Diagram · General

Virtual attraction

[A creative writing exercise]

The other day I was glad to see that I was far from alone in my puzzlement about how the so-called exchange of bosons (as “force carriers”) produces both repulsion and attraction. Modeled in classical mechanics as particles; in Quantum Field Theory (QFT) as gauge boson fields — force fields. In the case of electromagnetic (EM) fields, these are photons.

In physics a force field is a vector field that describes a non-contact force acting on a particle at various positions in space.

Other ways of asking this question in online forums are:

  • QED and electrostatic attraction and repulsion.
  • How does the exchange of virtual photons cause attraction?
  • How are virtual photons responsible for not just repulsive EM forces but attractive ones as well?
  • Can someone explain how [QFT] attraction works?
  • What corresponds to a static field in QFT?
  • What is the QFT picture of forces?
  • Deriving the Coulomb force equation from the idea of photon exchange?

The set-up for this question comes from statements like in this Wiki article (below) which cites the book The New Physics edited by Paul Davies (Cambridge University Press, Aug 28, 1992). A link to books.google.com displays Section 15.5 “Virtual particles,” which states that “The force between two charged particles results from exchange of virtual photons which carry momentum but not energy [zero mass]. … so that two charged particles can exchange virtual photons of arbitrarily small momentum and energy imbalance giving rise to the Coulomb force which has an infinite range” [an example of a Feynman diagram is included as well].

Photons mediate electromagnetic interactions between particles in quantum electrodynamics. An isolated electron at a constant velocity cannot emit or absorb a real photon; doing so would violate conservation of energy and momentum. Instead, virtual photons can transfer momentum between two charged particles. This exchange of virtual photons, for example, generates the Coulomb force.

The commentary on this topic in online forums is a wild ride from the early 2000’s to 2015 [1]. The earliest threads are sometimes wacky. Particle physics models. Later ones reference QFT more (this is a good trend, according to physicists like Sean Carroll). A much cited Feynman diagram on photonic transfer is sometimes analogized as like two people throwing basketballs back & forth — such exchange explains general repulsion between like charges (electrons). Hmm.

But how does an exchange of such balls explain attraction between unlike charges — an electron and proton? There’s quite a range of commentary on that point. Some commentators use analogies of basketballs attached to the “thrower” with an elastic string so the ball goes forward but with a negative momentum transfer. Others talk about negative energy or negative mass or even going backwards in time. Anti-particle pairs. Some delve into the math (e.g., perturbation theory) and how probabilistic vectors result in appropriate momentum transfer.

As noted above, virtual particles (virtual photons) are often cited as key to any “mechanism.” Lots of them. That introduces the notion of transient fluctuations and internal lines (intermediate terms) in Feynman diagrams. Do virtual “particles” play by the same rules as real particles?

In the quantum field theory view, actual particles are viewed as being detectable excitations of underlying quantum fields. Virtual particles are also viewed as excitations of the underlying fields, but appear only as forces, not as detectable particles.

So, perhaps that particular Feynman diagram isn’t so helpful. Talking about throwing balls back & forth might be too much of a reach.

Of course, I’ve thought about this situation before. Basically in either case — attraction or repulsion, there needs to be an appropriate field potential gradient. And any model (with or without any associated massless particles) needs to emerge classically as Coulomb’s law for charged particles and a proper extended electric (vector) field potential varying inversely with distance.

Let’s simplify.


  • Consider an isolated system (idealized as empty space) containing two free charged “particles” moving toward each other at non-relativistic velocities.
  • Two cases: (A) the particles are both electrons; (B) one particle is an electron and the other a proton.
  • The QFT model emerges at the classical level with Coulomb properties …
  • Such that field potential gradients are continuous and “smooth” and motion is from higher to lower potentials. (Displacement from lower to higher potential states is possible if the particles are constrained — work is performed on the system.)
  • Paths are defined by the action principle.

For each type of boson there’s a field. Within that field, bosons can “pile up” on each other. So, we might imagine a probabilistic density distribution of (virtual) photons such that the energy (momentum) gradient obeys (emerges as) the inverse square law. But that’s a lot of virtual photons, eh.

But let’s stay with a QFT model. A lot of field quanta. No big deal, eh. For two electrons the EM potential gradients result in a net repulsive force. The field potential (“resistance”) gets higher and higher as the distance narrows. The action principle “aligns” the force along the line joining the two charges. Perhaps there’s constructive interference — increasingly additive superposition of potential vector values (quantum vibrations in the wave function).

For an electron and proton there might be some type of destructive interference when their EM potential fields are superimposed; or the character of the respective fields is different (Coulomb field lines point radially inward for a negative charge and outward for a positive charge). The latter gets us to the first possibility anyway (if some property like spin or phase factors into superposition). A superposition of quantized vibrations may be additive or subtractive. A net lower field potential between the particles produces an attraction — much like the net force “pushing” on Casimir plates.

So, is there any colloquial analogy that might be useful? No more ballistics, eh.

