From the August 2009 paper (Progress of Theoretical Physics) by Takayuki Hori

Relativistic Particle in Complex Spacetime – A New Take on 4D Reality

The 2009 paper “Relativistic Particle in Complex Spacetime” by Takayuki Hori proposes a novel particle model
where spacetime coordinates are complex-valued. The ultimate aim? To explain why the universe appears to have
exactly four spacetime dimensions.

🌌 Core Idea

The particle’s position is written as a complex number:
zμ = xμ + i aμ. That is, it exists simultaneously in a real and imaginary
spacetime — a doubled universe of sorts. But gauge symmetries constrain the unphysical degrees of freedom.

🔍 Why This Matters

The model’s structure is such that only in four dimensions do the quantum constraints allow physical momentum eigenstates.
Thus, the model gives a mathematical reason for why our universe might have 4D spacetime.

🧪 Key Results

1. Lagrangian and Gauge Symmetry

The action for the particle includes complex terms:

∫ dτ (ẋ² / 2V + iλ ẋ·z + c.c.)

Here, V and λ are complex-valued gauge fields. The system shows SL(2, ℝ) symmetry and generates constraints through its dynamics.

2. Physical Equivalence and Dirac’s Conjecture

There are three first-class constraints, but only two gauge degrees of freedom — apparently violating Dirac’s conjecture.
Hori proposes a new criterion: states are physically equivalent if they have the same conserved charges, not just if they are
connected by gauge transformations.

3. Quantum Conditions Select 4D

Using BRST quantization, the model reveals that only in 4D does a consistent momentum eigenstate space exist.
This imposes a quantum-mechanical restriction on the dimension of spacetime.

4. Propagator and Scattering

Path integrals are computed to find a propagator and a toy scattering amplitude. Interestingly, the usual 1/k² behavior is
absent — suggesting new physics but also raising questions about how this model would connect to the Standard Model.

📉 Diagram: Complex Spacetime Particle Model

🧠 Significance

• Reformulates particle physics in a complexified spacetime background.
• Introduces a new way to think about gauge equivalence and constraints.
• Provides a possible explanation for why we live in a 4D universe.

⚖️ Strengths & Limitations

Pros:

• Mathematically consistent and gauge-invariant.
• Offers a dimensionality constraint from quantum principles.

Cons:

• No spin, internal quantum numbers, or Standard Model coupling.
• Propagator lacks a physical pole structure (no 1/k²).

📚 Summary

This paper offers a novel mathematical model where a particle lives in a complexified spacetime.
Quantization under constraints reveals that only four-dimensional spacetime yields viable physics — suggesting
our universe’s dimensionality may emerge from quantum geometry.