Transition radiation from a Dirac particle wave packet
traversing a mirror
P.O. Kazinski1 , G.Yu. Lazarenko1
e-mail: kpo@phys.tsu.ru
1Physics Faculty, Tomsk State University, Tomsk, 634050, Russia

20th Lomonosov Conference on Elementary Particle Physics,
Moscow State University, Moscow, August 19-25, 2021

References (see for more)
Phys. Rev. A 103, 012216 (2021); arxiv:2010.05236.
* The work is supported by the RFBR grant No. 20-32-70023.

P.O. Kazinski (TSU)

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1 / 19Outline1Problem statement and results

2Scheme3Model4

Probability of transition radiation

5Transition radiation from twisted particles

6General formula for radiation from 𝑁 -particle quantum state

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2 / 19Problem statement

Issue we are going to address
Does a quantum charged particle with the wave function of a nontrivial form
radiate as a charged fluid with the corresponding charged current?
What are the manifestations of the wave function profile in radiation fromcharged particles?
What are the quantum corrections to transition radiation due to quantum
recoil?
Whether the classical charged fluid picture is reproduced in the small
quantum recoil limit, as one may suspect.

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Outcomes

General results
1

The probability of radiation from a charged particle wave packet strongly
depends on the measurement of the state of the escaping charged particle.

2

The inclusive probability of radiation from a charged particle wave packet
cannot be described by the classical formula for radiation from thecorresponding current even in the small quantum recoil limit. In the case oftransition radiation, only the diagonal of the wave packet in the momentum
space contributes to the inclusive radiation probability.
The classical radiation from the current constructed with the aid of theparticle wave packet is reproduced in the case when:3

The escaping charged particle is measured in the state that is obtained fromthe initial one by a free evolution;
The coherent radiation from the 𝑁 -particle wave packet with identical
one-particle wave function profiles is considered for 𝑁 ≫ 1.

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Outcomes

Particular results
4The inclusive probability to record a photon created in transition radiationfrom a one Dirac particle wave packet traversing an ideally conducting plateis derived in the leading order of perturbation theory. The anomalous
magnetic moment of the Dirac particle is taken into account.

5

It is shown that the quantum corrections to transition radiation from anelectrically charged particle give rise to production of photons withpolarization vector orthogonal to the reaction plane. These corrections result
from both the quantum recoil and the finite size of a wave packet.

6

As for transition radiation produced by a neutron falling normally onto theconducting plate, the probability to detect a photon with polarization vector
lying in the reaction plane does not depend on the observation angle and theenergy of the incident particle.

7

The transition radiation produced by the wave packet of one twisted Dirac
particle is described.

8

The quantum formula for the inclusive probability to detect a photon
radiated by the 𝑁 -particle wave packet is derived.
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5 / 19Schemekgdz

pe-z

Figure : The schematic representation of the experimental installation for observation oftransition radiation. The electron wave packet falls onto the conducting plate resulting in theproduction of photons. The conducting plate is perforated in order to suppress the effects ofdiffraction and bremsstrahlung. The case of normal falling is depicted.
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6 / 19ModelLagrangian density of the system1¯ 𝜇 (𝑖𝜕𝜇 − 𝑒𝐴𝜇 ) − 𝑚]𝜓 − 𝜇𝑎 𝐹𝜇𝜈 𝜓𝜎¯ 𝜇𝜈 𝜓.
ℒ = − 𝐹𝜇𝜈 𝐹 𝜇𝜈 + 𝜓[𝛾42(1)𝜇𝑎 is the anomalous magnetic moment.
𝑒 is the particle charge.
𝑚 is the particle mass.

Quantum electromagnetic field in the interaction picture∑︁ ∫︁ 𝑉 𝑑k]︀01 [︀√a𝜆 (𝑘3 ; 𝑧)𝑒−𝑖𝑘0 𝑥 +𝑖k⊥ x⊥ 𝑐ˆ𝜆 (k) + H.c. .
Â(𝑥) =
32𝑘0 𝑉+ (2𝜋)𝜆(2)𝑘0 = |k|, 𝑉 characterizes the normalization volume. 𝜆 = ±1.
The integration region is restricted by the inequality 𝑘3 > 0.

