International Journal of Theoretical Physics

, Volume 54, Issue 6, pp 2042–2067 | Cite as

Macroscopic Observability of Spinorial Sign Changes under 2π Rotations

Article

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.

1 Introduction

It is well known that the state of a rotating body in the physical space depends in general not only on its local configuration but also on its topological relation to the rest of the universe [1]. While the former feature of space is familiar from everyday life, the latter feature too can be demonstrated by a simple rope trick, or Dirac’s belt trick [2, 3]. What is less easy to establish, however, is whether there are observable effects of the latter relation—often referred to as “orientation-entanglement relation” [1]—in the macroscopic domain [3]. The purpose of the present paper is to describe a macroscopic experiment in which orientation-entanglement has observable consequences.

The appropriate operational question in this context is: Whether rotating bodies in the physical space respect 2π periodicity or 4π? Consider, for example, a rock in an otherwise empty universe. If it is rotated by 2π radians about some axis, then there is no reason to doubt that it will return back to its original state with no discernible effects [3]. This, however, cannot be expected if there is at least one other object present in the universe [3]. The rock will then have to rotate by another 2π radians (i.e., a total of 4π radians) to return back to its original state relative to that other object, as proved by the twist in the belt in Dirac’s belt trick [2]. The twist shows that what is an identity transformation for an isolated object is not an identity transformation for an object that is rotating relative to other objects [3]. We can quantify this loss of identity by adapting a spinor representation of rotations.

Let the configuration space of all possible rotations of the rock be represented by the set S3 of unit quaternions (which, as we shall soon see, is a parallelized 3-sphere):
$$ S^{3}:=\left\{\textbf{q}(\psi, \textbf{a}):=\exp{\left[{\boldsymbol\beta}(\textbf{a}) \frac{\psi}{2} \right]} \Bigg| || \textbf{q}(\psi, \textbf{a}) ||^{2}=1\right\}, $$
(1)
where β(a) is a bivector rotating about a ∈ IR3 with the rotation angle ψ in the range 0 ≤ ψ < 4π. Throughout this paper we shall follow the notations, conventions, and terminology of geometric algebra [4, 5]. Accordingly, β(a) ∈ S2S3 can be parameterized by a unit vector a = a1e1 + a2e2 + a3e3 ∈ IR3 as
$$\begin{array}{@{}rcl@{}} {\boldsymbol\beta}(\textbf{a}) &:=& (I\cdot\textbf{a}) \\ &=& a_{1} (I\cdot\textbf{e}_{1}) + a_{2}(I\cdot\textbf{e}_{2}) + a_{3}(I\cdot\textbf{e}_{3}) \\ &=& a_{1} {\textbf{e}_{2}} \wedge {\textbf{e}_{3}} + a_{2} {\textbf{e}_{3}} \wedge {\textbf{e}_{1}} + a_{3} {\textbf{e}_{1}} \wedge {\textbf{e}_{2}}, \end{array} $$
(2)
with β2(a) = −1. Here the trivector I:=e1e2e3 (which also squares to −1) represents a volume form of the physical space [4]. Each configuration of the rock can thus be represented by a quaternion of the form
$$ \textbf{q}(\psi, \textbf{a}) = \cos\frac{\psi}{2} + {\boldsymbol\beta}(\textbf{a}) \sin\frac{\psi}{2}, $$
(3)
with ψ being its rotation angle from q(0, a) = 1. More significantly for our purposes, it is easy to check that q(ψ, a) respects the following rotational symmetries:
$$ \textbf{q}(\psi+2\kappa\pi, \textbf{a}) = - \textbf{q}(\psi, \textbf{a})\; \text{for}\;\kappa=1,3,5,7,{\dots} $$
(4)
$$ \textbf{q}(\psi+4\kappa\pi, \textbf{a}) = + \textbf{q}(\psi, \textbf{a})\;\text{for}\;\kappa=0,1,2,3,{\dots} $$
(5)
Thus q(ψ, a) correctly represents the state of a rock that returns to itself only after even multiples of a 2π rotation.

This is well and good as a mathematical representation of the orientation-entanglement [1], but can such changes in sign have observable consequences? The answer to this question turns out to be in the affirmative [6], provided the “rock” happens to be microscopic and can be treated quantum mechanically [7]. Despite the fact that physical quantities are quadratic in the wave function, Aharonov and Susskind were able to demonstrate that in certain circumstances sign changes of spinors under 2π rotations can lead to observable effects [6, 7, 8]. In particular, they were able to show that if fermions are coherently shared between two spatially isolated systems, then a relative rotation of 2π may be observable. They noted, however, that in classical physics relative as well as absolute 2κπ rotations are unobservable [6]. That is to say, in classical physics (4) has no observable consequences [3, 8].

It turns out, however, that this last conclusion is not quite correct. In what follows we shall demonstrate that in certain circumstances (4) does lead to observable consequences in classical physics. This is not surprising when one recalls the gimbal-lock singularity encountered in some representations of rotations, such as Euler angles. The singularity arises because the group SO(3) of proper rotations in the physical space happens to be a connected but not a simply-connected topological manifold [9, 10].

To visualize this, imagine a solid ball of radius π. Let each point in this ball correspond to a rotation with the direction from the origin of the ball representing the axis of rotation and the distance from the origin representing the angle of rotation. Since rotations by +π and −π are the same, the antipodes of the surface of the ball must be identified with each other. The ball therefore does not represent rotations in a continuous manner. Antipodal points of the ball are considered equivalent even though they are far apart in the representation [9]. This lack of continuity is the source of the gimbal-lock singularity.

This singularity can be eliminated [10] by considering a second ball superimposed upon the first with coordinates that are a mirror image of the first. The two surfaces are then glued together so that a rotation by +π on the one ball is connected to the corresponding rotation by −π on the other. This object is then equivalent to S3 in the same way that two discs glued together form S2. The topology is now represented by a simply-connected and continuous shape in four parameters, such as the set
$$ \left\{\cos\frac{\psi}{2}, a_{1} \sin\frac{\psi}{2}, a_{2} \sin\frac{\psi}{2}, a_{3}\sin\frac{\psi}{2}\right\} $$
(6)
appearing in definition (3). The gimbal-lock singularity is eliminated in this representation because each rotation is now quantified, not only by axis and angle, but also by the relative orientation of the gimbal with respect to its surroundings. Thus removal of a gimbal-lock provides a striking manifestation of orientation-entanglement in classical physics. In what follows we shall demonstrate a similar macroscopic scenario, giving rise to observable consequences of (4) in classical physics.

To this end, recall that the set of all unit quaternions satisfying (4) and (5) forms the group SU(2), which is homeomorphic to the simply-connected space S3 [11]. This group is relevant in the macroscopic world when rotations of objects relative to other “fixed” objects are important. On the other hand, purely local observations of rotations seem to be insensitive to the sign changes of the quaternions constituting SU(2). In other words, the points −q(ψ, a) and +q(ψ, a) within SU(2) seem to represent one and the same rotation in the physical space IR3. The group SO(3) is therefore obtained by identifying each quaternion −q(ψ, a) with its antipode +q(ψ, a) within SU(2)—i.e., by identifying the antipodal points of S3. The space that results from this identification is the real projective space, IRP3, which is a connected but not a simply-connected manifold. Consequently, the geodesic distances \({{\mathcal {D}}(\textbf {a}, \textbf {b})}\) between two quaternions q(ψa, a) and q(ψb, b) representing two different rotations within IR3 can be measurably different on the manifolds SU(2) ∼ S3 and SO(3) ∼ IRP3. These distances would thus provide a signature of spinorial sign changes between q(ψa, a) and q(ψb, b) within classical or macroscopic physics.

In the next two sections we therefore derive the relative geodesic distances on the manifolds SU(2) and SO(3). Then, in the subsequent section, we sketch a macroscopic experiment, which, if realized, would allow to distinguish the distances on SU(2) from those on SO(3) by observing correlations among a set of spin angular momenta. Then, in Section 5, we derive the correlation function from the first principles, before concluding the paper in Section 6.

2 Geodesic Distance on SU(2)

In this section the geometry of a parallelized 3-sphere will play an important role. To understand this geometry, consider the quadruple of real numbers defined in (6). These numbers may be used to define a three-dimensional surface embedded in IR4, homeomorphic to the 3-sphere of unit quaternions. This is possible because there exists a bi-continuous one-to-one correspondence between the parameter space (6) and the space of unit quaternions. These two spaces are thus topologically equivalent [11].

