Quaternions and Three Spatial Dimensions
Why Does Space Have Exactly Three Dimensions?
1. Quaternion Basics
1.1 Definition
A quaternion is a four-dimensional number system with one real unit and three imaginary units:
\[q = a + bi + cj + dk\]
where \(a, b, c, d \in \mathbb{R}\) and the imaginary units satisfy:
\begin{align}
i^2 = j^2 = k^2 &= -1 \\
ij &= k, \quad jk = i, \quad ki = j \\
ji &= -k, \quad kj = -i, \quad ik = -j
\end{align}
Key property: Quaternion multiplication is non-commutative (order matters).
1.2 Connection to Rotations
Quaternions naturally represent 3D rotations! A unit quaternion:
\[q = \cos(\theta/2) + \sin(\theta/2)(n_x i + n_y j + n_z k)\]
represents a rotation by angle \(\theta\) around axis \(\mathbf{n} = (n_x, n_y, n_z)\).
Key Insight: The three imaginary units \(i, j, k\) naturally correspond to the three spatial dimensions \(x, y, z\)!
2. Spacetime as Quaternions
2.1 Quaternionic Position Vector
Define a spacetime position quaternion:
\[Z = ct + xi + yj + zk\]
where:
- \(ct\) is the real (temporal) component
- \(xi, yj, zk\) are the three imaginary (spatial) components
2.2 Quaternionic Norm and Spacetime Interval
The quaternion norm (squared magnitude) is:
\[|Z|^2 = Z \bar{Z} = (ct)^2 - x^2 - y^2 - z^2\]
where \(\bar{Z} = ct - xi - yj - zk\) is the quaternion conjugate.
Remarkable Result: The quaternion norm automatically gives the Minkowski spacetime interval!
\[ds^2 = |dZ|^2 = c^2dt^2 - dx^2 - dy^2 - dz^2\]
No need to postulate the metricβit emerges from quaternion algebra!
3. Why Three Spatial Dimensions?
3.1 Division Algebras Theorem
Hurwitz's Theorem (1898): There are exactly four normed division algebras:
- Real numbers β: 1 dimension, 0 imaginary units
- Complex numbers β: 2 dimensions, 1 imaginary unit
- Quaternions β: 4 dimensions, 3 imaginary units
- Octonions π: 8 dimensions, 7 imaginary units
These are the ONLY possibilities!
3.2 Why Not Octonions?
Octonions have 7 imaginary units, suggesting 7 spatial dimensions. Why don't we observe this?
Problem with Octonions:
- Octonions are non-associative: \((ab)c \neq a(bc)\)
- This breaks causality: sequence of operations matters in weird ways
- Field equations become ill-defined
- No consistent way to define differential equations
Answer: Quaternions are the largest division algebra that is associative. This makes them the natural choice for physical spacetime where causality must be preserved!
