gleim200802
QED for fields obeying a square root operator equation 
Tobias Gleim 
Instead of using local field equations – like the Dirac equation for spin-1/2 and the Klein-
Gordon equation for spin-0 particles – one could try to use non-local field equations in order 
to describe scattering processes. The latter equations can be obtained by means of the 
relativistic energy together with the correspondence principle, resulting in equations with a 
square root operator. By coupling them to an electromagnetic field and expanding the square 
root (and taking into account terms of quadratic order in the electromagnetic coupling 
constant e), it is possible to calculate scattering matrix elements within the framework of 
quantum electrodynamics, e.g. like those for Compton scattering or for the scattering of two 
identical particles. This will be done here for the scalar case. These results are then compared 
with the corresponding ones based on the Klein-Gordon equation. A proposal of how to 
transfer these reflections to the spin-1/2 case is also presented. 
Free scalar particles are usually described by means of the well-known Klein-Gordon equation (see 
e.g. [4,5,6,10]): 
( ) ( ) 0,ˆ 222 =++∂ txmpt rr φ ,      (1) 
where we have used the momentum operator in configuration space ∇−=
ip̂  (and set the velocity of 
light as well as Planck’s constant h  to one). (1) can be regarded as an iteration of the following square 
root operator equation (see e.g. [1,2,4,5]): 
( ) ( )txpmtxi t ,ˆ, 22
 +=∂ φφ .     (2) 
Introducing an electromagnetic field with a 4-vector potential ( ) ( )AAA r,0=µ  and applying minimal 
coupling (with coupling constant e), i.e. replacing µµ xii ∂∂=∂  by ( )xeAxi µµ −∂∂ , the Klein-
Gordon equation yields (see e.g. [6]) 
( ) ( ) ( ) ( )( ) ( ) ( )[ ] ( )xxAxAexAxAiexmpt φφ µµµµµµ 2222 ˆ +∂+∂−=++∂ r ,  (3) 
where we have used the 4-vector notation ( ) ( ) ( )xtxxx rr ,,0 ==µ  and Einstein’s summation convention. 
Here, the coupling terms on the right hand side of (3) could easily be separated from the term with the 
free particle Hamiltonian on the left hand side of (3). This is unfortunately no longer possible, if one 
couples the non-local equation (2) to the electromagnetic field: 
( ) ( )( ) ( ) ( ) ( )xxeAxxAepmxi t φφφ 022 ˆ +
,    (4) 
because the vector potential A
 appears under the square root. But in a perturbation analysis of 
scattering processes, this property is useful, since such an analysis is based on the assumption that the 
coupling terms make only small contributions to the free particle solution due to the small value of the 
coupling constant e. By rewriting the Hamiltonian in (4), 
( )( ) ( ) ( )22220220 ˆˆˆˆˆ AepAeApepmeAxAepmeAH rrrrrrrr +⋅−⋅−+++=−++=′ ,  (5) 
one can split off a factor with the free Hamiltonian 
ˆ pmH +=′ ,       (6) 
which yields 
( )( ) ( )2122122220 ˆˆˆˆ1ˆ pmpmAepAeApeeAH rrrrrrr +++⋅−⋅−++=′ − .   (7) 
With the above-mentioned assumption, it is now very tempting to expand the first square root factor. 
A very similar approach has already been proposed by [3]. We would like to restrict ourselves to a 
series expansion of the kind  
...ˆˆ1ˆ1 2
1 +−+≈+ yyy       (8) 
containing only constant, linear and quadratic terms, where ŷ  denotes 
( )( ) 12222 ˆˆˆˆ −++⋅−⋅−= pmAepAeApey rrrrrr .     (9) 
Hamiltonian (7) is therefore approximated by 
ˆˆˆˆ HHHH ′+′+′≈′        (10) 
with 
( )( ) 
++⋅+⋅−=′
− 022
ˆˆˆˆ ApmpAApeH
rrrrr
,     (11) 
( ) ( )( ) ( )( ) 
+⋅+⋅+⋅+⋅−+=′
22122
ˆˆˆˆˆˆˆˆ pmpAAppmpAAppmAeH
rrrrrrrrrrrr
,  (12) 
where we have reordered the terms of expansion (9), retaining only terms to (and including the) 
quadratic order in e and recollected powers of ( )22 p̂m r+ . What we have won by (10) is a separation 
of the free Hamiltonian (6) from the coupling terms in (5) that approximately result in the sum of 1Ĥ ′  
and 2Ĥ ′  (i.e. (11) and (12) respectively). For we are only interested in corrections to the free 
Hamiltonian anyway, this approximation might not hurt very much. But however, this separation 
seems not to be a true one, because of the multiple factors of powers of ( )22 p̂m r+  in (11) and (12). 
That is, we need an interpretation of these operators. To this end, it is useful to know that for the free 
square root operator equation (2), an integral representation can be given (see  [1,2]): 
( ) ( ) ( ) ( )( )txtxxxxdtxi t ,:,, 3
rrrrr
φφφ Ω=′′−Ω′=∂ ∫ ,     (13) 
where Ω  denotes an energy distribution 
′−⋅−=′−Ω xxpip e
rrrrr
     (14) 
with 
22 pmpp
r +== ωω .     (15) 
(13) results from the fact, that one would expect to obtain the following momentum space 
representation of (2): 
( ) ( )tptpi pt ,
φωφ =∂  
with ( )tp,~ rφ  denoting the Fourier-transformed ( )tx,rφ . If the operator 22 p̂m r+  corresponds to 
( )∫ ′−Ω′ xxxd
rr3 , the operator ( ) 2122 ˆ −+ pm r  must correspond to ( )∫ ′−Ω′ − xxxd rr13  with 
′−⋅−−− =′−Ω xxpip e
rrrrr 1
,  (16) 
because 
( ) ( ) ( ) ( ) ( )xxxxxxxdxxxxxd ′−=′−′′Ω′′−Ω′′=′−′′Ω′′−Ω′′ −− ∫∫
rrrrrrrrrr 31313 δ   (17) 
with the Dirac distribution 
′−⋅−=′− xxpie
rrrrr
δ .      (18) 
(17) should be an integral representation of the “symbolic equation” 
( ) ( ) ( ) ( ) 1ˆˆˆˆ 21212121 22222222 =++=++ −− pmpmpmpm rrrr . 
Accordingly, terms with the nth power of ( )2122 p̂m r+ , 
( )222 ˆ npm r+ ,       (19) 
correspond to integrals over “the nth power of Ω “: 
′−⋅−=′−Ω xxpinp
rrrrr
.     (20) 
By replacing the operators of type (19) by integrals over “powers of Ω ” as given in (20), 1Ĥ ′  and 2Ĥ ′  
(see (11) and (12), respectively) can now be given a configuration space representation, too. 
With these preparations, we can now address to the quantisation of the scalar field with the aim to be 
able to calculate scattering matrix elements. 
Quantisation of the scalar field and the description of scattering processes 
Starting with Hamiltonian (10), it is now possible to describe scattering processes within the 
framework of quantum field theory. For free scalar particles, a quantum field theoretic ansatz is 
described e.g. in [2] and [4], using (2) and (13), respectively, as equations for a field operator ( )xφ . 
The latter one can be formulated with the help of creation and annihilation operators +pa rˆ  and pa rˆ , 
respectively: 
( ) p
xip ae
φ ,      (21) 
where as usual xptxpxp p
⋅−==⋅ ωµµ  with the 4-vector ( )pp p r,ω=  and the subsequent 
definitions are postulated: 
,00ˆ =pa r       (22) 
0ˆ0 =+pa r ,       (23) 
[ ] ( )ppaa pp ′−=+′ rrrr 3ˆ,ˆ δ ,      (24 a) 
[ ] 0ˆ,ˆ =′pp aa rr , [ ] 0ˆ,ˆ =+′+ pp aa rr      (24 b) 
with the vacuum state 0 . Since we are interested in a quantum theory for bosons, [ ]••,  in (24) 
must be a commutator (for fermions we would use here an anti-commutator instead, cf. e.g. [4]). 
Equations (21) to (24) are identical to those that one would postulate within a non-relativistic quantum 
field theory for bosons. 
For the density of a Hamiltonian, we make the usual ansatz (see e.g. [7]): 
( ) ( )xHxH φφ ′= + ˆˆ       (25) 
which one can retrieve from a density of a Lagrangian (see [2]): 
( ) ( ) ( ) φφφφφφφφ ++++ Ω−Ω−∂−∂=
iL .    (25 a) 
Substituting (10) into (25), we get 
ˆˆˆˆ HHHH ++≈       (26) 
with  
( ) ( )xHxH φφ 00 ˆˆ ′= + ,      (27) 
( ) ( )xHxH φφ 11 ˆˆ ′= + ,      (28) 
( ) ( )xHxH φφ 22 ˆˆ ′= + .      (29) 
(25) is (among other things) motivated by the fact that 
ppp aapdHxd rr ˆˆ
∫∫ = ω      (30) 
reproduces the relativistic analogue of the free non-relativistic Hamiltonian: 
pp aa
pd rr
∫ .      (31) 
(28) together with (29) are the densities of the Hamiltonian to (and including the) quadratic order in e. 
(21) to (24) are valid for free particles, but can also be used for interacting ones, if Dirac’s 
representation is used instead of the so far applied Heisenberg representation. Then, with (28) and (29) 
combined to a Hamiltonian density 
ˆˆˆ HHH I +=       (32) 
for the interaction of scalar bosons with photons, we can now start to calculate scattering matrix 
elements. To this purpose, we need the serial expansion of the S-operator (see e.g. [7]) to the order of 
2e : 
( ) ( ) ...ˆˆ1ˆ 21 +++= SSS        (33) 
with 
( ) ( )( )∫−= xHTxdiS Iˆˆ 41 ,      (34) 
( ) ( ) ( ) ( )( )∫ ∫−= 2124142212 ˆˆˆ xHxHTxdxdiS II ,   (35) 
where we have introduced a time ordering operator 
( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )1212010221020121 ˆˆˆˆˆˆˆˆ xHxHTxHxHxxxHxHxxxHxHT IIIIIIII =−+−= θθ   (36) 
with 
tθ .      (37) 
( )1Ŝ  does not only contribute to the expansion (33) with terms of order e, but also to order 2e . 
