Measuring the depth of ideas

In this philosophical text, I would like to open the question what it means for a mathematical or physical idea to be deep. Are there some rules that may be used to quantify the depth? At the very beginning, you should know that no exact and objective definition will be found. And I believe that no such a definition may ever be found, even in principle. But let us try to identify some qualitative features of the deep ideas.

In the commercial world or the world of applied physics, one may measure the depth by the amount of U.S. dollars that the idea will earn in five years, for example. (The Reference Frame considers 1 USD to be an international unit analogous to 1 meter, 1 kilogram, 1 second - and the speculations about 1 euro replacing 1 USD as the main unit in the next 20 years to be unrealistic.)

When this approximate concept of depth is combined with the invisible hand of free markets, a mechanism appears that looks for and helps the deep ideas to be created and to propagate. Capitalism works as many readers will agree. The exceptions may be those who call themselves communists in Europe or who call themselves - because of much better skills in the P.R. - progressives in the U.S.

We may leave this approximate notion of depth to the free markets, but I am sure that most of us will feel unsatisfied. We can't really believe the markets in assessing such complex questions as the depth of ideas in theoretical physics undoubtedly is. After all, the markets are driven by the people. This includes ordinary people. Clearly, the rich people - those who have shown their ability to earn money - have bigger statistical weight in the ensemble that controls the markets. While it is natural to believe that this fact may improve the ability of the markets to measure depth, it is obvious that it does not improve it enough. It is simply not the depth that is valued by the markets. We simply feel that an idea does not have to produce much money for it to be deep, and the more pure research we do, the more we feel so.




We need an objective - and more abstract - description of the deep ideas. One of their features is their unification power. For an explanatory idea to be deep, it should be able to explain many diverse phenomena or many ideas of a more concrete type. For a constructive idea to be deep, it should be vital for constructing a large group of new objects (either material objects or theoretical constructs).

Even if we accept this definition, we still face the task to determine how should one count the phenomena or concrete ideas and how should we assign them an appropriate weight. It's not shocking that if a more concrete idea is deep itself and it is explained by another, more general idea, it makes the more general idea deeper than if it had explained a shallow idea only. How do we solve this self-consistent problem to measure the depth?

This mathematical problem seems somewhat analogous to the question what is the importance of a web page; a basic question that the guys at Google and its competitors have faced for years. As Alex Wissner has told me, the solution is expressed in terms of eigenvectors.

Consider the matrix "M" whose size is "N times N" where "N" is the number of web pages you monitor. The matrix entry "M_{ij}" is equal to zero or one if the i-th page refers to the j-th page. If I have mistakenly exchanged "i" and "j", let me hope that a careful reader will correct me. Note that this distinction between "i" and "j" is important because the matrix "M" is not symmetric and because we are gonna work with its eigenvectors.

We want to determine which web pages are important etc. The best solution turns out to be the eigenvector of "M" corresponding to the highest eigenvalue; it is essentially the principal component. The coordinates of this eigenvector inform you about the importance of the individual web pages. Note that this importance is not simply the number of other web pages that link to a given web page; it is affected by the influence of your referrers and their referrers, and so forth.

A thoughtful reader should ask: Is the world of ideas truly analogous to the world of the web? Does this algorithm treat the repeated web pages correctly? How is it affected by the attempts of someone to promote his or her own false ideas or shallow web pages? Will we improve the counting if the entries of "M" won't be either 0 or 1, but will take one of many possible values instead? There are many other questions to be asked.

In our reasoning, we are still pretty close to the free markets. We still talk about the ability of general ideas to explain the phenomena that have actually been observed in the real world. It is understood that we want to formulate as compact and general idea as possible as long as it remains relatively straightforward to specialize the general idea and derive its consequences for particular situations and concrete examples.

Of course, we would like to attack an even more abstract question, namely the depth of the ideas in the world of mathematics - that does not have to be connected with the observed phenomena at all - or ideas from theoretical physics that are ambitious enough so that they can't be directly tested with the current equipment. How deep a given idea from this category is? Someone may offer the citation count - that would be analogous to the profit counting above and would be slightly unsatisfactory because of the very same reasons. The main point I would like to solve is the difference between a robust, general, far-reaching idea on one side; and a random idea or construction from recreational mathematics on the other side.

One can see that the main feature of the ideas from the former, deep category is their perfect or nearly perfect uniqueness. Consider chess. One must be pretty intelligent and a fast thinker to play the chess well. There are zillions of possible scenarios how a chess match may advance and a good player has to use many emergent ideas to make the right decisions.