Mode functions of the electromagnetic field∑︁ (𝜆)3a𝜆 (𝑘3 ; 𝑧) =f𝑟 (k)𝑒𝑖𝑟𝑘 𝑧 , f𝑟(𝜆) (k) = 𝑟f (𝜆) (k) + (1 − 𝑟)e3 𝑓 3(𝜆) (k),for 𝑧 > 0;

𝑟=±1

for 𝑧 < 0;(3)

a𝜆 (𝑘3 ; 𝑧) = 0,e3 is the unit vector along the 𝑧 axis.
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7 / 19ModelPolarization vector of a photon with helicity 𝜆√f (𝜆) (k) = (cos 𝜑 cos 𝜃 − 𝑖𝜆 sin 𝜑, sin 𝜑 cos 𝜃 + 𝑖𝜆 cos 𝜑, − sin 𝜃)/ 2,(4)𝑘⊥ = |𝑘1 + 𝑖𝑘2 |, 𝜑 = arg(𝑘1 + 𝑖𝑘2 ), sin 𝜃 := 𝑘⊥ /𝑘0 ≡ 𝑛⊥ , cos 𝜃 := 𝑘3 /𝑘0 ≡ 𝑛3 :(5)

k = 𝑘0 n = 𝑘0 (sin 𝜃 cos 𝜑, sin 𝜃 sin 𝜑, cos 𝜃).

State of the Dirac particle√︂∑︁ (2𝜋)3|𝜙⟩ :=𝜙˜𝛼 |𝛼⟩,𝑉𝛼∑︁ (2𝜋)3𝛼𝑉|𝜙˜𝛼 |2 =∑︁ ∫︁𝑑p|𝜙˜𝑠 (p)|2 = 1.

(6)𝑠𝜙
˜𝑠 (p) is concentrated at 𝑝3 < 0.

Density matrix
𝑅𝑠,¯𝑠 (p, p̄) := 𝜙𝑠 (p)𝜙*𝑠¯(p̄).

(7)

Spin density matrix
𝜌𝑠,¯𝑠 (p, p̄) := 𝑅𝑠,¯𝑠 (p, p̄)/𝑐(p),𝑐(p) :=∑︁∫︁𝑅𝑠,𝑠 (p, p),𝑑p𝑐(p) = 1.

𝑠(8)P.O. Kazinski (TSU)

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8 / 19ModelSpin vector𝑠𝜇 =(︁ 𝜁p𝑚,𝜁 +p(𝜁p) )︁
,𝑚(𝑝0 + 𝑚)

𝑠2 = −𝜁 2 .

𝑠𝜇 𝑝𝜇 = 0,(9)For pure states 𝜁 2 = 1.

General form of a pure state
𝜙0 (p) 𝑠(𝜅−𝑖𝜓)/2𝑒,𝜙𝑠 (p) = √2 cosh 𝜅𝜅, 𝜓 ∈ R.

(10)

𝜅 = 𝜅(p) and 𝜓 = 𝜓(p).

Diagonal of the density matrix and the spin density matrix𝑐(p) = |𝜙0 (p)|2 ,𝜌𝑠¯𝑠 =1𝑒𝜅(𝑠+¯𝑠)/2−𝑖𝜓(𝑠−¯𝑠)/2 .
2 cosh 𝜅(11)Rest frame spin vector𝜁 = 𝜏 tanh 𝜅 + 𝜁 0 / cosh 𝜅,𝜁 20 = 1,P.O. Kazinski (TSU)𝜁 0 𝜏 = 0,

𝜁 2 = 1.

𝑑
𝜁 = [𝜏 , 𝜁 0 ].
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(12)
(13)
9 / 19Probability of transition radiation

