Quasicanonical quantum field theory

Abstract

We call a quantum-field-theory model quasicanonical if it is defined by canonical equal-time field commutation relations (e.g., $\left[\dot{\phi }\mathrm{}\right(x), \phi \mathrm{}(0\left)\right]\delta \left(x^0\right)=-i{\delta }^4\mathrm{}\left(x\right)\mathrm{}$) and local field equations [e.g., $\square \phi \mathrm{}\left(x\right)=\lambda J\left(x\right)\mathrm{}$], and is locally invariant to scale transformations [e.g., $\phi \mathrm{}\left(x\right)\mathrm{}\to \rho \phi \left(\rho x\right)$]. [These requirements are not consistent if the model is purely canonical, i.e., if $J\left(x\right)\mathrm{}$ is the simple Wick product:${\phi }^3\mathrm{}\left(x\right)\mathrm{}$:.] Canonical Bjorken scaling is valid in such models provided that the field equations are also locally invariant to $R$ transformations [$\phi \mathrm{}\left(x\right)\mathrm{}\to \phi \mathrm{}\left(x\right)+r$] and the physical currents are $R$-invariant. We discuss here further properties and consequences of these models. (a) We incorporate positivity and $R$-invariance restrictions on light-cone expansions and deduce the form of the consequent bilocal operators [e.g., $\int \mathrm{da}\sigma \mathrm{}\left(a\right):\phi \mathrm{}\left(\mathrm{ax}\right)\mathrm{}\phi \mathrm{}\left(0\right):$]. (b) We exhibit a Hamiltonian formulation of the theory, both in the massless and massive cases. (c) We show that the theory is locally conformal- and inversion- [$\phi \mathrm{}\left(x\right)\mathrm{}\to {\left(x^2\right)}^{-1\mathrm{}}\phi (-\frac{x}{x^2})$] invariant. These symmetries are spontaneously broken. (d) We discuss the implications of the model for deep-inelastic electron-positron annihilation. Exact scaling is obtained. (e) We study the possible low-energy consequences of the (spontaneously broken) $R$ symmetry. These include the PCAC (partial conservation of axial-vector current) consistency conditions and the Gell-Mann charge algebra. (f) We consider the arguments for and consequences of a spontaneous breakdown of the dilatation symmetry.