On the theory of scalar pair production by a potential barrier
The problem of the scalar pair production by a one-dimensional vector-potential Am(x₃) is reduced to the S-matrix formalism of the theory with an unstable vacuum. Our choice of in- and out-states does not coincide with that of other authors and we argue extensively in favor of our choice. We show th...
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| Published in: | Вопросы атомной науки и техники |
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| Date: | 2001 |
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| Format: | Article |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | On the theory of scalar pair production by a potential barrier / A.I. Nikishov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 103-109. — Бібліогр.: 20 назв. — англ. |
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| author_facet | Nikishov, A.I. |
| citation_txt | On the theory of scalar pair production by a potential barrier / A.I. Nikishov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 103-109. — Бібліогр.: 20 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The problem of the scalar pair production by a one-dimensional vector-potential Am(x₃) is reduced to the S-matrix formalism of the theory with an unstable vacuum. Our choice of in- and out-states does not coincide with that of other authors and we argue extensively in favor of our choice. We show that the norm of a solution of the wave equation is determined by one of the amplitude of its asymptotic form for x₃→±∞. For the constant electric field we obtain the scalar particle propagator in terms of the stationary states and show that with our choice of in- and out-states it has the form dictated by the general theory.
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ON THE THEORY OF SCALAR PAIR PRODUCTION
BY A POTENTIAL BARRIER
A.I. Nikishov
I.E. Tamm Department of Theoretical Physics
P.N. Lebedev Physical Institute, Moscow, Russia
e-mail: nikishov@lpi.ru
The problem of the scalar pair production by a one-dimensional vector-potential Aµ(x3) is reduced to the S-matrix
formalism of the theory with an unstable vacuum. Our choice of in- and out-states does not coincide with that of
other authors and we argue extensively in favor of our choice. We show that the norm of a solution of the wave
equation is determined by one of the amplitude of its asymptotic form for x3→±∞. For the constant electric field we
obtain the scalar particle propagator in terms of the stationary states and show that with our choice of in- and out-
states it has the form dictated by the general theory.
PACS: 11.55Ds; 12.20Ds; 03.80+r
1. INTRODUCTION AND THE CHOICE OF
IN- AND OUT-STATES
Pair production by an external field can be treated
either in the framework of S-matrix formalism [1-5], or
equivalently by the Feynman method using the
propagators [1,6-8]. For the stationary potential the field
is not switched off for t→±∞. So the reduction to the S-
matrix formalism requires choosing the in- and out-
states. How to do this is briefly shown in [1]. Another
choice is made in [9] and accepted in later literature
[10,11]. The correct choice is especially important in
dealing with higher order processes, when the answer is
not known in advance from some other considerations.
In this paper we argue extensively in favor of our
choice. It is reasonable to consider the case of scalar
particle separately, because the complications due to the
spin are absent here. Besides it is useful to have all the
stages of a more simple case before eyes, when treating
the spinor case.
We consider at first the one-dimensional potential
A0(x3) and assume for the beginning that the
corresponding electrical field
3
0
3 x
AE
∂
∂−= disappears
for x3→±∞. We use the metric
ηµν=diag(-1,1,1,1). (1)
It is useful to introduce the kinetic energy π0(x3) and
momentum π3(x3) of a classical particle defined by the
expressions
π0(x3)=p0-eA0(x3), 2
3
2
033 )()( ⊥−= mxx ππ ,
2
2
2
1
22 ppmm ++=⊥ . (2)
The first relation in (2) merely expresses the total
energy conservation. We also use the notation
)()( 0
3
0
3
±=| ± ∞→ ππ xx
22
0333 )()( )(
3 ⊥± ∞→ −±=±= mx x πππ , (3)
In contrast to [1] and [5] we assume here that the
charge of a scalar particle e=-|e| in order the analogy
with the electron would be closer. We are interested
here mainly in the states that can be created by the field
(Klein region). Assuming for definiteness E3>0, we
have in this region
20 )( ⊥>− mπ , 20 )( ⊥−<+ mπ , (4)
i.e. large positive x3 are accessible only to antiparticles.
