Macroscopic Observability of Spinorial Sign Changes under 2π Rotations

Abstract

The question of observability of sign changes under 2π rotations is considered. It is shown that in certain circumstances there are observable consequences of such sign changes in classical physics. A macroscopic experiment is proposed which could in principle detect the 4π periodicity of rotations.

Appendix A: Parallelizing Torsion in S 3 and IRP3

In this appendix we show that both IRP³∼SO(3) and its covering space S ³ ∼ SU(2) can be characterized by torsion alone, since their Riemann curvatures vanish identically with respect to the Weitzenböck connection. To this end, we begin by defining possible basis vectors in IR⁴, which we have taken to be the embedding space:

$$ \left\{\begin{array}{llll} +{q}_{0} & +{q}_{1} & +{q}_{2} & +{q}_{3} \\ -{q}_{1} & +{q}_{0} & +{q}_{3} & -{q}_{2} \\ -{q}_{2} & -{q}_{3} & +{q}_{0} & +{q}_{1} \\ -{q}_{3} & +{q}_{2} & -{q}_{1} & +{q}_{0} \end{array}\right\}. $$

(A.1)

Either the four column vectors or the four row vectors of this array may be taken to form an orthonormal basis of IR⁴. If we take the first row of the array to represent the points of S ³ as in definition (1), then, as we derived in (41) to (43), the remaining rows of the array provide the basis of the tangent space T _q S ³ at each point of S ³:

$$\begin{array}{@{}rcl@{}} \beta_{1}(\textbf{q})&=&({\textbf{e}_{2}}\wedge{\textbf{e}_{3}})\textbf{q} =(-q_{1},+q_{0},+q_{3},-q_{2}), \end{array} $$

(A.2)

$$\begin{array}{@{}rcl@{}} \beta_{2}(\textbf{q})&=&({\textbf{e}_{3}}\wedge{\textbf{e}_{1}})\textbf{q} =(-q_{2},-q_{3},+q_{0},+q_{1}), \end{array} $$

(A.3)

$$\begin{array}{@{}rcl@{}} \beta_{3}(\textbf{q})&=&({\textbf{e}_{1}}\wedge{\textbf{e}_{2}})\textbf{q} =(-q_{3},+q_{2},-q_{1},+q_{0}). \end{array} $$

(A.4)

Similarly, in a dual description, if we take the leftmost column of the array (A.1) to represent the points of S ³, then the remaining columns of the array provide the basis of the cotangent space ${T^{\ast }_{\textbf {q}}S^{3}}$ at each point of S ³:

$$\begin{array}{@{}rcl@{}} \beta^{1}(\textbf{q}^{\dagger})&=&({\textbf{e}_{2}}\wedge{\textbf{e}_{3}})\textbf{q}^{\dagger} =(+q_{1},+q_{0},-q_{3},+q_{2}), \end{array} $$

(A.5)

$$\begin{array}{@{}rcl@{}} \beta^{2}(\textbf{q}^{\dagger})&=&({\textbf{e}_{3}}\wedge{\textbf{e}_{1}})\textbf{q}^{\dagger} =(+q_{2},+q_{3},+q_{0},-q_{1}), \end{array} $$

(A.6)

$$\begin{array}{@{}rcl@{}} \beta^{3}(\textbf{q}^{\dagger})&=&({\textbf{e}_{1}}\wedge{\textbf{e}_{2}})\textbf{q}^{\dagger} =(+q_{3},-q_{2},+q_{1},+q_{0}). \end{array} $$

(A.7)

For our calculations we shall use the index notation ${\beta ^{i}_{\mu }}$ for the basis elements and ${\beta _{i}^{\mu }}$ for their inverses, with the index μ = 1, 2, 3 representing the tangent space axes and the index i = 0, 1, 2, 3 representing the embedding space axes. Then the matrices ${\beta ^{i}_{\mu }}$ and their inverses ${\beta _{i}^{\mu }}$ satisfy

$$ \beta_{i}^{\mu}\beta^{i}_{\nu}=\delta_{\nu}^{\mu}\text{and} \beta^{i}_{\mu}\beta_{j}^{\mu}=\delta_{j}^{i}, $$

(A.8)

with ${\delta _{\nu }^{\mu }}$ as a 3 × 3 sub-matrix and ${\delta _{j}^{i}}$ as a 4 × 4 matrix.

