- Email: [email protected]
Zero-point oscillations and radiative-resonant interactions
23 August 1999
Physics Letters A 259 Ž1999. 393–398 www.elsevier.nlrlocaterphysleta
Zero-point oscillations and radiative-resonant interactions V.N. Tsytovich a
a,)
, R. Bingham
b
General Physics Institute, Russian Academy of Science Moscow, VaÕiloÕa str. 38, 117942, Moscow, Russia b Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, Ox11 0QX, UK Received 7 June 1999; accepted 12 June 1999 Communicated by M. Porkolab
Abstract We discuss new results describing the relationship between quantum radiation caused by zero-point oscillations and radiative resonant wave-particle interactions. We show that the formation of a power-law spectrum of fast particles produced by radiative resonant interactions wV.N. Tsytovich, Phys. Rep. 178 Ž1989. 261; Physica 210 Ž1981. 136; Physica Scripta 52 Ž1982. 54x is related to the interaction of zero-point oscillations with resonant particle acceleration. Possible experiments to measure the electromagnetic radiation produced by this process are suggested. q 1999 Published by Elsevier Science B.V. All rights reserved.
In this Letter we describe the relationship between two physical effects: the quantum radiation produced by zero-point oscillations, known examples are Hawking radiation w4,5x, Unruh radiation w6x and the Casimir w7x effect with radiative resonant wave-particle interactions w1–3x, which generates fast particles having properties of the observed cosmic ray spectra w1x. We propose several experimental verifications for radiative resonant interactions and radiation produced by zero-point oscillations during particle acceleration. Unruh w6x and Davies w8x point out that zero-point electromagnetic quantum fluctuations or virtual photons in an accelerating frame transform into real thermal photons with a temperature T s Ž "r2p k B .Ž arc . where a is the acceleration, c is the speed of light, and " and k B are Planck’s and
) Corresponding author. Tel. q7-095-1350247; fax: q7-0951350270; e-mail: [email protected], [email protected]
Boltzmann’s constants. In the Earth’s accelerating field this radiation is at a temperature of only 10y2 0 K. A detectable signal is therefore possible only from a system undergoing extreme acceleration. Yablonovitch w9x described an approach to observe radiation produced by this effect in a plasma which is rapidly ionized, suddenly changing its refractive index. Another possibility and perhaps more feasible approach is to use the acceleration induced by intense lasers in plasmas due to the transverse acceleration in the electromagnetic field of the laser itself or the longitudinal acceleration produced by plasma waves w10–12x. The influence of zero-point quantum fluctuations was investigated for quasilinear interactions w1x and it was shown that a power-law energy distribution of energetic particles is created. Although this effect is weak and the energy associated with these radiative resonance interactions is only approximately 10y4 of the energy given to create and support the plasma
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 4 5 4 - 5
394
V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
waves, the effect is measurable, as already pointed out by Doveil and Tsytovich w13x. The resonant interactions appear naturally independent on whether the electric field is random or coherent but in the case of random fields the effect is smaller since the interaction time between the wave and particle is shorter. Thus the effect should be more pronounced for coherent fields. At the present time large amplitude longitudinal plasma waves excited by lasers are used in laser plasma accelerators w10–12,14x, achieving plasma wave electric fields of the order of 1 G Vrcm for plasma densities n s 1.5 = 10 19 cmy3 , the laser intensity is 6 = 10 18 Wrcm2 . Using the formula for the thermal quantum radiation temperature w9x k B T s Ž "r2p .Ž arc . s Ž "r 2p m e c . eE, where a s eErm e is the acceleration of an electron of charge e and mass m e in the plasma wave field E, gives T f 0.006 eV which is approximately 100 K. In the near future it is possible that laser powers will be about 3 orders of magnitude greater reaching 10 22 Wrcm2 , with transverse fields ET Ž, 30'I Vrcm, I is the intensity. of order 3 = 10 12 Vrcm. The lasers excite the plasma waves which are initially coherent, they eventually randomize due to nonlinear effects. The radiation in the latter region is smaller but the size of the region can be much larger and the total emission can be increased by the increase in volume where the random waves exist. It is also possible to consider the quantum radiation emitted by the acceleration of particles in the field of the laser. The Casimir effect w7x describes the interaction between the two conducting plates in vacuum created by the force of zero-point fluctuations reflecting from them. The zero-point fluctuations impart momentum to the conductors. The effect also exist if the plates are partially transparent to zero-point oscillations. This is an important point showing that even if the particles form a transparent barrier, momentum, and energy exchange between the particles of the plates and the zero-point oscillations still exists. Resonant particles are accelerated coherently by the wave, forming an accelerated ‘‘mirror’’. Many ‘‘mirrors’’ of this kind are present in the ensemble of random resonant waves, they can propagate in many directions as wave packets or form coherent structures known as cavitons. The final result of particle acceleration by a random walk in an ensem-
