Why Spin Is Not a Relativistic Effect

hfwang132@gmail.com (hfwang132) — Wed, 09 Jul 2025 19:23:54 +0800

Conclusion: integer spin is a classical effect, whereas half-integer spin is a quantum effect and has little to do with relativity.

The reason is simple: spin-1 is the smallest faithful representation of SO(3), while spin-1/2 is the smallest faithful representation of SU(2).

So why can quantum mechanics lift SO(3) to SU(2)? Because quantum states are rays and are equivalent up to a global phase. In other words, quantum mechanics requires projective representations of SO(3). And projective representations of SO(3) are in one-to-one correspondence with representations of SU(2) (Bargmann’s theorem). This is why quantum mechanics considers SU(2) rather than SO(3).

What Is Curl in Higher-Dimensional Spaces?

hfwang132@gmail.com (hfwang132) — Fri, 26 May 2023 21:00:37 +0800

Differential Forms

Before introducing curl, we must first introduce differential forms and the exterior derivative operator.

An n-form can be defined as an alternating multilinear map \(\omega:(T_pM)^n\rightarrow \mathbb{R}\) . It maps multiple vectors to a real number. Moreover, it satisfies alternation: exchanging two input vectors introduces an additional minus sign in the output.

Thus, an n-form can be explicitly defined as follows:

\[\omega^1\wedge \omega^2\wedge\cdots\wedge \omega^n(v_1,v_2,\cdots,v_n)= \begin{vmatrix} \omega^1(v_1) & \cdots & \omega^{1}(v_n) \\ \vdots & \ddots & \vdots \\ \omega^n(v_1) & \cdots & \omega^n(v_n) \end{vmatrix}\in\mathbb{R}\]

The set of all n-forms on \(T_pM\) can be written as \(\bigwedge^n(T^*_pM)\) .

Mathematical Methods of Physics on Haifei's Home

Why Spin Is Not a Relativistic Effect

What Is Curl in Higher-Dimensional Spaces?

Differential Forms