Corrigendum to “Constraints on operator ordering from third quantization” [Ann. Phys. 365 (2016) 54–65]

In our previous paper (Ohkuwa et al., 2016) corrigendum was found in Eqs. (3.4) and (3.6). However, conclusions of our previous paper are not changed.     

Remarks on the quantum field theory in lattice space. I

We calculate the Gelfand functionsE(f,g;a) for quantized field in lattice space,a being the lattice constant. In the limita 0 the functionals take on two different forms depending upon the potentialF[] of the lattice Hamiltonian (coupling between different lattice sites not included). IfF[] is of a short-range type (see text for definition) the limit functional is Gaussian. The corresponding representation of CCR is reducible and its realization apparently non-unique unlessF[] is quadratic. The most natural realization is to represent the field as a linear combination of Fock fields whose masses are given by the excitation energies of the lattice Hamiltonian. IfF[] is of a long-range type, the limit functional takes the more general form once studied byAraki.     

Monopoles,baryon decay and charge conservation

The mechanism of baryon decay via monopoles is analyzed. For this purpose we quantize isodoublet fermion fields in the presence of a 't Hooft-Polyakov monopole. When the electromagnetic interactions are switched off, we find a condensation of a fermion pair \(\bar \psi _ - ^{(i)} \gamma ^0 \gamma ^5 \psi _ + ^{(i)} \) as well as that of \(\bar \psi _ - ^{(i)} \gamma ^0 \gamma ^5 \psi _ + ^{(i)} \bar \psi _ + ^{(j)} \gamma ^0 \gamma ^5 \psi _ - ^{(j)} \) . Here, the indices ± stand for the electric charge and (i,j) for the flavour. Hence, the charge symmetry is spontaneously broken. However, when the Coulomb interactions are switched on, it is proved that all fermion condensates carrying non-zero electric charges are removed; the condensates carrying zero electric charge, which induce baryon decay in the standardSU (5) model, are not removed by switching on the Coulomb interactions. In these analyses, the key element is the charge mixing boundary condition imposed on the fermion wave-function at the monopole center; the chiral anomaly does not play any role.