I. E. Segal

Postulates for General Quantum Mechanics

We present in this paper a set of postulates for a physical system and deduce from these the main general features of the quantum theory of stationary states. Our theory is strictly operational in the sense that only the observables of the physical system are involved in the postulates. The collection of all bounded self-adj oint operators on a Hilbert space, which has previously been used as a mathematical model for the observables in quantum mechanics, satisfy the postulates, as do a variety of considerably more general mathematical structures. We show that for any two observables there exists a pure state of the system (this term being defined essentially as done by von Neumann and Weyl) in which they have different (expectation) values. Each observable has a spectral resolution, and a state of the system induces, in a natural way, a probability distribution on the range of spectral values of each observable. A number of observables are simultaneously observable if and only if they commute (although the product of two observables is not defined in general, a reasonable notion of commutativity can be introduced). Inasmuch as Hilbert space plays no role in our theory, our proofs are necessarily of a different character from the proofs of these results for the case of the system of all bounded selfadjoint operators. Actually, Hilbert space appears to be somewhat inadequate as a state space even for the latter system, in that there exist pure states of the system which cannot be represented in the usual way by rays of the Hilbert space. The postulates are partly algebraic and partly metric. The algebraic postulates require essentially that an observable can be multiplied by real numbers and raised to integral powers, and that any two observables can be added. It is assumed that the usual algebraic laws are satisfied so that (1) the observables can be treated like the elements of a linear space, and (2) the usual rules for dealing with polynomials in one variable with real coefficients remain valid when the variable is replaced by an observable. It is not assumed that two observables have a product. The metric postulates require that for each observable there be defined a kind of maximum numerical value, which plays the part of a norm, and has various properties in accord with its physical significance. While this norm is quite essential to the development of the theory, an interesting consequence of the theory is that the norm can be (uniquely) defined in a purely algebraic fashion. This shows that the objective features of a physical system,-the spectral values and probability distributions of the observables, and the pure states,-are completely determined by the algebra of observables, i.e., by the rules for addition, scalar multiplication, and powers, of observables.

Introduction to Algebraic and Constructive Quantum Field Theory

The authors present a rigorous treatment of the first principles of the algebraic and analytic core of quantum field theory. Their aim is to correlate modern mathematical theory with the explanation of the observed process of particle production and of particle-wave duality that heuristic quantum field theory provides. Many topics are treated here in book form for the first time, from the origins of complex structures to the quantization of tachyons and domains of dependence for quantized wave equations. This work begins with a comprehensive analysis, in a universal format, of the structure and characterization of free fields, which is illustrated by applications to specific fields. Nonlinear local functions of both free fields (or Wick products) and interacting fields are established mathematically in a way that is consistent with the basic physical constraints and practice. Among other topics discussed are functional integration, Fourier transforms in Hilbert space, and implementability of canonical transformations. The authors address readers interested in fundamental mathematical physics and who have at least the training of an entering graduate student. A series of lexicons connects the mathematical development with the underlying physical motivation or interpretation. The examples and problems illustrate the theory and relate it to the scientific literature.Originally published in 1992.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Quantization of Nonlinear Systems

A direct method of quantization, applicable to a given nonlinear hyperbolic partial differential equation, is indicated. From such classical equations alone, without a given Lagrangian or Hamiltonian, or a priori linear reference system such as a bare or incoming field, a quantized field is constructed, satisfying the conventional commutation relations. While mathematically quite heuristic in part, local products of quantized fields do not intervene, and there are grounds for the belief that the formulation is free from nontrivial divergences.

Space-time decay for solutions of wave equations

Abstract The L 6 norm in space-time of a solution of the Klein-Gordon equation in two space-time dimensions is bounded relative to the Lorentz-invariant Hilbert space norm; the L p norms for p ≥ 6 are bounded relative to certain similar larger Hilbert space norms, including the energy norm.

The cauchy problem for the Yang-Mills equations

Abstract The Yang-Mills equation in Minkowski space with given initial data is considered. The gauge group is formulated in terms of Sobolev-Banach-Lie group, and the Cauchy problem for the equations thereby reduced to the temporal (hamiltonian) gauge. Given data for which are square-integrable over have respectively one and two derivatives which are square-integrable over space, a strong solution exists throughout space in a nontrivial time interval. If the initial data are infinitely differentiable in L2, the solution may be represented as a C∞ function on space-time satisfying the equations in the elementary sense. Strong solutions which agree at one time and have square-integrable derivatives as earlier agree throughout their regions of definition. For arbitrary finite-energy Cauchy data, there exists a quasi-solution (weak limit of solutions of truncated equations) which is global on Minkowski space. The solutions may take values in an arbitrary separable Hilbert Lie algebra.

