Vacuum-to-vacuum transition probability and the classic radiation theory

Radiation Physics and Chemistry 106 (2015) 268–270

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Short Communication

Vacuum-to-vacuum transition probability and the classic radiation theory Edouard B. Manoukian The Institute for Fundamental Study, Naresuan University, Phitsanulok 65000, Thailand

H I G H L I G H T S







Quantum viewpoint of radiation theory based on the vacuum-to-transition probabilities. Mathematical method in handling radiation for extended and point sources. Radiated energy and power for arbitrary source distribution obtained from the above. Explicit power of radiation for point relativistic sources from the general theory.

art ic l e i nf o

a b s t r a c t

Article history: Received 26 June 2014 Accepted 2 August 2014 Available online 11 August 2014

Using the fact that the vacuum-to-vacuum transition probability for the interaction of the Maxwell field Aμ ðxÞ with a given current J μ ðxÞ represents the probability of no photons emitted by the current of a Poisson distribution, the average number of photons emitted of given energies for the underlying distribution is readily derived. From this the classical power of radiation of Schwinger of a relativistic charged particle follows. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Electromagnetic radiation Quantum viewpoint of Maxwell's theory Classical and quantum probability Classic radiation theory

1. Introduction

(Manoukian, 2011). That is

The Maxwell Lagrangian density for the interaction of the vector potential Aμ ðxÞ with an external current J μ ðxÞ is given by

Prob½N ¼ n ¼

1 LðxÞ ¼  F μν ðxÞF μν ðxÞ þJ μ ðxÞAμ ðxÞ; 4

F μν ðxÞ ¼ ∂μ Aν ðxÞ  ∂ν Aμ ðxÞ:

ð1Þ

Prior to switching on of the current, as a source of photon production, one is dealing with a vacuum state, denoted by j0  〉, involving no photons. After switching on of the current, the state of the system may evolve to one involving any number of photons, or it may just stay in the vacuum state, involving no photons, with the latter state now denoted by j0 þ 〉. Quantum theory tells us that the vacuum-to-vacuum transition probability satisfies the inequality j〈0 þ j0  〉j2 o 1, due to conservation of probability, allowing the possibility that the system evolves to other states as well involving an arbitrary number of photons that may be created by the current source. A very interesting property of this system is that the probability distribution of the photon number N created by the current (Schwinger, 1970) is given by the Poisson distribution E-mail address: [email protected] http://dx.doi.org/10.1016/j.radphyschem.2014.08.003 0969-806X/& 2014 Elsevier Ltd. All rights reserved.

ðλÞn  λ e ; n!

n ¼ 0; 1; …; λ ¼ 〈N〉;

ð2Þ

where λ ¼ 〈N〉 denotes the average number of photons created by the current source, and exp½  〈N〉 ¼ j〈0 þ j0  〉j2 ;

ð3Þ

denotes the probability that no photons are created by the current source, i.e., it represents the vacuum-to-vacuum transition probability j〈0 þ j0  〉j2 as just stated. The purpose of this communication is by using the expression of the vacuum-to-vacuum transition amplitude, derive the exact expression of the average number of photons, of a given frequency, produced by a given general current distribution, from which the classic radiation theory of radiation emitted from a relativistic charge particle may be readily obtained for the power of radiation as well as for the power of radiation emitted along a given direction. Quantum viewpoint analysis, as discussed above, of electromagnetic phenomena and electromagnetic radiation, e.g., Manoukian and Viriyasrisuwattana (2006), and of the related applications (Feynman, 1985; Bialynicki-Birula, 1996; Manoukian, 1997; Kennedy et al., 1980;

E.B. Manoukian / Radiation Physics and Chemistry 106 (2015) 268–270

Deitsch and Candelas, 1979; Manoukian and Viriyasrisuwattana, 2006; Manoukian, 2013; Schwinger, 1973) turns out to be quite useful in applications and certainly in simplifying, to a large extent, derivations in this field as we will witness below. In particular, we note that due to the generality of the expressions leading to the total energy of radiation emitted, derived below in (18), and being valid for arbitrary current distributions, the analysis is expected to have further applications in the domain of synchrotron radiation as well as in considering quantum corrections to general physical problems in radiation theory. We are especially interested in generalizing the present analysis to radiation in the presence of obstacles as well in different media than just the vacuum and will be attempted in a future report.

