A Note on Penrose’s Spin-Geometry Theorem and the Geometry of ‘Empirical Quantum Angles’

In the traditional formalism of quantum mechanics, a simple direct proof of (a version of) the Spin Geometry Theorem of Penrose is given; and the structure of a model of the ‘space of the quantum directions’, defined in terms of elementary SU(2)-invariant observables of the quantum mechanical systems, is sketched.


Introduction
Penrose, in the pioneering work [1] (which appeared first already in 1966 as the appendix of his Adams Prize essay), suggested the so-called SU (2) spin network as a simple model for the quantum spacetime (see also [2][3][4][5]). The key idea was that, unifying and splitting quantum mechanical systems and measuring the probabilities of occurrence of the various total (i.e. the j) values of angular momenta obtained in this procedure, an empirical angle between the angular momentum vectors can be defined; and the key result was that, in the large j (i.e. in the classical) limit, these angles tend to be angles between directions in the Euclidean 3-space.
The significance of this result is that the (conformal structure of the) 'physical 3-space' that we use as an a priori given 'arena' in which the physical objects are thought to be arranged and the interactions between them occur is determined by the quantum physical systems themselves in the classical limit. Its proof was based on combinatorial/graphical techniques. The key idea and a sketch of the proof appeared in [3,6], but so far the detailed and complete proof has not been published [7]: the proof in [3] remained incomplete. Later, another (and mathematically different) version of the same physical result became known as the Spin Geometry Theorem [6].
The proof of the latter version was based on the more familiar formalism of quantum mechanics. (For some more historical remarks, see also [3]. ) However, soon after the appearance of the idea how the 'true' geometry of space(time) should be defined in an operational way, the emphasis was shifted from the systematic investigation of the consequences of the original ideas to the development of mathematical theories of possible a priori quantum geometries. The latter were only motivated by, but were not based directly on the original ideas. The twistor theory [8][9][10], and also canonical quantum theories of gravity (see e.g. Ref. [11]) are such promising mathematical models. In particular, in the latter, the area and volume [12] and also the angle [13], represented by appropriately regularized quantum operators, have been shown to be discrete; and, remarkably enough, the spin networks emerged as a basis of the states in this theory [14].
In the present note, we return to the original idea formulated in [1], and especially in [2,3]: while e.g. the electrons or the electromagnetic field are existing objects (i.e. 'things'), by their primary definition, the spacetime and its points, the events, are not. The events are phenomena, and the spacetime, the set of them, is only a useful notion by means of which the laws of Nature can be formulated in a convenient, simple way. Thus, we also share the positivistic, Machian view (see [2,3]) that spacetime and its geometry should be defined in an operational way by existing material systems.
Here, we adopt the idea of the algebraic formulation of quantum theory that the quantum system is specified completely if its algebra of (basic) observables (and, if needed, its representation) is fixed. Then the notion and all the structures of space/ spacetime should be defined in terms of the observables of the quantum physical subsystems of the Universe. In particular, the angles between 'directions' associated with quantum mechanical subsystems should also be introduced in this way even though the 'directions' themselves are not defined at all in the classical sense. Actually, these subsystems are chosen to be 'elementary' in the sense that the quantum observables are self-adjoint elements of the enveloping algebra of the su(2) Lie algebra of the angular momentum operators as the basic observables. We can use all the structures on this algebra, but, in addition to this, no a priori notion of space/ spacetime, as an 'arena' of events, is allowed to be used. It is this general strategy (but replacing su(2) by the Lie algebra e(3) of the Euclidean group) that we follow in [15] in deriving the metrical (rather than only the conformal) structure of the Euclidean three-space from elementary quantum systems.
As far as we know, no complete proof of the Spin Geometry Theorem, even the version given by Moussouris in [6], has been published. (The sketch of Moussouris's proof was summarized in [13].) The present note intends to make up this shortage, using an improved version of Moussouris's empirical angle between the angular momentum vectors of elementary quantum systems. We give a new proof of the theorem in the usual (but slightly more algebraic) formulation of quantum mechanics. In proving this theorem, no recoupling is needed. The present investigation also casts some light on the nature of the 'space of empirical quantum directions'.
