Poincare group operators with 4-vector position

0.12 MB·English

by  Shaun N Mosley

Poincar´e group operators with 4-vector position
Shaun N Mosley, Alumni, University of Nottingham, UK
∗
Abstract
We present a new set of massless Poincar´e group operators Hermitian with respect to the 1/r inner
productspace,whichhavequasi-planewaveenergy-momentumeigenfunctionshavingvelocityc along
4
their axis of propagation. These are constructed by means of a unitary transformation from a known
0
0 set of massless Poincar´egroup operators of helicitys=0, 1, 1...The position vectorr isthe space
2 ±2 ±
part of a null 4-vector.
n
a
J
I. Introduction
9
1
The standard relativistic quantization procedure [1] involves the integral operator
1
v ω √(m2 + p2)=√(m2 2)
4 ≡ − ∇
0
1 in units with h¯ =c=1, then
1
0 H p0 =ω, p= i∇ (1)
≡ −
4
0
which together with the Lorentz generators of boosts and rotations
/
h
1
p K= (rH + Hr), J= ir ∇ (2)
- 2 − ×
t
n make up the ten Poincar´e group operators. There are a number of well-known difficulties with this
a
procedure, of which we mentionthe following. 1) The conserved inner product in configuration space
u
q
is [2]
:
v
φ ψ = d3r[φ ωψ + (ωφ )ψ], (3)
Xi h | i Z ∗ ∗
r butthequantity[ψ∗ωψ+(ωψ∗)ψ] canbenegativeevenforsuperpositionsofpositiveenergysolutions
a
satisfying i∂ ψ = +ωψ. And 2) The position operator r is not Hermitian, and must be replaced by
t
the Newton-Wigner [3] position operator r the eigenfunctions of which are not delta functions.
NW
Wesetoutinthispaperanalternativequantizationprocedurewhichallowsthe‘natural’position
operator r. Our aims are
(A) to find an alternative set of Poincar´e group operators: while the momentum operator will not
have the simple form ( i∇), we require that
−
(B) the eigenfunctions of the new energy-momentum operators Pλ represent waves which propagate
at velocity c along their propagation axis,
(C) the Hamiltonian P0 will be a positive operator,
(D)thePoincar´egroupoperators willbeHermitianwithrespecttoainnerproductspace suchthat
L
ψ ψ is the integral of a positive definite density, and
h | i
(E) the position operator r must be Hermitian with respect to and be part of a 4-vector position
L
operator rλ.
E-mail: [email protected]
∗
1
In this paper we consider only mass zero representations with helicity s = 0, 1, 1..., leaving
±2 ±
the case with mass to a following paper. We start from a known [4] set of Poincar´e group operators
(aλ, Jλµ), the space part of aλ beingclosely relatedto the Runge-Lenz vectorfrom the theory of the
Schrodinger hydrogen atom. These operators are Hermitian with respect to the 1/r inner product
spaceof(7)below,andfurthermorethea0 operatorispositive. Astheeigenfunctionsofaλ arewaves
of variable velocity, we proceed in the next section to find a unitary transformation such that the the
transformed eigenfunctions have plane wave character - in that their velocity along their propagation
axis is c.
The operators [4]
a0 = r 2, a= 2(∂ r)∇ + r 2 (4)
r
− ∇ − ∇
K (J10,J20,J30)= ir∇, J (J23,J31,J12)= ir ∇ (5)
≡ − ≡ − ×
obey the necessary Poincar´e group commutation properties:
[Jλµ,aν]=i(ηµνaλ ηλνaµ), [aλ, aµ]=0, (6)
−
[Jλµ,Jνρ]=i(ηλρJµν +ηµνJλρ ηλνJµρ ηµρJλν).
