"Grandad, why is the air transparent?"

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The apparent transparency of air masks a deeper quantum field theoretical story. In QFT, the idea that a photon simply sails through air as if it weren’t interacting at all is a simplification—useful for optics, but incomplete. The more accurate picture is one of continual interaction, cancellation, and coherence.

Let’s break it down.

Photons in QFT Are Not Little Bullets

In quantum field theory, a "photon" is not a classical particle darting along a path, but an excitation of the quantised electromagnetic field. This field permeates all of spacetime. Its 'particle-like' properties emerge from its quantization, and it interacts with other charged quantum fields—like the electron field—wherever they are non-zero.

Atmospheric molecules (O₂, N₂, etc.) have electron clouds; these are regions where the probability density of the electron field is significant. 

So, from a QFT perspective, a photon moving through the air is continually 'sampling' the ambient electron field.

Interaction with Bound Electrons: Virtual Transitions

Photons passing through air don't typically have enough energy (in the visible range, approximately 2–3 eV) to ionise or excite the electrons in atmospheric molecules to higher real energy levels. However, they can still interact via virtual processes^*. That is:

• The photon's electromagnetic field couples to the electrons within the molecules.
• The molecule momentarily enters a virtual excited state—forbidden by energy conservation for real, observable transitions, but allowed within the time-energy uncertainty principle (ΔE Δt ≥ ℏ/2) of quantum mechanics for very short durations.
• The electron effectively reverts to its ground state, and the original photon is re-emitted.

This is not a real absorption and re-emission (as in fluorescence or phosphorescence, which involve real energy transfer and a time delay), but a coherent forward-scattering process—a transient polarisation of the electron clouds, which can be seen as a brief, virtual excitation of the molecules.

This is precisely what gives rise to the refractive index. The cumulative effect of all these virtual transitions across many molecules alters the phase velocity of the propagating light. In QFT terms, this is captured by radiative corrections to the photon propagator in a background of bound charged particles.

Photon Propagation in a Medium: Modified Propagator

The photon’s propagator—the function describing the amplitude for a photon to travel from one point in spacetime to another—is modified in the presence of a polarizable medium. The polarisation of the medium, arising from the response of its constituent charges to the electromagnetic field, effectively enters as a dielectric function.

This dielectric function is derived from quantum field theoretical calculations, often involving diagrams that represent the interaction of photons with the bound electrons of the medium.

This results in:

• A modified dispersion relation: the phase velocity of light in the medium becomes v_p = c/n, where n is the refractive index. This means the relation between angular frequency (ω) and wave number (k) changes from ω = ck (in vacuum) to ω = (c/n)k.
• Possibly a small attenuation (represented by an imaginary part of the refractive index) if the photon's energy is near an absorption band of the material (e.g., in the ultraviolet for air, where real transitions can occur).

Air Appears Transparent Because of Energy Gaps

The electrons in N₂ and O₂ are bound in quantised orbitals. The visible photon energy (approximately 2–3 eV) is too low to excite any real electronic transitions in these molecules. Therefore, the real part of the refractive index dominates, and the imaginary part (responsible for absorption) is tiny in the visible spectrum.

However, at a microscopic level, the photon is always interacting—continually probing and being reshaped by the polarisation fields of nearby electrons via these virtual processes. It’s not so much "bouncing off" as continuously interfering with virtual excitations.

Quantum Coherence

From the path integral point of view, all possible paths that a photon could take contribute to its propagation, including those where it virtually interacts (scatters elastically at tiny angles) off molecules. The net result, through precise destructive interference for off-forward paths and constructive interference for the effectively straight path with modified phase velocity, is the maintenance of a well-defined trajectory and phase.

This coherence is what allows light to propagate through air as a classical wave—even though its passage involves a ceaseless flurry of virtual exchanges.

Summary

When a photon transits the electron cloud of an atmospheric molecule, QFT describes this as:

• A coherent interaction with the quantised electron field via virtual excitations.
• No real energy transfer (for visible light), but a phase shift—a change in the dispersion relation encoded in the modified photon propagator.
• A collective, statistical effect of many such interactions gives rise to the macroscopic refractive index.
• Air appears transparent because visible photon energy is insufficient for real transitions, and the medium is non-absorbing in this frequency range.

In short, the photon interacts everywhere, but with such finesse and brevity that its path appears unperturbed to our macroscopic senses. Like a dancer gliding across a floor of invisible springs.

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* Appendix: Virtual Excited States and the Role of iε

The distinction between a virtual excited state and a real (physical) excited state is central to understanding how photons interact with matter at the quantum level, particularly in quantum field theory (QFT).

Real Excited States

These are genuine energy eigenstates of a molecule or atom. When a photon's energy precisely matches the energy gap between the ground and an excited state, a real transition can occur. This means:

• The photon is absorbed.
• The system is promoted to a higher, measurable energy level.
• Energy conservation holds strictly: ℏω = E_n − E₀.

Virtual Excited States

Virtual states, by contrast, are not eigenstates of the system’s Hamiltonian. They occur only as internal steps in quantum processes (e.g., second-order perturbation theory or Feynman diagrams). These states:

• Do not satisfy energy conservation: the energy difference ℏω − (E_n − E₀) is nonzero.
• Exist only briefly, allowed by the time-energy uncertainty relation: ΔE · Δt ≳ ℏ/2.
• Do not lead to population of excited states or observable emission; they are unmeasurable intermediates.

Their presence manifests through amplitude corrections—they affect the phase and scattering behaviour of light, giving rise to phenomena like the refractive index.

Mathematical Form

In perturbation theory, these virtual contributions appear in denominators like:

Amplitude ∼ ⟨ψ₀| Hint |ψn⟩ ⟨ψn| Hint |ψ₀⟩ / (E₀ + ℏω − En + iε)

This is nonzero even when ℏω does not match any real excitation energy (E_n − E₀). The result is a subtle modification of the overall scattering amplitude, not an actual jump into ψ_n.

The Meaning of iε

The term iε (where ε is a tiny positive number) is added to the denominator for deep mathematical reasons:

• It shifts the pole slightly off the real axis in the complex plane.
• It ensures causality—that effects do not precede causes in time.
• It tells the contour integral how to correctly pass around singularities.

After integration, the limit ε → 0⁺ is taken. The iε is not a physical constant, but a calculational device—small but mighty. It’s the QFT equivalent of a signpost: “this way to causal physics.”

Summary Table

Property	Real Excited State	Virtual Excited State
Energy Conservation	Exactly satisfied	Temporarily violated
Duration	Finite and measurable	Infinitesimal; unobservable
Appears in Final State?	Yes (can decay or emit light)	No (internal process only)
Mathematical Role	External line or eigenstate	Internal propagator denominator
Effect on Light	Absorption/emission	Phase shift, refraction

So the next time you see iε tucked into an equation, remember: it’s the ghost in the machine that keeps the whole quantum edifice logically and causally intact.