For brevity reasons we write only the wave function
factor depending on x3. Outside the field the particle is
free and we first classify the states by their asymptotic
form
[ ] [ ]33
2/1
3 )(exp)(2
3
xif xp −±−=| −
− ∞→± ππ
[ ] [ ]33
2/1
3 )(exp)(2
3
xif xp +±+=| −
∞→
± ππ (5)
The normalization factors are chosen in that way that
the density current along the third axis is equal to unity
up to a sign. Two sets of functions in (5) are connected
by the relations
ppppp fcfcf −+
+ += '
2
'
1 ,
ppppp fcfcf −+
− += '*
1
'*
2 . (6)
The second relation can be obtained from the first one
by complex conjugation. The current conservation
along the third axis gives (in Klein region)
1
2'
2
2'
1 =− pp cc . (7)
From (6) and (7) we find the reversed relations
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 103-109. 103
ppppp fcfcf −+
+ −= '
2
'*
1 ,
ppppp fcfcf −+
− −−= '
1
'*
2 . (8)
Now we have to classify solutions as in- and out-states.
Our choice is [1]
ppoutp f+
−− =≡ ψψ , ppoutp f+++ =≡ ψψ ,
p
p
p
pinp c
c
ψψψ ~
'*
2
'
2
−−− −=≡ , pp f−− =ψ~ ,
ppoutp f−
++ =≡ ψψ . (9)
Here the ± indexes before ψ-functions indicate the sign
of frequencies.
The heuristic argument in favor of this choice was
based on the fact that the description of a scattering
process in terms of pure states (unlimited vectors) is
only a way to a more realistic description by means of
wave packets. For the wave packets the field is
effectively switched off, when they leave the field
region. Consider, for example, the process described by
pf+ , see the first relation in (6). Initially we have the
antiparticle current with amplitude '
1pc moving from the
region of large positive x3 towards the field region near
x3=0. In the opposite direction from the region of large
negative x3 moves the particle current with unity
amplitude. It annihilates completely in the field region.
As a result we have the diminished antiparticle current
reflected from the barrier. So for t→∞ there is only
antiparticle packet, i.e. ppf ψ−
+ = . We would like to
remind you here that the momentum of the negative-
frequency wave function is opposite to the antiparticle
velocity.
In terms of in- and out-states the relations (6) and (8)
take the form
ppppp cc ψψψ −+
+ += 21 ,
ppppp cc ψψψ −+
− += *
1
*
2 ; (10)
ppppp cc ψψψ −+
+ −= 2
*
1 ,
ppppp cc ψψψ −+
− +−= 1
*
2 ; (11)
1
2
2
2
1 =− pp cc , '
2
'
1
1
p
p
p c
c
c −= ,
'
2
2
1
p
p c
c = . (12)
We note that in [1] the present f-functions were denoted
as ψ and it was explained that after using the
transformation indicated in (9) and (12) we get relations
(10-11) with (new) ψ-functions. The latter relations
coincide with those for the non-stationary solutions. Just
on the bases of these relations the S-matrix formalism is
build [1,5].
In terms of ψ-functions the field operator Ψ has the
usual form
∑
∑
−+
−+
+=
=+=Ψ
p
ppoutppout
p
ppinppin
ba
ba
)(
)(
†
†
ψψ
ψψ
(13)
Here pina is the destruction operator of a particle in the
state pψ+ and −
†
pinb is the creation operator of an
antiparticle in the state pψ− . The sum is over all p
including p in the Klein region.
In paper [9] the field operator is written as
∑
∑
+=
=+=Ψ
k
koutkoutkoutkout
k
kinkinkinkin
xnbxpa
xnbxpa
))()((
))()((
†
†
, (14)
see (31a), (31b) in [9]. Disregarding here the
normalization factors, the connections to our ψ-
functions are
kkinp ψ−= , kkinn ψ+= ,
kkoutp ψ−= , kkoutn ψ+= . (15)
The authors of [9] name their pk in-function the state of
incoming particle on the grounds that there are no other
waves for x3→±∞. In our nomenclature this is the
outgoing negative frequency state as explained above.
So the disagreement is in the sign of frequencies and in
the in- and out- labeling. The authors of subsequent
papers [10,11] accepted the classification of [9].
