[Paper Review] Fermionic scalar field

Yoshiharu Kawamura|arXiv (Cornell University)|Jun 24, 2014

Quantum Mechanics and Applications2 references3 citations

TL;DR

This paper re-examines the spin-statistics theorem by studying a complex scalar field quantized with anti-commutation relations—termed a 'fermionic scalar field'—which leads to negative-norm states despite positive energy and causality. The issue is resolved by introducing a dual scalar field to form a fermionic doublet, resulting in a consistent but empty theory. The same mechanism applies to spinor fields with commutation relations, showing that negative norms, not energy or causality, are the core obstacle to consistent quantization under abnormal statistics.

ABSTRACT

We reexamine the connection between spin and statistics through the quantization of a complex scalar field, using the formulation with the property that the hermitian conjugate of canonical momentum for a variable is just the canonical momentum for the hermitian conjugate of the variable. Starting from an ordinary Lagrangian density and imposing the anti-commutation relations on the field, we find that the difficulty stems from not the ill-definiteness (or unboundedness) of the energy and the breakdown of the causality but the appearance of states with negative norms. It is overcome by introducing an ordinary scalar field to form a doublet of fermionic symmetries, although the system becomes empty leaving the vacuum state alone. These features also hold for the system with a spinor field imposing the commutation relations on.

Motivation & Objective

• To investigate the foundational role of norm positivity in the spin-statistics theorem by studying non-standard quantization rules.
• To clarify why anti-commutation relations for scalar fields lead to inconsistencies, despite preserving energy positivity and causality.
• To propose a resolution to the negative-norm problem by introducing a dual scalar field forming a fermionic doublet.
• To extend the analysis to spinor fields quantized with commutation relations, identifying analogous issues and solutions.
• To develop a manifestly Hermitian formulation of analytical mechanics for systems with both bosonic and fermionic non-Hermitian variables.

Proposed method

• Starts from a standard complex scalar field Lagrangian and imposes anti-commutation relations on the field operators.
• Identifies the appearance of negative-norm states as the primary obstacle to consistent quantization, not unbounded energy or causality violation.
• Introduces a second ordinary scalar field to form a doublet under fermionic symmetries, restoring consistency at the cost of an empty spectrum.
• Applies the same formalism to spinor fields with commutation relations, showing analogous negative-norm problems.
• Constructs a new analytical mechanics framework where the Hermitian conjugate of the canonical momentum for a variable equals the momentum of its Hermitian conjugate, ensuring manifest Hermiticity.
• Uses Fock space with indefinite metric to analyze norm structure and derive conditions for consistency.

Experimental results

Research questions

• RQ1 What are the consequences of imposing anti-commutation relations on a complex scalar field in relativistic quantum field theory?
• RQ2 Why does a scalar field with anti-commutation relations lead to inconsistencies, and which standard assumptions (energy, causality, norm) are actually violated?
• RQ3 Can the negative-norm problem in such a system be resolved without breaking fundamental principles like causality or energy positivity?
• RQ4 How does the behavior of a spinor field quantized with commutation relations compare to that of a fermionic scalar field?
• RQ5 What role does the fermionic symmetry doublet play in restoring consistency in theories with abnormal statistics?

Key findings

• The primary obstacle to consistent quantization of a fermionic scalar field is the appearance of states with negative norms, not unbounded energy or causality breakdown.
• The negative-norm problem is resolved by introducing a second scalar field to form a fermionic doublet, leading to a consistent but empty theory where only the vacuum state remains.
• The system with a fermionic scalar field and a dual scalar field respects energy positivity, causality, and Lorentz invariance, but the spectrum is trivial due to the fermionic symmetry.
• For a spinor field quantized with commutation relations (a 'bosonic spinor field'), the same negative-norm issue arises, and it is resolved by introducing a dual spinor field obeying anti-commutation relations.
• The resulting theory with both fields is consistent and empty, preserving the vacuum, due to the invariance under fermionic symmetry transformations.
• A new formulation of analytical mechanics is developed where the Hermitian conjugate of the canonical momentum for a variable is the canonical momentum of its Hermitian conjugate, ensuring manifest Hermiticity in mixed bosonic-fermionic systems.