Triality, Biquaternion and Vector Representation of the Dirac Equation

Home , Biquaternion, Triality

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One wonders what three objects does the symmetric group Σ3 permutes; and the answer is that it permutes representations.

Theorem (Cartan’s principle of triality[1][2]): There exists an auto-

morphism J of order 3 of the vector space A = R S S− (dimen- × + × sion=8+8+8) which has following properties: J leaves the quadratic form

Ω and the cubic form F invariant; J maps R onto S+, S+ onto S−, and

S− onto R. The law of composition in the algebra A, is defined in terms of the quadratic forms Ω and cubic form F only. For the SO(8) case the defined algebra is the algebra of Cayley octonions[2], and it is clear that any automorphism of the vector space A which leaves the quadratic forms Ω and cubic form F invariant is an automorphism of this algebra. The Brioschi’s formula in the real domain (i.e. The product of two sums of eight squares is itself the sum of eight squares.), which deduced from the above considerations by Cartan[1], can be considered as a normed condition for division octonions. Generally, any division algebra gives a triality[4][5], and it follows that normed trialities only occur in dimensions 1,2,4 or 8. (Here we use a generalized concept of ’triality’ advocated by Adams[4].) This conclusion is quite deep. By comparison, Hurwitz’s classification of normed division algebras (in the real domain) is easier to prove[6]. The construction of the algebras from trialities has tantalizing links to physics. In the Standard Model, all particles other than the Higgs boson transform either as vectors or spinors. The interaction between matter (quarks and leptons) and the forces (gauge bosons) is described by a trilinear map involving two spinors and a vector. It is fascinating that the same sort of mathematics can be used both to construct a suitable algebra and to describe the interaction between matter and forces. One prima facie problem with the above speculation is that physics uses spinors associated to Lorentz groups rather than SO(8) rotational groups,

2 due to the fact that 4-dimension space-time has a Lorentzian rather than Eu- clidean metric. Luckily the dimension of spinor space, depends on both the dimension of vector space and the signature of the metric. In 4-Lorentzian dimensions, the gamma matrices γµ are (at least) 4 4, and thus the number × of complex spinor components are four too, or equivalently 4+4 independent real components. Thus, we have here three spaces each of four dimensions, that of vectors, that of semi-spinors of the first type, and that of semi-spinors of the second type. The quadratic form in the vector space is defined by means of η = diag(+1, 1, 1, 1), and the quadratic form in the spinor µν − − − space is defined by means of Dirac conjugate spinors (ψψ here ψ = ψ∗T γ0 ). We will prove that there exists a double covering vector representation of Dirac spinors, and that there exists an automorphism J of order 3 in the A = M 1+3 S4 S4 (vector space and two half-spinor spaces), which × + × − leaves quadratic forms and the special cubic form invariant up to the sign. (In Minkowski space we are faced with spacelike and timelike vectors, thus some- times it is convenient to use the term pseudoscalar, here the prefix ’pseudo’ referring to automorphism J.) The situation differs from the case of SO(8), and we propose to denominate this symmetry as a “ding” construction. (Chi- nese ding is an ancient vessel which has two loop handles and three legs. It is the metaphor of tripartite balance of forces.) The algebra associated with the above construction is the algebra of (complexified) biquaternion. In this sense the vector representation of Dirac spinor is equivalent to a biquaternion representation. Moreover we will prove that the massive Dirac equation in the vector representation is actually self-dual. It is important to stress that the triality transformation, i.e. the representations permutation, is not a symmetry of the theory. It maps one description of the theory to another description of the same theory.

