Exotic fermionic fields and minimal length

We investigate the effective Dirac equation, corrected by merging two scenarios that are expected to emerge towards the quantum gravity scale. Namely, the existence of a minimal length, implemented by the generalized uncertainty principle, and exotic spinors, associated with any non-trivial topology equipping the spacetime manifold. We show that the free fermionic dynamical equations, within the context of a minimal length, just allow trivial solutions, a feature that is not shared by the dynamical equations for exotic spinors. In fact, in this merging setup, the exoticity is shown to prevent the Dirac operator to be injective, allowing the existence of non-trivial solutions.

We investigate the effective Dirac equation, corrected by merging two scenarios that are expected to emerge towards the quantum gravity scale. Namely, the existence of a minimal length, implemented by the generalized uncertainty principle, and exotic spinors, associated with any non-trivial topology equipping the spacetime manifold. We show that the free fermionic dynamical equations, within the context of a minimal length, just allow trivial solutions, a feature that is not shared by the dynamical equations for exotic spinors. In fact, in this merging setup, the exoticity is shown to prevent the Dirac operator to be injective, allowing the existence of non-trivial solutions.

I. INTRODUCTION
Over the last decades, high energy physics has allowed us to experimentally look closer to the very structure of matter. It naturally raises concerns on the limits of such an approach. Rather this limit exists or not, fundamental concepts regarding the underlying spacetime geometry will be revisited, at high energy scales [1,2]. At least at energies as large as the Planck scale, quantum gravity effects are expected to set in. The non-renormalizability of quantum gravity consists of a major problem that physicists have been trying to overcome for decades. However, it could be effectively circumvented, by suggesting that gravity should lead to an ultraviolet cutoff, leading to a minimal observable length. The existence of a minimal length scale seems to be a model-independent feature of quantum gravity [3], from string theory [4][5][6] to loop quantum gravity [7], and quantum black holes [8][9][10][11][12][13]. Effectively, the Heisenberg uncertainty principle (HUP) is corrected by a minimal length, giving rise to the so-called Generalized Uncertainty Principle (GUP). See Refs. [14][15][16][17] for a comprehensive review. In particular, in the context of the Dirac equation, phenomenological bounds were explored in Ref. [18]. Formally, the minimal length can be implemented in quantum mechanics, through small quadratic corrections to the canonical commutation relations [19][20][21][22]. The relationship between the commutation relations and minimal length is not biunivocal, nevertheless, as investigated in Ref. [23]. The compatibility with relativistic covariance was accomplished in Ref. [24], with the so-called Quesne-Tkachuk algebra, leading to a generalization of the Heisenberg algebra. Some high energy physics phenomena were investigated in the scope of the Quesne-Tkachuk algebra [25,26], whereas generalizations of such an algebra were proposed in [27]. In Ref. [28], the relationship between a generalized algebra, induced by a minimal length, and higher derivative theories was established. Interacting fermionic fields were also studied in the context of a minimal length scale. Reciprocal effects of massive fermions were investigated in Ref. [29], while black hole tunnelling was treated in [30]. Interestingly, the investigation of free (classical) fermionic fields whose dynamics is dictated by the deformed Dirac operator, as shown in the next section, leads to a particularly odd result. The situation may be framed as follows: denoting by E the space encompassing solutions of the Dirac operator, D, it is clear that ker(D) = E. In the presence of a minimal length, as we will see, within the usual first-order approximation and considering massive particles, the Dirac operator becomes injective, collapsing its kernel into the null spinor. In other words, only the trivial solution is allowed.
A different perspective is reached by studying exotic spinors, along with the minimal length framework. As a matter of fact, provided that a given manifold has a non-trivial topology, more than one spinor bundle is then allowed. Such bundles belong to different equivalence classes, giving rise to the so-called exotic spin structures, which accommodate exotic spinors [31]. Inequivalent spin structures are felt, in a broad brush, through different spin connections [32,33], inferring a different dynamics, i. e., these spinors are annihilated by a modified (exotic) Dirac operator. Moreover, going towards Planck scale, besides the appearance of a minimal length, one also finds support for the existence of exotic spinors, as non-trivial topologies are also expected to appear in such regime [34,35]. Merging the two scenarios, we will show that the odd feature about free fermions, within dynamical equations corrected by minimal length, is solved in the scope of exotic spinors, being the doubly corrected Dirac operator not injective anymore. For completeness, and anticipating one of the results of Section III.A, we also study particular cases for which the trivial solution is also the only possibility, in the merging scenario. This paper is organized as follows: the next section is reserved to introduce the problem of free fermionic fields, within the minimal length scope. Section III is devoted to study exotic spinors, as well as the merging between exotic spinors and the minimal length setup. The formal aspects are analyzed and a correction in the topological term is proposed, after what the dynamical scenario is explored. In the final section, we summarize and comment our relevant results.

