fulltext

Classical electrodynamics of a particle with maximal

acceleration corrections (*)

A. FEOLI(1)(**), G. LAMBIASE(1)(***), G. PAPINI(2)(*

**) and G. SCARPETTA(

1

)(3) (1) Dipartimento di Scienze Fisiche “E. R. Caianiello”

Università di Salerno - 84081 Baronissi (SA), Italy INFN, Sezione di Napoli - Napoli, Italy

(2) Department of Physics, University of Regina - Regina, Sask. S4S 0A2, Canada (3) International Institute for Advanced Scientific Studies - Vietri sul Mare (SA), Italy (ricevuto il 26 Ottobre 1996; approvato il 26 Novembre 1996)

Summary. — We calculate the first-order maximal acceleration corrections to the

classical electrodynamics of a particle in external electromagnetic fields. These include additional dissipation terms, the presence of a critical electric field, a correction to the cyclotron frequency of an electron in a constant magnetic field and the power radiated by the particle. The electric effects are sizeble at the fields that are considered attainable with ultrashort TW laser pulses on plasmas.

PACS 41.20 – Electric, magnetic, and electromagnetic fields.

1. – Introduction

In an attempt to incorporate quantum aspects in the geometry of space-time, Caianiello wrote a series of papers about a theory called “Quantum Geometry” [1], where the position and momentum operators are represented as covariant derivatives and the quantization is interpreted as curvature of phase space. A novel feature of the model is the existence of a maximal acceleration [2]. Limiting values for the acceleration were also derived by several authors on different grounds and applied to many branches of physics such as string theory, cosmology, quantum field theory, black-hole physics, etc. [3].

The model proposed by Caianiello and his coworkers to include the effects of a maximal acceleration in a particle dynamics consisted in enlarging the space-time manifold to an eight-dimensional space-time tangent bundle TM. In this way the

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail: feoliHvaxsa.csied.unisa.it

(***) E-mail: lambiaseHvaxsa.csied.unisa.it (***) E-mail: papiniHcas.uregina.ca

standard invariant line element ds2

4 c2dt2_becomes

d tA2

4 gABdXAdXB, A , B 41, R , 8 ,

(1.1)

where the coordinates of TM are

XA₄

g

h

dt is the classical four velocity and

gAB4 gmn7 gmn.

(1.3)

The metric (1.1) can be written as

d tA2 4 gmn

g

h

and an embedding procedure can be developed [4] in order to find the effective space-time geometry in which a particle moves when the constraint of a maximal acceleration is present. In fact, if we find the parametric equations that relate the velocity field xNm _{to the first four coordinates x}m_{, we can calculate the effective}

four-dimensional metric gAmn on the hypersurface locally embedded in TM.

In sect. 2 we derive the quantum corrections to the metric for a charged particle moving in an electromagnetic field and the corresponding first-order corrections to the equations of motion. In sect. 3 we analyze some consequences of the corrections for particular instances. We derive the radiated power in sect. 4. Section 5 contains a discussion.

2. – Effective geometry

Consider the Minkowski metric hmn (of signature 22) as the background metric in

eq. (1.4). A particle of mass m and charge q moves in this background under the action of an externally applied electromagnetic field. The classical equation of motion of the particle is dxNm 4 2 q mcF m ndxn, (2.1)

where Fmn is the classical electromagnetic field tensor. Equation (2.1) may be taken as

the first-order approximation to the real velocity field of the particle. If we substitute eq. (2.1) into (1.4) we can calculate the correction of order A22 _{to the classical}

background metric. We can iterate this procedure, by calculating the new velocity field and substituting it again into the metric to obtain the correction A24_{, and so on.}

However, the value of the maximal acceleration is very high and we can neglect the

O( 1 OA4_{) terms. This leads to the new metric}

d tA2 C

g

h

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charged particle with the electromagnetic field and this curvature affects the motion of the particle itself.

The modified equation of motion of the charged particle may be obtained from the action S 4



k

and leads to the equations

xOs 1 Gsabx N_a xNb 1 q mc g Asr_F rgx N_g 4 0 , (2.4) where Gs ab4 q2 2 A2_m2 gA sn_[Fl n(Fla , b1 Flb , a) 1FlaFbn , l1 FlbFan , l] (2.5)

is the connection, and the dot means derivation with respect to tA.

The contravariant components of the metric tensor gAsn _{then follow from the}

equation gAsn_gA nr4 dsrin the approximation q2 A2_m2F s lFlnbhsnand are gAsn_{C h}sn₁ q 2 A2_m2F s lF ln . (2.6)

Hereafter the raising and lowering of the indices is made with the tensor hmn.

Obviously, the corrections disappear in the limit A KQ and we find the classical equation of motion.

