Pair production of Dirac particles in a d+1-dimensional noncommutative space-time

This work addresses the computation of the propability of fermionic particle pair production in $(d+1)-$ dimensional noncommutative Moyal space. Using the Seiberg-Witten maps that establish relations between noncommutative and commutative field variables, to first order in the noncommutative parameter $\theta$, we derive the probability density of vacuum-vacuum pair production of Dirac particles. The cases of constant electromagnetic, alternating time-dependent and space-dependent electric fields are considered and discussed.


I. INTRODUCTION
Noncommutative field theory (NCFT), arising from noncommutative (NC) geometry, has been the subject of intense studies, owing to its importance in the description of quantum gravity phenomenas. More precisely, the concepts of noncommutativity in fundamental physics has deep motivations originated from the fundamental properties of the Snyder space-time [1]. Further, the results by Connes, Woronowicz and Drinfeld [2][3][4] provided a clear definition of NC geometry, thus bringing a new stimulus in this area. The NC geometry arises as a possible scenario for the short-distance behaviour of physical theories (i. e. the Planck length scale λ p = G c 3 ≈ 1, 6 · 10 −35 meters), see [5][6][7] and references therein. This fundamental unit of length marks the scale of energies and distances at which the non-locality of interactions has to appear and a notion of continuous space-time becomes meaningless [5,6,8]. One of the important implications of noncommutativity is the Lorentz violation symmetry in more than two dimensional space-time [9][10][11], which, in part, modifies the dispersion relations [12]. It leds to new developments in quantum electrodynamics (QED) and Yang-Mills (YM) theories in the NC variable function versions [14,15]. The same observation appears in the framework of string theory [16,22]. Also, the quantum Hall effect well illustrates the NC quantum mechanics of space-time [23,24] (and references therein).
In this work, we use a NC star product obtained by replacing the ordinary product of functions by the Moyal star product as follows: where f , g ∈ C ∞ (R D ), m(f ⊗ g) = f · g; θ µν stands for a skew-symmetric tensor characterizing the NC behaviour of the space-time, and has the Planck's length square dimension, i.e.
[θ] ≡ [λ 2 p ]. The star product (1) satisfies the useful integral relation It provides the following commutation relation between the coordinate functions: For convenience, we choose the tensor (θ µν ) in the following form: The relation (4) means that the time does not commute with NC spatial coordinates. Recall that two main problems arise when one tries to implement the electromagnetism in a NC geometry: the loss of causality due to the appearance of derivative couplings in the Lagrangian density and, more fundamentally, the violation of Lorentz invariance exhibited by plane wave solutions [12,13].
Like in ordinary quantum mechanics, the NC coordinates satisfy the coordinate-coordinate version of the Heisenberg uncertainty relation, namely ∆x µ ∆x ν ≥ θ, and then make the space-time a quantum space. This idea leads to the concept of quantum gravity, since quantizing space-time leads to quantizing gravity. Apart from the overall results about QED and YM theory in NC space-time, it turns out to be important to understand how noncommutativity modifies the probability of pair production of fermionic particles. This is the task we deal with in this work.
A pair production refers to the creation of an elementary particle and its antiparticle, usually when a neutral boson interacts with a nucleus or another boson. Nevertheless a static electric field in an empty space can create electron-positron pairs. This effect, called the Schwinger effect [25], is currently on the verge of being experimentally verified. Recently, the vacuum-vacuum transition amplitude and its probability density were computed in four, three and two dimensional space-time within constant and alternating electromagnetic (EM) fields [26][27][28]31]. The related questions have been discussed and gained considerable attention in the researchers community.
In this work, we provide the NC version of pair production of Dirac particles. Specially, we derive the exact expression for the probability density of particle production by an external field. This establishes a relation with important analytical results previously obtained in the ordinary space-time, spread in the literature [25][26][27][28]31].
The paper is organized as follows. In section (II), we quickly review the Seiberg-Witten maps giving a relation between NC field variables and commutative ones [16,19,20]. Here also we expose the main result about gauge theory in NC space, that allows us to write the NC Lagrangian density of the Dirac particle (oupling to EM field) with the commutative field variables. In section (III) we compute the probability density of pair production of a Dirac particle in constant EM fields. In Section (IV) we give the discussions and the comments of our result. This section also contains a similar analysis in the case of an alternating (EM) field. Section (V) is devoted to concluding remarks. Appendices (A) and (B) are enriched by the proofs of key theorems set in the main part of this paper.

