A Theory of Scattering for Quasifree Particles

The following section is included:

• Contents

The following section is included:

• Introduction

In four space-time dimensions, there is no known interacting model obeying the Wightman axioms, which were published long ago, in 1956. Haag and Kastler suggested their axioms later, but there is no interacting model known to obey them either. The relation between the Wightman axioms and those of Haag and Kastler is not clear for a general Wightman theory, but for any free boson field a key result due to Slawny suggests a natural way to construct a set of local C*-algebras which obey the Haag–Kastler axioms. We briefly outline Slawny's construction, following Ref. [23]. Consider for example a free scalar quantised field of mass m > 0. In any Lorentz frame, and its time-derivative at constant time (say, time zero) can each be smeared in the space variable with any function in the class ; that space comprises continuous functions f, g, … of compact support; Segal proved that we then get self-adjoint operators on Fock space. Thus…

The following sections are included:

• Induced Representations of Groups
• Wigner's Theory of Symmetry
• Time-Reversal

The following sections are included:

• The Classical Electromagnetic Field
• The C*-norm for Electromagnetism

The following sections are included:

• The Tensor Product and Fock Space
• Segal's Form of the Weyl Relations
• Representations with Coherent States
• The Segal–Bargmann Transform

This chapter owes a lot to the presentation given by Simon. The free fields E, H obey the Wightman axioms, and so the energy operator in the theory is bounded below. Moreover, the vacuum expectation values at time t = 0 have an analytic continuation in t in the upper-half-plane, Im t > 0. The continuation from (0, x) to (it, x) yields a Euclidean tensor field which describes a non-quantum, that is, a classical field. We call it a Nelson field because it was he who showed us how to do the case for the scalar field. In this chapter, we shall first describe the massive relativistic free quantised scalar field. In this case, the Wightman functions of the field define, by analytic continuation to imaginary time, a Euclidean real field obeying Nelson's Markov property. We then show that a Hermitian, massless free Wightman field of spin one, namely the free electromagnetic field, defines a vector Euclidean field obeying a similar Markov property. In these cases, the Nelson field uniquely defines the corresponding Wightman field, up to unitary equivalence.

The construction of Wightman theories with interaction in 1 + 1 dimensions means that it has not seemed necessary to consider the idea introduced in this book for theories in two-dimensional space-time. However, in view of the difficulty, if not the impossibility, of constructing an interacting Wightman theory in four space-time dimensions, it is worthwhile pointing out the following model.

In this book, we have briefly studied quasifree quantum fields, and argued that they could provide non-zero scattering for some particles. This is surprising, since by definition, a quasifree quantum field is one with zero as the value of any connected time-ordered product of n quantised fields. We start with the Wightman functions of the free tranverse electromagnetic fields, and construct the C*-algebra of the Fock representation. We note that the algebra has some non-Fock representations. Some states in the Fock representation converge as time goes to infinity to states which are not in Fock space, but can be interpreted as products of a specific non-Fock pair of a particle and its antiparticle. The convergence is in the weak* topology, and the states are normalised positive linear forms on our C*-algebra . At infinite time, these lie in an inequivalent representation, possibly reducible. For all finite times, the free photon system is in a state which has a non-zero scalar product with some reducible exponential states on our algebra; these can be represented as an infinite sum of vacuum, one-photon states, two-photon states, and so on. Each n-photon state lies in Fock space, but the norm of the sum is infinite. The theory thus gives a prediction, for example, of the probability that two photons will produce the compound state being considered: it is the square of the two-photon contribution to the compound. The interesting fact is that there is no freedom to choose the effective coupling constant: it is uniquely determined by the theory. If we make the conjecture, that there do not exist Wightman theories that are not quasifree in four space-time dimensions, then it would follow that all types of particles might be produced by examining such models…

The following sections are included:

• Bibliography
• Index