Generalised uncertainty relations for angular momentum and spin in quantum geometry

We derive generalised uncertainty relations (GURs) for angular momentum and spin in the recently proposed smeared-space model of quantum geometry. The model implements a minimum length and a minimum linear momentum, and recovers both the generalised uncertainty principle (GUP) and the extended uncertainty principle (EUP) within a single formalism. In this paper, we investigate the consequences of these results for particles with extrinsic and intrinsic angular momentum, and obtain generalisations of the canonical ${\rm so(3)}$ and ${\rm su(2)}$ algebras. We find that, although ${\rm SO(3)}$ symmetry is preserved on three-dimensional slices of an enlarged phase space, individual subcomponents of the generalised generators obey nontrivial subalgebras. These give rise to GURs for angular momentum while leaving the canonical commutation relations intact except for a simple rescaling, $\hbar \rightarrow \hbar + \beta$. The value of the new parameter, $\beta \simeq \hbar \times 10^{-61}$, is determined by the ratio of the dark energy density to the Planck density. Here, we assume the former to be of the order of the Planck length and the latter to be of the order of the de Sitter momentum $\sim \hbar\sqrt{\Lambda}$, where $\Lambda$ is the cosmological constant, which is consistent with a finite cosmic horizon. In the smeared-space model, $\hbar$ and $\beta$ are interpreted as the quantisation scales for matter and geometry, respectively, and a quantum state vector is associated with the spatial background. We show that this also gives rise to a rescaled Lie algebra for generalised spin operators, together with associated subalgebras that are analogous to those for orbital angular momentum. Remarkably, consistency of the algebraic structure requires the quantum state associated with the background space to be fermionic. Finally, the modified spin algebra leads to GURs for spin measurements.


I. INTRODUCTION
GURs for position and linear momentum are motivated by gedanken experiments in phenomenological quantum gravity [1][2][3][4][5]. An advantage of this approach is that, being based on very general considerations, the resulting phenomenology is expected to be model-independent. Generalisations of the Heisenberg uncertainty principle (HUP) are, therefore, a fairly generic prediction of lowenergy quantum gravity, no matter how much individual models differ in their conceptual bases or mathematical structures. GURs for position and momentum have also been motivated by arguments in string theory, noncommutative geometry, and deformed special relativity, among others [6,7].
A closely related and very general prediction of quantum gravity models is the existence of a minimum length scale which is expected to be of the order of the Planck length ∼ G/c 3 [8,9]. Certain models also incorporate a minimum momentum scale, although there is less general agreement on whether this is an essential feature of any would-be quantum gravity theory, and, if so, on what value the minimum momentum should take [10][11][12][13]. Nonetheless, a minimum momentum scale is consistent with known physics as the existence of a positive cosmological constant Λ > 0, inferred from observations of type 1A supernovae, large-scale structure, and the cosmic microwave background (CMB) [14,15], implies a minimum space-time curvature. This, in turn, implies a maximum horizon distance of the order of the de Sitter length ∼ 1/ √ Λ [16]. This is approximately equal to the present day horizon radius, r U ≃ 10 28 cm [17]. Thus, in a universe with positive scalar curvature of order Λ, any uncertainty principle with a leading order Heisenberg term gives rise to a minimum momentum of order ∼ √ Λ since ∆x is bounded from above by the de Sitter scale.
Motivated by these considerations, two of the most intensively studied GURs in the quantum gravity literature are the GUP [1,2], and the EUP [3][4][5], where α and η are numerical constants of order unity. The former implements a minimum length scale, ∼ G/c 3 , but no minimum momentum, and may be obtained by extending the Heisenberg microscope thought experiment to include the gravitational attraction between the massive particle and the probing photon [1,2]. The latter implements a minimum momentum scale, ∼ √ Λ, but no minimum length, and may be obtained by modifying Heisenberg's argument to include the effects of repulsive dark energy [3][4][5]. Thus, taking the GUP or EUP separately breaks position-momentum symmetry in the uncertainty relations. However, taking into account the effects of both canonical gravitational attraction and repulsive dark energy motivates the extended generalised uncertainty principle (EGUP) [10]: Here,α andη are appropriate dimensionful constants, which may be obtained by comparing Eq. (3) with Eqs.
(1) and (2), respectively. In this way, the GUP and EUP may be obtained as appropriate limits of the EGUP. Unfortunately, the greatest strength of the phenomenological approach, namely, its modelindependence, is also its greatest drawback. Although the HUP may be motivated, heuristically, by Heisenberg's thought experiment, it may also be obtained rigorously from the canonical quantum formalism [18]. In the former, ∆x and ∆p represent, somewhat vaguely, unavoidable imprecisions in the position and momentum of a quantum particle. In the latter, ∆ ψ x and ∆ ψ p represent standard deviations of the probability density associated with the quantum wave function, |ψ| 2 . Thus, these quantities denote well defined measures of the width of the wave function in the position and momentum space representations, respectively. In this paper, we distinguish between heuristic and well defined uncertainties by labeling the latter with a subscript. This indicates the specific wave function from which the underlying probability distribution is derived.
By contrast with the HUP, there is currently no consensus on how to implement the GUP, EUP or EGUP within a well defined quantum formalism. One option, which, until recently, was the only possibility considered in the existing literature, is to modify the canonical commutation relations for position and momentum [6][7][8]. This approach leaves the canonical quantum wave function ψ, and, hence, its associated probability distribution |ψ| 2 , unchanged. However, additional (non-canonical) terms in the Heisenberg algebra generate additional terms in the uncertainty relations via the Schrödinger-Robertson relation [18]. In this scenario a rigorously defined version of the EGUP (3), with the heuristic uncertainties ∆x and ∆p replaced by ∆ ψ x and ∆ ψ p, respectively, may be obtained by modifying the canonical position and momentum operators such that [10]: Unfortunately, although such approaches yield the expected phenomenology (i.e., minimum-length GURs), they remain plagued by theoretical and mathematical difficulties, even after nearly 25 years of research [10,19]. Most notably, it may be shown that modifications of the Heisenberg algebra automatically imply violation of the equivalence principle [6,7]. Thus, the "correct" quantum gravity phenomenology is obtained at very high price, that is, by sacrificing the founding principle of classical general relativity. Even more seriously, modified commutators give rise to the so-called soccer ball problem for multiparticle states, and it is unclear whether a sensible macroscopic limit can be consistently defined within such a formalism [8,20,21].
An alternative approach, recently considered in [22], is to modify the canonical quantum wave function and, hence, the underlying probability distribution from which all operator uncertainties are derived. The basic idea, proposed in the so-called smeared-space model, is to associate quantum state vectors with spatial points in the classical background geometry. By the principle of quantum superposition, this allows a quantum state to be associated with the background space as a whole. The resulting extended phase space may be interpreted as a quantum superposition of geometries, and the canonical quantum state |ψ ∈ H is mapped such that |ψ → |Ψ ∈ H⊗H. The generalised state vector |Ψ describes the evolution of quantum matter on a quantum background, incorporating geometric superpositions. Here, points in the phase space of a classical system ( r, p) become "smeared" over finite minimum volumes in the position and momentum space representations of the corresponding quantum theory. Unlike in canonical QM, absolute limits are set to each [22].
Important consequences of the model include the existence of minimum position and momentum uncertainties, leading to rigorously defined analogues of the heuristic GUP, EUP and EGUP relations (1)- (3), and the emergence of a minimum energy density in nature. The latter is an unavoidable consequence of the theory and is of the order of the observed dark energy density if the minimum smearing scales scales are chosen to be of the order of the Planck length ∼ G/c 3 and the de Sitter momentum ∼ √ Λ. The canonical position and momentum operators must also be modified such that x i →X i andp j →P j , whereX i andP j act on the tensor product Hilbert space. However, crucially, the resulting commutation relations are simply a rescaled representation of the Heisenberg algebra, with → + β, where β ∼ 3 GΛ/c 3 . The model is therefore consistent with the equivalence principle and provides a neat solution of the soccer ball problem that plagues approaches based on modified commutators [21,22].
Despite these initial promising results, more general implications of the smeared-space model have not, so far, been extensively investigated. In this paper, we extend the analysis presented in [22] to include angular momentum and spin, which represents an important contribution to the development of the theory. Although our analysis remains non-relativistic, we note the close connection between the angular momentum generators of canonical QM and the Lorentz generators [23]. Thus, by generalising the operators for non-relativistic angular momentum and spin to incorporate the effects of smeared-space, we aim to lay the foundations for the construction of a relativistic theory of quantum matter evolving on a quantum background geometry.
As angular momentum is a pseudo-vector, relations involving angular momentum are, inherently, vector relations [24]. In canonical QM, this poses no fundamental difficulties: although tangent vectors in the background geometry define the classical metric [25,26], this is left unchanged by the quantisation of matter living "in" the classical space [27]. However, in the context of a wouldbe theory of quantum mechanical space, such relations must be handled with extreme care. For this reason, before presenting our main results, we discuss a number of somewhat subtle technical and conceptual problems that arise in this context. These naturally come to the fore when attempting to construct generalisations of vector relations to include the effects of quantum fluctuations of the background. In addition, we include a concise but thorough review of the smeared-space model, proposed in [22], which is intended as a self-contained introduction for readers not familiar with this work.
The structure of this paper is as follows. In Sec. II, we give a careful treatment of angular momentum in classical Euclidean space, for both classical and canonical quantum systems. Although the material presented is well known, we highlight subtleties that are relevant to the problem of quantum gravity in general, and to our later analysis in particular. In Sec. III, we review the smearedspace model and discuss the physical interpretation of the smearing function that gives rise to minimum length and momentum scales. We argue that, under a stricter interpretation than the one first presented in [22], the presence of smearing is compatible with the existence of flat Euclidean space in the weak-gravity limit. This allows us to circumvent many of the potential problems that would otherwise occur when attempting to generalise vector relations to include the effects of smearing. With this in mind, we present a simple proposal for the map to smeared-space in Sec. IV A. A useful alternative formalism is presented in Sec. IV B. In Sec. IV C, the new formalism is used to derive the generalised algebra and uncertainty relations for the smeared-space angular momentum operators. This concludes our treatment of orbital angular momentum. Sec. V deals with the generalisation of the spin operators. Here, we proceed by analogy with the historical development of the canonical QM spin-theory, by seeking coordinate-free representations of the angular momentum algebra. However, in this case, we seek coordinate-independent representations of the generalised algebra, presented in Sec. IV, instead of the canonical algebra, su (2). We conclude this section with a brief discussion of the implications of these results for the description of relativisitic spin and the physics of gravitons. Sec. VI contains a summary of our conclusions and a discussion of prospects for future work.