ble of random fields is insensitive to whether these fields form wave packets or cavitons w15x. The density boundaries formed by them will be more and more transparent as the frequency of the zero-point field increases. Finally Hawking w4,5x or Unruh w6x radiation reaches the thermal limit at infinite time and realistic conditions of finite time interactions results in partially optically thin conditions for radiation and fast particles created in the interaction. This needs a kinetic description for the interactions of zero-point fields with non-equilibrium particles forming accelerated ‘‘mirrors’’ or boundaries w1x. These comments are sufficient to point out the main physical effects occurring interactions of zero-point electromagnetic oscillations with a system of resonantly accelerated particles. Firstly, all the resonant interactions can be visualized as a Fermi type of acceleration Žsee w16,17x.. Assume that the size of the interacting region is l and the average distance between such regions is L, then on average the particle increases its energy e p at the rate e˙p s e 2 l 2 E 2rLmÕ s e 2 l ² E :2rmÕ where ² E :2 s E 2 lrL is the average field. This estimate gives exactly the rate of resonant particle acceleration which in a more exact description is given by the quasilinear operator Iˆp w18x: df p dt
s e 2p < f k < 2 k P
H
s Iˆp f p ,
ž
E Ep
/
ž
d Ž v yk P z. k P
E fp Ep
/
dk
Ž 1.
where f k is the electrostatic potential of the random waves, f p is the particle distribution function k s k, v 4 , Iˆp is the quasi-linear operator and the deltafunction describes the resonance. What happens when we consider the zero-point oscillations between these coherently accelerated density boundaries? It is possible to give physical arguments why their energy should on average decrease. Consider an extreme case where these boundaries completely reflect the zero-point oscillations as in the case of conducting boundaries and consider the case where two boundaries move with acceleration of opposite sign collide with each other and the case where the two boundaries move in opposite directions increasing the separation distance. In the
V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
first case the frequency of the zero-point oscillations will be up-shifted and their energy will increase while in the second case the frequency of the zeropoint oscillations will be down-shifted and their energy will decrease. But the time spent during the first type of collision will be less than that in the second type of collision since the distance travelled will be larger in the second case and thus on average the zero-point oscillations will lose energy. Real boundaries are partially transparent and for higher frequencies the refractive index decreases rapidly with frequency with the result that fewer particles receive energy. The larger the frequency the lower the refractive index and the energy of the zero-point oscillations can be transferred to fewer and fewer particles since the energy of each oscillation increases with frequency, forming a tail in the particle distribution. The tail will be a power-law in energy, because the refractive index falls with frequency according to a power-law Žsee below.. To find explicitly the rate of decrease of the zero-point energy for particle acceleration described by Ž1. we need to calculate not only the changes related to time varying particle distributions but additionally the nonlinear effects proportional both to the energy density of resonant waves and the spectral energy density of zero-point oscillations w1x. The same is true for real photons as described in detail in w19x. The difference is that for the dielectric constant of the zero-point oscillations not all the contributions to the dielectric constant that describes the propagating photons appear. This is in a sense trivial since the zero-energy oscillation can not go down in frequency and the corresponding matrix elements vanish Žsee w1x pp. 296–298.. Also the zero-point oscillations change their frequency adiabatically if their frequency is much larger than that related with the rate of particle acceleration described by Ž1.. We will demonstrate the effect for the zero-point oscillations of highest energy < k < 4 m e Ž " s c s 1, k is the momentum of zero-point oscillation.. Then this structure factor is equal to 1, i.e. under these conditions the change of the energy of the zero-point oscillations is determined by dispersion of the dielectric permittivity. We only take into account the fact that in the presence of particles the energy of a single zero-point oscillation is changed from < k
395
tric permittivity for electromagnetic transverse fields. For e t s 1 this expression indeed gives the energy < k
e ts1y
=
ž
2p e 2
v
2
f p dp
H Ž 2p .