To see this, denote the points of the parameter space (6) by the tips of the unit vectors Y(ψ, a)∈IR4 as
$$\begin{array}{@{}rcl@{}} \textbf{Y}(\psi, \textbf{a}) &:= &\left(\cos\frac{\psi}{2}\right) \textbf{e}_{0} +\left(a_{1} \sin\frac{\psi}{2} \right) \textbf{e}_{1} \\ &+&\left(a_{2} \sin\frac{\psi}{2} \right) \textbf{e}_{2} + \left(a_{3} \sin\frac{\psi}{2}\right)\textbf{e}_{3}, \end{array} $$
(7)
where \({{Y_{0}^{2}}+{Y_{1}^{2}}+{Y_{2}^{2}}+{Y_{3}^{2}}=1}\) because \({{a_{1}^{2}}+{a_{2}^{2}}+{a_{3}^{2}}=1}\). Y(ψ, a) thus sweeps the surface of a unit ball in IR4, and hence sweeps the points of a round 3-sphere of constant curvature and vanishing torsion. It can be transformed into a spinor q(ψ, a)∈IR4 representing a point of a “flat” 3-sphere as follows:
$$ \left(\begin{array}{l} \mathrm{q}_{0} \\ \mathrm{q}_{1} \\ \mathrm{q}_{2} \\ \mathrm{q}_{3} \end{array}\right)= \left(\begin{array}{llll} + \textbf{e}_{0} & 0 & 0 & 0 \\ 0 & +I & 0 & 0 \\ 0 & 0 & +I & 0 \\ 0 & 0 & 0 & +I \end{array}\right) \left(\begin{array}{l} \mathrm{Y}_{0} \\ \mathrm{Y}_{1} \\ \mathrm{Y}_{2} \\ \mathrm{Y}_{3} \end{array}\right). $$
(8)
This operation transforms the vector
$$ \textbf{q}(\psi, \textbf{a}):=\mathrm{q}_{0} \textbf{e}_{0}+\mathrm{q}_{1} \textbf{e}_{1}+\mathrm{q}_{2} \textbf{e}_{2}+\mathrm{q}_{3} \textbf{e}_{3}\in\text{I\!R}^{4} $$
(9)
into a unit quaternion:
$$\begin{array}{@{}rcl@{}} \textbf{q}(\psi, \textbf{a}) &=& \cos\frac{\psi}{2} \textbf{e}_{0} \textbf{e}_{0} + a_{1} (I\cdot\textbf{e}_{1}) \sin\frac{\psi}{2} \\ &+& a_{2}(I\cdot\textbf{e}_{2}) \sin\frac{\psi}{2} + a_{3}(I\cdot\textbf{e}_{3}) \sin\frac{\psi}{2} \\ &=& \cos\frac{\psi}{2} + a_{1}({\textbf{e}_{2}} \wedge {\textbf{e}_{3}}) \sin\frac{\psi}{2} \\ &+& a_{2} ({\textbf{e}_{3}} \wedge {\textbf{e}_{1}})\sin\frac{\psi}{2}+ a_{3} ({\textbf{e}_{1}}\wedge{\textbf{e}_{2}})\sin\frac{\psi}{2} \\ &=& \cos\frac{\psi}{2}+{\boldsymbol\beta}(\textbf{a}) \sin\frac{\psi}{2} \\ &=& \exp{\left[{\boldsymbol\beta}(\textbf{a})\frac{\psi}{2} \right]}, \end{array} $$
(10)
where q(ψ, a)q(ψ, a) = ||q(ψ, a)||2=1 since we have e0e0=e0e0 + e0e0=1 and β(a)β(a) = 1. As we shall see, q(ψ, a) represents a point of a 3-sphere that is “flat”, but exhibits non-zero and constant torsion.
The reciprocal relationship between the 4-vector (7) and the spinor (10) can now be succinctly described as
$$ \mathrm{Y}_{i}=\gamma^{\dagger}_{ij} \mathrm{q}_{j} \; \text{and}\; \mathrm{q}_{i}=\gamma_{ij}\mathrm{Y}_{j} , $$
(11)
where
$$ \gamma_{ij}^{\dagger}= \left(\begin{array}{llll} +\textbf{e}_{0} & 0 & 0 & 0 \\ 0 & -I & 0 & 0 \\ 0 & 0 & -I & 0 \\ 0 & 0 & 0 & -I \end{array}\right) $$
(12)
and
$$ \gamma_{ij}= \left(\begin{array}{llll} +\textbf{e}_{0} & 0 & 0 & 0 \\ 0 & +I & 0 & 0 \\ 0 & 0 & +I & 0 \\ 0 & 0 & 0 & +I \end{array}\right) $$
(13)
with \({\gamma _{ij}^{\dagger }\gamma _{ij}=\text {id}}\). The transformation-maps \({\mathrm {Y}_{i}=\gamma ^{\dagger }_{ij}\mathrm {q}_{j}}\) and qi=γijYj are thus smooth bijections \({\textbf {Y}: {S^{3}_{F}}\rightarrow {S^{3}_{R}}}\) and \({\textbf {q}: {S^{3}_{R}}\rightarrow {S^{3}_{F}}}\) mentioned above, with \({{S^{3}_{F}}}\) representing the flat 3-sphere and \({{S^{3}_{R}}}\) representing the round 3-sphere.

It is important to note here that the transformation of the 4-vectors Y(ψ, a) defined in (7) into the spinors q(ψ, a) defined in (10) induces dramatic differences in the geometry and topology of the 3-sphere. The round 3-sphere, charted by Y(ψ, a), is known to have constant curvature but vanishing torsion. This is the 3-sphere that appears in the Friedmann-Robertson-Walker solution of Einstein’s field equations [12]. The flat 3-sphere, charted by q(ψ, a), on the other hand, has vanishing curvature but constant torsion. This is the 3-sphere that appears in the teleparallel gravity [13]. In other words, the 3-sphere constituted by quaternions is parallelized—by the very algebra of quaternions—to be absolutely flat [14, 15].

To understand the topological difference between these two 3-spheres, let us bring out the round metric hidden in (7) by expressing the components of the vector a in the polar coordinates of an equatorial 2-sphere:
$$ a_{1} = \sin\theta \cos\phi, $$
(14)
$$ a_{2} = \sin\theta \sin\phi, $$
(15)
$$ a_{3} = \cos\theta. $$
(16)
In terms of these components the components of Y(ψ, a) take the form
$$ Y_{0} = \cos\chi, $$
(17)
$$ Y_{1}= \sin\chi \sin\theta \cos\phi, $$
(18)
$$ Y_{2} = \sin\chi \sin\theta \sin\phi, $$
(19)
$$ Y_{3} = \sin\chi \cos\theta. $$
(20)
where we have set ψ/2 = χ for convenience. Note that 𝜃 ranges from 0 to π, whereas χ and ϕ range from 0 to 2π. The corresponding line element can now be calculated from the differentials
$$ dY_{0} = - \sin\chi d\chi, $$
(21)
$$\begin{array}{@{}rcl@{}} dY_{1} &=& \cos\chi \sin\theta \cos\phi d\chi + \sin\chi \cos\theta \cos\phi d\theta \\ &-& \sin\chi \sin\theta \sin\phi d\phi \end{array}$$
(22)
$$ \begin{array}{@{}rcl@{}} dY_{2} &=& \cos\chi \sin\theta \sin\phi d\chi + \sin\chi \cos\theta \sin\phi d\theta \\ &+& \sin\chi \sin\theta \cos\phi d\phi, \end{array}$$
(23)
$$ dY_{3} = \cos\chi \cos\theta d\chi - \sin\chi \sin\theta d\theta. $$
(24)
Moreover, from the normalization condition
$$ {Y_{0}^{2}}+{Y_{1}^{2}}+{Y_{2}^{2}}+{Y_{3}^{2}}=1^{2}=1 $$
(25)
we also have
$$ 2Y_{0}dY_{0}+2Y_{1}dY_{1}+2Y_{2}dY_{2}+2Y_{3}dY_{3}=0, $$
(26)
which allows us to express dY0 in terms of dY1, dY2, and dY3. Then, in the hyper-spherical coordinates (χ, 𝜃, ϕ), the line element on the 3-sphere works out to be
$$\begin{array}{@{}rcl@{}} ds^{2}&=&g(d\textbf{Y},d\textbf{Y}) \\ &=&d{Y^{2}_{0}}+d{Y^{2}_{1}}+d{Y^{2}_{2}}+d{Y^{2}_{3}} \\ &=&d\chi^{2}+\sin^{2}\chi\left[d\theta^{2}+\sin^{2}\theta d\phi^{2}\right]. \end{array} $$
(27)
This is the Friedmann-Robertson-Walker line element [1] representing a 3-sphere embedded in a four-dimensional Euclidean space, with constant curvature and vanishing torsion:
$$ {\mathcal{R}}^{\alpha}_{\beta\gamma\delta}\not=0\text{but} {\mathcal{T}}_{\alpha\beta}^{\gamma}=0. $$
(28)
This metric, however, does not provide a single-valued coordinate chart over the entire 3-sphere [16]. In going from the north pole (χ = 0) to the equator (χ = π/2), the variable sinχ ranges from 0 to 1; however, in going from the equator to the south pole of the sphere (χ = π), it runs backwards from 1 to 0. Thus the space \({{S^{3}_{R}}}\) is charted by the coordinates that are not single-valued in sinχ. This can be seen more clearly by setting sinχ = r and rewriting the line element (27) as
$$ ds^{2}=\frac{dr^{2}}{1-r^{2}}+r^{2}\left[d\theta^{2}+\sin^{2}\theta d\phi^{2}\right]. $$
(29)
It is now easy to appreciate that there is a singularity in this metric at r = 1. This is of course the well known coordinate singularity which cannot be eliminated by a mere change of variables. It can be eliminated, however, by parallelizing or “flattening” the 3-sphere with respect to a set of quaternionic bases [15].
To understand this, let TqS3 denote the tangent space to S3 at the tip of the unit spinor q ∈ IR4, defined by
$$ T_{\textbf{q}}S^{3}:=\left\{(\textbf{q}, \textbf{t}_{q}) \Big| \textbf{q}, \textbf{t}_{q}\in \text{I\!R}^{4}, ||\textbf{q}||=1, \textbf{q}\cdot \textbf{t}_{\textbf{q}}^{\dagger}=0\right\}, $$
(30)
where \({\textbf {q}\cdot \textbf {t}_{\textbf {q}}^{\dagger }}\) represents the inner product between q and \({\textbf {t}_{\textbf {q}}^{\dagger }}\) with tq: = dq/dψ. Then, denoting the tip of q by q = qS3, the tangent bundle of S3 can be expressed as
$$ \mathrm{T}S^{3}:=\!\bigcup\limits_{q\in S^{3}}\{q\}\times T_{\textbf{q}}S^{3}. $$
(31)
Now this tangent bundle happens to be trivial:
$$ \mathrm{T}S^{3}\equiv S^{3}\times\text{I\!R}^{3}. $$
(32)
The triviality of the bundle TS3 means that the 3-sphere is parallelizable [14, 15]. A d-dimensional manifold is said to be parallelizable if it admits d vector fields that are linearly-independent everywhere. On a 3-sphere we can always find three linearly-independent vector fields that are nowhere vanishing. These vector fields can then be used to define a basis of the tangent space at each of its points. Thus, a global anholonomic frame can be defined on the 3-sphere that fixes each of its points uniquely.

The parallelizability of the 3-sphere, however, is not guaranteed for all of its representations, since there are more than one ways to embed one space into another. For example, one may consider embedding S3 into IR4 by means of the vector field Y(χ, 𝜃, ϕ) defined in (7), but as we saw above the resulting representation, namely \({{S^{3}_{R}}}\), would not be a parallelized sphere [15]. Given three linearly-independent vector fields forming a basis of the tangent space at one point of \({{S^{3}_{R}}}\), say at (χ, 𝜃, ϕ), it would not be possible to find three linearly-independent vector fields forming a basis of the tangent space at every other point of \({{S^{3}_{R}}}\). This turns out to be possible, however, if we switch from the vector field Y(χ, 𝜃, ϕ) to the spinor field q(χ, 𝜃, ϕ) defined by (10) with the property (4).

