Irreducible representations of the CPT groups in QED

0.09 MB·English

by  Brenda Carballo Perez

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n IRREDUCIBLE REPRESENTATIONS OF THECPTGROUPSIN QED
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J

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B. Carballo Pe´rez1, M.Socolovsky2

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h InstitutodeCienciasNucleares, UniversidadNacionalAuto´nomadeMe´xico,
p Circuitoexterior, CiudadUniversitaria, 04510, Me´xicoD.F.,Me´xico
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h 1e-mail: [email protected]
at 2e-mail: [email protected]
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3 Abstract: Weconstructtheinequivalentirreduciblerepresentations(IIR’s)oftheCPT
v groupsoftheDiracfieldoperatorψˆandtheelectromagneticquantumpotentialAˆ . The
µ
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results are valid both for free and interacting (QED) fields. Also, and for the sake of
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3 completeness,weconstruct theIIR’s oftheCPT groupoftheDiracequation.
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9 AMSSubject Classification: 20C30, 20C35.
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X KeyWords: CPT groups;Dirac-Maxwellfields; irreduciblerepresentations.
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1Introduction 2
1 Introduction
The CPT group G (ψˆ) of the Dirac quantum field ψˆ is isomorphic to the
Θˆ
direct product of the quaternion group Q and the cyclic group of two elements Z ([1]
2
and [2]):
G (ψˆ) ∼= Q×Z . (1)
Θˆ 2
Θˆ is the product Cˆ ∗ Pˆ ∗ Tˆ of the three operators: Cˆ (charge conjugation), Pˆ
(spaceinversion),Tˆ (timereversal);theoperationAˆ∗Bˆ,whereAˆandBˆ areanyofthe
operators Cˆ, Pˆ, Tˆ, is givenby (Aˆ∗Bˆ)·ψˆ = (AˆBˆ)†ψˆ(AˆBˆ). G (ψˆ) is one of the nine
Θˆ
non abelian groups of a total of fourteen groups with sixteen elements [3]; only three
ofthemhavethreegenerators.
TheisomorphismG (ψˆ) → Q×Z isgivenbytherelations(2):
Θˆ 2
1 7→ (1,1) −1 7→ (−1,1)
Cˆ 7→ (1,−1) −Cˆ 7→ (−1,−1)
Pˆ 7→ (ι,1) −Pˆ 7→ (−ι,1)
Tˆ 7→ (γ,1) −Tˆ 7→ (−γ,1)
Cˆ ∗Pˆ 7→ (ι,−1) −Cˆ ∗Pˆ 7→ (−ι,−1)
Cˆ ∗Tˆ 7→ (γ,−1) −Cˆ ∗Tˆ 7→ (−γ,−1)
ˆ ˆ ˆ ˆ
P ∗T 7→ (κ,1) −P ∗T 7→ (−κ,1)
Θˆ 7→ (κ,−1) −Θˆ 7→ (−κ,−1); (2)
whereι, γ, κare thethree imaginaryunitsdefining thequaternionnumbers.
On the otherhand, the action of theCˆ, Pˆ and Tˆ operators on the Maxwell electro-
magnetic 4-potential Aˆ is given by (3) ([4] and [5]), with the Minkowski space-time
µ
metricdiag(1,−1,−1,−1):
PˆAˆµ(x,t)Pˆ−1 = Aˆ (−x,t),
µ
CˆAˆµ(x,t)Cˆ−1 = −Aˆµ(x,t),
TˆAˆµ(x,t)Tˆ−1 = Aˆ (x,−t). (3)
µ
ThisleadstotheCPT groupoftheelectromagneticfieldoperator,G (Aˆ ),which
Θˆ µ
is an abelian group of eight elements with three generators, isomorphic to Z × Z ×
2 2
Z = Z3. Geometrically, Z3 is isomorphic to the group D , the symmetry group of
2 2 2 2h
theparallelepiped.