Therefore, we can split off ( )1Ŝ  into a term 1Ŝ  containing only terms in e,  
( ) ( ) ( ) ( )( )( )( ) ( ) ( ) ( )( )∫ +−+ +Ω⋅+⋅−−= xxAxxpxAxApxTxdieS φφφφ 012141 ˆˆˆ r
  (38) 
with 
( )( ) ( ) ( )∫ −Ω=Ω −− txxxxdx ,111131 rrr φφ      (39) 
and a part containing only terms in 2e : 
( ) ( ) ( )( )( )(
( ) ( ) ( )( ) ( ) ( ) ( )( )( )( ) )∫
Ω⋅+⋅−Ω⋅+⋅−
21212221
111118
,ˆ,,ˆˆˆ
txptxAtxApxxxdpxAxApx
xxAxTxdieS
rrrrrrrrrrrrr
(40) 
where the momentum operator 1p̂
 contains a gradient acting on 1x
 and 2p̂
 acting on 2x
. In (38) and 
(40), we have already substituted (32) into (34) and replaced powers of ( )2122 p̂m r+  by integrals over 
“powers of Ω ” (see (20)) in (11). Thus we can rewrite (34) as 
1 ˆˆˆ SSS += .       (41) 
The time ordering operator appearing in ( )1Ŝ  can be left out, because it contains only one time. In ( )2Ŝ  
we only want to retain terms of order 2e , therefore IĤ  can be approximated by 1Ĥ : 
( ) ( ) ( ) ( ) ( )( )( )( ) ( ) ( ) ( )(
( ) ( ) ( )( )( )( ) ( ) ( ) ( )( )( )( )
( ) ( ) ( ) ( ) ( ) ( ))22021101
222221
111114
111112
xxAxxxAx
xpxAxApxxpxAxApx
xxAxxpxAxApxTxdxdieS
Ω⋅+⋅Ω⋅+⋅+
Ω⋅+⋅−−= ∫ ∫
rrrrrrrr
. (42) 
Here, the time ordering operator must be taken into account, because there are two times: 1t  and 2t . 
We have already quantised the field of scalar bosons, but must do now the corresponding with the 
electromagnetic field. Since the scalar field in (38), (40) and (42) couple in a different way to the 
vector  potential A
 than to the scalar potential 0A , the choice of a Coulomb gauge seems to be 
appropriate: 
0=⋅∇ A
.       (43) 
Then the field equations take on the form 
( ) jAA tt r
=∇∂+∇−∂ 022 ,     (44 a) 
ρ−=∇ 02 A
       (44 b) 
with charge and current densities ρ  and j
, respectively. From (44 b) we can see that in this 
gauge the scalar potential is just a c-number: 
( ) ( )∫ ′−
xdtxA rr
,      (45) 
whereas the vector potential A
 becomes an operator when being quantised. For a free field, A
 can be 
chosen like (see e.g. [4,7,10]) 
( ) ( )λε
λλ ,ˆˆ~22
kecec
xA xik
rr∫ ∑
⋅+⋅− +=     (46) 
with the usual photon frequency 
   2~ kk
and with creation and annihilation operators λkc
rˆ  and +λkc
rˆ , respectively, for photons: 
[ ] ( )kkcc
rr −′= ′
3ˆ,ˆ δδ λλλλ , [ ] [ ] 0ˆ,ˆˆ,ˆ == ++ ′′′′ λλλλ kkkk cccc rrrr ,   (47) 
and 0A  would even vanish. The polarisation vectors ( )λε ,krr  fulfil the relation (see e.g. [4, 7]): 
( ) ( )
δλελε
.      (48) 
Due to the Coulomb gauge condition (43), 
( ) 0, =⋅ λε kk rrr          (49) 
is valid, too. 
In the following sections, we are going to calculate ( )1Ŝ  and ( )2Ŝ  by substituting the field operators 
( )xφ  and ( )xA
 from (21) and (46) as well as the distributions (20). Since we are considering 
electromagnetic interactions between (charged) spin-0 bosons, we have to take 0A  in (45) into 
account, too. Therefore, we first have to find out what the density of charge in (45) will be in this case. 
This can be done by coupling the Lagrangian density for free spin-0 bosons (25 a) to an 
electromagnetic field with the aid of the minimal coupling scheme µµµ eAii −∂→∂ . Then in that 
Lagragian density an extra term φφ +0eA  (the only one with an 0A ) emerges. If one subsequently 
regards the sum of that spin-0 and the electromagnetic Lagrangian, it is possible to obtain from it the 
electromagnetic field equations by means of the Euler-Lagrange equations. The former equations then 
contain a charge density φφ +e . That is why we can set ρ  in (45) to φφ + . 
With these results, one can then continue to calculate scattering matrix elements. 
The first term we calculate is 
( ) ( ) ( )( )( )( )∫ −+ Ω⋅+⋅= xpxAxApxxdI φφ 141 ˆˆ:ˆ r
     (50) 
appearing in ( )1Ŝ  (see (38)). To this end, it is useful to recognise that by means of (16) we get 
( )( )
⋅−−− =Ω p
φ .     (51) 
With (51) and integration by parts, (50) yields: 
( ) ( ) ( ) ( )( )++ −−++−+⋅= ∫ ∑ λλ
ωπ kk
ckppckppaa
pdpdkd
I rrrr
ˆˆˆˆ,
, (52) 
where 
( ) ( ) ( )kppkpp kpp
±−±−=±− 21
δωωωδδ     (53) 
and, of course, the a operators commute with the c operators. In 12Ŝ  (40), the first integral can be 
expressed in a similar way: 
( ) ( )( )( )
( ) ( )
( ) ( )(
( ) ( ) )
22112211
22112211
~2~22
ccppkkccppkk
ccppkkccppkk
pdpdkdkd
xxAxxdI
−+−−+−++−
+−+−+−++
   (54) 
After a quite lengthy but straightforward calculation, the second integral in (40) yields 
( ) ( ) ( )( ) ( )
( ) ( )( )( )( ) =Ω⋅+⋅
⋅−Ω⋅+⋅=
212122
111112
,ˆ,,ˆ
txptxAtxAp
xxpxAxApxxdxdI
rrrrrrr
rrrrrr
                                           (55) 
( ) ( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
−+−−+−++−
+−+−+−++−
11221122
11221122
~2~22
21222
ccppkkccppkk
ccppkkccppkk
ppkaa
pdpdkdkd
Finally, we want to determine the second term in ( )2Ŝ  (see (42)) which is not just like the product of 
two operators 1̂I  due to the time ordering operator as defined in (36). Unfortunately, we cannot use 
the famous Wick theorem, because the scalar field operator (21) contains only contributions to 
positive energy solutions: it does not consist of a sum of both positive and negative energy solutions as 
it would be the case for the field operator of the Klein-Gordon equation. Due to the symmetry of the 
time ordering operator (36) in its arguments, we may conclude 
( ) ( )( ) ( ) ( ) ( )2102012414212414 ˆˆ2ˆˆ xHxHxxxdxdxHxHTxdxd IIII ∫∫∫∫ −= θ .  (56) 
With this property, the calculation of the second term in (42) can be simplified a bit: 
( ) ( ) ( )( )( )( )(
( ) ( ) ( )( )( )( ) =Ω⋅+⋅
⋅Ω⋅+⋅=
22222
111112
xpxAxApx
xpxAxApxTxdxdI
     (57 a) 
( ) ( ) ( ) ( )
( ( ) ( )
( ) ( )( ) ( )( )
( ) ( )
( ) ( )( ) ( )( )
( ) ( )
( ) ( )( ) ( )( )
( ) ( )
( ) ( )( ) ( )( )
−−−−−−−
⋅−−′−′−′
++−−−−−−
⋅+−′−′−′
+−−−+−−−
⋅−−′+′−′
++−−+−−−
⋅+−′+′−′
21212
21212
21212
21212
21212
~exp~exp
~exp~exp
~exp~exp
~exp~exp
,,ˆˆˆˆ
~2~22
tititt
kppkppcc
tititt
kppkppcc
tititt
kppkppcc
tititt
kppkppcc
kaaaa
pdpdpdpdkdkd
kppkpp
kppkpp
kppkpp
kppkpp
ωωωωωωθ
ωωωωωωθ
ωωωωωωθ
ωωωωωωθ
rrrrrr
rrrrrr
rrrrrr
rrrrrr
The two integrals over the θ  function can be performed by means of the introduction of the two 
variables 
21: tt −=τ , 
    21: ttT +=         (58) 
with the Jacobian  
( ) 2
A linear combination of these variables  
21 tbtaTBA +=+τ       (59 a) 
can be expressed by means of 
( )baA −=
1 ,         (59 b) 
             ( )baB +=
(57) contains four terms of the subsequent type that can be simplified with the help of (58) and (59): 
( ) ( ) ( )
( ) ( )( ) ( )( )
( ) ( ) ( )( )τωωτθ
τωωτθ
21221
2122122
2211212
=+−−−
titixpett
   (60 a) 
The function θ  can be expressed by an integral in the complex plane (see e.g. [6,8]), 
( ) ∫
0      (61) 
with an ε  approaching zero. Substituting this into (60 a), we get: 
( ) ( ) ( )( ) ( )21
21221
τωωτθ
ωωδ +
=−−+ ∫ i
id i .  (60 b) 
With this result, (57) becomes: 
( ) ( ) ( ) ( )
( ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
⋅−−′−′−′−−+−−
⋅+−′−′−′+−+−−
⋅−−′+′−′−−++−
⋅+−′+′−′+−++−
′′′′′
δδωωωωωωδ
δδωωωωωωδ
δδωωωωωωδ
δδωωωωωωδ
kppkppcc
kppkppcc
kppkppcc
kppkppcc
kaaaa
pdpdpdpdkdkd
kppkppkk
kppkppkk
kppkppkk
kppkppkk
1~~ˆˆ
1~~ˆˆ
1~~ˆˆ
1~~ˆˆ
,,ˆˆˆˆ
~2~22
rrrrrr
rrrrrr
rrrrrr
rrrrrr
 (57 b) 
So far, we have only considered terms of the scattering matrix containing the electromagnetic vector 
potential A
. Now, we have to address to those terms containing the scalar potential 0A  too. (38) does 