3 2 Triality and ”Ding” construction

Let us ﬁrst introduce trinomial unit-basis (ϕβ, f α, jµ), where the basic unit vector jµ and two basic unit spinors ϕβ, f α are normalized so that

ϕ = j iγµf , f = j iγµϕ , jµ = ϕiγµf = fiγµϕ (1) µ µ −

µ ν ϕϕ = ff = j ηµν j =1 − µ − µ (2) ϕf = fϕ = jµ(ϕγ ϕ)= jµ(fγ f)=0 Here any two objects determine the third. (Raising and lowering indices µν by Lorentzian metric η and ηµν, the summation convention is assumed for repeated indices.) µ µ µ By using the quantities (jµiγ ), (iγ f), (iγ ϕ) we can translate between diﬀerent types of vectors and spinors indexes. In addition, we can introduce another unit vector kµ, which is determined by the above trinomial unit-basis, that will play a very important role in our theory.

µ def. µ µ µ µ k = ϕγ ϕ = fγ f , k kµ =1 , k jµ =0 (3)

k γµf = f , k γµϕ = ϕ (4) µ − µ The existence of such objects in Dirac theory is veriﬁed by the special case (28) below. Furthermore we need the following algebraic properties

ϕγµγνϕ = fγµγνf = ηµν + iǫµνλρk j − λ ρ (5) ϕγµγνf = fγµγνϕ = i(kµjν jµkν ) − ϕγµγνγλϕ = fγµγνγλf = iǫµνλρj + tµνλρk ρ ρ (6) ϕγλγνγµf = fγµγνγλϕ = ǫµνλρk itµνλρj − ρ − ρ here ǫµνλρ is the Levi-Civita symbol, with the deﬁnition ǫ0123 = 1, and

µνλρ i 5 µ ν λ ρ ǫ = 4 tr(γ γ γ γ γ ) def. (7) tµνλρ = 1 tr(γµγνγλγρ)=(ηµνηλρ + ηµρηνλ ηµληνρ) 4 −

4 (Comments: In calculating S-matrix elements and transition rates for pro- 1 cesses involving particles of spin 2 , we often encounters traces of products of Dirac gamma matrices. The above symbols ǫµνλρ and tµνλρ are not new for physics, we always use them in all such calculations.) We are now in a position to construct the suitable algebra, that are needed. Let def. cµνλ = (tµνλρ iǫµνλρ)k =(cλνµ)∗ − ρ µνλ def. µνλρ µνλρ λνµ ∗ cˇ = (t iǫ )jρ = (ˇc ) (8) def. − cµλ = cνλµj =ˇcνλµk = cλµ 5 − ν ν − 5 (in which asterisk represents complex conjugate.) then ∗ cµνση (cδρλ)∗ + cµρση (cδνλ)∗ =2ηµληνρ σδ σδ (9) cˇµνση (ˇcδρλ)∗ +ˇcµρση (ˇcδνλ)∗ = 2ηµληνρ σδ σδ − cµρη cσν = ηµν 5 ρσ 5 (10) cˇµνλ = cµνση cρλ =(cµσ)∗η cρνλ − σρ 5 5 σρ The Dirac operators are deﬁned as

def. Dˇ µσ =c ˇµνσ∂ , (Dˇ µλ)∗η (Dˇ ρν)= ηµν ν λρ − (11) µσ def. µνσ µλ ∗ ρν µν D = c ∂ν , (D ) ηλρ(D )= η

We deﬁne a bilinear law of composition (and ˇ ) for complex vectors ⊗ ⊗ λ def. µλν λ ρ def. µρν (G H) = Gµc Hν = S , (G ˇ H) = Gµcˇ Hν (12) ⊗ ⊗ def. The dot product is deﬁned by means of Minkowski metric, i.e. G H = µ ν · G ηµν H , and is not necessarily real; let alone positive. We can prove that