− −
These operators aλ, Jλµ are perhaps better knownas components of a SO(4,2) group representation,
see for example [5]. Also
aλ a =0
λ
·
so that Jλµ,aν of (4,5) are a massless spin-zero set of Poincar´e group operators. The a0 is a positive
operator. They are Hermitian with respect to the 1/r inner product space
1/r
L
1
: φ ψ φ (r) ψ(r)d3r= φ (r,θ,φ)ψ(r,θ,φ)rdrdΩ (7)
L1/r h | i1/r ≡Z ∗ r Z ∗
which is Lorentz-invariant [6]. These Jλµ,aν are to our knowledge the simplest (non-trivial) set
of configuration space Poincar´e operators containing only local (differential) operators. The density
ρ= 1ψ ψ is conserved provided that the ψ are solutions of i∂ ψ =a0ψ, as then
r ∗ t
1
∂ ( ψ ψ)=∇ i[ψ ∇ψ (∇ψ )ψ].
t ∗ ∗ ∗
r · −
Orthonormal eigenfunctions of aλ with respective eigenvalues kλ (k, k) (k, k) are the
≡ | | ≡
Bessel functions ([4] formula (92)):
aλuk =kλuk
where
1 1
uk(r)= J0 [ 2kλrλ]1/2 = J0 [2kr+2k r]1/2 (8)
4π (cid:16)− (cid:17) 4π (cid:16) · (cid:17)
whichmaybe provedbydirectdifferentiation. Thequantity(kr+k r) isnon-negativebySchwartz’s
·
rule. The orthogonality and completeness relations are [4,7]
d3r
uk(r)uk′(r)=kδ(k k′), (9)
Z r −
d3k
uk(r)uk(r′)=rδ(r r′). (10)
Z k −
2
Helicitys representationsof the Poincar´egroup whichreduce to(4,5) when s=0 werefound by
Derrick [8]. The operators a0, a , K , J which also satisfy the the Poincar´e group algebra (6) are
s s s s
i 1
a0 = r 2 + 2s ∂ + 2s2 , (11)
s − ∇ r z φ r z
− −
1
a = 2(∂ r)∇ + r 2 s4i(W ∇) s22ǫ (12)
s r 3
− ∇ − × − r z
K = ir∇ s(ˆr W), J = ir ∇ + s−W (13)
s s
− − × − ×
where
x y
W=( , , 1), ∂ =(r ∇) (14)
φ 3
r z r z − ×
− −
with s=0, 1, 1... The eigenfunctions of aλ are [8]
±2 ± s
usk(r)= 1 e2isf(ˆr,kˆ)J2s [2kr+2k r]1/2 . (15)
4π (cid:16) · (cid:17)
We give the angular phase factor e2isf(ˆr,kˆ) for reference in the appendix. And e2isf(ˆr,kˆ) e2isφ
→
when k=(0,0, k), where eiφ (rˆ +irˆ )/(rˆ2+rˆ2)1/2. Then the eigenfunction (15) is
± ≡ 1 1 1 2
1
u (r)= e2isφJ [2kr 2kz]1/2 . (16)
s(0,0,±k) 4π 2s(cid:16) ± (cid:17)
Theeigenfunctionsusk havethesameorthogonalityandcompletenessrelations(9,10)astheirhelicity
zero counterparts.
2. The unitary operator
V
The eigenfunctions usk are essentially waves in parabolic coordinates. As massless waves proceed
at the velocity of light c (which is unity in our units), we look for a unitary transformation of the
operators aλ, Jλµ defining new operators Pλ aλ 1, Jλµ Jλµ 1, such that the resulting
s ≡ V sV− s ≡ V s V−
eigenfunctionsVusk of the Psλ energy-momentumoperators represent wavesof velocityc along their
propagation axis.
Consider ψ(ur) with ( < u < ), which is the wavefunction along the line through the
′
−∞ ∞
origin which includes the point r . The unitary operators of (21) below map ψ(ur) onto itself,
′ ′
V
whichisclearlyseenbyinspectionof (21d). Inthe appendixweshow how canbe constructedfrom
V
the parity , inversion , and Fourier transform operators :
P N F
f(r,θ,φ) f(r,π θ,φ+π) (17)
P ≡ −
1 1
f(r,θ,φ) f( ,θ,φ) (18)
N ≡ r2 r
2 ∞ cos(rt)
c f(r,θ,φ) f(t,θ,φ)dt (19)
Fs ≡ rπ Z sin(rt)
0
( i ).