Inclusive radiation probability𝑑𝑃 (𝜆, k; 𝜙) =

∫︁
(𝜆)
*(𝜆)
𝑓𝑟𝑖 (k)𝑓𝑟′ 𝑗 (k)𝑑k𝑑p𝑐(p) ∑︁×32𝜋 3 𝑘0|𝑝′3 𝑝3 |(𝑝′3 − 𝑝3 + 𝑟𝑘3 )(𝑝′3 − 𝑝3 + 𝑟′ 𝑘3 )𝑟,𝑟 ′{︁ (︀)︀× 𝑒2 𝑝(𝑖 𝑝′𝑗) + 𝜂 𝑖𝑗 𝑞 2 /2 + 𝑖𝑚𝑞𝜇 𝑠𝜈 𝜀𝜇𝜈𝑖𝑗 +(︀+ 𝑒𝜇𝑎 2𝑚𝑞 2 𝜂 𝑖𝑗 − 2𝑚𝑞 𝑖 𝑞 𝑗 + 𝑖[2𝑚2 𝑞𝜇 𝑠𝜈 +)︀
+ (𝑞𝑠)𝑝𝜇 𝑞𝜈 + 𝑞 2 𝑞𝜇 𝑠𝜈 /2]𝜀𝜇𝜈𝑖𝑗 +(︀+ 𝜇2𝑎 𝑞 2 [2𝑚2 𝜂 𝑖𝑗 − (𝑝𝑖 + 𝑝′𝑖 )(𝑝𝑗 + 𝑝′𝑗 )/2] − 2𝑚2 𝑞 𝑖 𝑞 𝑗 +)︀}︁
+ 2𝑖𝑚[(𝑞𝑠)𝑝𝜇 𝑞𝜈 + 𝑞 2 𝑞𝜇 𝑠𝜈 /2]𝜀𝜇𝜈𝑖𝑗 .
𝜀0123 = 1,

𝜂 𝑖𝑗 = −𝛿𝑖𝑗 ,𝑞 2 = 2(𝑚2 − 𝑝𝑝′ ) = 𝑘32 − (𝑝3 − 𝑝′3 )2 < 0,𝑞𝜇 := 𝑝𝜇 − 𝑝′𝜇 ,𝑞 𝜇 = 𝑘𝑟𝜇 + 𝛿3𝜇 (𝑞 3 − 𝑟𝑘3 ).

(14)

(15)

𝑘𝑟𝜇 possesses the same components as 𝑘𝜇 but with 𝑘3 → 𝑟𝑘3 , 𝑟 = ±1.
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10 / 19Probability of transition radiation

Some properties of the general formula (14)1The expression in the curly brackets in (14) contracted with the polarization
vectors can be written in an explicitly Lorentz-invariant form.

2

The classical transition radiation from a point charged particle withoutmagnetic moment is described by the first term in the curly brackets standingat 𝑒2 in (14) provided one replaces 𝑝′𝑖 by 𝑝𝑖 and employs the small quantum
recoil approximation.

3

The terms depending on the effective particle spin contribute only to acircular polarization of radiated photons. They vanish when the polarization
vector f (𝜆) (k) is real and so corresponds to a certain linear polarization.

4

Only the momentum space diagonal of the density matrix contributes to theinclusive radiation probability. Hence, the inclusive radiation probability is notdetermined by the current constructed with the aid of the particle wave
function even in the small quantum recoil limit.

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Probability of transition radiation
Polarization vectorf ∼ [e3 , k]

(16)

The polarization vector (16) is normal to the reaction plane spanned by the vectors
p and k when the normal falling of the charged particle onto the conducting plateis considered.
It is known that transition radiation from a classical point particle with charge 𝑒 islinearly polarized in the plane spanned by the vectors p and k.

Inclusive radiation probability of photons with polarization vector (16) for normalfalling∫︁𝑑p𝑐(p)𝑛23 𝑑k(𝑒 + 2𝑚𝜇𝑎 )2𝑑𝑃 (𝜆, k; 𝜙) =.
(17)𝑚2
𝑛23 + 𝑛2⊥ 𝛾 2 (p) 16𝜋 3 𝑘0
The contribution (17) is entirely due to quantum effects (quantum recoil) and it isnot overlapped by the classical contribution which is zero.
The analogous effect exists in Vavilov-Cherenkov radiation [A. A. Sokolov, Iu. M.
Loskutov, Zh. Eksp. Teor. Fiz. 32, 630 (1957)].
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Transition radiation from twisted particles
Wave packet)︁
(︁ (𝑝 − 𝑝)2p2
3|𝑙|− ⊥2 + 𝑖𝑙𝜓 ,𝜙0 (p) = 𝑘𝑝⊥ exp −24𝜎34𝜎⊥

𝜙𝑠 (p) = 𝛿𝑠𝜎 𝜙0 (p),
𝜓 := arg(𝑝1 + 𝑖𝑝2 ),22 |𝑙|𝑘 −2 = (2𝜋)3/2 𝜎3 𝜎⊥(2𝜎⊥) |𝑙|!,(18)𝑙 ∈ Z.

𝜎 = ±1 characterizes the spin projection of the state onto the 𝑧 axis.