Another argument in favor of our choice (9) is as
follows [1]. In the case of a constant electric field we
can work either with stationary or non-stationary
solutions. For the latter the classification of states is
obvious. The experiment should be described in terms
of wave packets and using stationary or non-stationary
solutions should produce the same result. This agrees
with the fact that in both cases the relations (10) and
(11) have the same form with the same coefficients c1p,
c2p . By the way it is shown in [1] why the strict S-
matrix formulation is possible despite the fact that the
constant field is not turned off for t→±∞. The reason is
that for the given set of quantum numbers p the
formation length for pair production is finite. Outside
this length the field does not create pairs and does not
prevent the S-matrix formulation as does any field that
does not create pairs.
2. ORTHONORMALIZATION OF WAVE
FUNCTIONS
The choice (9) assumes that ψ+ and ψ− and also ψ+
and ψ− are orthogonal. We shall show in this section
that this is so. For definiteness we assume that (4) is
satisfied. The Klein- Gordon equation has the form
104
[ ] 0
)(2
0
2211
3
22
0
2
0
200
2
3
2
=×
×
−++−
−+
⊥
tpxpxpi
p
e
xfmpAepeA
dx
d
(16)
The J0 component of a transition current is given by the
expression
[ ]ψψψψψψ *'
00
'*'0 )(),( DDiJ −= ,
00 ieA
t
D −
∂
∂= . (17)
For our potential A0 and functions 'pf , pf we have
[ ]
[ ] pp
pp
ffxx
ffxeAppffJ
*
'3
0
3
0
*
'3
0000
)(')(
)(2'),'(
ππ +=
=−+=
. (18)
We consider also the J3 component of the transition
current
∂
∂−
∂
∂−≡
≡
∂
∂−=
*
'
33
*
'
3
*
'3
'
),'(
pppp
pp
f
x
ff
x
fi
f
x
ifffJ
(19)
and calculate its derivatives over x3 using (16)
000
3
3
)'( JppiJ
dx
d −= . (20)
From here we have
[ ]∫
−
−== −
−
=
u
d
du
L
L
LxLx JJ
pp
iffJdx
33 3300
0
3 '
),'( . (21)
For '
'
pff += , pff += we obtain (for details see [12])
∫ −−=++ )'(2),( 002'
2'
0
3 ppcffJdx ppp π δ . (22)
So for pf+ normalized as in (5) we have (22). One can
verify that ψ+ and ψ− are orthogonal.
3. SOLVABLE POTENTIAL
For the potential
A0(x3)=-tanh kx3 (23)
the solutions of the Klein-Gordon equation are
known [1]
);12;,(
)1()(
)(2
1)(
3
3
ziiiiiF
zzxf i
p
+++−+−−×
×−−
−
= −
−
µλνµλνµ
π
λµ
µπ kez kx 2)(, 3
2 3 =−=− , νπ k2)(3 =+ ,
eap +=± 00 )(π ,
λλ ~2/1 += ,
2
4
1~
−=
k
eaλ . (24)
Here F(α,β;γ;z) is the hypergeometric function. π3(±)
are real in the Klein region. Three other solutions with
quantum numbers p can be obtained from (24) by
employing the discrete symmetry of the Klein-Gordon
equation [13]. Thus )( 3xf p+ can be obtained from
)( 3xf p− by substitution μ→-μ,
);12;,(
)1()(
)(2
1)(
3
3
ziiiiiF
zzxf i
p
++++−×
×−−
−
=+
µλνµλνµ
π
λµ
(25)
)( 3xf p
+ ( )( 3xf p
− ) can be obtained from )( 3xf p− (
)( 3xf p+ ) by substitutions (not changing the Klein-
Gordon equation (16)) ,,,33 νµ →−→−→ aaxx
)()( 00 +↔− ππ :
);12;,(
)1()(
)(2
1)(
1
1
3
3
−
−+
+−+−+−−×
×−−
+
=
ziiiiiF
zzxf i
p
µλνµλνµ
π
λµ
(26)
);12;,(
)1()(
)(2
1)(
1
1
3
3
−
−−−
+++++−×
×−−
+
=
ziiiiiF
zzxf i
p
µλνµλνµ
π
λµ
(27)
The coefficients c'1p, c'2p in (6) and (8) have the form
)1()(
)2()12(
)(
)('
3
3
1 λνµλνµ
νµ
π
π
−++Γ++Γ
Γ+Γ
−
+=
iiii
iic p ,
)1()(
)2()12(
)(
)('
3
3
1 λνµλνµ
νµ
π
π
−+−Γ+−Γ
−Γ+Γ
−
+=
iiii
iic p . (28)
Two special cases are: the step potential (k→∞ in (23))
and constant electric field (k→0, ∞→a ,
ConstEak == ).