269

By making a simultaneous transformation ðx0 ; Q Þ2ðx;  Q Þ in the pair of the second exponentials within the square brackets, and taking into consideration the symmetry of the product J μ ðx0 ÞJ μ ðxÞ under the transformation x0 2x, the above expression simplifies to Z Z 3  1 d Q ω 0 ; ei Q ðx  xÞ δ jQ j ðdx0 ÞðdxÞ J μ ðx0 ÞJ μ ðxÞ NðωÞ ¼ 4 3 c ℏc 2jQ jð2π Þ Q 0 ¼ jQ j:

ð14Þ

We introduce the unit vector n, via the equation Q ¼ jQ jn;

ð15Þ

and use the fact that 2. Average number of photons emitted of a given frequency The vacuum-to-vacuum transition of the theory described by the Lagrangian density in (1) is given by (Schwinger, 1969)   Z i 〈0 þ j0  〉 ¼ exp ðdx0 ÞðdxÞ J μ ðx0 Þημν Dðx0 ; xÞJ ν ðxÞ ; 3 2ℏ c ðdxÞ ¼ dx0 dx1 dx2 dx3 ;

ð4Þ

½ημν  ¼ diag½  1; 1; 1; 1, the photon propagator is given by Z 0 2 ðdQ Þ ei Q ðx  xÞ Dðx0 ; xÞ ¼ ; Q 2 ¼ Q 2 Q 0 ; ϵ- þ 0; 4 2 ð2π Þ Q  iϵ

ð5Þ

and J μ ðxÞ is the conserved four-current ∂μ J μ ðxÞ ¼ 0, ðJ μ ðxÞÞn ¼ J μ ðxÞ. The vacuum-to-vacuum transition probability is then   Z 1 ð6Þ j〈0 þ j0  〉j2 ¼ exp  3 ðdx0 ÞðdxÞ J μ ðx0 Þημν ðIm Dðx0 ; xÞÞJ ν ðxÞ ; ℏc Im Dðx0 ; xÞ ¼ π

Z

ðdQ Þ ð2π Þ4

This gives from (6) " j〈0 þ j0  〉j2 ¼ exp 

π

ℏ c3

0

δðQ 2 ÞeiQ ðx  xÞ : Z

ð7Þ

ðdx0 ÞðdxÞ J μ ðx0 ÞJ μ ðxÞ

Z

ðdQ Þ ð2π Þ4

# 0

δðQ 2 ÞeiQ ðx  xÞ : ð8Þ

3    d Q ¼ dΩ Q  dQ j; jQ j

ð16Þ

to carry out the jQ j-integral in (14), using the property of the delta function δðjQ j ω=cÞ, giving the exact expression Z Z ω 0 00 0 ðdx0 ÞðdxÞ J μ ðx0 ÞJ μ ðxÞ dΩ eiωðx  xÞn=c e  iωðx  x Þ=c : NðωÞ ¼ 16π 3 ℏ c5 ð17Þ To obtain the total energy of radiation EðωÞ per unit angular frequency about ω, we simply have to multiply the above expression by ℏω, leading to Z Z ω2 0 00 0 ðdx0 ÞðdxÞ J μ ðx0 ÞJ μ ðxÞ dΩ eiωðx  xÞn=c e  iωðx  x Þ=c ; EðωÞ ¼ 3 5 16π c ð18Þ and is independent of ℏ as expected. The total energy is then given by Z Z 1 1 1 dω EðωÞ ¼ dω EðωÞ; ð19Þ E¼ 2 1 0 where in writing the last equality, we have used the reality condition on EðωÞ implying that the latter is an even function of ω.