Now in a parallelized 3-sphere a spinor v(p) ∈ T _p S ³ is said to be absolutely parallel to a spinor w(q) ∈ T _q S ³ if all of the components of v(p) at T _p S ³ are equal to those of w(q) at T _q S ³. That is to say, if v(p) = v ^μ β _μ (p) and w(q) = w ^μ β _μ (q), then v ^μ = w ^μ. For a parallelized 3-sphere we therefore require that the components of any spinor v(p) at a point p ∈ S ³ remain the same when it is parallel transported to a nearby point p+𝜖 ∈ S ³ [13]:

$$ v^{\mu}(p)\beta^{i}_{\mu}(p)=v^{\mu}(p+\epsilon)\beta^{i}_{\mu}(p+\epsilon). $$

(A.9)

By expanding the right hand side up to terms of order 𝜖 we obtain

$$\begin{array}{@{}rcl@{}} &&v^{\mu}(p+\epsilon)\beta^{i}_{\mu}(p+\epsilon) \\ &&=\left[v^{\mu}(p)-\epsilon^{\nu}v^{\alpha}(p){\Omega}_{\nu\alpha}^{\mu}(p)\right]\left[\beta^{i}_{\mu}(p)+ \epsilon^{\nu}\partial_{\nu}\beta^{i}_{\mu}(p)\right] \\ &&=v^{\mu}(p)\beta^{i}_{\mu}(p) \\ &&-\epsilon^{\nu}v^{\alpha}(p) \left[{\Omega}_{\nu\alpha}^{\mu}(p)\beta^{i}_{\mu}(p)-\partial_{\nu}\beta^{i}_{\alpha}(p)\right]+O(\epsilon^{2}), \end{array} $$

(A.10)

where ${{\Omega }_{\nu \alpha }^{\mu }}$ are the connection coefficients. Evidently, the second term of this equation must vanish for the relation (A.9) to hold. This gives the connection coefficients ${{\Omega }_{\nu \alpha }^{\mu }}$ in terms of the partial derivatives of the basis elements:

$$ {\Omega}_{\nu\alpha}^{\mu}(p)\beta^{i}_{\mu}(p)=\partial_{\nu}\beta^{i}_{\alpha}(p). $$

(A.11)

Contracting this relation with the basis elements ${\beta _{i}^{\sigma }(p)}$ then leads to the Weitzenböck connection [cf. (46)]:

$$ {\Omega}_{\nu\alpha}^{\mu}(p)=\beta_{i}^{\mu}(p)\partial_{\nu}\beta^{i}_{\alpha}(p). $$

(A.12)

Alternatively (but equivalently), absolute parallelism on S ³ can be defined by requiring that the basis elements ${\beta ^{i}_{\mu }}$ remain covariantly constant during parallel transport:

$$ \nabla_{\alpha}\beta^{i}_{\nu}:=\partial_{\alpha}\beta^{i}_{\nu}-{\Omega}_{\alpha\nu}^{\mu}\beta^{i}_{\mu}=0. $$

(A.13)

Solving this equation for ${{\Omega }^{\mu }_{\nu \alpha }}$ again gives the connection obtained in (A.12). We can now evaluate the curvature tensor of S ³ with respect to this asymmetric connection:

