# Tag Archives: P.W. Anderson

## Consistency in the Hierarchy

When writing on this blog, I try to share nuggets here and there of phenomena, experiments, sociological observations and other peoples’ opinions I find illuminating. Unfortunately, this format can leave readers wanting when it comes to some sort of coherent message. Precisely because of this, I would like to revisit a few blog posts I’ve written in the past and highlight the common vein running through them.

Condensed matter physicists of the last couple generations have grown up ingrained with the idea that “More is Different”, a concept first coherently put forth by P. W. Anderson and carried further by others. Most discussions of these ideas tend to concentrate on the notion that there is a hierarchy of disciplines where each discipline is not logically dependent on the one beneath it. For instance, in solid state physics, we do not need to start out at the level of quarks and build up from there to obtain many properties of matter. More profoundly, one can observe phenomena which distinctly arise in the context of condensed matter physics, such as superconductivity, the quantum Hall effect and ferromagnetism that one wouldn’t necessarily predict by just studying particle physics.

While I have no objection to these claims (and actually agree with them quite strongly), it seems to me that one rather (almost trivial) fact is infrequently mentioned when these concepts are discussed. That is the role of consistency.

While it is true that one does not necessarily require the lower level theory to describe the theories at the higher level, these theories do need to be consistent with each other. This is why, after the publication of BCS theory, there were a slew of theoretical papers that tried to come to terms with various aspects of the theory (such as the approximation of particle number non-conservation and features associated with gauge invariance (pdf!)).

This requirement of consistency is what makes concepts like the Bohr-van Leeuwen theorem and Gibbs paradox so important. They bridge two levels of the “More is Different” hierarchy, exposing inconsistencies between the higher level theory (classical mechanics) and the lower level (the micro realm).

In the case of the Bohr-van Leeuwen theorem, it shows that classical mechanics, when applied to the microscopic scale, is not consistent with the observation of ferromagnetism. In the Gibbs paradox case, classical mechanics, when not taking into consideration particle indistinguishability (a quantum mechanical concept), is inconsistent with the idea the entropy must remain the same when dividing a gas tank into two equal partitions.

Today, we have the issue that ideas from the micro realm (quantum mechanics) appear to be inconsistent with our ideas on the macroscopic scale. This is why matter interference experiments are still carried out in the present time. It is imperative to know why it is possible for a C60 molecule (or a 10,000 amu molecule) to be described with a single wavefunction in a Schrodinger-like scheme, whereas this seems implausible for, say, a cat. There does again appear to be some inconsistency here, though there are some (but no consensus) frameworks, like decoherence, to get around this. I also can’t help but mention that non-locality, à la Bell, also seems totally at odds with one’s intuition on the macro-scale.

What I want to stress is that the inconsistency theorems (or paradoxes) contained seeds of some of the most important theoretical advances in physics. This is itself not a radical concept, but it often gets neglected when a generation grows up with a deep-rooted “More is Different” scientific outlook. We sometimes forget to look for concepts that bridge disparate levels of the hierarchy and subsequently look for inconsistencies between them.

## Is it really as bad as they say?

It’s been a little while since I attended A.J. Leggett’s March Meeting talk (see my review of it here), and some part of that talk still irks me. It is the portion where he referred to “the scourge of bibliometrics”, and how it prevents one from thinking about long-term problems.

I am not old enough to know what science was like when he was a graduate student or a young lecturer, but it seems like something was fundamentally different back then. The only evidence that I can present is the word of other scientists who lived through the same time period and witnessed the transformation (there seems to be a dearth of historical work on this issue).

It was easy for me to find articles corroborating Leggett’s views, unsurprisingly I suppose. In addition to the article I linked last week by P. Nozieres, I found interviews with Sydney Brenner and Peter Higgs, and a damning article by P.W. Anderson in his book More and Different entitled Could Modern America Have Invented Wave Mechanics? In his opinion piece, Anderson also refers to an article by L. Kadanoff expressing a similar sentiment, which I was not able to find online (please let me know if you find it, and I’ll link it here!). The conditions described at Bell Labs in David Gertner’s book The Idea Factory also paint a rather stark contrast to the present status of condensed matter physics.

Since I wasn’t alive back then, I really cannot know with any great certainty whether the current state of affairs has impeded me from pursuing a longer-term project or thinking about more fundamental problems in physics. I can only speak for myself, and at present I can openly admit that I am incentivized to work on problems that I can solve in 2-3 years. I do have some concrete ideas for longer-term projects in mind, but I cannot pursue these at the present time because, as an experimentalist and postdoc, I do not have the resources nor the permanent setting in which to complete this work.