4. CONSTANT ELECTRIC FIELD
In this Section we obtain the propagator for the
scalar particle in a constant electric field and show that
in terms of our in- and out-states it has the form dictated
by the general theory. The vector-potential (23)
reduces to
30ExA µµ δ= , 0
0 AA −= . (29)
With this potential the Klein-Gordon equation (16)
takes on the form
0)(
24 3
2
2
2
=
−+ xfZ
dZ
d λ
(30)
)(2)(2 3
0
0
3 x
eEeE
pxeEZ π−=+= ,
eE
m2
⊥=λ . (31)
The solutions, normalized on unity current, are, see
Eq. (8.2.5) in [14],
)( 4/ ZeDcf i
ppp
π
νψ −== +
−
,
)( 4/
* ZeDcf i
ppp
π
νψ −
−− −== ,
105
)( 4/ ZeDcf i
ppp
π
νψ == −
+ ,
)( 4/
* ZeDcf i
ppp
π
νψ −++ == , (32)
2
1
2
−= λν i
, )
8
exp()2( 4/1 π λ−= eEc , 0
21 ,, pppp = . (33)
Now in the relations (6) and (8)
−
−Γ
= )(
4
exp
2
1
2'1 i
i
c p λπ
λ
π
,
= λπ
2
exp'2 ic p , (34)
and in relations (10) and (11)
−−
−Γ
=−= )(
4
exp
2
1
2
'
'
2
1
1 i
ic
c
c
p
p
p λπ
λ
π
,
−−== λπ
2
exp
'
1
2
2 i
c
c
p
p . (35)
We note that ψ− and ψ~− in (9) coincide in this
case. According to (22) to normalize ψp on
)'(2 00 pp −± π δ we have to replace cp in (32) and
(33) by
−= −
8
3exp)2(
'
4/1
2
π λeE
c
c
p
p
. (36)
Thus we may assume that ψ-functions in (10) and (11)
are normalized in this way. The same relations hold for
non-stationary states. The important thing is that the
relations (10) and (11) constitute all the necessary
ingredients for S-matrix theory [1,5].
We note now that the relation (18) takes on the form
pppp ffZZ
eE
ffJ *
''
0 )'(
2
),( +−= ,
)'(2'
0
3 eE
pxeEZ += , ee −= . (37)
On the other hand, using (30) we find
pppp ffZZpp
eE
f
dZ
df
dZ
d *
'
00*
' )')('(
8
1 +−−=
. (38)
Now it easy to verify that relations (19), (21) and (22)
remain valid as well as the orthogonality of ψ+ and
ψ− and also of ψ+ and ψ− .
