A Relationship Between Spin and Geometry

In a recent paper, algebraic descriptions for all non-relativistic spins were derived by elementary means directly from the Lie algebra $\specialorthogonalliealgebra{3}$, and a connection between spin and the geometry of Euclidean three-space was drawn. However, the details of this relationship and the extent to which it can be developed by elementary means were not expounded. In this paper, we will reveal the geometric content of the spin algebras by realising them within a novel, generalised form of Clifford-like algebra. In so doing, we will demonstrate a natural connection between spin and non-commutative geometry, and discuss the impact of this on the measurement of hypervolumes and on quantum mechanics.

The multipoles are defined recursively in terms of the adjoint action ad, ad(u * ucapptj@ucl.ac.uk, orcid.org/0000-0001-9938-8460 the left multiplication ∀A ∈ U (so(3, R)), L(A) := B → A⊗B, (5) and the Casimir element of U (so(3, R)), as ∀k ∈ Z + , α ∈ R, v ∈ so(3, R), B k ∈ so(3, R) ⊗k , The multipoles are important to the structure of U (so(3, R)), since ∀A k ∈ U (so(3, R)), ∀k ∈ N, for which ad(S 2 + k(k + 1))(A k ) = 0 may be written as an R[S 2 ]-linear combination of objects from Im(M (k) ), and all elements of U (so(3, R)) are linear combinations of such A k .For compactness, let us define ∀k ∈ Z + , M := M (0) (1) The spin algebras A (s) are real unital associative algebra of multipoles, ∀k ∈ Z + , A (0) ∼ = span R ({M}) within which, S 2 = −s(s + 1).(10) Since all multipoles M a1...a 2k are algebraic combinations of the {S a }, the A (s) encode the spin structures for arbitrary spins s entirely in terms of so (3, R).The Lie algebra so(3, R) generates the Lie group SO(3, R), which is the connected symmetry group of Euclidean three-space.In this way, the A (s) are connected to the geometry of Euclidean three-space, however the extent and consequences of this connection is unclear.

A Geometric Realisation of so(3, R) through Clifford Algebra
To understand the extent of the relationship between the A (s) and the geometry of Euclidean three-space, let us first attempt to understand the underlying geometric content of so(3, R), with which any geometric account of A (s) must be compatible.Towards this, we will explore the geometric structure of the more general so(p, q, R), the Lie algebra of the connected symmetry group SO + (p, q, R), which preserves the geometry of a (p+q)-dimensional space with indefinite signature.
Let (V, g) denote such a non-trivial finite-dimensional vector space V over R equipped with a symmetric, non-degenerate, bilinear map g : V ×V → R, which we shall follow relativity by referring to as a "metric".We may identify the Lie algebra so(p, q, R) with the set of all linear maps Such maps are closed under commutators, which serves as the Lie product.It has long been known that so(p, q, R) is in bijection [4] with Λ 2 (V ) ⊂ T (V ), the space of second-order antisymmetric tensors on where ∧ is the multilinear, totally antisymmetric, associative "wedge product", ∀k ∈ Z + ,∀v j ∈ V : j ∈ {1, . . ., k}, with S k is the set of all permutations of k objects, and sgn(σ) is the sign of the permutation σ.Explicitly, this bijection may be given, up to a scalar, as ∀v, w, x ∈ V , This bijection grants us an immediate geometric interpretation for the objects of so(p, q, R): linear combinations of planar elements.More generally, the "k-blade" [5] (13) can be interpreted as a hypervolume element of dimension k.For so(3, R), a 2-blade encodes both the plane and angle of the rotation it generates.We refer to an arbitrary element of Λ k (V ) as a "k-vector" [5] or "prefix-vector" e.g.2-vector and bivector are identical.For completeness, we consider 0-vectors and 0-blades to be the scalars of V .
With the objects of so(p, q, R) algebraically identified as bivectors Λ 2 (V ), we may find their Lie product by constructing the Clifford algebra [6] Cl(V, g): This quotient reduces all tensors of T (V ) to linear combinations of k-blades.The survival of the k-blades in T (V ) mark them as objects of geometric significance.Cl(V, g) is finite-dimensional, as all k-blades with k > dim(V ) are 0 by antisymmetry.Since the field of scalars of V is not of characteristic 2, this is equivalent to the construction of Cl(V, g) using a quadratic form [6].
The structure of the Clifford algebra reveals the Lie product between bivectors, turning Λ 2 (V ) into a Lie algebra.This Lie product is related to the usual one [7] by a scaling.The Clifford algebra also naturally defines an so(p, q, R)-action on vectors ∀a, b, c ∈ V , which is identical to (14).This enables a natural action of the symmetry group SO + (p, q, R) to be defined algebraically on Cl(V, g) [5].
Restricting our attention to three-dimensional Euclidean space (E, δ), we may introduce the transformation, ε abp e a ∧e b , where {e 1 , e 2 , e 3 } are a basis for E satisfying δ(e a ,e b ) = δ ab .Then, on basis bivectors the Lie product (16) becomes, consistent with (1b).Thus, we see that Cl(E, δ) algebraically realises so(3, R) in a geometrically meaningful way using bivectors.