(G H) K = G (H K) ⊗ ⊗ ⊗ ⊗ (13) (G ˇ H) ˇ K = G ˇ (H ˇ K) ⊗ ⊗ ⊗ ⊗ and (G G)(H H)=(G H) (G H) · · ⊗ · ⊗ (14) (G G)(H H)= (G ˇ H) (G ˇ H) · · − ⊗ · ⊗ 5 The algebra deﬁned here is associative and noncommutive. The last identities look like the modiﬁed normed condition. (It is worth a passing to mention that a somewhat vague formulation of normed condition for reals n n n n n 2 2 2 is ai bi = ci in which ci = ajγjikbk where γijk are i=1 i=1 i=1 i=1 k=1 constants.P ItP is natural toP ask whether or not doP analogousP identities involving more squares exist. It has occupied the mind of mathematicians for many years. Only in 1898 did Hurwitz[6] prove that the identities of interest to us are possible only for n = 1, 2, 4, 8. This result, of course, is intimately related to the well known result, due to ...... that there exist only four normed algebras over the reals : R, C, H, O.) Because the algebra above is associative, it can be considered in terms of the matrices. Let

µ µ ν· def. νσµ G = Gµe ; (e )·λ = c ησλ (15) µ here Gµ are complex and the matrices e can be considered as the hyper- complex basic units. We can prove that a bilinear law of composition (12) µ λ· ν σ· µ λ· is equivalent to matrix multiplication (Gµe )·σ(Hνe )·ρ = (Sµe )·ρ. In the special coordinate system (28), where kµ = δµ0, the basis element e0 = I and for ν =1, 2, 3, we have eν = √ 1ˆeν here − 0 i 0 0 −  i 0 0 0  eˆ1 = − (16)  0 0 0 1   −   0 010    0 0 i 0 −  0 0 01  eˆ2 = (17)  i 0 00   −   0 1 0 0   −  0 0 0 i −  0 0 1 0  eˆ3 = − (18)  010 0     i 0 0 0   −  6 One notices that they obey the multiplication law :

eˆ1eˆ1 =e ˆ2eˆ2 =e ˆ3eˆ3 =e ˆ1eˆ2eˆ3 = I − (19) eˆ1eˆ2 = eˆ2eˆ1 =e ˆ3 − (the last relation being cycle). These relations deﬁne a structure of the algebra. We shall call this algebra the algebra of complexiﬁed biquaternions[7]. Furthermore we can construct the commutative Jordan algebra in the following way

def. 1 G K = G (tµρνσk )K = (G K + K G) (20) ◦ µ σ ν 2 ⊗ ⊗ We can prove that

G K = K G ◦ ◦ (21) [(G G) K] G =(G G) (K G) ◦ ◦ ◦ ◦ ◦ ◦ Now let us study the problem of the vector representation of Dirac spinors. The most important thing to us is : the trinomial unit-basis (ϕβ, f α, jµ) which satisﬁes the above conditions exists. And it is easily veriﬁed from the special case (28) below. We know that a Dirac spinor has 4 complex components, or equivalently 4+4=8 independent real components. Any Dirac spinor can be decomposed into the sum of two ‘half-spinors’ Ψ = Ψ1 + Ψ2. Each half-spinor has 4 independent real components and can be constructed by means of f and ϕ, i.e. µ ν µ ν Ψ1 = B ηµν iγ f , Ψ2 = N ηµνiγ ϕ (22)

Bµ = 1 (fiγµΨ Ψiγµf) = 1 (fiγµΨ Ψ iγµf) 2 − 2 1 − 1 (23) N µ = 1 (Ψiγµϕ ϕiγµΨ) = 1 (Ψ iγµϕ ϕiγµΨ ) 2 − 2 2 − 2 Here Bµ and N µ are real vectors. They deﬁne the vector representation of the half-spinors Ψ1 and Ψ2. Sometime it is convenient to pass from trinomial unit-basis (ϕβ, f α, jµ) to α β µ the null-basis (r ,l ,k±), where the normalized ”right-handed” spinor r and

7 ”left-handed” spinor l (pure spinors) are determined in the following way

rl = lr =2 , f = i (r l) , ϕ = 1 (r + l) 2 − 2 (24) kµ = 1 (kµ jµ) , kµ η kν =0 , kµ η kν = 1 ± 2 ± ± µν ± ± µν ∓ 2 It is easy to prove that

