# Are the physical laws scale-dependent?

If you read the article "More Is Different", by P.W. Anderson (Science, 4 August 1972), you will find a deep question: are the physical laws dependent of the size of the system under study?

As an example, we can ask ourselves, are the description of a hundred of atoms more than simply one hundred times the description of one atom alone? Of course we have interactions, but, are these interactions dependent of the number of particles implied?

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It is important to digest appropriately Anderson's comments about scaling. In Physics, when one talks of "scaling phenomena", what's really being talked about are these two things:

And, as i mentioned above, conformal symmetry plays a leading role in all of this discussion.

Roughly, the bottom line is something like this: every physical theory has its domain of validity, ie, its "laws" are only valid within certain "conditions", which we usually express in terms of an Energy Scale.

So, e.g., GR is valid in certain regimes that we call "relativistic": if you're too slow compared to the speed-of-light, Newtonian gravity is a very good approximation. In this sense, Newtonian gravity is an "effective description" of GR (in the appropriate energy scale).

The same is true for Quantum Field Theories: you can start with a given description at a given energy (or length) scale and, as you change your scale, either increasing or decreasing the energy of the phenomena involved, you'll be lead to different theories, in order to describe the new, effective, phenomena that you'll see. For instance, you can describe the world in terms of protons and neutrons or in terms of quarks and gluons — the only change is in the energy scale used and, as such, in the "effective theory" you'll be using to describe the ingredients your experiments measure.

These are the concepts really behind Anderson's argument. In fact, when he says "more is different", he's alluding to a concept called emergent phenomena, which is basically described by the notion of Effective Field Theory i mentioned above. Here's a picture: you can describe a proton in terms of quarks and gluons, but it's very hard to describe a whole nucleus in terms of quarks and gluons — essentially because there are so many of them, that calculations become virtually impossible. So, what people do is to compute the Effective Field Theory of quarks and gluons, and use it instead to describe the whole nucleus.

A similar thing can be seen in Statistical Mechanics, when one observes that the behavior of a collection of particles is very different from that of a single particle — this is the prototypical "more is different": the physical properties of the collection of particles are not mirrored by the individual properties of each particle — this is "emergence", and that's why we use Effective Field Theories to describe the collection of particles.

What's really astounding is that, using Renormalization Group techniques, we can actually compute Effective Field Theories for several different energy scales! 8-)

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It's very hard to find non-trivial solutions to the Renormalization Group eqs; in the end of the day, they are the ones that end up determining how a certain phenomenon in a given scale will behave in another. But, i'm not sure i'd go as far as saying "never"… but it is pretty hard. –  Daniel Nov 13 '10 at 16:37
nice answer. Just a brief comment: in condensed matter, finding RG fixed points (using the epsilon expansion for instance) is a pretty well-developed technique for predicting new phases of matter. Perhaps that's not what you mean though. –  j.c. Nov 13 '10 at 16:54
@j.c.: you're correct, on both counts: i did have something else in mind; and the $\epsilon$-expansion is very good thing to keep in mind. ;-) –  Daniel Nov 13 '10 at 17:02

The question depends on what one's definition of "physical law" is.

Part of the point of Anderson's article is to argue that strict reductionism is not what scientists do in practice.

More explicitly, one caricature of pure reductionism is that only an explanation starting from the very bottom, i.e. involving strings and quarks, etc. counts as physical law, whereas Anderson would argue more that any reasonably quantitative workable theory of phenomena based at any scale is a physical law. I really doubt that any working physicists really hold the former position anymore, in part due to one of the great advances of post 1950's theoretical physics, the effective field theory and renormalization group philosophy that Daniel so nicely explained in his answer. We don't have to go that deep to see some illustrations of this though.

### examples

Let me give some easy examples and raise some discussion questions to help you decide what qualifies as a physical law.

Probably the first emergent insight that one learns in physics is the notion of "center of mass", that is, the idea that in many cases you can treat an object made out of $10^{27}$ atoms as a single point particle! Though it's not pointed out when you're a freshman, what's going on here is exactly the same as happens in all other applications of emergence - you ignore all the internal degrees of freedom of your sliding block, for instance, because they are much higher energy than the stuff that you want to talk about. Does the fact that we ignore all that information make the explanation of the acceleration of a block any less of a "physical law"?