F± ≡ Fc ± Fs
The operators , 1 are
−
V V
1 1 1
1g(r,θ,φ) ( √r) ( √r) g(r,θ,φ) (20a)
− +
V ≡ 2 √rF − √rF− P N
(cid:2) (cid:3)
1 1 1
f(r,θ,φ) ( √r) ( √r) f(r,θ,φ) (21a)
+
V ≡ 2N √rF− − √rF P
(cid:2) (cid:3)
= 1 1 ∞ dt e it/r √tf(t,θ,φ) eit/r √tf(t,π θ,φ+π) (21b)
√2π √r3 Z0 (cid:2) − − − (cid:3)
which can be written
3
f(r) = 1 ∞ du e iu √uf(ur) eiu √uf( ur) (21c)
−
V √2π Z − −
0 (cid:2) (cid:3)
= 1 ∞ du e iu u sgn(u)f(ur). (21d)
−
√2π Z | |
p
−∞
The kernel e iu u sgn(u) and its derivatives are continuous at u=0. We see that ( f)(r) is
− ′
| | V
essentiallya(cid:8)Fourieprtransform(cid:9)of√rf(r) alongthe liner=ur, ( <u< ).The 1 operator
′ −
−∞ ∞ V
can be simplified to (corresponding to the form (21d) for )
V
1g(r)= 1 ∞ du ei/u sgn(u) g(ur) (20b)
−
V √2π Z u
−∞ | |
p
If √rf(r) does not converge at infinity, the integral of (21) does not exist. For this case the
definitions of of (19) must be extended (see for example [9]) to the so called generalized cosine
F±
(sine) transform . In effect we write e iu = i∂ e iu in (21c) and then integrate by parts while
± u ±
∓
discarding the surface terms at infinity.
f(r) = i ∞ du e iu ∂ [√uf(ur)] + eiu ∂ [√uf( ur)]
′ − u u
V −√2π Z0 (cid:16) − (cid:17)
= i ∞ du e−iu √u∂ [√uf(ur)] + eiu √u∂ [√uf( ur)]
u u
−√2π Z0 (cid:16) √u √u − (cid:17)
= i √r∂ √r ∞ du e−iu f(ur) eiu f( ur)] . (21e)
r
−√2π n Z0 (cid:16) √u − √u − (cid:17)o
For the last line we have used the identity √u∂ √u f( ur) = √r∂ √r f( ur). The operator
u r
± ± ±
f(r) is equal to f(r) whenever the latter exists, so from now on we drop the prime on .
′ ′
V V V
3. The eigenfunctions usk
V
In this section we find the eigenfunctions of Pλ = aλ 1 which we call
s V s V−
wsk = usk (22)
V
and show that these represent waves which have velocity c along their propagation axis.