Solutions (32), in which the factor depending on x1,
x2, t is dropped for brevity, are characterized by the
quantum number p0 (and also p1 and p2). If instead of
vector-potential (29) we use in the Klein-Gordon
equation the vector-potential
ηδ µµµµ x
AExA
∂
∂−=−= −
3' , 3xtx ±=± ,
)2/( 2
33 xtxE −=η , (39)
we obtain the solutions −p'ψ characterized by the
quantum number −p see [15,16]. Of course, we can go
back to the potential (29) and obtain
−−− =≡ p
ie
pp eAx ')( ψψψ η , )'('' Axpp −− = ψψ . (40)
Making modifications due to the present assumption
e=-|e|, we have from the results in [15,16]
++−= −+−+
− zxeEixpi
eE
xp ln)(
42
exp
)4(
1)(' *2
4/1 νψ
eE
z
−
= π
, −−− −= eExpπ . (41)
The factor depending on x1, x2 is dropped for brevity. So
this is the positive-frequency out-state. To corroborate
this we may add to the arguments in [15,16] the
following physical justification. The classical particle
with the negative charge starts from the region with
large negative x3, is slowed down and is reflected back
to where it comes. Its kinetic momentum π3 is negative
and grows in magnitude for t→∞. Hence
∞→−= ∞→
−
t)( 3
0 πππ . (42)
Going back to the quantum state (41) we note that the
particles (antiparticles) are in the region where 0>−π (
0<−π ). Large −π indicates that we are far from the
region where pairs are created ( 0≈−π ). In the region
where 0<−π the wave function (41) must be small for
small probability of pair production. Thus in this region
)( zez i −= − π , 0<z . (43)
Similarly,
−++−×
×=
−+−
− −
)ln()(
42
exp
)4(
1)('
*2
4/1
zxeEixpi
eE
xp
ν
ψ
(44)
For positive z in (44) we have
zez iπ−=− , 0>z . (45)
The complete set of −pψ -solutions must satisfy the
relations (10) and (11). This gives
( ) −− ′=′ +−
+ ppp c ψπθψ 1 , ( ) 11 −−− ′−=′ −−
ppp c ψπθψ ,
( )
<
>
=
.0,0
0,1
x
x
xθ
(46)
The proper time representation of the scalar particle
propagator for the vector-potential (39) is 17,1]
( )
( )
( ) ( )
( ) ( )
−−++−×
+′−=′′ ∫
∞
−−
eEseEyyi
s
yyiism
eEss
dsxxyeEieEAxxG
coth
44
exp
sinh2
exp
4
|,
2
3
2
0
2
2
2
12
0
32π
,
xxy −′= . (47)
106
Now we multiply (47) by
−+− +− ypypypi
2
1exp 2211
and integrate over y1, y2 and y+, see (54). For the integral
over y+ we have
( ) ( )
( ) ( ) (48)
442
2
442
.eEseEyxxeEp
eEseEyyixxeEyiypidy
−′+−π δ=
∫
−′+−
−
−−
−
∞
∞−
+−−−++−+
In the expressions (47) and (48) the charge e can
have any sign. For e=-|e| we obtain for the right hand
side of (48)
( ) ( )00
2sinh8coth8 ττδτπτδπ −=−
−− eEy
R
eEy ,
−−
−−
−′
+′
=
ππ
ππR , (49)
seE ||=τ , −
−′
=
−
+=
π
πτ ln
2
1
1
1ln
2
1
0 R
R ,
−−− += xeEp ||π , −−− ′+=′ xeEp ||π ,
−−− −′= xxy . (50)
We note that the reversal of sign of π− and π′ does not
change R. In order to have the nonzero δ-function
argument, R must lie in the interval 1<R<∞ because s>0
in (47). This is possible only in two cases
0<π−<π′− , i.e. x−<x′− ; (51)
0>π−>π′− , i.e. x−>x′− . (52)
Taking into account that
−=
−∫
∞
∞−
2
111
2
11 4
exp2
4
exp ispisyipy
s
idy ππ (53)
and similarly for the integral over y2, we get for the
case (51)
( ) =′′∫ ∫ ∫
∞
∞−
∞
∞−
∞
∞−
−+−
+
+− ypypyp
eAxxGdydydy 2
1
21
2211
|,
( ) ( ) ( )
−′+
′
− −−
−
−
−− 22
2
1
4
ln
2
exp' xxeEiii
π
πλππ .(54)
The relation
−
−′
=
π
πτ 0
e is used here, see (50). We
note also that according to (41) and (44) both
( ) ( )xx pp
−
− ′′′ ++ *ψψ under condition (51) and
( ) ( )xx pp
−
− ′′′ −−
*ψψ under condition (52) can be written as
[ ]
−−+−− −
−
−−++
−
−− π
πλ
ππ
'ln
2
)()'(
4
)'(
2
exp
'2
1 22 ixxeEixxip
. (55)
Making the inverse Fourier transform of (54) i.e.