A. Classical mechanics
In classical mechanics, the angular momentum pseudovector of a point-particle in three-dimensional Euclidean space is where r is the position vector (relative to some origin) and p is the instantaneous linear momentum. The cross denotes the vector product which, for an arbitrary pair of vectors, is defined as a × b = | a|| b| sin θ n, where n is the unit vector perpendicular to the plane defined by a and b and θ is the angle between them. Strictly, both the vector and scalar products are defined between pairs of vectors at the same spatial point. Thus, we must parallel transport the linear momentum vector to the origin of the displacement vector, since, by convention, the angular momentum vector is defined at the centre of rotation [24]. More formally, if the position of the particle "x" is specified by the coordinates x i 3 i=1 , the true linear momentum is given by span the tangent (cotangent) space at x. [53] The vector " p" appearing in Eq.
Similar considerations hold when we take the scalar product, which is often written simply as a. b = | a|| b| cos θ (that is, without specifying the point x at which it is defined) for the sake of notational simplicity. For example, the p appearing in the usual dot product p. r, where r is the displacement vector, is in fact the parallel-transported vector p(0) = p i (0)e i (0), considered above.
In Euclidean space, parallel transport is pathindependent, and also preserves the inner product, i.e., Γ(γ) a, Γ(γ) b (x 1 ) = a, b (x 2 ) for arbitrary start and end points, x 1 and x 2 , on any path γ [26]. It follows that the vector product between any pair of vectors is also preserved. Thus, location-independent meanings can be ascribed to the quantities " a. b" and " a× b", which justifies the usual neglect of such subtleties for systems defined in a Euclidean background.
However, in curved geometries such nice properties do not, in general, hold. In fact, notions such as displacement can only be defined locally, e.g. via d r = dx i e i (x). Thus, expressions involving a displacement vector " r", such as " p. r" and " r × p", do not make sense. The integral of d r from x 1 to x 2 is path-dependent, and the result is not a genuine vector, so that curved geometries are not vector spaces [25,26]. Similarly, the canonical momentum p may be seen as a displacement vector in Euclidean momentum space, whereas only the local quantity d p = dp i (x)e i (x) can be consistently defined for particles in curved backgrounds. [54] In this case, angular momentum also becomes a local property, and is conserved only locally as a result of local (not global) rotational symmetry.
Recall that, for an arbitrary Riemannian geometry, the inner product between tangent vectors defines the metric, For Euclidean geometry, the metric takes a particularly simple form in Cartesian coordinates, i.e., η ij (x) = diag(1, 1, 1) for all x, so that there is no distinction be-tween covariant and contravariant components, or between the tangent and cotangent vectors. Such coordinate systems are extremely special. In particular, they are the only globally orthogonal coordinates that exist, even in Euclidean space, and, thus, the only coordinates that give rise to a set of globally orthonormal tangent vectors. [55] In three dimensions, these are defined by the algebra: i, j, k ∈ {1, 2, 3}, which holds for the vectors defined at all points x. Here, ǫ ij k is the Levi-Civita symbol. This is defined as ǫ ij k = 1 for cyclic permutations of ijk, ǫ ij k = −1 for non-cyclic permutations, and ǫ ij k = 0 otherwise, but it is not a tensor. Hence, even in three-dimensional Euclidean space, the only set of basis vectors satisfying Eq. (7) are tangent to the "curves" (i.e. straight lines) defined by the conditions are the Cartesian coordinates {x, y, z}. In this case, the relevant Poisson brackets (PB) are: and where the components of the angular momentum vector l := l i (0)e i (0) are given by The structures of the canonical Poisson brackets for the components of the position and linear momentum vectors (8)-(9), as well as for the components of the angular momentum (10)- (12), are therefore intimately related to both the geometric structure of Euclidean space, and, crucially, to the specific choice of coordinates used to describe physical systems (7). More generally, the vectors r and p satisfy: and the components of angular momentum along arbitrary vector directions, a and b, are related via where l n = n, l , etc. It is important to note that, although analogues of Eqs. (8)-(9) exist for any set of canonically conjugate phase space coordinates, q i , π j , analogues of Eqs. (10)-(13) do not. Specifically, any set of generalised position coordinates, together with their canonically conjugate momenta, where L(q,q) is the Lagrangian of the classical system, satisfy and The bracket structure (18)- (19) is then preserved by any canonical coordinate transformation [24]. When q i = x i are Cartesian coordinates, π j = p j are the components of the physical linear momentum, and the corresponding components of the angular momentum are given by Eq. (13). However, under a general canonical transformation, x i → q i , p j → π j , where the new phase space coordinates are not Cartesian, the transformed components of the physical angular momentum are not given by a formula analogous to (13). Although we may define the analogous quantities ξ i := ǫ ij k q j π k , these do not, in general, correspond to components of the angular momentum vector, unless q i , π j represent Cartesian phase space coordinates. Similarly, it is straightforward to show that {ξ i } 3 i=1 do not satisfy the algebras (10)- (12), unless q i = x i , π j = p j (ξ i = l i ).
In non-canonical coordinate systems, the canonical Poisson bracket structures (18)- (19) are also destroyed [24]. Nonethless, in general curved spaces, non-Cartesian canonical coordinates can always be defined (at least locally [26]) so that Eqs. (18)-(19) may still be satisfied for an appropriate choice of q i , π j . Despite this, the physical space displacement vector " r" and momentum space displacement vector " p" are not well defined for systems in curved geometries, so that Eq. (13) does not hold, regardless of our choice of coordinates.