While the above anecdote is personal and it may corroborate the viewpoints of the aforementioned scientists, I don’t necessarily perceive all these items as purely negative. I think it is important to publish a paper based on one’s graduate work. It should be something, however small, that no one has done before. It is important to be able to communicate with the scientific community through a technical paper — writing is an important part of science. I also don’t mind spending a few years (not more than four, hopefully!) as a postdoc, where I will pick up a few more tools to add to my current arsenal. This is something that Sydney Brenner, in particular, decried in his interview. However, it is likely that most of what was said in these articles was aimed at junior faculty.

Ultimately, the opinions expressed by these authors is concerning. However, I am uncertain as to the extent to which what is said is exaggeration and the extent to which it is true. Reading these articles has made me ask how the scientific environment I was trained in (US universities) has shaped my attitude and scientific outlook.

One thing is undoubtedly true, though. If one chooses to resist the publish-or-perish trend by working on long-term problems and not publishing, the likelihood of landing an academic job is close to null. Perhaps this is the most damning consequence. Nevertheless, there is still some outstanding experimental and theoretical science done today, some of it very fundamental, so one should not lose all hope.

Again, I haven’t lived through this academic transformation, so if anyone has any insight concerning these issues, please feel free to comment.

## What is Scientific Consensus?

When a theory is put forward, it takes time for the scientific community to evaluate its merits. Ultimately, one hopes that the theory is able to not only explain past data, but to be able to predict the outcome of future experiments as well. When the dust settles, we hope that we reach “scientific consensus” regarding a theory. But what does this mean?

Since this is a condensed matter blog, let us take BCS theory as an example. When BCS was formulated, it was able to explain numerous experimental observations, such as the evolution of the electronic gap as a function of temperature as well as the specific heat anomaly among several other observations. However, there were also apparent problems with BCS theory. Many physicists were concerned with the non-conservation of particle number and with some aspects of broken gauge symmetry (pdf!) in the theory. Notably also, there were materials that did not conform exactly to the BCS formulas, such as Pb (lead), where the predicted $2\Delta/k_BT_c=3.5$ relation and was found instead to be around 4.38.

So the question is, how were these issues resolved and how did the community reach the general consensus that BCS theory was applicable for the existing superconductors at that point in history?

This question actually leads to a more general scientific question: how do we reach a consensus concerning a theory? The answer to this question involves a Bayesian approach. We start with a prior probability based on our biases and update this prior probability as we begin to examine more and more data, making predictions as we go along. If physicist A had spent the past 10 years working actively on a theory of superconductivity and may secretly hope that BCS theory is wrong, s/he may start out with only 3% confidence that BCS theory is correct. On the other hand, physicist B may be completely neutral and would have a prior probability of 50%. Another physicist C would perhaps be swayed by the fact that Bardeen had just won a Nobel prize in physics for the invention of the transistor and therefore has a initial confidence level of 85% that BCS is correct. These constitute these physicists’ prior probabilities or “biases”.

What happens with time? Well, BCS predicted the existence of the Hebel-Slichter peak in the NMR spectrum, which was then observed shortly thereafter. Furthermore, Anderson showed that one could project out a particle-conserving part of the ground-state, which resolved some theoretical issues pertaining to particle-number conservation. Gorkov was also able to show that the phenomenological equations of Ginzburg and Landau were derivable from BCS theory (pdf!). McMillan and Rowell then conducted their famous experiments where they analyzed the second derivative of tunneling spectra, which exhibited phonon anomalies, to explain why lead did not obey the simple BCS formalism, but required a small extension.

As these data points accumulated, confidence in BCS theory grew for physicists A, B and C. In a Bayesian picture, we update our beliefs as we get more and more data points that are consistent with (or resolve questions pertaining to) a particular theory. Ultimately, the members of the scientific community would asymptotically approach a place where they understand the domain of validity of BCS theory and understand what it can predict. The picture I have in mind to represent this process is plotted below:

This plot is of a Bayesian updating scheme based on prior beliefs. The convergence of the viewpoints of physicists A, B and C is what is crudely meant by scientific consensus. Note that a person that starts out with a dogmatic 0% belief in the correctness of BCS theory will not change his/her mind with time.

It is important to emphasize that what I have called the 100% confidence level in my plot is meant to indicate a place where we understand the limitations and validity of a theory and how/when to apply this theory. For example, we can have 99.9% confidence that Newton’s theory of gravity will enable us to solve simple kinematics problems on the surface of the earth. While we know that Newton’s theory of gravity requires corrections from Einstein’s theory of general relativity, our confidence in Newton’s theory is not diminished when used in the correct limits. Therefore, in this Bayesian scheme, we get closer and closer to being 100% confident in a theory, but never quite reach it.

This is a rather Popperian view of scientific consensus and we know the limits of such a view in light of Kuhn’s work, but I think it nonetheless serves as a valuable guide as to how to think about the concept which is so often corrupted, especially in regard to the climate change discussion. Therefore, in the future, when people talk about scientific consensus, think convergence and think Bayes.

## Do We Effectively Disseminate “Gems of Insight”?

The content of this post extends beyond condensed matter physics, but I’ll discuss it (as I do most things) within this context.