To analyze the physical consequences of this correction, it is useful to write the equation of motion (2.4) in terms of the four-momentum pm

4 mcxNm and the four-velocity dxm

Od tA 4 um4 (c dtOd tA, vKdtOdtA), dpm d tA 4 q c gA mn_F nrur2 q2 2 A2_m gA mn_(Fl nFla , b1 FlaFbn , l) uaub. (2.7) 3. First-order corrections

The particle embedded in the external field experiences a curved manifold even if the background metric is flat. This curvature affects the motion of the particle and the rate of change of its energy e f cp0_{in time. We analyze first the simple case when only}

an electric field EKis present. We obtain from (2.7)

(3.1) de dt 4 q E K Q vK

u

v

y

u

v

z

The difference of (3.1) from the usual classical equation becomes evident when EKQ vK40. In this case we find

de dt 4 q2c3 2 A2md tA dt

y

z

which is an additional dissipation term. On the other hand, the orthogonality condition of EK and vK cannot be imposed in general because of the perturbing effect of the acceleration on the motion itself.

The quantity d tA dt in eq. (3.1) is defined by d tA dt 4 c

q

dt must be real, we have

N E K N E mA NqN

o

i.e. the intensity N EKN is limited by the critical electric field

Ec4 mA NqN

o

In the particle’s rest frame Ec becomes

Ec4

mA

NqN . (3.6)

If the particle is an electron, we find Ec(me2) C2.1 Q1016NOC when the maximal

acceleration is mass dependent, and EcC 5.6 Q 1039NOC when A refers to the Planck

mass. Similarly for the proton we find: Ec(mp1) C1024NOC and E_cC 1043NOC,

respectively.

It is interesting to observe that a similar critical value for the electric field was found for the open bosonic string propagating in an external electromagnetic field. In this case E ETONqaN (a 4 1 , 2 ), where T is the string tension and qaare the charges at

the ends of the string [5]. This condition is satisfied automatically when the background electromagnetic field is described by the Born-Infeld Lagrangian rather than by the Maxwell Lagrangian [6]. It is therefore possible to expect a connection between the Born-Infeld and the Caianiello Lagrangians [7]. The relation between acceleration and Ecwas also analyzed by Gasperini [8].

It is also instructive to calculate eq. (2.7) explicitly in the case of a particle with charge q 42 e, moving in a constant electromagnetic field of components EK4 (Ex, Ey, 0 ) and B K 4 ( 0 , 0 , B). We find de dt 4 2 e(Exx N 1 Eyy N ) 1 e 3 A2m2(Exx N 1 Eyy N )(N EKN22 B2) . (3.7)

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Finally, we consider the problem of charged particles moving in a constant magnetic field of components BK_{4 ( 0 , 0 , B). From eq. (2.7), the equations of motion for} a circular path are

. / ´ x(t) 4r0sin vt , y(t) 4r0cos vt , (3.8) where v 4 eB m 1 e3B3c2 A2_m3 4 vc1 Dv , (3.9) and Dv 4v3

cc2OA2 is the first correction to the classical frequency vc. This is the

same correction of ref. [9] if the “rigidity” correction b-term there obtained is calculated for v 40. Due to the existence of higher-derivative terms in the Lagrangian, the corrections to the cyclotron frequency found in [9] depend in fact on the velocity of the particle. From eq. (3.9) we obtain

Dv vc 4

g

h

The limit coming from the (g 22)O2 experiment for non-relativistic electrons trapped in a magnetic field B A50 KG is [10]

Dv

vc

G 2 Q 10210. (3.11)

From (3.10) we find DvOvcA 6 Q 10215 when the maximal acceleration is mass

dependent and DvOvcA 2 Q 10260 when the maximal acceleration is considered a

universal constant. Both results are in agreement with the experimental upper bound (3.11).

4. – Power radiated by a charged particle

The total four-momentum radiated by a charge moving in an electromagnetic field is given by the well-known formula [11]

D Pm_{4 2}2 e 2 3 c



If the contributions of the gradients of the electromagnetic fields can be neglected in (2.5), which is certainly legitimate at the highest gradients available at present or in a foreseeable future, one finds Gs

abB 0, and (2.4) becomes xOs 4 e mc g Asr_F rgx N_g B e mc

g

h

Substituting (4.2) into (4.1) we obtain

D Pm

4 D P( 0 )m 1 D P

m

(A),

where D P( 0 )m 4 2 2 e4 3 m2_c5



is the classical result, and

D P(A)m 4 2 2 e4 3 m2_c5 e2 A2_m2



represents the maximal acceleration contribution. When BK_{4 0 and E}Kis parallel to bK₄

v

K

Oc, Eq. (4.5) for m 4 0 becomes de(A) dx0 4 2 e4 3 m2_c3 e2 A2_m2N E K N4, (4.6) and (4.4) yields de dx0 4 2 e4 3 m2_c3N E K N2. (4.7)

