Kenneth Johnson

Differential regularization and renormalization: a new method of calculation in quantum field theory☆

Abstract Most primitively divergent Feynman diagrams are well defined in x -space but too singular at short distances for transformation to p -space. A new method of regularization is developed in which singular functions are written as derivatives of less singular functions which contain a logarithmic mass scale. The Fourier transform is then defined by formal integration by parts. The procedure is extended to graphs with divergent subgraphs. No explicit cutoff or counter-terms are required, and the method automatically delivers renormalized amplitudes which satisfy Callan-Symanzik equations. These features are thoroughly explored in massless φ 4 theory through 3-loop order, and the method yields explicit functional forms for all amplitudes with less difficulty than conventional methods which use dimensional regularization in p -space. The procedure also appears to be compatible with gauge invariance and the chiral structure of the standard model. This aspect is tested in extensive 1-loop calculations which include the Ward identity in quantum electrodynamics, the chiral anomaly, and the background field algorithm in non-abelian gauge theories.

INCONSISTENCY OF THE LOCAL FIELD THEORY OF CHARGED SPIN 3/2 PARTICLES

Abstract The relativistic quantum theory of Fermi Dirac fields of arbitrary spin is investigated and a general theorem is proved which aserts that for fields of half integral spin > 1 2 , the possibility of a consistent quantization requires that the equal-time anticommutators must be functions of the other fields to which the field in question is coupled. The case of spin 3 2 is studied in detail and the equivalence of various formulations of the theory is shown. The inconsistency of the relativistic local quantum theory of a charged spin 3 2 field in interaction with an external electromagnetic field is demonstrated by showing that the equal time commutation relations and relativistic covariance of the theory are not compatible. Finally, the mixed spin 3 2 - spin 1 2 (Bhabha) field is found to be characterized by the same inconsistency.

CURRENT-CHARGE DENSITY COMMUTATION RELATIONS

Abstract It is shown that the non-vanishing of the current-charge density commutator at equal times is required by and is compatible with the continuity equation.

Qualitative Features of the Glueball Spectrum

The low-energy meson spectrum is investigated. Quantum numbers of the lightest glueballs are obtained by studying low-dimension, gauge invariant, colorless operators. A general shell model of hadrons is formulated and its predictions are specified. A comparison and criticism of existing glueball models is included.

Conformal symmetry and differential regularization of the three-gluon vertex☆

Abstract The conformal symmetry of the QCD Lagrangian for massless quarks is broken both by renormalization effects and the gauge fixing procedure. Renormalized primitive divergent amplitudes have the property that their form away from the overall coincident point singularity is fully determined by the bare Lagrangian, and scale dependence is restricted to δ-functions at the singularity. If gauge fixing could be ignored, one would expect these amplitudes to be conformal invariant for non-coincident points. We find that the one-loop three-gluon vertex function Г μvp (x, y, z) is conformal invariant in this sense, if calculated in the background field formalism using the Feynman gauge for internal gluons. It is not vet clear why the expected breaking due to gauge fixing is absent. The conformal property implies that the gluon, ghost, and quark loop contributions to Г μvp are each purely numerical combinations of two universal conformal tensors D μvp ( x , y , z ) and C μvp ( x , y , z ) whose explicit form is given in the text. Only D μvp has an ultraviolet divergence, although C μvp requires a careful definition to resolve the expected ambiguity of a formally linearly divergent quantity. Regularization is straightforward and leads to a renormalized vertex function which satisfies the required Ward identity, and from which the beta function is easily obtained. Exact conformal invariance is broken in higher-loop orders, but we outline a speculative scenario in which the perturbative structure of the vertex function is determined from a conformal invariant primitive core by interplay of the renormalization group equation and Ward identities. Other results which are relevant to the conformal property include the following: 1. (1) An analytic calculation shows that the linear deviation from the Feynman gauge is not conformal invariant, and a separate computation using symbolic manipulation confirms that among D μ b μ background gauges, only the Feynman gauge is conformal invariant. 2. (2) The conventional (i.e., non-background) gluon vertex function is not conformal invariant because the Slavnov-Taylor identity it satisfies is more complicated than the simple Ward identity for the background vertex, and a superposition of D μvp and C μvp necessarily satisfies a simple Ward identity. However, the regulated conventional vertex can be expressed as a multiple of the tensor D μvp plus an ultraviolet finite non-conformal remainder. Mixed vertices with both external background and quantum gluons have similar properties.

Functional integrals for spin

Abstract A method based upon elementary quantum mechanics for constructing a path integral representation for spin amplitudes is described. A path integral representation obtained earlier for the unitary rotation matrices is rederived. In this system, the appropriate set of classical canonical variables and their quantum counterparts are identified. It is suggested how this method can be extended to more general systems by using the three-dimensional unitary group as another example.

A cutoff procedure and counterterms for differential renormalization

Explicit divergences and counterterms do not appear in the differential renormalization method, but they are concealed in the neglected surface terms in the formal partial integration procedure used. A systematic real space cutoff procedure for massless

An effective theory for the QCD vacuum

\phi^4

Gauge invariant geometric variables for Yang-Mills theory☆

theory is therefore studied in order to test the method and its compatibility with unitarity. Through 3-loop order, it is found that cutoff bare amplitudes are equal to the renormalized amplitudes previously obtained using the formal procedure plus singular terms which can be consistently cancelled by adding conventional counterterms to the Lagrangian. Renormalization group functions

Quantum electrodynamics in the infinite energy limit

\beta (g)