# Must a Hamiltonian be Hermitian

@article{Bender2003MustAH, title={Must a Hamiltonian be Hermitian}, author={Carl M Bender and Dorje C. Brody and H F Jones}, journal={American Journal of Physics}, year={2003}, volume={71}, pages={1095-1102} }

A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space–time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data. One would think that a quantum theory based on a non-Hermitian Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is possible to use a previously unnoticed…

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## References

SHOWING 1-10 OF 51 REFERENCES

### PT-symmetric quantum mechanics

- Physics
- 1999

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…

### Model of supersymmetric quantum field theory with broken parity symmetry

- Physics
- 1998

Recently, it was observed that self-interacting scalar quantum field theories having a non-Hermitian interaction term of the form g(i�) 2+δ , whereis a real positive parameter, are physically…

### Calculation of the hidden symmetry operator in -symmetric quantum mechanics

- Mathematics
- 2002

In a recent paper it was shown that if a Hamiltonian H has an unbroken symmetry, then it also possesses a hidden symmetry represented by the linear operator . The operator commutes with both H and .…

### Solution of Schwinger-Dyson equations for PT symmetric quantum field theory

- Physics
- 2000

In recent papers it has been observed that non-Hermitian Hamiltonians, such as those describing ig� 3 and −g� 4 field theories, still possess real positive spectra so long as the weaker condition of…

### Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian

- Physics
- 2001

We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in…

### Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry

- Mathematics
- 1998

The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of $\mathrm{PT}$ symmetry, one obtains new…

### A nonunitary version of massless quantum electrodynamics possessing a critical point

- Physics
- 1999

Recently, it has been observed that a quantum field theory need not be hermitian to have a real, positive spectrum. What seems to be required is symmetry under combined parity and time-reversal…