Theory Of Scattering For Quasifree Particles, A

Raymond F Streater

• Publisher: WSPC
• Year: 2014
• ISBN:
  9789814612098
• Topic: Scienze biologiche
• Subtopic: Scienza generale

In this book, the author presents the theory of quasifree quantum fields and argues that they could provide non-zero scattering for some particles. The free-field representation of the quantised transverse electromagnetic field is not closed in the weak*-topology. Its closure contains soliton–anti-soliton pairs as limits of two-photon states as time goes to infinity, and the overlap probability can be computed using Uhlmann's prescription. There are no free parameters: the probability is determined with no requirement to specify any coupling constant. All cases of the Shale transforms of the free field ϕ of the form ϕ→ϕ+φ, where φ is not in the one-particle space, are treated in the book. There remain the cases of the Shale transforms of the form ϕ → T ϕ, where T is a symplectic map on the one-particle space, not near the identity.

Contents:

• Introduction
• Haag–Kastler Fields
• Representations of the Poincaré Group
• The Maxwell Field
• Some Theory of Representations
• Euclidean Electrodynamics
• Models
• Conclusion

Readership: Graduate students and professional in particle and mathematical physics.

Chapter 1

Introduction

1.1Introduction

If anyone could explain ... the peculiar properties
of oxygen and hydrogen, ... he would have transferred
certain phenomena from Chemistry into Physics. Those
who have attempted to conceive of the chemical elements as
different arrangements of particles of one primitive kind
of matter, would, if they had been successful, have
reduced Chemistry to Mechanics.

Thus said J. C. Maxwell in his Inaugural Lecture, King’s College London, 1860.

It is fair to say that physicists’ attempts to describe the behaviour of elementary particles by relativistic quantum fields obeying the Wightman axioms [63] have not been a complete success. A similar statement is true for anyone trying to construct a theory obeying the Haag–Kastler axioms [36], which state physical properties that would hold for the C*-algebra of local bounded observables. In any Wightman theory in a Hilbert space

, with an isolated one-particle state of positive mass, the scattering states can be obtained by Haag–Ruelle theory [41] as strong limits of vectors in

, and so also lie in

. The same holds in a Haag–Kastler theory with an isolated one-particle state in the vacuum representation. This theory breaks down for models with particles of zero mass, such as quantum electrodynamics. It has been suggested [62] that the strong limit might be too strong, and that weak*-convergence is more appropriate. The use of a weaker form of convergence allows us to pass from one representation of the C*-algebra to another, separated from the first by a superselection rule: we just need to follow a weak*-convergent sequence of states in one representation- space to its limit in another. Then it often turns out that convergence in the weak vector topology does in fact hold in the direct sum of the two representations.

It has been extremely difficult to construct interacting solutions to renormalisable quantum field theories that satisfy the Wightman axioms, in four space-time dimensions; it might be that only free and generalised free fields exist. The received wisdom of present-day quantum-field models of particle physics indicates that the interacting field becomes free in the limit of high energy. The starting point is an approximate theory, for example with some cut-offs at large energy, so that the modes with energy larger than the cut-off do not interact. The approximate answer for the S-operator is obtained by putting in some large cut-off at a deep level, and for a more accurate answer, one must use an even larger cut-off at an even deeper level [72]. We shall see one reason why the interaction may become zero at large energy in one of our models, that of the Higgs mechanism.

Albeverio and Hoegh-Krohn [3, 4] have generalised the Wightman axioms to a theory which allows a scalar product for which the norm is not always positive. Albeverio, Gottschalk and Wu [2] have given a class of quantum field theories obeying the Wightman axioms except the positivity of the metric. This gives rise to a finite theory with non-linear interaction; we proceed along different lines.

It has been conjectured that quantum electrodynamics does not exist; only theories containing quantised, non-abelian gauge vector fields, it is claimed, could exist as well as giving a non-trivial S-matrix. In this book we suggest a way round this question, using unusual representations of the C*-algebra generated by the free transverse quantised electromagnetic field. Our work is inspired by some of Donaldson [29], and is applied to the cohomology of certain sheaves; we use the result that the differential structure of R⁴, given by adding a non-trivial, flat cocycle to the free-field, is different from the usual representation in Fock space. The importance of Donaldson’s result, that simply-connected four-space can have a different differential structure from that usually given to R⁴, has also been suggested by Mickelsson [51]; in topological field theory, it appears in the work of Witten [73], and of Asselmeyer [10–14]; our suggestion is slightly different from theirs. In Penrose’s book [53], as well as in Chapter 1 of Ref. 18, there is no discussion of the square-integrability of the solution, in the sense of whether or not it has finite norm as defined by Eq. (4.18). The paper [45] does impose square-integrability; thus these authors deal with the free theory. In the present book, we allow non-square-integrability; this gives rise to a flat connection on the sheaf of electromagnetic fields. This connection is a cocycle similar to the cocycles found by Doplicher and Roberts [30] in their analysis of superselection rules allowed by the Haag–Kastler axioms, and ensures the addition of a non-zero classical charge and current to the equations of motion. This leads us to our models given in Section 3 of Chapter 7.

The general theory of Haag fields has been analysed by Haag [30, 36, 37] and his coworkers, among them Kastler, Doplicher and Roberts. They obtain the result that the representation of the field might be reducible, having SU(n) as the group of superselection rules, for any positive integer n. This result holds for any interacting quantum field theory obeying Haag’s axioms. Now, in four dimensions, there is no known interacting quantum field theory: all known models are direct sums of free or quasifree quantum fields. However, these authors do not apply the theory to free fields, even though the results on the existence of superselection rules apply to these. We show that the theory is not trivial, when applied to the free electrodynamics of the photon field: there are quasifree representations of the photon field which possess charged states. More, there are quasifree representations in which particle–antiparticle pair has a non-zero overlap scalar product with a pair of photons: the transition probability of particle–antiparticle annihilation into a photon pair is positive.

Chapter 2

Haag–Kastler Fields

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 [63] and those of Haag and Kastler [37] is not clear for a general Wight-man theory, but for any free boson field a key result due to Slawny [59] 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

₀; that space comprises continuous functions f, g,... of compact support; Segal proved that we then get self...

Table of contents

• Cover
• HalfTitle Page
• Title Page
• Copyright
• Contents
• 1 Introduction
• 2 Haag–Kastler Fields
• 3 Representations of the Poincaré Group
• 4 The Maxwell Field
• 5 Some Theory of Representations
• 6 Euclidean Electrodynamics
• 7 Models
• 8 Conclusion
• Bibliography
• Index