Biorthogonal-Eigenstate Framework

• Biorthogonal-eigenstate framework is a quantum model that replaces conventional orthonormality with a dual set of nonorthogonal states, ensuring real eigenvalues.
• The framework revises probability assignment and measurement by generalizing the Born rule using biorthogonal inner products.
• It extends standard quantum mechanics with spectral decompositions and time evolution that preserve biorthogonal norms, offering insights for non-Hermitian systems.

A biorthogonal-eigenstate-based framework is a generalization of the quantum mechanical formalism that relaxes the Hermiticity condition traditionally imposed on observables and generators of dynamics. Instead of requiring operators to be Hermitian with respect to the canonical Dirac inner product (ensuring orthogonality and real spectral values), this framework admits a broader class of operators with real eigenvalues and complete, but generally nonorthogonal, eigenstate sets. Biorthogonality, the pivotal structural property, replaces Dirac orthogonality and solves foundational and practical issues associated with observables, probabilities, dynamics, and quantum measurements in non-Hermitian systems.

1. Biorthogonality: Structure and Mathematical Foundations

The central idea in the biorthogonal-eigenstate-based approach is the replacement of orthonormality with biorthogonality between two sets of eigenstates—$\{|\phi_n)\}$ for the (not necessarily Hermitian) operator $K$ and $\{|\chi_n)\}$ for its adjoint $K^\dagger$. These states satisfy

$(\chi_n|\phi_m) = \delta_{nm} \tag{1}$

and completeness is guaranteed by the resolution of the identity

$\sum_n |\phi_n)(\chi_n| = 1. \tag{2}$

This constitutes a pairing between the Hilbert space $\mathcal{H}$ and its dual $\mathcal{H}^*$. Notably, even if $\{|\phi_n)\}$ are nonorthogonal in $\mathcal{H}$, together with $\{|\chi_n)\}$ they provide a basis with maximal separation in projective Hilbert space. The biorthogonal structure ensures that the core quantum-theoretic notions—state separation, expansion, and projection—remain unambiguously defined even without self-adjointness of the underlying operators.

In this finite-dimensional setting, the spectral (projector) decomposition of the operator generalizes to

$K = \sum_n \kappa_n |\phi_n)(\chi_n|, \tag{8}$

where $\kappa_n$ are (real) eigenvalues, with $\kappa_n \in \mathbb{R}$ being a necessary condition for the physical viability of the framework.

2. Probability Assignment and Measurement

In standard quantum mechanics, the probability for outcome $n$ is $|(\psi|n)|^2$ if $|\psi)$ is normalized and $|n)$ is a normalized eigenstate. When the basis is nonorthogonal, this is replaced by a rule involving both basis sets. For any state $|\psi)$ expanded as $|\psi) = \sum_n C_n |\phi_n)$, the associated "dual" state is

$|\tilde{\psi}) = \sum_n C_n |\chi_n). \tag{3}$

Transition probabilities are assigned by

$$p_n = |C_n|^2 = (\chi_n|\psi)(\psi|\phi_n)/(\psi|\psi), \tag{5, 5a}$$

with the norm

$(\psi|\psi) = \sum_n |C_n|^2. \tag{4}$

This form, involving both $|\psi)$ and $|\tilde{\psi})$, ensures that probabilities are positive and sum to unity, generalizing the Born rule to the biorthogonal context. Importantly, measurement projection operators are defined as

$\Pi_n = |\phi_n)(\chi_n|, \tag{10}$

and the completeness property $\sum_n \Pi_n = 1$ is retained, but experimental outcomes now reflect only the projective (ray) structure in $\mathcal{H}$, not the nonorthogonality in the underlying vector space.

3. Observables, Expectation Values, and Operator Structure

Observables $F$ are treated analogously: matrix elements in the biorthogonal basis are

$f_{nm} = (\chi_n|F|\phi_m), \tag{6}$

and biorthogonal "Hermiticity" is enforced by $f_{nm} = f_{mn}$. The expectation value for an observable in state $|\psi)$ is

$$\langle F \rangle = (\psi|F|\psi) = \sum_{n,m} C_n^* C_m f_{nm}. \tag{7}$$

This ensures that physical observables, while not Hermitian under the canonical Dirac product, yield real expectation values and remain consistent with quantum measurement principles so long as the operator is Hermitian with respect to the biorthogonal pairing.

Spectral decomposition is written directly in terms of biorthogonal projectors (Eq. 8), mirroring the familiar structure in conventional quantum mechanics but governed by the dual basis as well.