Ly Ž p, k . e p y e pyk y v
3
q
Lq Ž p, k . e p q e pyk q v
/
,
Ž 2.
where 2
L . Ž p, k . s 1 .
m2 y Ž p P k . q Ž p P k . rk 2
e p e pyk
.
Ž 3. This is a fully relativistic and quantum expression as shown in Ref. w14x. We multiply Ž2. by v 2r4 differentiate the result with respect to the frequency, and integrate it over the whole phase volume of zero-point oscillations dkrŽ2p . 3 and calculate the change of their energy
V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
396
per unit time by using the change of the particle distribution in time described by Ž1. to get dW zero - point
1 s
dt
4
E
dk
H Ž 2p .
3
dkdp
H Ž 2p .
sy
y
6
Ev
ž
v2
de t dt
vs < k <
Ly Ž p, k .
Ž e p y e pyk y < k < .
Lq Ž p, k .
Ž e p q e pyk q < k < .
2
/
Iˆp f p .
2
Ž 4.
This result coincides with the expression found w1x for < k < 4 m e . In general there appears a factor w1x Ž e py k y e .r< k < which approaches 1 in the limit we considered here. The appearance of this factor is important for establishing the energy conservation law and to show that in general the decrease of the energy of zero-point oscillations has a consequence for particle creation. This simple result can be used to interpret the reaction of the particle distribution due to a lowering of the energy of zero-point oscillations. As discussed, the interaction of zero-point oscillations lead to an exchange of momenta with the particles, the higher the frequency of the zero-point oscillations the lower is the dielectric permittivity and therefore the lower are the potential barriers created for the zero-point oscillation by particles accelerated by the random waves. Therefore the energy of one high frequency oscillation can be transformed to a small number of particles, the number of particles decreases for higher frequencies. This is how the tail in the particle distribution is formed. The results of w1x are valid for any non-equilibrium distribution and take into account both the change of the zero-point oscillation energy and the change of the particle distribution function. The decrease of the energy of the zero-point oscillations is reflected not only in the particle distribution function but also in the rate of change of the energy of the resonant fields giving rise to overall conservation of energy found in w1x. This is important since the decrease of the zero-point oscillation energy does not mean an extraction of energy from the vacuum, in fact all the additional energy is extracted from the source which drives the resonant waves. The vacuum is only the catalyser of
the process and as soon as the acceleration disappear Iˆp s 0 the additional fast particle creation stops and the decrease of the zero-point oscillations also stops. An important question for the process we are considering here is how the vacuum returns to its initial stage when the acceleration stops. The change of the zero-point energy is accumulated during the whole process of acceleration. The source which excites the resonant waves should then produce the necessary emission to fill the decrease in energy of zero-point oscillations. It is important to find the spectral power density of the zero-point fluctuations. For non-relativistically accelerated particle and < k < 4 < pres < this distribution is described by the factor F Ž k .,Ž k s krm e .: FŽ k . s
'1 q k 2 y 1 3 k'1 q k 2 Ž k q '1 q k 2 . 4
.
Ž 5.
For k < 1 we have F Ž k . A k and for k 4 1 we find F A 1rk 2 . The spectrum of zero-point oscillations is shown on Fig. 1 which represents a wide spectrum around 0.5 MeV. Thus if the radiation emitted corresponds to the distribution Ž5. we should expect a hard x-ray emission from all regions where the particles are resonantly accelerated. This emission is measurable for most of the existing experiments w13x including laser-plasma interactions. Thus the regions of particle acceleration can be sources not only of fast particles but also the source of high frequency radiation. This prediction can easily be checked and we can give some estimate of the effect. The total change in energy of the electromagnetic field after time t between resonant particles and plasma waves with frequency v pe s 4p ne 2rm e can be estimated from w1x as
(
W ts
8 3p
ž
ln2 y
11
/
e2
24 "c
v pe t
n res ²< E res < 2 : n
4p
,
Ž 6.
where ²< E res < 2 :r4p is the average energy density of the accelerating resonant field and nres is the density of resonant particles much less than the plasma density n. We have assumed that the radiated energy after acceleration should be of the order of Ž4. as usual in transition emission processes. The numerical coefficient is obtained by integration with respect to k of the distribution of the type given by Ž5. but for
V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
397
Fig. 1. Wave number Žfrequency. dependence of the energy distribution of the changes to the zero-point oscillations caused by resonant particle acceleration.
