Properties of non-hermitian quantum field theories

Carl M. Bender[1]

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

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 4, page 997-1008
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Bender, Carl M.. "Properties of non-hermitian quantum field theories." Annales de l’institut Fourier 53.4 (2003): 997-1008. <http://eudml.org/doc/116072>.

@article{Bender2003,
abstract = {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 $\{\mathcal \{C\}\}\{\mathcal \{P\}\}\{\mathcal \{T\}\}$, but not symmetric under $\{\mathcal \{P\}\}$ and $\{\mathcal \{T\}\}$ 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 $-\phi ^4$ and $i\phi ^3$ 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.},
affiliation = {Washington University, Department of Physics, Campus Box 1105, St. Louis, MO 63130 (USA)},
author = {Bender, Carl M.},
journal = {Annales de l’institut Fourier},
keywords = {$\{\mathcal \{C\}\}\{\mathcal \{P\}\}\{\mathcal \{T\}\}$; non-hermitian; CPT},
language = {eng},
number = {4},
pages = {997-1008},
publisher = {Association des Annales de l'Institut Fourier},
title = {Properties of non-hermitian quantum field theories},
url = {http://eudml.org/doc/116072},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Bender, Carl M.
TI - Properties of non-hermitian quantum field theories
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 4
SP - 997
EP - 1008
AB - 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 ${\mathcal {C}}{\mathcal {P}}{\mathcal {T}}$, but not symmetric under ${\mathcal {P}}$ and ${\mathcal {T}}$ 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 $-\phi ^4$ and $i\phi ^3$ 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.
LA - eng
KW - ${\mathcal {C}}{\mathcal {P}}{\mathcal {T}}$; non-hermitian; CPT
UR - http://eudml.org/doc/116072
ER -

References

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  1. R. F. Streater, A. S. Wightman, PCT, (1964), Benjamin, New York Zbl0135.44305MR161603
  2. C. M. Bender, K. A. Milton, S. S. Pinsky, L. M. Simmons Jr., A New Perturbative Approach to Nonlinear Problems, J. Math. Phys 30 (1989), 1447-1455 Zbl0684.34008MR1002247
  3. C. M. Bender, S. Boettcher, Real Spectra in Non-Hermitian hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246 Zbl0947.81018MR1627442
  4. C. M. Bender, S. Boettcher, P. N. Meisinger, PT-Symmetric Quantum Mechanics, J. Math. Phys. 40 (1999), 2201-2229 Zbl1057.81512MR1686605
  5. 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 63 (2001), 45001-1--45001-10 
  6. C. M. Bender, S. Boettcher, H. F. Jones, P. N. Meisinger, M., Bound States of Non-Hermitian Quantum Field Theories, Phys. Lett. A 291 (2001), 197-202 Zbl0983.81045MR1876901
  7. C. M. Bender, G. V. Dunne, P. N. Meisinger, M., Quantum Complex Henon-Heiles Potentials, Phys. Lett. A 281 (2001), 311-316 Zbl0984.81042MR2046920
  8. C. M. Bender, S. Boettcher, P. N. Meisinger, Q. Wang, Two-Point Green's Function in PT-Symmetric Theories, Phys. Lett. A 302 (2002), 286-290 Zbl0998.81023MR1958668
  9. C. M. Bender, D. C. Brody, H. F. Jones, Complex Extension of Quantum Mechanics, quant-ph 0208076 (2002) Zbl1267.81234MR1950305
  10. C. M. Bender, M. V. Berry, A. Mandilara, Generalized 𝒫 𝒯 Symmetry and Real Spectra, J. Phys. A, Math. Gen. 35 (2002), 467-471 Zbl1066.81537MR1928842
  11. P. Dorey, C. Dunning, R. Tateo, The ODE/IM correspondence PT-symmetric quantum mechanics, J. Phys. A, Math. Gen. 34 (2002), 391-400 and 5679--5704 Zbl1002.82011MR1857169
  12. G. S. Japaridze, Space of state vectors in PT-symmetric quantum mechanics, J. Phys. A 35 (2002), 1709-1718 Zbl1010.81019MR1891621
  13. F. Pham, E. Delabaere, Eigenvalues of complex Hamiltonians with PT-symmetry, Phys. Lett. A 250 (1998), 29-32 MR1742960
  14. K. C. Shin, On the reality of the eigenvalues for a class of PT-symmetric oscillators, Comm. Math. Phys. 229 (2002), 543-564 Zbl1017.34083MR1924367

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