On The Pauli-Weisskopf Anti-Dirac Paper

0.2 MB·English

by  Walter Dittrich

On The Pauli-Weisskopf Anti-Dirac Paper.
Walter Dittrich
Institute for Theoretical Physics
5 University of Tu¨bingen
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Auf der Morgenstelle 14
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2 D-72076 Tu¨bingen
Germany
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a [email protected]
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Abstract
WereviewinthisarticletherolewhichtheworkofPauliandWeisskopf
played in formulating a quantum field theory of spinless particles. To
make our computations as transparent as possible, we offer a physicist’s
derivation of the Klein-Gordon-Fock equation. Since invariant functions
play a significant part in our paper, we will discuss them in great detail.
We emphasize Pauli’s and Weisskopf’s view that Dirac’s hole theory is
totally obsolete in formulating a consistent quantum field theory, be it
for scalar or spinor particles. As an important example we present the
calculation for producing charged scalar particles in an external electric
field.
1 Introduction
After Pauli and Weisskopf published their anti-Dirac paper[PW34], the Klein-
Gordon-Fock(K.G.F.)fieldtheorybecameawell-respectedconceptfordescrib-
ing the behavior of massive spinless scalar particles, like pions. Homogeneous
solutions of the K.G.F. equation as well as Green’s functions for the inhomoge-
neous K.G.F. equation were worked out in detail. It might not be well known
thatPauliandMajorananeverthoughtveryhighlyofDirac’sholetheory. More
recentlyapaperwaspublishedinwhichMajorana,severalyearsbeforePauliand
Weisskopf, studied the quantization of the relativistic K.G.F. equation[Esp07].
But the final blow to Dirac’s hole theory came from Pauli and Weisskopf.
Admittedly, it tooka longtime untilDirac’s ideawas relegatedto the cornerof
”merehistoricalinterest”. Withthediscoveryofpionsandother(pseudo-)scalar
particles it became clear that one could do without Dirac holes for antiparti-
cles. PauliandWeisskopfhadalwaysrejectedtheDiracequationasanequation
for a relativistic probability amplitude. They regarded the Dirac equation as
a relativistic matter field equation and not an equation of the probability am-
plitude in the (x,y,z) space. Although the idea of a second quantized field
operatorψ(~x,t)formany-particlesystemswascleartoJordaninthethree-man
paper[BHJ25] of 1925, it was Pauli and Weisskopf who insisted that a concept
like the probability of a particle to be found in ~x space does not make much
sense for relativistic particles, and this holds true for electrons, photons and
K.G.F. particles alike. Consequently, the Dirac equation as well as the K.G.F.
equation should be treated as matter-field equations for many particles rather
than as an equation for single-particle probability amplitudes[Tom97].
After having constructed the relativistic scalar field theory, Pauli wondered
”why nature had made no use of the possibility of the theory that there exist
spin-zerobosons[...]”. Needlessto say,Pauli’squestionwasansweredby nature
intheaffirmative. NotlongafterPauli’sandWeisskopf’spaperof1934,therel-
ativistic scalar wave field theory would be finally established in the appearance
of π mesons.
Admittedly, most of the facts in the present paper have been known for
quite some time. But what is probably new is the attempt to construct the
probability amplitude ~x,t~x ,t = 0 in relativistic single-particle quantum
0 0
h | i
mechanics(QM). We will discover that the probability amplitude of finding the
particle outside the light cone is not zero - it drops exponentially. So it can
be found in an area which is forbidden by special relativity. To get out of
this trouble we will use instead the the language of second quantization. Here
2
we will study the Pauli-Jordan commutator function ψ(~x,t),ψ (~x ) which is
† 0
related to the probability amplitude of detecting the particle at (~x,t) while it
(cid:2) (cid:3)
was created at (~x ,t =0).
0 0
Ourfirstattemptendsupinadisaster. Notonlyhaveweviolatedcausality-
meaningnosignalcantravelfasterthanlight. Inaddition,wefindabreakdown
of simultaneity, i.e., we cannot give ψ,ψ an invariant meaning outside the
†
light cone.