II. NC GAUGE THEORY AND SEYBERG-WITTEN MAPS
Like in an ordinary space-time, a gauge theory can be defined on a NC space-time [17]. In the sequel, the NC variables are denoted with a "hat" notation. Let A θ be a Moyal algebra of functions andX ∈ A θ be the covariant coordinate expressed in terms of gauge potentialÂ ∈ A θ as: For an arbitrary functionψ ∈ A θ , the infinitesimal gauge transformation with parameterΛ ∈ A θ isδψ = iΛ ψ . The infinitesimal variation of the gauge potential can be written aŝ Also the NC Faraday tensor is given bŷ Its infinitesimal variation iŝ Besides, the functional action for a Dirac particle on NC space-time can be defined as follows: In this expressionψ andψ are the Dirac spinor and its associated Hermitian conjugate, respectively. The γ's are the Dirac matrices which satisfy the Clifford algebra: {γ µ , γ ν } = 2η µν , and are given explicitly in terms of Pauli matrices σ i , i = 1, 2, 3, by: The covariant derivativeD µ is expressed as: We choose = c = 1 and take the charge of particle equal to the unit value, i.e. q e = 1. The Lagrangian L(ψ, ψ) describes the propagation of the massive fermion (electron in this case) and their interaction with photons via the covariant derivativê D µ . In this work, we treat in detail the case when the dimension of the space-time is equal to D = 3+1.
The results for the cases where D = 2+1, D = 1+1, and, more generally, D = d + 1, computed in a similar way, are given.
Using the Seiberg-Witten maps [16,19,20], we can write, at the first order of perturbation in θ, the NC field variables as function of commutative variables as follows:ψ Now by performing the path integral over the background fields ψ andψ, the vacuum-vacuum transition amplitude Z(A) is afforded by the expression: in which the normalization constant N is chosen such that Z(0) = 1. Note that B(θ, 0, 1, 1) = 0. Let M := iγ µ D µ − m + B(θ, A, 1, 1) + i . Then, we get a simpler form: Provided with the above quantity, we compute the probability density amplitude |Z(A)| 2 for various electromagnetic fields.

III. TRANSITION AMPLITUDE IN THE CASE OF A CONSTANT EXTERNAL EM FIELD
In this section, we consider the EM field, defined in x direction as B = Be x and E = Ee x , E > 0 and B ≥ 0. The position and momentum operators X µ = (X 0 , X 1 , X 2 , X 3 ) =: (X 0 , X, Y, Z) and P µ = i∂ µ = (P 0 , P 1 , P 2 , P 3 ) satisfy the commutation relation: The can be expressed as: with A µ = (−EX, 0, 0, BY ). Then, B(θ, A, 1, 1) is obtained as: Using the charge conjugation matrix C = iγ 2 γ 0 , the identity Cγ µ C −1 = −γ t µ , and taking into account the fact that the trace of an operator is invariant under a matrix transposition lead to The probability density is defined by the module of Z(A) as with The conjugate of B(θ, A, 1, 1), denoted by B t (θ, A, 1, 1), can be then written as: At this point it would be worth using the identity to get where the operator X (θ) should be Hermitian. We use the following commutation relations [X n , P 1 ] = −niX n−1 , [P n 1 , X] = niP n−1 , (30) also valid when one replaces X by Y and P 1 by P 2 . For an arbitrary operator A, we can define the associated Hermitian operator denoted by A H as In the rest of this paper, the H symbol indexing any operator A, e.g. A H , refers to the Hermitian operator associated with A. We then have the following: The Hermitian operator associated with X (θ), denoted X H (θ), is given by Further, Proof. Taking into account the fact that the trace is invariant under matrix transposition, and using the relation (31), the operator X H (θ) takes this form. Now we focus on the computation of the following quantity: where for σ µν = i 2 [γ µ , γ ν ], we use the relation and choose the 4-vectors |x = |x µ such that X µ |x = x µ |x . In the momentum representation, we get a similar relation for P µ |k = k µ |k , and obtain To achieve our goal, we use the Baker-Campbell-Hausdorff formula given by where the constants C n are given by the Zassenhaus formula [34,35]: with Explicitly, we get where the exponents of higher order in t are likewise nested. Then, take into account the first approximation of θ in the expansion of all quantities to arrive at the expression: e is[(P +A) 2 +X (θ)] = e is(P +A) 2 e isX (θ) e T (θ) = e is(P +A) 2 1 where, for t = is, U = (P + A) 2 , and V = X H (θ), we have The expectation value of the operator e is[(P +A) 2 +X H (θ)] is then evaluated as where J (θ) = 1 + isX H (θ) + T H (θ) . Now after expanding U as we can easily remark that U = U † . As it is the welcome, fortunately, we get the following statement. Proof. The proof of this proposition is simply obtained by using (31) and (32). Finally the quantity O is reduced to and We then come to the following result: Theorem 1. Let θ =: ℘ · θ 0 where ℘ is a dimensionless quantity which is bounded by two numbers a 1 , and a 2 and such that θ 0 << 1. The mass dimension of θ 0 is obviously θ 0 ≡ [M −2 ]. Let M ⊂ R 2 be the compact subset of R 2 in which the following integral is convergent.
The trace of the expectation value O is given by: where f (E, B) a being a positive function given by Proof. The proof of this theorem is given in Appendix (A). Remark that the quantity σ(θ 0 , E, B) leeds to the divergence in the limit where B = 0 and in the limit where θ 0 = 0. This expression do not contribute to the physical solution and then the trace of O is reduced to tr O = 1 − ℘ tr O c .