Suppose we are given a tangent space at the tip of a spinor q0 = (1, 0, 0, 0) ∈ IR4, spanned by the basis
$$\begin{array}{@{}rcl@{}} &&\left\{\beta_{1}(\textbf{q}_{0}),\beta_{2}(\textbf{q}_{0}),\beta_{3}(\textbf{q}_{0})\right\} \\ &\equiv& \left\{{\textbf{e}_{2}}\wedge{\textbf{e}_{3}},{\textbf{e}_{3}}\wedge{\textbf{e}_{1}},{\textbf{e}_{1}}\wedge{\textbf{e}_{2}}\right\}, \end{array} $$
(33)
with the base bivectors
$$ \beta_{1}(\textbf{q}_{0})=(0,1,0,0), $$
(34)
$$ \beta_{2}(\textbf{q}_{0})=(0,0,1,0), $$
(35)
$$ \text{and}\; \beta_{3}(\textbf{q}_{0})=(0,0,0,1) $$
(36)
satisfying the inner product
$$\begin{array}{@{}rcl@{}} \langle\beta_{\mu}(\textbf{q}_{0}),\beta^{\dagger}_{\nu}(\textbf{q}_{0})\rangle = \beta_{\mu}(\textbf{q}_{0})\cdot\beta^{\dagger}_{\nu}(\textbf{q}_{0})=\delta_{\mu\nu}, \end{array} $$
(37)
where μ, ν = 1, 2, 3 and \({\langle \beta _{\mu },\beta ^{\dagger }_{\nu }\rangle }\) is defined by the map
$$ \langle\cdot,\cdot\rangle: T_{p}S^{3}\times T^{\ast}_{p}S^{3} \rightarrow \text{I\!R}, $$
(38)
with p = pS3 (here \({T^{\ast }_{p}S^{3}}\) is the cotangent space at p [15]). This basis would allow us to express any arbitrary tangent bivector at the tip of q0 as
$$ I\cdot\textbf{n}=n_{1}{\textbf{e}_{2}}\wedge{\textbf{e}_{3}} +n_{2}{\textbf{e}_{3}}\wedge{\textbf{e}_{1}} +n_{3}{\textbf{e}_{1}}\wedge{\textbf{e}_{2}}. $$
(39)
Then the tangent bases {β1(q), β2(q), β3(q)} at the tip of any qS3 can be found by taking a geometric product of the basis (33) with q using the bivector subalgebra
$$ (I\cdot\textbf{e}_{\mu})(I\cdot\textbf{e}_{\nu})=-\delta_{{\mu}{\nu}} -\epsilon_{{\mu}{\nu}{\rho}}(I\cdot\textbf{e}_{\rho}) $$
(40)
(with repeated Greek indices summed over), which gives
$$\begin{array}{@{}rcl@{}} \beta_{1}(\textbf{q})&=&({\textbf{e}_{2}}\wedge{\textbf{e}_{3}})\textbf{q} \\ &=&-q_{1}+q_{0}({\textbf{e}_{2}}\wedge{\textbf{e}_{3}})+q_{3}({\textbf{e}_{3}}\wedge{\textbf{e}_{1}}) -q_{2}({\textbf{e}_{1}}\wedge{\textbf{e}_{2}}) \\ &=&(-q_{1},q_{0},q_{3},-q_{2}), \\ \end{array} $$
(41)
$$\begin{array}{@{}rcl@{}} \beta_{2}(\textbf{q})&=&({\textbf{e}_{3}}\wedge{\textbf{e}_{1}})\textbf{q} \\ &=&-q_{2}-q_{3}({\textbf{e}_{2}}\wedge{\textbf{e}_{3}})+q_{0}({\textbf{e}_{3}}\wedge{\textbf{e}_{1}}) +q_{1}({\textbf{e}_{1}}\wedge{\textbf{e}_{2}}) \\ &=&(-q_{2},-q_{3},q_{0},q_{1}), \\ \end{array} $$
(42)
$$\begin{array}{@{}rcl@{}} \beta_{3}(\textbf{q})&=&({\textbf{e}_{1}}\wedge{\textbf{e}_{2}})\textbf{q} \\ &=&-q_{3}+q_{2}({\textbf{e}_{2}}\wedge{\textbf{e}_{3}})-q_{1}({\textbf{e}_{3}}\wedge{\textbf{e}_{1}}) +q_{0}({\textbf{e}_{1}}\wedge{\textbf{e}_{2}}) \\ &=&(-q_{3},q_{2},-q_{1},q_{0}). \end{array} $$
(43)
It is easy to check that the bases {β1(q), β2(q), β3(q)} are indeed orthonormal for all q with respect to the usual inner product in IR4, with each of the three basis elements βμ(q) also being orthogonal to q = (q0, q1, q2, q3), and thus define a tangent space IR3 at the tip of that q. This procedure of finding orthonormal tangent bases at any point of S3 can be repeated ad infinitum, providing a continuous field of absolutely parallel spinorial tangent vectors at every point of S3. As a result, each point of S3 is characterized by a tangent spinor q of the form (10), representing a smooth flowing motion of that point, without any discontinuities, singularities, or fixed points hindering its coordinatization. One immediate consequence of the above construction of orthonormal basis for the tangent space at each point qS3 is that it renders the metric tensor on S3 “flat”,
$$ g(\beta_{\mu}(\textbf{q}){\hspace{1pt}},\beta^{\dagger}_{\nu}(\textbf{q}))=\beta_{\mu}(\textbf{q})\cdot\beta^{\dagger}_{\nu}(\textbf{q}) =\delta_{\mu\nu}, $$
(44)
extending the inner product (37) of the tangent basis at q0 to all points q of S3. Consequently (and contrary to the misleading impressions given by the Fig. 1), the Riemann curvature tensor of S3 vanishes identically,
$$ {\mathcal{R}}^{\alpha}_{\beta\gamma\delta}=\partial_{\gamma}{\Omega}_{\delta\beta}^{\alpha}- \partial_{\delta}{\Omega}_{\gamma\beta}^{\alpha}+{\Omega}_{\delta\beta}^{\rho}{\Omega}_{\gamma\rho}^{\alpha}- {\Omega}_{\gamma\beta}^{\rho}{\Omega}_{\delta\rho}^{\alpha}=0, $$
(45)
with respect to the a-symmetric Weitzenböck connection
$$ {\Omega}_{\alpha\beta}^{\gamma}={\Gamma}_{\alpha\beta}^{\gamma}+ {\mathcal{T}}_{\alpha\beta}^{\gamma}, $$
(46)
where \({{\Gamma }_{\alpha \beta }^{\gamma }}\) is the symmetric Levi-Civita connection and \({{\mathcal {T}}_{\alpha \beta }^{\gamma }}\) is the totally anti-symmetric torsion tensor. This vanishing of the curvature tensor renders the resulting parallelism on S3 absolute—i.e., it guarantees the path-independence of the parallel transport within S3. As a result, a parallel transport of arbitrary spinor (or tensor) within S3 is simply a translation of that spinor within S3.
Fig. 1

The relation between the angle χ between q(a) and q(a') and η between Ia and Ia' is non-linear in general

However, since S3 is not a Euclidean space [17], for the above path-independence to be possible the torsion within S3 must necessarily be non-vanishing. In fact it is straightforward to verify that the torsion within \({{S^{3}_{F}}}\) is indeed non-zero and constant. For two arbitrary spinors Ia = aμβμ and Ib = bνβν belonging to the tangent space TpS3, the torsion tensor \({\mathcal {T}}\) can be written as
$$ {\mathcal{T}}(I\cdot\textbf{a},I\cdot\textbf{b})=a^{\mu}b^{\nu}\left\{\nabla_{{\beta}_{\mu}}{\beta}_{\nu} -\nabla_{{\beta}_{\nu}}{\beta}_{\mu}-\left[{\beta}_{\mu},{\beta}_{\nu}\right]\right\}, $$
(47)
where the Lie bracket [⋅, ⋅] in the last term is the same as the commutator product bracket for bivectors. But since {β1(p), β2(p), β3(p)} defines a “frame field” at the tip of every p ∈ IR4, each base bivector βμ remains constant upon parallel transport relative to this frame field, giving \({\nabla_{\beta_\mu }}{\beta }_{\nu }\) = 0 = \({\nabla_{\beta_\nu }}{\beta }_{\mu }\). Consequently, the above expression of the torsion tensor simplifies to
$$\begin{array}{@{}rcl@{}} {\mathcal{T}}(I\cdot\textbf{a},I\cdot\textbf{b})&=&-a_{\mu}b_{\nu}\left[{\beta}_{\mu},{\beta}_{\nu}\right] \\ &=&-\left[(I\cdot\textbf{a}),(I\cdot\textbf{b})\right] \\ &=&I\cdot(\textbf{a}\times\textbf{b})=\textbf{a}\wedge\textbf{b}. \end{array} $$
(48)
In other words, instead of (28), we now have vanishing curvature but non-vanishing torsion [18]:
$$ {\mathcal{R}}^{\alpha}_{\beta\gamma\delta}=0\; \text{but}\; {\mathcal{T}}_{\alpha\beta}^{\gamma}\not=0. $$
(49)
This completes the transformation of the round 3-sphere \({{S^{3}_{R}}}\) of (27) into the flat 3-sphere \({{S^{3}_{F}}}\) of (44).
As noted above, for us the importance of parallelizing S3 and the corresponding vanishing of its Riemann tensor lies in the availability of the orthonormality preserving continuous transport of the basis of TpS3 to TqS3 in a path-independent manner. This enables us to introduce the notion of parallelity of vectors tangent to S3 at any two points p, q in S3 in an absolute manner, thus allowing unambiguous distant comparison between the directions of tangent spinors at different points of S3. Consequently, if {aμβμ(p)} and {bνβν(q)} are two bivectors belonging to two different tangent spaces TpS3 and TqS3 at two different points of S3, then their parallelity allows us to compute the inner product between them as follows:
$$\begin{array}{@{}rcl@{}} \langle\{a_{\mu}{\beta}_{\mu}(\textbf{p})\},\{b_{\nu}{\beta}_{\nu}(\textbf{q})\}\rangle&=& \langle\{a_{\mu}{\beta}_{\mu}(\textbf{q})\},\{b_{\nu}{\beta}_{\nu}(\textbf{q})\}\rangle \\ &=&-\beta_{\mu}(\textbf{q})\cdot\beta^{\dagger}_{\nu}(\textbf{q})a_{\mu}b_{\nu} \\ &=&-\delta_{\mu\nu}a_{\mu}b_{\nu} \\ &=&-\cos\eta_{\textbf{a}\textbf{b}} \\ &=&-\cos\frac{\psi_{\textbf{a}\textbf{b}}}{2}, \end{array} $$
(50)
where ψab is the amount of rotation required to align the bivector {aμβμ(q)} with the bivector {bνβν(q)}. Thus the inner product on S3 itself provides a unique distance measure between two rotations in the physical space with 4π periodicity (i.e., spinorial sensitivity). More precisely,
$$\begin{array}{@{}rcl@{}} {\mathcal{D}}(\textbf{a},\textbf{b})&=-\cos\eta_{\textbf{a}\textbf{b}} \\ &=\text{a geodesic distance on \({S^{3}\!\sim\text{SU(2)}}\)}, \end{array} $$
(51)
where ηab is half of the rotation angle described above.
For the discussion in Section 5 it is also important to recall here that the basis {β1(p), β2(p), β3(p)} of TpS3 at any pS3 satisfies the anti-symmetric outer product
$$ \frac{1}{2}\left[{\beta}_{\mu},{\beta}_{\nu}\right] =-\epsilon_{\mu\nu\rho}{\beta}_{\rho}, $$
(52)
which, together with the symmetry of the inner product
$$ \frac{1}{2}\left\{{\beta}_{\mu},{\beta}_{\nu}\right\} =-\delta_{\mu\nu}, $$
(54)
leads to the algebra
$$ {\beta}_{\mu}{\beta}_{\nu} =-\delta_{\mu\nu} -\epsilon_{\mu\nu\rho}{\beta}_{\rho}. $$
(54)
This is of course the familiar bivector subalgebra of the Clifford algebra Cl3,0 of the orthogonal directions in the physical space [4, 18]. What may be less familiar is that this algebra also represents the local structure—i.e., the tangent space structure—of a unit parallelized 3-sphere.

