The Distinction Between Quantum Electrodynamics (QED) and Quantum Field Theory (QFT): An Insight into Their Relationship and Key Differences

Quantum Field Theory (QFT) and Quantum Electrodynamics (QED) are two of the most fundamental and powerful frameworks in modern physics. QFT is a theoretical framework that combines quantum mechanics and special relativity to describe the behavior of subatomic particles. QED, a specific application of QFT, is the quantum field theory of electromagnetism. Both theories are interconnected but distinct in their formulations and implications. This article explores the differences and similarities between QED and QFT, clarifying the misconception about corrections from QED to QFT.

Introduction to Quantum Field Theory (QFT)

Quantum Field Theory (QFT) is a theoretical framework that aims to describe the behavior of particles and their interactions using the principles of quantum mechanics and special relativity. It is a generalization of quantum mechanics and classical field theory. In QFT, particles are considered as excitations of underlying fields. These fields are propagated by other fields, leading to the interactions between particles.

Quantum Electrodynamics (QED) as a Special Case of QFT

Quantum Electrodynamics (QED) is a specific application of Quantum Field Theory that describes the behavior of charged particles, particularly the interactions of photons, electrons, and positrons. It is a comprehensively successful theory that has been validated through numerous experiments. QED is an essential part of the Standard Model of particle physics.

Defining QFT through Lagrangian Density

At the heart of QFT lies the Lagrangian density, which is a function that describes the dynamics of the quantum field. In relativistic QFTs, the Lagrangian density must be a Lorentz scalar, meaning it transforms in a specific way under Lorentz transformations. Lorentz transformations describe how space and time are perceived by observers in different inertial frames, and they are particularly important in the context of special relativity.

QED through Dirac-Maxwell Coupling

To obtain QED, one can choose a specific Lagrangian density that includes the Dirac field (representing fermions like electrons and positrons) and the Maxwell field (representing photons, the quanta of the electromagnetic field). The minimal coupling, which is the form of the interaction between the Dirac field and the electromagnetic field, is used to couple these fields. The resulting Lagrangian density is given by:

L L electron L photon L interaction

where:

L electron (ψ-bar * (iγ^μ?_μ - m)ψ) and L photon -1/4 F_μν F^μν L interaction -e(ψ-bar * γ^μ A_μ ψ)

In the above, ψ is the Dirac field (fermion), A_μ is the electromagnetic four-potential, and e is the electric charge. The term e(ψ-bar * γ^μ A_μ ψ) represents the interaction term, where γ^μ are the Dirac gamma matrices that ensure the Lorentz invariance of the theory.

Differences and Specificities of QFT and QED

While QED is a specific example of QFT, it is important to note that not every Lagrangian density will result in QED. QED has a specific set of assumptions, including the existence of a Dirac field and minimal coupling to the electromagnetic field. Other choices of Lagrangian density can lead to other QFTs, such as Yang-Mills theory or scalar field theories.

No Corrections from QED to QFT

There is no question of corrections from QED to QFT. The relationship between QED and QFT is one of specialization, where QED is a specific instance of QFT with a particular choice of Lagrangian density. QFT is the broader framework that encompasses QED, as well as other quantum field theories.

The misconception of corrections from QED to QFT arises from a misinterpretation of the relationship between the two theories. QED is a well-established and validated theory within the broader context of QFT, but it is not correcting QFT. Instead, QFT provides the theoretical framework that encompasses QED and other quantum field theories.

Conclusion

In summary, Quantum Field Theory (QFT) is a general framework that allows for the description of particle interactions, while Quantum Electrodynamics (QED) is a specific instance of QFT that focuses on the interactions between charged particles and photons. The Lagrangian density in QFT plays a crucial role, and the minimal coupling of the Dirac field with the electromagnetic field in QED is a specific choice that sets it apart from other QFTs. Understanding the relationship between QED and QFT is essential for a deeper comprehension of the theoretical underpinnings of modern physics, and it clarifies the misconceptions surrounding the idea of corrections from QED to QFT.