Ordinary versus PT-symmetric ${\varphi }^3$ quantum field theory

Abstract

A quantum-mechanical theory is PT-symmetric if it is described by a Hamiltonian that commutes with PT, where the operator P performs space reflection and the operator T performs time reversal. A PT-symmetric Hamiltonian often has a parametric region of unbroken PT symmetry in which the energy eigenvalues are all real. There may also be a region of broken PT symmetry in which some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a PT-symmetric $ig{\varphi }^3$ quantum field theory. This quantum field theory is the analog of the PT-symmetric quantum-mechanical theory described by the Hamiltonian $H=p^2+i{x}^3$, whose eigenvalues have been rigorously shown to be all real. This paper compares the renormalization-group properties of a conventional Hermitian $g{\varphi }^3$ quantum field theory with those of the PT-symmetric $ig{\varphi }^3$ quantum field theory. It is shown that while the conventional $g{\varphi }^3$ theory in $d=6$ dimensions is asymptotically free, the $ig{\varphi }^3$ theory is like a $g{\varphi }^4$ theory in $d=4$ dimensions; it is energetically stable, perturbatively renormalizable, and trivial.

13 pages, 2 figures

References in corpus (7)

• Making Sense of Non-Hermitian Hamiltonians
• Unidirectional Invisibility induced by PT-Symmetric Periodic Structures
• PT-symmetry breaking and laser-absorber modes in optical scattering systems
• Experimental Study of Active LRC Circuits with PT-Symmetries
• PT symmetry and spontaneous symmetry breaking in a microwave billiard
• The ODE/IM Correspondence
• Bifurcation diagram and pattern formation in superconducting wires with electric currents

Cited by in corpus (23)

• Hermitian four-well potential as a realization of a PT-symmetric system
• Symmetries and conservation laws in non-Hermitian field theories
• $\mathcal{P}\mathcal{T}$-symmetric wave guide system with evidence of a third-order exceptional point
• Lee-Yang model from the functional renormalization group
• Coupling approach for the realization of a $\mathcal{P}\mathcal{T}$-symmetric potential for a Bose-Einstein condensate in a double well
• Discrete spacetime symmetries and particle mixing in non-Hermitian scalar quantum field theories
• Critical behavior of the PT-symmetric $i{\varphi }^3$ quantum field theory
• Vortices in Bose-Einstein condensates with $\mathcal{P}\mathcal{T}$-symmetric gain and loss
• Nonlinear Non-Hermitian Landau-Zener-Stückelberg-Majorana interferometry
• Direct Higgs-gravity interaction and stability of our Universe
• Poincaré symmetries and representations in pseudo-Hermitian quantum field theory
• Non-Hermitian Yukawa interactions of fermions with axions: potential microscopic origin and dynamical mass generation
• Simple models of three coupled $\mathcal{P}\mathcal{T}$-symmetric wave guides allowing for third-order exceptional points
• RG flows on $S^d$ and Hamiltonian truncation
• Universal Large-order asymptotic behavior of the Strong-coupling and High-Temperature series expansions
• Original study of the $g{\varphi }^2(i\varphi {)}^{\epsilon }$ theory. Analysis of all orders in $\epsilon$ and resummations
• RG and logarithmic CFT multicritical properties of randomly diluted Ising models
• Soft quantum vibrations of PT-symmetric nonlinear ion chain
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Purity oscillations in coupled Bose-Einstein condensates
• Boundary anomalous dimensions from BCFT: O($N$)-symmetric ${\varphi }^{2n}$ theories with a boundary and higher-derivative generalizations
• Non-linear and weak-coupling expansion in Quantum Field Theory
• Computing eigenfunctions and eigenvalues of boundary value problems with the orthogonal spectral renormalization method