3. The classic radiation theory

We introduce the resolution of the identity expressing the equality of the energy of a photon ℏω ¼ ℏjQ j c: Z 1 dω δðω  jQ jcÞ; ð9Þ 1¼

The current of a relativistic charged particle of charge e is given by the well known expressions

as well as the average of the photon number density NðωÞ of energy ℏω: Z 1 〈N〉 ¼ dω NðωÞ: ð10Þ

J 0 ðxÞ ¼ ec δ ðx  RðtÞÞ:

0

ð11Þ where we have used the fact that δðω  jQ jcÞ ¼ δðjQ j  ðω=cÞÞ=c. Upon using the relation

δðQ 2 Þ ¼

δðQ 0  jQ jÞ þ δðQ 0 þ jQ jÞ 2jQ j

;

ð12Þ

in (11), we obtain Z Z 3  1 d Q ω NðωÞ ¼ δ jQ j  ðdx0 ÞðdxÞ J μ ðx0 ÞJ μ ðxÞ 4 3 c 2ℏ c 2jQ jð2π Þ i Q ðx0  xÞ

½e

e

 ijQ jðx00  x0 Þ

þe

iQ ðx0  xÞ ijQ jðx00  x0 Þ

e

:

ð20Þ

ð3Þ

ð21Þ

This gives

0

Upon inserting the identity given in (9) in the Q-integral in the exponential in (8), we obtain the explicit expression for the photon number density to be given by Z Z π ðdQ Þ  ω 0 NðωÞ ¼ δ jQ j  δðQ 2 ÞeiQ ðx  xÞ ; ðdx0 ÞðdxÞ J μ ðx0 ÞJ μ ðxÞ 4 4 c ℏc ð2π Þ

d _ RðtÞ ¼ R; x0 ¼ c t; dt

ð3Þ JðxÞ ¼ eR_ δ ðx RðtÞÞ;

e2 E¼ 32π 3 c

Z

1 1

dω ω

Z 2

dΩ

eiω½Rðt Þ  RðtÞn=c e  i ωðt 0

0

Z

Z

1

dt 1

 tÞ

"

R_ ðt 0 Þ  R_ ðtÞ dt 1 c2 1 1

:

ð22Þ

For completeness and convenience of the reader we spell out the details in integrating this expression. To this end, we follow Schwinger and set   ½Rðt þ τÞ RðtÞ ¼ γ; ð23Þ t 0  t ¼ τ; τ 1  τc and note that   ½Rðt þ τÞ  RðtÞ   o c;   τ

τ- 71 ) γ - 7 1;

to obtain ð13Þ

#

0

E¼

e2 32π 3 c

Z

1 1

d ω ω2

Z

dΩ

Z

Z

1

1

dt 1

1

dγ e  i ωγ

ð24Þ



dτ dγ



270

E.B. Manoukian / Radiation Physics and Chemistry 106 (2015) 268–270

" R_ ðt þ τÞ  R_ ðtÞ c2

# 1 :

ð25Þ

Using the fact that  2 Z 1 d dω ω2 e  i ωγ ¼  ð2π Þ δðγ Þ: d γ 1

ð26Þ

Using the set of integrals Z dΩ 4π ¼ ; ½1  V^  n3 ½1  V^ 2 2 Z

ni dΩ 16π ¼ 4 ^ 3 ½1  V  n

gives E¼ 

e2 16π 2 c

Z

dΩ

Z

Z

1

1

dt 1

1

dγ δðγ Þ



d dγ

2 

 dτ ^ ½V ðt þ τÞ:V^ ðtÞ 1 ; dγ

R_ V^ ¼ : c

ð27Þ as a consequence of a property of the delta function. The chain rule d dτ d ¼ ; dγ dγ dτ

dτ 1 ¼  f ðt þ τÞ; dγ ½1  V^ ðt þ τÞ  n

4π

"

δij

2 ½1  V^ 3 3

;

