Attraction and Repulsion in Quantum Field Theory

R. Feynman: Not every simple question can be answered simply.

Dmitry V. Naumov

Joint Institute for Nuclear Research

2026-07-01

How QFT describes the attraction and repulsion of particles?

Repulsion like charges bend away + + Attraction opposite charges bend toward each other + -

What this lecture is based on

Cover of Quantum Field Theory for Experimentalists and Not Only

You always wanted to know but never dared to ask!

In QFT, where exactly do attraction and repulsion live?

  • Look for a Newtonian force!
    • There are no Newtonian forces in QFT
  • For like charges one expects repulsion, for opposite charges attraction. Look for the sign of charges in the cross-section!
    • But cross-sections \(\sigma \propto |A|^2>0\) do not depend on the sign of charges
  • Look at the Feynman diagram - electrons repel because they exchange a photon like a ball!
    • Really? Let us examine this ‘explanation’

How does QFT describe attraction and repulsion?

Repulsion picture: looks reasonable?

A qualitative picture for the repulsion

How to explain the attraction?

Attraction picture: looks reasonable too?

A qualitative picture for the attraction

Summary so far

  • QED does not switch from a “ball-like photon” for like charges to a “boomerang-like photon” for opposite charges.
  • The boomerang story is not just incomplete; it uses the wrong mechanism.
  • A real boomerang turns because it interacts with air. Its two wings move differently, aerodynamic forces tilt it, and angular momentum redirects the motion.
  • In empty space, with no air and no gravity, a “photon boomerang” would not curve back toward the second charge.
  • An analogy may be crude, but it must still work in principle. This one does not.

Key idea: interference between the initial and interaction-generated waves

Quantum interference picture

The same idea in formulas

Wave after interaction

\[ \Psi(\mathbf p,t)= \Phi(\mathbf p)e^{-i\mathbf p\cdot\mathbf R} +\Delta\Psi(\mathbf p,t) \]

Probability is not a sum of probabilities

\[ |\Psi|^2= |\Phi|^2 +2\,\operatorname{Re} \left[ \Phi^*(\mathbf p)e^{i\mathbf p\cdot\mathbf R} \Delta\Psi(\mathbf p,t) \right] +|\Delta\Psi|^2 \]

The force sits in the interference term

\[ \Delta G(\mathbf p,t)= 2\,\operatorname{Re} \left[ \Phi^*(\mathbf p)e^{i\mathbf p\cdot\mathbf R} \Delta\Psi(\mathbf p,t) \right] \]

Changing the sign of the interaction flips the redistribution

\[ q_1q_2\to -q_1q_2 \quad\Rightarrow\quad \Delta\Psi\to-\Delta\Psi, \qquad \Delta G\to-\Delta G \]

The packet moves because interference increases probability on one side of the momentum distribution and decreases it on the other.

Two waves turning a packet

\[ \begin{aligned} \psi(\mathbf r,t)&=\psi_0(\mathbf r,t)+\Delta\psi(\mathbf r,t), \qquad \Delta\psi=\psi_{\rm cl}-\psi_0,\\ \Delta\psi &\simeq -\delta\mathbf R\cdot\nabla\psi_0 +i\,\delta\mathbf P\cdot(\mathbf r-\mathbf R_0)\psi_0,\\ m\ddot{\mathbf R}_{\rm cl} &=\lambda q_1q_2 \frac{\mathbf R_{\rm cl}}{(|\mathbf R_{\rm cl}|^2+a^2)^{3/2}} . \end{aligned} \]

Do you miss equations?

Scalar Electrodynamics

The Lagrangian

  • Let us consider a theory with complex field \(\varphi\) and electromagnetic potential \(A^\mu\).
  • The Lagrangian for free fields reads \[ \mathcal L = (\partial_\mu \varphi)^*(\partial^\mu \varphi) -m^2\varphi^*\varphi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}, \]
  • Gauge invariance principle adds the interaction between fields with: \[ \partial_\mu \to D_\mu=\partial_\mu+iqA_\mu. \]
  • Then, \[ \boxed{ \mathcal L = (D_\mu \varphi)^*(D^\mu \varphi) -m^2\varphi^*\varphi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.} \]

Electromagnetic tensor \[ F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu. \]

Equations of Motion

  • In Lorenz gauge, \[ \partial_\mu A^\mu=0, \]
  • the equations become \[ \begin{aligned} (\partial^2+m^2)\varphi &= q^2A^2\varphi -2iqA_\mu\partial^\mu\varphi, \\ \partial^2 A^\mu &= iq(\varphi^*\partial^\mu\varphi-\varphi\partial^\mu\varphi^*) -2q^2A^\mu|\varphi|^2 . \end{aligned} \]
  • These are non-linear equations.
  • The first equation says that the electromagnetic potential shifts the scalar wave.
  • The second says that the scalar wave itself sources the electromagnetic field.
  • We are going to solve the equations of motion to see how attraction and repulsion appear.