II. INTRODUCTORY REMARKS ON MINIMAL LENGTH AND FREE FERMIONIC SYSTEMS
The generalized commutation relation between X µ and P µ , engendered by a minimal length [24] in the momentum space, reads 1 where the (square of the) minimal length is encoded into β. It is possible to show [24,28] that a quite suitable point in the parameter space may be chosen so that Eq. (1) holds in the configuration space, for P µ = (1 − βp ρ p ρ )p µ .
Hence a correction in the partial derivative operator reads With Eq. (2), the investigation of the profile of free spinors, in a base manifold endowed with minimal length, follows straightforwardly. The minimal length derivative prescription simply yields Inserting the operator [i(1 + β )/ ∂ + m1 4×4 ] into Eq. (3) from the left, and keeping the first order approximation in β, one has Replacing back ψ, coming from (4), into Eq. (3), implies that To make explicit the degenerate states corresponding to the two spin projections, the usual approach suggests the analysis of Eq. (5) in the rest frame formalism, in the momentum space (the Dirac representation of the gamma matrices is particularly clear in this regard), and proceed with a spinorial boost. Writing ψ = e ∓ipx w( p), where w( p) is a spinor, it is straightforward to see that, in the rest frame, Eq. (5) leads to for κ ≡ ±(1 − βm 2 ). Since κ = ±1 for β = 0, the unique solution consists of the trivial one, despite the matrix in Eq. (6) is not invertible. One could argue that the rest frame would not be attainable, however, then we are forced to conclude that the dynamical equation is insensitive to the minimal length (see, for instance, Eq. (5) for m = 0). For this reason, henceforth, we will no longer deal with the non-massive case. Following a different approach, and starting with a different corrected Dirac equation, in Ref. [36] the authors arrive at a similar conclusion. The existence of a minimal length does not allow the existence of free fermions solutions.
Returning to the analysis performed in the Introduction, let ψ be a section of the P Spin (1,3) × C 4 spin-Clifford bundle. Hence, denoting by E ⊂ P Spin (1,3) × C 4 the subset comprising solutions of the Dirac operator, D, we see that in the context of minimal length, the kernel of D is given by 0 ∈ E, and the operator becomes injective. It is difficult to encompass such behavior in a physical interpretation. On the one hand, it seems that the energy increment, to the point in which the minimal length becomes dominant, would implement a mapping ker(D) = E = {0}. However, as the energy increases, the second-order terms start to be relevant, at least comparing to the first-order terms, as these terms contrast to the usual terms. In this case, Eqs. (3 -5) would be modified accordingly, preventing ker(D) to collapse into the subset {0}. The peculiar point is that usual solutions would need second-order terms to survive, whereas non-trivial exotic fermionic solutions, as we will see, do exist in the first-order approximation. To make our claims clear, as we are looking for low energy effects, corrections of the dynamical equations will be restricted to the leading order.

III. EXOTIC SPINOR STRUCTURES AND THE MINIMAL LENGTH
We start with a discussion on pertinent details, concerning the foundations and construction of exotic spinors, which are particularly important for our analysis.
A spin structure on a 4-dimensional Lorentzian spacetime manifold M requires an orthogonal frame bundle P Spin(1,3) πs − − → M and a double cover s : P Spin(1,3) → P SO(1,3) such that π s = π • s, for both π : P SO(1,3) → M and π s denoting the bundle projection operator onto the M manifold. When the first cohomology group is not trivial, H 1 (M, Z 2 ) = {0}, exotic spin structures (P Spin(1,3) ,s), inequivalent to the usual ones, are then admitted to exist. Hence, one can construct exotic spinors as sections of the spinor bundle, associated to the principal bundleP Spin (1,3) .