Theorem 2. The vacuum-vacuum transition probability is |Z(
whose real part, denoted by e ω(x) = ω+ω * 2 , is given by Proof. The proof of this statement is given in the Appendix (B).

IV. DISCUSSION OF THE RESULTS
In this section we discuss the results of the theorems (1) and (2).
(1) Let (56) We get simply and therefore the corresponding serie, i.e. ∞ k=1 U k is obviously convergent. Note that there exist two positive constants a 1 and a 2 such that for a 1 ≤ ℘ ≤ a 2 , θ 0 << 1. Therefore there exist the bound on e ω(x) in which the solution (55) is well defined. (58) Then this expression correspond to the commutative limit derived by Q-G. Lin (see [26]) and given by: and we conclude that the noncommutativity increases the amplitude |Z(A)| 2 . This shows the importance of noncommutativity at high energy regime in which creation of particle is manifest. The same conclusion can be made in Ref [29] is which, pair production by a constant external field on NC space is also considered.
(3) Furthermore, the previous investigation [31], devoted to such EM field as E = E cos(t)e x and B = Be x , has been also considered here in the framework of the NCFT. Indeed, following step by step the approach displayed above in this work, after some algebra, we found that the probability density of the pair production of Dirac particle in NC spacetime with alternating EM field is given by from which, in the limit where the NC parameter θ = 0, we recover our formula [31]. A more compact form of the relation (61) in the case of arbitrary D = d + 1-dimensions can be also given in the same way. We get for B = 0 the following results Also, by replacing the vector field A µ by A µ = A µ + f µ , where A µ = (−EX, 0, 0, 0) and f µ = (−E sin(x), 0, 0, 0) corresponding to the plane wave function, we get All these results use the computations performed in the Appendices (A) and (B). In the limit where θ = 0, the relations (62) and (63) lead to the results of ref [31].

V. CONCLUDING REMARKS
In this paper, we have considered NC theory of fermionic field interacting with its corresponding boson. We have used the Seiberg-Witten expansion describing the relation between the NC and commutative variables, to compute the probability density of pair production of NC fermions. We have shown that, in the limit where the NC parameter θ = 0, we recover the result of Qiong-Gui Lin [26]. Our study has highlighted that the noncommutativity of space-time increases the density ω of the probability of pair creation of the fermion particle. Our results can be easily extended to take into account the cases where D = 2 + 1 and D = 1 + 1. Perimeter Institute for Theoretical Physics (Water We give the proof of Theorem (1). Consider the two quantities: θ 0 and ℘, and decompose the NC parameter θ as Then, re-express O nc (θ) as follows: where The expression (A3) is subdivided into three contributions, namely and Now note that Let us consider first the quantity G 1 , and the integral relation We get, respectively, (A13) Using the properties of the gamma matrices and the results of [26] and [31] we get the following: tr e is 2 σ µν Fµν = 4 cosh(Es) cos(Bs) (A14) We can evaluate the trace of relevant quantities in equation (A8). Before getting (50), we need the trace of dk exp isθ 0 k|G(E, B, θ)|k , i.e.
This is obtained by using the Gaussian integral. Using (A10), (A11), (A12) and (A13) we get: In this exponent, the term that contributes to the integration respect to x is Then the result after integration respect to x is . (A20) Taking into account (A20), the relation that contributes to the integration respect to y is After integration, we get: . (A22) Let f (E, B) is the positive defined function given by the following: we come to dx dy Note that the coordinates (t, z) ∈ R 2 need to be compactified in order to get a finite integral. We choose the subset M ∈ R 2 such that dtdz = b, (for example M must be a unit disk). We get the following expression: dt dz dx dy dk isθ 0 k|G(E, B, θ)|k Now let k(z) = e izm 2 z 4 coth(Ez) cot(Bz), z ∈ C. The integrand has singularities at point z = 0 (poles of order 6), at z = inπ E and z = nπ B (simple poles). Using the same argument like (B4), (B5) and (B6) we get, respectively, (B11) Now we come to the interpretation of the equations (B9), (B10) and (B11).
• The equations (B9) and (B11) lead to a complex probability density and then cannot be taken into account.
• As we have seen in (B5), the equation (B10) leads to a singularity at the limit B → 0. This pointless expression also will not contribute to e (ω). The same analysis can be provided, using the holomorphic function coth(Ez) cot(Bz) z 4 exp iz(m 2 + θ 0 G 0 ) .
Taking into account the above two comments, we conclude that the function σ(θ 0 , E, B) not contribute to the probality density, and therefore the positif integer n disappear in the equation (55). Finally, by taking into account only the relation (B8), the Theorem (2) is proved.