3. Feynman Diagrams and the Renormalization of Quantum Electrodynamics
At the 1947 Shelter Island Conference, Lamb reported his major discovery, now known as the Lamb shift: the 2S_{1/2} energy level of the hydrogen atom is approximately 1000 MHz higher than the 2P_{1/2} level. According to Dirac theory, for an electron-proton system with pure Coulomb interaction, these two energy levels should be degenerate. It was soon realized that this shift should be attributed to first-order radiative corrections [19]. Using a corrected mass for the electron, Bethe derived the shift for the nS energy levels of the hydrogen atom in a non-relativistic approximation:
\frac{8}{3\pi}\left(\frac{e^2}{\hbar c}\right)Ry \frac{Z^4}{n^3} \ln \frac{K}{\langle E_n - E_m \rangle_{AV}}
This result still diverges as K approaches infinity, although it is only a logarithmic divergence. However, Bethe wisely guessed that a relativistic treatment would provide an upper limit for K at mc^2. Based on this conjecture, he calculated the magnitude of the shift to be 1040 MHz [20]. This was an encouraging result. To verify this conjecture, after listening to Bethe’s report at Cornell University, Feynman immediately set out to calculate the self-energy problem. This was his first concrete calculation since he had first attacked the problem seven years earlier.
The self-energy problem in quantum electrodynamics is essentially a quantum physics problem. It is not a natural extension of the classical electromagnetic mass problem; Dirac’s multi-particle theory had already given it a brand-new meaning. In 1927, Dirac published “The Quantum Theory of the Emission and Absorption of Radiation,” specifically exploring the interaction between an atom and a radiation field [21]. For pure radiation fields, Dirac invented creation and annihilation operators and naturally derived the concept of photons and Bose statistics. Using these operators, Dirac also explained Einstein’s emission and absorption coefficients using first-order perturbation of the Hamiltonian. Later, Dirac developed second-order perturbation theory and used the concept of virtual states to explain photon scattering. In 1929, Oppenheimer discovered that the virtual state process was a new type of self-energy form without a classical counterpart. At the same time, Dirac’s positron theory derived another new form of self-energy—vacuum polarization. The Dirac equation was born in 1925, and its negative energy solutions forced him to propose the hole theory, which he initially viewed as protons until he adopted the positron hypothesis in 1931. However, Dirac’s theory did not include the Pauli principle. As early as 1925, only two months after the birth of matrix mechanics, Born, Jordan, and Heisenberg had already developed the second quantization method. In 1934, Heisenberg used this method to handle positron theory, thereby providing the modern form of Dirac-Maxwell theory ([9], pp. 334-337). Therefore, vacuum polarization and virtual state processes are both self-energy terms without classical counterparts.
Before the discovery of the Lamb shift, Feynman had never performed any calculations regarding self-energy. His ambition was to create a completely new theory to solve the unsatisfactory aspects of both classical and quantum electrodynamics in one stroke. He preferred the particle viewpoint rather than a field theory that merely treated particles as excited states. He said he could not understand how creation and annihilation operators could produce and annihilate particles, as it did not fit his physical intuition. His goal was to establish a relativistic particle theory within the framework of path integrals; therefore, he did not pay much attention to the second quantization theory. However, five years of hard thinking failed to achieve his intended goal. Now, to calculate the self-energy problem, he had to start learning standard quantum field theory from scratch. At this point, he abandoned the attempt to quantize the action A; the form of A served only a heuristic role in his mind. Similarly, he abandoned the hypothesis that electrons interact only with other particles through half-advanced and half-retarded waves, returning instead to the traditional view of retarded interaction. However, Dirac’s sea of negative energy still puzzled him. For this, he adopted an idea from Wheeler, viewing the positron as an electron traveling backward in time.
Although he abandoned these original ideas, he did not change the world-image he had contemplated for years. Compared to the path integral idea, the form of the action was merely a specific technical issue. In calculating the self-energy of the hydrogen atom, Feynman geniusly utilized the path integral method. Transitioning from the non-relativistic to the relativistic case, he directly adopted Dirac’s concept of pair creation and annihilation and used Dirac matrices to replace velocity operators. When performing the perturbation expansion, he invented a diagrammatic method, using intuitive images to mark each term in the expansion. Then, he appealed to physical experience and mathematical intuition to assign corresponding factors to each diagram; in this way, he could directly write down the matrix elements for electron self-energy. These are what we now call Feynman diagrams. In choosing convergence factors, he drew on his experience of modifying the \delta function with the function f(s^2). By renormalizing the electron’s mass m and charge e, he obtained finite results. At the 1948 Pocono Conference, Feynman presented his diagrams and rules to others, deriving results essentially identical to those of Schwinger.
Compared to the standard Hamiltonian method, Feynman’s method has obvious advantages. First, it places space and time on equal footing, so “relativistic invariance is self-evident” [22]; we no longer need to split an electromagnetic field into longitudinal and transverse components. Thus, Feynman said that once the Hamiltonian method is abandoned, relativity and quantum mechanics merge most naturally [22]. Second, it is “most suitable for discussing virtual quantum problems” [22] because it treats emission and absorption as a single entity. With the help of Feynman diagrams, we can write down the probability amplitude of the entire process as if playing with building blocks. In 1949, Feynman published “The Theory of Positrons” [23] and “Space-Time Approach to Quantum Electrodynamics” [22], providing the corresponding Feynman diagrams and rules for the interaction between electrons and photons. His later lectures, Quantum Electrodynamics [24], provided a more complete treatment of this subject.
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