(cid:2) (cid:3)
After we have remedied these failings, we begin a thorough study of the so-
calledinvariantfunctionsandpropagationfunctions. Herewerelyonthetotally
neglected but wonderful article by J. Schwinger[Sch49]. Finally, we present a
list of invariant functions of the K.G.F. theory in (~x,t) space.
2 The Free Klein-Gordon-Fock Theory - Parti-
cle Description
Our goal is to investigate the consequences of particles with relativistic energy
spectrum (~=c=1):
H := a†(p~) p~2+m2a(p~) . (1)
p~
X p
Now it is useful to recall that in the single-particle formalism we would start
with the transition amplitude (t =0):
0
1 i p~~x t√p~2+m2
~x,tp~ = e · − , (2)
h | i √V
(cid:0) (cid:1)
which satisfies the Schr¨odinger equation
∂
i ~x,tp~ = ~x,tH p~
∂th | i h | | i
= p~2+m2 ~x,tp~ . (3)
h | i
p
One might wonder as to whether it is reasonable to define a probability
amplitude ~x,t~x also in relativistic quantum mechanics. To find out let us
0
h | i
begin with
1
~x,t~x0 = ~x,tp~ p~~x0 , p~~x0 = e−i~x0·p~
h | i h | ih | i h | i √V
p~ (cid:18) (cid:19)
X
= 1ei p~·(~x−~x0)−t√p~2+m2 , ~r :=~x ~x0
V −
p~ (cid:0) (cid:1)
X
V→=∞ 1 ei p~·~r−t√p~2+m2 d3~p
(2π)3
Z (cid:0) (cid:1)
= 1 ∞p2dp 2π 1 eiprze it√p~2+m2dz
−
(2π)3
Z0 Z−1
z=c=osΘ 4π 1 ∂ 1 ∞ eipre it√p~2+m2dp .
−
−(2π)3r∂r2
Z−∞
3
Here we change variables: p=msinhφ, coshφ= 1+sinh2φ to obtain:
1 1 ∂ p
h~x,t|~x0i=−(2π)2r∂r mcoshφ ei(mrsinhφ−mtcoshφ)dφ
Z
1 1 ∂ ∂
= i ei(mrsinhφ mtcoshφ)dφ . (4)
−
−(2π)2r∂r ∂t
Z
If we believe in causality we want ~x,t~x to be zero for two points in a
0
h | i
space-like relation:
(~x ~x )2 t2 =r2 t2 >0, space-like(r >t) .
0
− − −
So let us compute the integral (4) for r >t, and since a particle of mass m>0
cannot travelwith the speed v c, we expect the integralto vanish. For r>t:
≥
mr =λcoshφ
0
λ:=m r2 t2 .
mt=λsinhφ0) −
p
In (4) we then obtain:
mrsinhφ mtcoshφ=m r2 t2(sinhφcoshφ coshφsinhφ )
0 0
− − −
=mpr2 t2sinh(φ φ ) .
0
− −
p
Changing φ φ φ we obtain for (4):
0
− →
eim√r2 t2sinhφdφ=2 ∞cos m r2 t2sinhφ dφ
−
−
Z Z0 (cid:16) p (cid:17)
cos m√r2 t2ψ
ψ:=sinhφ ∞
= 2 − dψ
Z0 (cid:0) 1+ψ2 (cid:1)
=2K0(m r2 t2)p.
−
This yields p
2i 1 ∂ ∂
~x,t~x = K (m r2 t2) . (5)
h | 0i −(2π)2r∂r∂t 0 −
p
ButK (m√r2 t2)andderivativesthereofareunequalzeroforr2 >t2,e.g.
0
−
if r t, then:
≫
π
K0(m r2−t2)∼= 2m√r2 t2e−m√r2−t2 . (6)
p r −
Hence, the probability amplitude for finding the particle outside the light cone
is non-zero - it drops exponentially. Hence it can be found in an area which is
forbidden by special relativity. We are in great trouble.
Now, let us use instead the language of second quantization. We create a
particle at (~x ,t =0) and detect it at (~x,t):
0 0
1
ψ†(~x0)= a†(p~′)e−ip~′·~x0
√V
p~′
X
ψ(~x,t)= 1 a(p~)ei(p~~x t√p~2+m2) .