þ2

ð34Þ i j V^ V^ 2

½1  V^ 

# ;

ð35Þ

" _ 2# 2 e2 1 _ 2 ðV^  V^ Þ ^ dΩ Pðn; tÞ ¼ V þ : 2 3 c 4π ½1  V^ 2 2 ½1  V^ 

ð36Þ

Acknowledgment The author wishes to thank his colleagues at the Institute for the interest they have shown in this particularly direct derivation. References

ð30Þ

ð31Þ

The power of radiation is independent of the surface term, and along the unit vector the Schwinger (1973) expression for it follows 8 9 2 _2 _ _ V^ e2 < n  V^ V^ _V^ ð1  V^ ÞðnV^ Þ2 = þ 2  : 16π 2 c:½1  V^  n3 ½1  V^  n5 ; ½1  V^  n4

2

½1  V^ 3

The simplicity of the derivations given above should be noticed. The results derived in Section 2 are quite general and are valid for arbitrary current distributions and are expected to be applicable in other problems as well. Some of such applications and further generalizations were mentioned at the end of Section 1, and will be attempted in a future report.

and is proportional to the acceleration for t- 7 1, while the remaining terms on the extreme right-hand side of (29) are given by

Pðn; tÞ ¼

i V^

gives the power of radiation PðtÞ ¼

ð28Þ

ð29Þ

2 _2 _ _ V^ n  V^ V^ _V^ ð1  V^ ÞðnV^ Þ2 þ 2  : ½1  V^  n4 ½1  V^  n3 ½1  V^  n5

ni nj dΩ ¼ ½1  V^  n5 Z

and the fact that γ ¼ 0 gives τ ¼ 0 allow one to write   Z 1 dτ d dτ d dτ ^ dγ δðωÞ ½V ðt þ τÞ:V^ ðtÞ  1 dγ dτ dγ dτ dγ 1 d d d d ^ ¼ V ðtÞ  f ðtÞ f ðtÞ f ðtÞV^ ðtÞ  f ðtÞ f ðtÞ f ðtÞ dτ dt dt dt d d _ _ ðtÞ d ðf ðtÞV^ Þ þ f ðtÞ _ 2 f ðtÞ; ¼ ½FððtÞ  V^  f ðtÞ2 ðf ðtÞV^ Þ  V^ f ðtÞf dt dt dt where F(t) is a surface term given by   d FðtÞ ¼ f ðtÞ2 V^  ðf ðtÞV^ Þ  f_ ðtÞ dt 2 ! ^ nð1  V^ Þ V _ ^  ¼V  : ½1  V^  n3 ½1  V^  n4

Z

ð33Þ

ð32Þ

Bialynicki-Birula, I., 1996. Photon wave function. Progress Opt. 36, 245–294. Deitsch, D., Candelas, P., 1979. Phys. Rev. D 20, 3063–3080. Feynman, R.P., 1985. QED: The Strange Theory of Light and Matter. Princeton University Press, Princeton New Jersey, pp. 17–35, 64–72. Kennedy, G., Critchley, R., Dowker, J., 1980. Ann. Phys. 125, 346. Manoukian, E.B., 1997. Quantum field theory of reflection and refraction of light II. Hadronic J. 20, 251–265. Manoukian, E.B., 2011. Modern Concepts and Theorems of Mathematical Statistics. Springer, New York. Manoukian, E.B., 2013. Red light blue light Feynman and a layer of glass: the causal propagator. J. Electromagn. Waves Appl. 27, 1986–1996. Manoukian, E.B., Viriyasrisuwattana, P., 2006. Quantum correction to the photon number emission in synchrotron radiation. Radiat. Phys. Chem. 76, 774–776. Schwinger, J., 1969. Particles and Sources. Gordon and Breach, New York, p. 14. Schwinger, J., 1970. Particles, Sources, and Fields. Addison-Wesley, Reading, Massachusetts, pp. 67–68. Schwinger, J., 1973. Classical radiation of accelerated electrons II. A quantum viewpoint. Phys. Rev. D 7, 1696–1701.