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?
Answer
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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