3 Geodesic Distance on SO(3)

Our goal now is to obtain the geodesic distance on SO(3) by projecting the geodesic distance (51) from SU(2) onto SO(3) [19]. Physically this means considering only those rotations that are insensitive to orientation-entanglement, or spinorial sign changes. Now, as we saw in the previous section, SU(2) is homeomorphic to a set of unit quaternions and can be embedded into the linear vector space IR4. A quaternion is thus a 4-vector within IR4, characterized by the ordered and graded basis
$$ \left\{1,\textbf{e}_{2}\wedge\textbf{e}_{3},\textbf{e}_{3}\wedge\textbf{e}_{1},\textbf{e}_{1}\wedge\textbf{e}_{2}\right\}. $$
(55)
There are, however, twice as many elements in the set S3 of all unit quaternions than there are points in the configuration space SO(3) of all possible rotations in the physical space. This is because every pair of quaternions constituting the antipodal points of S3 represent one and the same rotation in IR3. This can be readily confirmed by recalling how a quaternion and its antipode can rotate a bivector β(a) about a to, say, a bivector β(a′) about a:
$$ {\boldsymbol\beta}(\textbf{a'})=(+\textbf{q}){\boldsymbol\beta}(\textbf{a})(+\textbf{q})^{\dagger} =(-\textbf{q}){\boldsymbol\beta}(\textbf{a})(-\textbf{q})^{\dagger}. $$
(56)
Mathematically this equivalence is expressed by saying that S3—or more precisely the group SU(2)—represents a universal double covering of the rotation group SO(3) [11]. The configuration space of all possible rotations in the physical space is therefore obtained by identifying the antipodal points of S3i.e., by identifying every point +qS3 with its antipode −qS3. The space that results from this identification is a real projective space, IRP3, which is simply the set of all lines through the origin of IR4. There are thus precisely two preimages in S3, namely +q and −q, corresponding to each rotation Rq in SO(3). As a result, S3 is realized as a fiber bundle over IRP3 with each fiber consisting of exactly two points:
$$ S^{3}/\{-1,+1\}\approx \quad \text{I\!R}\mathrm{P}^{3}. $$
(57)
In other words, IRP3 is the quotient of S3 by the map q ↦ −q, which we shall denote by φ:S3→IRP3.

Although this quotient map renders the topologies of the spaces S3 and IRP3 quite distinct from one another (for example the space S3 is simply-connected, whereas the space IRP3 is connected but not simply-connected), it leaves their Lie algebra structure (i.e., their tangent space structure) unaffected. Therefore it is usually not possible to distinguish the signatures of the spinorial group SU(2) from those of the tensorial group SO(3) by local measurements alone. In fact the space IRP3—being homeomorphic to a Lie group—is just as parallelizable as the space S3. In more familiar terms this means that it is impossible to tell by local observations alone whether a given object has undergone even number of 2π rotations prior to observation or odd number of 2π rotations [3].

We are, however, interested precisely in distinguishing the global properties of S3 from those of IRP3 by means of local, albeit relative measurements [8]. Fortunately, it turns out to be possible to distinguish and compare the geodesic distances on the spaces S3 and IRP3 induced by the Euclidean metric δμν of the parallelized S3. This is possible because, for any Lie group, such as SO(3), with a left-invariant metric coinciding at the identity with a left-invariant metric on its universal covering group, such as SU(2), the geodesics on the covering group are simply the horizontal lifts of the geodesics on that group [19]. In other words, we can induce a metric on the base space IRP3 by projecting the Euclidean metric of the total space S3 by means of the quotient map φ:S3→IRP3. This induced metric then provides a measure of the respective geodesic distances within the manifolds S3 and IRP3.

To deduce this metric, let us recall the definitions of the tangent space TqS3 and the tangent bundle TS3 from the previous section [cf. (30) and (31)]. Under the map φ:S3 → IRP3 the tangent bundle TIRP3 is then the quotient of the tangent bundle TS3 by the involution
$$ (\textbf{q},\textbf{t}_{q})\mapsto(-\textbf{q},-\textbf{t}_{q}). $$
(58)
In other words, (q, tq)∈TqS3 and (−q,−tq)∈TqS3 have the same image in the tangent space T[q]IRP3 under the derivative map dφ:TS3TIRP3, where [q]∈IRP3 is a line through ±q in IR4. Since the quotient map φ(q) is a local diffeomorphism, its derivative map
$$ d\varphi(\textbf{q}):T_{\textbf{q}}S^{3}\rightarrow T_{[\textbf{q}]}\text{I\!R}\mathrm{P}^{3} $$
(59)
is an isomorphism for every qS3. Thus the tangent space T[q]IRP3 can be identified with the space of pairs:
$$ \left\{(\textbf{q},\textbf{t}_{q}),(-\textbf{q},-\textbf{t}_{q})\Big| \textbf{q},\textbf{t}_{q}\in \text{I\!R}^{4}\!,||\textbf{q}||=1,\textbf{q}\cdot\textbf{t}_{\textbf{q}}^{\dagger}=0 \right\} . $$
(60)
One such pair is depicted in Fig. 2, with the spinor +q making an angle η with a reference spinor +k. It is easy to see from this figure that at one point of S3 we have the differential relation
$$ d\textbf{q}=\textbf{t}_{\textbf{q}}d\eta, $$
(61)
whereas at its antipodal point we have the relation
$$ d\textbf{q}=-\textbf{t}_{\textbf{q}}d(\pi-\eta). $$
(62)
Identifying tq with −tq thus amounts to identifying dη with d(πη):
$$ d\eta=d(\pi-\eta). $$
(63)
This identification can be effected by the following change in the variable η:
$$ \eta\mapsto\alpha=-\pi\cos\eta. $$
(64)
Consequently, the infinitesimal measure of distance on S3 that projects down to IRP3 as a result of the quotient map φ:S3 → IRP3 is
$$ \pi\sin\eta d\eta\mapsto d\alpha, $$
(65)
where dα is the infinitesimal measure of distance on IRP3. Since sin(πη) = sinη, it is easy to see that the pair
$$ \{\pi\sin\eta d\eta,\pi\sin(\pi-\eta)d\eta\} $$
(66)
has the same image dα in T[q]IRP3. The corresponding finite interval [0, ηab] then projects down to IRP3 as
$$ \pi{\int}_{0}^{\eta_{\textbf{a}\textbf{b}}}\sin\eta d\eta\longmapsto {\int}_{0}^{2\eta_{\textbf{a}\textbf{b}}}d\alpha, $$
(67)
where the factor of 2 appearing in the upper limit on the RHS reflects the universal double covering of IRP3 by S3. These integrals can now be evaluated separately for the intervals 0 ≤ ηabπ and πηab ≤ 2π to give
$$\begin{array}{@{}rcl@{}} -\cos\eta_{\textbf{a}\textbf{b}}\mapsto&\left\{\begin{array}{lll} -1+\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} \text{if} & 0 \leq \eta_{\textbf{a}\textbf{b}} \leq \pi \\ \\ +3-\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} \text{if} & \pi \leq \eta_{\textbf{a}\textbf{b}} \leq 2\pi \end{array}\right. \\ &=-\cos\alpha_{\textbf{a}\textbf{b}}, \end{array} $$
(68)
which—as we shall soon confirm—is an expression of the fact that the map φ:S3→IRP3 is a local isometry [cf. (85) to (88)]. Thus the geodesic distance projected from the distance (51) in S3 onto IRP3 is given by
$$\begin{array}{@{}rcl@{}} {\mathcal{D}}(\textbf{a},\textbf{b})&= \left\{\begin{array}{lll} -1+\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} \text{if} & 0 \leq \eta_{\textbf{a}\textbf{b}} \leq \pi \\ \\ +3-\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} \text{if} & \pi \leq \eta_{\textbf{a}\textbf{b}} \leq 2\pi, \end{array}\right. \end{array} $$
(69)
where ηab is half of the rotation angle ψ, just as in (51).
Fig. 2

Projecting the metric tensor from TqS3 to \({T_{[\textbf {q}]}\text {I\!R}\mathrm {P}^{3}}\)