µ − µ + µ kµγ r = kµ γ r = l , kµ γ r =0 µ + µ − µ (25) kµγ l = kµ γ l = r , kµ γ l =0

The Dirac spinor can be equivalently decomposed into the sum of a ”right- handed” and a ”left-handed” spinors Ψ = R + L, here

def. 1 µ ν def. −1 ∗µ ν (26) R = 2 G ηµν γ l , L = 2 G ηµν γ r 1 Gµ = (rγµΨ Ψγµl)= Bµ + iN µ (27) 2 − In order to understand our idea easily, it is convenient to work in a special coordinate system such that

[ϕβ]T = [1, 0, 0, 0] [f α]T = [0, 0, i, 0] (28) jµ = (0, 0, 0, 1) kµ = (1, 0, 0, 0)

(here we use the Dirac representation of gamma matrices). In this special case B3 + iN 0  B1 + iB2  Ψ=Ψ1(B)+Ψ2(N)= . (29)  B0 + iN 3     N 2 + iN 1   −  Now let V µ M 1+3 be a vector in Minkowski space, Ψ (B) S and ∈ 1 ∈ 1 Ψ (N) S are two half-spinors referred to the trinomial unit-basis deﬁned 2 ∈ 2 above. The quadratic forms and the special cubic-form are deﬁned as

V µη V ν = V V ν , Ψ Ψ = B Bν , Ψ Ψ = N N µ (30) µν ν 1 1 − ν 2 2 µ 8 µ µ µ σ ν λ ρ V ηµν (Ψ1γ Ψ2 + Ψ2γ Ψ1)=2(ǫνλρσk )V N B (31) One realizes that there exists a ”ding” automorphism J of order 3 in M 1+3 × S S , which leaves quadratic two-forms (30) and the trilinear-form (31) 1 × 2 invariant up to the sign. J maps M onto S1, S1 onto S2, and S2 onto M. (Or equivalently V B N V .) → → →

3 Bosonization of Dirac equation.

The most interesting for physicists is: by passing from ordinary spinor representation to the vector representation, one can expresses Dirac Lagrangian in the Bosonic form

1 [Ψiγµ(∂ ieA )Ψ ((∂ + ieA )Ψ)iγµΨ] mΨΨ 2 µ − µ − µ µ − (32) = 1 [( G )∗icˇνµλG G∗icˇνµλ G + m(G∗ G∗ν + G Gν)] 2 ∇µ ν λ − ν ∇µ λ ν ν def. Here G = ∂ G ieA η cρσG andc ˇνµλ , cρλ are deﬁned by (8). ∇µ λ µ λ − µ λρ 5 σ 5 It is important to notice that from the abstract point of view, there is an arbitrariness in our Bosonization procedure since it depends on the choice of the neutral elements jµ end kµ. In the Lagrangian (32) they are arbitrary µ µ but must be ﬁxed. The only restrictions on them are that kµk = jµj = µ µνσ νµρ µν − 1 and kµj = 0. The remaining basis elements (c ,c ˇ , c5 , ...... ) can be reconstructed if desired, once jµ and kµ have been chosen. The corresponding massive Dirac equation in the vector-representation takes the form of cˇµνλi G mGµ∗ =0. (33) ∇ν λ − Or equivalently in the following self-dual form

µ µ∗ ∂µG imjµG =0 −i λρ (34) Gµν = 2 ǫµνλρG here (for simplicity) we take Aµ = 0 and

def. G = [(∂ G + imj G∗) (∂ G + imj G∗ )] (35) µν µ ν µ ν − ν µ ν µ 9 The Dirac equation (33) can be rewritten in the real form