When attending talks, lectures or reading a paper, sometimes one is struck by what I will refer to as a “gem of insight” (GOI). These tend to be explanations of physical phenomena that are quite well-understood, but can be viewed more intuitively or elegantly in a somewhat unorthodox way.

For instance, one of the ramifications of the Meissner effect is that there is a difference between the longitudinal and transverse response to the vector potential even in the limit that $\textbf{q}\rightarrow 0$. This is discussed here in the lecture notes by Leggett, an effect I find to be quite profound and what I would call a GOI. Another example is the case where Brian Josephson was famously inspired, by P.W. Anderson’s GOI on broken symmetry in superconductors, to realize the effect now known after him. Here is a little set of notes by P.B. Allen discussing how the collective and single-particle properties of the electron gas are compatible, which also contains a few GsOI.

My concern in this post, though, is how such information is spread. It seems to me that most papers today are not necessarily concerned with spreading GsOI, but more with communicating results. Papers are used for “showing” and not “explaining”. Part of this situation arises from the fact that the length of papers are constrained by many journals, limiting the author’s capacity to discuss physical ideas at length rather than just “writing down the answer”.

Another reason is that it sometimes takes a long time for ideas to sink in among the community, and the most profound way to understand a result is only obtained after a period of deep reflection. In this case, publishing a paper on the topic is no longer appropriate because the topic is already considered solved. Publishing a paper with only a physical explanation of an already understood phenomenon is “not new” and likely to be rejected by most journals. This is part of the reason why the literature on topological insulators contained the most clear expositions on the quantum hall effect!

So how should we disseminate GsOI? It seems to me that GsOI tend to be circulated in discussions between individual scientists or in lectures to graduate students, etc — mostly informal settings. It is my personal opinion that these GsOI should be documented somewhere. I had the privilege to learn superconductivity from Tony Leggett, one of the authorities on the subject. Many ideas he expressed in class are hardly discussed in the usual superconductivity texts, and some times not anywhere! However, it would probably be extremely fruitful for his lectures to be recorded and uploaded to a forum (such as YouTube) so that someone interested could watch them.

This is a difficult problem to solve in general, but I think that one of the ways we can rectify this situation is to include more space in papers for physical explanations while cleaning up lengthy introductions. Furthermore, we should not necessarily be discouraged from writing papers on topics that “aren’t new” if they contain important GsOI.

Do you agree? I’m curious to know what others think.

## Lessons from the Coupled Oscillator

In studying solid state physics, one of the first problems encountered is that of phonons. In the usual textbooks (such as Ashcroft and Mermin or Kittel), the physics is buried underneath formalism. Here is my attempt to explain the physics, while just quoting the main mathematical results. For the simple mass-spring oscillator system pictured below, we get the following equation of motion and oscillation frequency:

Simple harmonic oscillator

$\ddot{x} = -\omega^2x$

and      $\omega^2 = \frac{k}{m}$

If we couple two harmonic oscillators, such as in the situation below, we get two normal modes that obey the equations of motion identical to the single-oscillator case.

Coupled harmonic oscillator

The equations of motion for the normal modes are:

$\ddot{\eta_1} = -\omega^2_1\eta_1$      and

$\ddot{\eta_2} = -\omega^2_2\eta_2$,

where

$\omega_1^2 = \frac{k+2\kappa}{m}$

and   $\omega_2^2 = \frac{k}{m}$.

I should also mention that $\eta_1 = x_1 - x_2$$\eta_2 = x_1 + x_2$. The normal modes are pictured below, consisting of a symmetric and antisymmetric oscillation:

Anti-Symmetric normal mode

Symmetric normal mode

The surprising thing about the equations for the normal modes is that they look exactly like the equations for two decoupled and independent harmonic oscillators. Any motion of the oscillators can therefore be written as a linear combination of the normal modes. When looking back at such results, it seems trivial — but I’m sure to whoever first solved this problem, the result was probably unexpected and profound.

Now, let us briefly discuss the quantum case. If we have a single harmonic oscillator, we get that the Hamiltonian is:

$H = \hbar\omega (a^\dagger a +1/2)$

If we have many harmonic oscillators coupled together as pictured below, one would probably guess in light of the classical case that one could obtain the normal modes similarly.

Harmonic Chain

One would probably then naively guess that the Hamiltonian could be decoupled into many seemingly independent oscillators:

$H = \sum_k\hbar\omega_k (a^\dagger_k a _k+1/2)$

This intuition is exactly correct and this is indeed the Hamiltonian describing phonons, the normal modes of a lattice. The startling conclusion in the quantum mechanical case, though, is that the equations lend themselves to a quasiparticle description — but I wish to speak about quasiparticles another day. Many ideas in quantum mechanics, such as Anderson localization, are general wave phenomena and can be seen in classical systems as well. Studying and visualizing classical waves can therefore still yield interesting insights into quantum mechanics.