PT -Symmetric Quantum Mechanics

March 2, 2023
5:00pm to 6:00pm

Blue Room

Specialist level
Alexander Bernal González

Blue Room


Alexander Bernal González will present the following paper: 'PT -Symmetric Quantum Mechanics' (J.Math.Phys. 40 (1999) 2201-2229)

This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H‡=H, where ‡ represents combined parity reflection and time reversal PT, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H=p2+x2(ix)ϵ of the harmonic oscillator Hamiltonian, where ϵ is a real parameter. The system exhibits two phases: When ϵ≥0, the energy spectrum of H is real and positive as a consequence of PT symmetry. However, when −1<ϵ<0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken. The phase transition that occurs at ϵ=0 manifests itself in both the quantum-mechanical system and the underlying classical system. Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H=p2+x2N(ix)ϵ with N integer and ϵ>−N; each of these complex Hamiltonians exhibits a phase transition at ϵ=0. These PT-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.