3.3 The Unique Role of Quaternions
| Algebra |
Dimensions |
Imaginary Units |
Properties |
Physical Interpretation |
| β (Reals) |
1 |
0 |
Commutative, Associative |
Time only, no space |
| β (Complex) |
2 |
1 |
Commutative, Associative |
1 time + 1 space |
| β (Quaternions) |
4 |
3 |
Non-commutative, Associative |
1 time + 3 space β Our universe! |
| π (Octonions) |
8 |
7 |
Non-commutative, Non-associative |
Causality breaks down |
Conclusion: Three spatial dimensions may be a consequence of quaternions being the only normed division algebra with:
- More than one imaginary unit (needs space, not just time)
- Associativity preserved (causality requires this)
- Division property (all non-zero elements invertible)
4. Quaternions and Energy-Momentum
4.1 Quaternionic Energy-Momentum
Define energy-momentum quaternion:
\[\mathcal{P} = \frac{E}{c} + p_x i + p_y j + p_z k\]
The norm:
\[|\mathcal{P}|^2 = \mathcal{P}\bar{\mathcal{P}} = \frac{E^2}{c^2} - p_x^2 - p_y^2 - p_z^2 = m^2c^2\]
But wait! In our complex framework, mass is imaginary. So we need:
\[|\mathcal{P}|^2 = \frac{E^2}{c^2} - \mathbf{p}^2 = (imc)^2 = -m^2c^2\]
4.2 Reconciling Signs
The quaternion formulation gives \(+m^2c^2\) but we want mass to be imaginary, giving \(-m^2c^2\). Resolution:
Define the energy-momentum quaternion with imaginary mass component:
\[\mathcal{P} = \frac{E}{c} + \mathbf{p} \cdot \vec{\sigma}\]
where \(\vec{\sigma} = (i, j, k)\) are the quaternion imaginary units, and require:
\[\mathcal{P}\bar{\mathcal{P}} = -(imc)^2 = m^2c^2\]
5. Lorentz Transformations as Quaternion Rotations
5.1 Spatial Rotations
A rotation around the z-axis by angle \(\theta\):
\[Z' = q Z \bar{q}\]
where \(q = \cos(\theta/2) + k\sin(\theta/2)\).
5.2 Lorentz Boosts
A boost in the x-direction can be represented as a "hyperbolic rotation":
\[Z' = Q Z \bar{Q}\]
where \(Q = \cosh(\phi/2) + i\sinh(\phi/2)\) and \(\tanh\phi = v/c\).
This is like a rotation but with hyperbolic functions instead of circular ones!
Deep Connection: All Lorentz transformations (rotations + boosts) can be represented as quaternion operations! The Lorentz group is naturally encoded in quaternion algebra.
6. Electromagnetism in Quaternionic Form
6.1 Electromagnetic Field Quaternion
Maxwell's equations can be elegantly written using quaternions:
Define the electromagnetic field quaternion:
\[\mathcal{F} = 0 + (E_x/c)i + (E_y/c)j + (E_z/c)k + B_x i + B_y j + B_z k\]
Actually, this needs refinement. Better formulation:
\[\mathcal{F} = \frac{1}{c}(\mathbf{E} + i c\mathbf{B}) \cdot \vec{\sigma}\]
where \(\vec{\sigma} = (i, j, k)\).
6.2 Maxwell's Equations
All of Maxwell's equations can be written as a single quaternion equation:
\[\partial \mathcal{F} = \frac{4\pi}{c}\mathcal{J}\]
where \(\partial = \frac{1}{c}\frac{\partial}{\partial t} + i\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} + k\frac{\partial}{\partial z}\) is the quaternionic derivative operator, and \(\mathcal{J}\) is the current quaternion.
7. Spinors and Quaternions
7.1 Connection to Pauli Matrices
The quaternion imaginary units correspond to Pauli matrices:
\begin{align}
i &\leftrightarrow \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\
j &\leftrightarrow \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \\
k &\leftrightarrow \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\end{align}
Pauli matrices satisfy the same multiplication rules as quaternions!
7.2 Spin-1/2 Particles
Electrons are quaternionic! The spin-1/2 nature of electrons directly relates to quaternion algebra:
- Spin operators are represented by quaternion units
- A 360Β° rotation gives -1 (quaternion double cover of SO(3))
- Requires 720Β° for full rotation back to original state
8. Why Exactly Three Spatial Dimensions?
8.1 Physical Arguments
Stability Arguments:
- 2D space: No stable atoms (Coulomb potential too confining)
- 3D space: Stable atoms, planets, chemistry works β
- 4D+ space: No stable orbits (inverse-cube law too weak)
8.2 Mathematical Uniqueness
Quaternions provide a unique mathematical structure for 3+1 dimensional spacetime:
- Division algebra: Can always divide (invert operations)
- Associativity: Preserves causality
- Three imaginary units: Exactly three spatial dimensions
- Non-commutativity: Encodes rotation structure of 3D space
- Natural metric: Quaternion norm gives Minkowski interval
8.3 The Anthropic Connection
Perhaps the question isn't "why three dimensions?" but rather:
We exist in 3D space because:
- Quaternions are the unique associative division algebra with >1 imaginary unit
- This forces exactly 3 spatial dimensions
- Only in 3D can complex chemistry and life emerge
- We observe what we must, given the mathematical constraints of physics
9. Quaternions and Quantum Mechanics
9.1 Quaternionic Quantum Mechanics
Can quantum mechanics be formulated with quaternions instead of complex numbers?
