We present a formulation of quantum field theory based entirely on real numbers, which we call real quantum field theory (RQFT). The construction is obtained from the standard complex-number formulation by replacing the imaginary unit i with a matrix J throughout all formulas. Here J is the real 2x2 matrix satisfying the condition J^2=-1. This extends the recently developed formulation of real quantum mechanics based on the real K\"ahler space to systems with infinitely many degrees of freedom. As the basic example, we construct the RQFT for a scalar field. We define a field operator acting on the real K\"ahler analogue of the bosonic Fock space. This operator satisfies the Klein-Gordon equation and the J-form of canonical commutation relation. To construct the RQFT we develop the corresponding J-calculus. In particular, we introduce the direct and inverse J-Fourier transforms and the associated J-valued distributions. In RQFT, the usual unitarity condition for the complex S-matrix is replaced by the statement that the real scattering operator is both orthogonal and symplectic. The physical observables in RQFT coincide with those of ordinary QFT. Thus RQFT does not change the physical predictions of scalar QFT, but provides an equivalent formulation. We explore the possibility of purely real formulations of the Standard Model of elementary particles. We show that it does admit this formulation and the resulting theory has ortho-symplectic symmetry. The choice of the Standard Model as the testbed for exploring the possibility of purely real formulations is related with the fact it provides a realistic description of all known fundamental particles. The real formulation of the Standard Model naturally suggests a possible exit beyond the Standard Model. In particular, we consider the implications of breaking the J-symmetry as a marker for new physics.
Quantum computing provides a promising framework for the simulation of quantum field theories, where the computational cost depends both on the quantum algorithm employed and on the representation of the Hamiltonian. We investigate a formulation of the $\phi^4$ model based on the harmonic-oscillator basis in coordinate space. We derive the lattice $\phi^4$ Hamiltonian in this representation and analyze the structure of the resulting one-body matrix and interaction tensor. We show that both exhibit an effective band-diagonal structure, allowing controlled truncations of the Hamiltonian while preserving the low-energy spectrum. We validate this formulation by comparing low-energy observables obtained from numerical diagonalization with those computed in the standard harmonic-oscillator momentum-space representation. Finally, we estimate the resources required to encode the Hamiltonian on a quantum computer using both binary and unary boson-to-qubit mappings. By exploiting effective locality, the coordinate-space representation reduces the resources required for quantum simulation over a broad range of parameters.
Gaétan Bardy, M. Saubanère, A. Tanasa· 0 citations
This work presents a unified operator framework for quantum field theory based on a generalized relativistic wave differential operator defined on Minkowski space. Through the systematic selection of discrete parameters α, β, λ, m, and n ∈ N, this operator reproduces fundamental cases such as the Klein-Gordon, Dirac, and non-monogenic operators, revealing deep algebraic connections between seemingly disparate field theories.
We introduce a quantum field tensor constructed from scalar, spinor, and gauge fields, where binary indices act as activation signatures. This tensor product structure naturally generates all possible field combinations and their couplings, providing a first-principles derivation of interaction terms beyond the traditional Lagrangian formulation.
The framework is systematically extended to incorporate gauge interactions through a generalized minimal coupling prescription, yielding gauge-covariant operators that reproduce fundamental equations such as Proca and Maxwell. The formalism elegantly unifies gauge self-interactions, matter couplings, and free propagation within a single coherent equation, opening new avenues for exploring higher-spin fields and extensions beyond the Standard Model.
