Spinors, Helicity, Chirality and Parity

Question 1

Spinor Definition

Answer 1

-projected representation of some group as vector but with a different transformation law to vectors

Question 2

Representation Definition

Answer 2

-set of matrices with correct commutation relations

Question 3

Spinors and Rotation

Answer 3

• a full 360 degree rotation gives the negative of the original spinor
• a 720 degree rotation gets you back to the original

Question 4

Adjoint Spinor

Definition

Answer 4

Ψbar = Ψ†γo

• then have Ψbar Ψ is a scalar agreed on regardless of reference frame
• can also constrict vector as Ψbar γ^μ Ψ, a 4-vector

Question 5

Helicity

Definition

Answer 5

-spin of a particle as measured along axis of movement i.e. spin projected onto momentum

Question 6

Particle States and Helicity

Answer 6

• recall, 4 independent basis states for spinor; 2 particle states and 2 antiparticle states
• each of the two states (particle and antiparticle) are distinguished by their helicity

Question 7

Spinor Angular Momentum

Answer 7

• analysis of a spinor’s angular momentum shows that it separates into two parts:
• -orbital angular momentum
• -intrinsic angular momentum
• the operators for these, L_ and Σ_ do not individually commute with the Hamiltonian H^
• but (L_+Σ_) does so must represent total angular momentum

Question 8

Spinor Intrinsic Angular Momentum Operator and Spin

Answer 8

-the intrinsic angular momentum operator is given by:
Σ_ = 2x2 matrix, 1/2 () with entries σ, 0, 0, σ
-can show:
Σ² = 3/4 I
-where I is the identity matrix
=> any spinor has a Σ² eigenvalue of 3/4
s(s+1) = 3/4
=> s=1/2
-so the spinor describes spin 1/2 particles

Question 9

Helicity Operator

Definition

Answer 9

-can show that Σ_ . p_ commutes with H^
=> helicity is conserved
-by normalising this
=>
h(p_) = [Σ_.p_] / |p_|
-this projects the particle's spin in the direction of its motion

Question 10

Helicity Operator

Basis States u1,u2,v1,v2

Answer 10

• u1 and u2 are both eigenstates of h(p_) with eigenvalues +1 and -1 respectively
• we say that u1 solutions are right helical and u2 solutions are left helical

Question 11

Is helicity Lorentz invariant?

Answer 11

• helicity is an observed quantity, but is not Lorentz invariant
• > not all observers agree on its value
• e.g. consider a right helical particle from a stationary observers point of view, for an observer moving faster than the particle the helicity is actually reversed since the particles appears to move in the opposite direction but its spin is still aligned the same way

Question 12

Chirality

Definition

Answer 12

• a scalar, but NOT conserved

- related to how a spinor transforms under boosts

Question 13

Chirality Operator

Definition

Answer 13

γ^5 = i γo γ1 γ2 γ3

Question 14

Chirality Operator

Dirac Representation

Answer 14

γ^5 = 2x2 matrix; 0, I, I, 0

-where I is a 2x2 identity matrix and 0 is a 2x2 zero matrix

Question 15

Chirality Operator

Anticommutation Relations

Answer 15

{γ^μ, γ^5} = {γ^5,γ^μ} = 0

-anticommutates with all four gamma matrices

Question 16

Chirality

Eigenvalues

Answer 16

• the chirality operator has only two eigenvalues, +1 and -1, which correspond to the two distinct possibilities for how a spinor transforms
• an eigenstate with eigenvalue +1 is right chiral and a state with eigenvalue -1 is left chiral

Question 17

Chirality and Spinors

Answer 17

• since chirality is not conserved, the solutions we are considering cannot be eigenstates
• however, since there are only two possible chiralities, we can write any spinor as the sum of a left-chiral part and a right-chiral part

Question 18

Chirality

Projection Operators

Answer 18

-any spinor can be written as a sum of left chiral and right chiral parts
-can project these out with projection operators
-left projection operator:
L^ = 1/2(I - γ^5)
-right projection operator:
R^ = 1/2(I + γ^5)
-acting on a state Ψ=Ψl+Ψr:
L^Ψ = Ψl, R^Ψ=Ψr

Question 19

Chirality

Properties of Left and Right Projection Operators

Answer 19

-can show that L^ and R^ are idempotent:
L^² = L^ and R^²=R^
-also
L^R^ = R^L^ = 0

Question 20

Dirac Equation for Ψ = Ψl + Ψr

Answer 20

-sub in Ψ=Ψl+Ψr into the DIrac equation
-reduces to two coupled equations:
i ∂/ Ψl = m Ψr
i ∂/ Ψr = m Ψl
-i.e. the Dirac equation is equivalent to two coupled equations for left and right chiral parts only coupled by mass
-if mass=0, left chiral and right chiral parts are independent

