Flashcards in Spinors, Helicity, Chirality and Parity Deck (39)

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1

## Spinor Definition

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

2

## Representation Definition

### -set of matrices with correct commutation relations

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## 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

4

##
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

5

##
Helicity

Definition

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

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## 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

7

## 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

8

## 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

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##
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

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##
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

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## 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

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##
Chirality

Definition

###
-a scalar, but NOT conserved

-related to how a spinor transforms under boosts

13

##
Chirality Operator

Definition

### γ^5 = i γo γ1 γ2 γ3

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##
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

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##
Chirality Operator

Anticommutation Relations

###
{γ^μ, γ^5} = {γ^5,γ^μ} = 0

-anticommutates with all four gamma matrices

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##
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

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## 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

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##
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

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##
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

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## 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

21

## 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

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##
Dirac Spinors and Weyl Spinors

Relationship

###
Ψ = (Ψl Ψr)t

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

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##
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

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## Adjoint Dirac Equation

###
-take Hermitian adjoint of Dirac equation

Ψbar (i γ^μ ∂μ + m) = 0

-where ∂ acts to the left

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## 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

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## 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

27

## 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

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##
Parity

Definition

### -mirror symmetry

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##
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

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