Fuzzy de Sitter space
Abstract
We discuss properties of fuzzy de Sitter space defined by means of algebra of the de Sitter group \(\text {SO}(1,4)\) in unitary irreducible representations. It was shown before that this fuzzy space has local frames with metrics that reduce, in the commutative limit, to the de Sitter metric. Here we determine spectra of the embedding coordinates for \((\rho ,s=\frac{1}{2})\) unitary irreducible representations of the principal continuous series of the \(\text {SO}(1,4)\). The result is obtained in the Hilbert space representation, but using representation theory it can be generalized to all representations of the principal continuous series.
1 Introduction
Understanding of the structure of spacetime at very small scales is one of the most challenging problems in theoretical physics: more so as it is, as we commonly believe, related to the properties of gravity at small scales, that is to quantization of gravity. In the absence of a sufficient amount of experimental data, it is presently approached by mathematical methods: still there are basic tests which every model of quantum spacetime has to satisfy, as the mathematical consistency and the existence of a classical limit, usually to general relativity.
A feature very often discussed in relation to quantization is discreteness of spacetime. Discreteness can mathematically be implemented in various ways, for example by endowing spacetime with lattice or simplicial structure. When discreteness is introduced by means of representation of the position vector by noncommuting operators or matrices we speak of fuzzy spaces. Assumption that coordinates are operators comes from quantum mechanics: in fact, it is quite natural (perhaps even too elementary) to presume that generalization of \(\,[x^\mu ,x^\nu ] =0\,\) to \(\,[x^\mu ,x^\nu ] \ne 0\,\) describes the shift of physical description to lower length scales. Operator representation has a potential to solve various problems of classical gravity and quantum field theory: it introduces minimal length, which in the dual, momentum space, can in principle resolve the problem of UV divergences; singular configurations of gravitational field can potentially be dismissed as corresponding to nonnormalizable states, and so on. In addition, algebraic representation allows for a straightforward description of spacetime symmetries. Perhaps the main drawback of the assumption of discreteness is a loss of geometric intuition which is in many ways inbuilt in our understanding of gravity.
There are various ways to generalize geometry: one of the most important parts of any generalization is the definition of smoothness. In noncommutative geometry, derivatives are usually given by commutators; once they are defined, one can proceed more or less straightforwardly to differential geometry. We shall in the following use a variant of noncommutative differential geometry which was introduced by Madore, known as the noncommutative frame formalism [1]. It is a noncommutative generalization of the Cartan moving frame formalism and gives a very natural way to describe gravity on curved noncommutative spacetimes. In particular classical, that is commutative, limit of such noncommutative geometry is usually straightforward.
General features of the noncommutative frame formalism and many applications to gravity are known [2, 3, 4, 5]; the aim of our present investigation is to construct fourdimensional noncommutative spacetimes which correspond to known classical configurations of gravitational field. This means, to find algebras and differential structures which are noncommutative versions of, for example, black holes or cosmologies. One very important idea in this context is that spacetimes of high symmetry can be naturally represented within the algebras of the symmetry groups. The first model of such noncommutative geometry was the fuzzy sphere [6, 7]: it has a number of remarkable properties which make it a role example for understanding what fuzzy geometry should or could mean. Different properties of the fuzzy sphere were used as guidelines to define other fuzzy spaces [8, 9, 10, 11], including for us very important noncommutative de Sitter space in two and four dimensions [12, 13, 14]. In our previous paper [15] we analyzed differentialgeometric properties of fuzzy de Sitter space in four dimensions realized within the algebra of the \(\text {SO}(1,4)\) group. We found two different differential structures with the de Sitter metric as commutative limit. Here we analyze geometry of fuzzy de Sitter space that is the spectra of the embedding coordinates.
The plan of the paper is the following. In Sect. 2 we introduce notation for the \(\text {SO}(1,4)\), review some results of [15] and discuss the flat limit of fuzzy de Sitter space revealing its relation to the Snyder space. In Sect. 3 we solve the eigenvalue problem of coordinates in the unitary irreducible representation \((\rho , s=\frac{1}{2})\,\) of the principal continuous series. The obtained spectrum we compare to the known grouptheoretic result in Sect. 4.
2 Metric and scaling limits
3 Coordinates

principal continuous series, \(\rho \in \mathbf{R}\), \(\rho \ge 0\), \(s = 0, \frac{1}{2}, 1, \frac{3}{2},\dots \)
\({{\mathcal {Q}}} = s(s+1)+ \frac{9}{4} + \rho ^2\), \({{\mathcal {W}}} = s(s+1)( \frac{1}{4} + \rho ^2)\),

complementary continuous series, \(\nu \in \mathbf{R}\), \(\vert \nu \vert <\frac{3}{2} \), \(s = 0, 1, 2\dots \)
\({{\mathcal {Q}}} = s(s+1)+ \frac{9}{4}  \nu ^2\), \({{\mathcal {W}}} = s(s+1)( \frac{1}{4}  \nu ^2)\), and

