Theoretical Particle Physics 1

Prof. Dr. Th. Feldmann

20. Mai 2015

Summary – Lecture 9 (8.5.2015)

2.4 Particle Fields with non-trivial Spin

So far, we have considered scalar field operators φ(x) with a Fourier decomposition(here for a complex field/charged particle)

φ(x) =

∫dp(a+(p) e−i p·x + a†−(p) ei p·x

).

• For the description of particles with spin, we would have to generalize,

ai(p) −→ ai(p, s) , φ(x) −→ φα(x) ,

∫dp −→

∫dp∑s

.

Here the index α refers to some non-trivial representation of the Lorentz group.

2.4.1 Spinor Representations of the Lorentz Group

In the following we consider φ(+)α and φ

(−)α separately.

• We are looking for representations with a transformation law1

U(Λ, a) φ(±)α (x)U(Λ, a)−1 =

∑α′

Dαα′(Λ−1) φ(±)α′ (Λx+ a) (125)

• In order to describe the relation between field operators and creation/annihilationoperators in the Fourier decomposition, we thus need additional functions uα(p, s)that describe the connection between the spin orientation of the states and the re-presentation φα of the Lorentz group. We may anticipate the following ansatz:

φα(x) := φ(+)α (x) + φ(−)

α (x)

1Here, we define the Lorentz matrix Λ to act on the argument of the fields, i.e. a passive transformation,and therefore the argument of the representation matrix Dαα′ is given by the inverse of Λ.

1

=∑s

∫dp(a+(p, s) e−i p·x uα(p, s) + a†−(p, s) ei p·x vα(p, s)

). (126)

Because of translational invariance, D should neither depend on xµ nor on aµ.

– The representation matrix Dαα′ need not be unitary(as it does not act on physical state vectors now).

– The representations are finite dimensional.

– The scalar field corresponds to the trivial (1-dimensional) representation,Dαα′ ≡ δαα′ with α, α′ ∈ {1}.

Recapitulating the algebra for the generators Jµν for rotations and boosts,[Jµν , Jλσ

]= i

(gµλ Jνσ − gνλ Jµσ − gµσ Jνλ + gνσ Jµλ

)(127)

with ~J ≡ (J23, J31, J12) and ~K = (J10, J20, J30), we obtain e.g.

[K1, K2] = [J10, J20] = −i J12 = −iJ3 .

This can be summarized as[Ki, Kj

]= −i εijk Jk ,

[Ji, Kj

]= i εijk Kk ,

[Ji, Jj

]= i εijk Jk . (128)

• The important observation now is that by building complex linear combinations,

~A =1

2

(~J + i ~K

), ~B =

1

2

(~J − i ~K

), (129)

one can diagonalize the algebra,

[Ai, Aj] = i εijk Ak , [Bi, Bj] = i εijk Bk , [Ai, Bj] = 0 . (130)

• Therefore, the algebra of rotations and boosts is equivalent to the direct product

SU(2)A × SU(2)B .

Notice, however, that ~A and ~B are not hermitian operators anymore (and thereforethe representations we are going to construct are not unitary)!

• With this we can construct all finite-dimensional representation Dα′α from the ele-mentary spinor representations of SU(2). For the two elementary spinors for SU(2)Aand SU(2)B, respectively, we write the transformation law as

η −→ exp

[i (~θ + i~ϕ) · ~σ

2

]η ≡ N(~θ, ~ϕ) η ,

2

ξ −→ exp

[i (~θ − i~ϕ) · ~σ

2

]ξ ≡M(~θ, ~ϕ) ξ . (131)

Here ~σ = (σ1, σ2, σ3) are the familiar Pauli matrices, ~θ and ~ϕ collect the angular andboost angles, where the factor (±i) in front of the boost operator in the definition

of ~A and ~B (when starting from the infinitesimal Lorentz transformations) directly translatesinto the exponentiated formula,

~θA = ~θ + i~ϕ , ~θB = ~θ − i~ϕ . (132)

• The matrices M and N are general complex 2 × 2 matrices with det = 1, where Mand N are related by N(~θ, ~ϕ) = M(~θ,−~ϕ). (which makes 4 × 2 − 2 = 6 d.o.f. in line

with 3 boosts and 3 rotations). Mathematically, this corresponds to the group of speciallinear transformations on C2, denoted as SL(2,C).