In the next section we introduce the key notion, the improved version of the empirical angle between the angular momentum vectors of quantum systems. Based on this notion, in Sect. 3, we present the new proof and discuss the results. In Sect. 4 Clearly, both |J | and cos are SO(3)-invariant. If the direction of the angular momentum vector J a can be obtained from that of J a by an SO(3) rotation of angle in the plane spanned by J a and J a , then, clearly, = . Thus, by measuring the SO(3)-invariant observables cos we can recover the angle between the angular momentum vectors of the subsystems, defined in the space of the classical observables. Next we convey these ideas into the quantum theory in a systematic way. We will see that, for quantum systems, these two concepts of angle split with far reaching consequences.
Let S , = 1, 2, … , N , be quantum mechanical systems, whose (normalized) vector states (or, in the bra-ket notation, � ⟩ ) belong, respectively, to the Hilbert spaces H . The general (e.g. mixed) states are density operators on them: ∶ H → H . The basic quantum observables are the angular momentum vector operators a satisfying the familiar commutation relations [ a , b ] = iℏ ab c c , where abc is the alternating Levi-Civita symbol, and we lower and raise the Latin indices by the Kronecker delta ab and its inverse. 1 The action of SO(3) (or rather of SU(2)) on this algebra is given by a ↦ (R −1 ) a b b . If the SU(2) matrix U A B is parameterized by the familiar Euler angles ( , , ) according to then the corresponding rotation matrix, Here AA ′ a are the three non-trivial SL(2, ℂ) Pauli matrices (including the factor 1∕ √ 2 ) according to the conventions of [16,17]. (The minus sign in the expression of R a b is a consequence of the convention that, in the present note, we lower and raise the Latin indices by the positive definite metric ab and its inverse, respectively, rather than by the negative definite spatial part of the Minkowski metric. The spinor name indices A, B, … are lowered and raised by the anti-symmetric Levi-Civita symbol AB and its inverse.) For later use, note that Considering the systems S 1 ,...,S N to be a single system, the space of the vector (or pure) states of the resulting composite system will be H ∶= H 1 ⊗ ⋯ ⊗ H N . Its elements are general linear combinations of the tensor products 1 ⊗ ⋯ ⊗ N of the pure states of the subsystems, while a general state is given by a density operator ∶ H → H . The operators a define the operators 1 ⊗ ⋯ ⊗ a ⊗ ⋯ ⊗ N on H , denoted for the sake of simplicity also by a . Here is the identity operator acting on H . With these notations, the operators a b are well defined, and a and b are commuting if ≠ . Next, this will be denoted simply by ( ) 2 . Motivated by Eq. (1), we define the empirical (quantum) angle between the subsystems S and S in the pure tensor product state Since the absolute value of the expression on the right is not greater than one, this can, in fact, be considered to be the cosine of some angle ; and for the range of this angle it seems natural to choose [0, ]. Remarks: 1. cos depends only on the states of S and S , and it is independent of the states of the other subsystems. Moreover, using the transformation property † a = R a b b of the angular momentum vector operator, it is straightforward to check that cos is SU(2)-invariant. 2. Since for = one has that ⋅ = ab a b , which is just the Casimir operator ( ) 2 of su(2) on S , cos = 1 holds. Thus the empirical angle between any angular momentum vector and itself is always zero, as it could be expected. In the rest of this note, we assume that ≠ . 3. It is straightforward to define the angle between two subsystems even when the state of the composite system is an entangled state, when it is a general mixed state, represented by a density operator ∶ H → H . In the first case, it is still defined by (4), while in the second by cos cos − sin cos sin − sin cos − cos cos sin sin sin cos sin + sin cos cos − sin sin + cos cos cos − sin cos sin sin cos sin cos However, in these cases the states of the individual constituent subsystems would be mixed, and hence the interpretation of cos in these cases would not be obvious. In addition, this angle might depend on the state of the other subsystems, too. Nevertheless, this extended notion of the empirical angle may provide the appropriate mathematical formulation of Penrose's 'ignorance factor' [1,5] between S and S . 4. The operator ⋅ was introduced by Moussouris in [6]. However, in the definition of the empirical angle according to him the states had to belong to finite dimensional representation spaces of su (2). In fact, the denominator in his definition is the norm of the unbounded operator ⋅ , which is finite only on finite dimensional spaces. In our definition (4) the Hilbert spaces H and H are not required to be finite dimensional. Moreover, the geometric idea of angle given in the classical theory by (1) seems to be captured in the quantum theory more naturally if, in the denominator, the 'lengths' of the individual angular momentum vector operators in the given states are used, just according to (4), rather than the norm of ⋅ . Indeed, cos in the state �j , j ⟩ according to Moussouris would give (j + 1)∕j , which is always greater than 1. 5. In the theory of canonical quantum gravity, Major [13] defined the angle operator acting on two edges of the spin network states, labelled by two su(2) Casimir invariants, say j and j . That operator is ⋅ divided by the norm of a and of a . Thus, Major's angle operator is the correct 'operator version' of (1) (and hence of (4)). Nevertheless, since a and a are not bounded, this angle operator is well defined only on finite dimensional Hilbert spaces.