But on the other hand, there is a whole landscape of board games similar to chess you may imagine. That's not a big problem for those who want to train their minds and show others how powerful they are. Every board game in this landscape may be applied more or less in the same way. The detailed rules of chess are a matter of historical accidents; they are not arbitrary for the game to be meaningful, but we can certainly envision millions of other civilizations where the details of the rules differ but where the counterparts of Karpov and Kasparov enjoy more or less the same status.

But you should notice that when this status is discussed, we are focusing on the ability of the brain to deal with certain complex systems; we are not focused on the details of the ideas themselves. The rules of chess are not "true" in any profound way. Different rules would work, too. It is a matter of cultural heritage that we prefer the usual rules. This description is a typical feature of recreational mathematics: in this case, it is important that you train your mind but the details of the ideas are less crucial.

The terminology I used makes it clear that the analysis of a particular convoluted background of string theory is an example of recreational mathematics. In this case it is also true that one may (and has to) learn many things when she studies a particular background. She acquires skills that can be used to study other, similar backgrounds, too. But this fact is very different from the truth. It is very unlikely that one random vacuum from a huge landscape is "true". It should be obvious that a brute force analysis of a particular setup is not deep, and we don't need to discuss these things, I would say.

The real question is what conceptual ideas and visions are deep. For example, there are very deep ideas involving symmetries. It is by no means obvious that the observers in motion may be equivalent to observers at rest. Once you recognize this possible equivalence, you may postulate a Galilean symmetry. Then you may look for classical deformations of this group. You will only find a few examples and the Lorentz group is the only one that makes the speed of light finite and universal.

I don't only want to say that these symmetries are deep. Another observation is that the Lorentz group is deeper than the Galilean group. What do I mean? I mean not only that it is more correct in the actual world. It is more general given some assumptions we want to impose - such as the equivalence of observers in the state of uniform motion. The Galilean group and the Lorentz group have the same number of generators and respect the same basic consistency requirements.

Nevertheless, the Galilean group is a special example in which some commutators (of the boosts) are set to zero. This choice may be obtained from the Lorentz group by setting the defining speed - which is called the speed of light - equal to infinity; the Galilean group is a contraction of the Lorentz group. On the other hand, the speed of light is finite in relativity which allows one to unify the space and time; to unify electricity and magnetism; to identify the mass and energy by Einstein's most famous formula. The Lorentz symmetry also severely constrains the possible forms of the physical laws.

This comparison of the Galilean and Lorentz groups clarifies that I believe that the physical laws should be "as deformed as possible" as long as they are compatible with certain general aspects of beauty and physical consistency. If we continued to talk about symmetries, we would have to solve many other problems. For example, is quantum deformation of a group a very deep idea? My feelings are mixed. A quantum-deformed group only acts as a group in the context of quantum mechanics. The classical systems can't really have quantum symmetries. Is that a problem?

While my opinion about the quantum groups is uncertain, I truly feel that more deformed classical groups are deeper in a very similar sense in which the (technically) simple groups such as those in GUT theories are more profound than the non-simple groups.

I would also have to discuss the difference between global and local symmetries. The local symmetries are not true symmetries; they are redundancies of a description. Their representation theory is physically irrelevant because the trivial singlet representation is the only one that occurs in the physical spectrum. Moreover, the identity of a gauge symmetry is not uniquely given by a physical theory as we learned in the 1990s (S-duality, AdS/CFT, Matrix Theory). These facts reinforce the impression that the local, gauge symmetries are not really deep. Such a conclusion may be exaggerated a bit because the local symmetries remain an excellent guide to construct theories in which unphysical, ghostly modes of higher spin fields decouple (in Yang-Mills theory, general relativity, and elsewhere). Moreover, the global symmetries are typically a subgroup of the local symmetries.

This discussion would bring us too far from the main topic. Instead, let us ask: is quantum mechanics deep? Yes, I think that quantum mechanics is perhaps the deepest idea we know. It is once again a deformation of a conceptually simpler picture of classical physics. Much like the speed of light is finite in relativity and it unifies space and time, the Planck constant is finite in quantum mechanics which allows us to identify the energy with the frequency, among many other things - quantities that would otherwise remain as independent as space and time without relativity.

Quantum mechanics is the broad framework that controls thousands of interesting theories we know. Again, it is a deformation of the concept of a theory in classical physics. It is a generalization that may reduce to the previous, classical framework, but a generalization that still allows arbitrarily precise predictions of some measurable quantities, namely the probabilities. A non-trivial idea is that the total probability of all alternatives must be 100 percent which is guaranteed by the fact that the time evolution is a unitary operator. Because of the well-known relations between unitary operators and Hermitean operators (whose eigenvalues are real and eigenvectors are orthogonal), the whole picture makes a perfect physical sense. It is increasingly likely that the basic postulates of quantum mechanics will stay with us.