· −
√V
p~
X
4
We wish to calculate the commutator, which is related to the probability am-
plitude offindingthe particleat(~x,t)whenitcameintoexistenceat~x attime
0
t =0:
0
ψ(~x,t),ψ†(~x0) = 1ei p~·(~x−~x0)−t√p~2+t2 . (7)
V
p~ (cid:0) (cid:1)
(cid:2) (cid:3) X
This looks exactly like the expression we had for ~x,t~x in the single-particle
0
h | i
description. Therefore the commutator will not vanish if the two points ~x,~x
0
areinaspace-likerelation(c.f. Eq. (5)). Evenworse: forequaltimesweobtain:
ψ(~x,t),ψ†(~x0) t=0 =δ3(~x−~x0) .
(cid:2) (cid:3)
Not only have we violated causality - meaning no signal can travel faster than
light. In addition, we find a breakdownof simultaneity, i.e., we cannot give the
commutatoraninvariantmeaningoutsidethelightcone. Inordertoremedythe
whole situation, let us start anew by finding a relativistic invariant expression
for the commutator. The first step is to write
ψ(~x,t),ψ†(~x0) = (2π1)3ei p~·(~x−~x0)−t√p~2+m2 d3~p
Z (cid:0) (cid:1)
(cid:2) (cid:3)= d3p~ dp0ei(p~·(~x−~x0)−p0t)Θ(p0)δ(p0 p~2+m2)
(2π)3 −
Z Z
= d3p~ dp0ei(p~·(~x−~x0)−p0t)Θ(p0)δ (p0 p~2+m2)(p0+ p~2+m2) 2pp~2+m2
(2π)2 −
Z Z h p p i p
or
ψ(~x,t),ψ†(~x0) = (2dπ3p~)3 dp0Θ(p0)2 p~2+m2δ p02−p~2−m2 ei(p~·(~x−~x0)−p0t)
(cid:2) (cid:3) Z 1 Z p (cid:16) (cid:17)
= Θ(p0)2 p~2+m2δ(p2+m2)eip·(x−x0)d4p . (8)
(2π)3
Z
p
In our metric, pµ =(p~,p0),p2 =p~2 p02,(x x )=(~x ~x ,t).
0 0
− − −
Without the factor 2 p~2+m2 the integral in (8) is a Lorentz scalar, i.e.,
an invariant function under proper ortochronous Lorentz transformation:
p
1
Θ(p0)δ(p2+m2)eipyd4p=F(y2) .
·
(2π)3
Z
From our decomposition of the δ-function,
1
Θ(p0) δ(p0 p~2+m2)+δ(p0+ p~2+m2) =Θ(p0)δ(p2+m2) ,
2 p~2+m2 −
h p p i
p
we see that the integralin (8) picks up a contributionfrom the top sheet ofthe
mass shell hyperboloids.
Finally, to getridofthe factor 2 p~2+m2 in (8), we define a new operator:
1 p1
φ(~x,t):= ei(p~·~x−ωpt) a(p~), ωp := p~2+m2 . (9)
√V 1
p~ 2(p~2+m2)2
X p
q
5
The new commutator function is then given by
φ(~x,t),φ†(~x0) = (2dπ3p~)32 p~21+m2ei p~·(~x−~x0)−t√p~2+m2
Z (cid:0) (cid:1)
(cid:2) (cid:3)
d4p
= Θp(p0)δ(p2+m2)eip·(x−x0) , (10)
(2π)3
Z
which is an invariant function. Therefore φ and not ψ is the appropriate oper-
ator.
For later purposes it is useful to calculate the dp0 term in (10), which
reveals the invariant measure in momentum space:
R
1 d3p~
dω = , ω = p~2+m2 . (11)
p 2ω (2π)3 p
p
p
Havinggiven[φ,φ ]aninvariantmeaningwelookatitintheframewheret=0:
†
d3p~ 1
φ(~x,t),φ†(~x0) t=0 = (2π)32 p~2+m2eip~·(~x−~x0)
Z
(cid:2) (cid:3)
= eip~·(~x−~xp0)dωp , (12)
Z
which is not zero!
