Mathematical Structure and Physical Mechanism of the Difference between Positrons and Electrons: A Re-examination

Author: Zhang Suhang, Luoyang, Henan

Abstract

Standard quantum field theory attributes the physical distinction between electrons and positrons to charge conjugation transformation. Mathematically, the positron is treated as a time-reversed state of the electron. The boundary between this “mathematical equivalence” and physical difference has long remained ambiguous. Breaking away from the traditional view of charge as an intrinsic assignment, this paper develops a coherent logical chain: difference in phase orientation → difference in mathematical structure → difference in physical properties. The study shows that the primary distinction between electrons and positrons does not lie in the sign of charge, but in the opposing phase orientations of their quantum fields. This opposition of phase orientations manifests mathematically as the separation between positive-frequency and negative-frequency modes, and observationally as the difference in the sign of electric charge. The identical nature of electrons and positrons arises from their shared platform for electromagnetic interaction, while their difference originates from the bidirectional opposition of phase orientations. This framework strictly distinguishes mathematical description from physical reality and provides a self-consistent new physical interpretation for the core problem of identity and difference in particle physics.

Keywords: electron and positron; phase orientation; mathematical structure; charge conjugation; quantum field modes

1 Introduction

In conventional formulations of quantum field theory, the distinction between electrons and positrons is often simplified to the sign of electric charge, with charge regarded as an inherent static property of particles. At a deeper mathematical level, the Feynman–Stueckelberg interpretation defines the positron as an electron evolving backward in time. Through symmetric transformation of positive- and negative-frequency field modes, formal equivalence between the two is established, which serves as the foundation for scattering calculations and Feynman diagram analysis in quantum field theory.

Nevertheless, this mathematical symmetry has long been subject to interpretive ambiguity: is the time-reversal correspondence merely a mathematical tool for description, or a genuine physical process?

Existing explanations tend to use charge, a resulting property, to account for particle differences, and seldom discuss the origin of charge from the perspective of field phase configurations. On this basis, this paper takes the phase orientation of quantum fields as the entry point, sorts out the complete correlation from field configuration to field-theoretic mathematical structure and then to physical properties, and clarifies the boundary between mathematical symmetry and physical reality.

2 Phase Orientation: Starting Point for the Difference between Electrons and Positrons

2.1 Inherent Limitations of the Classical Static View

Classical physics and elementary quantum mechanics adopt a static understanding of electrons and positrons: electrons and positrons are presupposed as two separate particle species, and the sign of charge is taken as the core label distinguishing them.

This description only accounts for phenomena without discussing the mechanism behind the difference. It directly treats the sign of charge as an intrinsic feature requiring no further explanation, leaving the origin of particle differences unaddressed.

2.2 Phase Orientation: A Field Characteristic Distinguishing Particle Configurations

In quantum systems, the evolution of a field includes spatial momentum variation and temporal phase variation. Compared with observable spatial displacement, the temporal phase orientation of the wavefunction is an important field characteristic that distinguishes field configurations.

The time evolution of a quantum wavefunction follows the standard oscillatory phase form: e^{\mp iEt/\hbar}, which yields two opposing phase configurations:

1. Electron: phase rotates forward with time, corresponding to the positive-frequency evolution mode e^{-iEt/\hbar};
2. Positron: phase rotates backward with time, corresponding to the negative-frequency evolution mode e^{+iEt/\hbar}.

This paper argues that the core distinction between electrons and positrons arises from the opposition in phase orientation of their quantum fields. The sign of charge is not a pre-assigned intrinsic label but a physical effect manifested by phase orientation within the electromagnetic interaction framework.

3 Mathematical Structure: Field-Theoretic Formulation Corresponding to Phase Orientation

3.1 Structural Separation of Positive and Negative Frequency Modes

The standard field operator for the electron field can be decomposed rigorously into a superposition of positive and negative frequency components:

\psi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \sum_s \left( a_p^s u^s(p) e^{-ipx} + b_p^{s\dagger} v^s(p) e^{ipx} \right)

In this expression, the positive-frequency term e^{-ipx} corresponds to electron annihilation and represents forward phase orientation; the negative-frequency term e^{ipx} corresponds to positron creation and represents backward phase orientation.

The mathematical structural difference of the field operator is not merely an artificially set computational rule. It is the mathematical expression of two opposing phase configurations, reflecting the distinction between the field configurations of electrons and positrons.

3.2 Essential Meaning of Charge Conjugation Transformation

Charge conjugation transformation C is the core symmetry transformation connecting the electron field and positron field. Its transformation reads:

\psi \rightarrow \psi^C = C \bar{\psi}^T

From the perspective of field modes, charge conjugation transformation does not generate new physical properties. Its effect is to reverse the phase orientation of the quantum field and realize the mutual mapping between positive-frequency and negative-frequency modes.

This indicates that the symmetry corresponding to charge conjugation embodies the mathematical relation of bidirectional transformation of phase orientation, and the change in the sign of charge is brought about by the reversal of phase orientation.