B. Canonical QM
In quantum mechanics, canonical quantisation is always performed in Cartesian coordinates, x i , p j [28]. Specifically, one obtains the operators corresponding to the classical values x i and p j by performing the map: wherê and Here, we use the shorthand d 3 r = det g ij (x)dx 3 and d 3 p = detg ij (p)dp 3 , where g ij (x) andg ij (p) denote the metrics on the position and momentum space submanifolds of the classical phase space, respectively.
Eq. (22) is equivalent to the canonical de Broglie relation between momentum and wave number, p = k, which, together with Eq. (21), yields: and The canonical commutators (23)- (24) are the quantum counterparts of the classical Poisson brackets (8)- (9), and are consistent with the general correspondence [29] where O(x, p) is a function of the Cartesian coordinates of the classical phase space andÔ(x,p) has the same functional form with respect to the corresponding operators (assuming resolution of any potential ordering problems). However, we note that, in order to interpret the results of physical measurements, coordinate values alone are not enough: one must also know where in physical space the coordinate "axes" are located. For example, in order to reach the point in space labelled by the coordinates x i 3 i=1 , one must begin at the origin and travel x i units along the i th coordinate direction, keeping all other coordinate values fixed, for each i sequentially. This procedure is general, and holds regardless of whether each line x i = const. defines a linear or a curvilinear "axis". (Operationally, in order to detect a particle at "x", one must receive a signal emitted from the physical point defined by both the coordinates x i 3 i=1 and the associated coordinate directions.) Furthermore, since the tangent vectors {e i (x)} 3 i=1 are tangent to the curves x i = const. (in any coordinate system), it is clear that knowledge of the metric (6) is required in order to interpret coordinate values as positions in physical space. This is the case in classical mechanics and remains the case in canonical QM, in which the background space is assumed to be fixed and classical. Specifically, in threedimensional Euclidean space, the displacement vector and (parallel-transported) momentum vector of a classical particle may be written as: respectively, where e i (0) denote Cartesian tangent vectors defined at the origin. Thus, quantising the system by "promoting" coordinates, but not tangent vectors, to operators is equivalent to quantising matter (particles) while leaving the background geometry, which is defined by the classical metric (6), unchanged. Practically, this implies the de facto definition of a map: (note the hat above the vector arrow), wherê It follows immediately that which is the quantum counterpart of Eq. (14). Though such a definition does not form part of the abstract Hilbert space formalism of canonical QM, in which the spectral representation of the Hermitian operators (21) is agnostic to their physical interpretation [18], it is undoubtedly necessary in order for real-world experimentalists to connect the predictions of this formalism with the outcomes of real-world measurements. This implies a subtle link between the mathematical structure of quantum systems (abstract Hilbert spaces) and the mathematical structure of physical spaces (symplectic manifolds and their associated geometries) which is of vital importance for the problem of quantum gravity.
The existence of this link is especially highlighted when one considers the quantisation of angular momentum. Since the classical formula (5) is a relation between vector quantities, one would expect the tangent vector part of this expression to be affected by the quantisation of geometry, just as the component part is affected by the quantisation of matter. In any would-be theory of quantum matter living "in" a quantum geometry both aspects must be accounted for.
In relativistic quantum gravity, quantum fluctuations of the background geometry are expected to induce curvature fluctuations and, hence, fluctuations in the gravitational field strength over very small length scales comparable to the Planck length, l Pl ≃ 10 −33 cm. [56] This, in turn, is expected to give rise to a minimum resolvable length scale of the order of l Pl [31]. Assuming that fluctuations in the background space-time include fluctuations in the space-space part of the metric, g ij , a non-zero spatial (Riemannian) curvature, R ijk = 0, is also generated, in addition to the nonzero space-time (pseudo-Riemannian) curvature, R µνρσ = 0. In this scenario, such fluctuations give rise to two effects that are relevant to our previous discussion.
First, they destroy the Cartesian coordinate system on which the canonical quantisation of coordinates is based. (We recall, again, that global Cartesian coordinates do not exist in curved space.) Second, they destroy the physical rationale for the quantisation of coordinates alone, while leaving the tangent vectors with which they are associated, i.e., the coordinate axes and the geometry they define via the classical metric (6), unchanged. Equivalently, we may say that they destroy the coordinate system on which the map that defines the canonical quantisation of matter, (27)- (28), is based. In addition, their very existence implies the need to define a new map, between the classical tangent vectors {e i } 3 i=1 and a new set of operators {ê i } 3 i=1 , which represents the quantisation of the background geometry in which the quantum matter lives.
However, even in the study of quantum gravity phenomenology, such subtleties can easily be neglected, if we restrict our attention to the quantum counterparts of classical relations involving only coordinates. This is true for all studies of modified position-momentum commutators, in which one assumes the usual correspondence or both the left-and right-hand sides simultaneously [19,20]. It is also true of recent studies of the smeared-space model in which relations between vector quantities were similarly neglected [21,22].
Nonetheless, it is clear that such subtleties cannot be neglected when one explicitly considers the counterparts of vector relations, such as Eq. (5), on a quantum background. In this case, the coordinate-dependent ex- (13), and its associated algebra (11)- (12), emerge only after taking the inner product of (5) with the relevant tangent vector: l i = e i (0), l = l j e i (0), e j (0) = l j δ j i . Clearly, this relies on the definition of the classical metric (6).
In canonical QM, this poses no problems, since we are not required to quantise the background space. Thus, definingˆ r andˆ p via Eqs. (27)-(28), we may define the vector angular momentum operator as: It follows immediately that and where the Cartesian components ofˆ l :=l i e i (0) are given byl These are obtained asl i = e i (0),ˆ l =l j e i (0), e j (0) = l j δ j i , by complete analogy with the classical case, so that By the Schrödinger-Robertson relation, Eqs. (32) and (35) then give rise to the uncertainty relations respectively. Clearly, Eq. (37) is the more general relation, and the uncertainty relation for Cartesian components is recovered by taking θ = m(π/2) (m ∈ Z). Similarly, Eqs. (31) give rise to: We stress that in canonical QM one quantises r, p and l by quantising the relevant vector components, x i , p i and l i , respectively, but leaving the associated classical basis vectors unchanged. This is a subtle mathematical expression of the fact that canonical quantum systems are described by superpositions of eigenstates (e.g., position, linear momentum, or angular momentum eigenstates) that live on, or "in", a fixed classical background. More concretely, we may say that canonical quantum wave functions ψ( r) are defined as complex-valued fields on the metric space defined by the tangent vectors In a true quantum gravity scenario, in which the background itself is subject to quantum fluctuations associated with the minimum length scale [31], this picture must be radically revised. We now consider one scenario which may, however, be regarded as a more conservative solution to this problem. Below, we outline a model in which quantum fluctuations of the background give rise to a minimum length, but not to curvature fluctuations in the non-relativistic limit. (Though the model does not explicitly include gravity, we argue that it is consistent with existence of the Newtonian gravitational potential, viewed, in the standard way, as a scalar field in flat Euclidean space [24,32].) Thus, our quantisation procedure may be implemented via a map between classical coordinates and Hermitian operators, as in canonical QM. Furthermore, it may be implemented as a map between Cartesian coordinates in the classical phase space and generalisations of the canonical operatorsx i andp j . The new operators, denotedX i andP j , act on a composite quantum state incorporating both matter and geometry. In our model, points in the quantum background are subject to stochastic movements but these do not change the underlying flat geometry of the space.