In the present paper, we calculate the empirical angle between the subsystems only in pure tensor product states of the composite system according to (4). Thus, by the first remark above, it is enough to consider only two (and, at the end of Sect. 3, only three) subsystems. In the proof of the Spin Geometry Theorem, it will be enough to assume that these states are tensor products of eigenstates of the Casimir operators of the two subsystems, labelled by two Casimir invariants, say j 1 and j 2 . Let {�j 1 , m 1 ⟩} and {�j 2 , m 2 ⟩} be the canonical angular momentum bases in the corresponding eigenspaces. Note that here m is only an index labeling the vectors of an orthonormal basis in the 2j + 1 dimensional carrier space of the unitary representation of su (2), but it does not refer to any Cartesian frame in the 'physical 3-space'. The basis {�j, m⟩} is chosen to be adapted to the actual choice for the components of the vector operator a in the abstract space of the basic quantum observables. We choose 1 and 2 simply to be 1 �j 1 , m 1 ⟩ and 2 �j 2 , m 2 ⟩ , where the unitary operators 1 and 2 represent SU(2) matrices of the form (2) with some Euler angles ( 1 , 1 , 1 ) and ( 2 , 2 , 2 ) in the given representations, respectively.
Then, using † a = R a b b and how the angular momentum operators act on the canonical bases, (4) yields .
where, by (3), cos 12 ∶= (R −1 1 R 2 ) 33 = cos 1 cos 2 + cos( 1 − 2 ) sin 1 sin 2 . Remarks: 1. The expression of cos 12 above is a simple consequence of the well known addition formulae for the Euler angles, which can be read off directly from (3), too. 12 is just the angle between the unit vectors (R 1 ) a 3 and (R 2 ) a 3 , i.e. an angle between directions in the 3-space of the basic quantum observables. 12 depends only on the relative orientations of the two subsystems.
2. (5) shows that, for 12 ∈ [0, ∕2) , the empirical angle, 12 , is always greater than 12 , and for 12 ∈ ( ∕2, ] it is always smaller than 12 . 12 = 12 precisely when 12 = ∕2 . For given j 1 and j 2 , the range of cos 12 is the whole closed interval . If 12 is fixed, then the empirical angle is still not fixed and it can take different discrete values. Note that Planck's constant is canceled from its expression. 12 tends to 12 asymptotically when m 1 = j 1 , m 2 = j 2 and both j 1 and j 2 tend to infinity. It is this limit that is usually considered to be the classical limit of the spin systems (see e.g. [18]). 3. For 12 = 0 , the empirical angle is given by cos 0 12 = m 1 m 2 ∕ √ j 1 (j 1 + 1)j 2 (j 2 + 1) . Here, m∕ √ j(j + 1) is just the cosine of the 'classical' angle between the angular momentum vector of length √ j(j + 1) and its z-component with length m. Hence, for given m 1 m 2 ≠ 0 and 12 = 0 , the greater the product m 1 m 2 , the smaller the angle 0 12 , but it is never zero. Its minimum value corresponds to cos 0 12 = cos 1 cos 2 , where cos ∶= √ j∕(j + 1) . Hence 0 12 is greater than any of 1 and 2 . 0 12 tends to zero only asymptotically in the m 1 = j 1 → ∞ , m 2 = j 2 → ∞ (classical) limit.