Are there some generalizations of quantum mechanics? I doubt it. But there can be, on the contrary, special versions of quantum mechanics where some postulates are added. For example, the group U(N) acting on the Hilbert space is reduced to a subgroup. But let me not speculate too much.

The successful framework of quantum mechanics does not tell us what is the theory of everything yet. We must still learn the right "Hamiltonian" (or at least the "S-matrix" or whatever replaces them). In other words, we must learn what are the right degrees of freedom and what is their dynamics. What deep ideas do we know in this direction of research?

We know the deep basic idea of quantum field theory - a theory whose relation to classical field theory is analogous to the relation between quantum mechanics and classical mechanics; and moreover a theory that implies that the energy of the field is quantized which transforms every field into an arena for particles - the fields' quanta. Quantum field theory (QFT) is the simplest framework that unifies two previous deep ideas, special relativity and quantum mechanics.

Once we calculate loops etc., we encounter infinities and other things. The deepest idea about these facts is the idea of the Renormalization Group. A quantum field theory is an effective description valid below a certain energy scale. Moving from one scale to another generally requires to change your effective field theory or at least their parameters, according to a certain flow. There can be QFTs valid at all energy scales - UV complete ones - but it is not necessarily true that the world is an example. In fact, the existence of gravity makes it pretty likely that the language of QFT must be completely invalid above the Planck scale (or earlier); in this context, the whole procedure of organizing physics according to the scale must be revisited.

The Renormalization Group is deep because it allows us to understand many features of QFTs - the divergences; the relation between different theories; the methods to construct QFTs - in a unified fashion. Let us now assume that the full theory is not just a QFT, and let's think about the nature of the most fundamental degrees of freedom.

Perturbative string theory is an example of a generalization of the concepts of QFT. Point-like particles are replaced by one-dimensional strings. Is that a deep idea? A priori, I definitely think that it is not a deep idea. One could replace points by strings or little green men. When the founding fathers established string theory, the profound idea was not the sentence that you may invent in less than one minute. The profound aspect were the special mathematical features of the theory that are only manifest once you spend years to investigate it. The fundamental stringy worldsheet is the only kind of worldvolume of an object that is described by a quantum field theory whose ghosts, worldsheet UV divergences, and other problems are under control, which is also able to generate spacetime physics whose spacetime UV divergences are harmless and which permits finite spacetime interactions reducing to the usual QFTs.

This special status of the fundamental string follows from the huge size of the two-dimensional conformal group and other features. Are these technical things a deep idea? Well, definitely. But it is not quite clear whether they are a human idea. The people who kept on - and keep on - discovering all the miracles in string theory are more like cowboys who accidentally discovered a ton of gold. It is not necessarily their unique ingenuity that makes all this progress. It is a combination of their intelligence, good luck, and especially the objective fact that the ton of metaphorical gold is simply there - somewhere in the world of crucial mathematical ideas that could be used by Nature to make the world work.

Once we appreciate the fact that the string theorists are not inventing but rather discovering something that objectively exists, it is OK to admit that these technical features of two-dimensional conformal field theory are deep ideas. It is definitely worth asking what are exactly the abstract features of a two-dimensional conformal field theory that make it work and make it profound. These theories describe all weakly coupled string theories which is a significant fraction of the backgrounds of string theory we know of.

And it is natural to think that all backgrounds of string theory may be described by a system that follows the same requirement of beauty and physical consistency as two-dimensional conformal field theory, but one that is more general. The more general setup may avoid any explicit quantum field theory description of the worldvolume; it may be generated by self-consistent bootstrap conditions; the worldvolume may be a target space of some deeper string theory (or generalized string theory) which itself may be a target space of something else, and perhaps ad infinitum. This approach is also natural because the worldsheet has a two-dimensional theory of gravity on it, and string theory is a way to describe a quantum theory of gravity.

Unlike the very general framework of string theory, this line of reasoning has not led to huge progress so far, but I definitely believe that these are very deep ideas.

What about some other ideas that people think are deep? For example, what about discreteness of space and physics? Something that Stephen Wolfram, Ed Fradkin, and the majority of the loop quantum gravity community would consider deep? And they may even find a much more thoughtful support from people like Cumrun Vafa?

