Textbook proof of repulsion/attraction of equal charges according to odd/even spin

[arivero]

It is common lore that bosonic fields of odd spin, such as electromagnetism, cause equal charges to repel, while bosonic fields of even spin, as pions or gravitons, cause equal charges to attract.

Has anyone seen this argument in a textbook? And its proof? Or is it just internet lore, or classroom gossip?

I asked the question in one thread now closed,
https://www.physicsforums.com/threa...-still-work-for-non-abelian-theories.1017503/
but I forgot to ask for books or papers. That is the question now.

Note also https://www.physicsforums.com/threads/spin-parity-and-attractive-repulsive-forces.785087/ that also failed to find detailed references

 

[renormalize]

It is common lore that bosonic fields of odd spin, such as electromagnetism, cause equal charges to repel, while bosonic fields of even spin, as pions or gravitons, cause equal charges to attract.

Has anyone seen this argument in a textbook? And its proof? Or is it just internet lore, or classroom gossip?

Here's the link to a 1986 Physical Review paper discussing this: Attraction/repulsion between like charges and the spin of the classical mediating field.
Abstract
The question as to whether an even or odd spin of the mediating field results in an attractive or repulsive force between like sources is examined within the framework of classical theory. It is shown that the assertion holds in a Lorentz-invariant theory constructed in analogy with electrodynamics provided that the current does not depend on any intrinsic direction of the source. Counterexamples are presented to show that the connection does not exist when the current does depend on the intrinsic direction.

(I admit that I haven't actually read the article since I'm too lazy to drive to the college library to look this one up!)

 

[Vanadium 50]

As you know, at leading order the scattering is the same whether the potential is attractive or repulsive. So you need to go at least to NLO. That means regularization and renormalization, and it has to be done consistently across systems. I think you are more likely to find this as a text problem than a worked example.

It is also probably easier to do this for specific spins than odd and even, because then you can use properties of specific Ylm's.

 

[arivero]

(I admit that I haven't actually read the article since I'm too lazy to drive to the college library to look this one up!)

I will check tomorrow too. The spires citations are only about gravitation, but following this thread I found another reference to the lore, Nieto-Goldman, that mentions some history:

From the particle-physics point of view, general relativity is a theory of gravity where the force is mediated by a tensor (spin-two) particle with the charge being mass—energy [Fey63]. Therefore, the force is always attractive. On the other hand, classical and quantum electromagnetism both have two charges, positive and negative. The forces are mediated by a vector (spin-one) field which produces an attractive force between opposite charges and a repulsive force between like charges. Physicists were thus led to wonder if a more general statement could be made. This was especially true for people inthe late 1930’s and early 1940’s who were studying the nature of nuclear forces [Kem38;Mol40].

Many physicists who worked in that era [Bet82] have told us that it gradually came to be understoodthat charge-forces mediated by even-integer spin bosons are always attractive (scalar, tensor, etc.)whereas forces mediated by odd-integer spin bosons can be both attractive or repulsive, dependingupon whether the charges are opposite or alike. However, none were able to give us a “prime source.” The earliest publication we know of which made this statement was authored by Peaslee [Pea56],although more recent authors have also made this point [Fey63; Des67; Jag86].

[Bet82] is a list of private communications discussing the topic of the article: H.A. Bethe, F. Dyson, R.P. Feynman, D.C. Peaslee, F. Rohrlich and J.A. Wheeler are thanked here. [Jag86] is the article of Jagannathan and Singh, and the other ones are gravitational-centric:

• [Pea56]D.C. Peaslee, Nonexistence of gravity shields, Science 124 (1956) 1292.

• [Fey63]R.P. Feynman, Lectures on Gravitation 1962- 63, Lecture notes by F.B.Morinigo and W.G.Wagner (Caltech, Pasadena) Lecture 3.
• [Des67S]. Deser and F.A.E. Pirani, The sign of the gravitational force, Ann. Phys. (NY) 43 (1967) 436—451.

Note that the lecture notes from Dyson where I was expecting to find the argument -based on Born approximation- are contemporary to Peaslee; in fact a bit earlier, 1951, https://arxiv.org/abs/quant-ph/0608140 (My own copy is from Les Houches library, it could differ from the arxived version)

 

[renormalize]

(I admit that I haven't actually read the article since I'm too lazy to drive to the college library to look this one up!)

I still haven't read the original article by Jagannathan and Singh, but for completeness I want to display their result, as quoted in this more recent paper https://arxiv.org/pdf/1310.3539:

(Here $ee^{\prime}>0$ for like charges and $ee^{\prime}<0$ for opposite charges.)