Heisenberg Algebra in the Bargmann-Fock Space with Natural Cutoffs
We construct a Heisenberg algebra in Bargmann-Fock space in the presence of natural cutoffs encoded as minimal length, minimal momentum, and maximal momentum through a generalized uncertainty principle.
1. Introduction: The Generalized Uncertainty Principle and Fuzzy Spacetime
According to the equivalence principal in general relativity, gravitational field is coupled to everything. This means that photons in Heisenberg gedankenexperiment are actually coupled with electrons gravitationally and this leads to modification of the standard uncertainty principle. It has been characterized that gravity in very small length scales causes serious change in the structure of spacetime. It causes minimal uncertainty in positions of atomic and subatomic particles [1–15]. In fact, there is absolutely smallest uncertainty in position measurement of any quantum mechanical system and this feature leads nontrivially to the existence of a minimal measurable length in the order of Planck length. Existence of this natural cutoff requires deformation of the standard Heisenberg uncertainty principle to the so-called generalized uncertainty principle (GUP) (see, for instance, [13, 14, 16–20]). In one dimension of position and momentum operators, the deformed Heisenberg algebra can be represented as
In general, for two symmetric operators and , we have
So the generalized uncertainty principle can be deduced as
While in ordinary quantum mechanics can be made arbitrarily small by letting grows correspondingly, this is no longer the case if (3) holds. If for decreasing , increases, the new term on the right hand side of (3) will eventually grow faster than the left hand side. Hence can no longer be made arbitrarily small [16, 18]. To obtain this minimal uncertainty, we saturate inequality in (3) and solve the resulting equation for ,
The reality of solutions requires positivity of the term in square root, leading to
This being the smallest uncertainty in position measurement leads nontrivially to the existence of a minimal measurable length. In fact, a key characteristic of quantum theory is the emergence of uncertainties, and one might expect that the distance observable would also be affected by uncertainties. Actually, various heuristic arguments suggest that for such a distance observable the uncertainties might be more pervasive; in ordinary quantum theory one is still able to measure sharply any given observable, though at the cost of renouncing all information on a conjugate observable, but it appears plausible that a quantum-gravity distance observable would be affected by irreducible uncertainties. Quantum gravity suggests that in the Planck-scale regime there should be some absolute limitations on the measurability of distances. This restricted resolution of spacetime structure is referred to as spacetime fuzziness “foamy or fractal spacetime” . This picture replaces point-like structures with a smeared, distributional structure. The effect of smearing could be mathematically implemented as a substitution rule; the Dirac-delta function representing position of point-like particles is replaced everywhere with a Gaussian distribution with minimal width of the order of the Planck length.
On the other hand, in the context of the Doubly special relativity (DSR) theories (for review see [22–27]), one can show that a test particle’s momentum cannot be arbitrarily imprecise. In fact, there is an upper bound for momentum fluctuations [28–31]. As a nontrivial assumption, this may lead to a maximal measurable momentum for a test particle (see [20, 32–34]). In this framework, the GUP that predicts both minimal observable length and maximal momentum can be written (with ) as follows [32, 33]:
Since , by setting to obtain absolute minimal length, we find
This GUP contains both a minimal length and a maximal momentum. To see how a maximal momentum arises in this setup (see  for details), we note that with GUP (7) the absolute minimal measurable length is given by . Due to duality of position and momentum operators, it is reasonable to assume . By saturating the inequality in relation (7), we find
This results in
So we obtain
Now by using the value of , we find
The solution of this equation is
So, there is an upper bound on particle’s momentum uncertainty. As a nontrivial assumption, we assume that this maximal uncertainty in particle's momentum is indeed the maximal measurable momentum. This is of the order of Planck momentum.
After introducing minimal length and maximal momentum as natural cutoffs and also introduction of the notion of spacetime fuzziness, we introduce another cutoff, the minimal momentum. It is known that for large distances, where the curvature of space-time becomes important, there is no notion of a plane wave on a general curved spacetime  (see also ). This means that there appears a limit to the precision with which the corresponding momentum can be described. One can express this as a nonzero minimal uncertainty in momentum measurement. In this framework, we define new GUP with minimal length, minimal momentum, and maximal momentum as follows: where is a positive constant. By saturating this inequality and solving the resulting equation, we obtain as
So, the minimum uncertainty for position measurement is given by and minimum uncertainty for momentum measurement is
Now by setting the value of in (14), we attain the maximum uncertainty for momentum measurement as follows:
Thus we have shown that the uncertainty relation (13) encodes properly the existence of natural cutoffs.
