V–A Theory Fails to Properly Encode Chirality

I. Chirality is Not an Algebraic Tag

In modern particle physics, chirality is a central concept used to describe parity violation. It is a fundamental fact that weak interactions select "left-handed" fermions and exclude "right-handed" ones. On the surface, the V–A theory seems to have "solved" the encoding of chirality by introducing γ5 and projection operators PL, PR.

However, examining the mathematical structure of quantum theory more fundamentally reveals a critical conflict: spectral representation inherently describes "periodicity," whereas the essence of chirality involves "direction of spatial rotation." The former decomposes frequencies without direction, while the latter involves directed geometric motion.

Given this context, the "chirality encoding" of V–A theory exposes its inherent issue—it does not originate from real spatial rotations but instead uses an additional matrix label within a spectral framework to denote "left/right". Essentially, it acts as a mathematical patch rather than a reduction of the physical essence of chirality.

This argument will be unfolded in three steps:

1. Why spectral representation cannot depict rotational direction;

2. How V–A theory uses γ5 to affix a "pseudo-chirality" tag within this framework;

3. Why the true image of chirality must return to spatial geometry and topology.

II. Inherent Limitations of Spectral Representation: Seeing Periodicity But Not Direction

1. Fourier Spectrum: Decomposing Periods While Flattening Directions

The core of spectral analysis is the Fourier transform, which expresses evolutions in spacetime as superpositions of frequency modes. A typical form is:

f(t) = ∫ dω f~(ω)e^(-iωt).

Key points about this framework include:

• Spectra characterize "how fast something vibrates", not "which way it turns".

• Uniform clockwise and counterclockwise rotations share the same period; thus, they correspond to the same frequency peak in the spectrum—spectra see "speed", not "direction".

The symmetry of Fourier transforms flattens directional information:

• Both e^(iωt) and e^(-iωt) produce identical peaks in power spectra |f~(ω)|^2.

• Without considering specific complex phases, positive and negative frequencies are indistinguishable statistically.

Natural symmetries like time reversal and phase conjugation:

• Spectral techniques typically assume time-reversibility (within linear response frameworks), conflicting with genuine chirality (coupled to time-reversal and parity-breaking directional asymmetry).

Thus, mathematically:

Spectral decomposition breaks down "periodicity", not "rotational direction". It can indicate "an angular frequency ω is at play", but not whether it's left-handed or right-handed geometric rotation.

2. Limitations of Imaginary Structures: i Does Not Equal "Direction Arrow"

Quantum mechanics employs complex wave functions and operators such as:

ψ ~ e^(-iEt/ℏ),

seemingly allowing distinctions between "forward/backward". However, upon closer inspection, this distinction is conventional, not geometric.

Reasons include:

• The choice of imaginary unit i itself is conventional.

• Choosing i vs -i makes no difference for observable quantities.

Complex conjugate symmetry requires observables to be real:

• Any measurable quantity ultimately depends on ψ*ψ or similar real combinations, naturally erasing any directional information potentially carried by i.

Overall phase and gauge invariance:

• Transformations like ψ → eiαψ do not alter measurement outcomes, indicating that using a single global phase for "direction encoding" is ineffective physically.

Conclusion: Complex structures in spectra cannot provide genuine spatial rotation direction information. They introduce "phase" formally but do not address the ontological issue of geometric direction.

III. Pseudo-Chirality in V–A Theory: Labeling Within Spectral Framework

1. Construction of γ₅: Mathematically Elegant but Physically Suspended

V–A theory introduces:

γ5 = iγ0γ1γ2γ3

to characterize chirality, possessing familiar mathematical properties:

• Anticommutation relations {γ5, γμ} = 0;

• Eigenvalues ±1 distinguish "left/right":

  PL = (1 - γ5)/2, PR = (1 + γ5)/2;

Thus, fermion fields can be decomposed into:

ψL = PLψ, ψR = PRψ.

Formally, this is a neat algebraic operation using a matrix label to mark two complementary subspaces as "left-handed" and "right-handed". However, crucially, "left/right" here are purely internal space (spin space) algebraic labels, not explicitly derived from real spatial rotations.

2. V–A Remains Locked Within Spectral Perspective

The foundation of V–A theory remains the standard quantum field theory spectral framework:

• Still uses plane wave expansions:

  ψ(x) ~ u(p)e^(-ipx) + v(p)e^(ipx),

• Primarily operates in momentum space;

• Provides no geometric description of charge spiraling along specific trajectories in real 3D space.

Simplified, it does:

• Real physics: Charges exhibit specific spiral motions and chiral structures in space.

• ↓ (Spectralization: Losing geometric direction, retaining momentum and energy)

• Obtain pure periodic modes and spin indices.