discrete series, \(s = \frac{1}{2}, 1, \frac{3}{2},2 \dots \), \(q= s,s1,\dots 0\ \mathrm{or}\ \frac{1}{2}\) \({{\mathcal {Q}}} = s(s+1)  (q+1)(q2)\), \({{\mathcal {W}}} =  s(s+1)q(q1)\).
We therefore restrict to simpler problem: to find the eigenvalues of \(W^\alpha \) for a specific class of representations. The simplest possibility would be to consider Class I UIR’s (they are in the principal and complementary series): their Hilbert space representations are known, they have a lowest weight state so the coherent states can be constructed, etc. However, Class I is characterized by condition \(\,{{\mathcal {W}}}=0\): thus in our framework these UIR’s cannot be simply interpreted as de Sitter spaces: a fixed Open image in new window implies \(\varLambda \rightarrow \infty \).^{6} Another subset which is singled out mathematically and physically is the principal continuous series. As shown in [21], in the WignerInönü contraction limit these UIR’s contract to a sum of two representations of the Poincaré group with positive value of the masssquared. The Hilbert space representations of the principal continuous series were found in [22, 23, 24]: we shall perform the construction explicitly in the simplest nontrivial case, \(s=1/2\).
Higher spin representations \((\rho ,s)\) can be obtained from \((\rho , s=0)\,\) by adding spin generators \(\,S_{\alpha \beta }\) to orbital generators \(\,L_{\alpha \beta }\). Representation space will be again a direct sum of two spaces, each equivalent to the Hilbert space of the Bargmann–Wigner representation of the Poincaré group of a fixed spin s [25]. We shall here discuss the eigenvalue problem for \({s=\frac{1}{2}}\,\); the case of higher spins is more involved because of an additional projection to the highest spin states [27]. In addition, we will consider just a ‘half’ of the representation space, the other half being equivalent [24].
4 Grouptheoretic view
Our result for \(s=\frac{1}{2}\) is in accordance with this. There is only one summand in (75) corresponding to \(\, s_0=s=\frac{1}{2}\); the spectrum of \(W^0\) is the real axis, \(\, \lambda =\frac{\nu }{2}\in (\infty , +\infty )\, \). An analogous decomposition of unitary irreducible representations of the Poincaré group into a direct integral of UIR’s of the Lorentz group was done in [29]: as we here use the same representation space [22, 23], there are many parallels in two calculations.
5 Summary and outlook
In this paper we continued our investigation of fuzzy de Sitter space defined as a unitary irreducible representation of the de Sitter group \(\text {SO}(1,4)\), analyzing representations of the principal continuous series. In analogy with the commutative case, fuzzy de Sitter space in four dimensions is defined as an embedding in five dimensions: the embedding coordinates are proportional to components of the Pauli–Lubanski vector, \(\,x^\alpha =\ell W^\alpha \), and the embedding relation is the Casimir relation \(\, W_\alpha W^\alpha \)= const. By an explicit calculation in the \(\,(\rho ,s=\frac{1}{2})\) representation we found that the spectrum of time \(x^0\) is discrete while the spectra of spacelike coordinates \(x^4\) and \(x^i\) are continuous. This result is in fact general and holds for all principal continuous UIR’s \(\,(\rho ,s)\) of the \(\,\text {SO}(1,4)\), which can be proved by using the result [28] for the decomposition of representations of the principal series of \(\text {SO}(1,4)\) into the UIR’s its \(\text {SO}(1,3)\) subgroup.
There are other operators, that is other coordinates on fuzzy de Sitter space whose properties one would like to understand and physically interpret. First of them is certainly the cosmological time, \(\, \tau = \ell \log \, (W^0+W^4)\), and second are the isotropic coordinates. While it is, at least in the \(\,(\rho ,s=\frac{1}{2})\) representation, straightforward to write the eigenvalueproblem for \(\tau \), the corresponding differential equation turns out to be not easy to solve. This is one of the problems in the given setup which deserves additional work and which might give interesting results.
Footnotes
 1.
Differently from [15] we here use the fieldtheoretic signature. Indices \(\alpha \), \(\beta \), ... belong either to the set \(\{0,1,2,3,4 \}\) or \(\{ 0,1,2,3\}\); in cases when it is not completely obvious we specify explicitly one the two sets. Indices \(i,j =1,2,3\dots \) are spatial.
 2.
That is, it has a straightforward meaning; see a comment related to double scaling limits given below.
 3.
As we use units in which \(\,\hbar =1\), momenta have dimension of the inverse length.
 4.
See, however, comments given in the Appendix.
 5.
In comparison with [20], \(p=s\), \(\sigma = \frac{1}{4}+\rho ^2\).
 6.
It is on the other hand certainly possible to define specific double scaling limits, in order to interpret Class I representations as fuzzy de Sitter spaces; this point remains to be explored.
 7.
This very important observation is due to our referee, and it gives much better understanding of the construction of fuzzy de Sitter space and of its structure.
Notes
Acknowledgements
Authors are very much indebted to the referee for pointing out a mistake in the calculation of the spectrum (which was present in the first version of the paper) as well as for relating the given derivation to the decomposition of the UIR’s of \(\text {SO}(1,4)\) with respect to the UIR’s of its subgroups. This work was supported by the Serbian Ministry of Education, Science and Technological Development Grant ON171031, and by the COST action MP 1405 “Quantum structure of spacetime”.
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