Dirac particles Želectrons.. The only approximation in deriving Ž6. is made by letting kÕ s v pe for resonant particles. As an example we consider the turbulent region behind the laser pulse in laser plasma accelerators w11x. We take the resonant field in this region to be an order of magnitude less than that achieved for the coherent part of the excited plasma wave, i.e. we take E res f 10 8 Vrcm and ²< E res < 2 : f 3 = 10 15 V 2rcm 2 noting that the average field is at least 3 times less than the maximum field. Eq. Ž6. is in cgs units, therefore we divide E res by the permittivity of free space, i.e. 300. For a plasma density of the order of 10 19 cmy3 we have v pe f 2 = 10 14 sy1 , the duration of the random resonance field we take to be 3 times longer than the laser pulse duration, i.e. 3 = 10y1 2 s and the volume of the region where of the resonance field exist we take 10 times larger than the volume of the laser focus, i.e. we take this emitting volume to be f 10y9 cm3. We also assume that the density of the resonant particles is 10 times less than the plasma density. Then from Ž6. we find that their should be more than 10 3 hard x-ray photons of energy 300–500 keV emitted from this volume, is a measurable effect. Note that for this estimate we assume that the energy of the emitted photons be of order of magnitude equal to the change in the energy of the zero-point oscillations. This is taken from the analogy with the transition radiation processes including the time dependent refractive
index where the energy emitted is of the order of the change of the self-energy of the particles. Note also that this emission is related to the process where plasma boundaries reflect and decelerate the oscillations adiabatically. However, the temperature of Unruh w6x radiation due to the accelerating boundaries results in a lower frequency and emission rate is not included in the above estimate. The frequency of Unruh w6x radiation is given by " v f kT s Ž "r2p . arc and for arc s eErmc we get " v f Ž Õos rc .Ž " v 0r2p . where Õos is the classical quiver velocity in the laser radiation field and v 0 is the laser frequency. For Õos rc f 0.5 typical of present high power lasers the Unruh radiation frequency is v or4p i.e. about 10 6 times less than the effect considered in this paper. In laser plasmas the presence of stochastic fields is well established both experimentally, theoretically and numerically making the measurements easier than in ionization front experiments. The radiation described in this paper is due to the interaction of resonantly accelerated electrons in plasma waves with zero point quantum fluctuations. Many astrophysical sources are known to emit radiation with frequencies close to 0.5 MeV which is interpreted as electron–positron pair annihilation w21x. The observed line widths are in some cases much broader than Doppler broadening and the lines are down-shifted in frequency. It is worthwhile to con-
398
V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
sider the possibility of this emission being related to resonant particle acceleration since the curve given in Fig. 1 is peaked close to 0.5 MeV and is always rather broad and also down shifted in frequencies. In astrophysical plasmas the emitted volume and the time duration are much larger and the effect described here could lead to intense emission.
References w1x V.N. Tsytovich, Phys. Rep. 178 Ž1989. 261. w2x V.N. Tsytovich, Physica 210 Ž1981. 136. w3x V.N. Tsytovich, Physica Scripta 52 Ž1982. 54, The first paper where the quantum approach was given. w4x S.W. Hawking, Nature 248 Ž1974. 30. w5x S.W. Hawking, Commun. Math. Phys. 43 Ž1975. 199. w6x W.G. Unruh, Phys. Rev. D 14 Ž1976. 870.