• ↓ (Using γ5 to attach ±1 algebraic tags)

• Label these two algebraic subspaces as "left-handed state/right-handed state".

This leads to a fundamental consequence:

While V–A theory can fit decay rates and angular distributions, it fails to elucidate why left/right rotations differ in real space and the intrinsic connection between chirality and actual charge movement geometry. Formally, it achieves parameterization via tagging, not geometric-dynamical explanation.

3. Three Fundamental Deficiencies

Summarizing, V–A’s chirality encoding faces at least three fundamental issues:

• Discrete Labels vs Continuous Rotational Processes:γ5 provides only ±1 eigenvalues, discrete labels of "left/right", whereas true chiral processes involve continuous rotations and helical advancements.

• Mathematical Patch Rather Than First Principles Deduction:γ5 is constructed specifically to encode chirality within existing Dirac algebra, introduced to explain and fit observed parity violations, not derived from deeper geometric-topological structures.

• Using Mathematical Forms to Mask Fundamental Spectral Method Limitations: V–A addresses chirality within spectral-plane wave-operator algebra frameworks, masking the inability to express real-space rotational directions by simulating direction with labels.

From an ontological physics perspective, V–A’s chirality is "algebraic chirality", not true "spatial chirality"; it is highly consistent mathematically but represents "pseudo-chirality encoding" detached from real geometric processes.

IV. True Physical Image of Chirality: Returning to Spatial Geometry and Topology

To truly understand what chirality "is", one must return to space itself.

1. Essence of Chirality: Charge Rotation Direction in Space

True chirality includes:

• Geometric Structure in 3D Space: The core of chirality involves configurations that cannot coincide under mirror inversion—a geometric question about shapes and movements in R³, not just algebra in internal space.

• SO(3) and Its Topological Properties: Any rotation belongs to some path in SO(3); chirality involves not just rotation but asymmetric paths of rotation, related to winding numbers, covering spaces, and topological associations with spin representations.

• Real Vector Characteristics of Angular Momentum: Angular momentum connects directly with actual rotational motions, magnetic moments, and current loop configurations. Chirality breaking implies preferences for certain rotational vectors over their mirror counterparts.

These aspects are geometric-topological propositions, differing from spectral descriptions of frequency components.

2. Necessity of Topological and Geometric Tools

To genuinely comprehend chirality, mathematical tools must extend beyond spectral-operator algebras to geometric and topological structures:

• Introduction of Topological Invariants: Chern numbers, winding numbers, and topological charges describe properties where rotational structures cannot continuously revert to trivial states.

• Berry Phase and Geometric Information: Berry phases in parameter space directly relate to handiness by describing geometric information of traversal paths.

• Quantum Hall Effect Edge States: Unidirectional propagation in topologically protected edge states exemplifies spatial chirality.

• Weyl Semimetals: Chiral anomalies reveal deep connections between chirality and gauge field topology.

These phenomena all point to chirality being more of a geometric-topological concept than a property exhaustible by pure algebraic labels.

V. Physical Implications and Theoretical Prospects: From Instrumentalism to Ontology

1. Insights for Existing Quantum Theory

The inability of spectral representation to correctly encode chirality highlights a deep limitation of conventional quantum field theory formalism:

It excels at expressing spectral-energy-intensity-like quantities but is inherently inadequate in expressing real spatial geometrical structures, topological constraints, and rotational direction ontology.

2. Realism vs. Instrumentalism

If quantum theories are seen merely as computational tools, then V–A’s predictive success might suffice. However, adopting a realist stance demands theories reflecting independent physical reality structures. Thus, a theory encoding "left/right" through internal space labels falls short.

3. Future Directions: Building Truly Geometrized Chirality Theories

Future directions may involve:

• Developing Mathematical Frameworks Beyond Spectral Analysis: Incorporating field spatial configurations, rotational streamlines, and topological defects into core descriptions.

• Tracing Origins of Parity Violation and Chirality from First Principles: Seeking mechanisms generating chirality choices within deeper geometric structures.

• Unifying Topological Invariants and Dynamical Evolution: Explicitly incorporating topological terms and geometric terms into dynamical equations.

VI. Conclusion: From V–A’s Algebraic Tags Back to Geometric Reality of Chirality

In summary:

• Spectral representation inherently fails to distinguish rotational directions, capturing only periodicity.

• V–A theory uses γ5 and projection operators to label "left/right", encoding chirality algebraically rather than geometrically.

• While successfully fitting many experimental results of weak interactions, it masks fundamental limitations of spectral methods in representing real-space rotational structures.

True chirality involves actual rotations and helical motions of charges and fields in 3D space, a geometric and topological structure. Understanding chirality and parity violation necessitates moving beyond spectral-operator frameworks to new formulations centered on geometry and topology, representing both a technical enhancement of current mathematical forms and an ontological reconstruction of physical reality.