Krein Space Quantization

• Krein space quantization is a framework where the state space has an indefinite Hermitian inner product, enabling built-in regularization and symmetry preservation.
• It employs a metric structure that allows pseudo-Hermiticity and pseudo-unitary evolution, thereby canceling divergences in loop integrals without extra counterterms.
• Applications include gauge theories in curved spacetime, resolution of the Klein paradox, and innovative quantum algorithms using indefinite metrics.

Krein space quantization is a non-standard quantization formalism in mathematical physics in which the quantum field theory (QFT) state space is constructed as a vector space endowed with an indefinite, nondegenerate Hermitian inner product, rather than a strictly positive-definite one. This framework fundamentally extends the Hilbert space approach and enables novel regularization and symmetry considerations in quantum field theory, particularly in the presence of gauge symmetries, non-Hermitian Hamiltonians, and curved backgrounds.

1. Definition of Krein and Pontryagin Spaces

A complex Krein space $\mathbb{K}$ is a vector space equipped with an indefinite Hermitian inner product

$[\psi, \phi] := (\psi, \eta\phi)$

where $(\cdot, \cdot)$ is the ordinary positive-definite inner product on a “friendly” Hilbert space $\mathcal{H}^{(F)}$, and $\eta = \eta^\dagger$ is a bounded, invertible, self-adjoint operator whose spectrum consists of $N_+$ eigenvalues $+1$ and $N_-$ eigenvalues $-1$ (in finite dimensions),

$$\eta = P = \mathrm{diag}(+\underbrace{1, \ldots, 1}_{N_+}, -\underbrace{1, \ldots, 1}_{N_-})\,,$$

so that $\mathbb{K}$ decomposes as $K = K_+ \oplus K_-$ into orthogonal positive/negative norm subspaces.

A Pontryagin space $\tilde{\mathbb{K}}$ is a Krein space with the additional property that one of $N_+$ or $N_-$ is finite (in infinite dimension this means only finitely many “wrong-sign” directions), so that indefinite metrics of finite signature can be implemented within infinite-dimensional settings (Znojil, 2011).

2. Self-Adjointness, Pseudo-Hermiticity, and Spectral Structure

Within Krein or Pontryagin space, a non-Hermitian Hamiltonian $H$ in the auxiliary Hilbert space $\mathcal{H}^{(F)}$ is required to obey the $\eta$–self-adjointness (“pseudo-Hermiticity”) condition

$H^\dagger \eta = \eta H,$

which ensures $H$ is self-adjoint with respect to $[\cdot, \cdot]$. Equivalently,

$H = H^\ddagger := \eta^{-1} H^\dagger \eta\,,$

where $^\ddagger$ denotes the adjoint in $\mathbb{K}$ or $\tilde{\mathbb{K}}$.

For such $H$, the spectrum consists of either real eigenvalues, or complex-conjugate pairs. Under suitable circumstances (e.g., unbroken symmetry regions), the spectrum is entirely real and the corresponding evolution operator $e^{-iHt}$ is “pseudo-unitary,” i.e. it preserves the indefinite inner product (Znojil, 2011).

3. Constructive Quantization Scheme: Metric Construction and Physical Hilbert Spaces

The quantization process in Krein/Pontryagin contexts consists of the following procedure (Znojil, 2011):

• Step 1: Construction of the Indefinite Metric. Solve the linear constraint on potential metric operators

$H^\dagger \Theta - \Theta H = 0$

for invertible $\Theta$ with prescribed metric signature. In practice, expand

$\Theta = \sum_{k=1}^M \nu_k \Theta_{(k)}^{(N)}$

in a basis of sparse matrices and fix $\nu_k$ coefficients so that $H^\dagger \Theta = \Theta H$. If $\Theta$ is positive definite, a usual Hilbert space is obtained; otherwise, indefinite Krein or Pontryagin structures are realized.