not only contain (50), but also a Coulomb potential term: 
+= φφ 045ˆ AxdI .       (62 a) 
By substituting ρ  in (45) by φφ + , we obtain the following equation for (62 a), if we take into account 
that the Fourier transformed Coulomb potential looks like 
:       (63) 
43333
aaaakpkpkdkdpdpdI pkkp rr
−−′+′′′−= +′
.  (62 b) 
(42) contains two terms in 0A . The first term consists of a combination of (50) and (62 a), but taken at 
different times and therefore joined via the time ordering operator. That is why we have to use (60) as 
well as (45) and (63) again: 
( ) ( ) ( )( )( )( ) ( ) ( ) ( )( )
( ) ( ) ( )
( ) ( )[
( ) ( )
( ) ( )
( ) ( )
−−+−−+−−+
−−+−−+−−+
+−+−−++−+
+−+−−++−
−−′+′
=Ω⋅+⋅=
ωωωωωωωδδ
εωωωω
ωωωωωωωδδ
ωωωωωωωδδ
εωωωω
ωωωωωωωδδ
11121
11121
ˆˆˆˆˆˆ~
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ~
ˆˆˆˆˆˆ
111112
kppkkp
kpppkpk
kpkkpp
kpppkpk
kppkkp
kpppkpk
kpkkpp
kpppkpk
caaaaa
caaaaa
caaaaa
caaaaa
ppkkdkdpdpdpdpdkd
xxAxxpxAxApxTxdxdI
rrrrrr
rrrrrr
rrrrrr
rrrrrr
(64) 
The second term of (42) containing a scalar potential is even quadratic in 0A : 
( ) ( ) ( ) ( ) ( ) ( )( )∫∫ ++= 2202110124147ˆ xxAxxxAxTxdxdI φφφφ .    (65 a) 
(65 a) contains a factor of two integrands of the kind of (62 a), but taken at two different times. Thus 
the time ordering operator must be taken into account. With the same substitutions as in (62) and (64), 
we obtain the following result: 
( ) ( ) ( )
( ) ( ) εωωωω
δδωωωωωωωωδ
ikkkk
kkppkkpp
aaaaaaaapdkdkdpdpdkdkdpdiI
kpkpkpkp
pkkppkkp
+−−+−′−′
⋅−′+−′−′+−′−−++−−+
⋅′′′′=
11112222
22221111
ˆˆˆˆˆˆˆˆ
rrrrrrrr
rrrrrrrr
(65 b) 
The results (52), (54), (55), (57), (62), (64) and (65) substituted into (38) to (42) now enable us to 
evaluate scattering matrix elements for scattering processes to (and including) the order 2e . As two 
examples, we turn first to the scalar analogue of Compton scattering in order to address then to the 
scattering of two identical scalar bosons. These two scattering processes can be compared easily with 
the corresponding results of the well-known scalar QED dealing with the Klein-Gordon equation (3). 
Compton scattering 
For Compton scattering, we need one scalar boson and one photon each in the input and output 
channel. This means, we have to evaluate the element 
0ˆˆˆˆˆ0:ˆ ++′′′= µµ hqqh caSacS
rrrr ,      (66) 
with the Ŝ -operator (33). Firstly, we realise that the terms based on 1̂I  (see (52)) as well as 6Î  (see 
(64)) must vanish, because the subsequent two elements in the photon operators become zero: 
00ˆ00ˆˆˆ0 == ′′
′′ µλµµλµ δδ hhkhkh cccc
rrrrrr ,         (67) 
00ˆˆˆ0 =++′′ µλµ hkh ccc
rrr . 
There, we have used the commutation relations (47), the properties of creation and annihilation 
operators corresponding to those of (22) and (23) as well as abbreviated the delta functional by 
( )hk
rr −= 3δδ .      (68) 
The terms with 2Î  and 3Î  need 
qppqqppq aaaa rrrrrrrr 2121 0ˆˆˆˆ0 δδ ′
′ = ,    (69) 
whereas a term 
qppppqqppppq aaaaaa rrrrrrrrrrrr 12121212 0ˆˆˆˆˆˆ0 δδδ ′′′
′′ =  
belongs to 4Î . Here we have used again (22) to (24). 
For 2Î , 3Î  and 4Î  the following equations are necessary, too: 
  00ˆˆˆˆ0 =+′′′′ µλλµ hkkh cccc
rrrr ,                      
00ˆˆˆˆ0 =+++ ′′′′ µλλµ hkkh cccc
,       (70) 
µλλµµλλµ δδδδ ′′′′
′′′′ = khkhhkkh cccc
rrrrrrrr 0ˆˆˆˆ0 , 
λµλµµλλµ δδδδ ′′′′
′′′′ = khkhhkkh cccc
rrrrrrrr 0ˆˆˆˆ0 , 
where we have commutated the creation operators successively to the left and the annihilation 
operators to the right. 
Furthermore, we see that the term based on 5Î  from (62) becomes zero, due to 
00ˆˆˆˆˆˆ0 =++′
′′ qpkkpq aaaaaa
rrrrrr .       (71) 
A similar result holds true for the term based on 7Î  from (65). 
00ˆˆˆˆˆˆˆˆˆˆ0
22221111
′′ qpkkppkkpq aaaaaaaaaa
rrrrrrrrrr .     (72) 
Now, we can determine (66) by means of (33): 
( )( ) ( ) ( )762144134122215121 ˆˆˆˆˆˆˆ1ˆ IIIIiIiieIIieS +−+−−++−−≈− .   (73) 
For Compton scattering, 1̂I , 2Î , 3Î , 4Î , 5Î , 6Î  and 7Î  can be evaluated explicitly: 
0ˆˆˆˆ 7651 ==== IIII ,       (74) 
( ) ( ) ( )µεµε
δ ′′⋅
′−′−+
,     (75) 
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
′−⋅′′+′−⋅
++⋅′−+⋅
′−′−+
hqhhhqh
hqhhhqh
hqqhh
rrrrrrrrr
rrrrrrrrr
ωωωωπ
   (76) 
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
+⋅′+′⋅′′
′−⋅′′−′⋅′−′−+
′−′′−′
ωωωωπ
hqhhqh
hqhhqhhqhq
hqhqhq
hqhqhqqhh
rrrrrr
rrrrrr
rrrrrrrr
rrrrrrrr
2,2,1
2,2,1
   (77) 
The two terms 2Î  and 4Î  in (73) resemble the three terms in the corresponding formula of 
Compton scattering for scalar bosons, but this time being based on the Klein-Gordon equation (3) (see 
e.g. [6,8]): 
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )]µεµε
ωωωωπ
′−⋅′′
′+′⋅′′
⋅′−′−+
22~2~22
   (78) 
where ε  is the 4-dimensional generalisation of the three dimensional polarisation vector ε
 used so 
far. In (78), the first two terms correspond to 4Î . We realise that these terms in (78) look very much 
like those in (77), but also that especially the propagators of both theories are a bit different. (75) 
resembles the third term in (78). On the other hand, 3Î  in (76) could also be regarded as the 
analogue of the first two terms in (78) – at least after having used the Coulomb gauge condition (49) in 
(76).  
The first two terms in (78) vanish, if we choose the incoming scalar boson to be at rest, 
( ) ( )0,:,0 rr mqqq == ,      (79) 
and want to have transversally polarised photons in this laboratory system, 
( ) ( ) 0,, =⋅′′=⋅ qhqh µεµε ,     (80) 
and use the Lorentz gauge condition  
( ) ( ) 0,, =′⋅′′=⋅ hhhh µεµε .     (81) 
Accordingly, 3Î  and 4Î  in (73) vanish too, if we adopt (79) again and use the analogue of (80), 
( ) ( ) 0,, =⋅′′=⋅ qhqh rrrrrr µεµε ,     (82) 
as well as apply the Coulomb gauge condition (49). 
That is, under these conditions in both versions of scalar QED, only one term remains (i.e. (75) in (73) 
and, accordingly, the third term in (78)) which is a relativistically generalised version of the matrix 
element from which the well-known Thomson scattering cross section can be evaluated.  
Even though the propagators in both theories are rather different, for this example of scattering 
process, the results do not seem to differ very much from each other in the laboratory system chosen 
above. Therefore the question arises, whether this is also the case for further scattering processes. To 
this end, we are going to investigate what happens, if two identical scalar bosons interact with each 
other. 
Scattering of two identical scalar bosons 
If we want to calculate matrix elements of the S-operator (33) for the scattering of two identical scalar 
bosons, we need two scalar bosons in the input channel and two in the output channel: 
0ˆˆˆˆ0:ˆ
′′= qqqq aaSaaS rrrr .      (83) 
We can reuse (73), but have to evaluate 1̂I , 2Î , 3Î , 4Î , 5Î , 6Î  and 7Î  again. Firstly, 
we recognise that due to (52) and the analogue of (22) and (23) for the photon operators 
01̂ =I ,        (84) 
is valid. Furthermore, we need 
21212211
0ˆˆ0 λλλλ δδ kkkk cc
rrrr =+       (85) 
for calculating 2Î  with the help of (54), whereas all the other photon operator terms therein vanish. 
The same holds true for 3Î  with (55) and 4Î  with (57). 
As far as the scalar boson operators are concerned, for the evaluation of 2Î  and 3Î  
00ˆˆˆˆˆˆ0
121212
=+++′′ qqppqq aaaaaa rrrrrr . 