The operator sees through any angular variable, so recalling (15) we need to evaluate
V
J (2[kr+k r]1/2) where 2s iszero or an integer. We will specializeto the case when k=(0,0,k)
2s
V ·
and the eigenfunction is
1
u (r)= e2isφJ [2kr+2kz]1/2 , (23)
s(0,0,k) 4π 2s
(cid:16) (cid:17)
then from (21e) and the integrals (1.13.25), (2.13.27) of [10]
w (r)= u (r)
s(0,0,k) s(0,0,k)
V
= ie2isφ √r∂ √r ∞ du e−iu J [2ukr+2ukz]1/2 eiu J [2ukr 2ukz]1/2
r 2s 2s
−4π√2π n Z0 (cid:16) √u (cid:16) (cid:17) − √u (cid:16) − (cid:17)(cid:17)o
e2isφ kr+kz kr+kz
= (√r∂ √r) ei(1 2s)π/4 exp i( J ( )
r − s
− 4π√2 n { 4 } 4
kr kz kr kz
e−i(1−2s)π/4 exp i( − Js( − ) . (24a)
− {− 4 } 4 o
4
Omitting the e2isφ phase factor, the w are everywhere continuous, and are zero at the origin
s(0,0,k)
except for the spin zero case s = 0. The w have similar asymptotic behaviour for various s,
s(0,0,k)
and are of simpler form when s is half integer. When for example s=1/2
eiφ
w12(0,0,k)(r)=−4√π3 (cid:16)[kr+kz]1/2 exp(i[kr+kz]/2) − [kr−kz]1/2 exp(−i[kr−kz]/2)(cid:17) (25)
which is proportional to exp(ikz) along both halves of the z axis, so that e iktw is a
{ − 21(0,0,k)}
unidirectional wave proceeding in the +z direction at velocity c. For large r, the w are of
s(0,0,k)
order √r, the ‘extra’factor √r accountedfor by the 1/r innerproduct space. For referencewe write
out the w of (24a) in full below
s(0,0,k)
ws(0,0,k)(r)= 4e2πi√sφ2(cid:20)ei(1−42s)π exp{ikλ/2}n(2s−1)Js(kλ/2) − kλ(cid:16)Js−1(kλ/2) + iJs(kλ/2)(cid:17)o
e−i(1−42s)π exp ikµ/2 (2s 1)Js(kµ/2) kµ Js 1(kµ/2) iJs(kµ/2)
− {− }n − − (cid:16) − − (cid:17)o(cid:21)
(24b)
where λ = (r+z)/2, µ = (r z)/2. A contour plot of these functions reveals the planar nature of
−
the wave fronts.
4. The Lorentz operator K 1 simplified
−
V V
WeareinterestedtoseehowtheHermitianpositionoperatorr transformsunderboostsandrotations,
so we must evaluate the commutators [Ja, rb], [Ka, rb] where
s s
J J 1, K K 1.
s s − s s −
≡V V ≡V V
We will first simplify the J , K operators, and here we will only consider the s = 0 helicity zero
s s
operators J, K, because the extra helicity components commute with the position operator.
We first need to define the operators , 1 (we also write out again the , 1 of (21) for
− −
U U V V
comparison):
1 1 1 1
+ √r, 1 = + √r , (26)
+ − +
U ≡ 2N √r F− F P U 2 √r F F−P N
(cid:2) (cid:3) (cid:2) (cid:3)
1 1 1 1
√r, 1 = √r .
+ − +
V ≡ 2N √r F− − F P V 2 √r F − F−P N
(cid:2) (cid:3) (cid:2) (cid:3)
Also
1 1
[K,( √r)]=0, [J,( √r)]=0, (27)
N √r F± N √r F±
K = K, J = J (28)
P −P P P
K 1 = 1K, K =K (29)
− −
V U V U
[J, ]=[J, 1]=0. (30)
−
V V
The first relation (27) becomes apparent when we write out ( 1 √r)ψ(r) in the form
N √r F±
( 1 √r)ψ(r) 2 ∞ due iu √uψ(ur),
N √r F± ≡rπ Z ±
0
then one can see that K,J only act on the r in the argument of ψ. And (29) follows from (27), (28).
The relation (30) means that
J J 1 = J.