The ratio of the classical dissipation term (4.7) to the maximal acceleration term (4.6) for electrons is G 4 deOdx 0 de(A) Odx0 4 m2_A2 e2 N E K N2 A 1.21 10 36 N E K N2 , (4.8)

while the total dissipated power is

Ptot4 2 e4 3 m2 c3N E K N2

u

v

Even at their lowest order the maximal acceleration corrections to the classical electrodynamics of a particle offer interesting aspects. These include the existence of a critical field beyond which approximation and equations break down and corrections to the cyclotron frequency of an electron in a constant magnetic field. While the latter effect is compatible with present stringent experimental upper limits from the (g 22)O2 experiment, it may by its very size be difficult to be observed directly. On the contrary, the field EcA 2.1 Q 1016NOC for electrons is very close to values EA5Q1015NOC

that are presently considered as attainable by focusing TW laser beams in a plasma [12]. Radiation levels at these values of the electric field also become non-negligible as entailed by (4.6), (4.7) and (4.8). In fact the maximal acceleration contribution increases with the fourth power of N EKN and contributes sizeable amounts to the power radiated at the highest field intensities. Increased radiation should of course be expected on intuitive grounds. This typical functional increase would constitute the signature of the process once the promise of high electric fields became a reality.

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* * *

GP gladly acknowledges the continued research support of Dr. K. DENFORD, Dean

of Science, University of Regina. This research was supported by Ministero dell’Università e della Ricerca Scientifica.

R E F E R E N C E S

[1] CAIANIELLOE. R., Lett. Nuovo Cimento, 25 (1979) 225; 27 (1980) 89; Nuovo Cimento B, 59 (1980) 350; Riv. Nuovo Cimento, 15, No. 4 (1992) and references therein.

[2] CAIANIELLOE. R., Lett. Nuovo Cimento, 32 (1981) 65; CAIANIELLO E. R., DE FILIPPO S., MARMOG. and VILASIG., Lett. Nuovo Cimento, 34 (1982) 112; CAIANIELLOE. R., Lett. Nuovo Cimento, 41 (1984) 370.

[3] WOODW. R., PAPINIG. and CAIY. Q., Nuovo Cimento B, 104 (1989) 361, 727(E); PAPINIG., Math. Japonica, 41 (1995) 81; DASA., J. Math. Phys., 21 (1980) 1506; BRANDT H. E., Lett. Nuovo Cimento, 38 (1983) 522, 39 (1984) 192; Found. Phys. Lett., 2 (1989) 39 and references therein; TOLLERM., Nuovo Cimento B, 102 (1988) 261; Int. J. Theor. Phys., 29 (1990) 963; Phys. Lett. B, 256 (1991) 215; MASHOON B., Phys. Lett. A, 143 (1990) 176 and references therein; FROLOVV. P. and SANCHEZN., Nucl. Phys. B, 349 (1991) 815; PATIA. K., Europhys. Lett., 18 (1992) 285; FEOLIA., Nucl. Phys. B, 396 (1993) 261; SANCHEZN., in Structure: from Physics to General Systems, edited by M. MARINARO and G. SCARPETTA, Vol. 1 (World Scientific, Singapore) 1993, p. 118; CAPOZZIELLOS. and FEOLIA., Int. J. Mod. Phys. D, 2 (1993) 79; CAIANIELLOE. R., CAPOZZIELLOS.,DERITISR., FEOLIA. and SCARPETTAG., Int. J. Mod. Phys. D, 3 (1994) 485.

[4] CAIANIELLOE. R., FEOLIA., GASPERINIM. and SCARPETTAG., Int. J. Theor. Phys., 29 (1990) 131.

[5] BURGESSC. P., Nucl. Phys. B, 294 (1987) 427; BARBASHOV B. M. and NESTERENKOV. V., Introduction to the Relativistic String Theory (World Scientific, Singapore) 1990, p. 96. [6] FRADKINE. S. and TSEYTLINA. A., Nucl. Phys. B, 261 (1985) 1; Phys. Lett. B, 158 (1985) 516;

160 (1985) 69; 163 (1985) 123; ABOULSAOODA. A., CALLANC. G., NAPPIC. R. and YOSTS. A., Nucl. Phys. B, 280 (1987) 599.

[7] NESTERENKOV. V., FEOLIA. and SCARPETTAG., J. Math. Phys., 36 (1995) 5552. [8] GASPERINIM., Gen. Relativ. Gravit., 24 (1992) 219.

[9] FIORENTINIG., GASPERINIM. and SCARPETTAG., Mod. Phys. Lett. A, 6 (1991) 2033. [10] VANDYCKR. S. jr., SCHWINBERGP. B. and DEHMELTH. G., Phys. Rev. Lett., 59 (1987) 26. [11] LANDAUL. D. and LIFSHITZE. M., The Classical Theory of Fields (Pergamon Press, Oxford)

1969.