We consider a quantum system on the real line obeying a deformed Heisenberg algebra originally proposed by Wigner in 1950. Its explicit coordinate representation was provided by Yang in 1951 and in essence is identical in form with Dunkl's difference-differential operator introduced in 1989 in connection with roots systems of refection groups. Under certain conditions such quantum systems exhibit a supersymmetric (SUSY) structure where the reflection operator acts as the grading operator. We present a generalisation of Yang's representation by first considering only of one the two equations of motion in phase space. The corresponding non-interacting system is found to represent Witten's model of SUSY quantum mechanics. Imposing also the second equation of motion the original result of Wigner and Yang is reconsidered by extending their discussion to general symmetric potentials on the real line. As explicit example we discuss the harmonic oscillator and an attractive Coulomb-like potential $V(x)=-\gamma/|x|$. We also establish a Hooke-Newton duality between this Coulomb-like system and the original Wigner-Yang harmonic oscillator system.
We construct four bounded Hermitian operators, each one a finite polynomial in the unitary Weyl operators, for the Bell-CHSH inequality in a free massive scalar field in $1+1$ dimensions. The operators are localized in complementary wedges. Odd Weyl harmonics provide two exactly anticommuting axis observables in each wedge, while normalizable packets with compact support in the spectrum of the boost generator give exact modular inner products at nonvanishing bandwidth. The resulting Bell-CHSH correlator is a finite double sum. An operator with six Weyl terms per axis already gives $2.14885$. Using normalized Fej\'er approximants of $\operatorname{sgn}(\cos(x))$, we show that the supremum over the finite-polynomial family equals $2\sqrt{2}$, although no finite member attains it; a degree-$511$ example gives $2.80027$. We also point out that, in a centered quasifree state, a Bell-CHSH test whose four final settings are bounded functions of individual quadratures admits one common Gaussian representation and therefore stays below $2$. The finite-Weyl construction avoids this restriction because Bob's final settings mix two noncommuting axis observables.
J. G. Caribé, M. S. Guimaraes, I. Roditi et al.· 2 citations· ⚡1
This paper extends the Schrödinger quantum field—a wave function treated as a fundamental quantum field describing particle states and probabilities—to a fully relativistic framework within Parametrized Relativistic Quantum Theory (PRQT). A hallmark of parametrized theories is the use of an invariant evolution parameter, which addresses key conceptual challenges in standard quantum field theory (QFT). Feynman's path integral formulation is reviewed to illustrate the historical role of such parameters in early QFT developments. We then present a variational approach, based on the Pavšič–Barut action principle with a parametrized Lagrangian density, to derive field equations that naturally incorporate the invariant evolution parameter and yield a mass operator. The consistency of this formalism with physical measurements is demonstrated through applications to the Stueckelberg equation, including its implications for particle trajectories and mass states. The potential for resolving Standard Model anomalies is discussed.
J. Fanchi· American Journal of Physics· 0 citations
Recently, we introduced a geometric object analogous to an orthonormal frame in the Cartan formalism to study the parameter space of quantum systems; we called it N-bein, with N being the number of parameters that characterize the quantum system. Acting as the ‘square root’ of the quantum geometric tensor (QGT), the N-bein allows us to define new tensors to improve our understanding of the structure beneath the parameter space of quantum mechanics. In this work, we extend this mathematical framework surrounding the N-bein to analyze the parameter space of quantum systems with degenerate spectra. As in the non-degenerate case, we define a non-Abelian two-state QGT to identify possible transitions between degenerate states after two consecutive parameter variations. Additionally, using the Wilczek–Zee connection, we introduce a torsion-like tensor as the covariant derivative of the N-bein. This torsion captures the noncommutativity of successive parameter variations and coincides with the antisymmetric part of the two-state QGT. We also present a geometrical formulation using differential forms and discuss the physical implications of the newly defined tensors. Furthermore, we construct several gauge-invariant observables from the N-bein and its derivatives to highlight the utility of the new tensors. Finally, to illustrate the convenience and applications of this formalism, we apply the theoretical framework to a system of coupled harmonic oscillators immersed in an electric field. The coupling between the oscillators results in a degenerate system. Thus, using the new formalism, we found correlations among the quantum states quantified by the new invariants.
Jorge Romero, Carlos A. Velasquez, J. David Vergara· Journal of Physics A: Mathem...· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
