Lorentz Invariant Formulation

Quantum electrodynamics (QED) as a typical quantum field theory began with the famous work by Dirac [1]. Important works in the pioneering era of QED are found in [1-7]. Quantum field theory in a time-dependent canonical formalism was established by Heisenberg and Pauli in [4-5]. Tomonaga [8] and Schwinger [12] independently reformulated Heisenberg and Pauli's formalism in a Lorentz covariant way. Tomonaga and Schwinger's theory was called the super-many-time theory. It was applied to QED for the first time in [9,10]. Refs. [13-15] are a series of elaborate works by Schwinger along the line of [12]. A Lorentz covariant framework of QED was founded by Tomonaga and Schwinger's works. On the other hand, Feynman proposed the method of path integration in [16], on the basis of which he formulated Feynman rules with Feynman diagrams [17,18]. Mathmatical foundation of Feynman's theory was established in [19-21]. Tomonaga, Schwinger and Feynman's theories were dealt with in a unified way by Dyson in [68,69]. The notion of asymptotic fields is inevitable in view of the Heisenberg representation and it was shown in [22] that the notion of asymptotic fields actually leads to the Yang-Feldman equation. Bound state problems are investigated in [23,24]. In particular, it is shown in [23] that Green's functions in the Heisenberg representation are given by solutions to Gell-Mann and Low's equations. Ref's [25-37] have played important and fundamental roles in the development of the modern quantum field theory. Usefulness of Green's functions is discussed in [29,30]. Construction of quantum field theory by means of the variational principle was attempted in a series of Schwinger's works [31-37], in which invariance of the theory under Poincaré group and relation between spin and statistics are extensively investigated. Various considerations together with interesting memoirs and episodes in the development of QED are found in the Nobel lectures [38-40].