A. Basics
In [22], a new model of quantum geometry was proposed in which each point r in the classical background is associated with a vector |g r in a Hilbert space, where and g( r ′ − r) is any normalised function. For simplicity, however, we may imagine g( r) as a Gaussian centred at r = 0. Since each point in the background geometry may be associated (heuristically) with a Dirac delta δ 3 ( r ′ − r) or, equivalently, a ket | r , the background space may be "smeared" by mapping each point to a superposition of all points via: We may visualise the smearing map (40) as follows: for each point r ∈ R 3 in the classical geometry we obtain one whole "copy" of R 3 , thus doubling the size of the classical phase space. The resulting smeared geometry is represented by a six-dimensional volume, namely . This is interpreted as the quantum amplitude for the transition r → r ′ , and the higher-dimensional space is interpreted as a superposition of three-dimensional geometries [22].
Hence, in this model, "points" in the background exist in a superposition of states, and may undergo stochastic fluctuations as the result of measurements. This also affects the statistics of the canonical quantum matter living on (in) the space. Specifically, using (40), the canonical quantum state |ψ = ψ( r) | r d 3 r is mapped according to |ψ → |Ψ , where: The corresponding expansion in smeared momentum space is given by: where and In other words, the momentum space representation of the canonical quantum wave function ψ ( p) is given by the Fourier transform of ψ( x), which is transformed at the scale . (Here, we use the subscript to emphasise this point.) By contrast, the momentum space representation of the geometric part of the composite quantum state |Ψ ,g β ( p ′ − p), is given by the Fourier transform of g( r ′ − r), where the transformation is performed at a new scale β. This represents the quantisation scale for space (rather than matter) and must be fixed by physical considerations. In [22] it was shown that, in order to reproduce the observed vacuum energy density, [17], β must take the order of magnitude value: where ρ Pl ≃ 10 93 g . cm −3 is the Planck density. Consistency between Eqs. (41) and (42) requires: Hence, | p p ′ represents an entangled state in the rigged basis of the "enlarged" Hilbert space H ⊗ H, where H is the Hilbert space of canonical (three-dimensional) QM. [57] We emphasise this by not writing a comma between p and p ′ , by contrast with | r, r ′ := | r | r ′ . By complete analogy with the position space representation,g β ( p ′ − p) is interpreted as the quantum probability amplitude for the transition p → p ′ in smeared momentum space.
Since an observed value " r ′ " cannot determine which point(s) underwent the transition r → r ′ in the smeared superposition of geometries, we must sum over all possibilities by integrating the joint probability distribution |Ψ( r, r ′ )| 2 := |g( r ′ − r)| 2 |ψ( r)| 2 over d 3 r, yielding: Here, physical predictions are assumed to be those of the smeared-space theory and the canonical QM of the original (unprimed) degrees of freedom is only a convenient tool in our calculations. In this formulation of the model, only primed degrees of freedom represent measurable quantities, whereas unprimed degrees of freedom are physically inaccessible [22]. The variance of a convolution is equal to the sum of the variances of the individual functions, so that the probability distribution (46) gives rise to a GUR which is not of the canonical Heisenberg type. It is straightforward to verify that the same statistics can be obtained from the generalised position-measurement operatorX i , defined as:X where d 3P r ′ :=1 1 ⊗ | r ′ r ′ | d 3 r ′ . We then have: Analogous reasoning in the momentum space representation gives: It follows that: Note that the HUP, expressed here in terms of primed variables: (recall that the unprimed degrees of freedom are physically inaccessible) and the analogous relation: both hold, independently of Eqs. (48) and (51). We denote the position and momentum uncertainties by ∆ g x ′i = σ i g and ∆ g p ′ j =σ gj , respectively, when |g| 2 is chosen to be a Gaussian function. This saturates the inequality (53), yielding the definition of the Fourier transform scale β: The HUP contains the essence of wave-particle duality, which could also be called wave-point-particle duality, and is a fundamental consequence of the de Broglie relation p = k. This, in turn, is equivalent to the relation (43), which holds for particles propagating on a fixed (classical) Euclidean background. By contrast, Eq. (53) represents the uncertainty relation for spatial "points" (not point-particles "in" space). This follows directly from Eq. (45), which is equivalent to the modified de Broglie relation: The new relation holds for particles propagating in the smeared-space background and the non-canonical term may be interpreted, heuristically, as an additional momentum "kick" induced by quantum fluctuations of the geometry.
Combining Eqs. (48), (51) and (52)-(53), gives plus an analogous relation containing only (∆ ψ p ′ j ) 2 . Optimising the right-hand side of (56) with respect to ∆ ψ x ′i , and its counterpart with respect to ∆ ψ p ′ j , yields The same result is readily obtained by noting that the commutator of the generalised position and momentum observables is: whereÎ =1 1 ⊗1 1 is the identity matrix on the tensor product space and1 1 is the identity matrix on the Hilbert space of canonical three-dimensional QM. Eq. (58) then follows directly from the Schrödinger-Robertson relation [18]. Thus, the inequalities in all three uncertainty relations, (52)- (53) and (56), are saturated when |g| 2 is chosen to be a Gaussian, for which we denote ∆ g x ′i = σ i g and ∆ g p ′ j =σ gj , and when |ψ| 2 is chosen to be a Gaussian with (57). Importantly, the smeared-space model gives rise to minimum length and momentum uncertainties in the presence of commuting coordinates, i.e., Next, we note that setting where m dS c := /l dS := Λ/3 ≃ 10 −66 g is the de Sitter mass, yields the required value of β (44). We then have: where l Λ := 2 1/4 √ l Pl l dS ≃ 0.1 mm and m Λ := 2 −1/4 √ m Pl m dS ≃ 10 −3 eV. This gives rise to a minimum energy density of order: as required by current cosmological data [17]. In addition, using these values, Eqs. (48), (51) and (56) may be Taylor expanded to first order to yield the GUP, EUP and EGUP, respectively, expressed in terms of ∆ ψ x ′i and ∆ ψ p ′ j . As discussed in the Introduction, the GUP implements a minimum length scale of the order of the Planck length ∼ l Pl , but no minimum momentum scale, whereas the EUP implements a minimum momentum of the order of the de Sitter momentum ∼ m dS c, but no minimum length. The EGUP therefore accounts for the effects of both minimum length and minimum momentum scales in nature. However, in the smearedspace model, neither ∆ ψ x ′i nor ∆ ψ p ′ j are directly measurable, and only ∆ Ψ X i nor ∆ Ψ P j are physical. It is therefore useful to express the smeared-space GUR directly in terms of these quantities. Thus, directly combining Eqs. (48), (51) and (52), we obtain: Setting i = j, we may ignore dimensional indices. Substituting for σ g andσ g from Eq. (61), taking the square root and expanding to first order, and ignoring the subdominant term of order ∼ l Pl m dS c then yields: Finally, we note that the smeared-space model has important implications for the description of measurement in quantum mechanics. We now illustrate these by considering a generalised position measurement in detail.
Applying the generalised position operatorˆ R :=X i e i (0) to an arbitrary pre-measurement state |Ψ returns a random measured value, r ′ , and projects the state in the fixed background subspace of the tensor product onto: with probability (|g| 2 * |ψ| 2 )( r ′ ) [22]. The total state is then |ψ r ′ ⊗ | r ′ , which is non-normalisable, and therefore unphysical. This is analogous to the action of the canonical position measurement operator on |ψ , which projects onto the unphysical state | r with probability |ψ| 2 ( r). However, in the smeared-space formalism, we must reapply the map (40) to complete the description of the measurement process. Thus, although generalised position measurements, represented by the application of the map (40) to the state (67), yield precise measurement values, the post-measurement states are always physical, with well defined norms. Their position uncertainties, which may be determined by performing multiple measurements on ensembles of identically prepared systems, never fall below the fundamental smearing scale ∼ l Pl . Analogous considerations hold for generalised momentum measurements.

B. Physical interpretation of the smearing function, revisited
In [22], the smearing function g( r ′ − r) was interpreted as the probability amplitude for the transition r ′ → r. Importantly, this allows (at least in principle) for the smeared-space model to describe curvature fluctuations in the background geometry. The mechanism for this is as follows. As only primed variables are physically accessible, an arbitrary set of measured values ( r ′ ) determines a three-dimensional submanifold in the extended six-dimensional phase space, ( r, r ′ ). This may be described by an arbitrary vector function, r ′ ( r). Hence, if the metric on the ( r, r ′ ) hyperplane is known, a natural choice for the metric on the r ′ ( r) submanifold is the induced metric, which is obtained by performing the push-forward from the metric in the six-dimensional bulk space [25,26].
In the original analysis of the smeared-space model, it was argued that consistency requires the coordinates ( r, r ′ ) to label points in a flat pseudo-Riemannian manifold with 3 space-like dimensions and 3 time-like dimensions, i.e., a (3 + 3)-dimensional generalised Minkowski space with metric signature (+ + + − −−). Despite this, however, induced metrics on observable threedimensional submanifolds have positive signature, (+ + +), so that the model describes non-relativistic matter on a fluctuating spatial background [22]. In principle, these fluctuations can give rise to arbitrary Riemannian geometries, but, practically, the probability amplitudes for transitions with | r ′ − r| ≫ l Pl are vanishingly small. Indeed, if |g( r ′ − r)| is peaked at r ′ = r, as is the case for Gaussian smearing, the most probable geometry is isomorphic to the original, flat, Euclidean space. Transitions within one standard deviation of |g( r ′ − r)| remain relatively likely, but these correspond to small fluctuations of order | r ′ − r| l Pl , as expected phenomenologically [31].
In Sec. II B, we discussed the subtle ways in which canonical quantisation techniques encode assumptions about the nature of the background geometry in which material systems "live". In particular, we explained why the standard procedure of promoting classical coordinates to Hermitian operators is not applicable in the presence of spatial curvature. This is especially obvious for physical quantities that depend on the canonical displacement vectors, r and p, which are only well defined in Euclidean geometries, and, hence, is especially problematic for the quantisation of angular momentum in curved space.
To overcome this problem, we take a stricter inter-pretation of the smearing function in the present work. Instead of allowing arbitrary transitions r → r ′ in the extended ( r, r ′ ) phase space, we limit the available transitions to the pair-wise exchange of points. Thus, we interpret g( r ′ − r) as the probability amplitude for the transition r ↔ r ′ . This is a far more restrictive condition, but it is straightforward to verify that the results presented in Sec. III A are unaffected by our interpretation of g. Nonetheless, the new interpretation has several advantages. First, since Euclidean spaces of any dimension are maximally symmetric, all points are considered equivalent. It follows immediately that the pair-wise exchange of points, or even of whole neighbourhoods surrounding r and r ′ [34], cannot change the geometry of the underlying space. Thus, "quantising" Euclidean geometry in this way introduces an additional stochastic variation into the quantum measurement procedure, but the resulting fluctuations cannot generate spatial curvature. With this interpretation we are still able to generate GURs and to derive the dark energy density as the minimum energy density in nature [22]. However, we may also generalise canonical QM to include the effects of the smeared background by mapping only classical coordinates to Hermitian operators, as in canonical quantisation procedures.
Second, the new interpretation is compatible with the canonical treatment of the weak-gravity limit. In the non-relativistic approximation, we must distinguish between contributions to the local gravitational force induced by modifications of the time-time component of the metric, g 00 , and the space-space part, g ij . Indeed, in the weak-field limit of classical gravity, the vacuum Einstein field equations reduce to Laplace's equation, ∇ 2 √ g 00 ≃ 0 [17,32]. For a point source of mass m, g 00 = c 2 (1 − 2Gm/(c 2 r)), so that we recover the familiar Newtonian potential Φ = −Gm/r from the warping of time alone. [58] Although this point is not often highlighted in introductory texts on general relativity, we note that the spatial curvature of the background is formally set equal to zero in the Newtonian approximation, i.e., R ijk := 0. This allows us to to treat the Newtonian potential as a field on a flat Euclidean geometry, and to replace the radial coordinate r with the Cartesian distance r = η ij x i x j = x 2 + y 2 + z 2 . (Strictly, this substitution is not possible in the Schwarzschild geometry, in which the spatial components of the metric depend on r and global Cartesian coordinates cannot be consistently defined [32].) Hence, taking the stricter interpretation of g allows us to "smear" the Newtonian potential of canonical nonrelativistic gravity, which is given its standard interpretation as a scalar field defined on flat Euclidean space [24,32]. In this case, we may use the techniques outlined in [22] for the smearing of an arbitrary potential in the generalised Schrödinger equation. Nonetheless, we may consider the restricted smeared-space model, i.e., the model in which g( r ′ − r) is interpreted as the probability amplitude for the transition r ′ ↔ r, rather than the more general transition r ′ → r, as an approximation to a more general model in which genuine spatial curvature fluctuations are generated. This corresponds to the description of stronger gravitational fields, in which contributions to the field strength from spatial curvature cannot be neglected. However, the construction of such a model lies outside the scope of the present work and is left to future studies. Clearly, analogues of the above arguments apply equally well to Euclidean momentum space, and consistency requires us to reinterpretg β ( p ′ − p) as the probability amplitude for the transition p ′ ↔ p.

