Properties of non-hermitian quantum field theories

Bender, Carl M.

doi:10.5802/aif.1971

Properties of non-hermitian quantum field theories
[Propriétés des théories des champs quantiques non hermitiens]

Bender, Carl M.¹

¹ Washington University, Department of Physics, Campus Box 1105, St. Louis, MO 63130 (USA)

Annales de l'Institut Fourier, Tome 53 (2003) no. 4, pp. 997-1008.

Résumé
Abstract

Dans cet article, je traite des systèmes quantiques dont l’hamiltonien est non-hermitien mais dont les niveaux d’énergie sont tous réels et positifs. De telles théories doivent être symétriques sous $𝒞 𝒫 𝒯$ , mais pas sous $𝒫$ et $𝒯$ séparément. Récemment, des systèmes quantiques avec de telles propriétés ont été étudiés en détail. Dans cet article, j’étends ces résultats aux théories des champs quantiques. Parmi les systèmes dont je parle, se trouvent les théories $- φ^{4}$ et $i φ^{3}$ . Toutes ces théories ont des propriétés inattendues et remarquables. Je décris les fonctions de Green qui apparaissent dans ces théories et je présente de nouveaux résultats concernant les états liés, la renormalisation et les calculs non-perturbatifs.

In this paper I discuss quantum systems whose Hamiltonians are non-Hermitian but whose energy levels are all real and positive. Such theories are required to be symmetric under $𝒞 𝒫 𝒯$ , but not symmetric under $𝒫$ and $𝒯$ separately. Recently, quantum mechanical systems having such properties have been investigated in detail. In this paper I extend the results to quantum field theories. Among the systems that I discuss are $- φ^{4}$ and $i φ^{3}$ theories. These theories all have unexpected and remarkable properties. I discuss the Green’s functions for these theories and present new results regarding bound states, renormalization, and nonperturbative calculations.

MR Zbl

DOI : 10.5802/aif.1971

Classification : 34M60, 34B24, 34B40, 34L10
Keywords: ${\mathcal {C}}{\mathcal {P}}{\mathcal {T}}$, non-hermitian
Mot clés : ${\mathcal {C}}{\mathcal {P}}{\mathcal {T}}$, non-hermitien

Affiliations des auteurs :

Bender, Carl M. ¹

¹ Washington University, Department of Physics, Campus Box 1105, St. Louis, MO 63130 (USA)

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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
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     year = {2003},
     doi = {10.5802/aif.1971},
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Bender, Carl M. Properties of non-hermitian quantum field theories. Annales de l'Institut Fourier, Tome 53 (2003) no. 4, pp. 997-1008. doi : 10.5802/aif.1971. http://www.numdam.org/articles/10.5802/aif.1971/

Bibliographie
Cité par

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[4] C. M. Bender; S. Boettcher; P. N. Meisinger PT-Symmetric Quantum Mechanics, J. Math. Phys., Volume 40 (1999), pp. 2201-2229 | DOI | MR | Zbl

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[5] K. C. Shin On the reality of the eigenvalues for a class of PT-symmetric oscillators, Comm. Math. Phys., Volume 229 (2002), pp. 543-564 | DOI | MR | Zbl

[6] C. M. Bender; P. N. Meisinger; H. Yang Calculation of the One-Point Green's Function for a - $g p h i^{4}$ Quantum Field Theory, Phys. Rev. D, Volume 63 (2001), p. 45001-1--45001-10 | DOI

[7] C. M. Bender; S. Boettcher; H. F. Jones; P. N. Meisinger; M. \dSim\dsek Bound States of Non-Hermitian Quantum Field Theories, Phys. Lett. A, Volume 291 (2001), pp. 197-202 | DOI | MR | Zbl

[9] C. M. Bender; S. Boettcher; P. N. Meisinger; Q. Wang Two-Point Green's Function in PT-Symmetric Theories, Phys. Lett. A, Volume 302 (2002), pp. 286-290 | DOI | MR | Zbl

[10] C. M. Bender; D. C. Brody; H. F. Jones Complex Extension of Quantum Mechanics, Volume quant-ph 0208076 (2002) | MR

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[11] G. S. Japaridze Space of state vectors in PT-symmetric quantum mechanics, J. Phys. A, Volume 35 (2002), pp. 1709-1718 | DOI | MR | Zbl

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