Wave panel
The figure shows interference between two spherical waves. The wavelength increases from top to bottom, and the distance between the sources increases from left to right.

[1] A sampling of online commentary

Evidently this all may be explained in some way in the (expensive and technical) book Quantum Field Theory in a Nutshell: (Second Edition) by A. Zee, as one reviewer noted: “Now lest the reader be confused, I am a mathematician new to quantum field theory. But I have to confess I never expected to understand as I now do – and by p. 36 no less – why masses attract!”

General search: Coulomb field in QFT


How can exchange of photons generate attractive force?

Really, what happens is that both electron and proton have an electric field attached to them because they both carry electric charge. When two electrically charge objects are brought close to one another, their electric fields adds up, resulting in a change of energy of the electric field, or equivalently, a force between the charged objects. At least that’s how I like to think about it.


QED and electrostatic attraction and repulsion


Do force messenger particles really explain attraction between particles?

This has been described in detail by Deser, http://arxiv.org/abs/gr-qc/0411026

The bottom line is that particles of even spin mediate attractive forces, particles of odd spin mediate repulsive forces. One has to look a bit more carefully, in order  to understand the, apparent, exceptions and how they fit.


Re: How does the exchange of virtual photons cause attraction?

I think the problem with this analogy is that you really have to imagine a virtual basketball, and recognize that it has different properties than a real basketball. A real basketball moves in one direction and carries momentum in that direction. When you catch a basketball coming from your left, of course it must carry momentum to push you to your right. A virtual basketball could come at you from the left but carry momentum in either direction.

Richard Feynman was the first person to notice that the equation looks like the equation for particle exchange, so he (and others) came up with the “virtual particle” description that we use today. (That’s a useful line of scientific reasoning: it’s worth noting how closely the equation for ocean waves “looks like” the equation for a vibrating violin string, or a beam of light; or how the equation for a discharging capacitor looks like the draining of a water tank.) The description is a bit weird—that’s why you have to put up with the changing masses, and the backwards momenta—but that’s the only way we have thought of.


How are virtual photons responsible for not just repulsive electromagnetic forces but attractive ones as well?

This phenomenon is certainly not as intuitive a picture as the one described in the question (the photons being emitted and absorbed and thus transmitting momentum), but can be understood in the same way.  Consider for the moment a particle with Negative mass, and how that would respond to a “push” from the photon.  It would in fact move the particle in the opposite direction, causing the attraction referred to.  Now imagine that somehow the fact that the charge is opposite causes the virtual photon to behave as if the particle had negative mass.

Another commentator: Virtual photons are called “virtual” photons for a reason. They are not real. They are mathematical tricks to try to break down complex calculations into little chunks. Virtual “particles” describe static fields. As such, there is not really any definite particle with a definite location that shoots from one charge to another to deliver the electromagnetic force. Popular media gets this concept wrong all the time. There is just the static electromagnetic field. Like charges repel and opposite charges attract according to how the electromagnetic field operates.

Another: This is a great question!

If you look at Feynman’s 1949 paper where he introduces virtual photons, he explains them in a surprising way. Basically, he treats the reaction of two charged particles to be directly on each other. The only reason he explains virtual photons is to explain the delayed reaction. So his virtual photons don’t have any photon-like properties that you might expect. They just act as a conduit for the properties of the originating particle, and they introduce a delay.

Now virtual photons also appear in quantum field theory. But quantum field theory preserves the mathematics of the classical theory as much as possible. In this case it means that the electromagnetic interaction is described by a potential, which bends the particle motion much as a gravity potential well bends a planet’s orbit. Quantum field theory doesn’t add much to the classical description except to capture the fact that there is probability involved. For example, if the source of the radiation is an atom, there is a probability that the atom never emitted any radiation, so the atom is still in an excited state and the radiation field is zero.

It’s therefore useful to look at this classically. Suppose that the charges are a long way away (such as a light year), and are held in place so they don’t move. Wait for their emitted electric fields to propagate to each other. Now suppose you release one particle.

The particle starts to move the other particle, but it doesn’t “know” that there is another particle in that direction. All that it can know is that there is a gradient that is pushing is an that direction. It could just as easily be a particle of the opposite charge that pushing it away. No momentum is received from the other particle.

The released particle accelerates, and emits radiation according to the Larmor formula. This radiation is in all directions, but strongest perpendicular to the direction of acceleration. This radiation carries momentum, and if you add it up for all directions it balances the change in momentum of the accelerating particle.

Meanwhile, the other particle is one light-year away, and obviously is not affected at all by recent events such as the acceleration of its partner. If you were to release it, it too would start to move and emit radiation. Momentum and energy will be conserved.


This new theory was the quantum theory of *fields*, from which anti-particles are a necessary consequence if you want to have certain conserved quantum numbers…something that observation tells us we should have. This is the framework from which I was working in my repsonse to the original question. Using this framework, we *do* see how particles can attract (and repel) each other via mediators. In fact, you can construct a measurable (macroscopic) electrostatic potential with a basis of positive energy photons as we should be able to do if we are to make contact with the macroscopic world…this potential is the one coming from the classical Maxwell equations.


