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Dirac + QED Flashcards

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1
Q

What are spinors in the context of particle physics?

A

Quantum-mechanical objects called spinors rotating in SU(2) symmetry space

Spinors are essential for describing the properties of spin-1/2 particles in quantum mechanics.

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2
Q

What is the Lie group associated with ordinary 3D rotations?

A

SO(3)

SO(3) describes the group of rotations in three-dimensional space.

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3
Q

What is the commutation relation satisfied by the generators of SO(3) rotations?

A

[Ji, Jj] = i εijk Jk

This relation is fundamental in quantum mechanics, reflecting the angular momentum algebra.

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4
Q

What is the dimensionality of SU(2)?

A

3

This is derived from the formula dim [SU(n)] = n^2 - 1, where n=2 for SU(2).

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5
Q

What are the three traceless Hermitian matrices known as?

A

Pauli spin matrices

These matrices are fundamental in quantum mechanics and describe spin-1/2 particles.

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6
Q

What is the relationship between SU(2) generators and SO(3) rotations?

A

J ←→ 1/2 σ

This is an isomorphism that relates the two groups in the context of quantum mechanics.

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7
Q

What is the significance of the factor 1/2 in SU(2) rotations?

A

It indicates that a 360° rotation results in a sign change, U(2π) ξ = -ξ.

This is a unique property of spinors, which contrasts with ordinary vectors.

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8
Q

What does the term ‘chirality’ refer to in the context of SU(2) and Lorentz transformations?

A

A distinction between left-chiral and right-chiral Weyl spinors.

Chirality is analogous to handedness and is an inherent property of spin-1/2 particles.

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9
Q

How is parity defined in quantum mechanics?

A

The operation that transforms x → -x, flipping the sign of the boost β.

Parity conservation is an important symmetry in physics.

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10
Q

What is the form of the four-component spinor unifying left and right Weyl spinors?

A

ψ(W) = [ϕ, χ]^ op

This unification was proposed by Dirac to describe fermions like electrons.

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11
Q

What is the nonrelativistic energy-momentum relation for a free particle of mass m?

A

E = p^2 / 2m

This describes the kinetic energy of a particle in classical mechanics.

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12
Q

In the operator equation, what do the substitutions E → iℏ∂/∂t and p → -iℏ∇ represent?

A

They represent the correspondence between classical and quantum mechanics.

This is used to derive the Schrödinger equation for wavefunctions.

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13
Q

What is the form of the operator equation acting on a complex wavefunction, ψ(x, t)?

A

i ∂ψ/∂t = [-ℏ^2/2m ∇^2] ψ

This equation describes the time evolution of the wavefunction in quantum mechanics.

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14
Q

What is the nonrelativistic energy-momentum relation for a free particle of mass m?

A

E = p^2 / 2m

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15
Q

What does the operator substitution E → iℏ∂/∂t, p → -iℏ∇ lead to?

A

The time-dependent Schrödinger equation in the absence of an external potential.

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16
Q

What is the Klein-Gordon equation derived from?

A

The relativistic energy-momentum relation E^2 = |p|^2 c^2 + m^2 c^4.

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17
Q

What is the form of the Klein-Gordon equation?

A

−ℏ^2 ∂^2ψ/∂t^2 = −ℏ^2 c^2∇^2ψ + m^2 c^4ψ

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18
Q

What is the continuity equation for the wavefunction probability?

A

∂/∂t(ρ) + ∇·j = 0

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19
Q

What does the term ρ represent in the context of the wavefunction?

A

Probability density, defined as ρ = |ψ|^2.

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20
Q

True or False: The Klein-Gordon probability density can be negative.

A

True

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21
Q

What was Dirac’s main objective in formulating his equation in 1928?

A

To overcome the problem of negative probability in relativistic quantum mechanics.

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22
Q

What is the form of Dirac’s equation?

A

Eψ = iℏ∂ψ/∂t = (α·p)c + βmc^2

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23
Q

What are the conditions required for the matrices αi and β in Dirac’s equation?

A
  • α^2_i = 1
  • β^2 = 1
  • (α_iα_j + α_jα_i) = 0
  • (α_iβ + βα_i) = 0
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24
Q

What is the significance of the traceless condition for the matrices αμ?

A

It requires the dimension of αμ to be even.