The Classical Limit and the Spin Geometry Theorem
By (5), the empirical angles between the subsystems S and S of the composite system in the tensor product of the individual states � ⟩ = U �j , j ⟩ and � ⟩ = U �j , j ⟩ , respectively, are given by Thus, as a consequence of the discussion at the end of the previous section, we immediately obtain the following statement: cos .
Thus, in the large j limit, the empirical angles tend to the angles of the three dimensional Euclidean vector space of the basic quantum observables.
However, still we should check that, in this limit, the uncertainties do not grow. In fact, we show that these uncertainties tend to zero. First we calculate the square of the standard deviation of 1 ⋅ 2 . The expectation value of its square is Since the only non-zero matrix elements of a b in the states �j, j⟩ are (7) takes the form Using the explicit form (3) of the rotation matrix, a lengthy but elementary calculation yields that Hence, the square of the standard deviation of 1 ⋅ 2 in the state 1 ⊗ 2 is Since the state 1 ⊗ 2 is an eigenstate both of ( 1 ) 2 and ( 2 ) 2 , finally we obtain that the square of the uncertainty of cos in the state = 1 ⊗ ⋯ ⊗ N , defined by the first equality below, is (1 − cos ) 2 + 2(j + j ) sin 2 (j + 1)(j + 1) .
For given j and j this uncertainty is zero precisely at = 0 , and it takes its maximal value at cos = 1∕(1 − 2(j + j )) . However, independently of , this uncertainty tends to zero if j , j → ∞ . With this conclusion we have proven the next statement.
Proposition 2 For arbitrarily small > 0 there is a positive integer J such that, in the states above for any j , j > J , , = 1, … , N , Δ cos < hold.
With the Propositions 1 and 2 at hand we have already given a simple proof of (a version of) the Spin Geometry Theorem in the traditional framework of quantum mechanics: Theorem Let S be composed of the quantum mechanical systems S 1 , … , S N . Then there is a collection of pure tensor product states of S , 1 (j 1 ) ⊗ ⋯ ⊗ N (j N ) indexed by an N-tuple (j 1 , … , j N ) of non-negative integers or half-odd-integers, such that, in the j 1 , … , j N → ∞ limit, the empirical angles between these subsystems converge with asymptotically vanishing uncertainty to angles between directions of the three dimensional Euclidean vector space.

Remarks:
1. Since for any given and j ′ ≠ j the states �j , j ⟩ and �j ′ , j ′ ⟩ belong to orthogonal subspaces of the Hilbert space H of all the pure states of the system S , the states be recovered in the classical limit from quantum observables, is the same. Hence we can consider them only different versions of the same physical theorem. 4. According to expectations of certain recent investigations (see e.g. [20][21][22]), the geometry of 3-space/spacetime emerges from the entanglement of the states of the quantum subsystems of the Universe. However, the results of the present work do not seem to support this expectation, at least in the quantum mechanical approximation of the quantum world. In fact, the Spin Geometry Theorem could successfully be proven using only pure tensor product states of the subsystems. The entanglement of the states of the subsystems was not needed. On the other hand, the quantum operator ⋅ , by means of which the empirical angles were defined, is a sum of the products of observables, 1 1 + 2 2 + 3 3 , which structure is analogous to that of the entangled states. But, in contrast to local quantum field theory, in quantum mechanics there is no locality: the quantum operators 'feel' the whole wave function on the entire momentum/configuration space. Therefore, the entanglement can be considered to be built already into the structure of the quantum mechanical observables of the composite system, by means of which the geometry of 3-space/spacetime can be defined in an operational way.