What corresponds to a static field in QFT

For example for quantized EM field the Hilbert space is made up of photon states which correspond to EM waves classically. However what corresponds to static EM field in QFT? It can’t live in the Hilbert space of photons because any superposition of photon states will still correspond to a traveling EM wave (wave packet).

In the Hilbert space of free-particle photon states, all the photons are transverse. There are also photons with longitudinal and timelike polarizations. Even after imposing a gauge condition, these photons can exist as virtual particles, and do appear in the interaction jμAμ, and therefore can’t be completely dispensed with.

A static EM field has no real photons. The energy between two charges can be calculated by perturbation theory involving virtual photons. To order alpha, this gives the classical EM interaction potential. To higher order, it gives the Lamb shift, etc.

In quantum field theory, a static field is a stationary e/m field whose expectation is independent of time. Photon states arise from the quantization of the fluctuations of the e/m field and are always transverse and real. Longitudinal or virtual photons never enter the picture. Both are fictions stemming from an inadequate interpretation of the QFT formalism that takes perturbative expansions literally.


It sounds like your more general underlying question is: “How do we think about classical electromagnetic fields from the perspective of QED?” In particular, which configuration of quantum fields in QED do we associate with classical fields? How do we derive Maxwell’s equations for these classical fields, using the QED equations of motion for the corresponding quantum fields? How do we derive the Lorentz force law describing the force on electrons?

The basic answer is that classical fields are best represented by coherent states in QED. These states are an infinite superposition of photon number states — that is, there’s nonzero probability of finding any number of photons. (In particular, probabilities for different photon numbers obey Poisson statistics.) The expected value of the electromagnetic field operators for these states corresponds to what we’d classically call the values of the electromagnetic field, and these expected values obey classical Maxwell equations.

A more direct answer to your immediate question: the classical field produced by a static charge corresponds to a quantum field (a coherent state) with high photon number. You point out that single photons are associated with traveling waves, yet somehow we have a static field. There’s no contradiction because the coherent state is a superposition of many such traveling waves and thus has different behavior. Another charge placed nearby will interact with this photon field.

The Coulomb repulsion is a term in the Hamiltonian, not a quantum field.

The transverse part is in the field, the longitudinal part in the Hamiltonian. http://en.wikipedia.org/wiki/Coulomb_gauge#Coulomb_gauge

Thanks for the reference, this seems to answer my question:”It is particularly useful for “semi-classical” calculations in quantum mechanics, in which the vector potential is quantized but the Coulomb interaction is not.”


What is the QFT picture of forces?

Look at the Hamiltonian of QED. It has three terms, one for the electron, one for the EM field, and an interaction term that is like A ⋅ J A ⋅ J . The virtual photons that exchange energy between the field and the matter live in the interaction term.

But it would also be interesting to know how the Coulomb force acts through photons. At some point I will dive into QFT, but for know I was hoping to get a rough answer about what happens when two charges interaction. I mean where/when are there photons? How is momentum conserved?


Deriving the Coulomb force equation from the idea of photon exchange?


The exchange of photons gives rise to the electromagnetic force

What I am getting at is that a single electron doesn’t produce any EM field until it meets another charged particle! Actually, this should make sense if you think about it for a while. How do you detect there is an electron if nothing else at all is present? The simple answer is: you’re out of luck, you won’t detect it. You always need some test particles. So the classical picture of an electrostatic EM field of a point particle describes only what would happen if another particle would be inserted in that field.

Well, turns out all of the above talk about particles (although visually appealing and technically very useful) is just an approximation to the more precise picture of there existing just one quantum field for every particle type and the huge number of particles everywhere corresponding just to sharp local peaks of that field. These fields then interact by the means of quite complex interactions that reduce to the usual particle stuff when once look what those peaks are doing when they come close together.

In a classical framework yes the em force is “always there”. This is not true in a quantum description.

Wiki pages

  • https://en.wikipedia.org/wiki/Force_carrier
  • https://en.wikipedia.org/wiki/Virtual_particle
  • https://en.wikipedia.org/wiki/Polarization_(waves)
  • https://en.wikipedia.org/wiki/Interference_(wave_propagation)
  • https://en.wikipedia.org/wiki/Electric_potential
  • https://en.wikipedia.org/wiki/Force_field_(physics)
  • https://en.wikipedia.org/wiki/Potential_gradient
  • https://en.wikipedia.org/wiki/Conservative_force
  • https://en.wikipedia.org/wiki/Coulomb%27s_law#Electric_field
  • https://en.wikipedia.org/wiki/Action_(physics)
  • https://en.wikipedia.org/wiki/Casimir_effect
  • https://en.wikipedia.org/wiki/Elementary_charge
  • https://en.wikipedia.org/wiki/Gauge_theory
  • https://en.wikipedia.org/wiki/Introduction_to_gauge_theory