w7x G. Plunien, B. Muller, W. Greiner, Phys. Rep. 134 Ž1986. 87. w8x P C W Davies, J. Phys. A. 609 Ž1975.. w9x E. Yablonovitch, Phys. Rev. Lett. 62 Ž1989. 1742. w10x T. Tajima, J.M. Dawson, Phys. Rev. Lett. 43 Ž1979. 267. w11x D. Gordon, et al., Phys. Rev. Lett. 80 Ž1998. 2133. w12x R. Bingham, Nature 394 Ž1998. 619. w13x F. Doveil, V. Tsytovich, Comments in Plasma Physics and Controlled Fusion 16 Ž1996. 343. w14x M. Rabinivich, V. Tsytovich, Proceedings of the Lebedev Institute 66 Ž1973. 7. w15x L.I. Rudakov, V.N. Tsytovich, Phys. Rep. 40C Ž1978. 1. w16x V. Tsytovich, Radiophys. Quantum Electron. 8 Ž1966. 446. w17x V.N. Tsytovich, Sov. Phys. JETP 60 Ž1984. 631. w18x V.N. Tsytovich, Lectures on Nonlinear Plasma Kinetics, Springer-Verlag, Berlin, 1995. w19x S. Isakov, V. Krivitsky, V. Tsytovich, Sov. Phys - JETP 63 Ž1987. 545. w20x V. Tsytovich, Soviet Physics-JETP 13 Ž1961. 1249. w21x R Sunyaev, et al., Ap. J. 383 Ž1991. L49.
Physics Letters A 259 Ž1999. 393–398 www.elsevier.nlrlocaterphysleta
Zero-point oscillations and radiative-resonant interactions V.N. Tsytovich a
a,)
, R. Bingham
b
General Physics Institute, Russian Academy of Science Moscow, VaÕiloÕa str. 38, 117942, Moscow, Russia b Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, Ox11 0QX, UK Received 7 June 1999; accepted 12 June 1999 Communicated by M. Porkolab
Abstract We discuss new results describing the relationship between quantum radiation caused by zero-point oscillations and radiative resonant wave-particle interactions. We show that the formation of a power-law spectrum of fast particles produced by radiative resonant interactions wV.N. Tsytovich, Phys. Rep. 178 Ž1989. 261; Physica 210 Ž1981. 136; Physica Scripta 52 Ž1982. 54x is related to the interaction of zero-point oscillations with resonant particle acceleration. Possible experiments to measure the electromagnetic radiation produced by this process are suggested. q 1999 Published by Elsevier Science B.V. All rights reserved.
In this Letter we describe the relationship between two physical effects: the quantum radiation produced by zero-point oscillations, known examples are Hawking radiation w4,5x, Unruh radiation w6x and the Casimir w7x effect with radiative resonant wave-particle interactions w1–3x, which generates fast particles having properties of the observed cosmic ray spectra w1x. We propose several experimental verifications for radiative resonant interactions and radiation produced by zero-point oscillations during particle acceleration. Unruh w6x and Davies w8x point out that zero-point electromagnetic quantum fluctuations or virtual photons in an accelerating frame transform into real thermal photons with a temperature T s Ž "r2p k B .Ž arc . where a is the acceleration, c is the speed of light, and " and k B are Planck’s and
) Corresponding author. Tel. q7-095-1350247; fax: q7-0951350270; e-mail: [email protected], [email protected]
Boltzmann’s constants. In the Earth’s accelerating field this radiation is at a temperature of only 10y2 0 K. A detectable signal is therefore possible only from a system undergoing extreme acceleration. Yablonovitch w9x described an approach to observe radiation produced by this effect in a plasma which is rapidly ionized, suddenly changing its refractive index. Another possibility and perhaps more feasible approach is to use the acceleration induced by intense lasers in plasmas due to the transverse acceleration in the electromagnetic field of the laser itself or the longitudinal acceleration produced by plasma waves w10–12x. The influence of zero-point quantum fluctuations was investigated for quasilinear interactions w1x and it was shown that a power-law energy distribution of energetic particles is created. Although this effect is weak and the energy associated with these radiative resonance interactions is only approximately 10y4 of the energy given to create and support the plasma
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 4 5 4 - 5
394
V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
waves, the effect is measurable, as already pointed out by Doveil and Tsytovich w13x. The resonant interactions appear naturally independent on whether the electric field is random or coherent but in the case of random fields the effect is smaller since the interaction time between the wave and particle is shorter. Thus the effect should be more pronounced for coherent fields. At the present time large amplitude longitudinal plasma waves excited by lasers are used in laser plasma accelerators w10–12,14x, achieving plasma wave electric fields of the order of 1 G Vrcm for plasma densities n s 1.5 = 10 19 cmy3 , the laser intensity is 6 = 10 18 Wrcm2 . Using the formula for the thermal quantum radiation temperature w9x k B T s Ž "r2p .Ž arc . s Ž "r 2p m e c . eE, where a s eErm e is the acceleration of an electron of charge e and mass m e in the plasma wave field E, gives T f 0.006 eV which is approximately 100 K. In the near future it is possible that laser powers will be about 3 orders of magnitude greater reaching 10 22 Wrcm2 , with transverse fields ET Ž, 30'I Vrcm, I is the intensity. of order 3 = 10 12 Vrcm. The lasers excite the plasma waves which are initially coherent, they eventually randomize due to nonlinear effects. The radiation in the latter region is smaller but the size of the region can be much larger and the total emission can be increased by the increase in volume where the random waves exist. It is also possible to consider the quantum radiation emitted by the acceleration of particles in the field of the laser. The Casimir effect w7x describes the interaction between the two conducting plates in vacuum created by the force of zero-point fluctuations reflecting from them. The zero-point fluctuations impart momentum to the conductors. The effect also exist if the plates are partially transparent to zero-point oscillations. This is an important point showing that even if the particles form a transparent barrier, momentum, and energy exchange between the particles of the plates and the zero-point oscillations still exists. Resonant particles are accelerated coherently by the wave, forming an accelerated ‘‘mirror’’. Many ‘‘mirrors’’ of this kind are present in the ensemble of random resonant waves, they can propagate in many directions as wave packets or form coherent structures known as cavitons. The final result of particle acceleration by a random walk in an ensem-