2. Hilbert Space Representation with Natural Cutoffs
There are distinct approaches toward quantum gravity that all imply the presence of an observable minimal length belonging to the Planck length category. The minimal length makes serious problems in representation in the coordinate space of quantum mechanics. In case the minimal momentum is not taken into consideration, the representation of the momentum space would be sufficient to formulate the Hilbert space. But, whenever the minimal momentum is accounted for, the representation of the momentum space would lose the credibility it has in the standard quantum mechanics. Hence, modifications in Hilbert space representation with the help of natural cutoffs seem to be necessary. So far, the formulation of the Hilbert space has been done separately based on the minimal length , minimal length and minimal momentum , and minimal length and maximal momentum . The present paper aims to simultaneously treat the Hilbert space in the presence of all natural cutoffs, that is, the minimal length, the minimal momentum, and the maximal momentum, and the consequences are to be reviewed as well. This is going to be done through a new, generalized Hilbert space called the Bargmann-Fock space that includes -algebraic variables.
2.1. Heisenberg Algebra with Natural Cutoffs
Hinrichsen and Kempf in  defined the associative Heisenberg algebra with minimal length and minimal momentum addressed by the following commutation relation with :
Here we add a new ingredient: the existence of a maximal measurable momentum. With this extra ingredient, the associative Heisenberg algebra in the presence of all natural cutoffs contains the following commutation relation:
We are going to use the platform of  in our setup. For this purpose, we transform (19) in a manner that is comparable with (18) (or equation of ). In this viewpoint, (19) can be rewritten as follows:
The importance of this commutation relation lies in the fact that it contains all natural cutoffs. In fact, both UV and IR sectors of the underlying quantum theory are addressed properly in this commutation relation. By comparing (18) and (20), we see that these two relations are related through the transformations
So, the mathematical framework of Hinrichsen-Kempf pioneer work  can be applied to the present problem. We note that when one considers both minimal length and minimal momentum hypothesis, representation of position and momentum spaces breaks down. In this situation, there is no continuous Hilbert space representation and we have to build a generalized Hilbert space representation as follows.
2.2. Heisenberg Algebra in Bargmann-Fock Space
Existence of natural cutoffs requires a generalized Heisenberg algebra in Fock space developed in the context of quantum groups. In this framework, due to the fundamental structure of spacetime, all operators are anticommutative. In Bargmann-Fock space the following relations for and hold: where the constants , carry units of length and momentum and are related by and is the deformation parameter. Based on the deformed algebra in Fock space, we obtain the commutation relation with minimal length, minimal momentum, and maximal momentum, as follows: or through (21) and (22),
Note that these transform to ordinary quantum mechanics results where we set . The corresponding uncertainty relation is as follows: or simply as
Based on this uncertainty relation, there are uncertainties in position and momentum as follows:
Note that can be obtained through the procedure adopted in Section 1.
3. Some Analysis on Maximal Localization States
Now we consider the states of maximal localization around a position and we set the expectation value of the momentum to be zero
We note that we used (21) to arrive at this relation, but the terms have canceled each other in and . Now by setting in the above equation, we have
3.1. Maximal Localization States in Bargmann-Fock Space
We consider the states to be maximally localized around a position . Following , to calculate these states, we expand the based on -Hermite polynomials in Fock basis where is the normalization factor defined as follows:
Now by setting momentum and position operators in (32), we obtain a new relation in the Fock representation as where we have used the following relations: to define raising and lowering operators in Fock space. Such that and obey generalized commutation relations 
Comparing this result with the corresponding relation obtained by Hinrichsen and Kempf in  in the presence of minimal length and minimal momentum, we see that incorporation of the maximal momentum results in the extra term were given as in the recursion relation. This is the main deference of our setup with Hinrichsen-Kempf framework. The coefficients are given by -Hermit polynomials  as follows:
Having , the maximally localized states are given by (33). This gives the complete structure of generalized Hilbert space in the presence of all natural cutoffs.
Representation of states in quantum mechanics, in the presence of quantum gravity induced natural cutoffs, is an important issue. So far this issue has been studied separately in the presence of minimal length , minimal length and minimal momentum , and minimal length and maximal momentum . In , the authors have considered the Hilbert space representation in the presence of all natural cutoffs, simultaneously. Here we complete this study by further investigation of the scenario and its consequences with more details. This has been done through introduction of a generalized Hilbert space and the Bargmann-Fock space that includes -deformed algebraic variables.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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