A variational scheme for the derivation of generalized, symmetry-induced continuity equations for Hermitian and non-Hermitian quantum mechanical systems is developed. We introduce a Lagrangian which involves two complex wave fields and whose global invariance under dilation and phase variations leads to a mixed continuity equation for the two fields. In combination with discrete spatial symmetries of the underlying Hamiltonian, the mixed continuity equation is shown to produce bilocal conservation laws for a single field. This leads to generalized conserved charges for vanishing boundary currents and to divergenceless bilocal currents for stationary states. The formalism reproduces the bilocal continuity equation obtained in the special case of $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics and paraxial optics.A variational scheme for the derivation of generalized, symmetry-induced continuity equations for Hermitian and non-Hermitian quantum mechanical systems is developed. We introduce a Lagrangian which involves two complex wave fields and whose global invariance under dilation and phase variations leads to a mixed continuity equation for the two fields. In combination with discrete spatial symmetries of the underlying Hamiltonian, the mixed continuity equation is shown to produce bilocal conservation laws for a single field. This leads to generalized conserved charges for vanishing boundary currents and to divergenceless bilocal currents for stationary states. The formalism reproduces the bilocal continuity equation obtained in the special case of $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics and paraxial optics.

arXiv:1607.06730Physical Review ANov 21, 20166 citations

CONSERVATION LAWS IN QUANTUM MECHANICS

L.D. LANDAU, Efrat Lifshitz

Elsevier eBooksJan 1, 19748 citations

Quadratic Gauge Transformation

Sunita Singh, Akshit Sharma

Symmetries plays a significant role in understanding the conservation laws in Quantum field theories. Here, we attempted a quadratic type dimensionless gauge transformation to achieve the invariance in QFTs. We have shown the extensive study of invariance of complex scalar, Abelian and Non- Abelian theories and established the conservation laws. We included an explicit graphical analysis to invoke the invariance. This is studied in a physical context, where different field configurations correspond to the same physical state. The necessity of the covariant derivative is studied in detail, highlighting how it ensures consistent transformation under local symmetry operations. The meaning of covariance is clarified as the preservation of the form of physical laws under transformations.Symmetries plays a significant role in understanding the conservation laws in Quantum field theories. Here, we attempted a quadratic type dimensionless gauge transformation to achieve the invariance in QFTs. We have shown the extensive study of invariance of complex scalar, Abelian and Non- Abelian theories and established the conservation laws. We included an explicit graphical analysis to invoke the invariance. This is studied in a physical context, where different field configurations correspond to the same physical state. The necessity of the covariant derivative is studied in detail, highlighting how it ensures consistent transformation under local symmetry operations. The meaning of covariance is clarified as the preservation of the form of physical laws under transformations.

hep-thCross-listedarXiv:2606.30177Jun 30, 2026

Generalized Gibbs ensembles for quantum field theories

Fabian H. L. Eßler, Giuseppe Mussardo, Miłosz Panfil

We consider the nonequilibrium dynamics in quantum field theories (QFTs). After being prepared in a density matrix that is not an eigenstate of the Hamiltonian, such systems are expected to relax locally to a stationary state. In the presence of local conservation laws, these stationary states are believed to be described by appropriate generalized Gibbs ensembles. Here we demonstrate that in order to obtain a correct description of the stationary state, it is necessary to take into account conservation laws that are not (ultra)local in the usual sense of QFTs, but fulfill a significantly weaker form of locality. We discuss the implications of our results for integrable QFTs in one spatial dimension.We consider the nonequilibrium dynamics in quantum field theories (QFTs). After being prepared in a density matrix that is not an eigenstate of the Hamiltonian, such systems are expected to relax locally to a stationary state. In the presence of local conservation laws, these stationary states are believed to be described by appropriate generalized Gibbs ensembles. Here we demonstrate that in order to obtain a correct description of the stationary state, it is necessary to take into account conservation laws that are not (ultra)local in the usual sense of QFTs, but fulfill a significantly weaker form of locality. We discuss the implications of our results for integrable QFTs in one spatial dimension.

arXiv:1411.5352Physical Review AMay 14, 2015166 citations

\mathcal{P}\mathcal{T}

-symmetric quantum mechanics and paraxial optics.