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25
What are the four hermitian, traceless, 4 × 4 matrices that satisfy Dirac's equation?
* β = [[1, 0], [0, -1]] * αi = [[0, σi], [σi, 0]]
26
What is the Dirac four-current?
j^μ = [ρ, j]
27
What is the probability density in Dirac's formulation?
ρ = ψ†ψ
28
What commutation relation is required for angular momentum to commute with the Hamiltonian?
[H, L] = 0
29
What does the intrinsic property of the Dirac Hamiltonian imply?
Particles must have intrinsic spin-1/2.
30
What is the relationship between the Dirac equation and the Pauli spin matrices?
The rotations of spin-1/2 objects are generated by the Pauli spin matrices.
31
What does the operator S in the context of Dirac's equation represent?
It represents intrinsic angular momentum.
32
What is the form of the commutation relation for Pauli spin matrices?
[σi, σj] = 2iϵijkσk
33
What is the conclusion drawn from the Dirac equation regarding probability density?
The probability density is always positive.
34
Fill in the blank: The continuity equation can be expressed as ∂μ j^μ = _______.
0
35
What does the commutation relation [H, L + S] = 0 imply?
L + S commutes with the Hamiltonian ## Footnote This indicates the conservation of angular momentum.
36
What property must particles governed by the Dirac Hamiltonian possess?
An intrinsic property that behaves like and can add to orbital angular momentum.
37
What is the result of S zψ?
S zψ = ±ℏ/2 ψ.
38
What is the expression for S²ψ?
S²ψ = S(S + 1)ℏ²ψ for S = 1/2.
39
What are the properties of S in quantum mechanics?
S = 1/2 ℏ, S²ψ = S(S + 1)ψ, S zψ = 1/2 ℏψ.
40
What is the covariant form of the Dirac equation?
ℏ[iγ µ ∂µ - mc]ψ = 0.
41
What do the γ µ matrices represent?
Four matrices of constants, not four-vectors.
42
What is the anticommutation relation for γ µ matrices?
γ µ γ ν + γ ν γ µ = 2g µν.
43
What are the properties of γ 0 and γ a matrices?
γ 0 is hermitian; γ a are antihermitian.
44
What does the continuity equation ∂/∂t[ψ†γ 0 γ 0ψ] + ∇·[ψ†γ 0 γ 0αψ] = 0 represent?
Conservation of probability in quantum mechanics.
45
What is the form of the Dirac probability current?
j µ = ψ γ µ ψ.
46
What is the four-momentum definition in the Dirac equation?
i∂0 = p0 = E, i∂a = pa.
47
What is the expression for the wavefunction solution of the Dirac equation?
ψ(x µ) = u(p µ) exp[-ip µ x µ].
48
What is the condition for the wavefunction normalization factor N?
N = 1/√(E + m).
49
What does the equation E² - p² - m² = 0 signify?
It relates energy, momentum, and mass in the context of the Dirac equation.
50
What do the negative energy solutions of the Dirac equation represent?
Particles traveling backwards in time, perceived as antiparticles.
51
What is the form of the Dirac equation for antiparticles?
ℏ[γ µ p µ + m]ψ = 0.
52
What is the parity operator in the context of the Dirac equation?
P = γ 0.
53
What is the parity operator in the context of the Dirac equation?
P = γ^0 ## Footnote The parity operator modifies the Dirac equation to the parity-transformed coordinates.
54
How does the parity operator affect the Dirac spinor?
ψ(x, t) = γ^0ψ'(−x, t) ## Footnote This transformation reflects the change in coordinates under parity.
55
What is the result of applying the parity operator to the Dirac equation?
The Dirac equation is recovered in the parity-transformed coordinates: h iγ^μ∂'μ - m iψ' = 0 ## Footnote This validates the transformation of the equation under parity.
56
What does the anti-commutation relation γ^μγ^ν + γ^νγ^μ equal?
2g^μν ## Footnote This relation is fundamental in the context of the gamma matrices.
57
What are the components of the Dirac gamma matrices γ^0, γ^1, γ^2, and γ^3?
γ^0 = ∗ 1 1 1 1 γ^1 = ∗ -1 -1 1 1 γ^2 = ∗ i -i -1 i γ^3 = ∗ -1 1 1 -1 ## Footnote These matrices are used in the formulation of the Dirac equation.
58
What distinguishes the Weyl representation from the Dirac representation?
The Weyl representation groups entries according to chirality, while the Dirac representation does not ## Footnote This distinction affects how chiral information is encoded in the Dirac spinor.
59
What is the definition of the chirality operator γ^5?
γ^5 = iγ^0γ^1γ^2γ^3 ## Footnote It incorporates all gamma matrices and plays a crucial role in defining chirality.
60
What are the eigenvalues associated with the eigenstates of chirality?
±1 ## Footnote These eigenvalues indicate the chirality of the respective states.
61
What do the projection operators PR and PL do?
PR = 1/2(1 + γ^5) PL = 1/2(1 - γ^5) ## Footnote These operators project out the right-chiral and left-chiral components of a Dirac spinor.
62
True or False: Chirality is a quantum-mechanical observable.
False ## Footnote Chirality is a fundamental property of spin-1/2 particles but not a quantum-mechanical observable.