Attempts: Several physicists have tried, but face challenges:
- Non-commutativity complicates probability interpretation
- Measurement outcomes not naturally real-valued
- But: quaternions do appear in describing spin systems
9.2 Complex Numbers as Special Quaternions
Complex numbers \(\mathbb{C}\) are a subalgebra of quaternions \(\mathbb{H}\):
\[a + bi \subset a + bi + 0j + 0k\]
So quantum mechanics (built on \(\mathbb{C}\)) is actually using a restricted form of quaternions!
Interpretation: Perhaps quantum mechanics uses complex numbers (1 imaginary unit) for wave functions, while spacetime uses quaternions (3 imaginary units) for geometry. Both are different aspects of the same underlying quaternionic structure!
10. Octonions and Extra Dimensions
10.1 String Theory Connection
String theory predicts 10 or 11 dimensions. Could octonions play a role?
Octonions have 8 dimensions = 1 real + 7 imaginary
Some string theorists have explored octonionic formulations, but:
- Non-associativity makes them difficult to use
- The 7 imaginary units don't obviously map to 6 or 7 extra dimensions
- May play a role in exceptional structures (E8, G2)
10.2 Compactification
If extra dimensions exist, they must be "compactified" (curled up tiny). Maybe:
- Quaternions describe the 4 large spacetime dimensions
- Octonions describe the full 8-dimensional structure
- The 4 extra dimensions are compactified at Planck scale
11. Summary and Deep Questions
Key Conclusions
Yes, quaternions likely explain three spatial dimensions!
- Quaternions are the unique associative normed division algebra with 3 imaginary units
- These 3 imaginary units naturally correspond to x, y, z directions
- Quaternion norm automatically gives Minkowski spacetime interval
- All Lorentz transformations are quaternion operations
- Spin and angular momentum are quaternionic
- The mathematics forces exactly 3 spatial dimensions
11.1 Open Questions
- Why does nature "choose" quaternions over reals or complex numbers?
- Is there a deeper principle that selects associative division algebras?
- Do octonions describe physics at Planck scale or in extra dimensions?
- Can we formulate a fully quaternionic quantum field theory?
- Is the non-commutativity of quaternions related to uncertainty principle?
- Does the complex structure of QM + quaternion structure of spacetime reveal a hidden octonionic theory?
11.2 The Hierarchy
The Mathematical Hierarchy of Physics:
| Level |
Algebra |
Structure |
| Classical spacetime |
Real numbers β |
Time parameterization |
| Quantum mechanics |
Complex numbers β |
Wave functions, phases |
| Spacetime geometry |
Quaternions β |
3+1 dimensions, Lorentz group |
| Grand unification? |
Octonions π |
Exceptional groups, TOE |
Each level uses the next more complex division algebra!
12. Experimental Implications
12.1 Tests of Dimensionality
If quaternions are fundamental, we might expect:
- No evidence of extra spatial dimensions at any energy scale
- Perfect isotropy in all three spatial directions
- Rotation symmetry (SO(3)) is exact, not approximate
- No "leakage" into fourth spatial dimension
12.2 Quaternionic Corrections
If spacetime is fundamentally quaternionic rather than just having quaternionic symmetry, there might be tiny corrections to physics from non-commutativity:
\[\Delta E \sim \frac{\hbar c}{L}\left(\frac{L}{l_P}\right)^\alpha\]
where \(\alpha \approx 1-2\) and \(l_P\) is the Planck length.