The states do not need to be entangled. 4. Using the natural volume 3-form abc on the algebra su(2) of the basic quantum observables, a further potentially interesting geometric notion, viz. the 'empirical 3-volume elements' can be introduced. In the state ∈ H spanned by three subsystems this is defined by Then, in the tensor product of the states of the form �j, m⟩ with the unitary operator representing some SU(2) matrix U A B of the form (2), this expression gives where R a b is the rotation matrix (3) corresponding to U A B , and the second factor in (10) is just the Euclidean 3-volume of the tetrahedron spanned by the unit vectors (R ) a 3 , (R ) a 3 and (R ) a 3 . Thus, even if m = j , m = j and m = j , v is always smaller than its Euclidean counterpart: the former is only conformal to the latter, and it tends to the Euclidean 3-volume only in the m = j , m = j , m = j → ∞ limit.

A Classical Model of the 'Space of the Quantum Directions'
In [5], Penrose (quoting Aharonov, too) raises the possibility in the context of the quantum mechanical double slit experiment that the 'true' geometry that the electron 'sees' might be different from the 'classical' geometry of the two slits. Motivated by this idea, we may ask 'What kind of geometry should we have if we want to arrange the "empirical" geometric notions and quantities introduced via the observables of the quantum systems?' In particular, what could be the geometry in which the empirical angles and 3-volume elements, defined by the composite quantum system S = S 1 ∪ ⋯ ∪ S N , are angles and 3-volume elements?
To illustrate by a classical model how this 'true' (conformal) geometry might look like, for the sake of simplicity suppose that all the subsystems have the same total angular momentum j. Let us assume that the empirical angles are angles between pairs of unit vectors of (and the empirical 3-volumes are 3-volumes of tetrahedra formed by triplets of unit vectors in) some n dimensional real vector space. This vector space is modeled by ℝ n , which is endowed by some positive definite metric G , , = 1, … , n.
By (5) the empirical angle between the 'directions' of two subsystems cannot be smaller than min = arccos(j∕(j + 1)) and cannot be greater than max = arccos(−j∕(j + 1)) = − min . (For example, for j = 1∕2 these bounds are ≈ 70.53 • and ≈ 109.47 • ; and for j = 1 these are 60 • and 120 • , respectively.) Clearly, for any given N, these empirical angles can always be arranged in ℝ n for large enough but finite n. Nevertheless, the existence of the bounds min and max provides a lower bound for n. (For a given state 1 ⊗ ⋯ ⊗ N , the optimal value of n might be determined by a procedure analogous to that in the so-called sphere packing problem [23], see below.) Let the unit vector V represent the 'direction' associated with the th subsystem in this space. Then let us draw two solid cones in ℝ n with V as their common axis, their vertices at the origin, and with the opening angle min and max , respectively, such that the cone with opening angle max contains the cone with opening angle min . Then the result that the empirical angle between the 'directions' of any two subsystems cannot be greater than max and cannot be smaller than min implies that the inner cone with axis V and that with axis V intersect each other at most along one line in their lateral surface, and their outer cone intersect each other at least along one line in their lateral surface.
Let p be the point of the unit sphere S n−1 in ℝ n that the unit vector V defines, let C denote the intersection of the cone with axis V and opening angle max with the unit sphere, and let c be the intersection of the inner cone with the unit sphere. Thus, C and c are concentric spherical caps with the common centre p on S n−1 . Then the resulting spherical caps C and C , ≠ , must intersect each other at least in one point, but the corresponding inner spherical caps c and c may intersect each other at most in one point. Therefore, the set of the empirical angles between any two subsystems considered in Sect. 2 may be represented by the set of these configurations of the points p and the corresponding pairs (C , c ) , = 1, … , N . There are no distinguished directions in this space, i.e. the space does not have any naive lattice structure, but the angle between any two directions cannot be arbitrarily small or arbitrarily large.
This second realization of the classical model of the 'space of the quantum directions' makes it possible, at least in principle, to determine the minimal dimension n in which the directions of N subsystems, each with spin j, can be arranged: this is the minimal dimension for which N pairs of concentric (n − 1) dimensional spherical caps, or rather balls, with given radii can be packed into the unit sphere S n−1 satisfying the above conditions. This is a version of the 'sphere packing problem' of [23].
Requiring that the 3-volume elements determined by V , V and V in ℝ n be just v for the composite system, (10) might suggest to choose the metric G to be conformal to the Euclidean one, G = (j∕(j + 1)) .