Formulation of Electrodynamics with an External Source in the Presence of a Minimal Measurable Length

Abstract

In a series of papers, Quesne and Tkachuk (2006) presented a D + 1-dimensional (β, β′)-two-parameter Lorentz-covariant deformed algebra which leads to a nonzero minimal measurable length. In this paper, the Lagrangian formulation of electrodynamics in a 3 + 1-dimensional spacetime described by Quesne-Tkachuk algebra is studied in the special case of β′ = 2β up to the first order over the deformation parameter β. It is demonstrated that at the classical level there is a similarity between electrodynamics in the presence of a minimal measurable length (generalized electrodynamics) and Lee-Wick electrodynamics. We obtain the free space solutions of the inhomogeneous Maxwell’s equations in the presence of a minimal length. These solutions describe two vector particles (a massless vector particle and a massive vector particle). We estimate two different upper bounds on the isotropic minimal length. The first upper bound is near to the electroweak length scale (ℓ_electroweak  ~ 10⁻¹⁸ m), while the second one is near to the length scale for the strong interactions (ℓ_strong  ~ 10⁻¹⁵ m). The relationship between the Gaete-Spallucci nonlocal electrodynamics (2012) and electrodynamics with a minimal length is investigated.

1. Introduction

The unification between the general theory of relativity and the standard model of particle physics is one of the most important problems in theoretical physics [1]. This unification predicts the existence of a minimal measurable length on the order of the Planck length. Also, recent studies in perturbative string theory and loop quantum gravity suggest that there is a minimal length scale in nature [2].

Today we know that the existence of a minimal measurable length leads to an extended uncertainty principle. This extended uncertainty principle can be written as ħΔ𝑋Δ𝑃≥21+𝑎₁𝑙_𝑃ħ²(Δ𝑃)²+𝑎₂𝑙_𝑃ħ⁴(Δ𝑃)⁴+⋯, (1) where 𝑙_𝑃 is the Planck length and 𝑎_𝑖,∀𝑖∈{1,2,…}, are positive numerical constants [3, 4]. If we keep only the first two terms on the right-hand side of (1), we will obtain the usual generalized uncertainty principle (GUP) as follows: ħΔ𝑋Δ𝑃≥21+𝑎₁𝑙_𝑃ħ²(Δ𝑃)². (2) It is obvious that in (2), Δ𝑋 is always greater than (Δ𝑋)_min=√𝑎₁𝑙_𝑃. Many physicists believe that reformulation of quantum field theory in the presence of a minimal measurable length leads to a divergence free quantum field theory [5–7]. In the recent years, reformulations of quantum mechanics, gravity, and quantum field theory in the presence of a minimal measurable length have been studied extensively [4–18]. The first attempt to construct the electromagnetic field in quantized spacetime was made by Snyder [19]. In a previous work [14], we studied formulation of an electrostatic field with a charge density in the presence of a minimal length based on the Kempf algebra. In the present work, we study formulation of electrodynamics with an external source in the presence of a minimal measurable length based on the Quesne-Tkachuk algebra. The organization of our paper is as follows. In Section 2, the 𝐷+1-dimensional (𝛽,𝛽^′)-two-parameter Lorentz-covariant deformed algebra introduced by Quesne and Tkachuk is studied and it is shown that the Quesne-Tkachuk algebra leads to a minimal measurable length [20, 21]. In Section 3, the Lagrangian formulation of electrodynamics with an external source in a 3+1-dimensional spacetime described by Quesne-Tkachuk algebra is introduced in the special case of 𝛽^′=2𝛽, in which the position operators commute to the first order in 𝛽. We show that at the classical level there is a similarity between electrodynamics in the presence of a minimal measurable length and Lee-Wick electrodynamics. In Section 4, the free space solutions of the inhomogeneous Maxwell’s equations in the presence of a minimal measurable length are obtained. These solutions describe two different particles, a massless vector particle and a massive vector particle. In Section 5, we obtain two different upper bounds on the isotropic minimal length. One of these upper bounds on the isotropic minimal length is near to the electroweak length scale (ℓ_electroweak∼10⁻¹⁸m). The second upper bound on the isotropic minimal length is near to the length scale for the strong interactions (ℓ_strong∼10⁻¹⁵m). In Section 6, we investigate the relation between electrodynamics in the presence of a minimal measurable length and the concept of nonlocality in electrodynamics. Our conclusions are presented in Section 7. SI units are used throughout this paper.