ǫˇµνλ∂ B tˇµνλ∂ N eA (tµνλB + ǫµνλN ) mBµ =0 ν λ − ν λ − ν λ λ − (36) ǫˇµνλ∂ N + tˇµνλ∂ B eA (tµνλN ǫµνλB )+ mN µ =0 ν λ ν λ − ν λ − λ µνλ µνλρ µνλ µνλρ (hereǫ ˇ =ˇǫ jρ and ǫ = ǫ kρ) or equivalently

∂ N µ [mBµ + eA (B tµνλ + N ǫµνλ)]j =0 µ − ν λ λ µ ∂ Bµ [mN µ eA (N tµνλ B ǫµνλ)]j =0 (37) µ − − ν λ − λ µ ( ′ N ′ N )= 1 ǫ ( ′λBρ ′ρBλ) ∇µ ν −∇ν µ 2 µνλρ ∇ −∇ here

′ def. m ρ σλρ 1 σλρ µNν = ∂µNν + 2 ˇǫµνρN + e(jµηνσǫ 2 ˇǫµνσt )AλNρ ∇ def. − (38) ′ B = ∂ B m ǫˇ Bρ + e(j η ǫσλρ 1 ˇǫ tσλρ)A B ∇µ ν µ ν − 2 µνρ µ νσ − 2 µνσ λ ρ def. In addition, let us deﬁne the operator c = (∂ + imj C∗), such that ∇µ µ µ def. c G = ∂ G + imj G∗ (39) ∇µ νλ µ νλ µ νλ where C∗ is the operator of complex conjugation: C∗Φ = Φ∗. In this notation the identity c G + c G + c G =0 (40) ∇µ νλ ∇λ µν ∇ν λµ looks like the Bianchi identity, and

1 µνλρ ∗ µνλρ ∗ 4 [Gµνǫ Gλρ + Gµν ǫ Gλρ] 1 µνλρ ∗ ∗ = 2 ∂µ[ǫ (GνGλρ + GνGλρ)] (41) = 2∂ [ǫµνλρ(B ∂ B N ∂ N +2mB N j )] µ ν λ ρ − ν λ ρ ν λ ρ looks like the Chern-Pontryagin density and a total derivative of the Chern- Simons density. Thus the Dirac Lagrangian can be modiﬁed by the additional total derivative of the Chern-Simons density introduced above.

10 4 Lorentz rotations, U(1) and Chiral transformations

Because of the lack of commutativity there are two types of ˇ products ⊗ in the biquaternion algebra, the left and right multiplications. Thus for a given biquaternion G (and q such that q qµ = 1) we can consider the two µ − maps, i.e. two transformations[7]: G = q ˇ G and G = G ˇ q, depending as ⊗ ⊗ we multiply q on the left or the right.e In the vectore representation, these transformations take the form of

Gµ = Gν(η cˇλµσq )= Gν[S (q)]·µ νλ σ 1 ν· (42) µ σµλ ν µ· ν Hf =(qσcˇ ηλν )H = [S2(q)]·ν G From (modiﬁed normedf condition) (14) we ﬁnd that under the above trans- µν formations the dot product Gµη Gν is unchanged. Thus the two-form of 1 ∗ ∗ν ν the spinors ΨΨ = 2 (GνG + Gν G ) is invariant under the above ’complex − µ νµλ µνλ rotations’. (Similarly, if qµq = 1, thenc ˇ is replaced by c ). µ For the space-time (real) vector x (and ∂µ) the Lorentz transformation, in quaternionic terms, is characterized by the above unit-biquaternion q through the relation (considered as left-right mixed transformations[7]):

µ µ· ν ∗ µ· ν ·σ x = [Λ(q)]·ν x = [S2(q )]·σ(x [S1(q)]ν·) (43)