Analysis in space-time bundles. I. General considerations and the scalar bundle

Abstract Mathematical features involved in the systematic development of elementary particle theory on an alternative cosmos (space-time) are presented. Bundles representative of physical fields are studied in the unique variant of Minkowski space M 0 enjoying similar properties of causality and symmetry. The “universal cosmos” M is the universal cover of the causal compactification of M 0 . The bundles studied are induced from representations of the scale-extended Poincare group, which forms the isotropy group in M of the universal cover G ∽ = S ∽ U(2,2) of the connected component of the group of all causal transformations on M. Discrete symmetries and higher-dimensional cases are also discussed. The primary focus is on the temporal evolution, especially stability (involving positivity of the energy), wave equations (implicative of finite propagation velocity), and the unitarity and/or composition series of associated actions of G. General spin bundles on M are treated, parallelized, and correlated with bundles on M 0 . Associated covariant wave equations and the spectral resolution of fundamental quantum numbers are studied in detail in the scalar case.

Nonlinear functions of weak processes. I

Abstract It is well-known that powers of a (Schwartz) distribution generally fail to exist in nontrivial cases; the same is true of stochastic and operational distributions (i.e. linear mappings into random variables or operators in Hilbert space). However, in these latter cases a notion of “renormalization” is applicable to the powers, which in a number of interesting cases leads again to a distribution. Section 1 of this paper gives a general theory of renormalization; an intrinsic characterization of renormalized products; their existence in finite-dimensional situations; and a specialization to a certain quantum process underlying all “free Bose-Einstein quantum fields.” In the latter case the present renormalized product may be identified with the “Wick product” heuristically treated in connection with quantum field computations by means of a common recursion relation, and the “theorem of Wick” given a simple abstract formulation and treatment. Section 2 treats the renormalized powers in the case of a process constituting a mathematical formulation of the heuristic notion of “free neutral scalar field in a two-dimensional space-time.” An intrinsic characterization for the renormalized powers, as a self-adjoint operator-valued distribution in space, at a fixed time, is developed in terms of simple and natural transformation properties of the distributions, under certain unitary transformations. The existence of the distributions thus characterized is shown by an explicit limiting procedure which yields the commutativity as well as the self-adjointness of all the renormalized powers of the “field” at a fixed time, and provides their simultaneous spectral resolution. These results subsume Theorem 1 of [ 16 ].

Algebraic integration theory

There are a number of important analytical situations in which the approach to an integral through a measure space is unnatural or technically disadvantageous. In fact, we have developed a major part of the theory from the point of view of integration lattices, following the fundamental ideas of Daniell. However, in the examples of integration lattices considered so far, e.g., the real step functions on a basic measure space or the continuous real-valued functions of compact support on a locally compact space, there is another inherent element of structure which in many respects is more important; these particular lattices are also algebras, and there are definite advantages from a broad viewpoint to a formulation of integration theory which starts from a linear functional on an algebra rather than a lattice. This approach to the theory of integration, which might be called the algebraic approach, is not restricted to function algebras alone, arises naturally from the foundations of probability theory, and is explicitly or implicitly indicated in a variety of other situations, e.g., commutative spectral theory in Hilbert space, the theory of integration in infinite-dimensional linear spaces, harmonic analysis on Abelian and more general groups (in particular, the L2 theory), and developments closely paralleling integration theory in the theory of rings of operators.

The global Goursat problem and scattering for nonlinear wave equations

Abstract The Goursat problem for nonlinear scalar equations on the Einstein Universe M , with finite-energy datum, has a unique global solution in the positive-energy, Sobolev-controllable case. Such equations include those of the form □ ϑ + H ′( ϑ ) = 0, where H denotes a hamiltonian that is a fourth-order polynomial, bounded below, in components of the multicomponent scalar section ϑ. In particular, the conformally invariant equation (□ + 1) ϑ + λϑ 3 = 0 ( λ ⩾ 0) is included. In the higher-dimensional analog R × S n to the Einstein Universe the same result holds under the stronger conditions on H required for Sobolev controllability. Irrespective of energy positivity, there is a unique local-in-time solution for arbitrary finite-energy Goursat datum, for all n ⩾ 3, establishing evolution from the given lightcone to any sufficiently close lightcone. These results show the existence of wave operators in the sense of scattering theory, and their continuity in the (Einstein) energy metric, for positive-energy equations of the indicated type. They also permit the comprehensive reduction of scattering theory for conformally invariant wave equations in Minkowski space M 0 to the Goursat problem in M . In particular, any solution of the equation arising from a nonnegative conformally invariant biquadratic interaction Lagrangian on multicomponent scalar sections, having finite Einstein energy at any one time, is asymptotic to solutions of the corresponding multicomponent free wave equation as the Minkowski time x 0 → ± ∞. Thus given a finite-Einstein-energy solution of the equation □ƒ + λƒ 3 = 0 on M 0 (λ ⩾ 0) there exist unique solutions ƒ ± of the free wave equation which approach ƒ in the Minkowski energy norm as x 0 → ± ∞, and every finite-Einstein-energy solution of the free wave equation is of the form ƒ + (or ƒ − ) for a unique solution ƒ of the nonlinear equation. This generalizes, in part in maximality sharp form, earlier results of Strauss for this equation.

Construction of non-linear local quantum processes. II

Abstract : A species of singular perturbation theory applicable to a general class of nonlinear local quantum fields is developed. The theory is applied in detail to the arbitrary scalar relativistic field in two space time dimensions with positive-energy self-interaction. (Author)