2. Lorentz-Covariant Deformed Algebra with a Minimal Observable Distance

Recently, Quesne and Tkachuk have introduced a Lorentz-covariant deformed algebra which describes a 𝐷+1-dimensional quantized spacetime [20, 21]. The Quesne-Tkachuk algebra in a 𝐷+1-dimensional spacetime is specified by the following generalized commutation relations: 𝑋^𝜇,𝑃^𝜈=−𝑖ħ1−𝛽𝑃_𝜌𝑃^𝜌𝑔^𝜇𝜈−𝛽^′𝑃^𝜇𝑃^𝜈𝑋_𝜇,,𝑋^𝜈=𝑖ħ2𝛽−𝛽^′−2𝛽+𝛽^′𝛽𝑃_𝜌𝑃^𝜌1−𝛽𝑃_𝜌𝑃^𝜌𝑃^𝜇𝑋^𝜈−𝑃^𝜈𝑋^𝜇,𝑃^𝜇,𝑃^𝜈=0, (3)(4)(5) where 𝜇,𝜈,𝜌=0,1,2,…,𝐷 and 𝛽 and 𝛽^′ are two nonnegative deformation parameters (𝛽,𝛽^′≥0). In the above equations, 𝛽 and 𝛽^′ are constant parameters with dimension (momentum)⁻². Also, 𝑋^𝜇 and 𝑃^𝜇 are position and momentum operators in the deformed space and 𝑔_𝜇𝜈=𝑔^𝜇𝜈=diag(1,−1,−1,…,−1). In the special case where 𝐷=3 and 𝛽=0, the Quesne-Tkachuk algebra (3)–(5) reduces to the Snyder algebra [22].

An immediate consequence of relation (3) is the appearance of an isotropic minimal length which is given by Δ𝑋^𝑖₀=ħ𝐷𝛽+𝛽^′1−𝛽𝑃⁰²,∀𝑖∈{1,2,…,𝐷}. (6) In [23], Tkachuk introduced a representation which satisfies the generalized commutation relations (3)–(5) up to the first order in deformation parameters 𝛽 and 𝛽^′.

The Tkachuk representation is given by 𝑋^𝜇=𝑥^𝜇−2𝛽−𝛽^′4𝑥^𝜇𝑝_𝜌𝑝^𝜌+𝑝_𝜌𝑝^𝜌𝑥^𝜇,𝑃^𝜇=𝛽1−^′2𝑝_𝜌𝑝^𝜌𝑝^𝜇, (7) where 𝑥^𝜇 and 𝑝^𝜇=𝑖ħ(𝜕/𝜕𝑥_𝜇)=𝑖ħ𝜕^𝜇 are position and momentum operators in ordinary relativistic quantum mechanics. In this study, we consider the special case of 𝛽^′=2𝛽, in which the position operators commute to the first order in deformation parameter 𝛽, that is, [𝑋^𝜇,𝑋^𝜈]=0. In this linear approximation, the Quesne-Tkachuk algebra becomes 𝑋^𝜇,𝑃^𝜈=−𝑖ħ1−𝛽𝑃_𝜌𝑃^𝜌𝑔^𝜇𝜈−2𝛽𝑃^𝜇𝑃^𝜈,𝑋^𝜇,𝑋^𝜈𝑃=0,_𝜇,𝑃^𝜈=0. (8) The following representations satisfy (8), in the first order in 𝛽: 𝑋^𝜇=𝑥^𝜇,𝑃^𝜇=1−𝛽𝑝_𝜌𝑝^𝜌𝑝^𝜇. (9) Note that the representations (7) and (9) coincide when 𝛽^′=2𝛽.