A. A simple proposal
Considering the results presented in Secs. (II)-(III), it is clear that the simplest way to construct a model of angular momentum for particles propagating on the smeared-space background is by defining the map: whereˆ andX i ,P j are given by Eqs. (47) and (50), respectively. It follows immediately that and we may define the smeared-space angular momentum operator asˆ The Cartesian components ofˆ L :=L i e i (0) are given bŷ and may be obtained directly viaL i = e i (0),ˆ L = L j e i (0), e j (0) =L j δ j i . From Eqs. (59), (60) and (72), it also follows that: and Hence, By the Schrödinger-Robertson relation, Eqs. (74) and (76) give rise to the uncertainty relations: respectively. These are completely analogous to Eqs. (32)-(33) but with the rescaling → + β. Again, Eq.
(78) is the more general relation, and the uncertainty relation for Cartesian components is recovered by taking θ = m(π/2) (m ∈ Z). Similarly, Eqs. (73) give rise to: which are analogous to Eqs. (38). However, in order to gain deeper insight into the behaviour of angular momentum in the smeared-space model, it is worth investigating the origin of the relations (73)-(79) in more detail. We do this in the following section where we show explicitly that, despite their canonical form (except for the rescaling → + β), Eqs. (77), (78) and (79) are compatible with GURs for angular momentum. In this sense, they are analogous to Eq. (58) which, despite its canonical form (except for → + β), is compatible with the GURs (56) and (64)-(65).

B. Useful alternative formalism
To investigate the structure of the generalised commutator [L i ,L j ] it is useful to first rewrite the generalised position and momentum operators,X i andP j , as well as the smeared-state |Ψ , in a unitarily equivalent form. We begin by constructing the operator whose action on the smeared-space basis is defined as: Here, we again assume that, while sets the quantisation scale for the degrees of freedom in the first subspace of the tensor product, β sets the quantisation scale for the degrees of freedom in the second subspace. Hence, β −1 (1 1⊗ˆ p ′ ) generates translations on the second vector of the basis | r, r ′ , just as −1 (ˆ p ⊗1 1) generates translations on the first. This accounts for Eq. (81). Eq. (82) then follows by combining Eqs. (80)-(81) with (45), again assuming that β sets the Fourier transform scale for kets in the second subspace (as required for consistency).
Together, these considerations imply: where (as in canonical QM) and In Eqs. (83)-(85), we use the subscripts 1 and 2 to indicate which subspace of the tensor product state the brakets belong to. This is to avoid confusion since, in these expressions, the degrees of freedom in each subspace are no longer labelled exclusively by primed or unprimed variables, as they were previously. Nonetheless, they remain consistent with our convention that sets the quantisation scale for degrees of freedom in the first subspace of the tensor product, while β sets the quantisation scale for degrees of freedom in the second. We repeat that the former are associated with canonical quantum matter whereas the latter are associated with the quantum state of the background geometry.
Using these results, we map the smeared-space oper-atorsX i andP j , and the smeared-state |Ψ , according to: and Note that Eq. (88) implicitly defines the state |g , which is distinct from the state |g r defined in Eq. (39). From here on, we useX i ,P j and |Ψ to refer to the unitarily equivalent forms of the generalised position and momentum operators (86)-(87), and smeared-state (88), respectively, unless stated otherwise.
Next, we split each of the generalised operators (86) and (87) into the sum of two terms: wherê and In other words, we define the new classical variables Π := p ′ , π := p , π ′ := ( p ′ − p) , and construct their quantum operator counterparts. Note that, in this formulation of the smeared-space model, measurable quantities are no longer expressed in terms of primed variables only. That is, neither q i nor q ′i are measurable, individually, and only their sum q i +q ′i = x ′i is physical. Similarly, neither π j nor π ′ j is directly measurable, only π j + π ′ j = p ′ j . This has important physical consequences.
In the first formulation of the model [22], summarised in Sec. (III A), the wave functions corresponding to matter and geometry are entangled, as hypothesised in [36]. However, in the unitarily equivalent formulation, presented above, they are not. Nonetheless, physical measurements correspond to operators that act on both subsytems of the tensor product state |Ψ , regardless of our choice of basis. Furthermore, since the basis transformation (81) is a unitary operation, the effects of geometrymatter entanglement (in the first formulation) cannot be "undone" by this change. In other words, although the entanglement of states is basis-dependent, and therefore not fundamental, predictions for the results of physical measurements arise from the combination of both states and operators. These predictions are basis-independent, as required.
From Eqs. (91)-(92) it is straightforward to show that: and Together, Eqs. (95)-(96) recover Eq. (59). The remaining commutation relations are: and We then have: and Here, cov(X, Y ) = XY − X Y is the usual definition of the covariance of the random variables X and Y , and [ . , . ] + denotes the anti-commutator. The operator pairsQ i ,Q ′i andΠ j ,Π ′ j are uncorrelated since they act on separate subspaces of the total state |Ψ . Comparison of Eqs. (100) and (101) with Eqs. (48) and (51), respectively, suggests and it is straightforward to verify these equalities explicitly. Furthermore, in the position and momentum space representations of smeared-space wave mechanics, the new operators take the especially simple forms: respectively. Before concluding this section, we wish to point out that the unitary operator (80) is not a function of the physical position and momentum operators of the smeared-space model,Û β =Û β (ˆ R,ˆ P ). In terms of our new variables, (89)-(90), it may be written asÛ β = exp (i/β)ˆ Π ′ .ˆ Q , but neitherˆ Q norˆ Π ′ , alone, represents the physical (i.e., measurable) position or momentum of the particle. Hence, although we may constructÛ β mathematically, and utilise it to simplify our calculations, it is doubtful that it represents a viable physical transformation of the system that could actually be carried out in the "real" smeared-space universe. [59] Nevertheless, these conclusions could be revised by considering the following argument. If π ′ = p ′ − p is interpreted as an additional momentum "kick", imparted to the particle due to a quantum fluctuation in the background geometry, then − π ′ = p − p ′ is the corresponding recoil. In this case, p ′ + ( p − p ′ ) = p = π is the total momentum of the particle-plus-background system. This is obviously consistent with the canonical QM limit, in which the momentum of the background is simply zero. Similar considerations apply to − q ′ = x − x ′ and x ′ +( x− x ′ ) = x = q in the position space representation.
In this scenario the question of whether, for example, Π ′ represents a physically measurable quantity depends on whether there exists a way to measure the momentum carried by the background, or by the composite particleplus-background system, independently of the particle momentum. Philosophically, these considerations are analogous to those discussed in the early days of general relativity, when it was realised that there exists a similar problem for classical systems in a dynamical background (see [38] for a discussion of this point). In short, it was argued that it is not possible to determine space-time intervals between events without a material system, defined on the space-time, which acts as a reference frame. Since any material system necessarily induces a backreaction, there is, therefore, no physical mechanism capable of probing "pure" geometry [38]. Performing measurements on physical systems in any would-be model of quantum geometry, we face similar problems, since there is no way to subtract out the influence of background fluctuations. Hence, although we may treat our earlier assertion thatˆ Q,ˆ Q ′ ,ˆ Π andˆ Π ′ are not individually measurable with some caution, it is likely to remain valid for most experimental regimes.

Regardless of such subtleties, what is clear is that onlŷ
R =ˆ Q +ˆ Q ′ andˆ P =ˆ Π +ˆ Π ′ represent the the physical position and momentum of a particle on the smearedspace background. Thus, our results remain valid for measurements of material physical systems in which the back-reaction of the geometry is not directly probed.

