# Introduction to Algebraic and Constructive Quantum Field Theory

JOHN C. BAEZ
IRVING E. SEGAL
ZHOU ZHENGFANG
Pages: 310
https://www.jstor.org/stable/j.ctt7ztswr

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Preface
(pp. ix-xii)
4. Introduction
(pp. xiii-2)

Quantum field theory is quintessentially the algebra and analysis of infinite-dimensional dynamical systems, as constrained by quantum phenomenology, causality, and symmetry. Although it has a clear-cut central goal, that of the realistic description of particle production and annihilation in terms of the localized interactions of fields in space-time, it is clear from this description that it is a multifaceted subject. Indeed, both the relevant mathematical technology and the varieties of physical applications are extremely diverse. In consequence, any linear (i.e., sequential) presentation of the subject inevitably greatly oversimplifies the interaction between different parts of the subject.

At the level of...

5. 1 The Free Boson Field
(pp. 3-74)

Much of the quantum field theory is of a very general character independent of the nature of space-time. Indeed, a universal formalism applies whether or not there exists an underlying “space” in the usual geometrical sense. In its primary form, this universal part of quantum field theory depends only on a given underlying (complex) Hilbert space, say H. Colloquially, H is often called thesingle-particle space.

Thus, for a nonrelativistic particle in three-dimensional euclidean space R3, H is the spaceL2(R3) consisting of all square-integrable complex-valued functions on R3, in the usual formalism of elementary quantum mechanics. For a relativistic...

6. 2 The Free Fermion Field
(pp. 75-95)

The theory of the free fermion field is essentially distinct from but nevertheless parallel to that of the free boson field, and the underlying formal analogy is close and useful.

Definition. Anorthogonal spaceis a pair (L,S) consisting of a real topological vector space L and a given continuous symmetric nondegenerate bilinear formSon L.

Concerning the notions of nondegeneracy, symmetry, and of the algebraic topology on a real vector space, see Chapter 1.

Example 2.1. Let M be finite-dimensional real vector space and M*its dual. Let L denote the direct sum M⊕M*, and letS...

7. 3 Properties of the Free Fields
(pp. 96-117)

The free fields appear as a natural mathematical extension of classical lines of investigation. Notable among these are those of Schur, Weyl, and Brauer on the decomposition of tensor representations of the classical groups; Wiener’s work on functional integration and its further developments by Cameron and Martin, Kac, and others; and a variety of developments in number theory concerned with the symplectic group and theta-functions, along lines treated, e.g., by Cartier, Igusa, and Weil. Ideas connected with free field algebra are involved in works of Bernstein, Leray, Quillen, and Vergne, among others.

In this chapter we give mathematical characterizations of...

8. 4 Absolute Continuity and Unitary Implementability
(pp. 118-141)

A certain class of divergences in quantum field theory originates in the failure of the Stone-von Neumann uniqueness theorem. In general, there is no unitary equivalence between different irreducible canonical systems over the same space; or, put in terms of theoretical physical practice, the putative unitary equivalence in question is “divergent.”

A simple example may be provided by the canonical transformation$p_{j}^{\prime }={{p}_{j}}+{{q}_{j}},q_{j}^{\prime }={{q}_{j}}(j=1,\ldots ,n)$, and just what happens to it whennbecomes infinite. (We are working here modulo rigorous details, which follow from the Weyl relations; it is simpler conceptually to use the Heisenberg relations.) It is evident that the...

9. 5 C*-Algebraic Quantization
(pp. 142-152)

In Chapters 1 and 2, free fields have been established in a form dependent on an underlying complex Hilbert space. There may or may not be a given such structure in the underlying so-called single particle or “classical field” space L. Moreover, as seen in the preceding chapter, there are many unitarily inequivalent quantizations over a given space L, when L is infinite-dimensional. This creates a plethora of technical problems that are not clearly germane to the underlying physical ideas, but to which the precise mathematical situation is sensitive.

Such problems do not intervene when L is finite-dimensional, as a...

10. 6 Quantization of Linear Differential Equations
(pp. 153-173)

Quantum field theory originated in extremely intuitive and heuristic work, which in part appears to some physicists and mathematicians as almost fortuitously successful. Briefly, Dirac was inspired by the success of Heisenberg’s postulatepq-qp= -iℏfor treating systems of a finite number of degrees of freedom, and sought to extend it to the electromagnetic field, which has an infinite number of degrees of freedom. At a fixed time, the field values and their first time derivatives resemble analytically thepandqthat describe the kinematics of a nonrelativistic particle on the line, apart from the infinite-dimensionality,...

11. 7 Renormalized Products of Quantum Fields
(pp. 174-207)

Both multiplicative and additive renormalizations are used in practical quantum field theory, but only the latter is clearly involved at a foundational level. Additive renormalization in its practical form has sometimes been called “Subtraction Physics.” This refers to the isolation of apparently meaningless or “infinite” terms in an additive symbolic expression, followed by their removal as rationalized by such considerations as locality and invariance. In this chapter we formulate the basic theory necessary to give mathematical viability to the program of quantization of nonlinear wave equations and in particular avoid infinities.

More specifically, we are concerned here to give appropriate...

12. 8 Construction of Nonlinear Quantized Fields
(pp. 208-250)

The intuitive conceptual simplicity of nonlinear quantized field theory contrasts strikingly with the depth of the complications that arise from attempts at rigorous mathematical implementation of the underlying ideas. A priori, notwithstanding the formal simplicity and physical appeal of the proposed theory—rooted in the work of Dirac, Heisenberg, Pauli, et al.—there is no assurance that this theory actually exists, nor as yet is there a definitive mathematical interpretation for it. In direct formal terms, it involves nonlinear functions of quantized distributions, which are rightly looked on with considerable suspicion from a mathematical standpoint. In practical terms, the situation...

13. Appendix A. Principal Notations
(pp. 251-253)
14. Appendix B. Universal Fields and the Quantization of Wave Equations
(pp. 254-257)
15. Glossary
(pp. 258-280)
16. Bibliography
(pp. 281-288)
17. Index
(pp. 289-291)