ble of random fields is insensitive to whether these fields form wave packets or cavitons w15x. The density boundaries formed by them will be more and more transparent as the frequency of the zero-point field increases. Finally Hawking w4,5x or Unruh w6x radiation reaches the thermal limit at infinite time and realistic conditions of finite time interactions results in partially optically thin conditions for radiation and fast particles created in the interaction. This needs a kinetic description for the interactions of zero-point fields with non-equilibrium particles forming accelerated ‘‘mirrors’’ or boundaries w1x. These comments are sufficient to point out the main physical effects occurring interactions of zero-point electromagnetic oscillations with a system of resonantly accelerated particles. Firstly, all the resonant interactions can be visualized as a Fermi type of acceleration Žsee w16,17x.. Assume that the size of the interacting region is l and the average distance between such regions is L, then on average the particle increases its energy e p at the rate e˙p s e 2 l 2 E 2rLmÕ s e 2 l ² E :2rmÕ where ² E :2 s E 2 lrL is the average field. This estimate gives exactly the rate of resonant particle acceleration which in a more exact description is given by the quasilinear operator Iˆp w18x: df p dt
s e 2p < f k < 2 k P
H
s Iˆp f p ,
ž
E Ep
/
ž
d Ž v yk P z. k P
E fp Ep
/
dk
Ž 1.
where f k is the electrostatic potential of the random waves, f p is the particle distribution function k s k, v 4 , Iˆp is the quasi-linear operator and the deltafunction describes the resonance. What happens when we consider the zero-point oscillations between these coherently accelerated density boundaries? It is possible to give physical arguments why their energy should on average decrease. Consider an extreme case where these boundaries completely reflect the zero-point oscillations as in the case of conducting boundaries and consider the case where two boundaries move with acceleration of opposite sign collide with each other and the case where the two boundaries move in opposite directions increasing the separation distance. In the
V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
first case the frequency of the zero-point oscillations will be up-shifted and their energy will increase while in the second case the frequency of the zeropoint oscillations will be down-shifted and their energy will decrease. But the time spent during the first type of collision will be less than that in the second type of collision since the distance travelled will be larger in the second case and thus on average the zero-point oscillations will lose energy. Real boundaries are partially transparent and for higher frequencies the refractive index decreases rapidly with frequency with the result that fewer particles receive energy. The larger the frequency the lower the refractive index and the energy of the zero-point oscillations can be transferred to fewer and fewer particles since the energy of each oscillation increases with frequency, forming a tail in the particle distribution. The tail will be a power-law in energy, because the refractive index falls with frequency according to a power-law Žsee below.. To find explicitly the rate of decrease of the zero-point energy for particle acceleration described by Ž1. we need to calculate not only the changes related to time varying particle distributions but additionally the nonlinear effects proportional both to the energy density of resonant waves and the spectral energy density of zero-point oscillations w1x. The same is true for real photons as described in detail in w19x. The difference is that for the dielectric constant of the zero-point oscillations not all the contributions to the dielectric constant that describes the propagating photons appear. This is in a sense trivial since the zero-energy oscillation can not go down in frequency and the corresponding matrix elements vanish Žsee w1x pp. 296–298.. Also the zero-point oscillations change their frequency adiabatically if their frequency is much larger than that related with the rate of particle acceleration described by Ž1.. We will demonstrate the effect for the zero-point oscillations of highest energy < k < 4 m e Ž " s c s 1, k is the momentum of zero-point oscillation.. Then this structure factor is equal to 1, i.e. under these conditions the change of the energy of the zero-point oscillations is determined by dispersion of the dielectric permittivity. We only take into account the fact that in the presence of particles the energy of a single zero-point oscillation is changed from < k
395
tric permittivity for electromagnetic transverse fields. For e t s 1 this expression indeed gives the energy < k
e ts1y
=
ž
2p e 2
v
2
f p dp
H Ž 2p .