1Introduction 3
Theisomorphism
G (Aˆ ) → Z ×Z ×Z (4)
Θˆ µ 2 2 2
isgivenby
1 7→ (e ,e ,e )
1 2 3
Cˆ 7→ (a ,e ,e )
1 2 3
ˆ
P 7→ (e ,a ,e )
1 2 3
Tˆ 7→ (e ,a ,a )
1 2 3
Pˆ ∗Tˆ 7→ (e ,a ,a )
1 2 3
Cˆ ∗Pˆ 7→ (a ,a ,e )
1 2 3
Cˆ ∗Tˆ 7→ (a ,e ,a )
1 2 3
Θˆ 7→ (a ,a ,a ); (5)
1 2 3
where e and a (for i = 1,2,3) represent, respectively, the identity and the “minus 1”
i i
ofeach oftheZ groups.
2
AstheCPT transformationpropertiesoftheinteractingψˆ−Aˆ fieldsarethesame
µ
as for the free fields ([5] and [6]), it is then clear that the complete CPT group for
QED,G (QED),isthedirectproductofthetwoabovementionedgroups,G (ψˆ)and
Θˆ Θˆ
G (Aˆ ), i.e.,
Θˆ µ
G (QED) = G (ψˆ)×G (Aˆ ), (6)
Θˆ Θˆ Θˆ µ
whichturnsout tobeagroupoforder|G (QED)| = 16×8 = 128.
Θˆ
Thus,
G (QED) ∼= (Q×Z )×Z3. (7)
Θˆ 2 2
Since, as will be shown below, G (ψˆ) has ten inequivalent irreducible represen-
Θˆ
tations (IIR’s), while G (Aˆ ) has only eight IIR’s, then, the total number of IIR’s of
Θˆ µ
G (QED) iseighty: sixtyfourofthem1-dimensionaland sixteen2-dimensional.
Θˆ
It is interesting at this point to make a comment about the geometrical content of
G (QED),whichisasetofpointsinspheres. Infact,each factorZ isa0-sphereS0,
Θˆ 2
while Q can be thought as a subset of eight points in the 3-sphere S3. Thus, topologi-
cally
G (QED) ⊂ SU(2)×(U(1))4. (8)
Θˆ
In section 2 we construct by tensor products the ten IIR’s and character tables of
G (ψˆ). In section 3 we construct the eight IIR’s of G (Aˆ ); in this case, since all
Θˆ Θˆ µ
representations are 1-dimensional, the characters coincide with the representations. In
ˆ
2IIR’s ofGΘˆ(ψ) 4
section 4 we construct the IIR’s of the CPT group corresponding to the Dirac equa-
tion for the wave function ψ, G(2)(ψ) in [1], and compare the result with the case
Θ
for the Dirac field operator ψˆ. Finally, in section 5, we discuss the relation between
the ¨fermionic” and ¨bosonic” CPT groups and comment about the relation of these
groupswithsomeapproaches to quantumparadoxes.
ˆ
G
2 IIR’s of (ψ)
Θˆ
The irreducible representations (irrep’s) of the direct product of two groups,
G × H, are the tensor products of the irrep’s of each of the factors, namely the set
{r ⊗r }1 forallr′ s,irrep’sofGandallr′ s,irrep’sofH [7]. IfGisthequaternion
G H G H
group, which, as is well known, has five IIR’s ϕ (with ϕ ,...,ϕ 1-dimensional and
µ 1 4
ϕ 2-dimensional)[8] and characters tablegivenby table1;and H is Z (withIIR’s ψ
5 2 1
and ψ ), then G (ψˆ) hasten IIR’s:
2 Θˆ
φ = ϕ ⊗ψ , φ = ϕ ⊗ψ , α = 1,2,3,4, (9)
α α 1 α+4 α 2
φ = ϕ ⊗ψ , φ = ϕ ⊗ψ . (10)
9 5 1 10 5 2
φ ,...,φ are1-dimensional,whileφ and φ are2-dimensional.
1 8 9 10
Ch Q [1] [−1] 2[ι] 2[γ] 2[κ]
χ 1 1 1 1 1
1
χ 1 1 1 −1 −1
2
χ 1 1 −1 1 −1
3
χ 1 1 −1 −1 1
4
χ 2 −2 0 0 0
5
Table1: Characters tableofQ.