Hence, these two terms, 
0ˆ2 =I ,          (86) 
    0ˆ3 =I , 
are zero, too. For the evaluation of the non-vanishing term 4Î , the following result is useful: 
2122121121221211
212212112122121112121212
0ˆˆˆˆˆˆˆ0
pqpqpqpqpqpqpqpq
pqpqpqpqpqpqpqpqqqppppqq aaaaaaaa
′′′′′′′′
′′′′′′′′
rrrrrrrrrrrrrrrr
rrrrrrrrrrrrrrrrrrrrrrrr
δδδδδδδδ
δδδδδδδδ
. (87) 
With (85), (87) and the invariance of (48) under the transformation kk
−→ , we get 
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
21212112
11111122
21212121
11111111
Ainterms
qqqqqqqq
qqqqqqqqqqqqie
qqqqqqqq
qqqqqqqq
′+⋅′−′−⋅′+
 ′+⋅′−′−⋅′+′−′−+−
′−′′−′
′−′′−′
εωωωεωωω
εωωωεωωω
rrrrrrrrrr
rrrrrrrrrr
 (88 a) 
Here, we have omitted the terms 5Î , 6Î  and 7Î  containing terms in the scalar potential 
which are regarded later.  
(88 a) can be compared directly with the corresponding result of the Klein-Gordon equation (3) (see 
e.g. [9]): 
( ) ( )
( ) ( )
( ) 
′+⋅′+
′+⋅′+′−′−+
qqqqqqqqie
qqqq ωωωωπ
 , (89) 
if we use (48) and leave out the term ε  approaching zero in (88) : 
( ) ( )
( ) ( )
( ) ( )( ) ( )
( ) ( )
( ) ( )( ) ( )
( ) ( )
21212112
11111122
Ainterms
qqqqqqqq
qqqqqqqq
qqqqqqqqie
′+⋅′−′−⋅′+
′+⋅′−′−⋅′+
′+⋅′+
′+⋅′+′−′−+−
rrvrvrrr
rrvrvrrr
rrrrrrvr
(88 b) 
The first two terms in (88 b) correspond to (89) which contains the photon propagator in the Feynman 
gauge. But since we use a Coulomb gauge, only the space-like components of the linear 4-momenta 
appear in the numerators of the first two terms in (88 b). Moreover, in the second line of (88 b) two 
additional terms are present which can be reformulated by means of the delta distribution in (88 b): 
( )( )
( ) ( )
( )( )
( ) ( )221221
′−′−′
− .     (90) 
The structure of (88 b) is the same as that of (89). We have two terms: in the second term, the 
momenta of the two scalar bosons in the output channel have been exchanged compared to the first 
term. 
Now we address to the terms in the scalar potential 0A  in (88). The Coulomb term 5Î  contains a 
term 
pqkqqpqkkqpqqpqk
pqkqqkqpkqpqqkqpqqpkkpqq
aaaaaaaa
′′′′′′′′
′′′′′′′′
rrrrrrrrrrrrrrrr
rrrrrrrrrrrrrrrrrrrrrrrr
21122112
211221121212
0ˆˆˆˆˆˆˆ0
δδδδδδδδ
δδδδδδδδ
Therefore, 5Î  yields 
( ) ( ) 
−−′+′=
qqqqI rrrrδ
.    (91) 
The term 6Î  vanishes, since the annihilation operators for photons in (64) act directly on the 
vacuum states.  
On the other hand, 7Î  becomes a rather lengthy because of 
( )( )
( )( )
( )( )
( )( )11
12112121122222
21121121122222
121121211222
121121122122
122222111112
0ˆˆˆˆˆˆˆˆˆˆˆ0
aaaaaaaaaaaa
kqpqkppkqpqkqp
kkkqpqppqpqkqp
kqpqkppkqkqp
kqpqkkqkppqp
qqpkkppkkpqq
′↔′−+
′↔′−−
′↔′−−
′↔′−=
′′′′′′
′′′′′′
′′′′′′
′′′′′′
rrrrrrrrrrrrrr
rrrrrrrrrrrrrr
rrrrrrrrrrrr
rrrrrrrrrrrr
rrrrrrrrrrrr
δδδδδδδ
δδδδδδδ
δδδδδδ
δδδδδδ
where ( )11 kp ′↔′
 denotes the same term as the immediately preceding one, but with exchanged 
momenta p′
 and 1k ′
, respectively.  Thus, 7Î  gives 
−−′+′=
−′−−′−+′−+′−
22222222
12112221
qpqpqpqpqpqpqpqp
pqqpqq
rrrrrrrrrrrrrrrr
rrrrrr
ωωωωωωωω
ωωωωπ
.  (92) 
Hence in place of the time-like components (i.e. the energy terms) of the linear 4-momenta in the 
numerators of (89) derived from the Klein-Gordon equation, several terms arise: (90), (91) and (92). 
But (91) and (the non relativistic limit of) (92) would also have appeared, if we had started from the 
non-relativistic Schrödinger equation. Thus, these terms state the fact that the equation used for 
obtaining (88) (together with (90), (91) and (92)) is Schrödinger equation like.  
Conclusions and outlook 
For scalar bosons, we could see that it is possible to describe scattering processes by means of a square 
root operator equation being coupled to an electromagnetic field. We achieved this by splitting off a 
factor in the shape of the free square root operator from the equation and by a series expansion of a 
remaining square root factor containing terms in the electromagnetic vector potential and powers of 
the (inverse) free square root operator. The latter ones could be given an integral representation.  
Having quantised the fields involved, we could evaluate the scattering matrix elements for Compton 
scattering and for the scattering of two identical bosons (to – and including – the quadratic order of  e) 
which, on the one hand, resembled the results derived with the help of the corresponding Klein-
Gordon equation and, on the other hand, the results one would have obtained with a non-relativistic 
Schrödinger equation.  
Of course, now several questions arise, e.g.: 
• Can we formulate Feynman rules at all for our non-local scalar QED? 
• Do divergent terms appear and, if yes, can a renormalisation procedure be found? 
• Can the results of this non-local QED be confirmed (or refuted) by experiments? 
To the first question: if one tries to formulate Feynman rules, one must be aware that in each step of 
the approximation procedure, we have to expand the first square root factor (in the second term of) (7) 
to the desired order in e. Thus, the Hamiltonian we are using within that procedure must be adapted in 
each order of e of the approximation. Therefore we conclude that even if it were possible to formulate 
Feynman rules, they would be much more complicated than those for the Klein-Gordon theory. This is 
the price we have to pay for non-locality. But without Feynman rules, it is not very easy to analyse the 
renormalisability of that non-local scalar QED either.  
The answer to the last question listed above is negative due to a lack of elementary spinless bosons in 
nature. But that question would be sensible, if we had a corresponding non-local theory for spin-1/2 
particles. We could even make a guess, how this theory would look like: for free spin-1/2 particles, the 
wave functions in (2) should be 2-spinors. It would also be possible to couple this equation to an 
electromagnetic field, but the usual minimal coupling scheme does not work. Instead we would have 
to postulate an equation: 
ˆ ′=∂ ,       (93) 
with the Hamilton operator 
( ) ( ) 022 ˆˆ eAEiBeAepmH +⋅−−+=′ rmrrrr
σ ,    (94) 
which contains an additional term with the Pauli matrices σ
 and the magnetic field B
 as well as the 
electric field E
 under the square root. For this equation, it has already been shown that it can 
reproduce the gyromagnetic factor of 2 for the electron as well as, when being applied to a hydrogen 
atom, that it can reproduce correct binding energies of the electron at least to (and including) the 
quadratic order in 2e  (see [12]). We can apply the same approximation procedure to this equation as 
we have presented here for the scalar case. But of course, additional difficulties emerge: the 
Hamiltonian corresponding to the one shown in (25) should be Hermitian. And at least for the free 
spin-1/2 case, that Hamiltonian should be relativistically invariant. This means, that for the free case, 
(25) should be a Lorentz scalar with respect to the spin. To this end, we have to replace φ  and +φ  
therein by combinations of a mixture of left and right handed 2-spinors Lφ  and Rφ , respectively, and 
their Hermitian conjugates, because RLφφ
+  and LRφφ
+  are Lorentz scalars. Therefore, (25) could be 
replaced either by 
RLLR HHH φφφφ ±
++ ′+′= ˆˆˆ
       (95) 
or by 
RLLR HHH φφφφ mˆˆˆ 2
1 ′+′= +±
+       (96) 
so that the Hamiltonians (95) and (96) become Hermitian, because of the property 
±′=′ HH ˆˆ m .       (97) 
The results of a QED based on (95) or (96) could then be compared with the ones based on a 
corresponding Dirac equation.  
The author does not know, whether such an approach has already been performed for the spin-1/2 case 
or for the here presented spinless case. He does not know either, if an application for it can be found, 
where e.g. non-local properties are indispensable. But it seems to him that from the technical point of 
view, square root operator equations coupled to an electromagnetic field like (4) (or maybe even like 
(93) together with (94)) can be handled within the framework of a quantum field theory. Such 
equations were given up quite early in the history of quantum mechanics for several good reasons, e.g. 
due to their lack of relativistic invariance (see [13]), their non-local character accompanied by the 
difficulty of finding an appropriate mathematical interpretation and description, respectively (see e.g. 
[4,5,6]). While the first reason mentioned remains still valid, from today’s perspective, those non-local 
properties might not be refused as vehemently as in the past (e.g. when one looks out for 
approximations of the so called Bethe-Salpeter equation [11]). The author hopes to have shown, that at 
least answers to the question of possible descriptions of such non-local square root operator equations 
can be found. 
References 
[1] E. Trübenbacher, Z. Naturforschung 44a, 801-810 (1989). 
[2] C. Lämmerzahl, J. Math. Phys. 34 (9), 3918-3932 (1993). 
[3] T.L. Gill and W.W. Zachary, J. Phys. A: Math.Gen. 38 2479-2496 (2005). 
[4] W. R. Theis, Grundzüge der Quantentheorie (Teubner, Stuttgart, 1985). 
[5] A. Messiah, Quantenmechanik 2 (Walter de Gruyter, Berlin, 1990). 
[6] J.D. Bjorken and S.D. Drell, Relativistische Quantenmechanik (Bibliographisches Institut, 
Mannheim, 1966). 
[7] W. Greiner and J. Reinhardt, Feldquantisierung (Verlag Harri Deutsch, Thun/Frankfurt a.M., 
1993). 
[8] W. Greiner and J. Reinhardt, Quantenelektrodynamik (Verlag Harri Deutsch, Thun/Frankfurt a.M., 
1984. 
[9] W. Greiner and A. Schäfer, Quantenchromodynamik (Verlag Harri Deutsch, Thun/Frankfurt a.M., 
1989. 