Ly Ž p, k . e p y e pyk y v
3
q
Lq Ž p, k . e p q e pyk q v
/
,
Ž 2.
where 2
L . Ž p, k . s 1 .
m2 y Ž p P k . q Ž p P k . rk 2
e p e pyk
.
Ž 3. This is a fully relativistic and quantum expression as shown in Ref. w14x. We multiply Ž2. by v 2r4 differentiate the result with respect to the frequency, and integrate it over the whole phase volume of zero-point oscillations dkrŽ2p . 3 and calculate the change of their energy
V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
396
per unit time by using the change of the particle distribution in time described by Ž1. to get dW zero - point
1 s
dt
4
E
dk
H Ž 2p .
3
dkdp
H Ž 2p .
sy
y
6
Ev
ž
v2
de t dt
vs < k <
Ly Ž p, k .
Ž e p y e pyk y < k < .
Lq Ž p, k .
Ž e p q e pyk q < k < .
2
/
Iˆp f p .
2
Ž 4.
This result coincides with the expression found w1x for < k < 4 m e . In general there appears a factor w1x Ž e py k y e .r< k < which approaches 1 in the limit we considered here. The appearance of this factor is important for establishing the energy conservation law and to show that in general the decrease of the energy of zero-point oscillations has a consequence for particle creation. This simple result can be used to interpret the reaction of the particle distribution due to a lowering of the energy of zero-point oscillations. As discussed, the interaction of zero-point oscillations lead to an exchange of momenta with the particles, the higher the frequency of the zero-point oscillations the lower is the dielectric permittivity and therefore the lower are the potential barriers created for the zero-point oscillation by particles accelerated by the random waves. Therefore the energy of one high frequency oscillation can be transformed to a small number of particles, the number of particles decreases for higher frequencies. This is how the tail in the particle distribution is formed. The results of w1x are valid for any non-equilibrium distribution and take into account both the change of the zero-point oscillation energy and the change of the particle distribution function. The decrease of the energy of the zero-point oscillations is reflected not only in the particle distribution function but also in the rate of change of the energy of the resonant fields giving rise to overall conservation of energy found in w1x. This is important since the decrease of the zero-point oscillation energy does not mean an extraction of energy from the vacuum, in fact all the additional energy is extracted from the source which drives the resonant waves. The vacuum is only the catalyser of
the process and as soon as the acceleration disappear Iˆp s 0 the additional fast particle creation stops and the decrease of the zero-point oscillations also stops. An important question for the process we are considering here is how the vacuum returns to its initial stage when the acceleration stops. The change of the zero-point energy is accumulated during the whole process of acceleration. The source which excites the resonant waves should then produce the necessary emission to fill the decrease in energy of zero-point oscillations. It is important to find the spectral power density of the zero-point fluctuations. For non-relativistically accelerated particle and < k < 4 < pres < this distribution is described by the factor F Ž k .,Ž k s krm e .: FŽ k . s
'1 q k 2 y 1 3 k'1 q k 2 Ž k q '1 q k 2 . 4
.
Ž 5.