Properties of the twisted states (18)The state (18) is the eigenvector of the projection of the total angularmomentum operator on the 𝑧 axis with the eigenvalue 𝑙 + 𝜎/2. The case𝑙 = 0 describes a cylindrically symmetric Gaussian wave packet.
The momentum space diagonal of the density matrix(︁ (𝑝 − 𝑝)2p2⊥ )︁
32|𝑙|𝑐(p) = 𝑘 2 𝑝⊥ exp −−2 .
2𝜎322𝜎⊥(19)

As is seen from the general formula (14) for the inclusive radiation probability,
this probability does not depend on the phase of the wave function (18).
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13 / 19Transition radiation from twisted particles

Polarization vectors
f (1) = (cos 𝜃, 0, − sin 𝜃),

f (2) = (0, 1, 0).

(20)

n = (sin 𝜃, 0, cos 𝜃).

Inclusive radiation probability for 𝜇𝑎 = 0𝑑𝑃 (1, k; 𝜙) ≈{︁𝑒2 𝑝2
𝑛2 (𝑚2 + 𝑝2 )+(𝑚2 + 𝑛2⊥ 𝑝2 )2 ⊥2[︀ 4(|𝑙| + 1)𝜎⊥+𝑚 (1 − 10𝑛2⊥ + 10𝑛4⊥ )2222(𝑚 + 𝑛⊥ 𝑝 )]︀}︁ 𝑑k.
− 2𝑚2 𝑝2 𝑛2⊥ (5 − 7𝑛2⊥ + 𝑛4⊥ ) + 𝑝4 𝑛4⊥
4𝜋 3 𝑘03

𝑑𝑃 (2, k; 𝜙) ≈ 𝑒2

P.O. Kazinski (TSU)

2𝑝2 (|𝑙| + 1)𝜎⊥𝑛23 𝑑k.
(𝑚2 + 𝑛2⊥ 𝑝2 )2 4𝜋 3 𝑘03

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(21)

(22)

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Transition radiation from twisted particles

Properties of the inclusive radiation probability (21), (22)The leading contribution to (21) is the standard expression for the probabilityof transition radiation from the mirror [V. L. Ginzburg, V. N. Tsytovich, Transition
Radiation and Transition Scattering (Hilger, Bristol, 1990)]. The correction proportional to2𝜎⊥
is a non-paraxial quantum correction stemming from the wave packet
profile.
The non-paraxial quantum correction is the leading contribution to (22). The
2
non-paraxial corrections proportional to (|𝑙| + 1)𝜎⊥/𝑚2 are typical for theprocesses with twisted particles [see, e.g., D. Karlovets, Phys. Rev. A 98, 012137 (2018)].
These corrections are enhanced for the states with large projection of theorbital angular momentum.
Thus there are two quantum effects leading to the radiation of photons withpolarization vector f (2) lying out of the reaction plane. They are the quantum
recoil (17) and the divergence of momenta in the wave packet (22).

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Transition radiation from twisted particles
Inclusive radiation probability for 𝑒 = 0 (twisted neutron)(︁2 2 )︁(|𝑙| + 1)𝜎⊥𝑛3𝑑k𝑑𝑃 (1, k; 𝜙) = 𝜇2𝑎 1 −,2 22𝑚 + 𝑛⊥ 𝑝
4𝜋 3 𝑘0{︁2[︀ 4(|𝑙| + 1)𝜎⊥𝜇2𝑚 (1 − 3𝑛2⊥ )−𝑑𝑃 (2, k; 𝜙) = 2 𝑎 2 2 𝑚2 −2 2 2
2𝑚 + 𝑛⊥ 𝑝(𝑚 + 𝑛⊥ 𝑝 )]︀}︁ 𝑛23 𝑑k− 𝑚2 𝑝2 𝑛2⊥ (4 − 𝑛2⊥ ) − 𝑝4 𝑛4⊥.
4𝜋 3 𝑘0

(23)

(24)

Properties of the inclusive radiation probability (23), (24)The leading contribution to (23) does not depend on the form of the wave packet
provided 𝜎⊥ is sufficiently small. For example, this contribution has the same form
for the state (18) with two Gaussian humps with respect to the variable 𝑝3 . The
radiation described by this contribution is isotropic.
In the ultrarelativistic limit, the expressions for the leading contributions to (23) and(24) integrated over the azimuthal angle turn into classical formulas of [V. L.
Ginzburg, V. N. Tsytovich, Transition Radiation...] where one should take 𝜃0 = 𝜋/2. Inother words, in this limit the radiation is the same as from a classical magnetic
moment 𝜇𝑎 directed perpendicularly to the particle momentum.
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Radiation from 𝑁 -particle wave packet
Initial state of the system††
𝑐𝜙1in 𝛼1 · · · 𝜙𝑁in 𝛼𝑁 𝑎𝛼1 · · · 𝑎𝛼𝑁 |0⟩,(25)|𝑐|−2 = det(𝜙¯𝑖in 𝜙𝑗in ).
Inclusive radiation probability in the leading order of perturbation theory
𝑃 (𝛾) = |𝑐|2𝑁∑︁𝑑𝑙𝑘 ,𝑘,𝑙=1

