Abstract Dynamics of quantum field theory can be formulated by functional equations. To develop a complete functional quantum theory one has to describe the physical information by functional operations only. The most important physical information of elementary particle physics is the S'-matrix. In this paper the functional S'-matrix is constructed for relativistic spin 1/2 fermion scattering in nonlinear spinortheory with noncanonical relativistic Heisenberg quantization. With appropriate modifications the procedure runs on the same pattern as in the case of nonrelativistic potential scattering treated in I. Furthermore a calculational method for scattering functionals is proposed. In the appendices technical details are discussed.
Abstract Dynamics of quantum field theory can be formulated by functional equations. To develop a complete functional quantum theory one has to describe the physical information by functional operations only. The most important physical information of elementary particle physics is the S'-matrix. In this paper the functional S'-matrix is constructed for relativistic spin 1/2 fermion scattering in nonlinear spinortheory with noncanonical relativistic Heisenberg quantization. With appropriate modifications the procedure runs on the same pattern as in the case of nonrelativistic potential scattering treated in I. Furthermore a calculational method for scattering functionals is proposed. In the appendices technical details are discussed.
Abstract Dynamics of quantum field theory can be formulated by functional equations. To develop a complete functional quantum theory one has to describe the physical information by functional operations only. The most important physical information of elementary particle physics is the matrix. In this paper the functional S-matrix is constructed for nonrelativistic spin 1/2 fermions, as in this system a rigorous construction of operator representations is possible. The method of S-matrix derivation used in I and II is improved and the exact perturbation solution for the scattering functionals is given.
Functional quantum theory of free Fermi fields is treated for the special case of a free Dirac field. All other cases run on the same pattern. Starting with the Schwinger functionals of the free Dirac field, functional equations and corresponding many particle functionals can be derived. To establish a functional quantum theory, a physical interpretation of the functionals is required. It is provided by a mapping of the physical Hilbert space into an appropriate functional Hilbert space, which is introduced here. Mathematical details, especially the problems connected with anticommuting functional sources are treated in the appendices.
The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.
Abstract Chapter 4 has introduced the functional integral representation of the quantum statistical operators and thus, formally, evolution in imaginary or Euclidean time. By contrast, to calculate the evolution operator and the scattering S-matrix elements, quantities relevant to particle physics, it is necessary to make a continuation from imaginary to real time. However, the representation of the S-matrix follows from additional considerations. To illustrate the power of the formalism, we show how to recover the perturbative expansion of the scattering amplitude, some semi-classical approximations, and the eikonal approximation. When the asymptotic states at large time are eigenstates of the harmonic oscillator, instead of free particles, the holomorphic formalism becomes useful. A simple generalization of the path integral of Chapter 4 leads to the corresponding path integral representation of the S-matrix. In the case of the Bose gas, the evolution operator is then given by a holomorphic field integral. A parallel formalism leads to an analogous representation for the evolution operator of a system of non-relativistic fermions.
Starting from the non-relativistic field theory of spin- fermions interacting through the Abelian Chern-Simons term, we show that the quantized field theory leads, in the two-particle sector, to a two-particle Aharonov-Bohm-like Schrödinger equation with an antisymmetric (fermionic) wavefunction and without a delta function term. Calculating perturbatively the field-theoretic two-particle scattering amplitude up to one-loop order, we show that, in contrast to the scalar theory, the contribution of all the one-loop diagrams is finite and null, and that of the tree level ones coincides with the exact amplitude. Further, the Pauli matter-magnetic field interaction term is shown not to contribute to the amplitude to this order.
Functional integral representation of the partition functionThe customary approach to nonrelativistic many-body theory is to proceed with the method of second quantization begun in the first chapter.There is another approach, the method of functional integrals, which we shall follow here.Of course, what can be done in one formalism can always be done in another.Nevertheless, functional integrals seem to be the method of choice for most elementary particle theorists these days, and they seem to lend themselves more readily to nonperturbative phenomena such as tunneling, instantons, lattice gauge theory, etc.For gauge theories they are practically indispensable.However, there is a certain amount of formalism that must be developed before we can start to discuss physical applications.In this chapter, we shall derive the functional integral representation of the partition function for interacting relativistic non-gauge field theories.As a check on the formalism, as well as to obtain some feeling for how functional integrals work, we shall then rederive some well-known results on relativistic ideal gases for bosons and fermions.