On the Quantization of Length in Noncommutative Spaces

Balasundaram, Muthukumar; Rashid, Aamir

doi:https://doi.org/10.1155/2022/8009789

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Research Article | Open Access

Volume 2022 | Article ID 8009789 | https://doi.org/10.1155/2022/8009789

On the Quantization of Length in Noncommutative Spaces

Muthukumar Balasundaram¹and Aamir Rashid¹

Academic Editor: Gaetano Luciano

Received09 Sept 2022

Revised04 Nov 2022

Accepted30 Nov 2022

Published12 Dec 2022

Abstract

We consider canonical/Weyl-Moyal type noncommutative (NC) spaces with rectilinear coordinates. Motivated by the analogy of the formalism of the quantum mechanical harmonic oscillator problem in quantum phase-space with that of the canonical-type NC 2-D space, and noting that the square of length in the latter case is analogous to the Hamiltonian in the former case, we arrive at the conclusion that the length and area are quantized in such an NC space, if the area is expressed entirely in terms of length. We extend our analysis to the 3-D case and formulate a ladder operator approach to the quantization of length in 3-D space. However, our method does not lend itself to the quantization of spacetime length in and Minkowski spacetimes if the noncommutativity between time and space is considered. If time is taken to commute with spatial coordinates and the noncommutativity is maintained only among the spatial coordinates in and dimensional spacetime, then the quantization of spatial length is possible in our approach.

1. Introduction

Noncommutative (NC) spacetime, which was first introduced by Snyder [1] in an attempt to regulate the divergences in quantum field theories, has also been introduced in various contexts [2–7], and the literature in this area is quite replete. To name a few, field theories [8–10], gravity theories [11–15], and quantum mechanics [16–24] have all been considered with the background spacetime being noncommutative. In another development, the spectral manifolds in NC geometry are shown to exhibit the geometric analogue of Heisenberg commutation relation involving the Dirac operator and the Feynman slash operator for real scalar fields, leading to the quantization of volume [25]. In [15], it was shown from the pure geometrical point of view that the NC-Minkowski spacetime parametrized with spherical or cylindrical coordinates has nontrivial NC corrections to curvature tensors and curvature scalar.

Noncommutativity of spatial coordinates is related to the presence of a minimal length in a system. This minimal length in turn is usually related to the uncertainties in the distance measurements [3, 22]. Instead of relating the minimal length with uncertainties, we propose in this work an approach in which the actual square of the distance between any two points and in a commutative flat-spacetime is promoted as an operator with the introduction of the canonical/Weyl-Moyal type noncommutativity among the coordinate operators. Here, is taken to be a constant diagonal metric of spacetime, and is a constant and real antisymmetric matrix. The operators and may be taken either as the position operators of two particles or as the operators corresponding to the positions at which fields are considered. The idea of length as an operator has already been discussed in the literature in the context of canonical quantum gravity [26]. We set up an algebra of operators in such a way that the eigenvalues of the operator can be raised or lowered. The hint to such an approach is provided by an analogy of 2-D NC-space operator formalism with that of the quantum mechanical harmonic oscillator problem. Taking and assuming that the coordinate operators of different particles commute, i.e., , we can define the operator corresponding to the square of the distance as with where . If , then the operator in such a 2-D NC space can be related to the Hamiltonian of an appropriate harmonic oscillator. One important thing in such an analogy is the set of ladder operators , the lowering operator, and , the raising operator. If and , then the ladder operators have the following three important relations: where and are constants. If denotes the transformation from to , then Equation (5) implies that . With in 2-D, we can define the ladder operators as that satisfy Equations (3)–(5) with and . Here, is an arbitrary real number. The square of the length in this case is given by

In analogy with the quantum harmonic oscillator problem, it is clear that on the eigenstate of , with the minimum eigenvalue being . The operator plays the role of a raising operation, i.e., the eigenvalues of on the states and are apart. The operator does the lowering operation. The angle in Equation (6) corresponds to the orientation of the line segment that would connect the points and in the commutative 2-D space.