• A particular property of SU(2) is that the matrix N and the complex conjugate ofM are related by a similarity transformation,

N = εM∗εT , with ε = −iσ2 =

(0 −11 0

)(133)

and therefore the spinors εξ∗ and η transform the same way under Lorentz transfor-mations.

In summary, we can classify the field representations for different particles with spin as

scalar field invariant under SU(2)A × SU(2)B φ ∼ (0, 0)

spinor field η spinor under SU(2)A, invariant under SU(2)B ηα ∼ (12, 0)

spinor field ξ spinor under SU(2)B, invariant under SU(2)A ξα ∼ (0, 12)

· · · · · · · · ·higher representations X ∼ (jA, jB)

Parity and SU(2)A × SU(2)B

• We note that under parity transformations (P ) the vectors for the boost- and angular-momentum generators transform differently,

~K → − ~K and ~J → ~J ⇒ ~A↔ ~B , (134)

i.e. under parity the spinors ξ and η change their role. Standard convention:

η : right-handed spinor, ξ : left-handed spinor.

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• For parity-invariant theories (like QED) we thus need both Chiralities, such that aspin-1/2 electron corresponds to the SU(2)A × SU(2)B representation

electron: Ψα ∼ (1

2, 0)⊕ (0,

1

2) .

which consists of two 2-component complex spinors, written as

ψ ≡(ξη

)=

(ψLψR

)=

ψ1

ψ2

ψ3

ψ4

. (135)

in the so-called Chiral Representation of a Dirac 4-Spinor.

• The behaviour under Lorentz transformations follows as

ψ →

exp(i2~σ · (~θ − i~ϕ)

)0

0 exp(i2~σ · (~θ + i~ϕ)

) ψ ≡ S(

Λ(~θ, ~ϕ))ψ (136)

and under parity, we have

ψ →(ηξ

)=

(0 11 0

)ψ . (137)

2.4.2 Heuristic Derivation of the Dirac Equation

• We consider the spinor coefficients uα(p, s) (or analogously vα(p, s)) in the Fourierdecomposition of the Dirac spinor ψα (see previous lecture),

u(p, s) ≡(ξ(~p, s)η(~p, s)

)(138)

• Explicit Lorentz boost into the rest frame (for m 6= 0) yields

ξ(~p, s) = e−1/2~σ·~ϕ ξ(0, s) , η(~p, s) = e+1/2~σ·~ϕ η(0, s) (139)

• In the rest frame, the 2-component spinors ξ and η can be treated as the usual(non-relativistic) Pauli spinors, with s =↑, ↓ (see also below).

• The boost angle ~ϕ for a Lorentz boost into the rest frame is given by

~ϕ = ϕ~p

|~p|and coshϕ = γ = E/m

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With this one obtains by elementary properties of the Pauli matrices (see exercises)

e−1/2σ·ϕ = . . . =E +m− ~σ · ~p√

2m(E +m). (140)

And therefore

ξ(~p) =E +m− ~σ · ~p√

2m(E +m)ξ(~0) , η(~p) =

E +m+ ~σ · ~p√2m(E +m)

η(~0) (141)

• The crucial physical argument is now, that in the rest frame we cannot distinguishleft- or right-handed particles, and therefore ξ(~0) and η(~0) should be the same up toa (conventional) phase. We define:

ξ(~0) ≡ η(~0) in u(p, s)

In the moving frame this translates into

E +m+ ~σ · ~p√2m(E +m)

ξ(~p) =E +m− ~σ · ~p√

2m(E +m)η(~p) (142)

where we have used that (~σ · ~p) (~σ · ~p) = |~p|2 = E2 −m2 = (E +m)(E −m).

• Multiplying the last equation with (E +m− ~σ · ~p), one gets((E +m)2 − E2 +m2

)ξ(~p) =

((E +m)2 − 2 (E +m)~σ · ~p+ E2 −m2

)η(~p)

= 2m (E +m) ξ(~p) = 2 (E +m) (E − ~σ · ~p) η(~p)⇔ m ξ(~p) = (E − ~σ · ~p) η(~p) ,

and analogously m η(~p) = (E + ~σ · ~p) ξ(~p) . (143)

or, in matrix form,(−m1 E 1 + σi pi

E 1− σi pi −m1

)(ξ(~p)η(~p)

)= 0 (with σipi = −~σ · ~p) . (144)

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