3. Lagrangian Formulation of Electrodynamics with an External Source in the Presence of a Minimal Length Based on the Quesne-Tkachuk Algebra

The Lagrangian density for a massless vector field 𝐴^𝜇=((1/𝑐)𝜙,𝐀) with an external source 𝐽^𝜇=(𝑐𝜌,𝐉) in a 3+1-dimensional spacetime is [24] 1ℒ=−4𝜇₀𝐹_𝜇𝜈𝐹^𝜇𝜈−𝐽^𝜇𝐴_𝜇, (10) where 𝐹_𝜇𝜈=𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇 is the electromagnetic field tensor. In a 3+1-dimensional spacetime, the components of the electromagnetic field tensor 𝐹_𝜇𝜈 can be written as 𝐹_𝜇𝜈=0𝐸_𝑥𝑐𝐸_𝑦𝑐𝐸_𝑧𝑐−𝐸_𝑥𝑐0−𝐵_𝑧𝐵_𝑦−𝐸_𝑦𝑐𝐵_𝑧0−𝐵_𝑥−𝐸_𝑧𝑐−𝐵_𝑦𝐵_𝑥0. (11) The Euler-Lagrange equation for the vector field 𝐴^𝜇 is 𝜕ℒ𝜕𝐴_𝜆−𝜕_𝜌𝜕ℒ𝜕𝜕_𝜌𝐴_𝜆=0. (12) If we substitute the Lagrangian density (10) in the Euler-Lagrange equation (12), we will obtain the inhomogeneous Maxwell's equations as follows: 𝜕_𝜌𝐹^𝜌𝜆=𝜇₀𝐽^𝜆. (13) The electromagnetic field tensor 𝐹_𝜇𝜈 satisfies the Bianchi identity 𝜕_𝜇𝐹_𝜈𝜆+𝜕_𝜈𝐹_𝜆𝜇+𝜕_𝜆𝐹_𝜇𝜈=0. (14) Equation (14) leads to the homogeneous Maxwell's equations. Now, we obtain the Lagrangian density for electrodynamics in the presence of a minimal observable distance based on the Quesne-Tkachuk algebra. For this purpose, let us write the Lagrangian density (10) by using the representations (9), that is, 𝑥^𝜇→𝑋^𝜇=𝑥^𝜇,𝜕^𝜇→∇^𝜇∶=1+𝛽ħ²□𝜕^𝜇, (15) where □∶=𝜕_𝜇𝜕^𝜇 is the d’Alembertian operator. The result reads 1ℒ=−4𝜇₀∇_𝜇𝐴_𝜈−∇_𝜈𝐴_𝜇∇^𝜇𝐴^𝜈−∇^𝜈𝐴^𝜇−𝐽^𝜇𝐴_𝜇1=−4𝜇₀1+𝛽ħ²□𝜕_𝜇𝐴_𝜈−1+𝛽ħ²□𝜕_𝜈𝐴_𝜇×1+𝛽ħ²□𝜕^𝜇𝐴^𝜈−1+𝛽ħ²□𝜕^𝜈𝐴^𝜇−𝐽^𝜇𝐴_𝜇1=−4𝜇₀𝐹_𝜇𝜈𝐹^𝜇𝜈−14𝜇₀ħ2𝛽²𝐹_𝜇𝜈□𝐹^𝜇𝜈−𝐽^𝜇𝐴_𝜇+𝒪ħ2𝛽⁴. (16) The term −(1/4𝜇₀)(ħ2𝛽)²𝐹_𝜇𝜈□𝐹^𝜇𝜈 in the above Lagrangian can be considered as a minimal length effect.

