Embedding, simulation and consistency of $\mathcal{P}\mathcal{T}$ -symmetric quantum Theory

Abstract

A physical requirement on the Hamiltonian operator in quantum mechanics is that it must generate real energy spectrum and unitary time evolution. While the Hamiltonians are Dirac Hermitian in conventional quantum mechanics, they observe $\mathcal{P}\mathcal{T}$-symmetry in $\mathcal{P}\mathcal{T}$-symmetric quantum theory. The embedding property was first studied by Günther and Samsonov to visualize the evolution of unbroken $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians on ${\mathbb{C}}^2$ by Hermitian Hamiltonians on ${\mathbb{C}}^4$. This paper investigates the properties of $\mathcal{P}\mathcal{T}$-symmetric quantum systems including the embedding property. We provide a full characterization of the embedding property in the general case and show that only unbroken $\mathcal{P}\mathcal{T}$-symmetric quantum systems admit this property in a finite dimensional space. Furthermore, utilizing this property, we establish a physically realizable simulation process of the unbroken $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians. An observation that the unbroken $\mathcal{P}\mathcal{T}$-symmetric quantum systems can be viewed as open systems in the conventional quantum mechanics accounts for the consistency of $\mathcal{P}\mathcal{T}$-symmetric quantum theory.

14 pages, close to published version

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Cited by in corpus (10)

• Parity-Time Symmetry in Non-Hermitian Complex Optical Media
• Time-dependent $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics in generic non-Hermitian systems
• Simulating broken $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian systems by weak measurement
• Observing a Changing Hilbert-Space Inner Product
• Solvable dilation model of $\mathcal{P}\mathcal{T}$-symmetric systems
• Simulating non-Hermitian dynamics of a multi-spin quantum system and an emergent central spin model
• Internal nonlocality in generally dilated Hermiticity
• Extracting the internal nonlocality from the dilated Hermiticity
• Embedding of a non-Hermitian Hamiltonian to emulate the von Neumann measurement scheme
• To quantify the difference of $\eta$-inner products in $\mathcal{P}\mathcal{T}$-symmetric theory