Discreteness of area in noncommutative space
Abstract
We introduce an area operator for the Moyal noncommutative plane. We find that the spectrum is discrete, but, contrary to the expectation formulated by other authors, not characterized by a “minimum-area principle”. We show that an intuitive analysis of the uncertainty relations obtained from Moyal-plane noncommutativity is fully consistent with our results for the spectrum, and we argue that our area operator should be generalizable to several other noncommutative spaces. We also observe that the properties of distances and areas in the Moyal plane expose some weaknesses in the line of reasoning adopted in some of the heuristic analyses of the measurability of geometric spacetime observables in the quantum-gravity realm.
pacs:
Motivation for the study of spacetime noncommutativity comes primarily from its possible use in investigations of spacetime fuzziness, which in particular is expected mead ; padma ; ngmpla ; gacmpla ; ahluGUP ; garay to be relevant for the description of the quantum-gravity realm. However, most work focuses on establishing how a noncommutativity of spacetime would affect S-matrix/field-theory observables (such as the probabilities of occurrence of certain particle-physics processes) and still very little has been actually established concerning the fuzziness and other properties of geometric observables. While in Loop Quantum Gravity lqg1 ; lqg2 ; lqg3 ; lqg4 , another much-studied formalism with a quantum geometry, there is a deservedly renowned detailed analysis areaLQG of the properties of areas and volumes, very little is known about areas and volumes even for the simplest noncommutative spacetimes, the “canonical spacetimes” moyalrefs3 characterized by noncommutativity of the type ()
(1) |
(with a noncommutativity matrix that commutes^{1}^{1}1While over the last few years canonical spacetimes are indeed mostly studied assuming that commutes with the coordinates, the research programme that first led to the proposal of canonical noncommutativity dopli1994 is contemplating the possibility doplithetanontrivi of a that does not commute with the coordinates. with the coordinates ).
To our knowledge, the most explicit (and yet only tentative) investigation of any of these issues is the one of Ref. romero , which reported an attempt of characterization of the areas of discs in the (Groenewold-)Moyal plane moyalrefs1 ; moyalrefs2 , which is the case of canonical noncommutativity for only two spatial dimensions:
(2) |
We shall here take as starting point the strategy of analysis advocated in Ref. romero , but provide a more satisfactory characterization of areas in the Moyal plane, which ends up showing that, while some aspects of the methodology proposed in Ref. romero are indeed very valuable, the specific indications concerning the properties of areas that emerged in the study reported in Ref. romero were misleading.
It is convenient for us to start with a brief description of the analysis reported in Ref. romero . And we must immediately underline a first potentially worrisome aspect of that analysis concerning the fact that discs in the noncommutative plane were specified romero through the coordinates of a single point. The area of the disc was described by generalizing a familiar commutative-spacetime formula
(3) |
in terms of the noncommutative coordinates of that single point. So the discs considered in Ref. romero are centered around a “classical origin” (but in the Moyal plane no point, not even one chosen as the origin, can truly be treated classically) and their boundaries are identified by a single point with coordinates .
Ref. romero studies the spectrum of , on the basis of the fact that are governed by (2), and finds that
(4) |
which would amount to a discrete spectrum with “quanta of area” of and with a minimum eigenvalue .
Our motivation in setting up a more refined investigation of areas in the Moyal plane came from the observation that it seemed to us unreasonable that the Moyal-plane noncommutativity would give rise to a “minimum area principle” (independently of whether the minimum value was or some other value). From (2) one infers uncertainty relations of the type
(5) |
which do not prevent points from having “sharp” value of one of the coordinates, although this comes at the cost of uncontrolled uncertainties in the other coordinate. Therefore it should be possible to have situations in which sharply (of course, in an appropriate “state” of the Moyal quantum geometry agmStringspace ) all of the boundary points that identify a surface have the same value of, say, , which is the case in which the surface collapses sharply to having zero area.
This bit of intuition appeared to render more significant what could have been viewed as mere technical limitations of the characterization of area offered in Ref. romero : (I) the formula for makes reference (implicit but substantial) to a classical center of the disc, which, as stressed above, is a concept that is clearly foreign to the Moyal-plane quantum geometry, and (II) the formula for codifies the information on the boundary of the surface through the coordinates of a single point, which is the highest conceivable level of optimism (specifying a boundary with less than one point would require more than optimism).