If we neglect terms of order (ħ2𝛽)⁴ and higher in (16), we will obtain the following Lagrangian density: 1ℒ=−4𝜇₀𝐹_𝜇𝜈𝐹^𝜇𝜈−14𝜇₀ħ2𝛽²𝐹_𝜇𝜈□𝐹^𝜇𝜈−𝐽^𝜇𝐴_𝜇. (17) The Lagrangian density (17) is similar to the Abelian Lee-Wick model which was introduced by Lee and Wick as a finite theory of quantum electrodynamics [25–29]. Equation (17) can be written as follows: 1ℒ=−4𝜇₀𝐹_𝜇𝜈𝐹^𝜇𝜈+14𝜇₀ħ2𝛽²𝜕_𝛼𝐹_𝜇𝜈𝜕^𝛼𝐹^𝜇𝜈+𝜕_𝛼𝜒^𝛼−𝐽^𝜇𝐴_𝜇, (18) where 𝜒^𝛼1∶=−4𝜇₀ħ2𝛽²𝐹_𝜇𝜈𝜕^𝛼𝐹^𝜇𝜈. (19) After dropping the total derivative term 𝜕_𝛼𝜒^𝛼, the Lagrangian density (18) will be equivalent to the following Lagrangian density: 1ℒ=−4𝜇₀𝐹_𝜇𝜈𝐹^𝜇𝜈+14𝜇₀ħ2𝛽²𝜕_𝛼𝐹_𝜇𝜈𝜕^𝛼𝐹^𝜇𝜈−𝐽^𝜇𝐴_𝜇. (20) Using the Bianchi identity (14) and dropping the total derivative terms, the expression (20) can also be written as follows: 1ℒ=−4𝜇₀𝐹_𝜇𝜈𝐹^𝜇𝜈+12𝜇₀𝑎²𝜕_𝜎𝐹^𝜌𝜎𝜕^𝛽𝐹_𝜌𝛽−𝐽^𝜇𝐴_𝜇, (21) where 𝑎∶=ħ2𝛽. Equation (21) is the Lagrangian density originally introduced by Podolsky [30–33] and 𝑎 is called Podolsky’s characteristic length [34–38]. The Euler-Lagrange equation for the Lagrangian density (20) is [39, 40] 𝜕ℒ𝜕𝐴_𝜆−𝜕_𝜌𝜕ℒ𝜕𝜕_𝜌𝐴_𝜆+𝜕_𝜎𝜕_𝜌𝜕ℒ𝜕𝜕_𝜎𝜕_𝜌𝐴_𝜆=0. (22) If we substitute (20) into (22), we will obtain the inhomogeneous Maxwell’s equations in the presence of a minimal observable distance as follows: 𝜕_𝜌𝐹^𝜌𝜆+ħ2𝛽²□𝜕_𝜌𝐹^𝜌𝜆=𝜇₀𝐽^𝜆. (23) It should be mentioned that (23) have been previously obtained from a different perspective by Kober [41]. Equations (14) and (23) can be written in the vector form as follows: ∇⋅𝐄+ħ2𝛽²□()=𝜌∇⋅𝐄𝜀₀,∇×𝐄=−𝜕𝐁,ħ𝜕𝑡∇×𝐁+2𝛽²□1∇×𝐁−𝑐²𝜕𝐄𝜕𝑡=𝜇₀1𝐉+𝑐²𝜕𝐄,𝜕𝑡∇⋅𝐁=0. (24)(25)(26)(27) The generalized Maxwell’s equations (24)–(27) have been introduced earlier by Tkachuk in [23] with a different approach. In the limit ħ2𝛽→0, the generalized inhomogeneous Maxwell’s equations (24) and (26) become the usual inhomogeneous Maxwell’s equations.

4. Free Space Solutions of the Generalized Inhomogeneous Maxwell’s Equations

In this section, we obtain the plane wave solutions of the generalized inhomogeneous Maxwell’s equations (23) in a 3+1-dimensional spacetime.

In free space (𝜌=0,𝐉=0), (23) can be written as 𝜕_𝜌𝐹^𝜌𝜆+ħ2𝛽²□𝜕_𝜌𝐹^𝜌𝜆=0. (28) In the Lorentz gauge (𝜕_𝜌𝐴^𝜌=0), the field equations (28) become □𝐴^𝜆+ħ2𝛽²□□𝐴^𝜆=0. (29) Now, we try to find a plane wave solution of (29) as follows: 𝐴^𝜆(𝑥)=𝐴𝑒^{−(𝑖/ħ)𝑝⋅𝑥}𝜖^𝜆𝑝, (30) where 𝜖^𝜆(𝑝) is the polarization four-vector and 𝐴 is a normalization constant. In the above equation 𝑝^𝜇=(𝐸/𝑐,𝐩) is the momentum four-vector. If we substitute (30) in (29), we will obtain 𝑝²𝑎1−²ħ²𝑝²=0, (31) where 𝑝²=𝑝_𝜇𝑝^𝜇=(𝐸/𝑐)²−𝐩².

