# How fundamental is inverse square law?

#### guhan

A convincing way to reason inverse square law (of classical EM and gravity) is a conservation of the field lines as one proceeds away from a source...
http://hyperphysics.phy-astr.gsu.edu/HBASE/forces/isq.html

It appears as a reasonable and intuitive thing to expect... but, it does NOT seem to hold for weak forces. So, how to convince ourselves of this violation in weak interactions, in a classical sense? (I know it comes about 'naturally' in QFT, but I want a classical/intuitive reason).

As for the strong nuclear force, can we say the reason it does NOT obey inverse square law has got to do with the mass of the gluons and the strong gluon-gluon interaction? As in, can these negate any expectation of a conservation of (classical) 'field lines'?

PS: If I have not made myself clear, let me know... I will rephrase.

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#### confinement

It appears as a reasonable and intuitive thing to expect... but, it does NOT seem to hold for weak forces. So, how to convince ourselves of this violation in weak interactions, in a classical sense?
In classical language the weak-field is massive, just like the atmosphere, so radiation travels through this field with a longitudinal component (like sound) as well as with a transverse component (E&M waves). The speed of transmission is lower, and the field is "damped" in a sense.

As for the strong nuclear force, can we say the reason it does NOT obey inverse square law has got to do with the mass of the gluons and the strong gluon-gluon interaction? As in, can these negate any expectation of a conservation of (classical) 'field lines'?
Gluons are massless, and so the speed of glue corresponds to the speed of light. The behavior of the strong force is very complicated --- imagine that instead of one kind of charge that is positive or negative you have three kinds of charge (RGB), all of which can be positive or negative, and you have 8 kinds of photons (the gluons) all of which carry charge as well. As icing on the dessert of difficulty I will also point out that the color field cannot be represented by vectors, it must be represented by matrices and higher rank tensors.

#### guhan

Oh yes, I interchanged the mass factor!
So, can we say that the mass of the weak fields (mass of W and Z bosons) and the self-interaction of strong fields (gluon-gluon interaction) give rise to the 'dampening' effect (classically speaking) and hence a violation of the conservation of the field lines (again, classically speaking)?

#### confinement

Oh yes, I interchanged the mass factor!
So, can we say that the mass of the weak fields (mass of W and Z bosons) and the self-interaction of strong fields (gluon-gluon interaction) give rise to the 'dampening' effect (classically speaking) and hence a violation of the conservation of the field lines (again, classically speaking)?
First of all, we can have a perfectly well-defined classical theory of the strong and weak forces, it just doesn't correspond to reality at all --- these forces are only important on quantum scales.

In this classical theory of strong and weak forces there is no analogue of field lines for E&M. The reason we have field lines in E&M is that the electric field is a vector field, it assigns a 3D-vector (an arrow) to each point in space. The weak force assigns three 2x2 matrices to each point, and the strong force assigns three 3x3 matrices to each point in space. If you have a collection of arrows it is easy to draw a field line by connecting the arrows head to tail, but with a collection of matrices does not come with any obvious way for drawing field lines.

In other words, instead of arrow tips at each point in space you have something more like a multi-colored rotating lernaean hydra! Furthermore these hydra interact violently with their neighbors and exchange heads of different colors, etc.

If you want to learn more on a technical level, these kinds of fields are called 'Classical Yang-Mills fields', although the level of math is difficult. The amount of symmetry and structure in this theories opens the door for endless visualizations, all of which are incomplete because 8 gluons and three colors is just too much to imagine all at once (I hope I am wrong).

#### guhan

Thanks! :-)
I am reading QFT now and have come across some of them and should encounter the rest also soon.

#### Bob S

A convincing way to reason inverse square law (of classical EM and gravity) is a conservation of the field lines as one proceeds away from a source...It appears as a reasonable and intuitive thing to expect.
PS: If I have not made myself clear, let me know... I will rephrase.
The classical inverse square law does not hold at short distances when Coulomb fields are very strong. You begin to see un-renormalized fields (vacuum polarization), so the field drops off at greater than inverse square.

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