[10] L. H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1985). 
[11] W. Lucha  and F.F. Schöberl, Int. J. Mod. Phys. A 14 2309 (1999). 
[12] T. Gleim, quant-ph/0601211; quant-ph/0602047. 
[13] J. Sucher, J. Math. Phys. 4 (1), 17-23 (1963).

  Instead of using local field equations - like the Dirac equation for spin-1/2
and the Klein-Gordon equation for spin-0 particles - one could try to use
non-local field equations in order to describe scattering processes. The latter
equations can be obtained by means of the relativistic energy together with the
correspondence principle, resulting in equations with a square root operator.
By coupling them to an electromagnetic field and expanding the square root (and
taking into account terms of quadratic order in the electromagnetic coupling
constant e), it is possible to calculate scattering matrix elements within the
framework of quantum electrodynamics, e.g. like those for Compton scattering or
for the scattering of two identical particles. This will be done here for the
scalar case. These results are then compared with the corresponding ones based
on the Klein-Gordon equation. A proposal of how to transfer these reflections
to the spin-1/2 case is also presented.

gleim200802
QED for fields obeying a square root operator equation 
Tobias Gleim 
Instead of using local field equations – like the Dirac equation for spin-1/2 and the Klein-
Gordon equation for spin-0 particles – one could try to use non-local field equations in order 
to describe scattering processes. The latter equations can be obtained by means of the 
relativistic energy together with the correspondence principle, resulting in equations with a 
square root operator. By coupling them to an electromagnetic field and expanding the square 
root (and taking into account terms of quadratic order in the electromagnetic coupling 
constant e), it is possible to calculate scattering matrix elements within the framework of 
quantum electrodynamics, e.g. like those for Compton scattering or for the scattering of two 
identical particles. This will be done here for the scalar case. These results are then compared 
with the corresponding ones based on the Klein-Gordon equation. A proposal of how to 
transfer these reflections to the spin-1/2 case is also presented. 
Free scalar particles are usually described by means of the well-known Klein-Gordon equation (see 
e.g. [4,5,6,10]): 
( ) ( ) 0,ˆ 222 =++∂ txmpt rr φ ,      (1) 
where we have used the momentum operator in configuration space ∇−=
ip̂  (and set the velocity of 
light as well as Planck’s constant h  to one). (1) can be regarded as an iteration of the following square 
root operator equation (see e.g. [1,2,4,5]): 
( ) ( )txpmtxi t ,ˆ, 22
 +=∂ φφ .     (2) 
Introducing an electromagnetic field with a 4-vector potential ( ) ( )AAA r,0=µ  and applying minimal 
coupling (with coupling constant e), i.e. replacing µµ xii ∂∂=∂  by ( )xeAxi µµ −∂∂ , the Klein-
Gordon equation yields (see e.g. [6]) 
( ) ( ) ( ) ( )( ) ( ) ( )[ ] ( )xxAxAexAxAiexmpt φφ µµµµµµ 2222 ˆ +∂+∂−=++∂ r ,  (3) 
where we have used the 4-vector notation ( ) ( ) ( )xtxxx rr ,,0 ==µ  and Einstein’s summation convention. 
Here, the coupling terms on the right hand side of (3) could easily be separated from the term with the 
free particle Hamiltonian on the left hand side of (3). This is unfortunately no longer possible, if one 
couples the non-local equation (2) to the electromagnetic field: 
( ) ( )( ) ( ) ( ) ( )xxeAxxAepmxi t φφφ 022 ˆ +
,    (4) 
because the vector potential A
 appears under the square root. But in a perturbation analysis of 
scattering processes, this property is useful, since such an analysis is based on the assumption that the 
coupling terms make only small contributions to the free particle solution due to the small value of the 
coupling constant e. By rewriting the Hamiltonian in (4), 
( )( ) ( ) ( )22220220 ˆˆˆˆˆ AepAeApepmeAxAepmeAH rrrrrrrr +⋅−⋅−+++=−++=′ ,  (5) 
one can split off a factor with the free Hamiltonian 
ˆ pmH +=′ ,       (6) 
which yields 
( )( ) ( )2122122220 ˆˆˆˆ1ˆ pmpmAepAeApeeAH rrrrrrr +++⋅−⋅−++=′ − .   (7) 
With the above-mentioned assumption, it is now very tempting to expand the first square root factor. 
A very similar approach has already been proposed by [3]. We would like to restrict ourselves to a 
series expansion of the kind  
...ˆˆ1ˆ1 2
1 +−+≈+ yyy       (8) 
containing only constant, linear and quadratic terms, where ŷ  denotes 
( )( ) 12222 ˆˆˆˆ −++⋅−⋅−= pmAepAeApey rrrrrr .     (9) 
Hamiltonian (7) is therefore approximated by 
ˆˆˆˆ HHHH ′+′+′≈′        (10) 
with 
( )( ) 
++⋅+⋅−=′
− 022
ˆˆˆˆ ApmpAApeH
rrrrr
,     (11) 
( ) ( )( ) ( )( ) 
+⋅+⋅+⋅+⋅−+=′
22122
ˆˆˆˆˆˆˆˆ pmpAAppmpAAppmAeH
rrrrrrrrrrrr
,  (12) 
where we have reordered the terms of expansion (9), retaining only terms to (and including the) 
quadratic order in e and recollected powers of ( )22 p̂m r+ . What we have won by (10) is a separation 
of the free Hamiltonian (6) from the coupling terms in (5) that approximately result in the sum of 1Ĥ ′  
and 2Ĥ ′  (i.e. (11) and (12) respectively). For we are only interested in corrections to the free 
Hamiltonian anyway, this approximation might not hurt very much. But however, this separation 
seems not to be a true one, because of the multiple factors of powers of ( )22 p̂m r+  in (11) and (12). 
That is, we need an interpretation of these operators. To this end, it is useful to know that for the free 
square root operator equation (2), an integral representation can be given (see  [1,2]): 
( ) ( ) ( ) ( )( )txtxxxxdtxi t ,:,, 3
rrrrr
φφφ Ω=′′−Ω′=∂ ∫ ,     (13) 
where Ω  denotes an energy distribution 
′−⋅−=′−Ω xxpip e
rrrrr
     (14) 
with 
22 pmpp
r +== ωω .     (15) 
(13) results from the fact, that one would expect to obtain the following momentum space 
representation of (2): 
( ) ( )tptpi pt ,
φωφ =∂  
with ( )tp,~ rφ  denoting the Fourier-transformed ( )tx,rφ . If the operator 22 p̂m r+  corresponds to 
( )∫ ′−Ω′ xxxd
rr3 , the operator ( ) 2122 ˆ −+ pm r  must correspond to ( )∫ ′−Ω′ − xxxd rr13  with 
′−⋅−−− =′−Ω xxpip e
rrrrr 1
,  (16) 
because 
( ) ( ) ( ) ( ) ( )xxxxxxxdxxxxxd ′−=′−′′Ω′′−Ω′′=′−′′Ω′′−Ω′′ −− ∫∫
rrrrrrrrrr 31313 δ   (17) 
with the Dirac distribution 
′−⋅−=′− xxpie
rrrrr
δ .      (18) 
(17) should be an integral representation of the “symbolic equation” 
( ) ( ) ( ) ( ) 1ˆˆˆˆ 21212121 22222222 =++=++ −− pmpmpmpm rrrr . 
Accordingly, terms with the nth power of ( )2122 p̂m r+ , 
( )222 ˆ npm r+ ,       (19) 
correspond to integrals over “the nth power of Ω “: 
′−⋅−=′−Ω xxpinp
rrrrr
.     (20) 
By replacing the operators of type (19) by integrals over “powers of Ω ” as given in (20), 1Ĥ ′  and 2Ĥ ′  
(see (11) and (12), respectively) can now be given a configuration space representation, too. 
With these preparations, we can now address to the quantisation of the scalar field with the aim to be 
able to calculate scattering matrix elements. 
Quantisation of the scalar field and the description of scattering processes 
Starting with Hamiltonian (10), it is now possible to describe scattering processes within the 
framework of quantum field theory. For free scalar particles, a quantum field theoretic ansatz is 
described e.g. in [2] and [4], using (2) and (13), respectively, as equations for a field operator ( )xφ . 
The latter one can be formulated with the help of creation and annihilation operators +pa rˆ  and pa rˆ , 
respectively: 
( ) p
xip ae
φ ,      (21) 
where as usual xptxpxp p
⋅−==⋅ ωµµ  with the 4-vector ( )pp p r,ω=  and the subsequent 
definitions are postulated: 
,00ˆ =pa r       (22) 
0ˆ0 =+pa r ,       (23) 
[ ] ( )ppaa pp ′−=+′ rrrr 3ˆ,ˆ δ ,      (24 a) 
[ ] 0ˆ,ˆ =′pp aa rr , [ ] 0ˆ,ˆ =+′+ pp aa rr      (24 b) 
with the vacuum state 0 . Since we are interested in a quantum theory for bosons, [ ]••,  in (24) 
must be a commutator (for fermions we would use here an anti-commutator instead, cf. e.g. [4]). 
Equations (21) to (24) are identical to those that one would postulate within a non-relativistic quantum 
field theory for bosons. 
For the density of a Hamiltonian, we make the usual ansatz (see e.g. [7]): 
( ) ( )xHxH φφ ′= + ˆˆ       (25) 
which one can retrieve from a density of a Lagrangian (see [2]): 
( ) ( ) ( ) φφφφφφφφ ++++ Ω−Ω−∂−∂=
iL .    (25 a) 
Substituting (10) into (25), we get 
ˆˆˆˆ HHHH ++≈       (26) 
with  
( ) ( )xHxH φφ 00 ˆˆ ′= + ,      (27) 
( ) ( )xHxH φφ 11 ˆˆ ′= + ,      (28) 
( ) ( )xHxH φφ 22 ˆˆ ′= + .      (29) 
(25) is (among other things) motivated by the fact that 
ppp aapdHxd rr ˆˆ
∫∫ = ω      (30) 
reproduces the relativistic analogue of the free non-relativistic Hamiltonian: 
pp aa
pd rr
∫ .      (31) 
(28) together with (29) are the densities of the Hamiltonian to (and including the) quadratic order in e. 