So ournewcommutator,althoughrelativisticinvariant,stillviolatescausal-
ity (compare to the discussion of the transformation amplitude ~x,t~x ). Our
0
h | i
next goal is therefore to restore causality. By the way, we can use the operator
(9),
1 1
φ(~x,t)= ei(p~·~x−ωpt) a(p~)
√V 2√ωp
p~
X
to build a wave packet,
1 1
0φ(~x,t)= ei(p~·~x−ωpt) p~ 0a(p~)= p~ . (13)
h | √V 2√ωph | h | h |
p~
X (cid:0) (cid:1)
The inner product of two wave packets is
0φ(~x,t)φ (~x )0 = 0 φ(~x,t),φ (~x ) 0
† 0 † 0
h | | i h | | i
d4p
= (cid:2) Θ(p0)δ(p2(cid:3)+m2)eip·(x−x0) , (14)
(2π)3
Z
which means that the localization of a particle has an invariant meaning, i.e.,
again, looks the same for all observers.
However, the description is still complicated from the point of causality.
Although [φ,φ ] is Lorentz invariant, it does not vanish at t=0:
†
d3p~ 1
φ(~x,t),φ†(~x0) = (2π)32ω eip~·(~x−~x0)e−iωpt . (15)
p
Z
(cid:2) (cid:3)
Recall that this expression was obtained by using φ,φ given above and the
†
commutation relation [a,a ]=1.
†
6
In order to construct a commutator that vanishes for t = 0 we have to
subtractsomethingfromourformerexpression(15),namelythe secondtermin
d2~p 1
eip~·(~x−~x0)e−iωpt e−ip~·(~x−~x0)eiωpt . (16)
(2π)32ω −
p
Z (cid:16) (cid:17)
For the second term we use the lower sheet of the hyperboloid, i.e., take the
solutionp0 = p~2+m2. ThenΘ( p0)δ(p2+m2)selectsthe bottomsheetas
− −
Θ(p0)δ(p2+m2) picks out the top sheet - our situation so far.
p
Now comes the point: the minus sign in (16) plays a significant role. It
is the same minus sign that occurs if we look at [a,a ] = 1 and write instead
†
[a ,a] = 1. Therefore, if we interchange the role of creation and destruction
†
−
operators, we can convert the minus sign into a plus sign:
1 1
φ˜(~x,t)= ei(p~·~x−ωpt)a(p~)+e−i(p~·~x−ωpt)b†(p~) , (17)
√V 2ωp
Xp~ (cid:16) (cid:17)
p
with [a,a ] = 1,[a,b] = 0,[b,b ] = 1,ω = + p~2+m2. Then, with this new
† † p
object φ˜ φ we obtain:
→ p
d2~p
φ(~x,t),φ†(~x0) = (2π)3 eip~·(~x−~x0)e−iωpt−e−ip~·(~x−~x0)eiωpt ,
Z (cid:16) (cid:17)
(cid:2) (cid:3)
whichyieldsatlast[φ(~x,t),φ (~x )] =0for~x=~x and~x=~x . Theinvariant
† 0 t=0 0 0
6
form of [φ,φ ] becomes obvious when we write (y :=(~x ~x ,t)):
† 0
−
d3pdp0
φ(~x,t),φ (~x ) = Θ(p0)δ(p2+m2)eipy Θ( p0)δ(p2+m2)eipy
† 0 (2π)3 · − − ·
Z
(cid:2) (cid:3)=:i∆(y) . (cid:0) (18(cid:1))
This so-calledPauli-Jordaninvariantcommutatorfunctionis zerofort=0and
invariant for space-like distances y2 >0 with y2 =(~x ~x )2 t2. Using
0
− −
+1 , p0 >0
ǫ(p0)= =Θ(p0) Θ( p0) , (19)
1 , p0 <0 − −
(cid:26) − (cid:27)
we obtain the final form
d4p
φ(~x,t),φ (~x ,0) =i∆(y)= ǫ(p0)δ(p2+m2)eipy . (20)
† 0 (2π)3 ·
Z
(cid:2) (cid:3)
If we calculate 0φ(~x,t)φ (~x ,0)0 and insert the expressions for our old φ˜,φ˜
† 0 †
h | | i
from (17), the result becomes identical to our old calculation with the φ from
(14). Therefore the vacuum expectation value is the same - since we assume
that the particles associated with the operators a and b have the same mass.