Limitations of the Clifford Algebra Approach
Despite this natural emergence of so(3, R) within Cl(E, δ), this realisation is severely limited.To see this, we note that in Cl(E, δ), we have ∀a, b, c, d ∈ E, Applying (18), we find in Cl(E, δ), and the Casimir element S 2 of so(3, R) is, Together, (21) and (22) imply that the spin quadrupole M ′ pq = 0 in Cl(E, δ).By the multipole recurrence relationship (7), we also conclude that all spin multipoles M p1...,p k = 0 for k > 2. This shows that, unsurprisingly [5], the unital subalgebra {R, Λ 2 (E)} ⊂ Cl(E, δ) has spin- 1  2 structure, and is algebra isomorphic to A ( 1  2 ) .This is a direct result of the defining algebraic structure (15) of Cl(E, δ).Thus, Cl(E, δ) cannot support an arbitrary spin structure within it, and cannot be used to explore the geometric content of A (s) for s = 1 2 .Finding an algebra which can will be the focus of this paper.In section 2, we will define a "Spinless Weak Clifford Algebra" compatible with the structure of an arbitrary A (s) .We will present the "Spin-s Weak Clifford Algebras" derived from these in section 3, and show they may naturally entail spindependence in the measured sizes of hypervolumes.Finally, in section 4, we will discuss the connection between the spin-s Clifford algebras and non-commutative geometries, and contrast these new algebras with other higher-spin models.We will also consider the implications of these algebras for quantum mechanics.

Towards a weaker Clifford Algebra
As we saw in section 1.3, the incompatibility of the Clifford algebra with arbitrary spin algebras A (s) is inherent to its algebraic structure.This is unfortunate, since this algebraic structure enabled us to find both a natural Lie algebra action (17) and the geometric structure of so(3, R) (16).To study the A (s) , we will construct a weaker algebra with both of these features by elementary means.Such an algebra will have no spin structure at all, enabling any A (s) to be embedded within.We expect the Lie product of so(p, q, R) to follow from the so(p, q, R)-action on our algebra, as (16) follows from (17).Thus, to proceed we require only: an elementary derivation of an so(p, q, R)-action on vectors; and a determination of how this action must be implemented within an associative algebra.