On such admissible eigenstates, the square root of Equation (7) is also quantized. In 2-D Euclidean space, the commutative analogue of Equation (1) could also mean the area of a square of side , in which case the quantization of Equation (7) can also imply the quantization of area. In [25], it has been shown in the context of spectral manifolds in NC geometry that the area of a 2-D manifold is quantized. Here, we arrive at the same result in the case of 2-D plane with coordinate noncommutativity. In the following, we attempt to generalize this result to other NC spaces.

Before proceeding, note that the commutator Equation (4) is essential to construct a ladder of states which are all eigenstates of operator, and that the canonical commutator Equation (2) (or Equation (3)) can be used to reduce the degree of any polynomial by considering the commutator . Equation (4) is an example for such reduction of the degree of a polynomial. In this context, it is worth remarking that a generalized coordinate noncommutativity, which would involve a nonconstant in Equation (2), will pose problem to construct a ladder of eigenstates of using Equation (4). So, such generalized coordinate commutator structure (see for example [4]) is not considered in this work (The authors thank the referee for raising the issue of generalized coordinate commutator structure.).

2. A Generalization

It turns out that Equations (3)–(5) play a crucial role in the generalization of the idea of length quantization to other spaces with constant and symmetric metric . In an -dimensional space, we define the lowering operator and the operator corresponding to square of length, respectively, as where Einstein’s summation convention is implied, and values are complex constants to be determined. Substitution of Equation (9) in Equation (4) gives the relation

Also, Equations (3) and (10) lead to where is the complex conjugate of . In the following, we solve Equations (10) and (11) for the cases of 3-D space, dimensional spacetime, and dimensional spacetime and analyze the consequences.

2.1. The Case of 3-D

We assume that and that , in which case the set of three equations in Equation (10) have nontrivial solution only if the secular determinant which leads to the nontrivial value . The third trivial solution leads to a set of values such that , thereby violating Equation (3). Also, does not give any length-raising or lowering operation in the theory (see Equation (4)). Choosing the positive value for so that can be identified with a lowering operation and substituting it in Equations (10) and (11) gives a set of values for where and , and is a real constant. Note that . The lowering operator can then be expressed as

Since there are three independent operator variables , and , we need another operator in addition to and , to define the transformation in 3-D and to make it invertible. But and its Hermitian conjugate should not be independent of each other, else we would end up with four independent operator variables. We define in the following way: where values are complex numbers. Since is defined as a linear and homogeneous function of , its Hermitian conjugate is also linear and homogeneous in , and so , where is such that since . If and , then the analogue of Equation (5) is written as where is the matrix form of the metric tensor. The above can further be expressed as upon the condition that . Using the expressions for , , and in terms of as in Equations (9) and (15), we can get the elements of in terms of , , and from which the condition in indicial form is expressed as

For all the values , the above condition gives as

For and and we can, respectively, show that which in general leads to . Calculation of using Equations (20), (19), and (13) gives . Therefore, by writing , Equation (20) completely determines and in terms of and

Then the only free parameters in the theory would be and . These two parameters would correspond to the orientation of the line segment connecting the points and in the physical 3-D commutative space.

Note that the explicit values in Equation (13) of and values have the following properties:

Equation (22) can also be inferred from the . Equation (24) leads to . In fact, Equation (23) leads to , and thus commutes with all the operators of the form in the theory. In other words, the states characterized by any index corresponding to are unaffected by other operators of the form .

Using Equation (4), it is easy to show that . So it is possible to construct a complete set of simultaneous eigenstates of , , and .

Like in the case of quantum harmonic oscillator problem, the relations of Equations (3) and (4) ensure that the eigenvalues of should be nonnegative integers (since the eigenvalues of will also turn out to be nonnegative), and the eigenstate itself may be denoted by . Using the commutator we can show that

The above relation suggests that the state can be written as where , and are constants to be determined.