Already we see a generic fact about emergence -- at some point the explanation breaks down. For instance, there's no way from treating a block as a point particle to be able to understand the sound it makes when it hits the ground, though we can get a very good description of its motion (in a vacuum, say)!

Two discoveries at the start of the 20th century showed that the entire edifice of Newton's laws and Galilean relativity is an emergent phenomenon. Note that this understanding doesn't invalidate most applications of Newton's laws -- that the limit $c\rightarrow\infty$ and $\hbar\rightarrow0$ simplifies to something that high school students can calculate with is really amazing, isn't it? But, does the fact that they are "just" some limiting case mean that Newton's laws and Galilean relativity are not "physical laws"?

We can also turn these questions around and imagine ourselves in a universe where QM and SR were discovered first. In that case, if people then discovered the classical limits, would people think of them as less "physical"?

### conclusion

One could and should spend some time thinking about what our physical principles are and what "emergent" assumptions and approximations go into them; breaking down the assumption that mechanics ended with Newton was the work of the 1st half of the 20th century. On the other hand, Anderson points out that it's also useful to go in the opposite direction, in some ways, this is where the big problems in physics that I'm interested in are; to start with many-body descriptions which are impossible to calculate with and not very illuminating, and pick out the simplifications and insights that emerge! Whether you call it a physical law or not is sort of besides the point, right?

These are some of the things that come up when I think a bit about your question.

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As Sklivivvz already said, most Quantum Mechanical effects only show effectively on a "quantum scale", which is dependent on the variables involved. Although this scale is again dependent on the system itself, and so makes the reasoning moot.

If you mean scale dependent as in "the speed of light may well be different on a very small scale", or "the planck constant may be different on very large scale", I cannot answer as I do not know.

What I know is this: most physical laws as shown in textbooks are approximations in well-defined areas (e.g. the law of Newton for (not too) small r, the particle physics DGLAP evolution equations for their domain of relevance in kinetic space, etc...) To this effect, these laws in physics are scale dependent. The much more general derivations are (as far as I understand) not scale dependent.

As an example to "prove" the above statements:

Quantum effects in macroscopic systems: Fe as a conductor can be described as a Fermi sea of electrons, which is a very quantized state and obeys quantum mechanical laws on a macroscopic level. Properties like specific heat and conduction are direct consequences of the quantum mechanical description of this system, and to that effect, the macroscopic result is somewhat "the sum of quantum states". This is also exactly what statistical physics is based on: sums of states, and this is exactly what connects the "quantum world" with what we can observe.

If I misunderstood your question or make claims you think are false, please say so.

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It's a very interesting question.

Certainly, QM is scale dependent, as the planck-level world described is completely separated from the human scale world and can only be accessed through a disruptive measurement.

In other words, there must be a crossover point at which the quantum description, as described by the wave function $\psi$ collapses in a measured value.

General Relativity is not scale dependent. However, gravity is very weak and it's very hard to perform experiments about GR at a microscopic level, for example.

Other theories also tend to describe the very small, most of them, like string theory, do tend to reduce to QM in some appropriate limit and hence inherit its scale dependency.

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Sklivvz: the statement of "scale independence" in Physics is usually associated to Conformal symmetry, ie, whether or not the theory in have in hand has conformal symmetry. In this sense, GR is scale dependent, for not all solutions of Einstein's Field Equations are conformally symmetric. –  Daniel Nov 13 '10 at 13:54
QM also works in human-scale -- yet then its predictions are in agreement with Newton's dynamics. –  mbq Nov 13 '10 at 14:03
The effects of QM work at any scale, but $\psi$ is inherently hidden. I assume (and you may disagree) that the wavefunction collapse is related to scale. –  Sklivvz Nov 13 '10 at 14:05
@Sklivvz: this is far from being a matter of opinion: there are macroscopic quantum effects: Bose-Einsein condensates being a good example. –  Daniel Nov 13 '10 at 14:57
There is a point of connection between QM and CM, the Ehrenfest theorem. When the size of the wave packet is very tiny, you can use the classical Hamiltonian approach. My question is more in the sense of emergence, as Daniel pointed out. –  asanlua Nov 13 '10 at 15:18