Elementary Derivation of the so(p, q, R)-action on Vectors
Recall that V is a finite-dimensional real vector space and that g is a non-degenerate, symmetric, bilinear map on V .For any non-null element a ∈ V , i.e. g(a,a) = 0, we may always find a unique direct sum decomposition of V , where ∀w ∈ W a , g(a,w) = 0. Specifically, we can write ∀v ∈ V uniquely as, where g(a,b) = 0. We notice from (24) that the decomposition ( 23) is scale-invariant in two ways: ∀α ∈ R/{0}, a and a ′ = αa give the same decomposition, as do g and b = αg.This suggests that some of the structure imparted on V by g is independent of scale.Thus, let us explore this structure more easily by accepting scaling of the metric g in our arguments.Doing so will enable us to establish results using more mathematically convenient objects.From ( 23), we may use g to alter the scale of the component of v ∈ V parallel to a non-null vector accepting the overall scaling by g(a,a) that occurs.Note that ∀v, w ∈ V , precisely when k ± = ±1.Recognising the k + case as simply an overall scaling, we use the k − solution to define the "conformal reflection", The conformal reflections are a superset of the traditional reflections, but are similarly not closed under composition.
To see this, let us first define a "g-adjoint" of an endomorphism A ∈ End(V ) as an endomorphism Since g is symmetric and non-degenerate, B is unique and its g-adjoint is A; accordingly, we shall denote the g-adjoint of A as Ā with Ā = A. We also define "self-g-adjoint" to mean Ā = A, "anti-self-g-adjoint" to mean Ā = −A, and two maps a + and a − that respectively yield the self-g-adjoint and anti-self-gadjoint parts of an endomorphism A, We find all conformal reflections are self-g-adjoint, but the g-adjoint of R(a This indicates that there is structure in the product of two conformal reflections that is not itself a conformal reflection.
To identify this additional structure, we decompose R(a Noticing that there is a repeated pattern within (30a) and (30b), we define ∀a, b ∈ V , enabling us to write, Since a + (A) + a − (A) = A, we find that the map t(a,b) controls binary products of conformal reflections.It is also antisymmetric, t(b,a) = −t(a,b), and anti-self-g-adjoint.Furthermore, given (32a), and ∀v ∈ V , we see the {R(a), t(b, c)} forms a generating set for the algebra of conformal reflections.Therefore, we have derived a map with central importance to the conformal reflections, and whose image agrees with the image of (14).So far our derivation has only accounted for non-null vectors a, b ∈ V with g(a,b) = 0, so, Let us extend this definition to the whole of V .Since t(a,b + ǫa) = t(a,b), ∀ǫ ∈ R, t(a,b) for non-null vectors is defined without the need for limits.To define t(a,b) when b is null and g(a,b) = 0, by the non-degeneracy of g we may find a null c ∈ V such that g(b,c) = 0, and use this pair to construct a pair of vectors {p, n} such that, Thus, we may define, by the bilinearity of t.Similar arguments yield t(a,b) for a and b both null, regardless of the value of g(a,b).Thus, we have defined t(a,b) ∀a, b ∈ V and completed our derivation of the so(p, q, R)-action on V ; this was achieved in a coordinate-free, elementary way, without appealing to differential geometry or the theory of Lie groups [8].

The Algebraic Form of t(a,b)
Having acquired t(a,b), the so(p, q, R)-action on V , ∀a, b ∈ V , we must consider how to implement it algebraically within an associative algebra.More precisely, we seek a third-order tensor f (a, b, c) whose properties match those of t(a,b)(c), so that a quotient of T (V ) by the two-sided ideal ∀a, b, c ∈ V , yields the most general non-trivial algebra possible.We first note that t(a,b)(c) is antisymmetric in its first two arguments.The most general third-order tensor sharing this property is, We may constrain (38) even further by considering the commutator between two t maps, which is closed, From (39), we see ∀a, b, c, d, e ∈ V , from which we require, The only non-trivial solution to (41) is, defining f (a, b, c) up to an arbitrary scaling, which was expected.