Needless to say, this measure of geodesic distance can also be obtained directly from the geometry of SO(3) itself [9, 20]. Since it is a compact Lie group, SO(3) has a natural Riemannian metric—i.e., an inner product on its tangent space TpSO(3) at every point p [11]. At the identity of SO(3) this tangent space is isomorphic to the Lie algebra so(3) of skew-symmetric matrices of the form
$$ \left(\begin{array}{ccc} 0 & -a_{3} & +a_{2} \\ \\ +a_{3} & 0 & -a_{1} \\ \\ -a_{2} & +a_{1} & 0 \end{array}\right), $$
(70)
where a ∈ IR3. These elements of Lie algebra so(3) can be represented also by unit bivectors of the form ξ(a) ∈ S2 satisfying the anti-symmetric outer product as before,
$$ \frac{1}{2}\left[{\xi}_{\mu},{\xi}_{\nu}\right] =-\epsilon_{\mu\nu\rho}{\xi}_{\rho}, $$
(71)
with the basis bivectors {ξ1(p), ξ2(p), ξ3(p)} spanning the tangent spaces at every point p∈SO(3). These basis bivectors, however, respect a general symmetric product
$$ \frac{1}{2}\left\{{\xi}_{\mu},{\xi}_{\nu}\right\} =-J_{\mu\nu}, $$
(72)
with the metric tensor J now providing a non-Euclidean inner product on SO(3):
$$\begin{array}{@{}rcl@{}} \langle{\xi}_{\mu},{\xi}^{\dagger}_{\nu}\rangle =J_{\mu}^{\rho}{\beta}_{\rho}\cdot{\beta}^{\dagger}_{\nu}=J_{\mu}^{\rho}\delta_{\rho\nu} =J_{\mu\nu}. \end{array} $$
(73)
This metric can be verified by multiplying two matrices of the form (70), with the usual Euclidean inner product emerging as one-half of the trace of the product matrix. Physically J can be interpreted as a moment of inertia tensor of a freely rotating asymmetrical top [19]. For a spherically symmetrical top Jμν reduces to δμν and the Euclidean inner product (44) is recovered [21]. In general the orientation of the top traces out a geodesic distance in SO(3) with respect to its inertia tensor J [19, 22].
To calculate the length of this distance, recall that the inner product in the Riemannian structure provides an infinitesimal length on the tangent space so that the length of a curve can be obtained by integration along the curve [11]. Then the shortest path—i.e., the geodesic distance—from the identity of SO(3) to another point can be obtained by means of the exponential map
$$ \exp:T_{e}\text{SO(3)}\approx so(3)\rightarrow\text{SO(3)}, $$
(74)
which maps the line ξ(a)ta in the tangent space TeSO(3) at the origin e of SO(3) onto the group SO(3) such that
$$ {\boldsymbol\xi}(\textbf{a})\mapsto \exp{\left\{{\boldsymbol\xi}(\textbf{a})\right\}}\! :=\exp{\left\{{\boldsymbol\xi}(\textbf{a})t_{\textbf{a}}\right\}}\bigg|_{t_{\textbf{a}}=1}, $$
(75)
where ξ(a) ∈ so(3) with ξ(a)ξ (a) = 1, and
$$ t_{\textbf{a}}= \left\{\begin{array}{lll} -1+\frac{1}{\pi}\psi_{\textbf{a}}\text{if} & 0 \leq \psi_{\textbf{a}} \leq 2\pi \\ \\ +3-\frac{1}{\pi}\psi_{\textbf{a}}\text{if} & 2\pi \leq \psi_{\textbf{a}} \leq 4\pi. \end{array}\right. $$
(76)
Thus ta takes values from the interval [−1, +1], with ψa being the angle of rotation as before. More importantly, ta provides a measure of a geodesic distance between the identity of SO(3) and the element
$$ {\mathcal{R}}_{\textbf{a}}:=\exp{\left\{{\boldsymbol\xi}(\textbf{a})t_{\textbf{a}}\right\}}\in \text{SO(3)}. $$
(77)
Therefore, we can use this exponential map to define a bi-invariant distance measure on SO(3) as
$$ {\mathcal{D}}: \text{SO(3)}\times\text{SO(3)} \rightarrow \text{I\!R} $$
(78)
such that
$$ {\mathcal{D}}(\textbf{a},\textbf{b}) =\left|\left|\log\left\{{\mathcal R}^{~}_{\textbf{a}}{\mathcal R}^{\dagger}_{\textbf{b}}\right\}\right|\right|\!. $$
(79)
This distance measure calculates the amount of rotation required to bring \({{\mathcal R}_{\textbf {a}}\in \text {SO(3)}}\) to align with \({{\mathcal R}_{\textbf {b}}\in \text {SO(3)}}\) by finding \({{\mathcal R}_{\textbf {ab}}\in \text {SO(3)}}\) such that \({{\mathcal R}_{\textbf {a}}={\mathcal R}_{\textbf {ab}}{\mathcal R}_{\textbf {b}}}\), because then we have \({{\mathcal R}_{\textbf {ab}}={\mathcal R}^{~}_{\textbf {a}}{\mathcal R}^{\dagger }_{\textbf {b}}}\) since \({{\mathcal R}^{~}_{\textbf {b}}{\mathcal R}^{\dagger }_{\textbf {b}}=1}\). Thus, in view of definition (75), we arrive at the following distance between \({{\mathcal R}_{\textbf {a}}}\) and \({{\mathcal R}_{\textbf {b}}}\) (see also Fig. 11 of [5]):
$$\begin{array}{@{}rcl@{}} {\mathcal{D}}(\textbf{a},\textbf{b})= \left\{\begin{array}{lll} -1+\frac{1}{\pi}\psi_{\textbf{ab}}\text{if} & 0 \leq \psi_{\textbf{ab}} \leq 2\pi \\ \\ +3-\frac{1}{\pi}\psi_{\textbf{ab}}\text{if} & 2\pi \leq \psi_{\textbf{ab}} \leq 4\pi. \end{array}\right. \end{array} $$
(80)
But since ψab = 2ηab, this distance is equivalent to the one obtained in (69). In Fig. 3 both distance functions (51) and (69) are plotted as functions of the angle ηab.
Next, using the inner product (73) and the fact that orientations of a rotating body trace out a geodesic in SO(3) with respect to its inertia tensor, we can rewrite the above measure in terms of the metric tensor J as
$$\begin{array}{@{}rcl@{}} -J_{\mu\nu}a_{\mu}b_{\nu}&=:-\cos\alpha_{\textbf{a}\textbf{b}} \\ &= \left\{\begin{array}{lll} -1+\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} \text{if} & 0 \leq \eta_{\textbf{a}\textbf{b}} \leq \pi \\ \\ +3-\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} \text{if} & \pi \leq \eta_{\textbf{a}\textbf{b}} \leq 2\pi, \end{array}\right. \end{array} $$
(81)
with SO(3) counterpart of the inner product (50) being
$$\begin{array}{@{}rcl@{}} \langle\{a_{\mu}{\xi}_{\mu}(\textbf{p})\},\{b_{\nu}{\xi}_{\nu}(\textbf{q})\}\rangle&=& \langle\{a_{\mu}{\xi}_{\mu}(\textbf{q})\},\{b_{\nu}{\xi}_{\nu}(\textbf{q})\}\rangle \\ &=&-\langle\xi_{\mu}(\textbf{q}),\xi^{\dagger}_{\nu}(\textbf{q})\rangle a_{\mu}b_{\nu} \\ &=&-J_{\mu\nu}a_{\mu}b_{\nu} \\ &=&-\cos\alpha_{\textbf{a}\textbf{b}}. \end{array} $$
(82)
Here ηab is the angle between tangent bivectors aμβμ(q) and bνβν(q) within TqS3, whereas αab is the angle between tangent bivectors aμξμ(q) and bνξν(q) within T[q]IRP3. Consequently, using the algebraic identity
$$ {\xi}_{\mu}{\xi}_{\nu} = \frac{1}{2}\left\{{\xi}_{\mu},{\xi}_{\nu}\right\}+ \frac{1}{2}\left[{\xi}_{\mu},{\xi}_{\nu}\right] $$
(83)
and the equations (71) and (72), we arrive at the algebra
$$ {\xi}_{\mu}{\xi}_{\nu}=-J_{\mu\nu}-\epsilon_{\mu\nu\rho}{\xi}_{\rho}. $$
(84)
This induced algebra is then the SO(3) counterpart of the algebra (54). Note that it coincides with the algebra (54) at the identity (i.e., when ψab = π and ψab = 3π).
It is important to recognize here that the quotient map φ:S3→IRP3 we considered above is a local isometry. The infinitesimal map dφ(q):TqS3T[q]IRP3 is thus an isometry. The inner product in TqS3 defined by (37) is therefore locally preserved under the action of φ:
$$\begin{array}{@{}rcl@{}} &&\langle\{a_{\mu}{\beta}_{\mu}(\textbf{q})\},\{b_{\nu}{\beta}_{\nu}(\textbf{q})\}\rangle \\ &=&\langle\{a_{\mu}{\xi}_{\mu}[\varphi(\textbf{q})]\},\{b_{\nu}{\xi}_{\nu}[\varphi(\textbf{q})]\}\rangle. \end{array} $$
(85)
Note, however, that this equality between the metrics in TqS3 and T[q]IRP3 holds only for the values φ(q) = q, φ(q) = −q, φ(q) = tq, and φ(q) = −tq, which can be verified from the definition (60) of the tangent space T[q]IRP3 with the help of Fig. 2. For definiteness, let the quaternions q shown in Fig. 2 be parameterized as
$$\begin{array}{@{}rcl@{}} \textbf{q}_{a}&=&\cos\eta_{\textbf{a}}+{\boldsymbol\beta}(\textbf{c})\sin\eta_{\textbf{a}}, \\ \textbf{q}_{b}&=&\cos\eta_{\textbf{b}}+{\boldsymbol\beta}(\textbf{c})\sin\eta_{\textbf{b}}, \end{array} $$
(86)
etc., along with its tangent quaternions parameterized as
$$\begin{array}{@{}rcl@{}} \textbf{t}_{q_{a}}&=&-\sin\eta_{\textbf{a}}+{\boldsymbol\beta}(\textbf{c})\cos\eta_{\textbf{a}}, \\ \textbf{t}_{q_{b}}&=&-\sin\eta_{\textbf{b}}+{\boldsymbol\beta}(\textbf{c})\cos\eta_{\textbf{b}}, \end{array} $$
(87)
etc., where c = a×b/|a×b|. Then the angle between qa and qb (as well as between \(\textbf {t}_{q_{a}}\) and \(\textbf {t}_{q_{b}}\)) is given by
$$ \cos\eta_{\textbf{ab}}=|\textbf{q}_\textbf {a} \cdot\textbf{q}^{\dagger}_\textbf{b}|=|\cos\eta_{\textbf{a}}\cos\eta_{\textbf{b}} +\sin\eta_{\textbf{a}}\sin\eta_{\textbf{b}}|. $$
(88)
From Fig. 2 it is now easy to see that the projection φ(q) = q corresponds to ηab=0 and ηab=2π, whereas the projection φ(q) = −q corresponds to ηab = π. Two further possibilities, φ(q) = tq and φ(q) = −tq, which correspond to the projections (q, tq)↦(tq,−q) and (q, tq) ↦ (−tq, q), are also permitted by the definition (60) of T[q]IRP3, because TIRP3 is invariant under the rotation of q by π/2. They amount to swaps in the roles played by q and tq. From Fig. 2 it is easy to see that these possibilities correspond to ηab = π/2 and ηab=3π/2.
The argument above shows that the equality (85) holds only for the angles ηab = 0, π/2, π, 3π/2, and 2π. That is to say, the local isometry (85) under the map φ:S3 → IRP3 requires that the geodesic distances on S3 and IRP3 agree only in the infinitesimal neighborhoods of the angles ηab = 0, π/2, π, 3π/2, and 2π. They are not preserved globally. This is confirmed in Fig. 3, which is a plot of the distance functions (51) and (69). The geodesic distances on SU(2) and SO(3) are evidently different for the angles other than ηab = 0, π/2, π, 3π/2, and 2π. The underlying reason for the difference in the shape of the two geodesic distances is the difference in the topologies of S3 and IRP3. While S3 is simply-connected, IRP3 is connected but not simply-connected. As a result the map φ:S3→IRP3 is only a local and not a global isometry. Moreover, while both S3 and IRP3 are parallelized by the basis {βμ} and {ξν}, respectively, the torsion within these manifolds is different. And it is this difference in torsion that is reflected in the shapes of the geodesics depicted in Fig. 3. In the Appendix A below we spell out the parallelization process in more detail and bring out the crucial role played by torsion in these manifolds.
Fig. 3

Comparison of the geodesic distances on SU(2) and SO(3) as functions of half of the rotation angle ψ. The dashed lines depict the geodesic distances on SO(3), whereas the red curve depicts their horizontal lift to the covering group SU(2)

To sum up, we started out with a parallelized 3-sphere representing SU(2) and then identified each of its points with its antipodal point to obtain a parallelized IRP3 representing SO(3). But in doing so we lost the following spinorial symmetry satisfied by q(ψ, r), as described by equations (4) and (5) in the Introduction:
$$ \textbf{q}(\psi+2\kappa\pi,\textbf{r})=-\textbf{q}(\psi,\textbf{r})\; \text{for}\; \kappa=1,3,5,7,\dots $$
(89)
In other words, by identifying the antipodal points of S3 we lost the sensitivity of q(ψ, r) to spinorial sign changes. As a result, q(ψ, r) constituting IRP3 represents the state of a rotating body that returns to itself after any multiple of 2π rotation:
$$ \textbf{q}(\psi+2\kappa\pi,\textbf{r})=+\textbf{q}(\psi,\textbf{r})\; \text{for}\; \text{any}\;\kappa=0,1,2,3,\dots $$
(90)
A measurable difference between these two possibility is then captured by the geodesic distances on the manifolds SU(2) and SO(3), as depicted, for example, in Fig. 3.

