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Physica Scripta

On Some novel features of a Non-Hermitian multi-boson Hamiltonian : partial P T -symmetry and ghosts

Arindam Chakraborty

Published Nov 7, 2024• 3 citations
Publisher

Abstract

Abstract The eigenvalues and eigenstates of a three-boson Hamiltonian are considered. The Hamiltonian is found to be pseudo-Hermitian in relation to its various partial PT-symmetries understood in a Fock space setting. The possibilities of symmetry adopting and symmetry breaking states are also discussed. Ghost states are found in relation to different symmetry induced inner-products. At least two of the ghost states show some kind of transition to normal states depending on the values of certain parameter.

DOI
10.1088/1402-4896/ad8fdc

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Abstract The eigenvalues and eigenstates of a three-boson Hamiltonian are considered. The Hamiltonian is found to be pseudo-Hermitian in relation to its various partial PT-symmetries understood in a Fock space setting. The possibilities of symmetry adopting and symmetry breaking states are also discussed. Ghost states are found in relation to different symmetry induced inner-products. At least two of the ghost states show some kind of transition to normal states depending on the values of certain parameter.
A γγγ-deformed version of su(2)\mathfrak{su}(2)su(2) algebra has been obtained from a bi-orthogonal system of vectors in C2\bf{C^2}C2. Fusion of Jordan-Schwinger realization of complexified su(2)\mathfrak{su}(2)su(2) with Dyson-Maleev representation gives a 3-boson realization of Higgs algebra of cubic polynomial type. The non-hermitian Hamiltonian thus obtained is found to have real eigenvalues and eigen states with symmetry induced orthogonality. The notion of partial PT{\mathcal {PT}}PT-symmetry (henceforth ∂PT\partial_{\mathcal { PT}}∂PT​) has been introduced as a characteristic feature of these multi-boson realizations. The Hamiltonian along with its eigenstates have been studied in the light of ∂PT\partial_{\mathcal { PT}}∂PT​-symmetry. The possibility of ∂PT\partial_{\mathcal { PT}}∂PT​-symmetry breaking is also discussed. The deformation parameter γγγ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration.
We show that a diagonalizable (non-Hermitian) Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an antilinear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there is a pseudo-canonical transformation of the Hilbert space that maps H into a Hermitian operator.
Abstract A γ -deformed version of s u ( 2 ) algebra with non-Hermitian generators has been obtained from a bi-orthogonal system of vectors in C 2 . The related Jordan–Schwinger map is combined with boson algebras to obtain a hierarchy of fusion polynomial algebras. This makes possible the construction of Higgs algebra of cubic polynomial type which eventually leads to several multi-boson non-Hermitian Hamiltonians. Finally, the notion of global and partial PT -symmetry have been introduced in a typical Fock space setting. The possibility of PT -symmetry breaking is also discussed. The deformation parameter γ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration. The symmetry behavior has been conceptualized in terms of a couple of toy models.
Abstract The recognition that the eigenvalues of a non-Hermitian Hamiltonian could all be real if the Hamiltonian had an antilinear symmetry such as PT stimulated new insight into the underlying structure of quantum mechanics. Specifically, it led to the realization that Hilbert space could be richer than the established Dirac approach of constructing inner products out of ket vectors and their Hermitian conjugate bra vectors. With antilinear symmetry one must instead build inner products out of ket vectors and their antilinear conjugates, and it is these inner products that would be time independent in the non-Hermitian but antilinearly symmetric case even as the standard Dirac inner products would not be. Moreover, and in a sense quite remarkably, antilinear symmetry could address not only the temporal behavior of the inner product but also the issue of its overall sign, with antilinear symmetry being capable of yielding a positive inner product in situations such as fourth-order derivative quantum field theories where the standard Dirac inner product is found to have ghostlike negative signature. Antilinear symmetry thus solves the ghost problem in such theories by showing that they are being formulated in the wrong Hilbert space, with antilinear symmetry providing a Hilbert space that is ghost free. Antilinear symmetry does not actually get rid of the ghost states. Rather, it shows that the reasoning that led one to think that ghosts were present in the first place is faulty. Implications of our results for constructing unitary quantum theories of gravity are presented.
The η pseudo PT symmetry theory, denoted by the symbol η, explores the conditions under which non-Hermitian Hamiltonians can possess real spectra despite the violation of PT symmetry, that is the adjoint of H, denoted H^{†} is expressed as H^{†}=PTHPT. This theory introduces a new symmetry operator, η=PTη, which acts on the Hilbert space. The η pseudo PT symmetry condition requires the Hamiltonian to commute with the η operator, leading to real eigenvalues. We discuss some general implications of our results for the coupled non hermitian harmonic oscillator.