Symmetries and conservation laws in non-Hermitian field theories

Abstract

Anti-Hermitian mass terms are considered, in addition to Hermitian ones, for PT -symmetric complexscalar and fermionic field theories. In both cases, the Lagrangian can be written in a manifestly symmetric form in terms of the PT -conjugate variables, allowing for an unambiguous definition of the equations of motion. After discussing the resulting constraints on the consistency of the variational procedure, we show that the invariance of a non-Hermitian Lagrangian under a continuous symmetry transformation does not imply the existence of a corresponding conserved current. Conserved currents exist, but these are associated with transformations under which the Lagrangian is not invariant and which reflect the well-known interpretation of PT -symmetric theories in terms of systems with gain and loss. A formal understanding of this unusual feature of non-Hermitian theories requires a careful treatment of Noether's theorem, and we give specific examples for illustration.

Pseudo-Hermitian approach to Goldstone’s theorem in non-Abelian non-Hermitian quantum field theories

“…There is a well known problem that seems to suggest that non-Hermitian quantum field theories are inconsistent, see e.g. [22,23,8]. However, just as for non-Hermitian quantum theories [24,25,26,27] there are methods and techniques to overcome these issues to obtain a perfectly consistent theory.…”

PT -symmetric non-Hermitian quantum field theories with supersymmetry

“…We have shown that the free-field supersymmetric Dirac model is equivalent via a similarity transformation to a Hermitian supersymmetric model, but we have found that there is no such equivalence in the general Majorana case. As we have described in both models, there is an ambiguity in the definition of the supercurrent, and we have discussed the (non-)invariance of the Lagrangian and the (non-)conservation of the Noether current, which are analogous to the corresponding properties of PT -symmetric models with purely bosonic symmetries [20]. We have also extended the Dirac model to include Hermitian trilinear superpotential interactions, 5 in which case the model contains non-Hermitian trilinear bosonic interactions as well as non-Hermitian bilinear terms, whereas the dimension-4 interactions are Hermitian.…”

“…The latter is, however, not unexpected, since we know that conserved currents are not related to transformations that leave the Lagrangian invariant in the case of non-Hermitian theories, see Ref. [20]. 2 We have seen already that there is a four-fold freedom in choosing the on-shell condition for the auxiliary fields F a .…”

“…However, as first identified in Ref. [20], we are, in fact, free to choose any one of the Euler-Lagrange equations to define the equations of motion; each choice leads to the same physics. In the present case, we can readily convince ourselves that any choice leads to the same Lagrangian for the remaining scalar and fermionic fields:…”

“…It has also been suggested that non-Hermitian quantum field theory might also have applications in fundamental physics, e.g., to neutrino physics [8][9][10], dark matter [11] Higgs decays [12] and particle mixing [13]. It has been shown that it is possible to carry over to PT -symmetric non-Hermitian theories familiar concepts from Hermitian quantum field theory such as the spontaneous breaking of global symmetries [14][15][16] and the Englert-Brout-Higgs mechanism in gauge theories [17][18][19], despite the appearance of subtleties [20,21] in the relationship between current conservation, Lagrangian symmetries and Noether's theorem [22] in PT -symmetric theories.…”