C. Generalised algebra and GURs
In terms of our new operators (89)-(90) the components of the generalised angular momentum may be written as:L After straightforward (but tedious) algebraic manipulation it may be verified that the generalised commutator (74) is recovered from the following relations between the individual subcomponents ofL i ,L j : Eqs. (110a) and (110d) confirm thatL i andL ′ i represent genuine angular momentum operators since the sub- satisfy the required algebras, i.e., appropriately rescaled representations of so (3). According to our previous interpretation of the tensor product state (88),L i represents the angular momentum of the canonical quantum state vector |ψ (quantised at the scale ), whereasL ′ i represents the angular momentum associated with the quantum state of the background |g (quantised at the scale β). By contrast, Eqs. (110h) and (110j) show thatΛ i andΛ ′ i do not represent components of angular momentum in their own right. These "cross terms" determine the effect, on the angular momentum of a canonical quantum particle, of its interaction with the smeared background.
We also note that, since neitherΛ i norΛ ′ i commute with eitherL i orL ′ i , it is impossible for a smeared state |Ψ to be an eigenvector of all four subcomponents of L i simultaneously. Nonetheless, Eq. (75) demonstrates that the simultaneous eigenvectors ofL 2 andL i form a valid basis of the infinite-dimensional Hilbert space H ⊗ H ′ (H ′ ∼ = H). In other words, if both |ψ and |g represent angular momentum eigenstates (that is, , then the total state |Ψ = |ψ ⊗ |g is not an eigenstate ofL i . In this way, single-particle smearedstates differ starkly from unentangled bipartite states in canonical QM: |ψ tot = |ψ 1 ⊗ |ψ 2 .
It is also convenient to rewrite the relations between subcomponents as: It may be verified that (111a)-(111h) represent the most restrictive closed system of equations, for the subcomponents (109), that recover the appropriate rescaled Lie (74). By closed, we mean that only operators appearing on the left-hand sides of the equations appear on the right-hand sides. Thus, Eqs. (111a)-(111h) form an algebra, which is satisfied by the particular solution (109) (or, equivalently, by Eqs. (110)) but which may, in principle, admit other solutions as well. In this sense, the algebra (111) carries less information than either (109) or (110) since, although Eqs. (109) and (110a)-(110j) imply Eqs. (111a)-(111h), the converse is not true.
Alternatively, we may write the generalised operator L i as:L (Here,Λ i andΛ ′ i are defined as in Eq. (109).) The new subcomponents L i ,L ′ i ,L i then satisfy the algebra: This is less restrictive than (111) since, together, Eqs. The operators (*) are not equivalent to those defined in Eq. (113) and do not fully satisfy the generalised angular momentum algebra, but, despite this, they offer an important clue about generalised spin physics. As we will show explicitly in Sec. V, it is straightforward to construct finite-dimensional analogues of the subcompo- . However, it it is less obvious how to construct spin-operator counterparts of the commuting components Λ i exist. These take a form analogous to (*) and, if the background state |g is assumed to be spin-1/2, satisfy all the relevant equations of a generalised spin algebra. This algebra has the same formal structure as Eqs. (114).
Let us now consider the GURs generated by the generalised angular momentum operators (108). These are most elegantly expressed in terms of the the subcom- The first term on the right-hand side represents the contribution to the total uncertainty from the canonical QM wave function ψ, the second represents the pure geometric part (that is, the contribution from g), and the additional contributions are generated by operators that that cannot be decomposed as either1 1⊗(. . . ) or (. . . )⊗1 1. Thus, we see that Eq. (115) takes a form analogous to Eqs. (48) and (51), but with additional cross terms, i.e., terms that are generated by operators that do not act on one subspace of the composite state |Ψ = |ψ ⊗ |g (88) alone.
Multiplying Eq. (115) by a similar expression for (∆ Ψ L j ) 2 we obtain the GUR for orbital angular momentum implied by the smeared-space model. Though it is beyond the scope of this paper to investigate the consequences of this relation in detail, we note that it is of the general form: The leading contribution to the terms in the middle is of the form ( Ψ |, which is equivalent to the canonical uncertainty relation for angular momentum (36). All additional terms are non-canonical and appear only because of the smearing procedure (40).
In terms of the second set of subcomponents, (∆ Ψ L i ) 2 may be written as: Multiplying by the equivalent expression for (∆ Ψ L j ) 2 we obtain an alternative (and simpler) form of the GUR for smeared-space angular momentum.

Historical development of the theory
The phenomenon of quantum spin was first discovered empirically via the Zeeman effect [27]. The splitting of atomic energy levels in the presence of an external magnetic field suggested that electrons possessed a kind of "internal" angular momentum that was able to couple to (interact with) their quantised orbital angular momentum. This was later confirmed, explicitly, by the experiments of Stern and Gerlach [39]. As a possible mathematical description of this phenomenon, Pauli sought operators that satisfied the angular momentum Lie algebra (Eq. (32) in our text), but whose concrete representations contained only constant matrix elements. He reasoned that such operators represent the intrinsic (coordinate independent) rather than extrinsic (coordinate dependent) properties of quantum particles [27].
Later, the Pauli matrices were identified as the generators of the group SU(2) [40]. This is the double cover of the rotation symmetry group, SO(3), and shares the same Lie algebra, but its elements admit representations with both integer and half-integer eigenvalues [40]. These have no classical analogues and describe the internal angular momentum (now called spin) states of two very different types of fundamental particle. Particles with integer spin, called bosons, obey Bose-Einstein statistics and are able to condense freely into compact multi-particle states [41]. In short, identical bosons can "share space", since there are no obstructions to the spatial overlap of individual particle wave functions. By contrast, particles with half-integer spin, called fermions, obey Fermi-Dirac statistics and cannot condense in this way [29].
Formally, the spin-statistics theorem states that it is not possible for two fermions to have the same values of all four quantum numbers: n, the principle quantum number, l, the azimuthal quantum number, m l , the magnetic quantum number, and m s , the spin quantum number. Since the wave functions of any two fermions with the same four quantum numbers would overlap, the theorem forbids identical fermions from sharing the same region of physical space if the z-component of their spins are aligned. This is known as the Pauli Exclusion Principle. Mathematically, it arises from the fact that requiring a many-particle wave function to be single-valued is equivalent to requiring it to be antisymmetric with respect to the exchange of any two particles. It follows that bosons occupy symmetric quantum states while fermions occupy antisymmetric states [27,29].
Finally, the physical origin of quantum mechanical spin was discovered by Dirac, who showed that it arose as a necessary consequence of combining the principles of quantum theory, expressed via the de Broglie relations E = ω, p = k, with the principle of Lorentz invariance, expressed via the relativistic energy-momentum relation, E 2 = p 2 c 2 + m 2 c 4 or E = ± p 2 c 2 + m 2 c 4 . Roughly speaking, although the two forms of the energymomentum relation are classically equivalent, combining the former with the de Broglie relations leads to the Klein-Gordon equation, while combining these with the latter leads to the Dirac equation [23,27]. The first describes the dynamics of spin-1/2 fermions whereas the second describes the dynamics of bosons and is obeyed by all free quantum fields [23]. The Dirac equation is manifestly invariant under SU(2) symmetry and is expressed in terms of the Dirac "gamma" matrices {γ µ } 3 µ=0 . These, in turn, can be expressed in terms of the Pauli matrices, , and the two-dimensional identity matrix, 1 1 2 [23,27].
In the next section, we review the structure of the Pauli matrices and their associated algebras in more detail, highlighting the difference between representations with integer and half-integer spin. We then review the structure of the canonical gamma matrices and the Dirac equation. Our analysis of fermions is restricted to the treatment of spin-1/2 particles, since no fundamental particles with higher half-integer spin values are known to exist in nature [23].

Algebra and uncertainty relations
For s = 1/2, the Pauli matrices are: where σ 1 = σ x , σ 2 = σ y and σ 3 = σ z (by convention). These form the fundamental representation of the SU(2) group generators, but the Pauli matrices for all higher-order spins can be obtained, straightforwardly, using Kramer's method [42]. For arbitrary spin, s, the corresponding generators are (2s + 1)-dimensional square matrices. The Pauli matrices for all spin values satisfy the su(2) Lie algebra: For representations of spin-1/2, this follows from the identity: However, the Pauli matrices for other spin values do not satisfy this relation. Eqs. (119) imply both the canonical commutation relations (118) and the canonical anticommutation relations: also known as the SU(2) Clifford algebra [43]. We stress that the Pauli matrices for spin-1/2 fermions satisfy both the Lie and Clifford algebras, Eqs. (118) and (120), whereas those for other spin values satisfy only the Lie algebra (118). For any spin, s, the canonical spin-measurement operators are related to the corresponding Pauli matrices via:ŝ These have (2s + 1) eigenvectors, corresponding to the eigenvalues −s , −(s − 1) . . . (s − 1) , s , and obey the commutation relations In addition, for spin-1/2 particles, the spin operators obey the relation and, hence, the anti-commutation relations The total spin operatorŝ 2 is the SU(2) Casimir operator (scaled by 2 ) and the associated Casimir invariant is s(s + 1) 2 , giving [27,44]: It follows thatŝ 2 commutes with all group generators, so that the simultaneous eigenvectors ofŝ 2 andŝ z are chosen (by convention) as the basis vectors for the spin Hilbert space [27]. For s = 1/2, these are: and Eq. (125) is satisfied by the fact that the matrices

Relativistic spin and the gamma matrices
In (3 + 1)-dimensional Minkowski space, relativistic spin-1/2 fermions are described by the Dirac equation, where {γ µ } 3 µ=0 are the Dirac gamma matrices. In the Weyl, or chiral, representation these are given by [23]: It is straightforward to show that the gamma matrices (130) satisfy the canonical anti-commutation relations: where η µν is the Minkowski metric, expressed in terms of the "Cartesian" coordinates {t, x, y, z}.