Thecorrespondingcharacters arethefunctions
κ = χ ϕ : Q×Z → C, κ (q,h) = χ (q)ϕ (h), (11)
α α 1 2 α α 1
κ = χ ϕ : Q×Z → C, κ (q,h) = χ (q)ϕ (h), (12)
α+4 α 2 2 α+4 α 2
κ = χ ϕ : Q×Z → C, κ (q,h) = χ (q)ϕ (h), (13)
9 5 1 2 9 5 1
κ = χ ϕ : Q×Z → C, κ (q,h) = χ (q)ϕ (h). (14)
10 5 2 2 10 5 2
1⊗=⊗C isthetensorproductoverthecomplexnumbers
3IIR’s ofGΘˆ(Aˆµ) 5
Theserepresentationsand characters (for conjugateclasses)are explicitlygivenby
the tables 2 and 3, respectively, where λ = trφ , and the conjugate classes are given
i i
bytherelations(15):
[Iˆ] = [(1,e)] = {(1,e)},
ˆ
[C] = [(1,a)] = {(1,a)},
[−Iˆ] = [(−1,e)] = {(−1,e)},
[−Cˆ] = [(−1,a)] = {(−1,a)},
[Pˆ] = [(ι,e)] = {(ι,e),(−ι,e)},
[Cˆ ∗Pˆ] = [(ι,a)] = {(ι,a),(−ι,a)},
[Tˆ] = [(γ,e)] = {(γ,e),(−γ,e)},
[Cˆ ∗Tˆ] = [(γ,a)] = {(γ,a),(−γ,a)},
ˆ ˆ
[P ∗T] = [(κ,e)] = {(κ,e),(−κ,e)},
[Θˆ] = [(κ,a)] = {(κ,a),(−κ,a)}; (15)
whereeand aare theelementsofZ .
2
The character table of G (ψˆ) is the same as the character table of D , the group
Θˆ 4h
ofsymmetriesofaprismwithsquarebase, thoughQ×Z ≇ D .
2 4h
ˆ
G A
3 IIR’s of ( )
Θˆ µ
Since G (Aˆ ) ∼= Z3 is an abelian group, all its irrep’s are 1-dimensional.
Θˆ µ 2
Since |G (Aˆ )| = 8, then G (Aˆ ) has eight IIR’s, which can be identified with the
Θˆ µ Θˆ µ
corresponding characters. Using the isomorphism between G (Aˆ ) and Z3 (eq. (5)),
Θˆ µ 2
andthecharactertableofZ ,weobtainthetableofrepresentations(φ )orcharacters
2 ijk
(χ )forG (Aˆ ), whereφ = χ = ψ ψ ψ . See table4.
ijk Θˆ µ ijk ijk i j k
G(2)
4 IIR’s of (ψ)
Θ
In [1] it was shown that at the level of the Dirac equation, for the 4-spinor ψ,
exists two CPT groups, G(1)(ψ) and G(2)(ψ), whose elements are constructed with
Θ Θ
products of Dirac γ-matrices. Only G(2)(ψ), which turns out isomorphic to the semi-
Θ
direct productD ⋊Z , whereD ≡ DH isthedihedral groupof eightelements(the
4 2 4 8
groupofsymmetriesofthesquare), iscompatiblewithG (ψˆ).
Θˆ
4IIR’s ofG(2)(ψ) 6
Θ
ItisthenofinteresttoconstructtheIIR’sofG(2)(ψ),bothforcompletnessandalso
Θ
forcomparisonwiththeIIR’s oftheoperatorgroup.
The action of Z ∼= {1,−1} on D as a subgroup of S (the symmetric group of 4
2 4 4
elements)isgivenby
λ : Z → Aut(D ),
2 4
1 7→ λ(1) = Id ,
D4
λ(−1)(I) = I,
λ(−1)(1234) = (1234),
λ(−1)(24) = (13),
λ(−1)(13) = (24),
λ(−1)((12)(34)) = (14)(23),
λ(−1)((14)(23)) = (12)(34),
λ(−1)((13)(24)) = (13)(24),
λ(−1)(1432) = (1432). (16)
Here
D = {I;(1234),(1432);(13)(24);(12)(34),(14)(23);(24),(13)}, (17)
4
wherebetween semi-colonswehaveenclosedthefiveconjugationclasses.
