[Paper Review] Some New Unifications in Supersymmetry and Higher Dimensional Complex Space

Yi‐Fang Chang|ArXiv.org|Apr 1, 2008

Advanced Mathematical Theories and Applications3 references3 citations

TL;DR

This paper proposes a novel unification framework in supersymmetry using higher-dimensional complex space, where bosons are associated with real numbers and fermions with imaginary numbers. By leveraging this mathematical duality, the author derives unified formulations of supersymmetry, including commutation relations, propagators, and a unified partition function, suggesting a potential connection to statistics unification and possible violations of the Pauli exclusion principle.

ABSTRACT

Some new representations of the supersymmetric transformations are derived, and the supermultiplets are introduced. Based on these representations, various formulations (equations, commutation relations, propagators, Jacobi identities, etc.) of bosons and fermions may be unified. On the one hand, the mathematical characteristic of particles is proposed: bosons correspond to real number, and fermions correspond to imaginary number, respectively. Such fermions of even (or odd) number form bosons (or fermions), which is just consistent with a relation between imaginary and real number. The imaginary number is only included in the equations, forms, and matrixes of fermions. It is connected with relativity. On the other hand, the unified forms of supersymmetry are also connected with the statistics unifying Bose-Einstein and Fermi-Dirac statistics, and with the possible violation of Pauli exclusion principle; and a unified partition function is obtained. Therefore, one of the possible developments is the higher dimensional complex space.

Motivation & Objective

• To unify bosonic and fermionic formulations in supersymmetry using higher-dimensional complex space.
• To establish a mathematical correspondence between particle statistics and number systems—bosons as real numbers, fermions as imaginary numbers.
• To derive unified equations, commutation relations, and propagators for both particle types.
• To explore connections between supersymmetry, Bose-Einstein and Fermi-Dirac statistics, and possible Pauli exclusion principle violations.
• To propose a unified partition function as a key outcome of the framework.

Proposed method

• Derives new representations of supersymmetric transformations using complex number structures.
• Introduces supermultiplets based on the real-imaginary duality of bosons and fermions.
• Constructs unified equations and commutation relations by embedding fermionic components in imaginary number spaces.
• Applies Jacobi identities and propagator formulations within the complex space framework.
• Derives a unified partition function that incorporates both Bose-Einstein and Fermi-Dirac statistics.
• Proposes that higher-dimensional complex space provides a natural geometric setting for this unification.

Experimental results

Research questions

• RQ1 How can bosons and fermions be mathematically unified through a number-theoretic duality?
• RQ2 What role does higher-dimensional complex space play in formulating a unified supersymmetry?
• RQ3 Can the Pauli exclusion principle be naturally relaxed or generalized within this framework?
• RQ4 How do commutation relations and propagators unify across bosonic and fermionic sectors?
• RQ5 Can a single partition function describe both Bose-Einstein and Fermi-Dirac statistics?

Key findings

• Bosons are mathematically represented by real numbers, while fermions are represented by imaginary numbers, establishing a foundational duality.
• Fermions of even or odd count form bosons or fermions respectively, consistent with the algebraic relationship between real and imaginary numbers.
• The framework unifies equations, commutation relations, and propagators for both bosons and fermions within a single complex space formulation.
• A unified partition function is derived that incorporates both Bose-Einstein and Fermi-Dirac statistics.
• The model suggests a possible violation of the Pauli exclusion principle, indicating a new theoretical direction.
• The entire structure is embedded in a higher-dimensional complex space, providing a geometric foundation for the unification.