multiplying it by
−+ +−−− )'
2
1''(exp)2(2 2211
31 ypypypiπ
and integrating over p1, p2 and p− we get
( )
( ) ( )
( ) ( )∫
<′′′−−
>′′′
=′′
−−
−−
−−
−−++−−−
−
−
−
−
xxxx
xxxxdpdpdpi
AxxG
pp
pp
',)'()(
,',)'()(
)2(
|,
*
*
3
21
ψψπθπθ
ψψπθπθ
π
(56)
Here θ-functions take care of the conditions (51) and
(52). Besides we have
,'),()'()( −−−−− >= xxπθπθπθ
−−−−− <−=−− xx'),()'()( πθπθπθ . (57)
Taking into account (46) we obtain from (56) and (57)
the sought for representation, which for the potential
(29) has the form
( )
( ) ( )
( ) ( )∫
<′
>′
=′
−−−
−
−−
+
+−
−
−
−
−
xxxx
xxxx
c
dpdpdpiAxxG
pp
pp
p ',
,',
)2(
|,
*
*
*
1
3
21
ψψ
ψψ
π
(58)
One can verify [16] that the functions defined by the
upper and lower lines on the right hand side of (56) (and
(58)) coincide outside the light cones
02
3
2
0 <−=−+ yyyy . (59)
According to (40) and (41) we have
+−+−×
×=
−
+−
+
−
zxtieExpi
eE
xp
ln)
4
)(
2
(
2
exp
)4(
1)(
*
22
4/1
ν
ψ
, (60)
−− pψ is obtained from (60) by substitution z→-z under
the logarithm sign.
Now we note that the classification of −pψ -
functions was obtained in [15,16] from the obvious
classification of the non-stationary solutions 3pψ by an
integral transformation
∫
∞
∞−
−=− )(),()(
33
*
3 AxppMdpAx pp ϕψ , (61)
++=
−−
−−
eE
ppppieEppM
4
24)(exp)2(),(
2
33
2
2/1
3
* π .
(62)
Inserting )(
3
Axpϕ+ in the expression on the right hand
side of (61), we get )( Axp−
+ ψ on the left hand side and
so on. The reversed relations of (61) can be considered
as the definitions of 3pϕ . For the vector potential
EtA 3
~
µµ δ−= , considered in [15,16], we have
107
)(
)~()~(),(
4/
3
*
33
3
TeDeB
AxAxppMdp
ixip
p
pp
π
ν
ϕψ
=
== +
∞
∞−
+−−∫ −
)(
)~()~(),(
4/
3
*
33
3
TeDeB
AxAxppMdp
ixip
p
pp
π
ν
ϕψ
−=
== −
∞
∞−
−
−−∫ −
−−−= −
4
32ln
48
exp)2( 4/1 πλπ λ iieEBp ,
)(2 3
eE
pteET −= . (63)
By analogy we expect that
∫
∞
∞−
−−
−= pp ppKdp ψϕ ),( 0
0 (64)
++=
=
−−
−
−
eE
ppppieE
ppK
4
24)(exp)2(
),(
2
0
02
2/1
0
π (65)
where 0pϕ differs from 0pp ψψ = in (32) only by an
inessential phase factor, see (69). To see that this is true
we use first the relation
tipiizzi
xeEteExpi
eE
ppppi
0
22
222
0
02
24
)
4
)(
22
1
4
24)(
−−+−
=
−+−++− −
+−
−−
ςς
(66)
where
)(
0
3 eE
pxeE +−=ς ,
eE
z
−
= π
, ee −= , (67)
then formula 3.462(1) in [18], the prescriptions (43) and
(45) and the relations between the parabolic cylinder
functions, see 8.2(6)-(8) in [14]. Then we find
)(
8
32ln
48
exp)2(
),(
4/04/1
0
*
0
ZeDtipiieE
ppKdp
i
pp
π
ν
π λλπ
ψϕ
−−
∞
∞−
+−−+
−−−
== ∫ −
(68)
)(
8
32ln
48
exp)2(
),(
4/04/1
0
*
0
ZeDtipiieE
ppKdp
i
pp
π
ν
π λλπ
ψϕ
−−
∞
∞−
−
−−
−
−
−−−=
== ∫ −
(69)
and similar expressions for 0pϕ+ and 0pϕ− . Here Z is
the same as in (31).
Comparing these functions with the ones in (32)
together with (36), we see that 0pϕ coincide with 0pψ
up to an inessential phase factor. We could get rid of
this factor by modifying ),( 0 −ppK .