(21) to (24) are valid for free particles, but can also be used for interacting ones, if Dirac’s 
representation is used instead of the so far applied Heisenberg representation. Then, with (28) and (29) 
combined to a Hamiltonian density 
ˆˆˆ HHH I +=       (32) 
for the interaction of scalar bosons with photons, we can now start to calculate scattering matrix 
elements. To this purpose, we need the serial expansion of the S-operator (see e.g. [7]) to the order of 
2e : 
( ) ( ) ...ˆˆ1ˆ 21 +++= SSS        (33) 
with 
( ) ( )( )∫−= xHTxdiS Iˆˆ 41 ,      (34) 
( ) ( ) ( ) ( )( )∫ ∫−= 2124142212 ˆˆˆ xHxHTxdxdiS II ,   (35) 
where we have introduced a time ordering operator 
( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )1212010221020121 ˆˆˆˆˆˆˆˆ xHxHTxHxHxxxHxHxxxHxHT IIIIIIII =−+−= θθ   (36) 
with 
tθ .      (37) 
( )1Ŝ  does not only contribute to the expansion (33) with terms of order e, but also to order 2e . 
Therefore, we can split off ( )1Ŝ  into a term 1Ŝ  containing only terms in e,  
( ) ( ) ( ) ( )( )( )( ) ( ) ( ) ( )( )∫ +−+ +Ω⋅+⋅−−= xxAxxpxAxApxTxdieS φφφφ 012141 ˆˆˆ r
  (38) 
with 
( )( ) ( ) ( )∫ −Ω=Ω −− txxxxdx ,111131 rrr φφ      (39) 
and a part containing only terms in 2e : 
( ) ( ) ( )( )( )(
( ) ( ) ( )( ) ( ) ( ) ( )( )( )( ) )∫
Ω⋅+⋅−Ω⋅+⋅−
21212221
111118
,ˆ,,ˆˆˆ
txptxAtxApxxxdpxAxApx
xxAxTxdieS
rrrrrrrrrrrrr
(40) 
where the momentum operator 1p̂
 contains a gradient acting on 1x
 and 2p̂
 acting on 2x
. In (38) and 
(40), we have already substituted (32) into (34) and replaced powers of ( )2122 p̂m r+  by integrals over 
“powers of Ω ” (see (20)) in (11). Thus we can rewrite (34) as 
1 ˆˆˆ SSS += .       (41) 
The time ordering operator appearing in ( )1Ŝ  can be left out, because it contains only one time. In ( )2Ŝ  
we only want to retain terms of order 2e , therefore IĤ  can be approximated by 1Ĥ : 
( ) ( ) ( ) ( ) ( )( )( )( ) ( ) ( ) ( )(
( ) ( ) ( )( )( )( ) ( ) ( ) ( )( )( )( )
( ) ( ) ( ) ( ) ( ) ( ))22021101
222221
111114
111112
xxAxxxAx
xpxAxApxxpxAxApx
xxAxxpxAxApxTxdxdieS
Ω⋅+⋅Ω⋅+⋅+
Ω⋅+⋅−−= ∫ ∫
rrrrrrrr
. (42) 
Here, the time ordering operator must be taken into account, because there are two times: 1t  and 2t . 
We have already quantised the field of scalar bosons, but must do now the corresponding with the 
electromagnetic field. Since the scalar field in (38), (40) and (42) couple in a different way to the 
vector  potential A
 than to the scalar potential 0A , the choice of a Coulomb gauge seems to be 
appropriate: 
0=⋅∇ A
.       (43) 
Then the field equations take on the form 
( ) jAA tt r
=∇∂+∇−∂ 022 ,     (44 a) 
ρ−=∇ 02 A
       (44 b) 
with charge and current densities ρ  and j
, respectively. From (44 b) we can see that in this 
gauge the scalar potential is just a c-number: 
( ) ( )∫ ′−
xdtxA rr
,      (45) 
whereas the vector potential A
 becomes an operator when being quantised. For a free field, A
 can be 
chosen like (see e.g. [4,7,10]) 
( ) ( )λε
λλ ,ˆˆ~22
kecec
xA xik
rr∫ ∑
⋅+⋅− +=     (46) 
with the usual photon frequency 
   2~ kk
and with creation and annihilation operators λkc
rˆ  and +λkc
rˆ , respectively, for photons: 
[ ] ( )kkcc
rr −′= ′
3ˆ,ˆ δδ λλλλ , [ ] [ ] 0ˆ,ˆˆ,ˆ == ++ ′′′′ λλλλ kkkk cccc rrrr ,   (47) 
and 0A  would even vanish. The polarisation vectors ( )λε ,krr  fulfil the relation (see e.g. [4, 7]): 
( ) ( )
δλελε
.      (48) 
Due to the Coulomb gauge condition (43), 
( ) 0, =⋅ λε kk rrr          (49) 
is valid, too. 
In the following sections, we are going to calculate ( )1Ŝ  and ( )2Ŝ  by substituting the field operators 
( )xφ  and ( )xA
 from (21) and (46) as well as the distributions (20). Since we are considering 
electromagnetic interactions between (charged) spin-0 bosons, we have to take 0A  in (45) into 
account, too. Therefore, we first have to find out what the density of charge in (45) will be in this case. 
This can be done by coupling the Lagrangian density for free spin-0 bosons (25 a) to an 
electromagnetic field with the aid of the minimal coupling scheme µµµ eAii −∂→∂ . Then in that 
Lagragian density an extra term φφ +0eA  (the only one with an 0A ) emerges. If one subsequently 
regards the sum of that spin-0 and the electromagnetic Lagrangian, it is possible to obtain from it the 
electromagnetic field equations by means of the Euler-Lagrange equations. The former equations then 
contain a charge density φφ +e . That is why we can set ρ  in (45) to φφ + . 
With these results, one can then continue to calculate scattering matrix elements. 
The first term we calculate is 
( ) ( ) ( )( )( )( )∫ −+ Ω⋅+⋅= xpxAxApxxdI φφ 141 ˆˆ:ˆ r
     (50) 
appearing in ( )1Ŝ  (see (38)). To this end, it is useful to recognise that by means of (16) we get 
( )( )
⋅−−− =Ω p
φ .     (51) 
With (51) and integration by parts, (50) yields: 
( ) ( ) ( ) ( )( )++ −−++−+⋅= ∫ ∑ λλ
ωπ kk
ckppckppaa
pdpdkd
I rrrr
ˆˆˆˆ,
, (52) 
where 
( ) ( ) ( )kppkpp kpp
±−±−=±− 21
δωωωδδ     (53) 
and, of course, the a operators commute with the c operators. In 12Ŝ  (40), the first integral can be 
expressed in a similar way: 
( ) ( )( )( )
( ) ( )
( ) ( )(
( ) ( ) )
22112211
22112211
~2~22
ccppkkccppkk
ccppkkccppkk
pdpdkdkd
xxAxxdI
−+−−+−++−
+−+−+−++
   (54) 
After a quite lengthy but straightforward calculation, the second integral in (40) yields 
( ) ( ) ( )( ) ( )
( ) ( )( )( )( ) =Ω⋅+⋅
⋅−Ω⋅+⋅=
212122
111112
,ˆ,,ˆ
txptxAtxAp
xxpxAxApxxdxdI
rrrrrrr
rrrrrr
                                           (55) 
( ) ( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
−+−−+−++−
+−+−+−++−
11221122
11221122
~2~22
21222
ccppkkccppkk
ccppkkccppkk
ppkaa
pdpdkdkd
Finally, we want to determine the second term in ( )2Ŝ  (see (42)) which is not just like the product of 
two operators 1̂I  due to the time ordering operator as defined in (36). Unfortunately, we cannot use 
the famous Wick theorem, because the scalar field operator (21) contains only contributions to 
positive energy solutions: it does not consist of a sum of both positive and negative energy solutions as 
it would be the case for the field operator of the Klein-Gordon equation. Due to the symmetry of the 
time ordering operator (36) in its arguments, we may conclude 
( ) ( )( ) ( ) ( ) ( )2102012414212414 ˆˆ2ˆˆ xHxHxxxdxdxHxHTxdxd IIII ∫∫∫∫ −= θ .  (56) 
With this property, the calculation of the second term in (42) can be simplified a bit: 
( ) ( ) ( )( )( )( )(
( ) ( ) ( )( )( )( ) =Ω⋅+⋅
⋅Ω⋅+⋅=
22222
111112
xpxAxApx
xpxAxApxTxdxdI
     (57 a) 
( ) ( ) ( ) ( )
( ( ) ( )
( ) ( )( ) ( )( )
( ) ( )
( ) ( )( ) ( )( )
( ) ( )
( ) ( )( ) ( )( )
( ) ( )
( ) ( )( ) ( )( )
−−−−−−−
⋅−−′−′−′
++−−−−−−
⋅+−′−′−′
+−−−+−−−
⋅−−′+′−′
++−−+−−−
⋅+−′+′−′
21212
21212
21212
21212
21212
~exp~exp
~exp~exp
~exp~exp
~exp~exp
,,ˆˆˆˆ
~2~22
tititt
kppkppcc
tititt
kppkppcc
tititt
kppkppcc
tititt
kppkppcc
kaaaa
pdpdpdpdkdkd
kppkpp
kppkpp
kppkpp
kppkpp
ωωωωωωθ
ωωωωωωθ
ωωωωωωθ
ωωωωωωθ
rrrrrr
rrrrrr
rrrrrr
rrrrrr
The two integrals over the θ  function can be performed by means of the introduction of the two 
variables 
21: tt −=τ , 
    21: ttT +=         (58) 
with the Jacobian  
( ) 2
A linear combination of these variables  
21 tbtaTBA +=+τ       (59 a) 
can be expressed by means of 
( )baA −=
1 ,         (59 b) 
             ( )baB +=
(57) contains four terms of the subsequent type that can be simplified with the help of (58) and (59): 
( ) ( ) ( )
( ) ( )( ) ( )( )
( ) ( ) ( )( )τωωτθ
τωωτθ
21221
2122122
2211212
=+−−−
titixpett
   (60 a) 
The function θ  can be expressed by an integral in the complex plane (see e.g. [6,8]), 
( ) ∫
0      (61) 
with an ε  approaching zero. Substituting this into (60 a), we get: 
( ) ( ) ( )( ) ( )21
21221
τωωτθ
ωωδ +
=−−+ ∫ i
id i .  (60 b) 