$$\begin{array}{@{}rcl@{}} {\mathcal R}^{\sigma}_{\alpha\mu\nu}[S^{3}]&=&\partial_{\mu}{\Omega}^{\sigma}_{\nu\alpha}-\partial_{\nu}{\Omega}^{\sigma}_{\mu\alpha}+ {\Omega}^{\lambda}_{\nu\alpha}{\Omega}^{\sigma}_{\mu\lambda} - {\Omega}^{\lambda}_{\mu\alpha}{\Omega}^{\sigma}_{\nu\lambda} \\ &=&\partial_{\mu}(\beta_{i}^{\sigma}\partial_{\nu}\beta^{i}_{\alpha})-\partial_{\nu}(\beta_{i}^{\sigma} \partial_{\mu}\beta^{i}_{\alpha}) \\ &+&\beta_{k}^{\sigma}\partial_{\mu}\beta^{k}_{\lambda}\beta_{i}^{\lambda}\partial_{\nu}\beta^{i}_{\alpha} -\beta_{k}^{\sigma}\partial_{\nu}\beta^{k}_{\lambda}\beta_{i}^{\lambda}\partial_{\mu}\beta^{i}_{\alpha} \\ &=&\partial_{\mu}\beta_{i}^{\sigma}\partial_{\nu}\beta^{i}_{\alpha}+\beta_{i}^{\sigma}\partial_{\mu}\partial_{\nu}\beta^{i}_{\alpha} -\partial_{\nu}\beta_{i}^{\sigma}\partial_{\mu}\beta^{i}_{\alpha}-\beta_{i}^{\sigma}\partial_{\nu}\partial_{\mu}\beta^{i}_{\alpha} \\ &-&\beta_{i}^{\sigma}\beta^{i}_{\lambda}\partial_{\mu}\beta_{k}^{\lambda}\partial_{\nu}\beta^{k}_{\alpha}+ \beta_{i}^{\sigma}\beta^{i}_{\lambda}\partial_{\nu}\beta_{k}^{\lambda}\partial_{\mu}\beta^{k}_{\alpha} \\ &=&\partial_{\mu}\beta_{i}^{\sigma}\partial_{\nu}\beta^{i}_{\alpha}-\partial_{\nu}\beta_{i}^{\sigma}\partial_{\mu}\beta^{i}_{\alpha} -\partial_{\mu}\beta_{i}^{\sigma}\partial_{\nu}\beta^{i}_{\alpha}+\partial_{\nu}\beta_{i}^{\sigma}\partial_{\mu}\beta^{i}_{\alpha} \\ &=&0. \end{array} $$

(A.14)

Thus the curvature of S ³ with respect to Weitzenböck connection vanishes identically. The geometric properties of the quaternionic 3-sphere are thus entirely captured by the parallelizing torsion, which is evaluated in (48). In the present index notation it can be expressed as

$$ {\mathcal T}_{\mu\nu}^{\sigma}[S^{3}]={\Omega}_{\mu\nu}^{\sigma}-{\Omega}_{\nu\mu}^{\sigma}= \beta_{i}^{\sigma}\left(\partial_{\mu}\beta^{i}_{\nu}-\partial_{\nu}\beta^{i}_{\mu}\right). $$

(A.15)

It is important to recognize here that the quotient map φ:S ³ → IRP³ we discussed in Section 3 to obtain IRP³ from S ³ is a local isometry. The infinitesimal map d φ(q):T _q S ³→T _[q]IRP³ is therefore an isometry. The inner product in T _q S ³ defined by (37) at each point of S ³ is thus preserved only locally under the action of φ:

$$\begin{array}{@{}rcl@{}} \langle\{a_{\mu}{\beta}_{\mu}(\textbf{q})&\},\{b_{\nu}{\beta}_{\nu}(\textbf{q})\}\rangle \\ &=\langle\{a_{\mu}{\xi}_{\mu}[\varphi(\textbf{q})]\},\{b_{\nu}{\xi}_{\nu}[\varphi(\textbf{q})]\}\rangle, \end{array} $$

(A.16)

where {ξ _μ [φ(q)]} are the basis defining T _[q]IRP³. Thus, because of the presence of torsion, the rule for parallel transporting a spinor from one point to another on IRP³ is not preserved by the map φ:S ³ → IRP³. It is given by a different Weitzenböck connection, defined by ξ _μ :

$$ \widehat{\Omega}_{\nu\alpha}^{\mu}(p)=\xi_{i}^{\mu}(p)\partial_{\nu}\xi^{i}_{\alpha}(p). $$

(A.17)