4 Proposed Experiment

We now sketch a classical experiment which could, in principle, distinguish the geodesic distance on SU(2) from that on SO(3), as calculated in the previous sections (cf. Fig. 3). It can be performed either in the outer space or in a terrestrial laboratory. In the latter case the effects of gravity and air resistance would complicate matters, but it may be possible to choose experimental parameters sufficiently carefully to compensate for such effects.

With this assumption, consider a “bomb” made out of a hollow toy ball of diameter, say, three centimeters. The thin hemispherical shells of uniform density that make up the ball are snapped together at their rims in such a manner that a slight increase in temperature would pop the ball open into its two constituents with considerable force. A small lump of density much greater than the density of the ball is attached on the inner surface of each shell at a random location, so that, when the ball pops open, not only would the two shells propagate with equal and opposite linear momenta orthogonal to their common plane, but would also rotate with equal and opposite spin momenta about a random axis in space. The volume of the attached lumps can be as small as a cubic millimeter, whereas their mass can be comparable to the mass of the ball. This will facilitate some 106 possible spin directions for the two shells, whose outer surfaces can be decorated with colors to make their rotations easily detectable.

Now consider a large ensemble of such balls, identical in every respect except for the relative locations of the two lumps (affixed randomly on the inner surface of each shell). The balls are then placed over a heater—one at a time—at the center of the experimental setup [23], with the common plane of their shells held perpendicular to the horizontal direction of the setup. Although initially at rest, a slight increase in temperature of each ball will eventually eject its two shells towards the observation stations, situated at a chosen distance in the mutually opposite directions. Instead of selecting the directions a and b for observing spin components, however, one or more contact-less rotational motion sensors—capable of determining the precise direction of rotation—are placed near each of the two stations, interfaced with a computer (for further details see the end of this section). These sensors will determine the exact direction of the spin angular momentum sk (or −sk) for each shell in a given explosion, without disturbing them otherwise so that their total angular momentum would remain zero, at a designated distance from the center. The interfaced computers can then record this data, in the form of a 3D map of all such directions, at each observation station.

Once the actual directions of the angular momenta for a large ensemble of shells on both sides are fully recorded, the two computers are instructed to randomly choose a pair of reference directions, say a for one station and b for the other station—from the two 3D maps of already existing data—and then calculate the corresponding pair of numbers sign(ska) = ±1 and sign(−skb) = ±1. The correlation function for the bomb fragments can then be calculated as
$$ {\mathcal E}(\textbf{a},\textbf{b})=\!\!\lim\limits_{n\gg1}\!\left[\frac{1}{n}\!\sum\limits_{k=1}^{n} \{{sign}(+\textbf{s}^{k}\cdot\textbf{a})\} \{{sign}(-\textbf{s}^{k}\cdot\textbf{b})\}\right]\!, $$
(91)
together with
$$ {\mathcal{E}}(\textbf{a})=\!\!\lim\limits_{n\gg1}\!\left[\frac{1}{n}\!\sum\limits_{k=1}^{n}\{{sign}(+\textbf{s}^{k}\cdot\textbf{a})\}\right]\!=0=\!\!\lim\limits_{n\gg1}\!\left[\frac{1}{n}\!\sum\limits_{k=1}^{n}\{{sign}(-\textbf{s}^{k}\cdot\textbf{b})\}\right]\!={\mathcal E}(\textbf{b}), $$
(92)
where n is the total number of experiments performed.

In the next section we shall see how the correlation function (91) provides a measure of geodesic distances on the manifolds SU(2) and SO(3). It can thus serve to detect spinorial sign changes even classically in a manner similar to how they are detected quantum mechanically. Recall that—as Aharonov and Susskind noted long ago—even quantum mechanically spinorial sign changes cannot be detected in a direct or absolute manner [6]. They showed that if fermions are shared coherently between spatially isolated systems, then relative rotations of 2π between them may be observable [3, 7, 8]. Similarly, in the above experiment two macroscopic bomb fragments would be rotating in tandem, relative to each other. Consequently, the observables sign(+ska) and sign(−skb) would be sensitive to the relative spinorial sign changes between these fragments. The correlation function \({{\mathcal E}(\textbf {a},\textbf {b})}\) would then provide a measure of such sign changes in terms of geodesic distances on the manifolds SU(2) and SO(3).

It is also worth noting here that the observability of the spinorial sign changes in this manner is intimately tied up with the fact that the quaternionic 3-sphere we defined in (1) is naturally parallelized. In other words, the metric on SU(2) is Euclidean, gμν = δμν, and hence the geodesic distances within it are dictated by the function −cosηab, as we noted in (51) and depicted in Fig. 3. By contrast, correlation between the points of S3 with a round metric and zero torsion cannot be stronger than the linear correlation between the points of SO(3) [18]. Thus it is the discipline of parallelization in the manifold S3 ∼ SU(2) which dictates that the correlation between sign(+ska) and sign(−skb) will be as strong as −cosηab. Fortunately, parallelization happens to be the natural property of the manifold S3 ∼ SU(2), and hence the correlation function (91) would provide us a natural means to detect the signature of spinorial sign changes.

One may wonder why such a non-trivial experiment is necessary when we can infer the topological properties of S3 ∼ SU(2) by means of a simple classical device like Dirac’s belt [3]. It should be noted, however, that what the belt trick provides is only an indirect indication—by means of an external, material connection (namely, the belt, or a cord)—of the rotational symmetry of SU(2). What it does not provide is a direct physical evidence that the symmetry as non-trivial as SU(2) is indeed respected by our physical space. To establish that SU(2) is indeed the symmetry group respected by our physical space even classically, what is needed is a demonstration of spinorial sign changes between rotating objects even in the absence of any form of material interaction between them.

The experiment considered above would allows us to accomplish just such a “belt-free” demonstration. Once the bomb has exploded and the two fragments are on their way towards the observation stations, there would be no material object (such as a belt) connecting them. That is to say, once they have separated the stress-energy tensor between them would be zero everywhere. The only physical link between them would be the orientation of the manifold S3, which would provide a local standard of reference for their mutual rotations and spinorial sign changes [8]. In other words, only the orientation λ of the 3-sphere would act as a non-material belt connecting the two fragments. In the following section we formalize this “belt without belt” scenario in a systematic manner.

Let us now turn to the practical problem of determining the direction of rotation of a bomb fragment. In order to minimize the contributions of precession and nutation about the rotation axis of the fragment, the bomb may be composed of two flexible squashy balls instead of a single ball. The two balls can then be squeezed together at the start of a run and released as if they were two parts of the same bomb. This will retain the spherical symmetry of the two constituent balls after the explosion, reducing their precession and nutation effects considerably. Consequently, we assume that during the narrow time window of the detection process the contributions of precession and nutation are negligible. In other words, during this narrow time window the individual spins will remain confined to the plane perpendicular to the horizontal direction of the setup. This is because we will then have s = r × p, with r specifying the location of the massive lump in the constituent ball and p being the ball’s linear momentum. We can now exploit these physical constraints to determine the direction of rotation of a constituent ball unambiguously, as follows.

Since only the directions of rotation are relevant for computing the correlation function (91), it would be sufficient for our purposes to determine only the direction of the vector s at each end of the setup. This can be accomplished by arranging three (or more) successive laser screens perpendicular to the horizontal path of the constituent balls, say about half a centimeter apart, and a few judiciously situated cameras around them. To facilitate the detection of the rotation of a ball as it passes through the screens, the surface of the balls can be decorated with distinctive marks, such as dots of different sizes and colors. Then, when a ball passes through the screens, the entry points of a specific mark on the ball can be recorded by the system of cameras. Since the ball would be spinning while passing through the screens, the entry points of the same mark on the successive screens would be located at different relative positions on the screens. The rotation axis of the ball can therefore be determined unambiguously by determining the plane spanned by the entry points and the right-hand rule. In other words, the rotation axis can be determined as the orthogonal direction to the plane spanned by the entry points, with the sense of rotation determined by the right-hand rule. This procedure of determining the direction of rotation can be followed through manually, or it can be automated with the help of a computer software. Finally, the horizontal distance from the center of the setup to the location of the middle of the screens can be taken as the distance of the rotation axis from the center of the setup. This distance would help in establishing the simultaneity of the spin measurements at the two ends of the setup.

5 Distinguishing \({{\mathcal D}(\textbf {a},\textbf {b})}\) on SU(2) from \({{\mathcal {D}}(\textbf {a},\textbf {b})}\) on SO(3) by Calculating \({{\mathcal E}(\textbf {a},\textbf {b})}\)

The prescription for \({{\mathcal E}(\textbf {a},\textbf {b})}\) in the previous section is how one would calculate the correlations in practice. Let us now derive them theoretically, for the two cases considered in the Sections 2 and 3. To this end, we first note the following definition of the orientation of a vector space [17, 24]:

Definition V.1

An orientation of a finite dimensional vector space \({\mathcal {U}}_{d}\) is an equivalence class of ordered basis, say {f1,…,fd}, which determines the same orientation of \({\mathcal {U}}_{d}\) as the basis \({\{f^{\prime }_{1},\dots ,f^{\prime }_{d}\}}\) if \({f^{\prime }_{i} = \omega _{ij} f_{j}}\) holds with det(ωij) > 0, and the opposite orientation of \({\mathcal {U}}_{d}\) as the basis \({\{f^{\prime }_{1},\dots ,f^{\prime }_{d}\}}\) if \({f^{\prime }_{i} = \omega _{ij} f_{j}}\) holds with det(ωij) < 0.

Thus each positive dimensional real vector space has precisely two possible orientations, which we shall denote as λ = +1 or λ = −1. More generally an oriented smooth manifold such as S3 consists of that manifold together with a choice of orientation for each of its tangent spaces.

It is important to note that orientation of a manifold is a relative concept [24]. In particular, the orientation of a tangent space \({{\mathcal {U}}_{d}}\) of a manifold defined by the equivalence class of ordered basis such as {f1,…,fd} is meaningful only with respect to that defined by the equivalence class of ordered basis \({\{f^{\prime }_{1},\dots ,f^{\prime }_{d}\}}\), and vice versa. To be sure, we can certainly orient a manifold absolutely by choosing a set of ordered bases for all of its tangent spaces, but the resulting manifold can be said to be left or right oriented only with respect of another such set of ordered basis.

