Particle exchange forces

**da_willem** · Feb27-05, 06:44 AM

In Yukawas model the force between nucleons comes from the exchange of a pion. How precisely does this lower the energy? Has it something to do with confinement energy?

I know an exchange force can be understood in terms of a momentum exchange by the force carrying particle. But there must be a reason this particle is emitted. How does this lower the energy?

**marlon** · Feb27-05, 07:09 AM

Originally Posted by da_willem

In Yukawas model the force between nucleons comes from the exchange of a pion. How precisely does this lower the energy? Has it something to do with confinement energy?

I know an exchange force can be understood in terms of a momentum exchange by the force carrying particle. But there must be a reason this particle is emitted. How does this lower the energy?

$LaTeX Code: E = \\frac{- e^{-mr}}{4 \\pi r}$

This is the expression for the energy that QFT gives us when we consider a massive free field theory (starting from the ordinary Klein Gordon equation) which we will submit to a perturbation expressed by two delta dirac functions. You can think of this as two massive lumps sitting on a matress (the field F to which those massive objects couple). This coupling means that the two massive objects will make the matress vibrate and this vibration (also called fluctuation or perturbation) exactly represents a certain particle of mass m and with integer spin (this will be the pi-meson). Them massive lumps are the nucleons.

This energy is negative, which means that the presence of the two delta dirac functions has lowered the energy of the free field theory. This really means that the two objects attract.

Also dE/dr > 0, two massive lumps sitting on the mattress can lower the energy by getting closer to each other.

The potential drops off exponentially over a distance 1/m : the range of the attractive force generated by the field F, is determined inversely by the mass m of the particle described by that field (ie the pi meson).

These are the reasons for attraction : the lowering of energy. The pi meson (also called Ykawa meson) is only emitted (NOT TO LOWER THE ENERGY) but because of the influence of the two massive objects (ie two nuclei for example) on the field F (of which the fluctuation describes a particle of mass m : the pi-meson.)

regards
marlon

**da_willem** · Feb27-05, 07:23 AM

Originally Posted by marlon

$LaTeX Code: E = \\frac{- e^{-mr}}{4 \\pi r}$

This is the expression for the energy that QFT gives us when we consider a massive free field theory (starting from the ordinary Klein Gordon equation)

I know the (stationary spherically symmetric) solution to the KG equation is:

$LaTeX Code: \\psi = C \\frac{e^{-\\frac{mc}{\\hbar} r}}{r}$

[Only considering the negative signed exponent as the physical one]
What is the interpretation of psi? I know it is not susceptible to a statistical interpretation, but why can you interpret it as a (Yukawa) potential? And the sign of C, isn't that something you put in? So why attraction?

**marlon** · Feb27-05, 07:41 AM

Hold on, you are forgetting a very important aspect here : namely the fact that you need to incorporate the nucleons. You just solved the KG equation.

I am gonna assume you know the Path integral formalism because this is the easiest way to prove the attraction mediated by a Yukawa meson.

$LaTeX Code: W(J) = \\frac{-1}{2} \\int \\int d^4x d^4y J(x) D(x-y) J(y)$

The J's express the two massive lumps i talked about. The D is the propagator of the free field (coming from the KG equation.) It's equation is
$LaTeX Code: D(x-y)= \\int \\frac{d^4x}{(2 \\pi)^4} \\frac {e^{ik(x-y)}}{k^2-m^2 + i \\epsilon}$

Now plug this into the expression for W and make a Fouriertransform. Then write each J(x) = J1 + J2 and only look at the contribution of J1 and J2. So we neglect selfcouplings. You will get :

$LaTeX Code: W(J) = \\frac{-1}{2} \\int \\frac{d^4k}{(2 \\pi)^4} J_2^*(k) \\frac {1}{k^2-m^2 + i \\epsilon} J_1(k)$

In order to compute the energy just realize that iW=iET (this comes from the path integral formalism), you will get :

$LaTeX Code: - \\int \\frac{d^3 k}{(2 \\pi)^3} \\frac{e^{ik.(x_1 -x_2)}}{k^2 +m^2}$

In the above formula, the x and k are VECTORS where as they are 4-vectors in the first formula's

Now combine this with the insights of my first post and you will have all that you need

marlon

**da_willem** · Feb27-05, 07:55 AM

Thanks for the effort, but I am not too familiar with the path integral formalism. I would like to know 2 things:

1) Why can the solution to the KG equation be interpreted as a potential?

