Spinors, Helicity, Chirality and Parity Flashcards
Spinor Definition
-projected representation of some group as vector but with a different transformation law to vectors
Representation Definition
-set of matrices with correct commutation relations
Spinors and Rotation
- a full 360 degree rotation gives the negative of the original spinor
- a 720 degree rotation gets you back to the original
Adjoint Spinor
Definition
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
Helicity
Definition
-spin of a particle as measured along axis of movement i.e. spin projected onto momentum
Particle States and Helicity
- 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
Spinor Angular Momentum
- 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
Spinor Intrinsic Angular Momentum Operator and Spin
-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
Helicity Operator
Definition
-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
Helicity Operator
Basis States u1,u2,v1,v2
- 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
Is helicity Lorentz invariant?
- 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
Chirality
Definition
- a scalar, but NOT conserved
- related to how a spinor transforms under boosts
Chirality Operator
Definition
γ^5 = i γo γ1 γ2 γ3
Chirality Operator
Dirac Representation
γ^5 = 2x2 matrix; 0, I, I, 0
-where I is a 2x2 identity matrix and 0 is a 2x2 zero matrix
Chirality Operator
Anticommutation Relations
{γ^μ, γ^5} = {γ^5,γ^μ} = 0
-anticommutates with all four gamma matrices
Chirality
Eigenvalues
- 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
Chirality and Spinors
- 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
Chirality
Projection Operators
-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
Chirality
Properties of Left and Right Projection Operators
-can show that L^ and R^ are idempotent:
L^² = L^ and R^²=R^
-also
L^R^ = R^L^ = 0
Dirac Equation for Ψ = Ψl + Ψr
-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
Weyl Representation of the Gamma Matrices and Chirality Operator
γo = 2x2 : 0, I, I, 0 γi = 2x2: 0, σi, -σi, 0 γ^5 = 2x2: -I, 0, 0, I
Dirac Spinors and Weyl Spinors
Relationship
Ψ = (Ψl Ψr)t
-where Ψ is the dirac spinor and Ψl and Ψr are Weyl spinors
Dirac Spinors and Weyl Spinors
Massless Case
- 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
Adjoint Dirac Equation
-take Hermitian adjoint of Dirac equation
Ψbar (i γ^μ ∂μ + m) = 0
-where ∂ acts to the left
Conserved Current for the Dirac Equation
-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
Quantum Electrodynamics Equations
-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
Discrete Symmetries in Particle Physics
- parity, P - reverse all spatial coordinates
- charge conjugation, C - replace all particles with antiparticles and vice versa
- time reversal, T - reverse direction of time
Parity
Definition
-mirror symmetry
Parity
Operator and Eigenvalues
-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
P and the Hamiltonian
-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)
Parity of Fermions
-define fermions of the standard model to have positive parity, P=1 and their antiparticles to have negative parity, P=-1
Parity of Photons
- photon must have P=-1
- it is a vector valued boson so the mirrors image must be minus what you started with
Parity and Symmetry
- 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
Parity and the Weak Interaction
- 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
Charge Conjugation Operator
Eigenvalues and Eigenstates
- 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
Charge Conjugation and the Weak Interaction
- 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
Combined C^P^ Operation
- 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
Time Reversal Symmety, T^
-no associated quantum number
CPT Theorem
- 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