For k < 1 we have F Ž k . A k and for k 4 1 we find F A 1rk 2 . The spectrum of zero-point oscillations is shown on Fig. 1 which represents a wide spectrum around 0.5 MeV. Thus if the radiation emitted corresponds to the distribution Ž5. we should expect a hard x-ray emission from all regions where the particles are resonantly accelerated. This emission is measurable for most of the existing experiments w13x including laser-plasma interactions. Thus the regions of particle acceleration can be sources not only of fast particles but also the source of high frequency radiation. This prediction can easily be checked and we can give some estimate of the effect. The total change in energy of the electromagnetic field after time t between resonant particles and plasma waves with frequency v pe s 4p ne 2rm e can be estimated from w1x as
(
W ts
8 3p
ž
ln2 y
11
/
e2
24 "c
v pe t
n res ²< E res < 2 : n
4p
,
Ž 6.
where ²< E res < 2 :r4p is the average energy density of the accelerating resonant field and nres is the density of resonant particles much less than the plasma density n. We have assumed that the radiated energy after acceleration should be of the order of Ž4. as usual in transition emission processes. The numerical coefficient is obtained by integration with respect to k of the distribution of the type given by Ž5. but for
V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
397
Fig. 1. Wave number Žfrequency. dependence of the energy distribution of the changes to the zero-point oscillations caused by resonant particle acceleration.
Dirac particles Želectrons.. The only approximation in deriving Ž6. is made by letting kÕ s v pe for resonant particles. As an example we consider the turbulent region behind the laser pulse in laser plasma accelerators w11x. We take the resonant field in this region to be an order of magnitude less than that achieved for the coherent part of the excited plasma wave, i.e. we take E res f 10 8 Vrcm and ²< E res < 2 : f 3 = 10 15 V 2rcm 2 noting that the average field is at least 3 times less than the maximum field. Eq. Ž6. is in cgs units, therefore we divide E res by the permittivity of free space, i.e. 300. For a plasma density of the order of 10 19 cmy3 we have v pe f 2 = 10 14 sy1 , the duration of the random resonance field we take to be 3 times longer than the laser pulse duration, i.e. 3 = 10y1 2 s and the volume of the region where of the resonance field exist we take 10 times larger than the volume of the laser focus, i.e. we take this emitting volume to be f 10y9 cm3. We also assume that the density of the resonant particles is 10 times less than the plasma density. Then from Ž6. we find that their should be more than 10 3 hard x-ray photons of energy 300–500 keV emitted from this volume, is a measurable effect. Note that for this estimate we assume that the energy of the emitted photons be of order of magnitude equal to the change in the energy of the zero-point oscillations. This is taken from the analogy with the transition radiation processes including the time dependent refractive
index where the energy emitted is of the order of the change of the self-energy of the particles. Note also that this emission is related to the process where plasma boundaries reflect and decelerate the oscillations adiabatically. However, the temperature of Unruh w6x radiation due to the accelerating boundaries results in a lower frequency and emission rate is not included in the above estimate. The frequency of Unruh w6x radiation is given by " v f kT s Ž "r2p . arc and for arc s eErmc we get " v f Ž Õos rc .Ž " v 0r2p . where Õos is the classical quiver velocity in the laser radiation field and v 0 is the laser frequency. For Õos rc f 0.5 typical of present high power lasers the Unruh radiation frequency is v or4p i.e. about 10 6 times less than the effect considered in this paper. In laser plasmas the presence of stochastic fields is well established both experimentally, theoretically and numerically making the measurements easier than in ionization front experiments. The radiation described in this paper is due to the interaction of resonantly accelerated electrons in plasma waves with zero point quantum fluctuations. Many astrophysical sources are known to emit radiation with frequencies close to 0.5 MeV which is interpreted as electron–positron pair annihilation w21x. The observed line widths are in some cases much broader than Doppler broadening and the lines are down-shifted in frequency. It is worthwhile to con-
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V.N. TsytoÕich, R. Binghamr Physics Letters A 259 (1999) 393–398
sider the possibility of this emission being related to resonant particle acceleration since the curve given in Fig. 1 is peaked close to 0.5 MeV and is always rather broad and also down shifted in frequencies. In astrophysical plasmas the emitted volume and the time duration are much larger and the effect described here could lead to intense emission.
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