Then, thecompositionlawin D ⋊Z is
4 2
(g′,h′)(g,h) = (g′λ(h′)(g),h′h) : (18)
(g′,1)(g,h) = (g′g,h) (19)
(g′,−1)(g,h) = (g′λ(−1)(g),−h), (20)
where, as aset,
D ×Z = {(I,1),(I,−1),(−I,1),(−I,−1),(P,1),(P,−1),(−P,1),
4 2
(−P,−1),(CT,1),(CT,−1),(−CT,1),(−CT,−1),(Θ,1),
(Θ,−1),(−Θ,1),(−Θ,−1)}, (21)
withinverses
(g,h)−1 = (λ(h)(g−1),h). (22)
4IIR’s ofG(2)(ψ) 7
Θ
From eq. (55)in[1], thereis theisomorphism
D → {I,−I,P,−P,CT,−CT,Θ,−Θ} (23)
4
givenby
I 7→ I,
(1234) 7→ P,
(1432) 7→ −P,
(13)(24) 7→ −I,
(12)(34) 7→ Θ,
(14)(23) 7→ −Θ,
(24) 7→ −CT,
(13) 7→ CT (24)
whilefrom eq. (60)in [1]and eq. (17), theisomorphismbetween G(2) and D ⋊Z ,
Θ 4 2
G(2) → D ⋊Z , (25)
Θ 4 2
isgivenby:
I 7→ (I,1), −I 7→ (−I,1),
C 7→ (−Θ,−1), −C 7→ (Θ,−1),
P 7→ (P,1), −P 7→ (−P,1),
T 7→ (P,−1), −T 7→ (−P,−1),
CP 7→ (CT,−1), −CP 7→ (−CT,−1),
CT 7→ (CT,1), −CT 7→ (−CT,1),
PT 7→ (−I,−1), −PT 7→ (I,−1),
Θ 7→ (Θ,1), −Θ 7→ (−Θ,1). (26)
4IIR’s ofG(2)(ψ) 8
Θ
It iseasy to verifythatG(2) hasten conjugationclasses,namely
Θ
[I] = = {I},
[−I] = = {−I},
[C] = = {C,−C},
[T] = = {T},
[−T] = = {−T},
[P] = = {P,−P},
[CP] = = {CP,−CP},
[CT] = = {CT,−CT},
[PT] = = {PT,−PT},
[Θ] = = {Θ,−Θ}. (27)
Then, being a finite group, G(2) has as many IIR’s as conjugation classes. Since
Θ
thesum ofthe squares ofthe dimensionsof theserepresentations mustbe 10, a simple
calculationleadstotheexistenceofeight1-dimensionalirrep’sϕ ,k = 1,...8,andtwo
k
2-dimensionalirrep’s,ϕ and ϕ .
9 10
ϕ is the trivial representation g 7→ 1, for all g ∈ G(2). The three 1-dimensional
1 Θ
irrep’sϕ ,ϕ andϕ areobtainedtakingintoaccount:
2 3 4
1. D , C ×Z andQ areinvariantsubgroupsofD ⋊Z .
4 4 2 4 2
2. The well known theorem by which if H is an invariant subgroup of G and ϕ :
G/H → K < GL(V) is a representation of G/H over the vector space V, then
ϕ◦p is a degenerate representation of G, where p : G → G/H is the canonical
projectionp(g) =< g >= gH [9]. Thus,wehavethecommutativediagram:
K
ϕ◦p ր տ ϕ . (28)
G −→ G/H
p
D isgivenby ther.h.s. ofeq. (23)and C =< {P} >,so
4 4
C ×Z = {(I,1),(I,−1),(−I,1),(−I,−1),(P,1),(P,−1),(−P,1),(−P,−1)},
4 2
(29)
and
Q ∼=< {C,P} >= {I,C,P,CP,−I −C,−P,−CP}. (30)
In diagram (28), we choose G = D ⋊ Z and H equal to D , C × Z and Q,
4 2 4 4 2
respectively. Then
4IIR’s ofG(2)(ψ) 9
Θ
D ⋊Z
4 2 ∼= Z = {e,a} (31)
2
H
with
e = D , a = {C,T,CP,PT,−C,−T,−CP,−PT},
4
e = C ×Z , a = {C,CP,CT,Θ,−C,−CP,−CT,−Θ},
4 2
e = Q, a = {T,CT,PT,Θ,−T,−CT,−PT,−Θ}, (32)
respectively. Theelementsinthee’sarerepresentedby1,whiletheelementsinthea’s
arerepresented by −1 (seetable5).
