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In my physics textbook, while talking about the scope of physics, it mentions the four fundamental forces of nature which are Gravitational force, Electromagnetic force, Strong Nuclear force and Weak Nuclear force, of these the most astonishing one I feel is the weak nuclear force. My book states that it is responsible for $\beta$-decay and other kinds of radioactivity. Now as far as I know:

$\mathbf {Force}$ is an interaction between an object and it's surrounding which causes the object to change it's momentum while the interaction is happening.

Now for gravitational and electromagnetic force I know about their classical model and, as far as I know, there they just are cause change in momentum of the object though one acts through charge and other acts through mass. But for weak nuclear force there is something more to it, it causes the particles on which it acts to change it's properties (like proton turning into neutron and vice versa) which is more than just a change in momentum.

So my questions is:

  • How do physicists account for these properties of weak interaction mathematically?

Thanks in advance.

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  • $\begingroup$ Related: physics.stackexchange.com/q/4428/2451 $\endgroup$ – Qmechanic Oct 30 '19 at 15:21
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    $\begingroup$ See physics.stackexchange.com/q/405411/44126 and links therein for an example of a scattering interaction which is modified by the weak force. In such experiments it's more convenient to discuss "cross sections" for an interaction than bare forces. $\endgroup$ – rob Oct 31 '19 at 2:35
  • $\begingroup$ Feedback to the post (v3): Asking How do physicists account for fundamental forces mathematically? is still too broad. Try to focus on your main subquestion. If the answers do not solve your doubts, submit a new question. Look for duplicates. $\endgroup$ – Qmechanic Oct 31 '19 at 9:40
  • $\begingroup$ @Qmechanic I have now specifically asked for weak interaction can you consider reopening the question. $\endgroup$ – user238497 Oct 31 '19 at 10:17
  • $\begingroup$ Why don't you read up on a book or Wikipedia? Looks like you are inviting a tutorial session. $\endgroup$ – Cosmas Zachos Oct 31 '19 at 18:33
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According to quantum field theory all the interactions are described in a very similar way.

  • Gravitational interaction is the exchange of gravitons between massive particles.
  • Electromagnetic interaction is the exchange of photons between charged particles.
  • Strong interaction is the exchange of gluons between quarks (the constituents of protons and neutrons).
  • Weak interaction is the exchange of $W$ and $Z$ bosons between leptons (i.e. electrons and neutrinos) and quarks.

The weak interaction is different from the other 3 interactions because of the special properties of the $W$ and $Z$ bosons:

  • Unlike the other massless bosons (gravitons, photons, gluons) the $W$ and $Z$ bosons have quite a big mass. Therefore they can exist only for a very short time ($\approx 10^{-25}$ s), and hence the weak interaction is very short-ranged ($\approx 10^{-17}$ m). This also means that the weak interaction has a quantum model only, but there is no classical model for it.
  • Unlike the other electrically neutral bosons (gravitons, photons, gluons) the $W^+$ and $W^-$ boson have an electric charge. Therefore a particle emitting or absorbing a $W^+$ and $W^-$ boson will also change its charge, i.e. it turns into a different kind of particle (e.g. a proton turns into a neutron).
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Actually one should single out electromagnetic force as special, since its gauge group $U(1)_{EM}$ is Abelian.

All the other 3 forces are non-Abelian, thus capable of changing an interacting particle's properties (with the understanding that the particle transforms non-trivially with regard to the non-Abelian gauge symmetry). For example, strong force could mutate the color of a quark, and gravity could flip the spin of a fermion.

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All the four forces do other curious things besides transfer momentum. Gravity will change the shape of a large object, pulling it into a ball, or when strong enough will even collapse spacetime inside an event horizon. Electromagnetism is the force which makes glue work, and glue is not renowned for its momentum change. Unlike the others the strong nuclear force actually increases with distance and if you try to pull two quarks apart it will suddenly snap apart and create new quarks.

In the case of the weak force, as another answer noted, it carries electric charge and so tends to change the charge as well as the momentum of particles.

The electromagnetic and weak nuclear forces are in fact described by a unified electroweak theory. We even strongly suspect that in the very early Universe they were one and the same force.

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