Conformal Invariant Theory Research Articles

Conformal invariant self-interacting theory

Some time ago Gt~RSE~ (1) investigated the possibility of extending Heisenberg's spinor theory of elementary particles (2) to a conformal invariant theory. Although the kinetic term of the equation is natural ly conformal invariant for the canonical field considered, the interaction terms only exhibit the invariance when fractional powers of the field are involved. From the vantage point of the progress in the spinor theory in recent years, a new starting point for constructing a unified self-interacting theory of elementary particles, based on eonformal invariance, is considered. The basic motivation for requiring an underlying eonformal invariance derives from the circumstance that the substratum of the theory which describes electrodynamic and neutrino phenomena, contained in a unified theory of matter, should possess conformal symmetry. The results of physical interest obtained in the Heisenberg theory at'e a consequence of the noneanonieal quantization (s) of the theory, as implied by the scale invariance (2.4) of the Heisenberg equation of motion for subcanonical dimension of the spinor field (spinor potential (5)). These features manifest themselves physically in the emergence of an attractive core (e) in fermion-antifermion interactions which generate the primary composite bosons in the theory. Furthermore, it has recently been found by DURR and the author (v) that the natural habitat for conformal invariant theories for fields of subcanonical scale dimension resides in higher-order equations, as do Lagrangian theories for such fields (s). With a view to construct a self-interacting theory, invariant under the conformal transformation

CONFORMAL INVARIANT BOOTSTRAP FIELD THEORY

The construction of an exactly conformal invariant theory directly without having to consider first nonscale-invariant theories is considered using the bootstrap approach. Only strong interactions are treated, allowing for weak and electromagnetic ones only to lowest order. 30 references. (JFP)

CONFORMAL INVARIANT QUANTUM FIELD THEORY

We shall present here some attempts to analyze quantum field theory (QFT) nonperturbatively. The main tool in this is a partially summed up and supposedly convergent version of Wilson operator product expansions on the vacuum. For realistic quantum field theory with mass and without conformal symmetry we conjecture that they are true. For conformal invariant theories we are able to prove that they hold, given only the existence of Wilson expansions as asymptotic expansions at short distances. Given these vacuum expansions (as we shall call them), arbitrary n-point Wightman functions can also be expanded; the expansion is completely fixed if all the two and three point functions of all the fields (including composite ones) in the theory are known. Given that these two and three point functions satisfy the axioms themselves, the expansion formula provides an Ansatz which satisfies all the Wightman axioms automatically if it converges, except locality. The locality constraint amounts to a crossing relation for the four point functions.

Solutions of the conformal invariant spinor equations and theory of the electron–muon system

Solutions of the conformal invariant spinor equations and theory of the electron–muon system

Conformal Invariance and Field Theory in Two Dimensions

The relevance of the representation theory of the conformal group to quantum field theory is illustrated in two dimensions by the Thirring model. Space-time position operators and their complex extensions are defined as operator-valued functions of the generators. These complex position variables coincide with the Gel'fand-Naimark labels and can be interpreted as labels for nonorthogonal coherent states. A Hilbert-space metric then becomes necessary. It is given by the matrix elements of the metric operator $G$ and is nontrivial for the nonanalytic supplementary series and the analytic representations. In this case ${G}^{\ensuremath{-}1}$ gives the two-point function for the Thirring model. For nonanalytic representations only weak (infinitesimal) conformal invariance holds for interacting fields if causal and spectral properties are imposed, while those properties become compatible with strong (global) conformal invariance in the case of analytic representations which lead either to free fields or to interacting fields with a quantized value of the coupling constant.

Currents, stress tensor and generalized unitarity in conformal invariant quantum field theory

Matrix elements of internal symmetry currents and energy momentum density tensor are constructed in Migdal Polyakov conformal invariant bootstrap field theory. Their 3-point functions satisfy Bethe Salpeter equations which determine any free coefficients that may still occur in the conformal invariant Ansatz. Ward identities are verified for alln-point functions. They imply correct equal time current commutation relations. A proof of generalized unitarity is also given. Various equivalent forms of the propagator bootstrap are discussed. Our algebraic techniques also yield an eigenvalue equation for first order correction to the exactly conformal invariant theory, assuming the latter is Gell-Mann Low large momentum asymptote of a renormalizable finite mass theory.

On calculations in conformal invariant field theories

On calculations in conformal invariant field theories

Canonical scaling and conformal invariance

Arguments are given to show that a strictly conformal invariant skeleton theory is compatible with the observed scaling only if it possesses an infinite number of local tensors of spin n = 2, 4, … and scale dimension l n = 2+ n, which are all conserved. The conclusion suggests that conformal invariance is spontaneously broken.