• Step 2: Inner Product and Hilbert Space Definition. For each metric $\Theta$,

$$\langle \psi | \phi \rangle_\Theta := (\psi, \Theta \phi)$$

defines the scalar product in $\mathcal{H}^{(S)}$. The nature of $\Theta$ (positive-definite, indefinite with finite negative sector, etc.) sets whether the physical space is Hilbert, Krein, or Pontryagin.

• Step 3: Spectral and Evolution Structure. When $H$ is $\eta$-self-adjoint and $\Theta$ is positive definite, time-evolution is unitary in $\mathcal{H}^{(S)}$. For indefinite $\Theta$, pseudo-unitary evolution prevails; the spectrum may acquire complex pairs but reality is guaranteed under appropriate conditions.

Example: For the tridiagonal $N\times N$ Hamiltonian $H^{(N)}(\lambda)$, explicit sparse metrics realizing positive-definite (Hilbert), Krein, or Pontryagin spaces can be constructed and correspond to different physical/auxiliary spaces (Znojil, 2011).

4. Field Quantization and Indefinite Metric: Structure of Fock-Krein Spaces

In field theory, Krein quantization systematically includes both positive- and negative-norm modes in the field operator decompositions,

$\phi(x) = \phi^{(+)}(x) + \phi^{(-)}(x)$

where $\phi^{(+)}$ contains the usual annihilation/creation operators acting on positive-norm modes, and $\phi^{(-)}$ is built with independent operators acting on negative-norm modes. The crucial commutators are

$$[a_{\mathbf{k}}, a_{\mathbf{k}'}^\dagger] = +\delta^3(\mathbf{k} - \mathbf{k}'), \quad [b_{\mathbf{k}}, b_{\mathbf{k}'}^\dagger] = -\delta^3(\mathbf{k} - \mathbf{k}'),$$

with all others vanishing. The Krein-Fock vacuum is defined by $$a(\mathbf{k})|0\rangle = b(\mathbf{k})|0\rangle = 0$$, and negative-norm states decouple from physical asymptotic amplitudes (Pejhan et al., 2012).

In the context of de Sitter space or with gauge fields, the full Fock-Krein (or Krein-Gupta-Bleuler) space admits covariant field operators and vacuum, and the physical Hilbert space is defined as a positive-definite quotient by the null-norm/gauge sector (Gazeau et al., 2010, Rahbardehghan et al., 2014).

5. Regularization, Divergences, and Renormalization

A central feature of Krein quantization is the automatic cancellation of ultraviolet and infrared divergences. Because singular contributions to propagators and Green's functions from positive- and negative-norm sectors have opposite sign, pointwise divergences cancel: $G(x,x') = G^{(+)}(x,x') + G^{(-)}(x,x'),$ with $G^{(-)}$ providing a built-in subtraction which renders the expectation values of composite operators and vacuum energies finite without the need for normal ordering, zeta-function, or other ad hoc regularization (Sojasi et al., 2012, Pejhan et al., 2012, Payandeh et al., 2012).

Loop integrals—such as the one-loop QED electron and photon self-energies, and vertex corrections—become finite due to the altered propagator structure, which is given by the principal-part prescription rather than the Feynman $i\epsilon$ regularization,

$G_T(p) = \mathrm{P.V.} \frac{1}{p^2 - m^2}.$

All would-be on-shell divergences are symmetrically avoided, and quantum-metric fluctuation can smooth out remaining light-cone singularities (Takook, 2021).