2) In the case of the hydrogen molecular ion eg I can understand where the exchange energy comes from. The electron is no longer confined to surrounding one proton but can smear out thus lowering the total energy, constiting the binding force between the two protons. But in the case of a proton and a neutron where does the exchange energy you mention come from?

**marlon** · Feb27-05, 08:22 AM

Originally Posted by da_willem

Thanks for the effort, but I am not too familiar with the path integral formalism. I would like to know 2 things:

1) Why can the solution to the KG equation be interpreted as a potential?

Because the fluctuations of this field (generated by two nuclei for example) correspond to a particle of mass m ; ie the Yukawa meson. The interaction between the nuclei is described by the way this KG field vibrates due to the presence of the nuclei. Therefore the KG field really expresses the ongoing interaction between the two nuclei...

2) In the case of the hydrogen molecular ion eg I can understand where the exchange energy comes from. The electron is no longer confined to surrounding one proton but can smear out thus lowering the total energy, constiting the binding force between the two protons. But in the case of a proton and a neutron where does the exchange energy you mention come from?

The example i gave you contains two nuclei expressed by J1 and J2. each of these terms is written as a delta direc function. You need to look at this as two massive lumps "sitting on a KG-mattress". So the interaction really comes from the delta dirac functions expressing the two massive lumps. this has nothing to do with charge...However incorporating charge is not that difficult : just split up the J's in a positive and negative part.

I don't get the example you gave on the hydrogen molecular ion, though ??? Two protons do not interact with each other by the exchange of electrons !!! Their EM-interactions are mediated by virtual photons. You can prove this by using the eact same way that i described above. However you WILL need to know your introductory QFT/QED in order to both prove and understand what is going on.

marlon

**da_willem** · Feb27-05, 01:47 PM

Originally Posted by marlon

I don't get the example you gave on the hydrogen molecular ion, though ??? Two protons do not interact with each other by the exchange of electrons !!!

A hydrogen molecular contains aside from two protons also an electron. This electron can reside on either of the protons. But when the electron is shared between the two the configuration has a lower energy than a hydrogen molecule and a proton separately. This lowering in energy comes from the fact that the electron is no longer confined to one proton but is smeared out between the two thus lowering its kinetic energy. You can view this as the protons resonating between a proton and a hydrogen molecule.

In Yukawas theory a proton can resonate between being a proton or a neutron and a positive pion. Does the energy lowering have a similar origin to the case of the hydrogen molecular ion?

**marlon** · Feb27-05, 06:42 PM

Originally Posted by da_willem

A hydrogen molecular contains aside from two protons also an electron. This electron can reside on either of the protons. But when the electron is shared between the two the configuration has a lower energy than a hydrogen molecule and a proton separately. This lowering in energy comes from the fact that the electron is no longer confined to one proton but is smeared out between the two thus lowering its kinetic energy. You can view this as the protons resonating between a proton and a hydrogen molecule.

Ok, it is clear to me now

In Yukawas theory a proton can resonate between being a proton or a neutron and a positive pion. Does the energy lowering have a similar origin to the case of the hydrogen molecular ion?

No, the above example is a QM-thing. The Yukawa meson is a QFT thing so they are very different in nature and very different in theoretical explanaition

marlon

**da_willem** · Feb28-05, 02:35 AM

But can you give me an intuitive physical reason (pls without advanced QFT...?!) why:

1) a particle would continuously emit gauge bosons?

["because of the influence of the two massive objects (ie two nuclei for example) on the field F" doesn't say much to me]

2) why this sometimes gives an attractive and sometimes repulsive force?