Equation (31) leads to the following energy-momentum relations: 𝐸²=𝑐²𝐩²,𝐸²=𝑚²_eff𝑐⁴+𝑐²𝐩², (32)(33) where 𝑚_effħ∶=.𝑎𝑐 (34) Equation (32) describes a massless vector particle whereas (33) describes a massive vector particle with the effective mass 𝑚_eff.

5. Upper Bound Estimation of the Minimal Length in Generalized Electrodynamics

Substituting 𝛽^′=2𝛽 into (6) and remembering 𝑎=ħ2𝛽, we have Δ𝑋^𝑖₀=𝐷+22𝑎²1+𝒪𝑎²,∀𝑖∈{1,2,…,𝐷}. (35) If we neglect terms of order 𝑎⁴ and higher in (35), the isotropic minimal length in a 3+1-dimensional spacetime becomes Δ𝑋^𝑖₀≃√102𝑎,∀𝑖∈{1,2,3}. (36) Now we are ready to estimate the upper bounds on the isotropic minimal length in generalized electrodynamics.

5.1. Upper Bound on the Isotropic Minimal Length Based on the Anomalous Magnetic Moment of the Electron

In a series of papers, Accioly and coworkers [27, 29, 34] have estimated an upper bound on Podolsky’s characteristic length 𝑎 by computing the anomalous magnetic moment of the electron in the framework of Podolsky’s electrodynamics. This upper bound on 𝑎 is [27, 29, 34] 𝑎≤4.7×10⁻¹⁸m. (37) Inserting (37) into (34) and (36), we find 𝑚_eff≥41.8GeV𝑐²,Δ𝑋^𝑖₀≤7.4×10⁻¹⁸m. (38)(39)

5.2. Upper Bound on the Isotropic Minimal Length Based on the Ground State Energy of the Hydrogen Atom

In [37], Cuzinatto and coworkers have studied the influence of Podolsky's electrostatic potential on the ground state energy of the hydrogen atom. In their study, the upper limit on 𝑎 is 𝑎≤5.56×10⁻¹⁵m. (40) Inserting (40) into (34) and (36), we find 𝑚_eff≥35.51MeV𝑐²,Δ𝑋^𝑖₀≤8.79×10⁻¹⁵m. (41)(42) It should be noted that the upper bound (42) is about three orders of magnitude larger than the upper bound (39) that is, Δ𝑋^𝑖_{0GroundStateEnergyoftheHydrogenAtom}∼10³Δ𝑋^𝑖_{0AnomalousMagneticMomentoftheElectron}, (43) while the lower bound (41) is about three orders of magnitude smaller than the lower bound (38) that is, 𝑚_{eff---GroundStateEnergyoftheHydrogenAtom}∼10⁻³𝑚_{eff---AnomalousMagneticMomentoftheElectron}. (44)

6. Relationship between Nonlocal Electrodynamics and Electrodynamics in the Presence of a Minimal Length

In a series of papers, Smailagic and Spallucci [42–44] have introduced an approach to formulate noncommutative quantum field theory. Using the Smailagic-Spallucci approach, Gaete and Spallucci introduced a nonlocal Lagrangian density for the vector field 𝐴^𝜇 with an external source 𝐽^𝜇 as follows: 1ℒ=−4𝜇₀𝐹_𝜇𝜈exp(𝜃□)𝐹^𝜇𝜈−𝐽^𝜇𝐴_𝜇, (45) where 𝜃 is a constant parameter with dimensions of (length)² [45]. We assume that the function exp(𝜃□) in (45) can be expanded in a power series as follows: exp(𝜃□)=^∞_𝑙=0𝜃^𝑙□𝑙!^𝑙, (46) where □^𝑙 denotes the □ operator applied 𝑙 times [46].