On the basis of these considerations (both intuitively originating from and technically originating from the structural inadequacies, as a candidate area operator, of the operator ) we concluded that of Ref. romero cannot be a meaningful tool of investigation of the spectrum of areas. And in devising an alternative strategy we found guidance also in the observation that even in elementary commutative geometry one could reasonably argue klein that areas are most transparently characterized for polygons with vertices, with the case of surfaces with smooth boundaries (such as discs) discussed as a careful limit in which the differential calculus plays a central role. In noncommutative spaces one does have a notion of differential calculus majidbook , but not immune from certain counter-intuitive peculiarities (see, e.g., Ref. sitarz ), which in turn may affect nontrivially the description of surfaces with smooth boundaries as a limit of polygons. Moreover, the Loop-Quantum-Gravity results on the spectrum of volume in a 3-dimensional spatial quantum geometry guided us to expect that triangles would have to be the most natural framework for area investigation in the Moyal plane: in the description of volumes in the 3-dimensional quantum geometry of Loop Quantum Gravity tetrahedra play a pivotal role lqg1 ; areaLQG , which we may well expect to be similar to the role played by triangles in the description of areas in two-dimensional quantum geometries, such as the Moyal plane.
We therefore set out to establish whether or not the spectrum of areas is discrete in the Moyal plane, and whether or not this spectrum is characterized by a “minimum-area principle” (a minimum, nonzero, allowed sharp value of area), by focusing primarily on triangles. The starting point for our proposal of a triangle area operator for the Moyal plane is an elementary formula in commutative spacetime (whose points have commutative coordinates ) which describes klein the area of a triangle with vertices , and in the following way
(6) |
in terms of
(7) |
The formula (7) is exactly of the type we should deem desirable for the purposes of proposing a quantum version. It depends exclusively on the coordinates of the vertices, without any reference to “special” external points (such as an origin, which might then generate embarrassment in the quantum version, in the sense discussed above) . And the fact that this formulation as the absolute value of expresses the area as a function of the coordinates of the vertices in a way that is invariant under permutations of the vertices allows us to avoid a potential difficulty: formulas that depend on identifying the sequence of vertices would be unpleasant in a fuzzy geometry, since in principle the fuzziness should prevent us from being able to establish such sequences in general.
In light of these observations we promote to the status of a quantum-geometry observable, obtained from the quantum coordinates of three points in the Moyal plane, by posing
(8) |
and we shall describe the area of a triangle in the Moyal plane as the absolute value of the expectation of in a quantum state of the Moyal-plane geometry.
We introduced a dedicated notation for the j-th coordinate of the m-th point also as a reminder of the fact that in order to consider at once three points in a noncommutative space a couple of steps of formalization must be taken. The relevant issues may be discussed rigorously at the abstract algebraic level (see, e.g., Ref. majidbook ), but as physicists we find particularly comfortable to rely on representations. and in (2) can be described as operators on a Hilbert space majidbook ; gacmajid with structure that exactly reproduces the Hilbert space of a particle in nonrelativistic quantum mechanics. Denoting the “state of the point” by , so that the “wave functions” of the first coordinate of the point is , in light of (2) we can prescribe that and act as follows
(9) |
(10) |
Our observable is a 3-point observable, so it acts on states , , are of course intended as follows: . And the operators such that
(11) |
where .