In my opinion, the very bare statement that everything in physics should be discrete is not deep at all. It is a program of fundamentalism. It seems that we have known discrete and continuous features of the real world for quite some time. There are many observables in the real world that look completely continuous and the discrete people can't offer any replacement - at least not a replacement that would preserve all physical consistency rules as well as the "amount of beauty" - as represented by the symmetries, for example. In this situation, the call to abandon all continuous theoretical constructs is a call to throw away a huge portion of our vital current knowledge. It is unrealistic fundamentalism, not unsimilar to the Islamic one among others - not a deep idea.

Discrete systems may approximate continuous ones but in the actual systems we know, the continuous description is the more correct one.

There are more detailed reasons why I think that such calls are shallow. One of the deep abilities of quantum mechanics is to construct a discrete spectrum of observables that are defined as continuous functions of variables with continuous spectrum. The spectrum of a Hamiltonian can simply be discrete (or have a discrete sector) because of the uncertainty principle and the miraculous abilities of the theory of linear operators. Quantum mechanics admits many bases of the Hilbert space. The discrete bases are more appropriate to teach the truly new interpretational features of quantum mechanics - which is what Feynman did in his lectures. But if the task is to find the right Hamiltonian, it seems completely clear to me that the continuous language is deeper and more fundamental. The continuous language based on "x" and "p" (or their generalizations) as the elementary operators is better in explaining symmetries (both global as well as local ones) and locality.

The old quantum mechanics of Bohr was a system which implied some discreteness by its very assumptions. The new quantum mechanics allows us to derive these things - the spectrum of the Hydrogen atom, among millions of others. The old quantum theory was a naive phenomenological theory that captured some features of the real world; the transition to the new quantum mechanics was an amazing progress, I think. Those who want to present the discreteness as a fundamental idea are returning us to the age of the old quantum theory, to say the least. More likely, they want to return us to the age of luminiferous aether or even Democritus' atoms. I don't think that they have made much progress since Maxwell's and Fitzgerald's aether or Democritus' atomic school.

There exists an opposite fundamentalist approach in which everything must be derived from a continuous framework. In my opinion, it is much much deeper than its main enemy ("discrete physics"). Why do I think so? In some sense, the analysis of number theory (especially the prime integers) using zeta functions is one of the achievements of this way of thinking in mathematics. The distribution and properties of prime integers are encoded in a continuous (holomorphic) functions and the continuous tools of functional analysis may shed a lot of new light on questions from number theory. The discrete interpretation of the number-theoretical facts may count as the set of emergent phenomena. Yes, we need to know them, but it is still legitimate to consider the zeta functions to be the more fundamental description of the same data. Analogous statements hold for various generating functions etc.

In a similar fashion, the continuous theories such as topological string theories may be used to calculate many discrete and combinatorial features of Calabi-Yau manifolds. Also, Chern-Simons theories may generate many features of knots in knot theory. I believe that these continuous descriptions are deeper than the original, purely discrete form of the questions. In some sense, I would like to believe that all of theoretical physics and mathematics is about emergent phenomena found in some theoretical structure that is as continuous as you may imagine.

How can you construct highly continuous theories? What does it mean? Spaces of higher dimensions are "more continuous", in a sense. For example, the description of a background in terms of M-theory is more "geometrical" and "continuous" than in terms of perturbative string theory because more degrees of freedom are "geometrized" (into 11 dimensions). Whether or not you also say that it is more fundamental is a matter of taste. I would not say so - they are just different pictures to organize an infinite number of continuous degrees of freedom.

And the infinite-dimensional spaces are "more continuous" still. For example, when you quantize a system in classical physics, you obtain a smoother system because the sharp, discrete, point-like particle is replaced by a smooth wavefunction. If you perform a second quantization, the amount of continuity increases again. This could lead you to try to make a third quantization (something mathematically isomorphic to the situation that appears in string field theory) or a fourth, fifth quantization; infinitely many, if you wish. Can we learn something from these "higher order" hyper-functional methods (for example, from functionals defined on the space of functionals), or is it a generalization of successful ideas that was dragged too far?

At any rate, if a discrete, combinatorial, finite conclusion is derived from a continuous starting point, in my opinion, the depth of our understanding increases.

What are other deep ideas in string theory? For example, Polchinski's framework to calculate with D-branes is a very profound discovery. The broader message is that strings with more general boundary conditions can have oscillations that are connected with fields that describe a large spectrum of physical observables of a large number of new kind of objects (D-branes, in this example). That's definitely a deep idea because it shows that some previous ideas (fields from vibrating strings) are much more general and far-reaching than what was thought previously once the right subset of their features is extracted and imposed. Note that the previous sentence reflects a rather successful template to search for new profound ideas: preserve most features of a successful theoretical construct from the past, sacrifice some unimportant ones, and find new classes of solutions.