The Spinless Weak Clifford Algebra
We now have everything we need to construct the "Spinless Weak Clifford Algebra" Cl s w (V, g): choosing whose defining identity, (a∧b agrees with the Clifford algebra's so(p, q, R)-action (17) up to a scaling.We use the term "weak" Clifford algebra to contrast the "strong" Clifford algebra in the sense used in logic: the defining relationship of Cl(V, g) is stronger than the defining relationship of Cl s w (V, g).The bivector a∧b necessarily appearing whole in (43) demonstrates its significance to the properties of g, and that t(a,b) is truly a bivector-action on a vector.We may capture this by defining, u(a∧b As the unique embedding of t(a,b) in Cl s w (V, g), we may consider the properties of u(a ∧ b) to be the natural extension of t(a,b) to Cl s w (V, g).In particular, we find that u(a∧b) is naturally a derivation This agrees with the standard prescription for a Lie algebra-action on a tensor product of representations, so that exponentiation yields a representation of a Lie group [9].To ensure consistency with the structure of the A (s) , we shall extend u to T (Λ 2 (V )) as an associative algebra-action, ∀α ∈ R, As with ( 17), (43) determines the Lie product of so(p, q, R), which in Cl s consistent with the standard Lie product [7] with Ĵµν = iJ µν = i(e µ ∧ e ν ), and Ĵµν the generators in the physics convention.This is also consistent with the spin generator convention Ŝa = iS a in [1].By properties (47b) and (45), we may use (48) to write the commutator of u compactly, showing that u is indeed an so(p, q, R)-action.

The Spinless Weak Clifford Algebra for (E, δ)
Restricting our attention to the present problem, we consider the three-dimensional Euclidean space (E, δ) from earlier, and its spinless weak Clifford algebra Cl s w (E, δ).Using an orthonormal basis {e a }, we define, ε abp e a ∧e b , (50) which in Cl s w (E, δ) turns (48) into the standard Lie product of so(3, R) (1b).Therefore, U (so(3, R)) ⊂ Cl s w (V, g), and we identify, u| where ad was defined in (4).We note the inverse transformation of ( 50), and recognise that it enables us to write the multipoles of the A (s) in the language of bivectors, ∀k ∈ Z + , M Significantly, though U (so(3, R)) ⊂ Cl s w (E, δ), Cl s w (E, δ) has no spin structure whatsoever.This makes it the ideal basic structure with which to realise the A (s) and explore their geometric content.

k-Volumes in Cl s
w (V, g) In Cl(V, g), geometric measurements about its objects are conveyed by the scalar part [5] • , for example: w (E, δ), there is a similar notion.Recall from section 1.1, that all elements A 0 ∈ U (so(3, R)) for which ad(S 2 )(A 0 ) = 0 are R[S 2 ]-linear combinations of the monopole M. Thus, given an element A ∈ U (so(3, R)) on which ad(S 2 has minimal polynomial m(x), we define its "Monopole Part" Mon(A), We may use u to identify geometrically meaningful objects within Cl s w (V, g) with objects forming simple u(U (so(3, R)))-modules; this is consistent with how the multipole tensors were identified [1].The k-blades are such modules, demonstrating their significance even in Cl s w (V, g).

k-Volumes in Cl s
w (E, δ) In Cl s w (E, δ), we wish to capture the sizes of bivectors and trivectors in some natural way.Accordingly, the invariant tensors at second-order, Mon(S p ⊗S q ) = 1 3 δ pq S 2 (56) and third-order, with all other combinations zero, and •, • is the usual Cauchy-Binet metric [3] for (E, δ), Note that ∆ is not (yet) scalar-valued, 3 Results