We note here that
∫
∞
∞−
−−− −= )'(),(),'( 0000* ppppKppKdp δ (70)
and
∫
∞
∞−
−−−− −= )'(),()',( 00*0 ppppKppKdp δ , (71)
So the relation (64) can be reversed
∫
∞
∞−
−=− 0),( 0*0
pp ppKdp ϕψ (72)
Using this formula we rewrite the integrals over −p in
(58) as follows
ttxxdpxxdp pppp >= ∫∫
∞
∞−
+
+
∞
∞−
+
+−
−− ',)(')()(')( *0*
00 ψψψψ
(73)
ttxxdpxxdp pppp <= ∫∫
∞
∞−
−
−
∞
∞−
−
−
−
−
− ',)(')()(')( *0*
0
0 ψψψψ .
The conditions on t and t' are written on account of the
remark after (58). Thus
( )
( ) ( )
( ) ( )
<′
>′
×
×=′
−
−
+
+
∞
∞−
∞
∞−
∞
∞−
∫ ∫ ∫
ttxx
ttxx
c
dpdpdpiAxxG
pp
pp
p
',
,',
)2(
1|,
0
0
0
0
*
*
*
1
3
0
21
ψψ
ψψ
π
(74)
By the way it is clear from (74) and (40) that
( ) ( )AxxGeAxxG xxie |,'|, ))'()(( ′=′ − ηη . (75)
The expression (74) has the form dictated by the general
theory [1,19].
Finally we note that it follows from (61) and (64)
that
∫
∞
∞−
=
3
0 ),( 3
0
3 pp ppNdp ϕϕ
(76)
where
∫
∞
∞−
−−− == ),(),(),( 3
*0
3
0 ppMppKdpppN
−−= −
eE
pipieE 3
0
2/1
4
exp)2( ππ (77)
The relation (76) can be checked with the help of
formula 2.11.4(7) in [20] which can be adjusted as
follows
108
)
1
()1(2)(
2
1exp 2/2
−
−=
∫
∞
∞−
cyD
c
cxDxycdx κ
κ
κ
π
,
4
3π<cphase . (78)
Here )1(1 ii −=−=− for )( *νκνκ == . So we
insert )(
3
Axpϕ+ into the right hand side of (76) and
take into account that
)~()(
3
3
3
AxeAx p
txeEi
p ϕϕ +−+ = ,
3
0
33
3
0
2
txeEitipZTixp
eE
ppi +−−=+− , (79)
Then we get
∫
∞
∞−
++ == )(),()(
3
0 3
0
3 AxppNdpAx pp ϕϕ
)(
2
exp
)2(2
4/
4/1
2ln
488
*
0
TeDZTidT
eE
e i
tipii
π
ν
λπ λπ
π
−= ∫
∞
∞−
−−−
. (80)
Now using (78) we obtain the right hand side of (68).
For )(
3
Axpϕ we proceed similarly, but use the
substitution xT −= instead of xT = , when
employing (78).
It is shown in [15,16] that for example −
+
pψ can be
obtained from 3pψ+ by changing continuously the
gauge of the electric field potential. The same is true for
−
+
pψ and 0pψ+ . For this reason these functions are
indistinguishable and only the wave packets are
observable.
It is clear that ),( 3
0 ppN and ),( 3
−ppM have the
orthogonality properties of ),( 0 −ppK , see (70) and (71).
ACKNOWLEDGMENTS
I am greatly indebted to Prof. V.L. Ginzburg, who
showed me article [11], discussed it and urged me to
expound my understanding of the problem and points of
disagreement with other authors. This work was
supported in part by the Russian Foundation for Basic
Research (projects №00-15-96566 and №01-02-30024).
REFERENCES
1. A.I. Nikishov. Intensive-field problems in
quantum electrodynamics // Tr. Fiz. Inst. Akad.
Nauk SSSR. 1979, v. 111, p. 152; J. Sov. Laser Res.
1989, v. 6, p. 619-678.
2. A.I. Nikishov. The role of connection
between spin and statistics in quantum
electrodynamics with pair creating external field
(in Problems in Theoretical Physics). Moscow:
“Nauka”, 1972, p. 229-305 (in Russian).
3. A.I. Nikishov. S-matrix in quantum
electrodynamics with external field // Teor. Mat.
Fiz. 1974, v. 20, p. 48-56.
4. A.A. Grib, S.G. Mamaev, and
V.M. Mostepanenco. Vacuum Quantum Effects in
Strong Fields. Moscow: “Energoatomizdat”, 1988.