Holography is another deep idea; this statement is strengthened once mathematically controllable examples of holography are found in the context of the AdS/CFT correspondence. Holography tells you that a certain physical system should be described in terms of another physical system that lives in a space whose dimension is smaller. If formulated in this fashion, the idea is essentially unique. The details of the idea - the fact that the first system is gravitational, the second system is non-gravitational, and the dimensions differ by one, among other observations - are not unique a priori, but the correct answer may be derived mathematically. These are the features of a research direction that often leads to deep conclusions.

What about some ideas that seem much more shallow? For example, rewrite your metric in terms of some new variables in such a way that some observables will have discrete eigenvalues. It's not hard to see that I talk about loop quantum gravity. These new variables seem shallow because of many reasons: they are not unique at all. Writing a system in new variables may be a good way to solve it if the change of variables is legitimate, and if you're lucky to choose the right one that will help you to solve a well-defined question. That's not the case of the new variables in loop quantum gravity. In that case, you don't solve anything you wanted to solve. What you derive is the discrete spectrum of the areas - which is not a consequence of quantum gravity but a consequence of your particular choice of the new variables (that is not legitimate globally on the configuration space, and therefore it implies some discreteness that does not really exist in quantum gravity itself).

Some people may like the discrete spectrum of the areas for some reasons that I find irrational; one of them is the confusion between discretization and quantization. In a deep contrast with the beliefs of many people, a quantum theory does not have to have a discrete spectrum of all observables.

There are zillions of other globally invalid field redefinitions that could lead to discreteness of other observables - various integrals of the Riemann tensor, for example. There is no reason why one of these observables should have a well-defined discrete spectrum while another one should not. There is nothing deep about one particular choice compared to others. It's just recreational mathematics, and because it has led to no consistency checks or insights that are independent of the particular choices, it is unsuccessful recreational mathematics. One of the completely critical features of a deep idea worth scientific investigation is that it must explain more than the input - measured by the extent to which the assumptions are non-trivial and "unlikely" - that you used to construct it. My huge respect for this principle may partially reflect the influence of Feynman's books on me.

If you explain a seemingly unlikely value of an observable - call it Lambda, for example - by postulating an equally unlikely network of new objects, a whole new decoupled sector of new fields, or a plethora of Universes - call it a landscape - whose only achievement is to explain a single thing, then you have not really explained anything. Occam's razor dictates that these unnecessary quasiexplanations should be jettisoned as excess baggage in order to make progress. (Thanks to Murray Gell-Mann and his cute commercial for Enron.)

The observation that Lambda is small and has an unlikely value that we can incorporate into effective field theory is a deeper description of the situation than a new system of theoretical constructs that can only describe one thing. However, some of us simply like recreational mathematics. Imagine that you want to explain one number (Lambda) and in order to do that, you add a new mostly decoupled sector to your theory with a lot of new degrees of freedom and new parameters (c_1, c_2, ... , c_N). Then your Lambda is a function of these numbers "c_i" and perhaps other things, and it may be fun for you to calculate what Lambda would be in this hypothetical situation.

You may call it physics, but it is not a scientific explanation of anything because you actually *increased* the number of things that remain unexplained. It is a mental image that someone may like to have in mind, much like luminiferous aether that was thought to underlie electromagnetic phenomena. But until there is an independent prediction of these things, you can't think that it is an intriguing or deep idea. And once the experimental tests are doable, the history teaches us that these ideas whose only ability was to explain one thing that was used to construct them are nearly always falsified at the very end. The aether wind does not exist, for example.

Let me summarize. Deep ideas are those that are unique among conceivable similar statements at comparable levels of complexity and that are able to cover a large set of particular examples (models, phenomena, metaphenomena) and explain a large number of patterns using a small number of independent assumptions and parameters, especially if the deep ideas are inevitable. Whether or not a given idea is unique among ideas that a priori look analogous, may often require hours or years of calculations. These calculations are crucial because we must choose our deep ideas not only according to the impression they make in the first 3 minutes, but also according to their ability to offer us true insights in the long term.

In order to make progress, we must not be too dogmatic - so that we would believe that an idea that looks intriguing in the first 3 minutes must be studied despite decades of failures. But we should not sacrifice all of our principles either - so that we would study whatever requires some calculations. But we should know that being in the middle is not a sufficient condition either.