−
≡V V
5
Following from (29)
K K = ( )K = K( ). (31)
† † †
≡V V VU UV
We can simplify the ( ), ( ) unitary operators of (31) above which are adjoints of each
† †
VU UV
other. We need the further identities (referred to in the appendix)
= i( ), = i( ),
+ + e o e o
F F − H −H F−F− H −H (32)
=2 i( + ), =2 + i( + )
+ e o + e o
F F− − H H F−F H H
where , are the Hilbert transforms of even, odd functions:
e o
H H
2r ∞ f(t,θ,φ) 2 ∞ f(ur)
f(r,θ,φ) dt, f(r)= du,
He ≡ − π Z r2 t2 He −π Z 1 u2
0 − 0 − (33)
2 ∞ tf(t,θ,φ) 2 ∞ uf(ur)
f(r,θ,φ) dt, f(r)= du,
Ho ≡ −π Z r2 t2 Ho −π Z 1 u2
0 − 0 −
and also we note that
1 1 1 1
( √r) =( √r), ( √r) =( √r). (34)
e o o e
N √r H N √r H N √r H N √r H
Then with the aid of (32), (34)
1 1
( ) = + √r
† + +
VU 4N √r F− − F P F F−P N
(cid:2) (cid:3)(cid:2) (cid:3)
1 1
= i( + ) + i( ) √r
e o e o
2N √r H H H −H P N
(cid:2) (cid:3)
1 1 1
= i ( + ) ( ) √r i √r (35)
e o e o
2 √r H H − H −H P ≡ √r G−
(cid:2) (cid:3)
and similarly
1 1 1
( ) = i ( + ) + ( ) √r i √r (36)
† e o e o +
UV 2 √r H H H −H P ≡ √r G
(cid:2) (cid:3)
where
1
[( + ) ( ) ] (37a)
G± ≡ 2 He Ho ± He−Ho P
1 ∞ f(ur) f( ur)
f(r)= − du. (37b)
G± π Z0 (cid:16) u−1 ∓ u+1 (cid:17)
Finally substituting (35), (36) into (31) we have the transformed boost operator
1 1
K K = K(i √r) = (i √r)K (38)
† +
≡ V V √r G √r G−
where K ( ir∇). Note that K is not a local (differential) operator like K.
≡ −
5. The position 4-vector operator
The simplification (38) allows us to to calculate the commutator [Ka, rb]. For the position operator
r to be the space part of a 4-vector rλ (r0,r), then r0 must satisfy both of the following:
≡
[Ka, rb] =iδabr0, [K, r0] =ir. (39)
6
To calculate [Ka, rb] we must first evaluate [ , r]. Recalling (37b) then
G±
1 ∞ urf(ur) urf( ur)
rf(r) = − du
G±(cid:0) (cid:1) π Z0 (cid:16) u−1 ± u+1 (cid:17)
1 ∞ (u 1+1)f(ur) (u+1 1)f( ur)
= r − − − du
π Z0 (cid:16) u−1 ± u+1 (cid:17)
1 ∞
= r f(r) + r (f(ur) f( ur)) du, (40)
G± π Z0 ± −
and multiplying(37b) from the left by r and then subtracting from (40) yields the operator identity
[ , r] = r , (41)
G± Z±
where
1 ∞
f(r) (f(ur) f( ur))du. (42)
Z± ≡ π Z0 ± −
Also with similar methods to the above we find
1 ∞
rf(r) = r f(r) + r (f(ur) f( ur))du
G±(cid:0) (cid:1) G∓ π Z0 ∓ −
or
r = r + r . (43)
G± G∓ Z∓
With the commutator (41) we can now evaluate [Ka, rb] :
[Ka, rb] =[Ka(i 1 √r), rb]
+
√r G
1 1
=[Ka , rb](i √r) + Ka(i [ , rb]√r)
+ +
√r G √r G
1 1 1
= i δabr(i √r) + Ka( rb √r) i δabr(i √r) (44)
+ + +
− √r G √r Z → − √r G
whereforthelastlineweput √r =0.Thisisaboundary conditiononthewavefunctionrequiring
+
Z
that
1 ∞
√rψ(r) √r du√u[ψ(ur) + ψ( ur)] =0. (45)
+
Z ≡ π Z −
(cid:0) (cid:1) 0
The above is automatically satisfied if ψ(r) is odd under the parity transformation. When ψ(r) is
even under parity then (45) is equivalent to ∞dr √rψ(r,θ,φ) =0.
0
R
Recalling (39), the formula (44) suggests that
1
r0 = r(i √r)
+
− √r G
1 1 1
= (i √r)r + ir(i √r) = (i √r)r
+
− √r G− √r Z − √r G−
where the second line results from (43) and (45). So we can write two equivalent forms for r0 :
1 1
r0 = r(i √r) = (i √r)r.
