A simple derivation of a meaningful, manifestly covariant inner product for real Klein-Gordon (KG) fields with positive semi-definite norm is provided which turns out - assuming a symmetric bilinear form - to be the real-KG-field limit of the inner product for complex KG fields reviewed by A. Mostafazadeh and F. Zamani in December, 2003, and February, 2006 (quant-ph/0312078, quant-ph/0602151, quant-ph/0602161). It is explicitly shown that the positive semi-definite norm associated with the derived inner product for real KG fields measures the number of active positive and negative energy Fourier-modes of the real KG field on the relativistic mass-shell. The very existence of an inner product with positive semi-definite norm for the considered real, i.e. neutral, KG fields shows that the metric operator entering the inner product does not contain the charge-conjugation operator. This observation sheds some additional light on the meaning of the by C.M. Bender, D.C. Brody and H.F. Jones identified C operator in the CPT inner product of PT-symmetric Quantum Mechanics.A simple derivation of a meaningful, manifestly covariant inner product for real Klein-Gordon (KG) fields with positive semi-definite norm is provided which turns out - assuming a symmetric bilinear form - to be the real-KG-field limit of the inner product for complex KG fields reviewed by A. Mostafazadeh and F. Zamani in December, 2003, and February, 2006 (quant-ph/0312078, quant-ph/0602151, quant-ph/0602161). It is explicitly shown that the positive semi-definite norm associated with the derived inner product for real KG fields measures the number of active positive and negative energy Fourier-modes of the real KG field on the relativistic mass-shell. The very existence of an inner product with positive semi-definite norm for the considered real, i.e. neutral, KG fields shows that the metric operator entering the inner product does not contain the charge-conjugation operator. This observation sheds some additional light on the meaning of the by C.M. Bender, D.C. Brody and H.F. Jones identified C operator in the CPT inner product of PT-symmetric Quantum Mechanics.

Jan 1, 200611 citations

Klein–Gordon equation within the real Hilbert space formalism

Cristiano Rosa, Sergio Giardino

arXiv:2510.16602Annals of PhysicsOct 21, 20252 citations

Quantum Mechanics of Klein-Gordon Fields I: Hilbert Space, Localized States, and Chiral Symmetry

Alí Mostafazadeh, Farhad Zamani

We derive an explicit manifestly covariant expression for the most general positivedefinite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. This expression involves a one-parameter family of conserved current densities J µ a, with a ∈ (−1,1), that are analogous to the chiral current density for spin half fields. The conservation of J µ a is related to a global gauge symmetry of the Klein-Gordon fields whose gauge group is U(1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symmetry is responsible for the conservation of the total probability of the localization of the field in space. This provides a simple resolution of the paradoxical situation resulting from the fact that the probability current density for free scalar fields is neither covariant nor conserved. Furthermore, we discuss the implications of our approach for free real scalar fields offering a direct proof of the uniqueness of the relativistically invariant positive-definite inner product on the space of real Klein-Gordon fields. We also explore an extension of our results to scalar fields minimally coupled to an electromagnetic field.We derive an explicit manifestly covariant expression for the most general positivedefinite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. This expression involves a one-parameter family of conserved current densities J µ a, with a ∈ (−1,1), that are analogous to the chiral current density for spin half fields. The conservation of J µ a is related to a global gauge symmetry of the Klein-Gordon fields whose gauge group is U(1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symmetry is responsible for the conservation of the total probability of the localization of the field in space. This provides a simple resolution of the paradoxical situation resulting from the fact that the probability current density for free scalar fields is neither covariant nor conserved. Furthermore, we discuss the implications of our approach for free real scalar fields offering a direct proof of the uniqueness of the relativistically invariant positive-definite inner product on the space of real Klein-Gordon fields. We also explore an extension of our results to scalar fields minimally coupled to an electromagnetic field.

Jan 1, 200631 citations

Generalized PT-, C-, and CPT-Symmetries, Position Operators, and Localized States of Klein-Gordon Fields

Mostafazadeh, A

We construct a concrete realization of the generalized parity (P), time-reversal (T), and charge-conjugation (C) operators, that were initially defined in the study of the PT-symmetric and pseudo-Hermitian Hamiltonians, for Klein-Gordon fields. We show that PT and C operators, that signify certain symmetries of the system, correspond to the ordinary time-reversal and charge-conjugation transformations, respectively. Furthermore, we construct a positive-definite and relativistically invariant inner product on the space of Klein-Gordon fields and show that the generator h of time-translations viewed as acting in the Hilbert space ${cal H}$ defined by this inner product is Hermitian. The quantum system having ${cal H}$ as its Hilbert space and h as its Hamiltonian is unitarily equivalent to the one defined by a Hermitian Hamiltonian acting in the Hilbert space $L^2(R^3)oplus L^2(R^3)$ . We give the explicit form of the corresponding unitary transformation and use it to construct position operators, localized and coherent states, and a set of wave functions for the Klein-Gordon fields. Our approach provides a consistent quantum mechanical description of the Klein-Gordon fields that admits a genuine probabilistic interpretation.We construct a concrete realization of the generalized parity (P), time-reversal (T), and charge-conjugation (C) operators, that were initially defined in the study of the PT-symmetric and pseudo-Hermitian Hamiltonians, for Klein-Gordon fields. We show that PT and C operators, that signify certain symmetries of the system, correspond to the ordinary time-reversal and charge-conjugation transformations, respectively. Furthermore, we construct a positive-definite and relativistically invariant inner product on the space of Klein-Gordon fields and show that the generator h of time-translations viewed as acting in the Hilbert space ${cal H}$ defined by this inner product is Hermitian. The quantum system having ${cal H}$ as its Hilbert space and h as its Hamiltonian is unitarily equivalent to the one defined by a Hermitian Hamiltonian acting in the Hilbert space $L^2(R^3)oplus L^2(R^3)$. We give the explicit form of the corresponding unitary transformation and use it to construct position operators, localized and coherent states, and a set of wave functions for the Klein-Gordon fields. Our approach provides a consistent quantum mechanical description of the Klein-Gordon fields that admits a genuine probabilistic interpretation.

Jul 8, 20032 citations

{cal H}

defined by this inner product is Hermitian. The quantum system having

{cal H}

as its Hilbert space and h as its Hamiltonian is unitarily equivalent to the one defined by a Hermitian Hamiltonian acting in the Hilbert space

L^2(R^3)oplus L^2(R^3)

. We give the explicit form of the corresponding unitary transformation and use it to construct position operators, localized and coherent states, and a set of wave functions for the Klein-Gordon fields. Our approach provides a consistent quantum mechanical description of the Klein-Gordon fields that admits a genuine probabilistic interpretation.