If we take b=a, then
1 1
φ(~x,t)= ei(p~·~x−ωpt)a(p~)+e−i(p~·~x−ωpt)a†(p~) , (21)
√V 2√ωp
Xp~ (cid:16) (cid:17)
meaning φ =φ, i.e., φ is Hermitean.
†
7
If we consider (17) again and calculate the derivative i∂φ, we find that it
∂t
does not satisfy a first-order differential equation but a second-order one:
∂2
φ(~x,t)= ~2 m2 φ(~x,t) , (22)
∂t2 ∇ −
(cid:16) (cid:17)
which is local in space-time; a and b correspond to the two constants of inte-
†
gration. The equation
2
∂
φ= ~2 m2 φ , (23)
∂t ∇ −
(cid:18) (cid:19)
(cid:16) (cid:17)
iscalledtheKlein-Gordon-Fock-Schr¨odinger(K.G.F.Sch.) equation. Itisalocal
relativistic equation.
Remark When we constructed the local operator φ we used [a,a ]=1. At
†
one point itwas necessary- inorderto restorecausality- to use the minus sign
in [b ,b]= 1. If we now would go back and use instead an anti-commutation
†
−
relationaa +a a= a,a =1,ourproceduretoconstructalocalfieldoperator
† † †
{ }
would fail. The particles have to have spin.
Conventional textbooks start with the K.G.F.Sch. equation
2
∂
∂2+m2 φ(~x,t)= ~2+m2 φ(~x,t)=0 (24)
− "(cid:18)∂t(cid:19) −∇ #
(cid:0) (cid:1)
and look for solutions with the two constants of integrationspecified by φ(~x,0)
and φ˙(~x,0) and find that (17) is the local solution to the K.G.F.Sch. equation.
Thisisthepaththatmathematicianswouldtake. Ourprocedureismoresuited
to physics-minded students and teachers.
Wefoundin(20)amostimportantinvariantfunction∆(Pauli-Jordan). Let
us write it as
∆(x,κ2)= i(2π) 3 (dk)eikxǫ(k0)δ(k2+κ2),
0 − − 0
Z
where (dk)=dk dk dk dk . But there is another invariant function:
0 1 2 3
∆(1)(x,κ2)=(2π) 3 (dk)eikxδ(k2+κ2) .
0 − 0
Z
Evidently, both are solutions to the K.G. equation:
(κ2 ∂2) ∆,∆(1) =0 .
0− { }
The two functions fulfill different boundary conditions and different symmetry
properties. In fact,
∆( x)= ∆(x), since ǫ is odd
− −
∆(1)( x)=∆(1)(x) .
−
They bothshareinvarianceunderproper ortochronousLorentztransformation:
∆,∆(1) (Λx)= ∆,∆(1) (x) .
(cid:16) (cid:17) (cid:16) (cid:17)
8
The basic fact about ∆ is that it is zero outside the light cone while ∆(1)
reachesintothespace-likesectorwhereitdiesoutexponentially. Inotherwords,
microcausality is realized by ∆ and not determined by ∆(1)! It is due to the
behavior of ∆, i.e., the disappearance of the commutator of the fields, that
measuring the field at x can have no consequence on measuring the field at
0
x since the points are not causally connected. However both ∆ and ∆(1) are
the basic functions for constructing the remaining invariant functions. More
abouttheirexplicitexpressionincoordinatespacewillbegivenandthoroughly
discussed in the remaining chapters.