With this result, (57) becomes: 
( ) ( ) ( ) ( )
( ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
⋅−−′−′−′−−+−−
⋅+−′−′−′+−+−−
⋅−−′+′−′−−++−
⋅+−′+′−′+−++−
′′′′′
δδωωωωωωδ
δδωωωωωωδ
δδωωωωωωδ
δδωωωωωωδ
kppkppcc
kppkppcc
kppkppcc
kppkppcc
kaaaa
pdpdpdpdkdkd
kppkppkk
kppkppkk
kppkppkk
kppkppkk
1~~ˆˆ
1~~ˆˆ
1~~ˆˆ
1~~ˆˆ
,,ˆˆˆˆ
~2~22
rrrrrr
rrrrrr
rrrrrr
rrrrrr
 (57 b) 
So far, we have only considered terms of the scattering matrix containing the electromagnetic vector 
potential A
. Now, we have to address to those terms containing the scalar potential 0A  too. (38) does 
not only contain (50), but also a Coulomb potential term: 
+= φφ 045ˆ AxdI .       (62 a) 
By substituting ρ  in (45) by φφ + , we obtain the following equation for (62 a), if we take into account 
that the Fourier transformed Coulomb potential looks like 
:       (63) 
43333
aaaakpkpkdkdpdpdI pkkp rr
−−′+′′′−= +′
.  (62 b) 
(42) contains two terms in 0A . The first term consists of a combination of (50) and (62 a), but taken at 
different times and therefore joined via the time ordering operator. That is why we have to use (60) as 
well as (45) and (63) again: 
( ) ( ) ( )( )( )( ) ( ) ( ) ( )( )
( ) ( ) ( )
( ) ( )[
( ) ( )
( ) ( )
( ) ( )
−−+−−+−−+
−−+−−+−−+
+−+−−++−+
+−+−−++−
−−′+′
=Ω⋅+⋅=
ωωωωωωωδδ
εωωωω
ωωωωωωωδδ
ωωωωωωωδδ
εωωωω
ωωωωωωωδδ
11121
11121
ˆˆˆˆˆˆ~
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ~
ˆˆˆˆˆˆ
111112
kppkkp
kpppkpk
kpkkpp
kpppkpk
kppkkp
kpppkpk
kpkkpp
kpppkpk
caaaaa
caaaaa
caaaaa
caaaaa
ppkkdkdpdpdpdpdkd
xxAxxpxAxApxTxdxdI
rrrrrr
rrrrrr
rrrrrr
rrrrrr
(64) 
The second term of (42) containing a scalar potential is even quadratic in 0A : 
( ) ( ) ( ) ( ) ( ) ( )( )∫∫ ++= 2202110124147ˆ xxAxxxAxTxdxdI φφφφ .    (65 a) 
(65 a) contains a factor of two integrands of the kind of (62 a), but taken at two different times. Thus 
the time ordering operator must be taken into account. With the same substitutions as in (62) and (64), 
we obtain the following result: 
( ) ( ) ( )
( ) ( ) εωωωω
δδωωωωωωωωδ
ikkkk
kkppkkpp
aaaaaaaapdkdkdpdpdkdkdpdiI
kpkpkpkp
pkkppkkp
+−−+−′−′
⋅−′+−′−′+−′−−++−−+
⋅′′′′=
11112222
22221111
ˆˆˆˆˆˆˆˆ
rrrrrrrr
rrrrrrrr
(65 b) 
The results (52), (54), (55), (57), (62), (64) and (65) substituted into (38) to (42) now enable us to 
evaluate scattering matrix elements for scattering processes to (and including) the order 2e . As two 
examples, we turn first to the scalar analogue of Compton scattering in order to address then to the 
scattering of two identical scalar bosons. These two scattering processes can be compared easily with 
the corresponding results of the well-known scalar QED dealing with the Klein-Gordon equation (3). 
Compton scattering 
For Compton scattering, we need one scalar boson and one photon each in the input and output 
channel. This means, we have to evaluate the element 
0ˆˆˆˆˆ0:ˆ ++′′′= µµ hqqh caSacS
rrrr ,      (66) 
with the Ŝ -operator (33). Firstly, we realise that the terms based on 1̂I  (see (52)) as well as 6Î  (see 
(64)) must vanish, because the subsequent two elements in the photon operators become zero: 
00ˆ00ˆˆˆ0 == ′′
′′ µλµµλµ δδ hhkhkh cccc
rrrrrr ,         (67) 
00ˆˆˆ0 =++′′ µλµ hkh ccc
rrr . 
There, we have used the commutation relations (47), the properties of creation and annihilation 
operators corresponding to those of (22) and (23) as well as abbreviated the delta functional by 
( )hk
rr −= 3δδ .      (68) 
The terms with 2Î  and 3Î  need 
qppqqppq aaaa rrrrrrrr 2121 0ˆˆˆˆ0 δδ ′
′ = ,    (69) 
whereas a term 
qppppqqppppq aaaaaa rrrrrrrrrrrr 12121212 0ˆˆˆˆˆˆ0 δδδ ′′′
′′ =  
belongs to 4Î . Here we have used again (22) to (24). 
For 2Î , 3Î  and 4Î  the following equations are necessary, too: 
  00ˆˆˆˆ0 =+′′′′ µλλµ hkkh cccc
rrrr ,                      
00ˆˆˆˆ0 =+++ ′′′′ µλλµ hkkh cccc
,       (70) 
µλλµµλλµ δδδδ ′′′′
′′′′ = khkhhkkh cccc
rrrrrrrr 0ˆˆˆˆ0 , 
λµλµµλλµ δδδδ ′′′′
′′′′ = khkhhkkh cccc
rrrrrrrr 0ˆˆˆˆ0 , 
where we have commutated the creation operators successively to the left and the annihilation 
operators to the right. 
Furthermore, we see that the term based on 5Î  from (62) becomes zero, due to 
00ˆˆˆˆˆˆ0 =++′
′′ qpkkpq aaaaaa
rrrrrr .       (71) 
A similar result holds true for the term based on 7Î  from (65). 
00ˆˆˆˆˆˆˆˆˆˆ0
22221111
′′ qpkkppkkpq aaaaaaaaaa
rrrrrrrrrr .     (72) 
Now, we can determine (66) by means of (33): 
( )( ) ( ) ( )762144134122215121 ˆˆˆˆˆˆˆ1ˆ IIIIiIiieIIieS +−+−−++−−≈− .   (73) 
For Compton scattering, 1̂I , 2Î , 3Î , 4Î , 5Î , 6Î  and 7Î  can be evaluated explicitly: 
0ˆˆˆˆ 7651 ==== IIII ,       (74) 
( ) ( ) ( )µεµε
δ ′′⋅
′−′−+
,     (75) 
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
′−⋅′′+′−⋅
++⋅′−+⋅
′−′−+
hqhhhqh
hqhhhqh
hqqhh
rrrrrrrrr
rrrrrrrrr
ωωωωπ
   (76) 
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
+⋅′+′⋅′′
′−⋅′′−′⋅′−′−+
′−′′−′
ωωωωπ
hqhhqh
hqhhqhhqhq
hqhqhq
hqhqhqqhh
rrrrrr
rrrrrr
rrrrrrrr
rrrrrrrr
2,2,1
2,2,1
   (77) 
The two terms 2Î  and 4Î  in (73) resemble the three terms in the corresponding formula of 
Compton scattering for scalar bosons, but this time being based on the Klein-Gordon equation (3) (see 
e.g. [6,8]): 
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )]µεµε
ωωωωπ
′−⋅′′
′+′⋅′′
⋅′−′−+
22~2~22
   (78) 
where ε  is the 4-dimensional generalisation of the three dimensional polarisation vector ε
 used so 
far. In (78), the first two terms correspond to 4Î . We realise that these terms in (78) look very much 
like those in (77), but also that especially the propagators of both theories are a bit different. (75) 
resembles the third term in (78). On the other hand, 3Î  in (76) could also be regarded as the 
analogue of the first two terms in (78) – at least after having used the Coulomb gauge condition (49) in 
(76).  
The first two terms in (78) vanish, if we choose the incoming scalar boson to be at rest, 
( ) ( )0,:,0 rr mqqq == ,      (79) 
and want to have transversally polarised photons in this laboratory system, 
( ) ( ) 0,, =⋅′′=⋅ qhqh µεµε ,     (80) 
and use the Lorentz gauge condition  
( ) ( ) 0,, =′⋅′′=⋅ hhhh µεµε .     (81) 
Accordingly, 3Î  and 4Î  in (73) vanish too, if we adopt (79) again and use the analogue of (80), 
( ) ( ) 0,, =⋅′′=⋅ qhqh rrrrrr µεµε ,     (82) 
as well as apply the Coulomb gauge condition (49). 
That is, under these conditions in both versions of scalar QED, only one term remains (i.e. (75) in (73) 
and, accordingly, the third term in (78)) which is a relativistically generalised version of the matrix 
element from which the well-known Thomson scattering cross section can be evaluated.  
Even though the propagators in both theories are rather different, for this example of scattering 
process, the results do not seem to differ very much from each other in the laboratory system chosen 
above. Therefore the question arises, whether this is also the case for further scattering processes. To 
this end, we are going to investigate what happens, if two identical scalar bosons interact with each 
other. 
Scattering of two identical scalar bosons 
If we want to calculate matrix elements of the S-operator (33) for the scattering of two identical scalar 
bosons, we need two scalar bosons in the input channel and two in the output channel: 
0ˆˆˆˆ0:ˆ
′′= qqqq aaSaaS rrrr .      (83) 
We can reuse (73), but have to evaluate 1̂I , 2Î , 3Î , 4Î , 5Î , 6Î  and 7Î  again. Firstly, 
we recognise that due to (52) and the analogue of (22) and (23) for the photon operators 
01̂ =I ,        (84) 
is valid. Furthermore, we need 
21212211
0ˆˆ0 λλλλ δδ kkkk cc
rrrr =+       (85) 
for calculating 2Î  with the help of (54), whereas all the other photon operator terms therein vanish. 