3.3 Strict Demarcation between Mathematical Equivalence and Physical Reality

This paper distinguishes two layers of concepts to clarify long-standing interpretive biases:

1. At the mathematical level: positive/negative frequency modes, particle/antiparticle states, and time-reversed states can achieve formal equivalence and be derived from one another via gauge transformations. These serve as concise theoretical descriptive tools.
2. At the physical level: the electron field with forward phase orientation and the positron field with backward phase orientation are two fixed and distinguishable field configurations. There is no physical picture of a single particle traveling back and forth along the time axis.

4 Formation Mechanism of the Sign of Electric Charge

4.1 Physical Effects of Phase Orientation under Electromagnetic Coupling

The interaction between particles and electromagnetic fields is described by the covariant derivative D_\mu = \partial_\mu + iqA_\mu. The sign of charge directly determines the phase evolution rule of gauge coupling, establishing the link between field configuration and observable properties:

- The coupling term for negative charge corresponds to the forward phase evolution rule, matching the field phase configuration of the electron;
- The coupling term for positive charge corresponds to the backward phase evolution rule, matching the field phase configuration of the positron.

It follows that the sign of charge is not an inherent label carried by the particle. It is an observable physical property emerging from the coupling between field phase configuration and the electromagnetic gauge field, determined by the field phase configuration.

4.2 Logical Chain of the Difference between Electrons and Positrons

In summary, this paper establishes a clear relationship:

Opposition in quantum field phase orientation → structural separation of positive/negative frequency mathematical modes → difference in the physical property of charge sign

All observable particle differences can be traced back to the field phase configuration, abandoning the static interpretation where intrinsic properties define field morphology.

5 Unified Interpretation of Identity and Difference

5.1 Identity: A Consequence of the Shared Electromagnetic Interaction Platform

Although electrons and positrons have opposite phase orientations, both are generated and evolve on the same global unified platform for electromagnetic interaction. The coupling strength, gauge structure and quantum constraints of this platform remain consistent across the universe. As a result, electrons and positrons share identical rest mass, spin, gauge coupling strength and other core parameters, manifesting particle identity.

Particle identity originates from the consistency of the interaction platform and is independent of the phase orientation of field configurations.

5.2 Difference: Distinction of Field Configurations from Bidirectional Phase Orientation

The unified electromagnetic interaction platform permits two symmetric phase orientations: the forward phase configuration corresponds to the electron, and the backward phase configuration corresponds to the positron.

The consistency of the platform guarantees identical fundamental particle parameters, while the bidirectional opposition of phase orientation produces the difference in charge sign. The two features coexist in a self-consistent logical framework.

5.3 Unified Understanding of Identity and Difference

This paper provides a unified explanation for the “identity–difference” characteristics of electrons and positrons:

1. Identity arises from the overall consistency and stability of the electromagnetic interaction platform;
2. Difference arises from the existence of two alternative opposing phase orientations for quantum field configurations.

Identity is a global constraint imposed by the interaction platform, while difference is a morphological distinction at the level of field configurations, forming a complete physical picture of particles.

6 Discussion

6.1 Feature of the Approach: Reordering the Understanding of Properties and Field Configurations

This paper adjusts the conventional way of viewing particles. Instead of defining particles first by their intrinsic properties, it interprets particle physical characteristics starting from field phase configurations. Observable particle properties are effects exhibited by field configurations under interaction, with field configurations coming first and properties as consequences.

This perspective offers a new angle for understanding the origin of fundamental particle properties and the symmetry relations between them.

6.2 Theoretical Boundaries and Scope of Application

1. This paper retains the existing mathematical framework of quantum field theory without modifying field-theoretic computational methods, and only supplements the physical interpretation connecting field modes and particle properties.
2. The work belongs to conceptual theoretical clarification. It does not propose new experimental predictions or revise existing physical formulas.
3. The discussion is confined to the system of electrons, positrons and electromagnetic interactions. It may serve as a reference for research on symmetric differences of similar gauge particles.

7 Conclusion

Starting from the phase orientation of quantum fields, this paper re-examines the symmetric difference mechanism of electrons and positrons and arrives at a self-consistent physical understanding. The core conclusions are as follows:

1. The key distinguishing feature between electrons and positrons is not the sign of charge, but the opposition in phase orientation of their quantum fields. Charge is a physical effect arising from field configurations.
2. The bidirectional opposition of phase orientation manifests mathematically as the structural separation between positive-frequency and negative-frequency quantum field modes, and physically as the difference in the sign of electric charge. This forms the logical chain: phase orientation — mathematical structure — physical property.
3. The identity of electrons and positrons comes from the globally unified electromagnetic interaction platform, while their difference originates from the bidirectional feature of phase orientation. This interpretation can simultaneously account for particle identity and difference.

This paper presents a new reading of the correspondence between quantum field modes and particle charge, providing a reference perspective for research on symmetry mechanisms of elementary particles.

References

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