3 Selection of invariant commutation and prop-
agation functions
In the last chapter, the Pauli-Jordan commutation function was constructed,
starting from a scalar field:
1
∆(x)= i eikxǫ(k )δ(k2+µ2)(dk) . (25)
− (2π)3 · 0
Z
We also mentioned a second invariant function:
1
∆(1)(x)= eikxδ(k2+µ2)(dk) . (26)
·
(2π)3
Z
AlthoughtheybothsatisfytheK.G.F.equation,theyplayatotallydifferentrole
in scalar quantum field theory (Q.F.T.). While ∆(x) vanishes if x2 >0 (space-
like argument), ∆(1)(x) does not vanish for space-like distances(c.f. Appendix
A).Instead,∆(1) extendsintothespace-likeregion,droppingoffonthescaleof
theComptonwavelength 1. However,itisalsothefamousFeynmanpropagator
µ
function that reaches into the space-like region.
Inourconvention,∆ (x)=∆(1)(x)+iǫ(x)∆(x), ∆ =:2i∆ . Notethatthe
F F c
inhomogeneous∆ isconstructedfromthetwohomogeneousinvariantfunctions
F
∆ and ∆(1). For our purposes we will use the momentum representation of
∆ (x):
c
1 eikx
·
∆ (x)= (dk) . (27)
c (2π)4 k2+µ2 iǫ
Z −
Employing
1 =i ∞e is(k2+µ2 iǫds ,
− −
k2+µ2 iǫ
− Z0
we obtain:
∆ (x)= (dk) i ∞e is(k2+µ2)eikxds
c (2π)4 − ·
Z Z0
= i ∞ds e−isµ2 e−isk2+ik·x(dk) . (28)
(2π)4
Z0 Z
The k-integral in the previous equation is given by
e−isk2+ik·x(dk)= iπ2eix42s . (29)
− s2
Z
9
Therefore
∆c(x)= (2π1)4 ∞e−isµ2πs22eix42sds= 161π2 ∞ s12e−isµ2eix4s2ds
Z0 Z0
Hankel function 1 ,propagation,x2 <0 inside light cone
∆ (x): ∼ ∼ √ x2
c ( K1 function e−cons−t.√x2,not propagation,x2 >0 outside light cone .
∼ ∼
Here is the result for the causal Green’s function ∆ (x):
c
1 iµ µ
∆ (x)= δ(x2)+Θ(x2) K (µ√x2) Θ( x2) H(2)(µ x2)
c 4π 4π2√x2 1 − − 8π√ x2 1 −
1 µ − p
= δ(x2) Θ( x2) J (µ x2) iN (µ x2)
1 1
4π − − 8π√ x2 − − −
iµ − h p p i
+Θ(x2) K (µ√x2) .
1
4π√x2
Forthe Pauli-Jordanfunction (25)we use, fromthe detailedcalculationsin the
appendix (∆(x):= 2∆¯(x)ǫ(x)), the result (53):
−
1 µ2 H(1)(µ√ x2)
∆¯(x)= δ(x2) Re 1 −
4π − 8π " µ√ x2 #
−
Re H1(1)(µ√−x2) = J1µ(µ√√−xx22) x2 <0
" µ√ x2 # (0 − x2 >0 .
−
∆¯(x) and therefore ∆(x) vanishes if x2 > 0 (space-like distance), which is how
we constructed the Pauli-Jordancommutator function. Finally,
1 µ
∆(x)= ǫ(x )δ(x2)+ ǫ(x )Θ( x2)J (µ x2) .
0 0 1
−2π 4π√ x2 − −
− p
Unlikethesituationfor∆¯(x)thereisnodiscontinuityatx2 =0for∆(1)(x)and
therefore∆(1) doesnotvanishforspace-likedistances(x2 >0). Toshowthis we
refer to the appendix:
1 1
∆(1)(x)= ∆(x ǫτ) dτ .
πP − τ
Z
Here we insert
i
∆(x)= eikxδ(k2+µ2)ǫ(k )(dk) ,
−(2π)3 · 0
Z
such that
i 1 1
∆(1)(x)= (dk)ǫ(k) e ikǫτdτ eikxδ(k2+µ2) ,
− · ·
−π(2π)3 P τ
Z (cid:20) Z (cid:21)
iπǫ(k)
| {z }
which yields:
1
∆(1)(x)= eikxδ(k2+µ2)(dk) ,
·
(2π)3
Z
10