6. Applications and Physical Implications

• Quantum Field Theory and Gauge Theories: Krein quantization provides a means to quantize minimally coupled massless fields, photons, and linearized gravitons in de Sitter or curved spacetime, solving the “zero-mode problem” and restoring covariance otherwise lost in Hilbert space frameworks (Gazeau et al., 2010, Rahbardehghan et al., 2014). For QED and $\lambda \phi^4$ theory, Krein regularization yields finite one-loop effective actions that agree with renormalized results, such as the Schwinger and Euler-Heisenberg effective actions, but without explicit counterterms (Refaei et al., 2011, Refaei, 2013, Refaei et al., 2011). The beta function and running coupling constants coincide with standard results (Refaei et al., 2011).
• Casimir Effect and Boundary Problems: In problems with boundaries (e.g., a scalar field on a spherical shell), only the positive-norm sector “sees” the boundary; negative-norm modes supply a continuum subtraction, yielding the standard Casimir energy directly and finitely (Pejhan et al., 2012, Payandeh et al., 2012).
• Resolution of the Klein Paradox: By quantizing in Krein space, all four Dirac solutions (positive/negative energy, particles/antiparticles) are treated symmetrically. Negative-norm states prevent paradoxical reflection/transmission probabilities without resorting to spontaneous pair creation, and ensure current conservation (Payandeh et al., 2013).
• Quantum Gravity and Axiomatic Yang-Mills: Krein quantization in the ambient-space formalism leads to a well-defined, renormalizable quantum linear gravity at all orders, and an axiomatic confinement scenario with mass gap in Yang-Mills theory (Takook, 26 May 2025, Takook, 2021).
• Quantum Algorithms: The indefinite-metric structure facilitates Krein dilation schemes for embedding ill-conditioned or singular linear systems, or non-unitary evolution, into a larger “Krein-unitary” framework, extending techniques such as quantum singular value decomposition (Takook, 26 May 2025).

7. Comparison to Other Regularization Schemes and Open Directions

Krein regularization is distinct from Pauli–Villars: rather than introducing massive regulator fields, Krein quantization algebraically subtracts identical-mass negative-norm states, producing similar ultraviolet convergence properties but requiring no new scales or limits (Payandeh et al., 2012). In the context of curved backgrounds or gauge theories, it provides unique features such as de Sitter covariance, automatic vacuum energy cancellation, and resolves pathologies associated with zero modes.

Future research concerns include the extension to interacting theories beyond one loop, maintenance of unitarity and causality in the presence of negative-norm states, the interpretation and physical status of negative-norm (“ghost”) sectors, and the exploration of phenomenological implications—such as connections to fermion doubling and dark matter in discrete-time quantum mechanics (Jaroszkiewicz, 2020).

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Selected References

• Znojil, “PT-symmetric quantum models living in an auxiliary Pontryagin space” (Znojil, 2011)
• Gazeau, Siegl, Youssef, “Krein Spaces in de Sitter Quantum Theories” (Gazeau et al., 2010)
• Pejhan, Tanhayi, Takook, “Casimir Effect For a Scalar Field via Krein Quantization” (Pejhan et al., 2012)
• Takook et al., “Krein Regularization of QED” (Forghan et al., 2012)
• Sedehi, Takook, “Scalar effective action in Krein space quantization” (Refaei et al., 2011)
• Forghan, Refaei, Takook, “QED effective action in Krein space quantization” (Refaei et al., 2011)
• Takook, Pejhan, “Trans-Planckian Scale and Krein Space Quantization” (Sojasi et al., 2012)
• Jaroszkiewicz, “Hyperspace fermions, Möbius transformations, Krein space, fermion doubling, dark matter” (Jaroszkiewicz, 2020)
• Pejhan, Tanhayi, Takook, “A brief review on the Problem of Divergence in Krein Space Quantization” (Payandeh et al., 2012)
• Zarei, Forghan, Takook, “Euler-Heisenberg lagrangian through Krein regularization” (Refaei, 2013)
• Gazeau, Seddigh, Takook, “The Krein-Gupta-Bleuler Quantization in de Sitter Space-time” (Rahbardehghan et al., 2014)
• Sedehi, Takook, “Spectrum of Gravitational Waves in Krein Space Quantization” (Mohsenzadeh et al., 2012)
• Sedehi, Takook, “Krein regularization method” (Takook, 2021)
• Zarei, Forghan, Takook, “A Krein Quantization Approach to Klein Paradox” (Payandeh et al., 2013)
• Forghan, Takook, “Krein space quantization and New Quantum Algorithms” (Takook, 26 May 2025)