'Coordinate Independence, and Quantum Mechanics' - by ChatGPT

Complex Multiplication, Coordinate Independence, and the Nature of Quantum Mechanics

Complex numbers are not merely a convenient mathematical shorthand. They encode geometric and algebraic ideas which run surprisingly deep. One of those depths is the distinction between coordinate independence in vector spaces and the structured role that complex multiplication plays in quantum mechanics. These two concepts are often assumed to cohere, but on inspection they live in quite different philosophical neighbourhoods.

Vector Addition: Coordinate-Free Geometry

Let us begin with something uncontroversial. The addition of complex numbers corresponds geometrically to the addition of 2D vectors. This operation is coordinate-independent in the usual Euclidean sense. Rotate the plane, reflect it, translate the origin—none of this affects the outcome of adding two vectors. What matters are relative displacements. The parallelogram rule works regardless of your choice of coordinate axes.

In mathematical terms, vector addition is intrinsic to the affine structure of the plane. It respects the full symmetry group of Euclidean geometry. It is, in short, a democratic operation.

Multiplication: Structure Dependent and Orientation-Specific

Complex multiplication, however, is a very different animal. To multiply two complex numbers is to rotate and scale—an operation heavily reliant on the fixed identification of the imaginary unit i as a direction orthogonal to the real line, and on a specific orientation (counter-clockwise in the standard plane). This is not coordinate-free. Rotate the plane, and unless you redefine what you mean by i, the operation itself changes.

Multiplication is not just a function of relative displacement; it is a structured linear transformation with a preferred basis. As such, it fails the test of coordinate independence as understood in the geometry of real vector spaces.

So Why Is It Central to Quantum Mechanics?

Quantum mechanics is famously fond of symmetries. Coordinate independence, gauge invariance, unitary transformations—these are sacred cows. So why does it build its entire formalism on an operation (complex multiplication) that seems so structurally rigid?

Because in quantum theory, complex numbers do not live in the Euclidean plane. They are not geometric vectors. They are scalars in a field, and the vectors live in a Hilbert space defined over that field. Complex multiplication doesn’t rotate arrows—it governs interference, phase, and time evolution. These are internal symmetries, not spatial ones.

Indeed, what looks like a structural bug in the complex plane becomes a feature in Hilbert space. The imaginary unit i enables the Schrödinger equation to generate unitary evolution via the exponential map. It is because multiplication by eiθ induces oscillatory behaviour—interference patterns, phase shifts—that quantum mechanics becomes predictive and rich. The theory is not invariant under arbitrary redefinitions of i—but it is invariant under the symmetry transformations it deems physically meaningful: unitaries, not Euclidean rotations.

Gauge, Not Geometry

The invariance that matters in quantum mechanics is not under spatial coordinate transformations, but under U(1) gauge transformations. Multiply a quantum state by a global phase factor and nothing changes. That’s a symmetry. This is what complex multiplication encodes in practice—not geometry, but a freedom in representation.

To demand coordinate independence of complex multiplication is to ask the wrong question. The complex structure is not meant to be arbitrary. It is the skeleton upon which the theory is built. We do not float freely among field choices in quantum theory. We live in a complex Hilbert space, and we deal in amplitudes, not Euclidean vectors.

Conclusion

So yes, complex multiplication is not coordinate-independent in the naive geometric sense. But quantum mechanics is not built on the Euclidean plane. It is built on Hilbert spaces and symmetry groups. The appearance of rigidity in complex multiplication is in fact the source of the theory’s expressive power—its ability to encode phase, time, and interference. The coordinate-independence of quantum theory lies elsewhere: not in the mutability of the imaginary unit, but in the invariance of amplitudes under transformation. A different sort of freedom, but no less real.

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Appendix: The Matrix Representation of Complex Multiplication

Multiplying a complex number

z = a + ib

by another complex number

w = x + iy

can be represented as a matrix transformation on ℝ2. In this interpretation, z becomes a two-dimensional real vector:

    z  ≡  [ a ]
            [ b ]

Multiplication by w corresponds to applying the 2×2 real matrix:

    Mw = [  x  -y ]
              [  y   x ]

The product wz is then:

    Mw · z = [ xa - yb  ]
                   [ ya + xb ]

Giving exactly the real and imaginary parts of the complex product wz.

Geometric Interpretation

• Scaling: The matrix scales by the modulus |w| = √(x² + y²).
• Rotation: It rotates by the argument arg(w) = tan-1(y/x).

Properties of the Multiplication Matrix

• Determinant: det(Mw) = x² + y² = |w|²
• Unitary when |w| = 1: The transformation is a pure rotation.
• Composition: Mw₁ · Mw₂ = Mw₁w₂, preserving complex multiplication.

This matrix representation reveals why complex multiplication is not coordinate-independent in the geometric sense: the matrix Mw presumes a fixed orientation of the real and imaginary axes. It is a structured linear transformation defined relative to those axes — and thus carries a preferred direction and handedness.