Euler-Lagrange equations \[ \frac{\partial \mathcal L}{\partial \varphi} = \partial_\mu\frac{\partial\mathcal L}{\partial\partial_\mu \varphi} \]

Weak-Coupling

  • Use the perturbative expansion \[ \varphi = \varphi^{(0)} +q^2\varphi^{(2)} +O(q^4), \qquad A_\mu = qA_\mu^{(1)} +q^3A_\mu^{(3)} +O(q^5). \]

  • Then \[ (\partial^2+m^2)\varphi^{(0)}=0, \qquad \partial^2 A_\mu^{(1)}=J_\mu, \] with \[ J_\mu = i\left( \varphi^{(0)*}\partial_\mu\varphi^{(0)} -\varphi^{(0)}\partial_\mu\varphi^{(0)*} \right). \]

What We Need From the Expansion

  • First: a free packet \(\varphi^{(0)}\).
  • Second: the electromagnetic field \(A_\mu^{(1)}\) generated by that packet.
  • Third: the \(O(q^2)\) phase correction to the packet \(\varphi^{(0)}\).
  • Finally: two packets and their interaction.

The plan 1. \(\varphi^{(0)}\).

  1. \(J^\mu\).

  2. \(A^{(1)}_\mu\).

  3. \(\varphi^{(2)}\)

  4. Enjoy interaction!

One Gaussian Packet

Zero order for \(\varphi\) and first order for \(A_\mu\)

  • In the packet rest frame, take \[ \widetilde\psi(\mathbf k) = (2\pi)^{3/4}\sigma_p^{-3/2} \exp\!\left[-\frac{\mathbf k^2}{4\sigma_p^2}\right], \qquad \sigma_p\ll m . \]
  • The exact free solution is \[ \varphi_{\rm rest}^{(0)}(x) = \int\frac{d^3k}{(2\pi)^3} \widetilde\psi(\mathbf k) e^{i\mathbf k\cdot\mathbf x-iE_{\mathbf k}t}, \qquad E_{\mathbf k}=\sqrt{m^2+\mathbf k^2}. \]

Using

\[ E_{\mathbf k} =m+\frac{\mathbf k^2}{2m} +O\!\left(\frac{\sigma_p^4}{m^3}\right), \]

the packet is approximately

\[ \varphi_{\rm rest}^{(0)}(x) \approx \left(\frac{2\sigma_p^2}{\pi}\right)^{3/4} \frac{1}{(1+i\tau)^{3/2}} \exp\!\left[-\frac{\sigma_p^2\mathbf x^2}{1+i\tau}\right] e^{-imt}, \]

\[ \tau=\frac{2\sigma_p^2t}{m}, \qquad t_{\rm disp}=\frac{m}{2\sigma_p^2}. \]

Define

\[ s_p(t)=\frac{\sigma_p}{\sqrt{1+\tau^2}}, \qquad \sigma_x(t)=\frac{1}{2s_p(t)}. \]

Then

\[ n(t,\mathbf x)=|\varphi_{\rm rest}^{(0)}(x)|^2 = \left(\frac{2s_p^2(t)}{\pi}\right)^{3/2} \exp[-2s_p^2(t)\mathbf x^2]. \]

To leading order in \(\sigma_p/m\),

\[ J^\mu(t,\mathbf x) \approx \left( 2mn(t,\mathbf x), \; 4\frac{\sigma_p^2\tau}{1+\tau^2}\mathbf x\,n(t,\mathbf x) \right). \]

The spatial current is suppressed by \(\sigma_p/m\), so

\[ A_{\rm rest}^{(1)i}=O\!\left(\frac{\sigma_p}{m}\right), \qquad -\Delta A_{\rm rest}^{(1)0}(t,\mathbf x)\approx J^0(t,\mathbf x). \]

Thus

\[ A_{\rm rest}^{(1)0}(t,\mathbf x) \approx 2m\int d^3y\, \frac{n(t,\mathbf y)}{4\pi|\mathbf x-\mathbf y|}. \]

A finite packet replaces the point Coulomb source by a smooth charge cloud.

For a Gaussian packet,

\[ \boxed{ A_{\rm rest}^{(1)0}(t,r) \approx \frac{m}{2\pi r} \operatorname{erf}\!\left(\sqrt{2}\,s_p(t)r\right) } \]

or

\[ A_{\rm rest}^{(1)0}(t,r) \approx 2mV_{s_p(t)}(r), \qquad V_s(r)= \frac{1}{4\pi r} \operatorname{erf}\!\left(\sqrt{2}sr\right). \]

At large distance this is Coulomb. At the center it is finite.

Gaussian self-potential profile compared with the Coulomb limit

Dimensionless profile \(4\pi V_s(r)/(\sqrt{2}s)\) as a function of \(\rho=\sqrt{2}sr\). The dashed curve is the point Coulomb limit \(1/\rho\).

