Zhengfang Zhou

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.

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.

An inverse problem for scattering by a doubly periodic structure

Consider scattering of electromagnetic waves by a doubly periodic structure S = {X3 = f(Xl,x2)} with f(xl +niAl,x2 +n2A2) = f(Xl,x2) for integers nl, n2. Above the structure, the medium is assumed to be homogeneous with a constant dielectric coefficient. The medium is a perfect conductor below the structure. An inverse problem arises and may be described as follows. For a given incident plane wave, the tangential electric field is measured away from the structure, say at X3 = b for some large b. To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of the following problem in a convex domain Q: --L =Au in Q, { V u = 0 in Q, n xu=0 on &Q.

Stability of the Scattering from a Large Electromagnetic Cavity in Two Dimensions

This work is concerned with a time harmonic scattering problem of electromagnetic waves from a two-dimensional open cavity embedded in the infinite ground plane. Because of the highly oscillatory nature of the solution for large or deep cavity, the model scattering problem is challenging both mathematically and computationally. A variational formulation reduces the scattering problem into a bounded domain (the cavity) problem. The stability of the solution is established for the bounded domain problem in the energy space. Moreover, our stability estimates provide the explicit dependence on the high wave number and the depth of the cavity.

Singular operators on boson fields as forms on spaces of entire functions on Hilbert space

Abstract Invariant scales of entire analytic functions on Hilbert space are introduced and applied. Singular operators represented by sesquilinear forms on spaces of regular vectors are given explicit integral representations via kernels that are entire functions on the direct sum of the Hilbert space with its dual. The Weyl (or, exponentiated boson field) operators act smoothly and irreducibly on corresponding spaces of entire functions. Arbitrary symplectic operators on a single-particle Hilbert space are shown to be implementable on the corresponding boson field by appropriate generalized operators.

An acoustic intensity-based method for reconstruction of radiated fields

An acoustic intensity-based method is proposed for the reconstruction of acoustic radiation pressure. Unlike the traditional inverse acoustic methods, the proposed method includes the acoustic pressure gradient as an input in addition to its simultaneous, co-located acoustic pressure in a radiated field. As a result, the reconstruction of acoustic radiation pressure from the input acoustic data over a portion of a surface enclosing all the acoustic sources, i.e., an open surface, becomes unique due to the unique continuation theory of elliptic equations. Hence the method is more stable and the reconstructed acoustic pressure is less dependable on the locations of the input acoustic data. Furthermore, the proposed method can be applied for both inverse and forward problems up to the minimum sphere enclosing the sources of interest. The effectiveness of the method is demonstrated by the results of several acoustic radiation examples with single or multi-frequency source in a two-dimensional configuration. The results from the method also show a measurable improvement in accuracy and consistency of reconstructed acoustic radiation pressure, in particular when the effect of the signal-to-noise ratio is included.

The contractivity of the free Hamiltonian semigroup in the Lp space of entire functions

Abstract A semigroup of operators T(t), t ⩾ 0, generated by the number operator in the complex wave representation is given by T(t)ƒ(z) = ƒ(e −t z) for any entire function on C n. The problem is to find all possible r, p ϵ [1, ∞) such that T(t):HLr(Cn,dgn)→(Cn,dgn) is contractive or bounded. It turns out that T(t):HLr(Cn,dgn)→(Cn,dgn) is contractive if and only if p ϵ [1, re2t], and is unbounded if and only if p > re2t. Furthermore all maximizers are identified; they are constant functions if p ϵ [1, re2t) and exponential functions if p = re2t, t > 0.

Statistically efficient testing of the Hubble and Lundmark laws on IRAS galaxy samples

The local redshift-distance (z-r) power laws, z∞r p , of exponents p=1, 2, which are predicted respectively by generic big bang cosmology and chronometric cosmology, are tested on flux- and redshift-limited subsamples of the sample of Strauss et al. (1992). The completeness of this sample at 60 μm makes possible statistically optimal nonparametric estimates of the corresponding luminosity functions, which are used for the tests. The present analysis is basically free of assumptions regarding the spatial distribution of the galaxies; only the absence of selection on flux down to its limiting value, at each given redshift, is used

Massless φdq quantum field theories and the nontriviality of φ44

Abstract The integrated quantized interaction lagrangian L for the massless φ4m2q theory (m ⩾ 1, q ⩾ 2) or the φ4m + 2q theory (m ⩾ 1, q ⩾ 3) exists rigorously as a hermitian operator on a dense domain in the free field Hilbert space, and has a self-adjoint extension. The proof uses a conformal mapping technique to eliminate infrared singularities, and applies also to multicomponent fields. In the case φ44, the leading term in the S-matrix is rigorously self-adjoint and nontrivial, and the total hamiltonian H(φint) becomes a nontrivial self-adjoint operator when φfree is substituted for πint in a suitably desingularized expression for H(φint). In the case of φ63, L vanishes identically.

Stability of weakly almost conformal mappings

We prove a stability of weakly almost conformal mappings in W 1,p(Ω;Rn) for p not too far below the dimension n by studying the W 1,pquasiconvex hull of the set Cn of conformal matrices. The study is based on coercivity estimates from the nonlinear Hodge decompositions and reverse Holder inequalities from the Ekeland variational principle.