The constrain q qµf= 1 (implying two real conditions) ensures that this µ − transformation is 6-dimensional. This transformation preserves the value of the dot product and the reality of the space-time vector. Thus we have here two types of vectors with one-to-one distinct transformation operations ·µ µ· [S1(q)]ν· and [Λ(q)]·ν . In contrast with an ordinary real space-time vector, we shall use the name ’s-vector’ for Gµ which transforms as (42). We can prove that under the above Lorentz transformations (with ﬁxed jµ and kν) ν· ∗ µ· σρδ ·λ µνλ [Λ(q)]·ρ[S2(q )]·σcˇ [S1(q)]δ· =ˇc ν· ∗ µ· σρδ ·λ µνλ (44) [Λ(q)]·ρ[S2(q )]·σc [S1(q)]δ· = c

11 and this ensures the Lorentz invariance of the Dirac Lagrangian (32) which is written in the s-vector form. Also the Lagrangian (32) is invariant under U(1) gauge transformations

iα Ψ=Ψe = Ψ(cos α + i sin α) , Aµ = Aµ + ∂µα (45)

In the vectore representation it is equivalent to f

B = B cos α [ˇǫ kλBν (k j k j )N ν ] sin α µ µ − µνλ − µ ν − ν µ (46) N = N cos α [ˇǫ kλN ν +(k j k j )Bν] sin α fµ µ − µνλ µ ν − ν µ or in complexf form

µ µν µν µν G =(η cos α + ic5 sin α)Gν = (exp iαc5) Gν (47)

f µ µσ ρν µν It looks like a chiral transformation for G , here (c5 ησρc5 )= η , and from n which the De Moivre theorem is deduced (cos α + ic5 sin α) = (cos nα + ic5 sin nα). We know that the massless Dirac Lagrangian is invariant under the chiral transformation Ψ eiaγ5 Ψ; Ψ Ψeiaγ5 (48) → → In the s-vector representation it is equivalent to

G eiaG (49) µ → µ It means that a chiral transformation for Ψ is equivalent to a U(1) transformation for G . The mass term mΨΨ = m (G∗G∗ν + G Gν) is not invariant µ − 2 ν ν under the above chiral transformation, but is invariant under U(1) transformation (45)-(47). Therefore we must distinguish between the plan wave solutions for Ψ and for Gµ. Now, we deﬁne the ”dual” transformation

B N , N B , m m (50) → → − → − µ ν which leaves the quadratic form mΨΨ = m(NµN BνB ) , the cubic form νµλ − VµBν Nλǫ and thus the above Lagrangian and Dirac equation invariant.

12 (It looks like the electro-magnetic duality which is accompanied by e 99K q and q 99K e). In this sense the half-spinor of the ﬁrst type is dual to the − half-spinor of the second type. One may have many choices of ”representations” of the neutral elements (f,ϕ,jµ,kµ) andc ˇµνλ. For example, we can change the representations in the following way f = 1 (a + 1 )f + i (a 1 )ϕ , r = ar 2 a 2 − a (51) ϕ = 1 (a + 1 )ϕ i (a 1 )f , l = 1 l e 2 a − 2 − a e a e jµ = 1 (aa + 1 )jµ + 1 (aa 1e)kµ 2 aa 2 − aa (52) kµ = 1 (aa + 1 )kµ + 1 (aa 1 )jµ e 2 aa 2 − aa here a represents thef complex conjugate of a. Correspondingly 1 1 1 1 Gµ = [ (a + )ηµν + (a )cµν]G (53) 2 a 2 − a 5 ν The Dirac Lagrangianf (32) is invariant under this change of representation. In fact the U(1) transformation (47), can be considered as the special case of (53).

5 The behavior of the mass term. Theorem There exists a suitable complex vector Kµ associated with spinor Ψ= R + L (and completely determined by it) such that

µ − µ ∗ µ Kµγ R = Kµ γ R = L, RKµγ = L µ + µ ∗ µ (54) Kµγ L = Kµ γ L = R, LKµγ = R

The explicit form of these vectors are

K = RγµR + LγµL = K+ + K− µ 2RL 2LR µ µ (55) µν Kµη Kν =1

(Comments: the existence of such vector is due to the existence of kµ = + − kµ + kµ which has been introduced in the previous section (see eq.(25)), and thus has close relation with the ”ding” construction.)

