Biorthogonal quantum mechanics

By Dorje C. Brody. 2013

575 citations recorded by the source corpus.

The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called ‘biorthogonal quantum mechanics’, is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/be4b8d4be8a280ab86d92e232aa95876d463a522

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems. (2018, background): “To address this situation one can choose the normalization 〈ΨL,i|ΨR,j〉 = δi,j to obtain a biorthogonal set [33], which is complete away from EPs, such that N ∗ LNR = (r∗ LrR)(r LrR − 1)/[(r∗ LrR) − 1] for the states described in Eqs.”
Non-Hermitian systems and topology: A transfer-matrix perspective (2018, unknown): “expectation value [97] of an observable O as 〈φ|O|ψ〉.”
Anomalous helical edge states in a non-Hermitian Chern insulator (2018, unknown): “Here En (k) is a complex eigenenergy and |φn (k)〉 (|χn (k)〉) is the corresponding right (left) eigenstate [130, 131], which satisfy”
Symmetry-protected nodal phases in non-Hermitian systems (2018, unknown): “[51] D.”
Exceptional topology of non-Hermitian systems (2019, unknown): “To discuss the bio-rthogonal BBC in a self-contained manner, we recall basic elements of biorthogonal quantum mechanics (QM); see Brody (2014) for a pedagogical review.”
Topological acoustics (2022, background): “are usually characterized by topological invariants defined by wavefunctions under a biorthogonal basis [177].”
Symmetry and Topology in Non-Hermitian Physics (2018, background): “Non-Hermitian matrices exhibit unconventional characteristics compared with Hermitian ones [35]: eigenstates are, in general, nonorthogonal [36], and a complex-energy spectrum can possess exceptional points [37,38].”
Non-Bloch band theory of non-Hermitian Hamiltonians in the symplectic class (2020, background): “This indicates that |φ〉 and T |χ〉, which belong to the same eigenenergy, are biorthogonal [76] to each other and linearly independent of each other.”
Geometry and Response of Lindbladians (2015, background): “[144] D. Viennot, A. Leclerc, G. Jolicard, and J. P. Killingbeck, Consistency between Adiabatic and Non-Adiabatic Geometric Phases for Non-Self-Adjoint Hamiltonians, J. Phys.”