# 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

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topBender, 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

top- R. F. Streater, A. S. Wightman, PCT, (1964), Benjamin, New York Zbl0135.44305MR161603
- 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
- C. M. Bender, S. Boettcher, Real Spectra in Non-Hermitian hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246 Zbl0947.81018MR1627442
- C. M. Bender, S. Boettcher, P. N. Meisinger, PT-Symmetric Quantum Mechanics, J. Math. Phys. 40 (1999), 2201-2229 Zbl1057.81512MR1686605
- C. M. Bender, P. N. Meisinger, H. Yang, Calculation of the One-Point Green’s Function for a -$g\phantom{\rule{4pt}{0ex}}ph{i}^{4}$ Quantum Field Theory, Phys. Rev. D 63 (2001), 45001-1--45001-10
- 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
- C. M. Bender, G. V. Dunne, P. N. Meisinger, M., Quantum Complex Henon-Heiles Potentials, Phys. Lett. A 281 (2001), 311-316 Zbl0984.81042MR2046920
- 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
- C. M. Bender, D. C. Brody, H. F. Jones, Complex Extension of Quantum Mechanics, quant-ph 0208076 (2002) Zbl1267.81234MR1950305
- C. M. Bender, M. V. Berry, A. Mandilara, Generalized $\mathcal{P}\mathcal{T}$ Symmetry and Real Spectra, J. Phys. A, Math. Gen. 35 (2002), 467-471 Zbl1066.81537MR1928842
- 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
- G. S. Japaridze, Space of state vectors in PT-symmetric quantum mechanics, J. Phys. A 35 (2002), 1709-1718 Zbl1010.81019MR1891621
- F. Pham, E. Delabaere, Eigenvalues of complex Hamiltonians with PT-symmetry, Phys. Lett. A 250 (1998), 29-32 MR1742960
- 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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