The Spin-s = 0 Weak Clifford algebras
Using the spinless weak Clifford algebra for Euclidean three-space Cl s w (E, δ) of (42), we may define the "Spin-s Weak Clifford Algebra" Cl (s)  w (E, δ) for spin-s = 0, Since U (so(3, R)) ⊂ Cl s w (E, δ), this is equivalent to embedding the structure of the A (s) within Cl s w (E, δ).Within this algebra, we may positively identify the A (s) as algebras of bivectors in general, whose action on E respects its Euclidean geometry δ.Thus, we have finally explicated the structure of the A (s) in both geometric and algebraic terms.However, the embedding of their spin structures also has an impact on the geometry itself.
As before, the quotient (63) entails, within Cl (s)  w (E, δ).This ensures that the metric ∆ becomes scalar valued for bivectors and trivectors, ∆(a∧b∧c, a∧b∧c) = s(s + 1) 3 a∧b∧c, a∧b∧c , and naturally spin dependent.The metric on vectors and on scalars remains spin independent.Besides this feature, the values of the metric ∆ are quite different from those of the usual Clifford algebra on (E, δ).It has more consistency with Cl(E, − 1 2 δ), for example the bivectors have the same size and sign, and the trivector has the same sign.However, the vectors and trivector are too large by a factor of 2, and the vectors are also positive-definite.There is freedom in scaling ∆ in these sectors for consistency, but the author can see no mathematical reason for doing so at time of writing.
Despite this change to square norms, the algebraic structure imparted on Cl (s)  w (E, δ) does not affect the action of u(a∧b) nor the rotational behaviour of E ⊂ Cl (s)  w (E, δ), since (63) constrains only totally symmetric combinations of bivectors.However, the structure of Cl (s)  w (E, δ) is significantly affected by the spin structure from A (s) .Recall that A (s) is defined completely from U (so(3, R)) by Im(M (2s+1) ) = {0}.Taking spin-1 2 as an example, this is equivalent to a series of tensor identities, In the language of bivectors this condition is equivalent to ∀a, b, c, d Recognising ∆(a∧b, c∧d) as a scalar, we may break up each bivector on the left-hand side according to (12), revealing (67) to be an constraint on fourth-order tensors in Cl (1/2)   w (E, δ).For spin-s, these identities constrain order 2(2s + 1) tensors.Interpreting E as physical Euclidean space, these embeddings of the spin structures of A (s) within Cl s w (E, δ) constitute non-commutative geometries for E, in the sense of non-commuting position observables [10,11,12].

The Spin-0 Weak Clifford algebra
The case of the spin-0 algebra is an edge-case requiring separate treatment.A (0) contains only the monopole M = 1, and is defined by Im(M (1) ) = {0}, which entails S 2 = 0.In the bivector language, this means that ∀a, b ∈ E, a∧b = 0.
Trying to apply this identity to Cl s w (E, δ) as in (63) results in the trivial algebra R.This is because in (42) we associate Λ 2 (E), E with the whole of E, so quotienting all bivectors in Cl s w (E, δ) sets all vectors in the algebra to 0. Really, the identity (43) is only reasonable when the bivectors are non-zero in the algebra, otherwise we seek an action of zero mapping any vector to any other.To avoid this, we must impose the structure of A (0) directly on T (E), This algebra is unique amongst the spin-s weak Clifford algebras: Im(M (1) so Cl (0) w (E, δ) ∼ = Sym(E) is commutative.In fact, a spin-s weak Clifford algebra is commutative iff it has a spin-0 structure.Additionally, S 2 = 0 implies that, ∆(a∧b, a∧b) = 0 (71a) ∆(a∧b∧c, a∧b∧c) = 0, (71b) which is consistent with our expectations from the other spin-s weak Clifford algebras, and the commutative nature of Cl (0)  w (E, δ).

