# What is the definition of “force” in quantum field theory?

In quantum field theory, there are certain interactions that we seem to associate with the action of "forces." For example, the exchange of a gauge boson between two matter particles is associated with the electromagnetic/strong/weak force, and the exchange of a pion between two nucleons is associated with the residual strong nuclear force. There are also certain interactions that we do not associate with "forces." For example, electron-positron annihilation does not seem to be associated with the action of a force, and neither does matter-antimatter oscillation. Is there any systematic way to tell whether a given interaction is associated with the action of a "force"?

• You've defined it as an exchange of a boson. Is there a reason you're not satisfied with your own definition? – Ben Crowell Jan 20 at 15:14
• @BenCrowell For a few reasons: 1) Defining it as such would mean that Compton scattering would not be associated with the action of the electromagnetic force, since it doesn't involve the exchange of a photon; 2) Defining it as such would include the exchange of a Higgs boson as associated with the action of a force, which is an idea I'm not sure has universal support; 3) Not knowing much about effective field theories, I didn't think it out of the question that such a definition was violated in some obscure effective field theory with a fermion-exchange interaction associated with a force. – probably_someone Jan 20 at 15:22
• @DanYand Makes sense, but "forces" are still talked about in QFT, despite these points. So, suppose we drop the assumption that the "force" we talk about in QFT has any connection to the Newtonian definition. Is there anything close to a definition of "force" in QFT, or is the association of certain interactions with "forces" done completely arbitrarily? – probably_someone Jan 20 at 16:54
• A force at the feynman diagram level is given by the basic defintion of F=dp/dt. For fundamental forces see my answer here physics.stackexchange.com/q/455480 – anna v Jan 20 at 18:20
• @annav So matter-antimatter oscillation, as seen in $B$ mesons, is a force that changes the momentum of something? Higgs decay is a force? – probably_someone Jan 20 at 18:22

This started as a comment, but then it grew...

The word "force" is used loosely in QFT. The traditional "$$F=ma$$" meaning is less useful in quantum theory, because particles typically don't have well-defined locations, so the "$$a$$" is not well-defined. Even "particles" tend to be ill-defined in QFT, as during scattering.

But if we had to choose one definition that best fits the way most physicists most often use the word "force" in QFT, it might be this one:

• Provisional definition: A "force" is any interaction between matter fields (fermion fields) that is mediated by a gauge field.

The rest of this answer is a collection of thoughts about why this definition isn't perfect.

First, consider the Schwinger model, which is massless QED in $$1+1$$ dimensional spacetime. The model is constructed in terms of fermion field ("electrons") and an EM gauge field, just like QED in $$1+3$$ dimensions but without the mass term for the fermions. However, the $$1+1$$ dimensional model is exactly solvable, and it turns out to be equivalent to a free, massive, non-interacting scalar field. This model is reviewed in reference [1], which says this in the conclusion of chapter 2:

... QED$$_{1+1}$$ already teaches us a lesson. As a matter of fact, in the Schwinger model, the perturbative intuition is misleading. Namely, it is one of the few models (in the absence of supersymmetry) whose non-perturbative solution is known. As announced in the preamble, the dynamics of massless QED$$_{1+1}$$ is that of a free massive boson...

Should we say that this result is due to the EM "force"? That might make sense when looking at the QED-like construction of the model, but then the meaning of "force" would depend on how the model is constructed rather than only on the model's physical content.

More generally, different-looking QFTs (with different lagrangians involving different sets of fields) can all make the same predictions at sufficiently low resolution. So a definition of "force" that refers specifically to the field-content and structure of the lagrangian might be too superficial. For example, consider the low-energy effective description of the nuclear force that was mentioned in the OP, where the nucleon-nucleon interaction (at least the longest-range part) is described as being mediated by the pion field. The pion field is not a gauge field (it's closer to being a Goldstone boson), so according to the provisional definition of "force" highlighted above, this would not be a "force." Hmmm, that's awkward. Maybe we could argue that the pion-mediated interaction still counts as a "force" because it's ultimately due to the gluon-mediated interaction in QCD, but that's still awkward because it means that in order to use the definition, we have to know the underlying theory (QCD). The whole Standard Model is only a low-resolution effective QFT, and we don't yet know the underlying theory, so a strict definition of "force" that requires such knowledge is not a very useful definition.

To drive this last point home, note that gauge fields aren't always "fundamental." The utility of gauge fields in physics is often (maybe always) associated with the need to implement constraints — specifically constraints that the initial conditions must satisfy. Examples occur in condensed matter physics, where various constraints (such as constraints on the number of particles that can occupy a given site in a lattice) can often be conveniently enforced by introducing a new gauge field, as mentioned in [2] and also in this post:

Can there be an emergent non-compact gauge field?

Such gauge fields are often called "emergent." So the "forces" in the Standard Model may be quite different than the "forces" in the underlying theory.

An even more extreme example is the AdS/CFT correspondence (assuming it's correct). On the CFT side, it's a QFT without gravity. One the AdS side, it's a theory with gravity living in a higher-dimensional emergent spacetime. Granted, the CFT does involve gauge fields, but they are vector-like gauge fields, not metric-like gauge fields. So the provisional definition of "force" highlighted above fails to adequately represent the all-important gravitational force that lives in the higher-dimensional emergent spacetime. On the other hand, we could argue (as in classical general relativity) that "force" is not the right word for gravity (because gravity is "just geometry"); but that's a whole different semantic issue, and I think it's beside the point here anyway, because we can include gravity (perturbatively) as an explicit gauge field in the Standard Model, and then it does satisfy the provisional definition highlighted above. That isn't how gravity works in AdS/CFT, but it is how gravity can be perturbatively shoe-horned into the Standard Model. So according to the provisional definition of "force" shown above, gravity either is or is not a force depending on exactly how it arises from the model. Seems like an unnatural distinction.

The bottom line seems to be that QFT is too rich for conventional language. We may choose whatever definition of "force" we like, but it probably won't work the way we want it to under all circumstances. That's probably why the word is used so loosely in QFT.

Is the Standard Model's weak interaction a "force"? It is a force according to the provisional definition highlighted above, but it's not a force according to any "$$F=ma$$"-like definition. I still think this is an interesting question, though, despite the ambiguity — actually I think it's interesting because of the ambiguity. This one simple question, ill-posed though it may be, has led me into a lot of interesting physics that I'm still struggling to master.

References:

[1] Michaël Fanuel, "Non-perturbative Quantum Electrodynamics in low dimensions," https://cp3.irmp.ucl.ac.be/upload/theses/phd/These%20Michael%20Fanuel.pdf

[2] Witten (2017) "Symmetry and Emergence," http://arxiv.org/abs/1710.01791