Historical analogy as a guide to generalisation
To construct a mathematical model of spin measurements in smeared-space we proceed by analogy with the historical development of canonical QM. Hence, we seek a set of constant-valued matrices Ŝ i 3 i=1 that satisfy the same algebraic structures as the components of angular In the canonical theory, the relevant algebra for the angular momentum operators is simply the well known so(3) algebra, scaled by a factor of (32). However, in the smeared-space model, the situation is more complicated. In Sec. IV C, we showed how the smearedspace angular momentum operators can be decomposed into the sum of four terms: a canonical quantum term L i acting on the first subspace of the tensor product state |Ψ (88), a "pure" geometric partL ′ i acting on the second, and two "cross terms",Λ i andΛ ′ i , acting on both subspaces (108)-(109). The subcomponents were found to obey the subalgebra defined by Eqs. (111a)-(111h). Together, these equations ensure that the rescaled so(3) Lie algebra, with → +β . In addition, we used the alternative definitionL i :=Λ i +Λ ′ i (113), leading to the subalgebra (114a)-(114e) for L i ,L ′ i ,L i . Hence, when searching for a generalised spin algebra, we have three options to choose from. First, we may search for exact analogues Eqs. (108)-(109) and, therefore, of Eqs. (110a)-(110j). This requiresŜ i to be decomposed asŜ In this case,α i andβ j are required to be finite-dimensional constant-valued matrices, acting on the first spin-subspace of the smeared tensor product state, that satisfy the -scaled Heisenberg algebra:  [45]. This group has one central element (z) and two sets of generators, usually denoted x i and p j by analogy with the canonical commutation relations, that satisfy the following algebra: In other words, while the central element z commutes with all other generators, it is not the identity element. Perhaps confusingly, the previous commutation relations are also typically referred to as the "Heisenberg algebra" in the mathematical literature, since they are the algebra of the Heisenberg group. However, they are not equivalent to the position-momentum commutation relations of canonical QM [45].
Second, we may search for a set of finite-dimensional constant-valued matrices, In this case, we must again require thatŜ i act on the first subspace of the tensor product state, thatŜ ′ i act on the second subspace, and thatΣ i ,Σ ′ i act on both subspaces simultaneously. With this in mind, we note that the most natural operator acting on both spin subspaces is of the Based on our previous considerations, this is clearly the most promising route. In the following section, we explore this possibility and construct explicit representations of the generator sub-componentsŜ i ,Ŝ ′ i andŜ i .

Generalised algebra and GURs
Considering the arguments presented above, we define the generalised spin operatorŜ i as: whereŜ i andŜ ′ i are given by: and for any s, s ′ . Next, we defineŜ i as: also hold for any values of s and s ′ . Hence, in order to recover a rescaled spin Lie algebra for the generalised (with → + β), we require the following commutation relations between the cross terms: In this section, our main aim is to describe the generalised spin physics of electrons in smeared-space. Hence, since the situation in which s = 1/2 is of greatest physical interest, we restrict ourselves to this from now on. We then have: and and These relations hold only for s ′ = 1/2. Hence, consistency of the generalised spin structure implies that the quantum state associated with the background geometry must be fermionic, with spin-1/2. The generalised spin algebra for the subcomponents Ŝ i ,Ŝ ′ i ,Ŝ i is, therefore: Together, Eqs. (145a)-(145e) give rise to the rescaled su(2) Lie algebra: and the rescaled Clifford algebra: for the generalised spin-measurement operators Ŝ i (132). From (146), it also follows that We note that, in the limit → β, theŜ i term is not necessary to maintain the canonical Lie algebra structure.
are representations of the su(2) generators, and these representations commute with each other (133), the combinationŜ i +Ŝ ′ i =:Ŝ i also satisfies the su(2) algebra if both sets of generators are weighted by the same scale factor. In this case, we may pull a single factor of outside the sum of terms on right-hand sides of the commutation relations, yielding [Ŝ i ,Ŝ j ] = i ǫ ij k (Ŝ k +Ŝ ′ k ) =: i ǫ ij kŜ i . However, in the presence of a two-scale theory, which is an essential feature of the smeared-space model [22], the presence of S i is unavoidable. Without it, it is not possible to construct an operatorŜ i that includes commuting representations of su(2) weighted by different scale factors, i.e., , and which also satisfies a canonical-type commutation relation. In this case, it is not possible to pull a single factor (with units of action) outside the expression on the right-hand side of the relation [Ŝ i ,Ŝ j ] = (. . . ) without includingŜ i (137) in the definition ofŜ i (132). Written explicitly, the generalised spin matrices take the form: andŜ 2 is given by: This follows from the fact the matrices are involutions. Hence, in the smeared-space model, are the analogues of the canonical . However, unlike the canonical Pauli matrices, depend explicitly on both quantisation scales, and β. It is straightforward to verify that all three spin oper- (149) have the eigenvalues: which, forŜ z , correspond to the following (unnormalised) eigenvectors: (1, 0, 0, 0), 0, The normalised eigenvectors ofŜ z may then be written interval, ∆s. Consistency of the position and momentum space pictures should then imply a corresponding minimum uncertainty in the length of the 4-momentum vector, ∆m. This is possible if the classical parameter m is promoted to the status of a Hermitian operator, m →m.
In [22], it was shown how to incorporate the effects of smearing directly into the definitions of observables. The resulting "smeared" Hermitian operators then act on the canonical quantum state |ψ ∈ H. This formulation of the model yields exactly the same predictions as the smeared-state picture in which the fundamental state is |Ψ ∈ H ⊗ H. However, in the smeared-operator picture, classical isometries are mapped to superpositions of isometries in the extended phase space of theory [22]. Thus far, this method has only been applied to the translation generators of classical Euclidean space (see [22]), but may in principle be extended to generators of other symmetries. Hence, we will address ways to implement smeared Lorentz symmetry, using this method, in a future publication. It may be hoped that such an approach may yield a natural definition of the mass operatorm. According to general relativity, classical gravity is the curvature of space-time [17]. Hence, space-time exists even when gravity does not. This assumption is the conceptual basis of special relativity and a cornerstone of quantum field theory, which is the quantum theory of matter in flat space-time [23,46]. The non-relativistic limits of these theories yield Newtonian mechanics and canonical QM, respectively, both of which are formulated in flat Euclidean space [24,27].
Thus, in any classical or canonical quantum theory, a background geometry exists, even if it is not curved. In relativistic theories, where the gravitational field is the curvature of the background, assuming zero curvature is equivalent to switching off gravity. (This is the case for the standard model of particle physics, which is formulated in the limit G → 0.) In the Newtonian approximation, however, this is no longer true. As discussed in Sec. III B, taking the static weak-field limit of the vacuum Einstein equations yields the familiar Laplace equation, in which the role of the gravitational potential is played by the time-time component of the metric: ∇ 2 √ g 00 ≃ 0, g 00 = c 2 (1 + 2Φ/c 2 ) [17,32]. Newtonian gravity is therefore due, solely, to the warping of time. In this limit, the Riemmanian curvature of the spatial part of the metric has a negligible effect on the total gravitational field strength and, formally, is set equal to zero. It is this fact that allows us to describe the Newtonian gravitational potential as a scalar field on a flat Euclidean background. This is a subtle but important aspect of the procedure used to recover the Newtonian limit from general relativity. It is important, for our purposes, since "gravitons" are usually identified with quantised perturbations of the metric [31]. In the standard treatment, due originally to Pauli and Fierz [47], this is decomposed as: where η µν is the Minkowski metric and |h µν | ≪ 1. Crucially, it is assumed that the Minkowski piece remains classical and only h µν is quantised: The quantised perturbationsĥ µν obey the Pauli-Fierz equations which describe the dynamics of spin-2 particles in flat Minkowski space [47]: Clearly, we may recover the classical Newtonian limit from Eq. (159) by setting h ij = 0 and h 00 ∝ Φ. In principle, we may also recover the "quantum" Newtonian limit from Eqs. (159)-(161) by performing the same classical approximations before mapping h 00 →ĥ 00 ∝Φ. However, one may ask the question: what is the quantum description of the background space when gravity is negligible?
Put another way, if gravitons describe quantised curvature (i.e., the quantised gravitational field), of what is the quantum space-time composed when it is flat? In this case there is no spatial curvature and no warping of time. It stands to reason that the resulting space-time cannot be composed of gravitons.
Thankfully, established physics suggests an answer to this question. In quantum field theories forces are mediated by virtual bosons and the fundamental particles that "feel" these forces are real fermions [23]. It is therefore possible that the fundamental quanta of space-time are fermionic in nature. The exchange of virtual spin-2 bosons (gravitons) between space-time quanta could then describe quantised curvature, by analogy with the gauge field description of other fundamental forces. [60] In this scenario both the flat space-time of relativistic theories, and the flat Euclidean space of non-relativistic models, should admit quantum descriptions. The smearing procedure, proposed in [22] and extended in the present work, is intended as a first step towards their construction. As shown in V B 2, the consistent description of angular momentum and spin in the smeared-space model requires the quantum state associated with the background space to be fermionic. However, based on the arguments considered here, this does not put it in conflict with the known physics of gravitons.

