org/wiki/Troll_%28Internet%29#Trolling.2C_identity.2C_and_anonymity

and flaming

http://en.wikipedia.org/wiki/Flaming_%28Internet%29

In summary, the concept of a "common denominator" between the non physicist PW and any reasonable honest physicist that deserves this identifier, is zero by definition. So one should avoid trying to apply such ill defined (common) denominators, because they lead to very bad infinities that can not be renormalized away ;-) and that would seriously threat science.

I have written this because I personally had some "fun" with PW...

Lee Smolin I know less well, so I leave this issue to others to explain.

Yeah, I newly have the Zwiebach book in my bookshelf too :-), since I have noted that not everything Lenny Susskind talks about in his lectures is covered in my demystified book too.

Cheers

Just curious, can you name some of the young, emerging below 30 physicists that you think have a chance to bring a significant conceptual development development of revolution to theoretical physicists?

I can. But if you asked me whether I will, the answer would be different.

Hello Lubos,

I disagree, even if the Planck scale is not directly accessible, clever phenomenologists may be capable to give advice on how to indirectly detect QG or other higher energy BSM effects and theoretical mathematical and conceptual studies of at the moment not directly accessible higher energy physics is worthwhile and interesting.

There is no reason to give such negative and sourballish advice to young people and throw in the towel.

Vladimir, are you joking? They both wrote books that were destructive to theoretical physics, and have been sarcastic and dismissive of the only theory so far that has a chance of uniting QM and gravity. Woit is NOT an active physicist, and seems to delight in sitting back and taking uninformed potshots at those who are, and Smolin is all over the map, jumping from one discipline to another (now economics) with his crackpot theories. Both are masters at insinuating themselves into the main stream media and popular science journals as "experts". If you actually read Lubos' posts, you would see that real physicists don't take Smolin seriously---at a seminar/conference, science journalist, George Johnson piped up when he mentioned something Smolin had proposed "Well, no one here thinks that Lee Smolin is a crackpot".
I think it was David Gross who said something like "Really? Hands up everyone who thinks he is a crackpot" and hands sprung into the air everywhere. ( my version was improvised on the fly, but is reasonably accurate.)
And you want us to "get in the same denominator in order to sing as a chorus."??
Really? In order to do this, real scientists would have to first sustain serious brain damage.

Dear Dilaton,

You, as almost all physicists these days, seem to completely missing the
point (Matt seems to
have left the sinking ship) and I'm curious what Lubos has to say about this. It has nothing to do with giving
"sourballish advice" and also with not believing in BSM physics. My
replay is full of BSM physics. But it is an honest advice to starting
physicist to not waist their lives on trying to develop new theories
without experimental data since the current theory has been developed by
legendary physicists who had a pile of phenomenological data to their
disposal or their new theories could be easily tested proving or
disproving their theories and also because the current theory is
indicating that such new theories are not likely to be seen. It's an
honest advice, it's easier to give a false advice. I'm afraid the LHC will prove my point, but we can wait
for that, you never know :-)

Dear Marcel van Velzen,

I am interested too in what Lumo has to say :-)

In the meantime, to the question what it would mean if for example the LHC sees no SUSY, he said this on Physics SE:

http://physics.stackexchange.com/a/6439/2751

Concerning the interplay between theory and experiment in advancing physics (or science generally), I stick to my point of view that it is perfectly legitimate too if theory is ahead of experiment at times and concerning certain topics and that not only the converse situation (experiment comes before theory) is allowed.

Cheers