5. Α.Ι. Nikishov. S-matrix in quantum
electrodynamics with pair creating external field //
Tr. Fiz. Inst. Akad. Nauk SSSR, 1985, v. 168,
p. 157, in Issues Intensive-Field Quantum
Electrodynamics, Ed. by V.L. Ginzburg (Nova
Science, Commack, 1987).
6. A.I. Nikishov. Pair production by a constant
external field // Zh. Eksp. Teor. Fiz. 1969, v. 57,
p. 1210 [Sov. Phys. JETP. 1969, v. 30, p. 660-662.
7. N.B. Narozhny and A.I. Nikishov. The simp-
lest processes in the pair creating field // Yad. Fiz.
1970, v. 11, p. 1072-1077.
8. A.I. Nikishov. Barrier scattering in field
theory. Removal of Klein paradox // Nucl. Phys.
1970, B 21, p. 346-358.
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resolution // Physica Scripta. 1981, v. 23, p. 1036-
1042.
10. W. Greiner, B. Müller, J. Rafelski.
Quantum Electrodynamics of Strong Field.
Springer-Verlag 1985, 594 p.
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production by a potential barrier. hep-th/0111137,
19 p.
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109
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110
1. INTRODUCTION AND THE CHOICE OF IN- AND OUT-STATES
hmn=diag(-1,1,1,1). (1)
. (2)
, (3)
, , (4)
(5)
,
. (6)
. (7)
. (8)
, ,
,,
. (9)
; (10)
; (11)
,,
. (12)
, (14)
, ,
, . (15)
2. ORTHONORMALIZATION OF WAVE FUNCTIONS
(16)
. (17)
. (18)
(19)
. (21)
3. SOLVABLE POTENTIAL
(25)
(26)
4. CONSTANT ELECTRIC FIELD
ACKNOWLEDGMENTS
|
| id | nasplib_isofts_kiev_ua-123456789-79435 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:49:11Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Nikishov, A.I. 2015-04-01T19:51:30Z 2015-04-01T19:51:30Z 2001 On the theory of scalar pair production by a potential barrier / A.I. Nikishov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 103-109. — Бібліогр.: 20 назв. — англ. 1562-6016 PACS: 11.55Ds; 12.20Ds; 03.80+r https://nasplib.isofts.kiev.ua/handle/123456789/79435 The problem of the scalar pair production by a one-dimensional vector-potential Am(x₃) is reduced to the S-matrix formalism of the theory with an unstable vacuum. Our choice of in- and out-states does not coincide with that of other authors and we argue extensively in favor of our choice. We show that the norm of a solution of the wave equation is determined by one of the amplitude of its asymptotic form for x₃→±∞. For the constant electric field we obtain the scalar particle propagator in terms of the stationary states and show that with our choice of in- and out-states it has the form dictated by the general theory. I am greatly indebted to Prof. V.L. Ginzburg, who showed me article [11], discussed it and urged me to expound my understanding of the problem and points of disagreement with other authors. This work was supported in part by the Russian Foundation for Basic Research (projects №00-15-96566 and №01-02-30024). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics of high energies in matter and strong fields On the theory of scalar pair production by a potential barrier К теории образования скалярных пар потенциальным барьером Article published earlier |
| spellingShingle | On the theory of scalar pair production by a potential barrier Nikishov, A.I. Electrodynamics of high energies in matter and strong fields |
| title | On the theory of scalar pair production by a potential barrier |
| title_alt | К теории образования скалярных пар потенциальным барьером |
| title_full | On the theory of scalar pair production by a potential barrier |
| title_fullStr | On the theory of scalar pair production by a potential barrier |
| title_full_unstemmed | On the theory of scalar pair production by a potential barrier |
| title_short | On the theory of scalar pair production by a potential barrier |
| title_sort | on the theory of scalar pair production by a potential barrier |
| topic | Electrodynamics of high energies in matter and strong fields |
| topic_facet | Electrodynamics of high energies in matter and strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79435 |
| work_keys_str_mv | AT nikishovai onthetheoryofscalarpairproductionbyapotentialbarrier AT nikishovai kteoriiobrazovaniâskalârnyhparpotencialʹnymbarʹerom |