A. Conclusions
We have constructed generalised operators for angular momentum and spin in the smeared-space model of quantum geometry, originally proposed in [22]. In this model, the canonical state |ψ ∈ H is mapped to the generalised "smeared" state, |Ψ ∈ H ⊗ H. This represents the state of quantum matter, described by the wave function ψ, on a quantum background geometry. The latter is associated with an additional quantum state, g, so that Ψ depends on both functions.
In the original formulation of the smeared-space model, |ψ and |g are entangled, as proposed in the mattergeometry entanglement hypothesis [36]. However, in Sec. IV B, we defined a unitary operation that renders the smeared-state separable, yielding |Ψ = |ψ ⊗ |g (88). The transformation was inspired by the treatment of quantum reference frames (QRFs), considered in [37], in which entanglement between subsystems of a composite state is frame-dependent.
In the new formalism, the generalised angular momentum operators can be written as the sum of three subcomponents,L i =L i +L ′ i +L i (112). The first,L i , which acts on the first subspace of the tensor product state, represents the angular momentum of a canonical quantum particle described by |ψ . The second,L ′ i , which acts on the second subspace, represents the angular momentum of the quantum state associated with the background geometry, |g . The third subcomponent,L i , includes cross terms that act on both subspaces. This determines how quantum fluctuations of the background affect the angular momentum of particles propagating in the smeared geometry.
The subcomponents (112) were found to obey a generalised algebra, defined by Eqs. (111a)-(111e). These equations depend on two parameters, and β, where the new parameter β ≃ ×10 −61 is interpreted as the quantisation scale for geometry [22]. Crucially, the generalised algebra implies the existence of GURs for angular momentum but recovers the canonical so(3) Lie algebra up to a simple rescaling, → +β. In this respect, the angular momentum GURs are analogous to those for position and momentum, found in [22], in which the associated commutation relations are simply a rescaled representation of the Heisenberg algebra.
Having constructed the generalised operators for orbital angular momentum, we considered the status of spin in the smeared background geometry. We argued, by analogy with the historical development of canonical QM, that the generalised spin operators should be finitedimensional constant-valued matrices satisfying the same algebra as the components of angular momentum. Thus, we split the generalised spin operators into the sum of three terms,Ŝ i =Ŝ i +Ŝ ′ i +Ŝ i (132). By analogy with the subcomponents ofL i (112), we requiredŜ i to act on only the first spin-subspace of the tensor product smearedstate andŜ i to act on only the second spin-subspace. The third subcomponentŜ i , representing the interaction between the spin of the canonical quantum particle and the spin of the quantum state associated with the background geometry, was permitted to act nontrivially on both subspaces.
We then required Ŝ i ,Ŝ ′ i ,Ŝ i to satisfy the algebra defined by Eqs. (145a)-(145e), which are completely analogous to Eqs. (111a)-(111e) under the interchangê L i ↔Ŝ i ,L ′ i ↔Ŝ ′ i ,L i ↔Ŝ i . We found that, assuming spin-1/2 fermions as the matter component of the composite state, Eqs. (145a)-(145e) can be satisfied, in general, if and only if the quantum state of the background is also spin-1/2. Remarkably, therefore, consistency of the smeared-space spin algebra implies that the quantum state of the background space must be fermionic in nature. The implications of this for the description of relativistic spin and for the physics of gravitions were briefly discussed in Secs. V B 3 and V C, respectively, where it was argued that this does not contradict existing results in quantum gravity theory.
For electrons in a spin-1/2 background, the explicit forms of the generalised spin operatorsŜ x ,Ŝ y andŜ z were also determined (149). The composite smearedbackground plus matter spin-state was found to have four eigenvectors, corresponding to two sets of 2-fold degenerate eigenvalues, 3( + β) 2 /4, ±( + β)/2 . By analogy with the angular momentum case, the generalised spin algebra gives rise to GURs for spin measurements but recovers the canonical su(2) Lie algebra up to a simple rescaling, → + β.

B. Future work
To conclude our present analysis, we briefly review its limitations and consider ways in which they may be overcome in future studies. Firstly, due to limitations of time and space, several key questions have not been addressed in the current work. These include the following: • We have not determined the spectral representations of the generalised angular momentum opera- , or the explicit form of their associated eigenstates. This is crucial for the smearedspace model since, without these, we are unable to determine how the re-smearing procedure (discussed in Sec. III A) affects the form of the postmeasurement states. In [22], it was shown how resmearing via the map (40) yields physical states as the outcomes of generalised position and momentum measurements. This also ensures that the minimum uncertainties, ∆ Ψ X i l Pl and ∆ Ψ P j m dS c, hold for states prepared by such measurements. Thus, successive measurements can never violate these bounds. Naïvely, we would expect a similar result to hold for measurements of angular momentum, e.g., such that ∆ Ψ L i l Pl m dS c ≃ β. This is in accordance with our intuition that perfectly sharp rotations cannot be performed on an unsharp background geometry.
Furthermore, if such a fundamental limit to ∆ Ψ L i exists due to re-smearing, it would be especially instructive to contrast this with our results for generalised spin measurements. In Sec. V B 2, the explicit forms of the generalised spin opera- were determined, and their eigenvalues and eigenvectors were found. In principle, we may use these to rewrite the spin-measurement operators in spectral form. However, in this case, there is no "re-smearing" procedure, since the map (40) applies only to the position-dependent part of the wave vector. Thus, states for which ∆ Ψ S i = 0 certainly exist. This is in accordance with our intuition that, as an internal property of the quantum particle, spin is not affected by the smearing of the external space in the same way as angular momentum. Unfortunately, in the present work, we were not able to demonstrate the existence of nonzero minimum bound on ∆ Ψ L i .
• We have not considered multiparticle states, or attempted to generalise the Pauli exclusion principle (PEP) or the spin statistics theorem. This is a crucial step in the construction of a complete smeared-space generalisation of canonical QM. In particular, we note that the prediction of degenerate spin eigenstates, | 3( +β) 2

4
, ± ( +β) 2 δ , is potentially problematic for the model. For example, if the entangled and unentangled states in the spin "up" and spin "down" doublets are empirically indistinguishable, via measurements ofŜ z andŜ 2 , yet the spatial overlap of their associated wave functions is not forbidden by the generalised PEP, the model could be in immediate conflict with existing experimental data. That said, this may not be the case if the production of entangled states is extremely rare. This is not such an unreasonable assumption since the interaction between the background and the canonical quantum fermions is characterised by √ β ≃ × 10 −30 .
• We have not investigated the potential consequences of our results for cosmology. In this respect, it is especially intriguing that the consistency of the generalised spin algebra requires the quan-tum state associated with the background space to be fermionic. In [48][49][50][51][52], it was shown how the pair-production of fermionic dark energy particles can generate the expansion of space ad infinitum. Remarkably, the particle mass required to generate the observed expansion rate is m Λ ≃ √ m Pl m dS ≃ 10 −3 eV. This is the unique mass scale that minimises the smeared-space GUR, Eq. (56). In this scenario, there exists a space-filling "sea" of dark energy fermions so that additional pair-production goes hand-in-hand with a concomitant production of space. This drives eternal universal expansion as the positive rest mass of the new particles is exactly cancelled by their negative gravitational energy (see [48][49][50][51][52] for further discussions). Hence, it is clear that, if the fundamental quanta of space-time are fermionic, universal expansion can also be viewed as a result of their continuous pair-production. Such a view is consistent with the model of particulate dark energy proposed in [48][49][50][51][52].
• Finally, we note that, given the close connection between the canonical angular momentum operators, the rotation generators in three-dimensional Euclidean space, and the Lorentz generators in (3+1)dimensional Minkowski space [23,27,40], the next logical step is to extend our analysis to the smearing of relativistic quantum field theories. This should include "smeared" generalisations of the Maxwell, Klein-Gordon and Dirac equations, and, ultimately, of the QED Lagrangian. Clearly, many conceptual and mathematical problems must be resolved before smeared-space QFTs can be rigorously defined, but the results presented herein represent a first step towards their construction.