In order to evince the difference between the usual and the exotic spin structures, and making explicit the nontrivial elements of H 1 (M, Z 2 ), let one remembers that two spin structures P := (P Spin(1,3) , s) andP := (P Spin(1,3) ,s) are equivalent as long as a Spin(1, 3)-equivariant mapping q : P →P , compatible to s ands, exist. Namely, this equivalence holds if the diagram 3) commutes. The group homomorphism σ : such that a ij • a jk = a ik on U i ∩ U j ∩ U k and a jj = id. For a spin structure (P Spin(1,3) , s) on M , a system of transition functions has the properties [32,38] σ In this way, two spin structures (P Spin(1,3) , s) and (P Spin(1,3) ,s) are, respectively, portrayed by the mappings b ij and with c ij • c jk = c ik . These cocycles c ij are the non-trivial elements of the first cohomology group H 1 (M, Z 2 ). This construction is shown to define an one-to-one correspondence, between inequivalent spin structures and H 1 (M, Z 2 ) [42]. This shows that the very existence of the cocycles c ij is necessary to have exotic spinors on M . Yet, notice that c ij requires the existence of an open covering for M . Now, we will address the exotic spinors construction on a manifold, M , endowed with both non-trivial topology and minimal length. We have already seen that to establish exotic spinors on a manifold, one needs an open cover ∪ i∈I U i forM . On the other hand, in the current case it is necessary to proceed with some caution, due to the minimal length. More precisely, we need to have |U i ∩ U j | > L, for all i, j, where L accounts for the minimal length. SinceM is a Hausdorff paracompact topological space, then every open cover has an open refinement that is locally finite, which ensures the possibility of such open sets to exist. In other words, as the arbitrary union of open sets is also an open set, it is always possible to construct a collection W of open sets, W k , implemented by an arbitrary union of U i as W k = ∪ i U i , for all k, such that W = ∪ k W k . The W k are chosen such that |W k | > L, for all k, and |W k ∩ W l | > L, for all k, l, covering the manifold. Hence, the construction of the transition functions a ij : Here, σ is a group homomorphism between Spin(1, 3) = Spin(1, 3) = Spin(1, 3) and SO (1, 3).
As a conclusion of this construction, it is always possible to find cocycles c ij : h ij (x)c ij , which are non-trivial elements of H 1 ( M , Z 2 ), allowing the existence of exotic spinors on M .
We will study, in the next section, the behavior of exotic spinors,ψ, along with the minimal length corrections in their dynamics. Before doing so, we call attention to the fact that the correction due to the non-trivial topology is implemented when one replaces the Dirac operator as Therefore this topological correction encompasses a derivative term which could, in principle, also be corrected by the minimal length. Let us look at this issue more closely. Following Refs. [33,[37][38][39][40], one assumes a set of scalar fields χ i : U i → C such that χ i (x) ∈ U(1), and where ω denotes a faithful irreducible representation from the Minkowski spacetime Clifford algebra to the space M (4, C) of 4 × 4 matrices with complex entries. If the case of a 2-torsionless second cohomology group H 2 (M, Z 2 ), the χ i (x) scalar fields always exist, and χ 2 i (x) = χ 2 j (x), for x ∈ U i ∩ U j [37,38,40,41]. Consequently, the local functions χ i define a unique unimodular scalar field χ : M → C such that, for all x ∈ U i , the equality χ(x) = χ 2 i (x) holds. Besides, the χ i scalar field form a local root system for χ [38], and are the generators of the cocycles c ij , which are the non-trivial elements of H 1 (M, Z 2 ). As we have presented, such cocycles do exist and, equivalently, exotic spinors exist in this new spacetime manifold, so it seems natural to think that their generators carry information on the minimal length that characterizes M . Yet, the unimodular scalar fields may be arranged as χ = e iθ(x) , where θ : M → R [38]. Usually, without a minimal length, the θ(x) represent a strongly continuous function that reflects the non-trivial topology. Within the minimal length context, however, |χ(x ′ ) − χ(x)| = 2 − 2 cos(θ(x ′ ) − θ(x)), and one is certainly forbidden to make x ′ infinitely close of x. Equialently, |χ(x ′ ) − χ(x)| ∼ O(β), where β has L 2 order of magnitude, and we are forced to conclude that the derivative operator should also be corrected. Nevertheless, the formal and exact functional form of such a correction is a quite difficult task. Up to the best of our knowledge, there is no completely satisfactory answer to this issue, culminating into a manageable operator. Here we will adopt, instead, an effective approach, motivating the corrected derivative operator. Our proposal goes as follows: since the minimal length is responsible for this correction in the derivative operator, it must be small compared to the identity, i. e.