Since the basis {ξ _μ } are covariantly constant with respect to ${\widehat {\Omega }_{\nu \alpha }^{\mu }}$, the curvature of IRP³ also vanishes identically:

$$ {\mathcal R}^{\sigma}_{\alpha\mu\nu}[\text{I\!R}\mathrm{P}^{3}]=0. $$

(A.18)

Despite the fact that the metric tensor J _{μ ν} on IRP³ is no longer Euclidean [cf. (73)], the steps in the derivation analogous to (A.14) go through because, just as in (A.8), the matrices ${\xi ^{i}_{\mu }}$ and their inverses ${\xi _{i}^{\mu }}$ continue to satisfy

$$ \xi_{i}^{\mu}\xi^{i}_{\nu}=\delta_{\nu}^{\mu}\text{and} \xi^{i}_{\mu}\xi_{j}^{\mu}=\delta_{j}^{i}. $$

(A.19)

These are simply reciprocal relations between matrices and their inverses and not the orthonormality relations for the basis. Consequently, the manifold IRP³ remains as parallelized as S ³ under the map φ:S ³ → IRP³. This state of affairs, in fact, forms the basis of some powerful theorems in the mathematics of division algebras [18].

What does change under the map φ:S ³ → IRP³ is the characteristic torsion within the parallelized manifold:

$$ {\mathcal T}_{\mu\nu}^{\sigma}[\text{I\!R}\mathrm{P}^{3}]=\widehat{\Omega}_{\mu\nu}^{\sigma} -\widehat{\Omega}_{\nu\mu}^{\sigma}= \xi_{i}^{\sigma}\left(\partial_{\mu}\xi^{i}_{\nu}-\partial_{\nu}\xi^{i}_{\mu}\right). $$

(A.20)

The difference between the two expressions of the torsion, namely (A.15) and (A.20), is seen more transparently in the notation of geometric algebra used in the derivation of (48). In this notation the torsion within S ³ is given by

$$ {\mathcal T}[{\boldsymbol\beta}(\textbf{a}),{\boldsymbol\beta}(\textbf{b})]=\textbf{a}\wedge\textbf{b} ={\boldsymbol\beta}(\textbf{c})\sin\eta_{\textbf{ab}}, $$

(A.21)

whereas that within IRP³ is given by

$$ {\mathcal T}[{\boldsymbol\xi}(\textbf{a}),{\boldsymbol\xi}(\textbf{b})]=\textbf{a}\wedge\textbf{b} ={\boldsymbol\xi}(\textbf{c})\sin\alpha_{\textbf{ab}}. $$

(A.22)

Here c = a×b/|a×b|, and—as we discussed in Section 3—the angles η _ab and α _ab are non-linearly related as

$$\begin{array}{@{}rcl@{}} \sin\alpha_{\textbf{a}\textbf{b}}= \left\{\begin{array}{lll} +\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} &\text{if}-\frac{\pi}{2} \leq \eta_{\textbf{a}\textbf{b}} \leq \frac{\pi}{2} \\ \\ +2-\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} &\text{if}\frac{\pi}{2} \leq \eta_{\textbf{a}\textbf{b}} \leq \frac{3\pi}{2}. \end{array}\right. \end{array} $$

(A.23)

But since the map φ:S ³ → IRP³ is a local isometry, the bivectors β(c) and ξ(c) representing a binary rotation about c are the same. Consequently, the torsion within IRP³ in terms of the Euclidean angle η _ab is given by

$$\begin{array}{@{}rcl@{}} {\mathcal T}_{\textbf{a}\textbf{b}}={\boldsymbol\beta}(\textbf{c})\times \left\{\begin{array}{lll} +\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} &\text{if}-\frac{\pi}{2} \leq \eta_{\textbf{a}\textbf{b}} \leq \frac{\pi}{2} \\ \\ +2-\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} &\text{if}\frac{\pi}{2} \leq \eta_{\textbf{a}\textbf{b}} \leq \frac{3\pi}{2}. \end{array}\right. \end{array} $$

(A.24)

Comparing this expression with (A.21) we now clearly see the difference between the parallelizing torsions within S ³ and IRP³. It is this difference that is reflected in Fig. 3.