For a moving packet with mean four-momentum \(p_0^\mu=(E_0,\mathbf p_0)\),

\[ \mathbf v_0=\frac{\mathbf p_0}{E_0}, \qquad \gamma_0=\frac{E_0}{m}, \qquad \mathbf R=\mathbf x-\mathbf v_0t, \]

\[ t_*=\gamma_0(t-\mathbf v_0\cdot\mathbf x)=\frac{p_0\cdot x}{m}, \]

\[ r_*^2 = |\mathbf R|^2+ (\gamma_0^2-1)(\hat{\mathbf v}_0\cdot\mathbf R)^2. \]

Then

\[ \boxed{ A^{(1)\mu}(x) \approx 2p_0^\mu V_{s_p(t_*)}(r_*) }. \]

Second order for \(\varphi\)

The \(O(q^2)\) scalar correction obeys

\[ (\partial^2+m^2)\varphi^{(2)} = -2iA_\mu^{(1)}\partial^\mu\varphi^{(0)}. \]

For a narrow packet,

\[ \varphi^{(2)}(x) = -2im\,\mathcal I_{\sigma_p}(t_*,r_*)\, \varphi^{(0)}(x), \]

where

\[ \mathcal I_{\sigma_p}(t,r) = \int_0^t d\lambda\,V_{s_p(\lambda)}(r). \]

To this order,

\[ \varphi(x) \approx \varphi^{(0)}(x) \exp\!\left[-2iq^2m\,\mathcal I_{\sigma_p}(t_*,r_*)\right]. \]

In the packet rest frame this is

\[ \varphi_{\rm rest}(x) \approx \varphi_{\rm rest}^{(0)}(x) \exp\!\left[-i\int_0^t d\lambda\,U_{\rm eff}(\lambda,r)\right], \]

\[ \boxed{ U_{\rm eff}(t,r) = 2q^2m\,V_{s_p(t)}(r) }. \]

At this level, the self-interaction is primarily a local phase shift.

A lonely packet has no opinion

  • For one packet, the correction knows only \[ q^2 . \]

  • So the packet can acquire a self-phase, but it cannot decide whether the world is attractive or repulsive. \[ q \to -q \qquad\Rightarrow\qquad q^2 \to q^2 . \]

  • To ask about attraction or repulsion, we need a second packet.

A single lonely Gaussian packet first, then the full picture with two interacting packets

Two Packets

Eikonal form and 2D animations

  • We now take two well separated Gaussian packets.
  • Each packet is the same object we have just constructed: a free wave packet plus the smooth electromagnetic field it sources.
  • The only new label is the sign of its charge.

\[ Q_a=\pm |Q_a|,\qquad a=1,2. \]

  • At first order, the electromagnetic fields simply add:

\[ A^\mu \simeq A_1^\mu+A_2^\mu. \]

  • This is not yet a Newtonian force.
  • It is the background in which each wave accumulates phase.

For packet \(a\),

\[ \Phi_a \simeq \Phi_a^{(0)} \exp[-iq^2\Theta_a]. \]

In the first non-trivial approximation, attraction and repulsion live in the phase of the wave packet.

  • A position-dependent phase changes the local momentum.
  • The mutual part of the phase is the part created by the other packet.
  • The center bends because the phase gradient is different on the two sides of the packet.

\[ \text{field} \longrightarrow \text{phase} \longrightarrow \text{phase gradient} \longrightarrow \text{deflected center}. \]

  • The construction is the same for both cases.
  • Changing one packet from particle to antiparticle changes the sign of \(Q_1Q_2\).
  • The phase gradient flips direction.

\[ Q_1Q_2>0 \Rightarrow \text{repulsion}, \qquad Q_1Q_2<0 \Rightarrow \text{attraction}. \]

Like charges: the same eikonal construction bends the packet centers apart.

Opposite charges: only the sign changes, so the phase gradient points the other way.

Attraction and Repulsion in Quantum Field Theory. Once More in Pictures

The chain of ideas

Repulsion like charges bend away + + Attraction opposite charges bend toward each other + -

The exchange picture is familiar. The question is sharper: where does the sign of attraction or repulsion enter?

Qualitative boat picture for repulsion

Qualitative boat picture for attraction

The analogy changes the mechanism between the two cases. Scalar QED does not.

\[ \Psi(p,t)=\Phi(p)e^{-ipR}+\Delta\Psi(p,t) \]

\[ |\Psi|^2 = |\Phi|^2 +2\,\mathrm{Re}\!\left[\Phi^*(p)e^{ipR}\Delta\Psi(p,t)\right] +|\Delta\Psi|^2 \]

Repulsion constructive on the left, destructive on the right velocity v amplitude incoming + produced position x amplitude <v> shifts left The force is not inserted as a Newtonian push. It appears as a phase-sensitive sum of waves.
Attraction destructive on the left, constructive on the right velocity v amplitude incoming + produced position x amplitude <v> shifts right Changing the sign of the interaction flips the interference pattern and the drift.

Like charges: the centers bend apart.

Opposite charges: the same construction bends them together.