Phases of quartic scalar theories and PT symmetry

Leqian Chen; Sarben Sarkar

doi:10.1103/PhysRevD.111.025001

Phases of quartic scalar theories and PT symmetry

Leqian Chen, Sarben Sarkar

King’s College London

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Abstract

For quantum mechanical anharmonic oscillator-type Hamiltonians, it is shown that there is a relation between the energy eigenvalues of parity symmetric and PT-symmetric phases for weak coupling. The possibility of such a relation was conjectured by Ai, Bender, and Sarkar on examining the imaginary part of the ground state energy using path integrals. In the weak coupling limit, we show that the conjecture is true also for the real part of the ground state energy and of the excited state energies. However, the conjecture is false for strong coupling. The analogous relation for partition functions in zero spacetime dimensions is valid for many cases. However O(N) symmetric multicomponent scalar fields, with N>1 and a quartic interaction, do not satisfy the conjecture for zero and one dimensional spacetime. The possibility that the conjecture is valid, for a single component field theory in higher dimensional spacetimes, is discussed in a simplified model.

Original language	English
Article number	025001
Journal	Physical Review D
Volume	111
Issue number	2
DOIs	https://doi.org/10.1103/PhysRevD.111.025001
Publication status	Published - 3 Jan 2025

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Phases of quartic scalar_CHEN_2025_GOLD VoR (CC BY)Final published version, 378 KBLicence: CC BY

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abstract = "For quantum mechanical anharmonic oscillator-type Hamiltonians, it is shown that there is a relation between the energy eigenvalues of parity symmetric and PT-symmetric phases for weak coupling. The possibility of such a relation was conjectured by Ai, Bender, and Sarkar on examining the imaginary part of the ground state energy using path integrals. In the weak coupling limit, we show that the conjecture is true also for the real part of the ground state energy and of the excited state energies. However, the conjecture is false for strong coupling. The analogous relation for partition functions in zero spacetime dimensions is valid for many cases. However O(N) symmetric multicomponent scalar fields, with N>1 and a quartic interaction, do not satisfy the conjecture for zero and one dimensional spacetime. The possibility that the conjecture is valid, for a single component field theory in higher dimensional spacetimes, is discussed in a simplified model.",

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N2 - For quantum mechanical anharmonic oscillator-type Hamiltonians, it is shown that there is a relation between the energy eigenvalues of parity symmetric and PT-symmetric phases for weak coupling. The possibility of such a relation was conjectured by Ai, Bender, and Sarkar on examining the imaginary part of the ground state energy using path integrals. In the weak coupling limit, we show that the conjecture is true also for the real part of the ground state energy and of the excited state energies. However, the conjecture is false for strong coupling. The analogous relation for partition functions in zero spacetime dimensions is valid for many cases. However O(N) symmetric multicomponent scalar fields, with N>1 and a quartic interaction, do not satisfy the conjecture for zero and one dimensional spacetime. The possibility that the conjecture is valid, for a single component field theory in higher dimensional spacetimes, is discussed in a simplified model.

AB - For quantum mechanical anharmonic oscillator-type Hamiltonians, it is shown that there is a relation between the energy eigenvalues of parity symmetric and PT-symmetric phases for weak coupling. The possibility of such a relation was conjectured by Ai, Bender, and Sarkar on examining the imaginary part of the ground state energy using path integrals. In the weak coupling limit, we show that the conjecture is true also for the real part of the ground state energy and of the excited state energies. However, the conjecture is false for strong coupling. The analogous relation for partition functions in zero spacetime dimensions is valid for many cases. However O(N) symmetric multicomponent scalar fields, with N>1 and a quartic interaction, do not satisfy the conjecture for zero and one dimensional spacetime. The possibility that the conjecture is valid, for a single component field theory in higher dimensional spacetimes, is discussed in a simplified model.

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Phases of quartic scalar theories and PT symmetry

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