Discussion
Interpreting the meaning of the spin-s weak Clifford algebras depends heavily on our interpretation of the Euclidean three-space E. The simplest and most relevant view of E, is as the non-relativistic configuration space for a point-like particle.We may then interpret each vector as a position in threespace, or as the underlying algebraic object for a position operator in a quantum mechanical system.In this setting, we see that each spin-s weak Clifford algebra describes a different algebra of position variables according to its spin structure: the spin-0 weak Clifford algebra is commutative and corresponds to the usual position operator algebra in quantum mechanics; and the higher spin algebras are all noncommutative.
In the sense of non-commuting position operators, we see a correspondence between the spin-s weak Clifford algebras, and non-commutative geometries whose structures are determined by their spin.Though the meaning of the spin dependence of area and volume is unclear, especially when identically zero, these phenomena further indicate that the spin structure affects the geometry of, or perhaps experienced by, the system.These non-commutative geometries are, in general, much weaker than those common to the literature [10,11,12], which typically place the position operators into a Heisenberglike [13,14] algebra.
From these observations, we see the Cl (s)  w (E, δ) as a new way to incorporate spin into quantum mechanical theories: directly as certain non-commutative algebras of position (and perhaps, by symmetry, momentum) operators.It is viable to extend the Cl (s)  w (E, δ) to such a more phenomenologically complete model, since they contain the totally symmetric tensors, which are essential to algebraically perform dynamical (symplectic) transformations [4,6].Aside from these considerations, Cl (1/2)   w (E, δ) and Cl (1)  w (E, δ) are also weak enough that the Euclidean Clifford and Duffin-Kemmer-Petiau [15,16,17] algebras respectively may be derived from them.Thus, the Cl (s)  w (E, δ) may form the basis for a generalised spin-s theory of such algebras.Furthermore, relativistic versions of this formalism may prove useful in the construction of theories of quantum gravity which incorporate both non-commutative geometry and spin.
With our interpretation of the Cl (s) w (E, δ) laid out, it is instructive to compare it against other arbitrary spin models.The most relevant such comparison is with the standard tensor product of center of mass and "internal" spin degrees of freedom [18].An immediate similarity is that both models include the spin algebra A (s) as a subalgebra, originating from their spin structures.An immediate difference is that the traditional model incorporates the Heisenberg algebra [13,14], and therefore the notion of momentum, whereas the spin-s weak Clifford algebras do not.However, the most significant difference is that in the traditional model the position and spin degrees of freedom are commuting, and thus independent of each other; they are not in Cl (s)  w (E, δ) by construction.This implies that there is phenomenology between position and spin in a dynamical model containing Cl (s)  w (E, δ) which the tensor product model cannot describe.The tensor product model should however be recoverable within this richer formalism as an approximation in some suitable setting.
Another standard approach to higher spin in non-relativistic physics is to consider a subspace of Cl(E, δ) ⊗k with the appropriate spin structure [19].However, due to the strength of the algebra, it lacks totally symmetric tensors, and so cannot easily form part of a model which algebraically encodes symplectic transformations.Such algebras also have interpretational issues regarding the underlying substructure of Cl(E, δ) ⊗k when applying it to fundamental particles; Cl (s)  w (E, δ) does not suffer from this.Beyond the realm of non-relativistic physics are the Bargmann-Wigner [20] and Joos-Weinberg [21,22] equations.Since the former equations do not have definite spin in general [23], we shall focus on the latter.The γ µ1...µ2s for a particle of spin-s in the Joos-Weinberg equation are comprised of objects which bear close, but not exact, resemblance to the multipole tensors of order s [1], revealing a link to the spin structure of the theory.Much like the tensor product model however, the spin sectors of the Joos-Weinberg equations and their center of mass sectors commute, as do the position operators.In this way the comparisons made between Cl (s)  w (E, δ) and the tensor product model are valid for the Joos-Weinberg equations also.

Conclusion
In this paper, we demonstrated the incompatibility of the Clifford algebra with arbitrary non-relativistic spin structures, and defined a family of generalised Clifford-like algebras which support arbitrary spin structures.To do this, we presented a novel algebraic derivation for the structure of the Lie algebra so(p, q, R), without the need to appeal to the theory of differential geometry or Lie groups.We also defined an even more general Clifford-like algebra with no spin structure at all, which underpins the structure of the arbitrary spin Clifford-like algebras.In so doing, we explicated the geometric structure of the spin algebras, and showed that they each define a unique non-commutative geometry on Euclidean three-space.We found that this structure induces a spin-dependence on the measured notions of area and volume, and compared these new arbitrary spin algebras with existing models of arbitrary spin.Some applications and avenues for further enquiry were discussed.