+
− √r G − √r G−
(46)
We must still check the commutator [K, r0], recalling from (38) and (46) that both the operators
K, r0 can be written in either of two ways:
[K, r0]=Kr0 r0K
−
1 1 1 1
= K(i √r)(i √r)r + r(i √r) (i √r)K
+ +
− √r G √r G− √r G √r G−
= Kr + rK = ir (47)
−
7
which is the required result, and we have used the identity
i i
( √r)( √r) = ( )( ) =1 (48)
+ † †
√r G− √r G VU UV
recalling (35), (36). The results (44), (47) show the 4-vector property of
1
rλ = r(i √r), r
+
(cid:16) − √r G (cid:17)
only assuming (45). Furthermore we see from (46) that
1 1
(r0)2 = r(i √r)(i √r)r =r2 (49)
+
√r G √r G−
so that rλ is a null 4-vector. Finally the r0, r components of rλ commute as
1 1
[r0, r] = r(i [ , r]√r)= r(i r √r) = 0
+ +
− √r G − √r Z
with (45).
5. Discussion
We have achieved our aims (A) to (E) of the Introduction. And we have shown that the position
operator r has covariant meaning, in that it is the space part of a null 4-vector (as discussed in the
last section, this 4-vector property of rλ requires the wavefunction to be subject to the boundary
condition (45)). The difficulties with the position operator have been well known since the inception
of relativistic quantum mechanics: the eigenfunctions of the Newton-Wigner [3] operator (r ) are
NW
smearedout inspace, whichrendersproblematicthe exactmeaningofr when, forexample,potential
terms are included in the Hamiltonian. From our viewpoint the difficulties in the usual theory arise
fromtheinnerproduct(3),whichisadirectconsequenceofthemomentumoperatorbeingp=( i∇).
−
Then the inner product (3) disallows the ‘natural’ position operator r.
Ourapproachhasbeentoallowotherpossibilitiesforthemomentumoperator,onlyrequiringthat
it is the space part of a 4-vector and that the eigenfunctions of the energy-momentum operator have
planewavecharacter. Becausethemomentumoperators presentedhereareHermitianwithrespectto
the simple1/r innerproduct space of (7), thisinturnallowsfor the natural position operator r. The
‘cost’ we have to pay for this transparency of the position operator is that the momentum operators
as well as their eigenfunctions are more complicated.
8
Appendix
The angular phase factor e2isf(ˆr,kˆ)
In spherical coordinates
ˆr=(cosφ sinθ, sinφ sinθ, cosθ)
kˆ =(cosφ sinθ , sinφ sinθ , cosθ )
k k k k k
then we give Derrick’s formula ((39) of [8])
eif(ˆr,kˆ) =[2/(1+kˆ·ˆr)]1/2(eiφ cos21θ cos21θk + eiφk sin12θ sin21θk) (A1)
and in the particular case when kˆ =(0,0,1) so that θ =0, then
k
eif(ˆr,kˆ) →[2/(1+rˆ3)]1/2(eiφ cos21θ)=eiφ (A2)
The unitary operators
V
The unitary operators of (21) can be constructed from the parity , inversion , and Fourier
V P N
transform operators :
F
f(r,θ,φ) f(r,π θ,φ+π) (A3)
P ≡ −
1 1
f(r,θ,φ) f( ,θ,φ) (A4)
N ≡ r2 r
2 ∞ cos(rt)
c f(r,θ,φ) f(t,θ,φ)dt (A5)
Fs ≡ rπ Z sin(rt)
0
( i ) (A6)
F± ≡ Fc ± Fs
with the adjoint properties
1 1
† = , † = , ( c√r)† = ( c√r). (A7)
P P N N √r Fs √r Fs
(The self-adjoint property = can be shown by a change of variablesr 1/u withinthe scalar
†
N N →
product L1/r, and the adjoint property of (√1r Fsc√r) follows by changing the order of integration
within the scalar product.) Also
= = = = 1. (A8)
c c s s
PP N N F F F F
For the latter property f(r) = f(r) = f(r) to hold always, the definitions of
c c s s c s
F F F F F F
(cid:0) (cid:1) (cid:0) (cid:1)
must be extended [9] to the so called generalized cosine (sine) transform when the integral of (A5)
does not exist. Then
1 1 i
∂ , ∂ , ∂ (A9)
Fc →− r Fs r Fs → r Fc r F± →±r F± r
whichiseffectivelyanintegrationbypartswithintheintegral(A5)whilediscardingthesurfaceterms.