As is well known, D has five IIR’s (one 2-dimensional and four 1-dimensional).
4
The2-dimensionalirrep isgivenby
1 0
I 7→ = 1 ≡ I
(cid:18)0 1(cid:19) 2×2
−1 0
(13)(24) 7→ = −1 ≡ −I
(cid:18) 0 −1(cid:19) 2×2
0 −1
(1234) 7→ = −iσ ≡ P
(cid:18)1 0 (cid:19) 2
0 1
(1432) 7→ = iσ ≡ −P
(cid:18)−1 0(cid:19) 2
−1 0
(12)(34) 7→ = −σ ≡ Θ
(cid:18) 0 1(cid:19) 3
1 0
(14)(23) 7→ = σ ≡ −Θ
(cid:18)0 −1(cid:19) 3
0 1
(24) 7→ = σ ≡ −CT
(cid:18)1 0(cid:19) 1
0 −1
(13) 7→ = −σ ≡ CT, (33)
(cid:18)−1 0 (cid:19) 1
whereinthelastidentificationweused therelations(24).
WiththechoiceC = iσ , weobtain,
1
i 0 i 0 0 −i
T = = iI, CP = = iσ , PT = = σ , (34)
(cid:18)0 i(cid:19) (cid:18)0 −i(cid:19) 3 (cid:18)i 0 (cid:19) 2
5Final comments 10
which completes the 2-dimensionalirrep ϕ of G(2); whilewith the choice C = −iσ ,
9 Θ 1
weobtain,
−i 0 −i 0 0 i
T = = −iI, CP = = −iσ , PT = = −σ ,
(cid:18) 0 −i(cid:19) (cid:18) 0 i(cid:19) 3 (cid:18)−i 0(cid:19) 2
(35)
which completes the 2-dimensional irrep ϕ of G(2). It can be easily verified that ϕ
10 Θ 9
and ϕ are inequivalent, i.e., it does not exist a matrix S such that SMS† = M′ (or
10
SMS−1 = M′)forM ∈ ϕ and M′ ∈ ϕ .
9 10
The remaining four 1-dimensional IIR’s, ϕ , ϕ , ϕ and ϕ , are obtained from
5 6 7 8
the orthogonality between the columns of the characters table for conjugation classes
(completeness relation). Then, the orthogonality between the rows of the complete
characters tableisverified.
TheIIR’s and characters forG(2)(ψ) ∼= D ⋊Z are summarizedintables5 and 6.
Θ 4 2
ComparingwiththetablesforG (ψˆ), weseethatthe1-dimensionalirrep’scoincide:
Θˆ
φ = ϕ , φ = ϕ , φ = ϕ , φ = ϕ ,
1 1 2 3 3 4 4 2
φ = ϕ , φ = ϕ , φ = ϕ , φ = ϕ . (36)
5 8 6 6 7 7 8 5
However,φ is notequivalenttoeitherϕ orϕ andthesameholdsforφ .
9 9 10 10
5 Final comments
It is of interest to ask the question if, from the point of view of group theory,
thebehaviourofthephotonfieldAˆ undertheC,P,T transformationisindependentof
µ
thebehaviouroftheelectron-positronfieldψˆ. Theanswertothisquestionisafirmative.
In fact, in G (ψˆ) there is only one Z subgroup, namely that generated by Cˆ : Z ∼=
Θˆ 2 2
{Iˆ,Cˆ}. Then,G (Aˆ ) ∼= Z3 is notasubgroupofG (ψˆ):
Θˆ µ 2 Θˆ
G (Aˆ ) 6< G (ψˆ). (37)
Θˆ µ Θˆ
ThesamehappensinrelationtoG(2)(ψ). InthisgrouptherearethreeZ -subgroups:
Θ 2
Z(1) = {I,CT}, Z(2) = {I,PT}, Z(3) = {I,Θ}. (38)
2 2 2
However, the table of Z3 has eight 1’s, while the table of G(2)(ψ) has only seven
2 Θ
1’s. So
G (Aˆ ) 6< G(2). (39)
Θˆ µ Θ