13 Corollary The mass quadratic form equal to the interaction trilinear form µ ∗ µ RL = KµRγ R = KµLγ L µ ∗ µ (56) LR = KµLγ L = KµRγ R If Ψ= R + L , then µ mΨΨ = mΨKµγ Ψ (57) These identities have close relation with the modiﬁed normed condition (14).

Next we use the Dirac idea of non-integrable phases[8]. Theorem The charged massive Dirac equation can be rewritten in the following ”uncharged massless” forms

iγµ(∂ ieA )R mL µ − µ − (58) = iγµ(∂ ieA + imK )R = [iγµ∂ R ]Θ−1 =0 µ − µ µ µ 0 iγµ(∂ ieA )L mR µ − µ − (59) = iγµ(∂ ieA + imK )L = [iγµ∂ L ]Θ−1 =0 µ − µ µ µ 0 µ d −i (eAµ−mKµ)dx where R0 = R e = R Θ. · R · It is equivalent to

iγµ(∂ ieA )Ψ mΨ µ − µ − (60) = iγµ[∂ i(eA mK )]Ψ = [iγµ∂ Ψ ]Θ−1 =0 µ − µ − µ µ 0

µ d −i (eAµ−mKµ)dx here Ψ0 =Ψ e satisﬁes the massless Dirac equation. · R One knows that a charged particle must be massive. Thus in our formalism the mass m was introduced in the same way as the charge e! It is mathematically beautiful and physically natural.

It is important to notice that in general Kµ = [Re(Kµ)+ iIm(Kµ)] is complex : 5 5 (RγµR−LγµL)(LR−RL) πµ(Ψγ Ψ) iIm(Kµ) = = ν 4(RL)(LR) (πν π ) (61) (RγµR+LγµL)(RL+LR) πµ(ΨΨ) Re(Kµ) = = ν 4(RL)(LR) (πν π )

14 here πµ = ΨγµΨ= G∗ cλµν G λ ν (62) πµ = Ψγµγ5Ψ= G∗ cˇλµν G 5 − λ ν It means that Re(Kµ) is associated with unit vector kµ and Im(Kµ) is associated with unit vector jµ.

Only the real part Re(Kµ) is associated with the phase, and the imaginary µ −m [Im(Kµ)]dx part Im(Kµ) is associated with the “scale factor” σ = e of a R spinor. (A similar scale change can be found in the Weyl’s earlier work.[9]). Furthermore [Re(K )] [Im(Kµ)] = kµη jν =0 (63) µ · µν µ Ψ[Im(Kµ)]γ Ψ=0 µ µ (64) Ψ[Re(Kµ)]γ Ψ= ΨKµγ Ψ= ΨΨ

One can realize that the trilinear form is independent of Im(Kµ). Thus, if

d µ Ψ =Ψ e−i (eAµ−mKµ)dx =ΨΘ (65) 0 · R d d Ψ = Ψ∗T γ0 ; Ψ = Ψ Θ−1 (66) 0 · then the Lagrangian

Ψiγµ(∂ Ψ ieA )Ψ mΨΨ µ − µ − = Ψiγµ[∂ Ψ ieA + imRe(K )]Ψ (67) µ − µ µ µ = Ψ0iγ ∂µΨ0

µ is independent of the “scale factor” σ = e−m [Im(Kµ)]dx . R

6 Non-integrable exponential factors

A physical spinor ﬁeld Ψ which satisﬁes the massive charged Dirac equation (i.e. all physical solutions of massive Dirac equation) can be expressed in the following form