These clarifications, besides giving a precise characterization of the observable , also set the stage for a rather straightforward derivation of the spectrum of . For this purpose it is convenient to introduce the notation for the first coordinate of the i-th point and for the second coordinate of the i-th point
(12) |
This allows us to rewrite as follows
(13) |
and the correspondence with a Hamiltonian interaction term which is familiar to physicists becomes evident upon noticing that
(14) |
since from (2) it follows that . Clearly has all the properties of the familiar operator describing the interaction between the angular momentum observable and an external (classical) homogeneous magnetic field , but with the Planck constant denoted by (see (14)). This allows us to deduce that the spectrum of is , with , and therefore the spectrum of the area of the triangle is
(15) |
We conclude that in the Moyal plane the allowed sharp values of area of a triangle are characterized by a regular discretization with “quantum of area” , and most importantly the Moyal plane is not an example of quantum space in which a “minimum area principle” holds: the lowest eigenvalue of is . And in particular, as expected (see the considerations guided by which we offered above), our area of the triangle in the Moyal noncommutative plane vanishes if the three vertices have the same coordinates. In fact, it is easy to verify that
(16) |
if
Concerning our intuition that triangles should play a role in the general description of areas in the Moyal plane that is just as central as the role of tetrahedra in the Loop-Quantum-Gravity description of general volumes (perhaps a different role, but equally central), we can only report some very preliminary observations, which however appear to confirm our expectations. A starting point is provided by the (very elementary) observation that in an ordinary (commutative) plane the areas of all polygons can be obtained as a sum of areas of triangles. And a formula giving the area of the polygon as function of the coordinates of the vertices can be arranged klein as a sum of functions of the type
then for each triplet of vertices (labeled , , ) of the -vertices polygon one has
So, even though, as stressed above, for the relationship between the area and a certain sum of terms of the type requires a corresponding choice of ordering of the vertices, it seems we can conclude that the area is zero on , since each term in the sum vanishes on , independently of the ordering of the vertices.
Both in our detailed analysis of the area of triangles and in these brief preliminary remarks on general polygons we placed particular significance on the existence of zero-area states because this feature (already rather significant intrinsically within the exploration of the Moyal plane) has some broader implications. To see this we must first briefly summarize the simpler result on the squared-distance observable which we reported in Ref. agmStringspace (within an analysis of distance observables in a few different examples of quantum spaces). Unsurprisingly in Ref. agmStringspace we adopted as squared-distance observable the operator
(18) |
where , .
Introducing here and one of course can rewrite as follows
(19) |
for , . And observing that from (2) it follows that , one easily concludes agmStringspace that the spectrum of is of the harmonic-oscillator type:
(20) |
with integer and nonnegative.
So also for one finds a discrete spectrum, with quanta of squared-distance of , but interestingly here there is a “minimum-distance principle” at work: the lowest eigenvalue is .
We conclude that the Moyal plane is an example of noncommutative space in which there is a minimum distance but no minimum area. A posteriori one can see that this should have been expected, considering once again the uncertainty relations implied by Moyal-plane noncommutativity . As already stressed above these relations do not obstruct the case of zero area, because they allow the possibility that all points on the boundary sharply have the same, say, coordinate. But they do exclude the case of zero distance: for the distance between two points to be zero one should demand that all coordinates of the two points coincide, which is very clearly incompatible with .
These observations suggest that, in spite of its relative simplicity, the study of the Moyal plane may contribute some valuable indications for the debate, within the quantum-gravity community, that indeed concerns the possible emergence of features such as minimum area and minimum distance at the Planck scale. In that debate a significant role is played by semi-heuristic arguments that combine quantum mechanics and general relativity and find rather robust (as robust as heuristic arguments can be) evidence of the necessity to implement a Planckian minimum-distance principle. Once the minimum-distance is (heuristically) established then it is not uncommon that, with somewhat looser use of logics, the authors make the assumption that one would have to reach similar conclusions also for areas and volumes, as if in a quantum spacetime the presence of a minimum-length bound would necessarily imply corresponding bounds for areas and volumes. The Moyal plane, in light of the results we obtained, is a quantum space that provides a counter-example for the assumptions that guide this type of line of reasoning.
And it should be possible to investigate how common it is to find this specific scenario for the spectra of distances and areas in noncommutative spaces. In fact, many aspects of our analysis of the Moyal plane are immediately adaptable to other, more complex, noncommutative spaces. In particular, our formalization of the observable “area of a triangle”, based on the novel operator , should be applicable without modification to a rather large class of noncommutative spaces.
Acknowledgments
G. A.-C. is supported by grant RFP2-08-02 from The Foundational Questions Institute (fqxi.org).
G. G. is supported by ASI contract I/016/07/0 ”COFIS”.
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