Whentheconventionalcosine(sine)transformof(A5)doesexist,itagreeswiththeextendeddefinition
(A9). The extension procedure of (A9) can be repeated. Then the operators ( 1 √r), ( 1 √r)
√r Fc √r Fs
as well as , , are unitary.
P N
The operators , do not commute but combine as follows [11]:
c s
F F
= , = (A10)
s c e c s o
F F −H F F H
9
where , are the Hilbert transforms of even, odd functions:
e o
H H
2r ∞ f(t,θ,φ) 2 ∞ f(λr)
f(r,θ,φ) dt, f(r)= dλ,
He ≡ − π Z r2 t2 He −π Z 1 λ2
0 − 0 − (A11)
2 ∞ tf(t,θ,φ) 2 ∞ λf(λr)
f(r,θ,φ) dt, f(r)= dλ.
Ho ≡ −π Z r2 t2 Ho −π Z 1 λ2
0 − 0 −
Then it follows that
= i( ), = i( ),
+ + e o e o
F F − H −H F−F− H −H (A12)
=2 i( + ), =2 + i( + ).
+ e o + e o
F F− − H H F−F H H
The identities (A12) allow us to construct the unitary operator and its adjoint:
V
1 1 1 1
√r, 1 = √r , (A13)
V ≡ 2N √r F− − F+P V− 2 √r F+ − F−P N
(cid:2) (cid:3) (cid:2) (cid:3)
then the necessary properties 1 = 1 etc follow from (A8), as the parity operator commutes
−
VV P
with and , also 2 = 2 =1. Writing out the operator f(r,θ,φ) :
F± N P N V
1 1 1
f(r,θ,φ) ( √r) ( √r) f(r,θ,φ) (A14a)
+
V ≡ 2N √rF− − √rF P
(cid:2) (cid:3)
= 1 1 ∞ dt e it/r √tf(t,θ,φ) eit/r √tf(t,π θ,φ+π) (A14b)
√2π √r3 Z0 (cid:2) − − − (cid:3)
= 1 ∞ du e iu √uf(ur,θ,φ) eiu √uf(ur,π θ,φ+π) (A14c)
−
√2π Z − −
0 (cid:2) (cid:3)
where we have substituted t=ur. Then changing to cartesian coordinates we can write
f(r) = 1 ∞ du e iu u sgn(u)f(ur) (A14d)
−
V √2π Z | |
p
−∞
which is (21d). Then with the operator substitutions of (A9)
f(r) i ∞ du e iu ∂ [√uf(ur)] + eiu ∂ [√uf( ur)]
− u u
V → −√2π Z0 (cid:16) − (cid:17)
which is (21e).
References
[1] Foldy L.L., Phys. Rev. 102, 568-561 (1956)
[2] Schweber S.S., An Introduction to Relativistic Quantum Field Theory (Harper and Row, New
York, 1961) p57
[3] Newton T. D. and Wigner E. P. Rev. Mod. Phys. 21, 400-406 (1949)
[4] Derrick G. H., J. Math. Phys. 28 , 1327-1340 (1987)
[5] Barut A. O. and Ramsmussen W., J. Phys. B 6, 1695-1712 (1973)
[6] Peres A., J. Math. Phys. 9, 785-789 (1967)
[7] Kastrup H. A., Phys. Rev. B 140, 183-186 (1965)
[8] Derrick G. H., J. Math. Phys. 29 , 636-641 (1988)
[9] ZemanianA.H.,GeneralizedIntegral Transformations(IntersciencePublishers,NewYork,1968)
p163
[10] Bateman H., Tables of Integral Transforms, Vol 1 (McGraw-Hill, New York, 1954) Formulae
(1.13.25) and (2.13.27)
[11] Rooney P. G., Can. J. Math. 24, 1198-1216 (1972) Sec.8
10