By acting on Equation (27) and comparing the result with Equation (26), we have

Projecting the state Equation (27) onto , , and , respectively, and in those projections comparing the action of on the ket vector with its action on the bra vector leads to

Making use of the commutator in the calculation of , we can show that , and therefore is independent of , which we denote by . To be consistent with the relations like , needs to satisfy which is possible if we choose anyone of the following set for : which gives for any particular , where is some real constant. The values in Equation (31) are motivated by the values of , Equation (21), because both and have the same properties with (compare Equations (30) and (24)).

If we choose the states to normalize to 1, where is not summed over and values are dimensionless constants to be determined, then using Equation (27) and its Hermitian conjugate, the normalization leads to

We denote the above expression by since it is the same for all and it depends on . Also, using Equations (24), (28), and (31), it is straightforward to show that and therefore the states may be properly denoted by instead of . Since the operator commutes with all other operators, it is not possible to determine through the operator algebra method.

Finally, the eigenvalues of in Equation (17) are worked out to be

Essentially, we have worked out the eigenvalues of in terms of the eigenvalues corresponding to and . Since is real because of Equation (29), the minimum value for is , and therefore we have,

In the Euclidean 3-D space, the commutative analogue of Equation (17) can also be interpreted as the area of a square of side in the plane formed by and , and so the quantization of Equation (17) implies the quantization of area with the minimum value being . But the actual uncertainty relation would yield the minimum which is lower than . So the minimum value of the quantized area is not violating the uncertainty principle. If a volume is written as a function of the side , then the volume is also quantized along the dimensions of and .

2.2. Dimensional Spacetime

If and , then the relation in Equation (10) leads to the purely imaginary value for . Also, the substitution of this value back into Equation (10) implies that , which will contradict Equation (3). Therefore, the quantization of spacetime length in this approach is not feasible.

2.3. Dimensional Spacetime

If we take and , then the solution to the secular equation corresponding to Equation (10) gives only the purely imaginary number as the nontrivial value. Putting the value in Equation (10) gives no nontrivial solution to , and the value results in , contradicting Equation (3). So, in this case also, our method is not feasible to quantize the spacetime length. However, if , i.e., if the time commutes with the spatial coordinates, then the method outlined in the Introduction can be used to quantize the spatial part of in a particular rectilinear system of coordinates.

2.4. Dimensional Spacetime

In this case, if the time is taken to commute with the spatial coordinates, then the same method for the 3-D case can be used to quantize the spatial length in a particular rectilinear system of coordinates. However, if time is assumed to noncommute with spatial coordinates, it requires a separate elaborate treatment and it will be published elsewhere.

3. Concluding Remarks

In this work, we have proposed a length operator in Weyl-Moyal type noncommutative (NC) spaces and analyzed the consequences on the quantization of length. By comparing the operator formalism of 2-D NC space with that of quantum harmonic oscillator problem, we have deduced that the operator corresponding to the square of length is analogous to the Hamiltonian of the oscillator, and hence the length and area are quantized in 2-D NC space. This result is in conformity with and a special case of the already established result in the context of spectral manifolds in NC geometry [25]. We have also succeeded in showing that the length is quantized in a 2-D NC subspace of a 3-D NC space. Since the length quantization is more fundamental, the quantization of area and volume [25] can be inferred from it for the special cases in which they directly depend on the length along the 2-D subspace. But our method does not work for the cases of and spacetime dimensions if the time is taken to noncommute with the spatial coordinates. This is because the eigenvalue equation (Equation (10)) which is the result of the commutator (Equation (4)) gives imaginary eigenvalues for —the quantum of . This essentially implies that the length is increased or decreased in steps of imaginary values which is unphysical. When the metric is changed to an Euclidean metric, the quanta of become real.

Data Availability

No new data were created or analyzed in this study.

Disclosure

A version of this work has been published as an arXiv preprint: https://arxiv.org/abs/2206.07972 [27].

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this work.

Acknowledgments

The authors gratefully acknowledge the funding (No. PU/PS2/PHYS/Minor-Equip/21-22/241) by Pondicherry University under the Minor Equipment Grant No. PU/PD2/Minor-Equip/2022/536.

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Copyright

Copyright © 2022 Muthukumar Balasundaram and Aamir Rashid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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