63
What is the form of the Dirac equation expressed with γ matrices?
h γ^μpμ - m iψ = 0 ## Footnote This equation describes the motion of spin-1/2 particles in relativistic quantum mechanics.
64
Fill in the blank: The Dirac equation requires _______ objects.
spin-1/2 ## Footnote This requirement stems from the commutation of angular momentum with the Dirac Hamiltonian.
65
What transformation is used to switch between Weyl and Dirac representations?
γ^μ(W)(D) = A^−1γ^μ(D)(W) ## Footnote This transformation helps in relating the two different representations of the Dirac spinor.
66
What does the Dirac equation describe?
The Dirac equation describes the behavior of fermions, such as electrons, incorporating quantum mechanics and special relativity.
67
What are the components of the Dirac equation expressed in terms of γ matrices?
The components include terms like γ^µ p_µ - m_i ψ = 0 and γ^0 E - γ^0 α · p - m I_i u = 0.
68
What is helicity?
Helicity is the scalar product of a particle’s normalized momentum and spin, defined as h = S · p / |S||p|.
69
What are the eigenvalues of the helicity operator?
The eigenvalues of the helicity operator are ±1.
70
True or False: Helicity is Lorentz invariant for all particles.
False. Helicity is Lorentz invariant only for massless particles traveling at the speed of light.
71
Define the term 'chiral nature' as used in the context of the Dirac equation.
Chiral nature refers to the property of particles that can be classified into right-chiral and left-chiral states.
72
What does the operator γ^5 represent in quantum field theory?
The operator γ^5 is used to project out chirality states and is associated with the concept of helicity.
73
What is the significance of the Dirac probability four-current?
The Dirac probability four-current j^µ = ψ γ^µ ψ represents the flow of probability density and is crucial in understanding particle interactions.
74
Fill in the blank: The Dirac equation separates into doublets A and B forming coupled relations, characterized by the matrix ___ .
[σ · p]
75
What happens to chirality in the context of the Dirac current?
Chirality is not conserved in certain interactions; the chirality non-conserving currents vanish.
76
Describe the (V ± A) structure of the four-current.
The (V ± A) structure refers to the separation of the Dirac current into vector and axial vector components.
77
What is the role of the spinor transformation S in the Dirac equation?
The spinor transformation S relates the transformed spinor to the original spinor under spacetime transformations.
78
How does the helicity operator relate to the helicity basis?
The helicity operator allows us to reformulate the Dirac solutions as eigenstates of helicity.
79
What does the equation h(u) = +1 indicate?
It indicates that the spin state u is aligned with the momentum direction.
80
What is the significance of the eigenstates of helicity in particle physics?
They allow experiments to probe the dependence of fundamental interactions on chirality.
81
True or False: The eigenstates of helicity and chirality are always aligned.
False. The alignment can depend on the specific conditions of the particle's momentum and spin.
82
What is the form of the invariant in the exponent?
ψ(x) = u(pµ) exp(−ipµxµ)
83
How is S related to the coordinate transformation?
By premultiplying by S−1 and requiring consistency with the Dirac equation
84
What is the form of the equation consistent with the Dirac equation?
S−1hγµ∂′µ − m i S ψ = 0
85
What condition must hold for the transformed Dirac equation?
S−1γµS∂/∂xµ′ = γµ∂/∂xν
86
What is Λµν in the context of spacetime transformation?
Λµν = ∂xµ′/∂xν
87
What must be true for the Dirac probability density to be invariant?
ψ′ = ψ S−1
88
What is the Weyl representation of the Dirac spinor?
Sϕχ = (exp(h1/2 iσ·θ - 1/2σ·ρ)i 0 0 exp(h1/2 iσ·θ + 1/2σ·ρ)i)ϕχ
89
What is the form of the parity transformation matrix Λνµ?
Λνµ = (1 -1 -1 -1)
90
What is the spinor transformation for a parity transformation of coordinates?
S−1P γ0 SP = +γ0, S−1P γa SP = -γa for a = 1, 2, 3
91
What 4 × 4 matrix represents the spinor transformation SP?
SP = γ0 = (I 0 0 -I)
92
What happens to the spatial components of a vector under a parity transformation?
They flip sign
93
What happens to the spatial components of an axial vector under a parity transformation?
They do not flip sign
94
What is the transformed Dirac current under parity?
P(jµ) = ψ′γµψ′ = ψγ0γµγ0ψ
95
How does the term ψγµψ transform under parity?
As a Vector
96
How does the term ψγµγ5ψ transform under parity?
As an Axialvector
97
What is preserved in the current (V + A + V - A) under parity?
Both left- and right-handed components

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