Understanding Feynman Diagrams and Virtual States in Quantum Electrodynamics

In this MIT OpenCourseWare lecture, the instructor guides through the core ideas of quantum electrodynamics, focusing on how time evolution is built from perturbation theory, how Feynman diagrams correspond to mathematical terms, and how virtual intermediate states contribute to physical processes. The talk highlights a microwave Cross-Phase shift using a Josephson junction, the interpretation of photon interactions in cavities, and the emergence of Casimir forces from vacuum fluctuations. It also discusses how single mode light behaves in different contexts and how gauge choices relate to observable outcomes.

Overview and Context

The lecture presents foundational tools for understanding light and matter interactions in quantum electrodynamics. It shows how a cross phase between photons can be mediated by nonlinear media in the microwave domain using a Sapphire substrate and a Josephson junction. The talk connects these ideas to their optical counterparts and emphasizes that the course aims to enable readers to follow current research papers in the field. The discussion also introduces that a coherent state can lose phase information when propagating through a medium and then re phase later, illustrating revival phenomena.

Single Mode Light and Intensity Fluctuations

The instructor revisits the G two function for bosons and fermions and then explains the special case of a perfectly single mode field. In this limit the field takes the form e to the i omega t and the intensity fluctuations disappear, leading to a G two function equal to one. A subtle distinction is drawn between looking at a single cavity in thermal contact with a reservoir and the ensemble average over many cavities. In the first case the cavity mode remains with fixed phase and fluctuating photon number within the thermal distribution; in the second case the ensemble shows fluctuations due to different photon numbers in each cavity. The limit of long time averages introduces broadening of the mode, which is interpreted as mixing with environmental modes.

From Time Evolution to Scattering Theory

The talk then bridges the time dependent evolution operator with standard perturbation theory. It defines the S matrix as the limit of the time evolution operator at large times and shows how the T matrix arises after factoring out the energy conserving delta function. This yields the second order transition rates that align with Fermi Golden Rule results. The connection between the formalism and measurable transition probabilities is emphasized as a core part of how one derives observable rates in quantum systems.

Virtual States and Energy Uncertainty

A central theme is the interpretation of intermediate states that appear in higher order diagrams. If an intermediate state has energy lower than the initial state, its propagation carries a phase factor that suppresses its contribution when integrated over time. The system can briefly access these virtual states in a time that obeys a form of Heisenberg uncertainty. The discussion stresses that energy conservation does not fail in quantum dynamics; it is restored in the long time limit, while short times allow virtual excursions that are essential to the full time evolution.

Gauge Representations and Thomson Scattering

The lecture contrasts dipole and other representations for light atom interaction, noting that a square term arises in some gauges which permits photon scattering without changing the atomic state. It is explained that different gauges yield identical observable results, illustrating gauge invariance. Thomson scattering serves as a simple example of scattering in the far off resonance regime where the intermediate energy defects are large and the corresponding states contribute only briefly.

Van der Waals, Casimir Forces and Vacuum Structure

The final section connects the diagrammatic time evolution to the physics of Van der Waals forces and the Casimir effect. It explains that two neutral atoms can exchange virtual photons that mediate forces, and that this perspective can be derived purely from Schr digestion without quantizing the field, or by considering photon exchange in quantum electrodynamics. The Casimir force is introduced as a manifestation of vacuum fluctuations of the electromagnetic field. Three pictures are presented as equivalent: a semi classical dipole interaction, exchange of virtual photons, and zero point fluctuations of the field. The talk ends by previewing how these ideas extend to more complex interactions in quantum electrodynamics.

Concluding Remarks

The lecturer closes with reflections on the physical meaning of diagrams as terms in the Schr equation solution, the reality of virtual processes, and the necessity of gauge consistency for observable quantities. The next topic, Wanderwaals interactions, is introduced as a bridge between atomic forces and vacuum phenomena.