Question 21

Weyl Representation of the Gamma Matrices and Chirality Operator

Answer 21

γo = 2x2 : 0, I, I, 0
γi = 2x2: 0, σi, -σi, 0
γ^5 = 2x2: -I, 0, 0, I

Question 22

Dirac Spinors and Weyl Spinors

Relationship

Answer 22

Ψ = (Ψl Ψr)t

-where Ψ is the dirac spinor and Ψl and Ψr are Weyl spinors

Question 23

Dirac Spinors and Weyl Spinors

Massless Case

Answer 23

• if m=0, have two independent particles each with 2DoF

- can also show that in the massless case, helicity and chirality coincide, LH=LC and RH=RC

Question 24

Adjoint Dirac Equation

Answer 24

-take Hermitian adjoint of Dirac equation
Ψbar (i γ^μ ∂μ + m) = 0
-where ∂ acts to the left

Question 25

Conserved Current for the Dirac Equation

Answer 25

-multiply the original Dirac equation to left by Ψbar and the adjoint equation to the right by Ψ
-add them together
=>
j^μ = Ψbar γ^μ Ψ

Question 26

Quantum Electrodynamics Equations

Answer 26

-use substitution i∂μ -> i∂μ + eqAμ in Dirac equation
=>
(i∂/ - m)Ψ = eq A/ Ψ
-also need a current / source term for photon
∂² A^μ - ∂μ ∂.A = eq Ψbar γ^μ Ψ
-from these two coupled equation, can derived the Feynman rules for QED which agree with experiment to 14sf

Question 27

Discrete Symmetries in Particle Physics

Answer 27

• parity, P - reverse all spatial coordinates
• charge conjugation, C - replace all particles with antiparticles and vice versa
• time reversal, T - reverse direction of time

Question 28

Parity

Definition

Answer 28

-mirror symmetry

Question 29

Parity

Operator and Eigenvalues

Answer 29

-for quantum state in system |Ψ⟩ take mirror image P|Ψ⟩
-if we take mirror image twice, should get back to |Ψ⟩
=>
PP|Ψ⟩ = P²|Ψ⟩ = |Ψ⟩
-if |Ψ⟩ is a P eigenstate with eigenvalue p
=>
p² = 1
p = ±1

Question 30

P and the Hamiltonian

Answer 30

-if we expect the mirror image to behave exactly as the original, then P must commute with H^and all states would be P eigenstates
(this is NOT true)

Question 31

Parity of Fermions

Answer 31

-define fermions of the standard model to have positive parity, P=1 and their antiparticles to have negative parity, P=-1

Question 32

Parity of Photons

Answer 32

• photon must have P=-1

- it is a vector valued boson so the mirrors image must be minus what you started with

Question 33

Parity and Symmetry

Answer 33

• the definitions of parity for fermions and photons assume that parity is a symmetry of the universe
• this is true if we restrict our attention to the electromagnetic and strong nuclear forces
• if the weak interaction is also included, parity does NOT commute with the Hamiltonian and is not a genuine symmetry

Question 34

Parity and the Weak Interaction

Answer 34

• parity IS NOT conserved in weak interactions
• in particular, in any experiment involving neutrinos, the conservation of momentum and angular momentum allows us to calculate the helicity of the neutrino
• -every neutrino ever measured in this way is found to be left-handed (handed not helical since the neutrino is close to massless allowing approximate identification of helicity and chirality states)
• however the practice of assigning parity to particles is not invalidated since the weak interaction is so weak it can often be neglected, in these situations parity is conserved

Question 35

Charge Conjugation Operator

Eigenvalues and Eigenstates

Answer 35

• a similar argument to parity shows that the eigenvalues of C^ must be C=±1
• for a particle to be an eigenstate of C^ it must be its own antiparticle
• e.g. photons, some neutral mesons
• for these particles we can define a C value

Question 36

Charge Conjugation and the Weak Interaction

Answer 36

• as with partity, C is found to be conserved in electromagnetic and strong interactions but not in weal
• experimental evidence for this is in weak decays
• just as all neutrinos are found to be left-chiral, all anit-neutrinos are found to be right chiral
• only a process that violates both P^ and C^, the weak interaction, could be capable of producing such a result

Question 37

Combined C^P^ Operation

Answer 37

• after the discovery of C^ and P^ violation of the weak interaction, it was initially postulated that the combined C^P^ may still be a symmetry
• this transformation would involve the simultaneous inversion of spatial coordinates and the replacement of particles with antiparticles
• given that neutrinos are left-chiral and antineutrinos are right-chiral, this would seem to suggest that weak interactions may satisfy this weaker symmetry
• in fact, even the combined C^P^ symmetry is violated by the weak interaction

Question 38

Time Reversal Symmety, T^

Answer 38

-no associated quantum number

Question 39

CPT Theorem

Answer 39

• the most fundamental principles of QFT automatically imply CPT theorem
• CPT combined HAVE to be a symmetry of the universe
• since CPT is an exact symmetry, CP violation implies T violation