µ d i (eAµ−mKµ)dx Ψ =Ψ0 e R · µ µ (68) i (eAµ−mKµ)dx i (eAµ−mKµ)dx = R0 e + L0 e · R · R 15 Here Ψ0 satisﬁes the massless Dirac equation and

RγµR LγµL R0γµR0 L0γµL0 Kµ = + = + (69) 2RL 2LR 2R0L0 2L0R0 The plane-wave solution of an electron is the most important solution in the quantum field theory. In this special case, the real time-like vector mKµ is nothing but the energy-momentum of a massive Dirac particle. The connection between non-integrability of phase and electromagnetic potential Aµ given here is not new, which is essentially just Weyl’s (1929) principle of gauge invariance[10] in its modern form. C.N.Yang has refor- mulated the concept of a gauge field in an integral formalism and extended Weyl’s idea to the more general non-Abelian case[11]. The non-integrable phase for the wave functions was also discussed by Dirac in 1931[8], where the problem of monopole was studied. He emphasizes that “non-integrable phases are perfectly compatible with all the general principles of quantum mechanics and do not in any way restrict their physical interpretation.” Dirac conjectured that : ”The change in phase of a wave function round any closed curve may be different for different wave functions by arbitrary multiplies of 2π”. There is the famous Dirac relation: eq/(4π)=(n/2) . This means that if the quantization of electric charge (the universal unit e ) is accepted, then the above relation is the law of quantization of the magnetic pole strength. Notice, in our case Kµ = [Re(Kµ)+ iIm(Kµ)] is complex. Thus only µ i [eAµ m(Re(Kµ))]dx is associated with the phase change, and the imag- − µ inaryR part Im(Kµ)dx is associated with the scale change. BecauseR of the single-valued nature of a quantum mechanical wave function, we naturally conjecture that: 1. The phase change of a wave function round any closed curve must be close to 2nπ where n is some integer, positive or negative. This integer will be a characteristic of possible singularity in mRe(Kµ). 2. The scale change of a wave function round any closed curve must be close to zero. As being mentioned in the previous section, the Lagrangian (67) is independent of this scale change. This Lagrangian in the massless

16 form is conformally invariant[12]. Thus this scale-factor can be gauged away by conformal rescaling. This is a new (very strong) assumption, and cannot be proved, not de- rived. It is a conjecture of the overall consistency among all the solutions to the same equation. The existence of magnetics monopole is an open question µ yet, thus in our case it means that the change in mKµdx round any closed curve, with the possibility of there being singularityH in mRe(Kµ) , will lead to the law of quantization of physical constants, including mass.

References

[1] E. Cartan, (in French Edition) Lecons sur la Theorie des Spineurs I & II, Herman, Paris (1938); (in English translation by R. Streater) The theory of spinors, Herman, Paris (1966).

[2] C.C. Chevalley, The Algebraic Theory of Spinors, Columbia U.P., New York (1954).

[3] J.F. Adams, Superspace and Supergravity, ed. S.W. Hawking and M. Rocek (Cambridge University Press, Cambridge). (1981) p.435.

[4] J.F. Adams, Lectures on Exceptional Lie Groups, eds. Zafer Mahmud and Mamoru Mimira, University of Chicago Press, Chicago, 1996.

[5] J.C. Baez, May 2001, arXiv:math.RA/0105155.

[6] A. Hurwitz, Nachr. Ges. Wiss. Gottingen, (1898), pp.309-316.

[7] J.P. Ward, Quaternions and Cayley Numbers, Kluwer Academic Pub- lishers, (1997).

[8] P.A.M. Dirac, Proc. Roy. Soc. A133, 60 (1931)

[9] H. Weyl, Raum, Zeit and Materie, 1920. 3rd edit. Springer Verlag. Berlin-Heidelberg. New York.

17 [10] H. Weyl, Z. Physik 56 (1929) 330.

[11] T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 3845.

[12] Liu Y.F., Commun. Theor. Phys. (Beijing, China), 34 (2000), p.705.