𝜙¯1in 𝜙1in⎢..
⎢
⎢ 1 . 𝑙𝑑𝑙𝑘 := det ⎢⎢ 𝜙¯in 𝐴𝜙in⎢..
⎣.
𝜙¯1in 𝜙𝑁in⎡···

𝜙¯𝑘in 𝐴† 𝜙1in..
.

······

𝜙¯𝑘in 𝐴† 𝐴𝜙𝑙in······..
.
𝜙¯𝑘in 𝐴† 𝜙𝑁in···⎤1𝜙¯𝑁in 𝜙in⎥..
⎥.
⎥𝑙 ⎥𝜙¯𝑁
𝐴𝜙inin ⎥ .
⎥..
⎦.
𝑁 𝑁
𝜙¯in 𝜙in(26)𝐴𝛼′ 𝛼 is the one-particle transition amplitude.
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Radiation from 𝑁 -particle wave packet
Approximation(𝜙¯𝑖in 𝜙𝑗in ) ≈ 𝛿𝑖𝑗 ,(27)𝑖, 𝑗 = 1, 𝑁 .

Inclusive radiation probability
𝑃 (𝛾) ≈𝑁∑︁𝜙¯𝑘in 𝐴† 𝐴𝜙𝑘in +𝑘=1𝑁∑︁′𝑁∑︁]︀
′ [︀(𝜙¯𝑘in 𝐴† 𝜙𝑘in )(𝜙¯𝑙in 𝐴𝜙𝑙in ) − (𝜙¯𝑘in 𝐴† 𝜙𝑙in )(𝜙¯𝑙in 𝐴𝜙𝑘in ) ,𝑘,𝑙=1𝑁𝑁⃒ ∑︁⃒2 ∑︁⃒⃒ 𝑙⃒⃒⃒𝜙¯in 𝐴𝜙𝑙in ⃒2 .
(𝜙¯𝑘in 𝐴† 𝜙𝑘in )(𝜙¯𝑙in 𝐴𝜙𝑙in ) = ⃒𝜙¯𝑙in 𝐴𝜙𝑙in ⃒ −𝑘,𝑙=1

𝑙=1

𝑙=1

(28)

Interpretation of the terms in (28)
The first term in (28) is the incoherent sum of the quantum contributions toradiation from the wave packets 𝜙𝑘 .
The second term in (28) is the classical contribution to radiation produced by thesum of the classical currents corresponding to the wave packets 𝜙𝑘 , the incoherent
contribution to the classical radiation should be excluded.
The third term in (28) is the exchange term.
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Radiation from 𝑁 -particle wave packet
Neglecting the exchange term, we have𝑃 (𝛾) ≈𝑁∑︁(︀2𝜙¯𝑘in 𝐴† 𝐴𝜙𝑘in − ⃒𝜙¯𝑘in 𝐴𝜙𝑘in ⃒⃒⃒ )︀

𝑁⃒ ∑︁
⃒2⃒⃒
+⃒
𝜙¯𝑘in 𝐴𝜙𝑘in ⃒ .

𝑘=1(29)𝑘=1

Some properties of (29)The classical contribution with the total current of 𝑁 wave packets
determines the inclusive probability of coherent radiation from the 𝑁 -particle
wave packet.
This term dominates provided the classical radiation amplitudes
corresponding to the wave packets 𝜙𝑘 add up coherently and 𝑁 is large.
For example, the transition radiation from a bunch train of twisted electrons
with the wave functions obtained from each other by a parallel transport
contains coherent harmonics. The radiation at these harmonics isapproximately the coherent radiation from point charged magnetic momentswith 𝜇 ≈ 𝑙𝜇𝐵 , where 𝑙 is a large projection of the orbital angular momentumof a one twisted electron [I. P. Ivanov, D. V. Karlovets, Phys. Rev. Lett. 110, 264801 (2013)].
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