Now the natural configuration space for the experiment considered above is a unit parallelized 3-sphere, which can be embedded in IR4 with a choice of orientation, say λ = ±1. This choice of orientation can be identified with the initial state of the bomb fragments with respect to the orientation of the detector basis as follows. We first characterize the embedding space IR4 by the graded basis
$$ \left\{1,L_{1}(\lambda),L_{2}(\lambda),L_{3}(\lambda)\right\}, $$
(93)
with λ = ±1 representing the two possible orientations of S3 and the basis elements Lμ(λ) satisfying the algebra
$$ L_{\mu}(\lambda)L_{\nu}(\lambda) =-g_{\mu\nu}-\epsilon_{\mu\nu\rho}L_{\rho}(\lambda). $$
(94)
As the notation suggests, we shall take the unit bivectors {aμLμ(λ)} to represent the spin angular momenta of the bomb fragments. These momenta can then be assumed to be detected by the detector bivectors, say {aμDμ}, with the corresponding detector basis {1, D1, D2, D3} satisfying the algebra
$$ D_{\mu}D_{\nu} =-g_{{\mu}{\nu}}-\epsilon_{{\mu}{\nu}{\rho}}D_{\rho} $$
(95)
and related to the spin basis {1,L1(λ), L2(λ), L3(λ)} as
$$ \left(\begin{array}{c} 1 \\ L_{1}(\lambda) \\ L_{2}(\lambda) \\ L_{3}(\lambda) \end{array}\right)= \left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda \end{array}\right) \left(\begin{array}{c} 1 \\ D_{1} \\ D_{2} \\ D_{3} \end{array}\right)\!. $$
(96)
Evidently, the determinant of this matrix works out to be det(ωij) = λ. Since λ2 = +1 and ω2 is a 4 × 4 identity matrix, this relation can be more succinctly written as
$$ L_{\mu}(\lambda)=\lambda D_{\mu}\; \text{and}\; D_{\mu}=\lambda L_{\mu}(\lambda), $$
(97)
or equivalently as
$$ \left\{1,L_{1}(\lambda),L_{2}(\lambda),L_{3}(\lambda)\right\}= \left\{1,\lambda D_{1},\lambda D_{2},\lambda D_{3}\right\} $$
(98)
and
$$ \left\{1,D_{1},D_{2},D_{3}\right\}= \left\{1,\lambda L_{1}(\lambda),\lambda L_{2}(\lambda),\lambda L_{3}(\lambda)\right\}. $$
(99)
These relations reiterate the fact that orientation of any manifold is a relative concept. In particular, orientation of S3 defined by the spin basis {1, Lμ(λ)} is meaningful only with respect to that defined by the detector basis {1, Dμ} with the orientation λ = +1, and vice versa. Thus the spin basis are said to define the same orientation of S3 as the detector basis if Lμ(λ = +1) = +Dμ, and the spin basis are said to define the opposite orientation of S3 as the detector basis if Lμ(λ = −1) = −Dμ.
We are now in a position to identify the formal counterparts of the measurement variables sign(ska) = ±1 and sign(−skb) = ±1 defined in the previous section:
$$\begin{array}{@{}rcl@{}} {sign}(+\textbf{s}^{k}\cdot\textbf{a})\equiv {\mathcal A}(\textbf{a},{\lambda^{k}})&=\{-a_{\mu}D_{\mu}\}\{a_{\nu}L_{\nu}(\lambda^{k})\} \\ &= \left\{\begin{array}{lll} +1\text{if} &\lambda^{k}=+1 \\ -1\text{if} &\lambda^{k}=-1 \end{array}\right. \end{array} $$
(100)
and
$$\begin{array}{@{}rcl@{}} {sign}(-\textbf{s}^{k}\cdot\textbf{b})\equiv {\mathcal B}(\textbf{b},{\lambda^{k}})&=\{+b_{\mu}D_{\mu}\}\{b_{\nu}L_{\nu}(\lambda^{k})\} \\ &= \left\{\begin{array}{lll} -1\text{if} &\lambda^{k}=+1 \\ +1\text{if} &\lambda^{k}=-1, \end{array}\right. \end{array} $$
(101)
where the relative orientation λ is now assumed to be a random variable, with 50/50 chance of being +1 or −1 at the moment of the bomb-explosion considered in the previous section. We shall assume that the orientation of S3 defined by the detector basis {1,Dν} has been fixed before hand [18]. Thus the spin bivector {aμLμ(λ)} is a random bivector with its handedness determined relative to the detector bivector {aνDν}, by the relation
$$ \textbf{L}(\textbf{a},\lambda) \equiv\{a_{\mu}L_{\mu}(\lambda)\}=\lambda\{a_{\nu}D_{\nu}\}\equiv\lambda\textbf{D}(\textbf{a}). $$
(102)
Using this relation the spin detection events (100) and (101) follow at once from the algebras (94) and (95).
It is important to note that the variables \({{\mathcal A}(\textbf {a},{\lambda })}\) and \({{\mathcal B}(\textbf {b},{\lambda })}\) given in equations (100) and (101) are generated with different bivectorial scales of dispersion (or different standard deviations) for each measurement direction a and b. Consequently, in statistical terms these variables are raw scores, as opposed to standard scores [25]. Recall that a standard score, z, indicates how many standard deviations an observation or datum is above or below the mean. If x is a raw (or unnormalized) score and \(\overline {\mathrm {x}}\) is its mean value, then the standard (or normalized) score, z(x), is defined by
$$ \mathrm{z}(\mathrm{x})=\frac{\mathrm{x} - \overline{\mathrm{x}}{\sigma(\mathrm{x})}}, $$
(103)
where σ(x) is the standard deviation of x. A standard score thus represents the distance between a raw score and population mean in the units of standard deviation, and allows us to make comparisons of raw scores that come from very different sources [18, 25]. In other words, the mean value of the standard score itself is always zero, with standard deviation unity. In terms of these concepts the correlation between raw scores x and y is defined as
$$ {\mathcal E}(\mathrm{x},\mathrm{y})=\frac{{\lim_{n \gg 1}}\left[{\frac{1}{n}} {\sum\limits\limits_{k=1}^{n}}(\mathrm{x}^{k}-{\overline{\mathrm{x}}}) (\mathrm{y}^{k}-{\overline{\mathrm{y}}})\right]}{\sigma(\mathrm{x})\sigma(\mathrm{y})} $$
(104)
$$ = \lim_{n \gg1}\left[\frac{1}{n} \sum\limits_{k = 1}^{n}\mathrm{z}(\mathrm{x}^{k})\mathrm{z}(\mathrm{y}^{k})\right]. $$
(105)
It is vital to appreciate that covariance by itself—i.e., the numerator of (104) by itself—does not provide the correct measure of association between the raw scores, not the least because it depends on different units and scales (or different scales of dispersion) that may have been used in the measurements of such scores. Therefore, to arrive at the correct measure of association between the raw scores one must either use (104), with the product of standard deviations in the denominator, or use covariance of the standardized variables, as in (105).
Now, as noted above, the random variables \({{\mathcal A}(\textbf {a},{\lambda })}\) and \({{\mathcal B}(\textbf {b},{\lambda })}\) are products of two factors—one random and another non-random. Within the variable \({{\mathcal A}(\textbf {a},{\lambda })}\) the bivector L(a, λ) is a random factor—a function of the orientation λ, whereas the bivector −D(a) is a non-random factor, independent of the orientation λ. Thus, as random variables, \({{\mathcal A}(\textbf {a},{\lambda })}\) and \({{\mathcal B}(\textbf {b},{\lambda })}\) are generated with different standard deviations—i.e., different sizes of the typical error. More specifically, \({{\mathcal A}(\textbf {a},{\lambda })}\) is generated with the standard deviation −D(a), whereas \({{\mathcal B}(\textbf {b},{\lambda })}\) is generated with the standard deviation +D(b). These two deviations can be calculated as follows. Since errors in the linear relations propagate linearly, the standard deviation \({\sigma ({\mathcal A})}\) of \({{\mathcal A}(\textbf {a},{\lambda })}\) is equal to −D(a) times the standard deviation of L(a, λ) [which we shall denote as σ(A) = σ(La)], whereas the standard deviation \({\sigma ({\mathcal B})}\) of \({{\mathcal B}(\textbf {b},{\lambda })}\) is equal to +D(b) times the standard deviation of L(b, λ) [which we shall denote as σ(B) = σ(Lb)]:
$$\begin{array}{@{}rcl@{}} \sigma({\mathcal A})&=-\textbf{D}(\textbf{a})\sigma({A}) \\ \; \text{and}\; \sigma({\mathcal B})&=+\textbf{D}(\textbf{b})\sigma({B}). \end{array} $$
(106)
But since the bivector L(a, λ) is normalized to unity, and since its mean value m(La) vanishes on the account of λ being a fair coin, its standard deviation is easy to calculate, and it turns out to be equal to unity:
$$\begin{array}{@{}rcl@{}} \sigma({A})&=&\sqrt{\frac{1}{n}\sum\limits_{k=1}^{n}\left|\left|A(\textbf{a},{\lambda}^{k})- {\overline{A(\textbf{a},{\lambda}^{k})}}\right|\right|^{2}} \\ &=&\sqrt{\frac{1}{n}\sum\limits_{k=1}^{n} \left|\left|\textbf{L}(\textbf{a},\lambda^{k})-0\right|\right|^{2}}=1, \end{array} $$
(107)
where the last equality follows from the normalization of L(a, λ). Similarly, we find that σ(B) is also equal to 1. Consequently, the standard deviation of \({{\mathcal A}(\textbf {a},{\lambda })=\pm 1}\) works out to be −D(a), and the standard deviation of \({{\mathcal B}(\textbf {b},{\lambda })=\pm 1}\) works out to be +D(b). Putting these two results together, we arrive at the following standard scores corresponding to the raw scores \({\mathcal A}\) and \({\mathcal B}\):
$$\begin{array}{@{}rcl@{}} A(\textbf{a},{\lambda})&=\frac{{\mathcal A}(\textbf{a},{\lambda})- {\overline{{\mathcal A}(\textbf{a},{\lambda})}}}{\sigma({\mathcal A})} \\ &=\frac{-\textbf{D}(\textbf{a})\textbf{L}(\textbf{a},\lambda)-0}{-\textbf{D}(\textbf{a})} =\textbf{L}(\textbf{a},\lambda) \end{array} $$
(108)
and
$$\begin{array}{@{}rcl@{}} B(\textbf{b},{\lambda})&=&\frac{{\mathcal B}(\textbf{b},{\lambda})- {\overline{{\mathcal B}(\textbf{b},{\lambda})}}}{\sigma({\mathcal B})} \\ &=&\frac{+\textbf{D}(\textbf{b})\textbf{L}(\textbf{b},\lambda)-0}{+\textbf{D}(\textbf{b})} =\textbf{L}(\textbf{b},\lambda), \end{array} $$
(109)
where we have used identities such as −D(a)D(a) = +1.
Now, since we have assumed that initially there was 50/50 chance between the right-handed and left-handed orientations of the 3-sphere (i.e., equal chance between the initial states λ = +1 and λ = −1), the expectation values of the raw scores \({{\mathcal A}(\textbf {a},{\lambda })}\) and \({{\mathcal B}(\textbf {b},{\lambda })}\) vanish identically. On the other hand, as discussed above, the correlation between these raw scores can be obtained only by computing the covariance between the corresponding standardized variables A(a, λ) and B(b, λ), which gives
$$\begin{array}{@{}rcl@{}} {\mathcal E}(\textbf{a},\textbf{b})&=&\lim\limits_{n\gg1}\left[\frac{1}{n}\sum\limits_{k=1}^{n}A(\textbf{a},{\lambda}^{k})B(\textbf{b},{\lambda}^{k})\right] \\ &=&\lim\limits_{n\gg1}\left[\frac{1}{n}\sum\limits_{k=1}^{n} \left\{a_{\mu}L_{\mu}(\lambda^{k})\right\}\left\{b_{\nu}L_{\nu}(\lambda^{k})\right\}\right] \\ &=&-g_{\mu\nu}a_{\mu}b_{\nu}-\lim\limits_{n\gg1}\left[\frac{1}{n}\sum\limits_{k=1}^{n} \left\{\epsilon_{\mu\nu\rho}a_{\mu}b_{\nu}L_{\rho}(\lambda^{k})\right\}\right] \\ &=&-g_{\mu\nu}a_{\mu}b_{\nu}-\lim\limits_{n\gg1}\left[\frac{1}{n}\sum\limits_{k=1}^{n} \lambda^{k}\right]\left\{\epsilon_{\mu\nu\rho}a_{\mu}b_{\nu}D_{\rho}\right\} \\ &=&-g_{\mu\nu}a_{\mu}b_{\nu}-0, \end{array} $$
(110)
where we have used the algebra (94) and relation (102). Consequently, as explained in the paragraph just below (105), when the raw scores \({{\mathcal A}=\pm 1}\) and \({{\mathcal B}=\pm 1}\) are compared, their product moment will inevitably yield
$$\begin{array}{@{}rcl@{}} {\mathcal E}(\textbf{a},\textbf{b})&=&\lim\limits_{n\gg1}\left[\frac{1}{n}\sum\limits_{k=1}^{n} {\mathcal A}(\textbf{a},{\lambda}^{k}){\mathcal B}(\textbf{b},{\lambda}^{k})\right] \\ &=&-g_{\mu\nu}a_{\mu}b_{\nu}, \end{array} $$
(111)
since the correlation between the raw scores \({{\mathcal A}}\) and \({\mathcal B}\) is equal to covariance between the standard scores A and B.
So far in this section we have put no restrictions on the metric tensor, which, in the normal coordinates centered at p would be of the form
$$ g_{\mu\nu}(x)=\delta_{\mu\nu}-\frac{1}{3} {\mathcal R}_{\alpha\mu\nu\gamma}x^{\alpha}x^{\gamma}+O\left(|x|^{3}\right). $$
(112)
In other words, the algebra (94) we have used in the derivation of the correlation (111) is a general Clifford algebra, with no restrictions placed on the quadratic form 〈⋅, ⋅〉 [26]. On the other hand, in the previous sections we saw that there are at least two possibilities for the metric tensor gμν corresponding to the two geometries of the rotation groups SU(2) and SO(3) (cf. Fig. 3):
$$ -g_{\mu\nu}a_{\mu}b_{\nu}=-\delta_{\mu\nu}a_{\mu}b_{\nu}=-\cos\eta_{\textbf{a}\textbf{b}}, $$
(113)
which manifests sensitivity to spinorial sign changes, and
$$\begin{array}{@{}rcl@{}} -g_{\mu\nu}a_{\mu}b_{\nu}&=&-J_{\mu\nu}a_{\mu}b_{\nu}=-\cos\alpha_{\textbf{a}\textbf{b}} \\ &=& \left\{\begin{array}{lll} -1+\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} \text{if} & 0 \leq \eta_{\textbf{a}\textbf{b}} \leq \pi \\ \\ +3-\frac{2}{\pi}\eta_{\textbf{a}\textbf{b}} \text{if} &\!\! \pi \leq \eta_{\textbf{a}\textbf{b}} \leq 2\pi, \end{array}\right. \end{array} $$
(114)
which manifests insensitivity to spinorial sign changes. These two possibilities amount to identifying the spin basis {1, Lμ(λ)} either with the basis {1, βμ(λ)} of TqS3 defined in (54) or with the basis {1,ξμ(λ)} of T[q]IRP3 defined in (84). The two metrics δμν and Jμν, respectively, are therefore measures of the geodesic distances on the manifolds S3 and IRP3, as we discussed in the previous sections. Thus, the correlation \({{\mathcal E}(\textbf {a},\textbf {b})}\) we derived in (110) can serve to distinguish the geodesic distances \({{\mathcal D}(\textbf {a},\textbf {b})}\) on the groups SU(2) and SO(3).

