Abstract This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.Abstract This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.

Nov 1, 20150 citations

Paul Dirac and The Principles of Quantum Mechanics

Helge Kragh

Edition Open Access eBooksSep 10, 201317 citations

Quantum Mechanics of Dirac Particles

World Scientific lecture notes in physicsSep 1, 20200 citations

Real Dirac Dynamics and Conservation Laws in Unified Quantum Mechanics

Pushpendra Singh

We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit $i$ is systematically replaced by the temporal Hilbert transform operator $\mathcal{H}_{t}$ , which satisfies the anti-involution property $\mathcal{H}_{t}^{2}=-\mathcal{I}$ and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field $\psi$ and its Hilbert-transformed partner $\phi=\mathcal{H}_{t}[\psi]$ , whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation $\psi_{a}=\psi-i\phi$ , under which the standard $U(1)$ gauge symmetry emerges naturally as a global $SO(2)$ rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density $\rho=\psi^{\dagger}\psi+\phi^{\dagger}\phi=|\psi_{a}|^{2}$ is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit $i$ is systematically replaced by the temporal Hilbert transform operator $\mathcal{H}_{t}$, which satisfies the anti-involution property $\mathcal{H}_{t}^{2}=-\mathcal{I}$ and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field $\psi$ and its Hilbert-transformed partner $\phi=\mathcal{H}_{t}[\psi]$, whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation $\psi_{a}=\psi-i\phi$, under which the standard $U(1)$ gauge symmetry emerges naturally as a global $SO(2)$ rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density $\rho=\psi^{\dagger}\psi+\phi^{\dagger}\phi=|\psi_{a}|^{2}$ is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.

Jan 27, 20260 citations

Implementing Dirac Approach to Quantum Mechanics in a Hungarian Secondary School

Marisa Michelini, Alberto Stefanel, Kristóf Tóth

Quantum mechanics is included in many curricula across countries because of its cultural value and technological application. In the last decades, two-state approaches to quantum mechanics became popular because of the age of quantum computers. This article presents an experiment with 24 Hungarian high school students on teaching/learning quantum mechanics according to Dirac’s approach to concepts and basic formalism developed in the context of light polarization. Tutorials, pre/post-tests, and oral interviews are the main monitoring tools used to collect data on the students’ learning path. From the qualitative and quantitative data analysis, learning progressions emerged in the phenomenology exploration and on the probabilistic nature of single quantum measurement. The students’ conceptions of quantum state are enriched, confirming the importance to focus educational approaches on fundamental topics. For one section of students, the complex relationship between quantum state and property remained problematic, but the students’ interpretations of a quantum state can be categorized. Two lines of reasoning emerged regarding the impossibility to attribute a trajectory to a quantum system, one more orthodox and one that seeks to avoid the probabilistic nature of the quantum world.Quantum mechanics is included in many curricula across countries because of its cultural value and technological application. In the last decades, two-state approaches to quantum mechanics became popular because of the age of quantum computers. This article presents an experiment with 24 Hungarian high school students on teaching/learning quantum mechanics according to Dirac’s approach to concepts and basic formalism developed in the context of light polarization. Tutorials, pre/post-tests, and oral interviews are the main monitoring tools used to collect data on the students’ learning path. From the qualitative and quantitative data analysis, learning progressions emerged in the phenomenology exploration and on the probabilistic nature of single quantum measurement. The students’ conceptions of quantum state are enriched, confirming the importance to focus educational approaches on fundamental topics. For one section of students, the complex relationship between quantum state and property remained problematic, but the students’ interpretations of a quantum state can be categorized. Two lines of reasoning emerged regarding the impossibility to attribute a trajectory to a quantum system, one more orthodox and one that seeks to avoid the probabilistic nature of the quantum world.

Education SciencesSep 6, 202224 citations

Dirac's Quantum Mechanics

WORLD SCIENTIFIC eBooksMay 1, 20000 citations

i

is systematically replaced by the temporal Hilbert transform operator

\mathcal{H}_{t}

, which satisfies the anti-involution property

\mathcal{H}_{t}^{2}=-\mathcal{I}

and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field

\psi

and its Hilbert-transformed partner

\phi=\mathcal{H}_{t}[\psi]

, whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation

\psi_{a}=\psi-i\phi

, under which the standard

U(1)

gauge symmetry emerges naturally as a global

SO(2)

rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density

\rho=\psi^{\dagger}\psi+\phi^{\dagger}\phi=|\psi_{a}|^{2}

is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.