Article

# Cryptoreality of nonanticommutative Hamiltonians

University of Nantes, Naoned, Pays de la Loire, France

Journal of High Energy Physics (Impact Factor: 6.11). 03/2007; 2007(7). DOI: 10.1088/1126-6708/2007/07/036 Source: arXiv

**ABSTRACT**

We note that, though nonanticommutative (NAC) deformations of Minkowski supersymmetric

theories do not respect the reality condition and seem to lead to non-Hermitian Hamiltonians H, the

latter belong to the class of ``cryptoreal'' Hamiltonians considered recently by Bender and

collaborators. They can be made manifestly Hermitian via the similarity transformation

H???eRHe?R with a properly chosen R. The deformed model enjoys the same supersymmetry

algebra as the undeformed one, though being realized differently on the involved canonical variables.

Besides quantum-mechanical models, we treat, along similar lines, some NAC deformed field models in

4D Minkowski space.

theories do not respect the reality condition and seem to lead to non-Hermitian Hamiltonians H, the

latter belong to the class of ``cryptoreal'' Hamiltonians considered recently by Bender and

collaborators. They can be made manifestly Hermitian via the similarity transformation

H???eRHe?R with a properly chosen R. The deformed model enjoys the same supersymmetry

algebra as the undeformed one, though being realized differently on the involved canonical variables.

Besides quantum-mechanical models, we treat, along similar lines, some NAC deformed field models in

4D Minkowski space.

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**ABSTRACT:**The $$ \mathcal{N} $$-extended Supersymmetric Quantum Mechanics is deformed via an abelian twist which preserves the super-Hopf algebra structure of its Universal Enveloping Superalgebra. Two constructions are possible. For even $$ \mathcal{N} $$ one can identify the 1D $$ \mathcal{N} $$-extended superalgebra with the fermionic Heisenberg algebra. Alternatively, supersymmetry generators can be realized as operators belonging to the Universal Enveloping Superalgebra of one bosonic and several fermionic oscillators. The deformed system is described in terms of twisted operators satisfying twist-deformed (anti)commutators. The main differences between an abelian twist defined in terms of fermionic operators and an abelian twist defined in terms of bosonic operators are discussed. -
##### Article: Supersymmetry vs ghosts

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**ABSTRACT:**Final version to be published in JMP (April, 2008) - [Show abstract] [Hide abstract]

**ABSTRACT:**The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.