6 Conclusion

In their landmark textbook on gravitation Misner, Thorne, and Wheeler noted that there is something about the geometry of orientation that is not fully taken into account in the usual concept of orientation [1, 2, 3]. They noted that rotations in space by 0, ±4π, ±8π,…, leave all objects in their standard orientation-entanglement relation with their surroundings, whereas rotations by ±2π, ±6π, ±10π,…, restore only their orientation but not their orientation-entanglement relation with their surroundings. The authors wondered whether there was a detectable difference in physics for the two inequivalent states of an object. Earlier Aharonov and Susskind had argued that there is a detectable difference for such states in quantum physics, but not classical physics, where both absolute and relative 2π rotations are undetectable [6].

In this paper we have argued that there is, in fact, a detectable difference between absolute and relative 2π rotations even in classical physics. In particular, we have demonstrated the observability of spinorial sign changes under 2π rotations in terms of geodesic distances on the group manifolds SU(2) and SO(3). Moreover, we have proposed a macroscopic experiment which could infer the 4π periodicity in principle [6]. The proposed experiment has the potential to transform our understanding of the relationship between classical and quantum physics [18].

Notes

Acknowledgments

I am grateful to Martin Castell for his kind hospitality in the Materials Department of Oxford University where this work was completed, and to Manfried Faber and Christian Els for comments on the earlier versions of this paper. Christian Els also kindly carried out parts of the calculation in Appendix A especially the derivation of the curvature in (A.14). This work was funded by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-MGA-1215 to the Foundational Questions Institute. I thank Jurgen Theiss of Theiss Research for administering the grant on my behalf.

References

  1. 1.
    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman and Company, New York (1973)Google Scholar
  2. 2.
    Penrose, R.: The Road to Reality: A Complete Guide to the Laws of the Universe. Jonathan Cape, London (2004)Google Scholar
  3. 3.
    Hartung, R.W.: Am. J. Phys. 47 (900) (1979). See also P.O. Brown and N.R. Cozzarelli, Science 206, 1081 (1979); and T.R. Strick, V. Croquette, and D. Bensimon, Nature London 404, 901 (2000)Google Scholar
  4. 4.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  5. 5.
    Hestenes, D.: Am. J. Phys. 71, 104 (2003)CrossRefADSGoogle Scholar
  6. 6.
    Aharonov, Y., Susskind, L.: Phys. Rev. 158 (1237) (1967). see also H.J. Bernstein, Sci. Res 18 33 (1969)Google Scholar
  7. 7.
    Werner, S.A., Colella, R., Overhauser, A.W., Eagen, C.F.: Phys. Rev. Lett. 35, 1053 (1975)CrossRefADSGoogle Scholar
  8. 8.
    Weingard, R., Smith, G.: Synthese 50, 213 (1982)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Penrose, R., Rindler, W.: Spinors and Space-Time, vol. 1. Cambridge University Press, Cambridge (1987)Google Scholar
  10. 10.
    Koks, D.: Explorations in Mathematical Physics: The Concepts Behind an Elegant Language Springer (2006)Google Scholar
  11. 11.
    Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M.: Analysis, Manifolds and Physics: Parts I and II, Revised Edition, North Holland, Amsterdam (2000)Google Scholar
  12. 12.
    D’Inverno, R.: Introducing Einstein’s Relativity. Oxford University Press, Oxford (1992)MATHGoogle Scholar
  13. 13.
    Aldrovandi, R., Pereira, J.: Teleparallel Gravity: An Introduction (2013). T. Ortín, Gravity and Strings (Cambridge University Press, 2004); K. Hayashi and T. Shirafuji, Phys. Rev. D 19, 3524 (1979)Google Scholar
  14. 14.
    Eisenhart, L.P.: Amer. Bull. Math. Soc. 39, 217 (1933)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Nakahara, M.: Geometry, Topology and Physics. Adam Hilger IOP Publishing Ltd, Bristol and New York (1990)CrossRefMATHGoogle Scholar
  16. 16.
    Ryder, L.H.: J. Phys. A 13, 437 (1980)CrossRefADSMATHMathSciNetGoogle Scholar
  17. 17.
    Eberlein, W.F.: Am. Math. Monthly 69, 587 (1962); See also Eberlein, W.F.: Am. Math. Monthly 70, 952 (1963)Google Scholar
  18. 18.
    Christian, J.: Disproof of Bell’s Theorem: Illuminating the Illusion of Entanglement, 2nd Edition. BrownWalker Press, Boca Raton, Florida (2014)Google Scholar
  19. 19.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics (AMS Chelsea Publishing, Providence, RI (2008)Google Scholar
  20. 20.
    Du, Q.H.: J. Math. Imaging Vision 35, 155 (2009)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Kambe, T.: Geometrical Theory of Dynamical Systems and Fluid Flows, 2nd Revised Edition. World Scientific Publishing Company, Singapore (2010)MATHGoogle Scholar
  22. 22.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer-Verlag, New York (1989)CrossRefGoogle Scholar
  23. 23.
    Peres, A.: Quantum Theory: Concepts and Methods, p. 161. Kluwer, Dordrecht (1993)Google Scholar
  24. 24.
    Milnor, J.W.: Topology from the Differentiable Viewpoint. Princeton University Press, Princeton, NJ (1965)MATHGoogle Scholar
  25. 25.
    Rodgers, J.L., Nicewander, W.A.: Am. Stat 42, 59 (1988)CrossRefGoogle Scholar
  26. 26.
    Frankel, T.: The Geometry of Physics: An Introduction, p. 501. Cambridge University Press (1997)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Wolfson CollegeUniversity of OxfordOxfordUK

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