4. State Characterization: Pure, Mixed, and Composite Systems

Pure states are superpositions in the biorthogonal basis, $|\psi) = \sum_n C_n |\phi_n)$, normalized such that $(\psi|\psi) = 1$. Mixed states are described by biorthogonal density matrices of the form

$\rho = \sum_{n,m} p_{nm} |\phi_n)(\chi_m|, \tag{9}$

with $p_{nm} = p_{mn}$ to ensure "Hermiticity" in the biorthogonal sense and $\operatorname{tr} \rho = 1$.

In the case of spin systems (including spin-1/2 and spin-1), even when the biorthogonal eigenstates are nonorthogonal in $\mathcal{H}$, their mapping to projective (ray) space, or the Bloch sphere $S^2 \subset \mathbb{R}^3$, remains unique. This Pauli correspondence provides a direct physical interpretation of spin states: although Hilbert-space vectors may not be orthogonal, the associated points on the Bloch sphere are antipodal, maintaining the integrity of angular momentum representations.

Composite systems are treated via tensor products, $|\psi,\phi) = |\psi)\otimes|\phi)$, with entangled states, such as the singlet, exhibiting structure

$$|\Psi_0) \propto |\phi_1)\otimes|\chi_2) - |\chi_1)\otimes|\phi_2), \tag{11}$$

ensuring correct antisymmetry and projective equivalence of physically identical (but biorthogonally distinct) vectors.

5. Dynamics, Perturbations, and Time Evolution

Small perturbations are addressed via a biorthogonal modification of Rayleigh–Schrödinger theory. For $K(\epsilon) = K + \epsilon K'$, the first-order eigenvalue correction is

$$\epsilon \kappa_n^{(1)} = (\chi_n|K'|\phi_n). \tag{13}$$

Orthogonality conditions are replaced by the biorthogonality constraint; i.e., state corrections are chosen such that $(\chi_n|\phi_n^{(1)}) = 0$. The biorthogonal norm $(\chi|\phi)$ is preserved under perturbation, even as the standard Dirac norm may not be.

Time evolution is generated by

$\widehat{U}(t) = \exp(-i K t), \tag{14}$

which is not unitary unless $K$ is Hermitian. The key result is that for operators with real eigenvalues, the biorthogonal norm is preserved: $(\chi(t)|\phi(t)) = \text{constant}, \tag{15}$ ensuring conservation of probability (in the biorthogonal sense) despite general nonunitarity of the evolution operator.

6. Measurement, Entanglement, and Physical Interpretation

Measurement theory in the biorthogonal framework uses the projectors $\Pi_n$ defined above and depends crucially on the projective geometry of the ray space. Experimentally measurable quantities, such as "distances" between quantum states or overlaps, depend only on the biorthogonal pairing, not on the inner products within the nonorthogonal basis of $\mathcal{H}$. As a consequence, distinct $|\phi_n)$ (though nonorthogonal) are maximally distinguishable in the ray sense, ensuring that the probabilistic structure of the theory is compatible with observed outcomes.

Entanglement in composite biorthogonal systems is formulated such that antisymmetry and quantum indistinguishability are respected at the projective level, with the biorthogonal partners ensuring the correct transformation properties under particle exchange.

7. Limitations in Infinite Dimensions

While the biorthogonal-eigenstate-based framework is rigorous and complete in finite-dimensional Hilbert spaces, extending the formalism to infinite dimensions introduces additional challenges. Biorthogonal completeness (i.e., the existence of a complete dual basis) may fail, the metric operator mapping between $\mathcal{H}$ and its dual can be unbounded, and completeness of the biorthogonal resolution of the identity may only hold in a weak or distributional sense. An example provided in the source highlights the existence of states with vanishing biorthogonal norm in infinite-dimensional settings, which can lead to pathologies absent in finite-dimensional cases. These limitations underline the necessity of careful mathematical analysis when generalizing biorthogonal quantum mechanics to infinite-dimensional systems, such as those occurring in field theory or quantum optics.

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The biorthogonal-eigenstate-based framework thus generalizes the axioms of quantum mechanics by replacing Hermiticity with the weaker requirement of real spectra and completeness under a biorthogonal basis, introducing new probability rules, and redefining observables and dynamics in a way that remains fully consistent with experimental phenomena, at least for finite-dimensional systems. This framework clarifies the position of PT-symmetric quantum mechanics as a subcase and provides the tools to systematically treat non-Hermitian systems with real spectra, including mixed and entangled states, spin mappings, and dynamical evolution, with prospects for applications in quantum information, open system dynamics, and beyond (Brody, 2013).