The same holds true for 3Î  with (55) and 4Î  with (57). 
As far as the scalar boson operators are concerned, for the evaluation of 2Î  and 3Î  
00ˆˆˆˆˆˆ0
121212
=+++′′ qqppqq aaaaaa rrrrrr . 
Hence, these two terms, 
0ˆ2 =I ,          (86) 
    0ˆ3 =I , 
are zero, too. For the evaluation of the non-vanishing term 4Î , the following result is useful: 
2122121121221211
212212112122121112121212
0ˆˆˆˆˆˆˆ0
pqpqpqpqpqpqpqpq
pqpqpqpqpqpqpqpqqqppppqq aaaaaaaa
′′′′′′′′
′′′′′′′′
rrrrrrrrrrrrrrrr
rrrrrrrrrrrrrrrrrrrrrrrr
δδδδδδδδ
δδδδδδδδ
. (87) 
With (85), (87) and the invariance of (48) under the transformation kk
−→ , we get 
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
21212112
11111122
21212121
11111111
Ainterms
qqqqqqqq
qqqqqqqqqqqqie
qqqqqqqq
qqqqqqqq
′+⋅′−′−⋅′+
 ′+⋅′−′−⋅′+′−′−+−
′−′′−′
′−′′−′
εωωωεωωω
εωωωεωωω
rrrrrrrrrr
rrrrrrrrrr
 (88 a) 
Here, we have omitted the terms 5Î , 6Î  and 7Î  containing terms in the scalar potential 
which are regarded later.  
(88 a) can be compared directly with the corresponding result of the Klein-Gordon equation (3) (see 
e.g. [9]): 
( ) ( )
( ) ( )
( ) 
′+⋅′+
′+⋅′+′−′−+
qqqqqqqqie
qqqq ωωωωπ
 , (89) 
if we use (48) and leave out the term ε  approaching zero in (88) : 
( ) ( )
( ) ( )
( ) ( )( ) ( )
( ) ( )
( ) ( )( ) ( )
( ) ( )
21212112
11111122
Ainterms
qqqqqqqq
qqqqqqqq
qqqqqqqqie
′+⋅′−′−⋅′+
′+⋅′−′−⋅′+
′+⋅′+
′+⋅′+′−′−+−
rrvrvrrr
rrvrvrrr
rrrrrrvr
(88 b) 
The first two terms in (88 b) correspond to (89) which contains the photon propagator in the Feynman 
gauge. But since we use a Coulomb gauge, only the space-like components of the linear 4-momenta 
appear in the numerators of the first two terms in (88 b). Moreover, in the second line of (88 b) two 
additional terms are present which can be reformulated by means of the delta distribution in (88 b): 
( )( )
( ) ( )
( )( )
( ) ( )221221
′−′−′
− .     (90) 
The structure of (88 b) is the same as that of (89). We have two terms: in the second term, the 
momenta of the two scalar bosons in the output channel have been exchanged compared to the first 
term. 
Now we address to the terms in the scalar potential 0A  in (88). The Coulomb term 5Î  contains a 
term 
pqkqqpqkkqpqqpqk
pqkqqkqpkqpqqkqpqqpkkpqq
aaaaaaaa
′′′′′′′′
′′′′′′′′
rrrrrrrrrrrrrrrr
rrrrrrrrrrrrrrrrrrrrrrrr
21122112
211221121212
0ˆˆˆˆˆˆˆ0
δδδδδδδδ
δδδδδδδδ
Therefore, 5Î  yields 
( ) ( ) 
−−′+′=
qqqqI rrrrδ
.    (91) 
The term 6Î  vanishes, since the annihilation operators for photons in (64) act directly on the 
vacuum states.  
On the other hand, 7Î  becomes a rather lengthy because of 
( )( )
( )( )
( )( )
( )( )11
12112121122222
21121121122222
121121211222
121121122122
122222111112
0ˆˆˆˆˆˆˆˆˆˆˆ0
aaaaaaaaaaaa
kqpqkppkqpqkqp
kkkqpqppqpqkqp
kqpqkppkqkqp
kqpqkkqkppqp
qqpkkppkkpqq
′↔′−+
′↔′−−
′↔′−−
′↔′−=
′′′′′′
′′′′′′
′′′′′′
′′′′′′
rrrrrrrrrrrrrr
rrrrrrrrrrrrrr
rrrrrrrrrrrr
rrrrrrrrrrrr
rrrrrrrrrrrr
δδδδδδδ
δδδδδδδ
δδδδδδ
δδδδδδ
where ( )11 kp ′↔′
 denotes the same term as the immediately preceding one, but with exchanged 
momenta p′
 and 1k ′
, respectively.  Thus, 7Î  gives 
−−′+′=
−′−−′−+′−+′−
22222222
12112221
qpqpqpqpqpqpqpqp
pqqpqq
rrrrrrrrrrrrrrrr
rrrrrr
ωωωωωωωω
ωωωωπ
.  (92) 
Hence in place of the time-like components (i.e. the energy terms) of the linear 4-momenta in the 
numerators of (89) derived from the Klein-Gordon equation, several terms arise: (90), (91) and (92). 
But (91) and (the non relativistic limit of) (92) would also have appeared, if we had started from the 
non-relativistic Schrödinger equation. Thus, these terms state the fact that the equation used for 
obtaining (88) (together with (90), (91) and (92)) is Schrödinger equation like.  
Conclusions and outlook 
For scalar bosons, we could see that it is possible to describe scattering processes by means of a square 
root operator equation being coupled to an electromagnetic field. We achieved this by splitting off a 
factor in the shape of the free square root operator from the equation and by a series expansion of a 
remaining square root factor containing terms in the electromagnetic vector potential and powers of 
the (inverse) free square root operator. The latter ones could be given an integral representation.  
Having quantised the fields involved, we could evaluate the scattering matrix elements for Compton 
scattering and for the scattering of two identical bosons (to – and including – the quadratic order of  e) 
which, on the one hand, resembled the results derived with the help of the corresponding Klein-
Gordon equation and, on the other hand, the results one would have obtained with a non-relativistic 
Schrödinger equation.  
Of course, now several questions arise, e.g.: 
• Can we formulate Feynman rules at all for our non-local scalar QED? 
• Do divergent terms appear and, if yes, can a renormalisation procedure be found? 
• Can the results of this non-local QED be confirmed (or refuted) by experiments? 
To the first question: if one tries to formulate Feynman rules, one must be aware that in each step of 
the approximation procedure, we have to expand the first square root factor (in the second term of) (7) 
to the desired order in e. Thus, the Hamiltonian we are using within that procedure must be adapted in 
each order of e of the approximation. Therefore we conclude that even if it were possible to formulate 
Feynman rules, they would be much more complicated than those for the Klein-Gordon theory. This is 
the price we have to pay for non-locality. But without Feynman rules, it is not very easy to analyse the 
renormalisability of that non-local scalar QED either.  
The answer to the last question listed above is negative due to a lack of elementary spinless bosons in 
nature. But that question would be sensible, if we had a corresponding non-local theory for spin-1/2 
particles. We could even make a guess, how this theory would look like: for free spin-1/2 particles, the 
wave functions in (2) should be 2-spinors. It would also be possible to couple this equation to an 
electromagnetic field, but the usual minimal coupling scheme does not work. Instead we would have 
to postulate an equation: 
ˆ ′=∂ ,       (93) 
with the Hamilton operator 
( ) ( ) 022 ˆˆ eAEiBeAepmH +⋅−−+=′ rmrrrr
σ ,    (94) 
which contains an additional term with the Pauli matrices σ
 and the magnetic field B
 as well as the 
electric field E
 under the square root. For this equation, it has already been shown that it can 
reproduce the gyromagnetic factor of 2 for the electron as well as, when being applied to a hydrogen 
atom, that it can reproduce correct binding energies of the electron at least to (and including) the 
quadratic order in 2e  (see [12]). We can apply the same approximation procedure to this equation as 
we have presented here for the scalar case. But of course, additional difficulties emerge: the 
Hamiltonian corresponding to the one shown in (25) should be Hermitian. And at least for the free 
spin-1/2 case, that Hamiltonian should be relativistically invariant. This means, that for the free case, 
(25) should be a Lorentz scalar with respect to the spin. To this end, we have to replace φ  and +φ  
therein by combinations of a mixture of left and right handed 2-spinors Lφ  and Rφ , respectively, and 
their Hermitian conjugates, because RLφφ
+  and LRφφ
+  are Lorentz scalars. Therefore, (25) could be 
replaced either by 
RLLR HHH φφφφ ±
++ ′+′= ˆˆˆ
       (95) 
or by 
RLLR HHH φφφφ mˆˆˆ 2
1 ′+′= +±
+       (96) 
so that the Hamiltonians (95) and (96) become Hermitian, because of the property 
±′=′ HH ˆˆ m .       (97) 
The results of a QED based on (95) or (96) could then be compared with the ones based on a 
corresponding Dirac equation.  
The author does not know, whether such an approach has already been performed for the spin-1/2 case 
or for the here presented spinless case. He does not know either, if an application for it can be found, 
where e.g. non-local properties are indispensable. But it seems to him that from the technical point of 
view, square root operator equations coupled to an electromagnetic field like (4) (or maybe even like 
(93) together with (94)) can be handled within the framework of a quantum field theory. Such 
equations were given up quite early in the history of quantum mechanics for several good reasons, e.g. 
due to their lack of relativistic invariance (see [13]), their non-local character accompanied by the 
difficulty of finding an appropriate mathematical interpretation and description, respectively (see e.g. 
[4,5,6]). While the first reason mentioned remains still valid, from today’s perspective, those non-local 
properties might not be refused as vehemently as in the past (e.g. when one looks out for 
approximations of the so called Bethe-Salpeter equation [11]). The author hopes to have shown, that at 
least answers to the question of possible descriptions of such non-local square root operator equations 
can be found. 
References 
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