Commutator

Abstract: Consequences of the local commutation relations of vector and axial currents proposed by Gell-Mann are explored: (1) A recipe for detecting and isolating Schwinger terms in the commutators, proportional to derivatives of the δ function, is discussed. (2) Under assumptions of smooth asymptotic behavior of form factors for forward scattering of the isovector current from a proton, we show that the U(3)⊗U(3) algebra for the time components of the currents implies the U(6)⊗U(6) algebra for space components, at least for spin-averaged diagonal single-particle states. (3) The derivation of the Adler-Weisberger formula for GAGV is sharpened by giving arguments that, at fixed energy, the forward π−p Green's function satisfies an unsubtracted dispersion relation in the pion mass. (4) A lower bound for inelastic electron-nucleon scattering at high momentum transfer is derived on the basis of U(6)⊗U(6). (5) The contribution of very virtual photons to the hyperfine anomaly in hydrogen is shown to be related to an equal-time commutator of currents; this contribution is crudely estimated to be <4 parts per million (ppm). (6) The logarithmically divergent part of electromagnetic mass differences of hadrons is shown to be proportional to matrix elements of the equal-time commutator of the electromagnetic current with its time derivative. It is suggested that this "divergent" part be identified with the Coleman-Glashow "tadpoles"; this suggestion is discussed in the framework of a simple quark model. (7) The logarithmically divergent part of the electromagnetic correction to the process π−→π0+e−+ν¯ is, on the basis of the U(6)⊗U(6) current algebra, shown to be nonvanishing, and is computed. (8) A speculative argument is presented that the rate e++e−→hadrons is comparable to the rate e++e−→μ++μ− in the limit of large energies.

Abstract: It has long been believed that no Hermitian optical phase operator exists. However, such an operator can be constructed from the phase states. We demonstrate that its properties are precisely in accord with the results of semiclassical and phenomenological approaches when such approximate methods are valid. We find that the number-phase commutator differs from that originally postulated by Dirac. This difference allows the consistent use of the commutator for inherently quantum states. It also leads to the correct periodic phase behaviour of the Poisson bracket in the classical regime.

Abstract: Preface.- Comments on notations.- 1 Some Spaces of Functions and Distributions.- 2 Real Interpolation of Banach Spaces.- 3 C0-Groups and Functional Calculi.- 4 Some Examples of C0-Groups.- 5 Automorphisms Associated to C0-Representations.- 6 Unitary Representations and Regularity.- 7 The Conjugate Operator Method.- 8 An Algebraic Framework for the Many-Body Problem.- 9 Spectral Theory of N-Body Hamiltonians.- 10 Quantum-Mechanical N-Body Systems.- Bibliography.- Notations.- Index.