[repulsive I can understand by momentumexchange, attractive I can understand because by the uncertainty principle you cant really tell from wich direction a gauge boson comes. But why in certain cases atrractive and in certain cases repulsive. If this lowers the energy, why?]

**arivero** · Feb28-05, 03:02 PM

Originally Posted by da_willem

But can you give me an intuitive physical reason (pls without advanced QFT...?!) why:
2) why this sometimes gives an attractive and sometimes repulsive force?

Spin of the mediator has a role. Even spin happens to be attractive between identical particles, odd spin happens to be repulsive between identical particles. The proof is done -SHOULD BE DONE, teachers usually ignore the whole point- by comparing Born approximation (convolution of a potential) with tree level feymann diagrams.

Of course identical particles have equal charge.

Electron-Positron, on the other hand, hmm, let me to think, could be seen as two electrons, one coming from the future, other from the past, and then its repulsive force should be seen as attractive. Uff, I can not picture this. Surely it is easier to get a minus sign multiplying the potential got in the previous step.

**arivero** · Feb28-05, 03:07 PM

Between electron and quark, I have never seen a calculation of the potential. But I believe I have seen it done between electron and pion, considered both as elementary particles.

**arivero** · Feb28-05, 03:15 PM

Originally Posted by da_willem

But can you give me an intuitive physical reason (pls without advanced QFT...?!) why:

1) a particle would continuously emit gauge bosons?

Because it can.

**marlon** · Feb28-05, 04:37 PM

Originally Posted by da_willem

But can you give me an intuitive physic al reason (pls without advanced QFT...?!) why:

1) a particle would continuously emit gauge bosons?

Particles emit gauge bosons because they interact with other particles. The energy of this interaction is "used" to generate such gauge-bosons. Don't take this too litterally, it is just a way of speaking. The gauge bosons really express the fact that there is an interaction going on. That is all there is to it.

2) why this sometimes gives an attractive and sometimes repulsive force?

[repulsive I can understand by momentumexchange, attractive I can understand because by the uncertainty principle you cant really tell from wich direction a gauge boson comes. But why in certain caskes atrractive and in certain cases repulsive. If this lowers the energy, why?]

Well why do two positive charges repel ?

A more thorough study can be made BUT NOT without QFT, so this is as intuitive as you can get

regards
marlon

**marlon** · Feb28-05, 06:21 PM

http://www.physicsforums.com/journal...90&action=view

Here is another vision on attracion and repulsion. I wrote it my journal

regards
marlon

**da_willem** · Mar1-05, 04:37 AM

Spin of the mediator has a role. Even spin happens to be attractive between identical particles, odd spin happens to be repulsive between identical particles. The proof is done -SHOULD BE DONE

Any references on that? Or is the proof very difficult?

Originally Posted by arivero

Between electron and quark, I have never seen a calculation of the potential. But I believe I have seen it done between electron and pion, considered both as elementary particles.

How is a pion an elementary particle if it is composed of a quark and an antiquark?

Originally Posted by Marlon

Now if you realize that at each point where the foton and the charged particle "meet" you need energy conservation, you can easily see that this picture works.

Does this mean that the charged particle emits as many virtual photons as it absorbes?

**arivero** · Mar1-05, 06:59 AM

Originally Posted by da_willem

Any references on that? Or is the proof very difficult?

It is. It appears in old handbooks on advanced quantum mechanics, under "Born approximation". I do not know if there is another way.

How is a pion an elementary particle if it is composed of a quark and an antiquark?

I said "considered". Ie, the mathematics in the argument is the one corresponding to a charged point particle different of the electron (but perhaps excessively different. Beware spins)

Does this mean that the charged particle emits as many virtual photons as it absorbes?

Hmm, it does not mean, nor the contrary. It is an interesting thing to think about. For instance, a fermion loop (vacuum polarisation). can adjust the number of virtual particles. It is a pity that research in this kind of semiclassical approaches is currently deprecated.