∂ → (1 4×4 +βO)∂, where henceβ ∼ β, and O is an operator acting on sections of P Spin(1,3) × C 4 . Now, nothing ensures that O is a differential operator. If it is not, one would have to take special care ensuring covariance of the dynamical equation. Being O a derivative operator, however, covariance imposes that either O = a 0 γ µ ∂ µ or it contains only even powers in the derivative. In this last case, its more complete form would be given by O = a 1 + a 2 2 + · · · , where a 0 , a 1 , a 2 , · · · are arbitrary parameters with suitable dimensions. Anticipating Sec. III.A, in the first order approximation there is no substantial difference in the net result. We will present our calculations with O being taken as a derivative operator of the form O = , with a 1 already absorbed intoβ, and just mention the other case by passing. This motivated functional form of correction is, then, quite similar to the minimal length correction in the derivative operator obtained, however, by other arguments.
As a parenthetical remark, we notice that the topological term is commonly accepted to be eliminated by a gauge redefinition. We will end this section making explicit that this cannot be accomplished [43,44]. (For this remark we will not pay attention to minimal length issues.) The explicit form of Eq. (11), along with a gauge interaction term (obtained for instance by the minimal coupling prescription), is given by (iγ µ ∂ µ + γ µ A µ + iγ µ ∂ µ θ(x) − m1 4×4 )ψ = 0. As a connected group, the U (1) representations are unitary and may be disposed as e iΛ(x) , with real Λ(x). By means of a gauge transformation A µ → A µ − ∂ µ Λ(x) it would be necessary the identification of Λ(x) with iθ(x), to eliminate the topological term. Nevertheless, θ(x) is also real and, therefore, such an elimination cannot be accomplished.

A. Exotic fermions and minimal length
Bearing in mind the discussion performed in the previous section, we will investigate the free fermionic exotic case, with a minimal length correction in the usual derivative and also with the correction in the exotic term. This last correction is not so determinant after all, and its absence would not significantly modify the final result. At the end of the analysis, we also comment about non-derivative corrections.
The exotic spinor dynamical equation, in the context here explored reads It is helpful to remember that the corrected derivative terms are already saturated, that is ∂ µ · = ∂ ρ [∂ ρ (∂ µ ·)]. For further use, we write Eq. (13) in the form Inserting from the left the operator i(1 + β )/ ∂ + i 1 +β / ∂θ(x) + m1 4×4 into Eq. (14), we arrive at the following awkward expression, Notice that, in Eq. (14), ψ is already multiplied by β. Therefore, in the first-order limit, and considering ββ → 0, as discussed, the relevant terms (in advance of what will be used back into Eq. (14)) read 2 The only tricky term is / ∂(/ ∂θ(x)ψ) = θ(x)ψ + γ µ γ ν ∂ ν θ(x)∂ µψ . Using {γ µ , γ ν } = 2η µν we may write Inserting back this expression into Eq. (16) we find Now we are able to return to Eq. (14) inserting there β ψ obtained above. A bit of simple, but tedious, algebra leads to Before going any further, we remark that Eq. In view of the results pointed here and the problem previously described, one would be tempted to think of the minimal length setup as being the natural ambience to find out physical signatures exclusively inplemented to exotic spinors. Nevertheless, keeping in mind that our investigation was carried out focusing on free spinors, such assertion is not as straightforward. What seems plausible to conclude is that physical signatures regarding exotic spinors will be dominant (though not exclusive) in a limit of the interacting terms going to zero. However, this phenomenological bridge needs a deeper investigation. In any case, the spectrum arising from Eq. (19) will furnish the clues for a typical signature. Here, we will restrict ourselves to further investigate Eq. (19).