If we insert (46) into (45), we will obtain the following Lagrangian density: 1ℒ=−4𝜇₀𝐹_𝜇𝜈𝐹^𝜇𝜈−14𝜇₀𝜃𝐹_𝜇𝜈□𝐹^𝜇𝜈−14𝜇₀^∞_𝑙=2𝜃^𝑙𝐹𝑙!^𝜇𝜈□^𝑙𝐹^𝜇𝜈−𝐽^𝜇𝐴_𝜇. (47) After neglecting terms of order 𝜃² and higher in (47), we obtain 1ℒ=−4𝜇₀𝐹_𝜇𝜈𝐹^𝜇𝜈−14𝜇₀𝜃𝐹_𝜇𝜈□𝐹^𝜇𝜈−𝐽^𝜇𝐴_𝜇. (48) A comparison between (17) and (48) shows that there is an equivalence between the Gaete-Spallucci electrodynamics to the first order in 𝜃 and the Lee-Wick electrodynamics (or electrodynamics in the presence of a minimal length). The relationship between the noncommutative parameter 𝜃 in (48) and 𝑎=ħ2𝛽 in (17) is 𝑎=√𝜃. (49) Inserting equation (49) into (34) and (36), we find 𝑚_eff=ħ√,𝜃𝑐Δ𝑋^𝑖₀≃√10𝜃2,∀𝑖∈{1,2,3}. (50) Using (40) in (49), we obtain the following upper bound for the noncommutative parameter 𝜃: 𝜃_{GroundStateEnergyoftheHydrogenAtom}≤3.09×10⁻²⁹m². (51) The above upper bound on the noncommutative parameter 𝜃, that is, 3.09×10⁻²⁹m², is near to the neutron-proton scattering cross section (10⁻²⁵cm²) [47]. It is necessary to note that the electrodynamics in the presence of a minimal observable distance is only correct to the first order in the deformation parameter 𝛽, while the Gaete-Spallucci electrodynamics is valid to all orders in the noncommutative parameter 𝜃.

7. Conclusions

Heisenberg believed that every theory of elementary particles should contain a minimal observable distance of the magnitude ℓ₀∼10⁻¹³cm [47–50]. He hoped that the introduction of a minimal length would eliminate divergences that appear in quantum electrodynamics. Today we know that every theory of quantum gravity predicts the existence of a minimal measurable length which leads to a GUP. An immediate consequence of the GUP is a generalization of position and derivative operators according to (15) for 𝛽^′=2𝛽. It was shown that the Lagrangian formulation of electrodynamics with an external source in the presence of a minimal measurable length leads to the inhomogeneous fourth-order field equations. We demonstrated the similarity between electrodynamics in the presence of a minimal length and Lee-Wick electrodynamics. We have shown that the free space solutions of the inhomogeneous Maxwell's equations in the presence of a minimal length describe two particles, a massless vector particle and a massive vector particle with the effective mass 𝑚_eff=ħ/𝑎𝑐. Now, let us compare the upper bounds on the isotropic minimal length in this paper with the results of [47–51]. The upper limit on the isotropic minimal length in (39) is near to the electroweak length scale (ℓ_electroweak∼10⁻¹⁸m) [51], while the upper limit (42) is near to the minimal observable distance which was proposed by Heisenberg (ℓ₀∼10⁻¹³cm) [47–50]. It is interesting to note that the lower bound on the effective mass 𝑚_eff in (38), that is, 41.8(GeV/𝑐²), is of the same order of magnitude as the mass of the 𝑊^± and 𝑍⁰ vector bosons (𝑀_𝑊=80.425±0.038(GeV/𝑐²),𝑀_𝑧=91.1876±0.0021(GeV/𝑐²)) [52]. Finally, we have investigated the relationship between the Gaete-Spallucci nonlocal electrodynamics and electrodynamics with a minimal length.

Note Added. After this work was completed, we became aware of an interesting article by Maziashvili and Megrelidze [53], in which the authors study the electromagnetic field in the presence of a momentum cutoff. For their discussion, they use the following modified Heisenberg algebra: 𝑋^𝑖,𝑃^𝑗=𝑖ħ2𝛽𝐏²1+4𝛽𝐏²𝛿−1^𝑖𝑗+2𝛽𝑃^𝑖𝑃^𝑗,𝑋^𝑖,𝑋^𝑗𝑃=0,_𝑖,𝑃^𝑗=0, (52) where 𝑖,𝑗=1,2,3 and 𝛽 is a deformation parameter [54]. In our work we have formulated electrodynamics in the framework of Quesne-Tkachuk algebra which is a Lorentz-covariant deformed algebra whereas the authors of [53] have studied electrodynamics in the framework of (52) algebra which is not a Lorentz-covariant algebra.

Acknowledgment

The authors would like to thank the referees for their useful comments.