Derivation of Equal-Time Commutators Involving the Symmetric Energy-Momentum Tensor and Applications

Abstract

The use of covariance and the Jacobi identity in the study of equal-time commutators is investigated. Denoting by ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ the conserved and symmetric tensor density of Poincar\'e transformations and by $X$ any of the operators $\ensuremath{\varphi}$, ${\ensuremath{\partial}}_{0}\ensuremath{\varphi}$, ${J}_{0}$, ${J}_{l}$, or ${J}_{0l}$, we use the most general form of the equal-time commutators [$i{T}_{0\ensuremath{\mu}}(x)$, $X(y)$] and [$i{T}_{00}(x)$, $i{T}_{00}(y)$] compatible with covariance, together with the Jacobi identities for [[$i{T}_{00}(x)$, $i{T}_{00}(y)$], $X(z)$], to derive relations between the equal-time commutators [$i{T}_{0m}(x)$, $X(y)$] and [$i{T}_{00}(x)$, $Y(y)$], where $Y$ is any of the operators denoted by $X$ or $\ensuremath{\square}\ensuremath{\varphi}$, ${\ensuremath{\partial}}^{\ensuremath{\mu}}\overline{\ensuremath{\psi}}{\ensuremath{\gamma}}_{\ensuremath{\mu}}$, ${\ensuremath{\partial}}^{\ensuremath{\mu}}{J}_{\ensuremath{\mu}}$, and ${\ensuremath{\partial}}^{0}{J}_{0m}$. This information is first used in deriving equal-time commutators in canonical models. We then show that the assumption of $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ charge-current commutators together with ${[{{A}_{0}}^{\ensuremath{\alpha}}(x), \overline{\ensuremath{\psi}}(y)]}_{{x}_{0}={y}_{0}}\ensuremath{\propto}\overline{\ensuremath{\psi}}(x){\ensuremath{\tau}}^{\ensuremath{\alpha}}{\ensuremath{\gamma}}_{5}\ensuremath{\delta}(\mathrm{x}\ensuremath{-}\mathrm{y})$ (where ${{A}_{\ensuremath{\mu}}}^{\ensuremath{\alpha}}$ denotes the axial-vector current and $\ensuremath{\psi}$ denotes a spinor field) implies (as obtained earlier by the authors under different assumptions) ${[{{A}_{k}}^{\ensuremath{\alpha}}(x), \overline{\ensuremath{\psi}}{(y)}_{0}]}_{{x}_{0}={y}_{0}}=\frac{1}{2}\overline{\ensuremath{\psi}}(x){\ensuremath{\gamma}}_{5}{\ensuremath{\gamma}}_{k}{\ensuremath{\tau}}^{\ensuremath{\alpha}}\ensuremath{\delta}(\mathrm{x}\ensuremath{-}\mathrm{y})+i{(y\ensuremath{-}x)}_{k}{[{{A}_{0}}^{\ensuremath{\alpha}}(x), {{f}_{m}}^{\ifmmode\dagger\else\textdagger\fi{}}(y){\ensuremath{\gamma}}_{0}]}_{{x}_{0}={y}_{0}}$ [where $f$ denotes $(i{\ensuremath{\gamma}}^{\ensuremath{\mu}}{\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{-}m)\ensuremath{\psi}$]. For the conserved vector current an analogous relation holds. The incompatibility of field-algebra current commutators with $\ensuremath{\int}{d}^{3}x{[{{A}_{k}}^{\ensuremath{\alpha}}(x), \overline{\ensuremath{\psi}}(y){\ensuremath{\gamma}}_{0}]}_{{x}_{0}={y}_{0}}\ensuremath{\propto}\overline{\ensuremath{\psi}}(y){\ensuremath{\gamma}}_{5}{\ensuremath{\gamma}}_{k}$ is noted. Taking $\ensuremath{\psi}$ to be the nucleon field, it is shown that a certain form of the nucleon current leads to the above unless the right-hand side vanishes. Imposing this requirement, one then obtains ${g}_{{A}_{1}}={g}_{\ensuremath{\rho}}$, where ${g}_{{A}_{1}}{{a}_{\ensuremath{\mu}}}^{\ensuremath{\alpha}}(x){\ensuremath{\gamma}}_{5}{\ensuremath{\gamma}}^{\ensuremath{\mu}}(\frac{{\ensuremath{\tau}}^{\ensuremath{\alpha}}}{2})\ensuremath{\psi}(x)$ [${g}_{\ensuremath{\rho}}{{v}_{\ensuremath{\mu}}}^{\ensuremath{\alpha}}(x){\ensuremath{\gamma}}^{\ensuremath{\mu}}\ifmmode\times\else\texttimes\fi{}(\frac{{\ensuremath{\tau}}^{\ensuremath{\alpha}}}{2})\ensuremath{\psi}(x)$] denotes the contribution of ${A}_{1}$ ($\ensuremath{\rho}$) to ${f}_{m}$ in terms of the renormalized field ${{a}_{\ensuremath{\mu}}}^{\ensuremath{\alpha}}$ (${{v}_{\ensuremath{\mu}}}^{\ensuremath{\alpha}}$). From this and the usual saturation of the Weinberg spectral-function sum rules by single-particle intermediate states, we obtain the universality relations ${g}_{\ensuremath{\rho}}=\frac{{{m}_{\ensuremath{\rho}}}^{2}}{{f}_{\ensuremath{\rho}}}$ and ${g}_{{A}_{1}}=\frac{{(\frac{{m}_{\ensuremath{\rho}}}{{m}_{{A}_{1}}})}^{2}{{m}_{{A}_{1}}}^{2}}{{f}_{{A}_{1}}}$, where ${f}_{{A}_{1}}$ (${f}_{\ensuremath{\rho}}$) is defined by ${\ensuremath{\rho}}_{{A}_{1}}({m}^{2})={{f}_{{A}_{1}}}^{2}\ensuremath{\delta}({m}^{2}\ensuremath{-}{{m}_{{A}_{1}}}^{2})$ [${\ensuremath{\rho}}_{\ensuremath{\rho}}({m}^{2})={{f}_{\ensuremath{\rho}}}^{2}\ensuremath{\delta}({m}^{2}\ensuremath{-}{{m}_{\ensuremath{\rho}}}^{2})$]. For currents obeying the algebra-of-fields commutators, we obtain restrictions on Schwinger terms contained in equal-time commutators involving time derivatives of the currents. These relations show, for example, that in canonical realizations of current-field identities one needs derivative couplings of the spin-1 field.