We have asserted that Eq. (19) admits non-trivial solutions. While it is generally true, we would like to present here a counterexample, making explicit the fact that for some particular cases of (19), non-trivial solutions are not allowed. This is important, since it points to the necessity of some caution in drawing conclusions from this fermionic system. In order to make our point clear here, we notice that it is somewhat usual to think of the additional terms in Eq. (19) as effective interaction terms, reason by which no plane waves are expected. Here we will not adopt this point of view. Instead, we reinforce that the fermionic fields investigated here are completely free, but in a rather unusual (although physically justified) base manifold. If one is willing to accept this approach, then it is plausible to look for plane wave solutions, with possibly a changed energy spectrum. Hence by writing, as before,ψ = e ∓ipxw ( p), Eq. (19) in the rest frame yields In order to make explicit our claim that not every (topologically non-trivial) scenario leads to non-trivial solutions, we investigate the particular case θ = θ(t), i. e., the topological term is only a smooth function of time. In this particular case, Eq. (20) may be recast as (from now on θ(t) ≡ θ) where in the last term the multiplication by the identity is implicit. Defining f ± (θ) = ±m(1 − βm − 3βθ + βθ 2 ) and g ± (θ) = (±2m 2 β + 1)θ + (β − β) ... θ , we may recast Eq. (21) in 2 × 2 blocks as Notice that in the θ = constant limit, Eq. (22) recovers (6), as expected. Moreover, besides assuming θ = constant, if β andβ both vanish, then Eq. (22) turns into the usual Dirac case, recovering the two usual degenerate states for the free particle case. As θ(t) is a real function, reflecting the unusual topology of the spacetime, it is clear that g ± (θ) must be zero, as a necessary (although not sufficient) condition to non-trivial solutions. It is straightforward to see that g ± (θ) = 0 leads to (β − β)θ + (±2m 2 β + 1)θ = c, where c is constant 3 . The θ * function, that solves g ± (θ * ) = 0, is then given by a simple combination of sines and cosines. Nevertheless, by inspecting the terms f ± (θ) ± m, it is readily seen that to have a situation as in the usual Dirac case, it is also necessary to have 4 3θ −θ 2 + m = 0. The solution of this last equation, however, is given by a Weierstrass elliptic function, with no superposition with θ * for all frequences. Hence, it is not possible to satisfy all the necessary conditions to have a non-trivial solution. This ends the counterexample.
We finish by stressing, in the light of the discussion made at the end of Sect. III, that had we investigated the minimal length correction in the topological term as an operator proportional to γ µ ∂ µ , let us say 1 4×4 +β / ∂ ∂ µ , the only difference (keeping the same approximation in vogue) would be to replace the termβ / ∂θ(x) byβ θ(x), in Eq. (19).

IV. CONCLUSIONS
We have shown that the very existence of a minimal length makes the Dirac operator D to be injective, i.e., its kernel turns into {0}, which is not interesting at all for fermionic physics. One can understand, thus, the existence of the minimal length encoded in β as a kind of realization of an operation that squeezes the space of solutions of the Dirac equation, ker(D) = E ⊂ P Spin (1,3) × C 4 , into a space having only the null spinor. It could be a clue of a feature presented in approaches to quantum gravity. However, when the spacetime admits the existence of exotic spinors, the Dirac equation may have non-trivial spinors as solutions. Thus, assuming that both effects are present in high energy phenomena, the existence of free fermions should depend on which one is effectively dominant, when the energy is raising from TeV order towards the Planck scale. It is important to notice, though, that not every scenario merging exotic spinors and a minimal length allows non-trivial solutions of the Dirac equation. As we have shown in the last section, when one takes the particular case of θ = θ(t), for instance, the kernel ofD shrinks to the null spinor again. In other words, it is necessary to have a local exotic term θ (with minimal length), to have non-null spinors.
The discussions above indicate an unexpected relationship between the topologies of the spacetime and the space of solutions of the Dirac equation, when contributions of a minimal length scale are relevant. This helps to understand which are the feasible physical scenarios for non-null fermionic fields to exist in a spacetime with minimal length. Besides, not only the Dirac equation in other contexts [45,46], but other first order equations of motion can be explored, driving flagpole and flag-dipole spinor fields [47][48][49], in the exotic context [50].
Further studies on the relation between these spaces are being carried on by the authors, aiming to expand our understanding about exotic spinors dynamics.