If magnetic monopoles existed it would not be the case - the forces on an electron and a magnetic monopole passing by each other would be at right angles to the vector connecting the two particles!

But is there some reason that real forces should always be in the direction of the particles they connect (that is, attractive or repulsive)?

Edited to add: I guess I should limit the discussion to forces mediated by virtual particles, otherwise an oscillating (or just accelerating) charged particle can produce a wave which acts on another charged particle in a direction orthogonal to the wave's propagation.

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    $\begingroup$ Force is a classical concept which is meaningless with relation to particle interactions. One can't attach a force gauge to an electron, neither can one measure an electron's acceleration close to another strongly interacting particle. The action of a classical magnetic monopole on a charge particle can be analyzed, but such a thing does not exist in the real world, except in a fairly poor approximation as the end of a long magnetized rod. $\endgroup$
    – CuriousOne
    Dec 20 '14 at 1:32
  • $\begingroup$ @CuriousOne, you can always obtain classical behaviour in the limit. Consider the question to be about "an electrically charged large object" passing by "a magnetically charged large object". $\endgroup$
    – Hugh Allen
    Dec 20 '14 at 1:37
  • $\begingroup$ So you're asking for examples of when Newtons third law doesn't apply? :) $\endgroup$
    – jabirali
    Dec 20 '14 at 1:56
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    $\begingroup$ You mean a potential like discussed here? arxiv.org/abs/physics/0701232 $\endgroup$
    – CuriousOne
    Dec 20 '14 at 2:23
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    $\begingroup$ Theory can not tell you if something is possible or not. At most it can tell you if something fits into an existing theoretical framework. Do magnetic monopoles fit into a classical framework? Sure. You can plug ANY potential into classical physics and turn the cranks. Do elementary magnetic monopoles fit into the standard model of particle physics? Absolutely, or we wouldn't be searching for them. $\endgroup$
    – CuriousOne
    Dec 20 '14 at 2:56

The answer is no, we cannot rule out forces not directed along the line joining particles with current theory. As you point out, magnetic monopoles are a counterexample, and, as CuriousOne points out, magnetic monopoles both fit into a classical framework, are consistent with the Standard Model and are actively, experimentally sought.

Some historical comments on this question are in order though, I believe.

One reason that you might classically give (i.e. in Newtonian mechanics) for a "yes" answer to your question is that an assumption of acting of force along the line joining the two particles is needed, otherwise the system's angular momentum is not conserved. Consider two particles, equidistant from the origin and with equal and opposite forces on each other at right angles to the line joining them. The orbital angular momentum about the whole origin steadily and linearly increases with time! So in Newtonian, action-at-a-distance thinking, this is a problem. Hence, I believe Newton himself postulated that the force between particles is always along the line joining them. Notice that, contrary to some of the comments, this is not a breach of Newton's third law. A postulate of force along the line joining particles is something further to and independent from Newton's third law.

Another way of thinking about this from a Noether's Theorem standpoint is that our proposed, spontaneous angular momentum-generating system above has physics that are not invariant with respect to a global rotation of our co-ordinate system. You need to choose a preferred direction to define the force direction in our example. So our system's description is not rotationally invariant, hence the breakdown of the law of conservation of angular momentum.

A flaw in all of the above (assumption of force along the line joining particles) as we now know is that the World is not just particles and "empty space" (in the sense of a void). There are fields too, and these are strongly believed to have an objective reality in modern physics (indeed I would say that "empty space" is made of ground state quantum fields). Physics is local. So in tallying up angular momentum, we need to take account of the momentum and angular momentum of fields. Thus, in the classical sense Newton's third law does not hold, but its abstraction and generalisation to the law of conservation of momentum most certainly does when we tally up the field contribution. Likewise for angular momentum.


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