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EXAMS Summer 2023 > Gauge Theories > Flashcards

Gauge Theories Flashcards

1
Q

A group G is a set of elements endowed with composition law * that has properties:

A

i) CLOSURE (the product of any two elements of the group is also an element of the group)

ii) ASSOCIATIVITY (the composition law is associative)

iii) EXISTENCE OF IDENTITY (there is an identity element e that, when applied to element a, returns a)

iv) EXISTENCE OF INVERSE (for each element a, there is an inverse element a^-1, the product of these two is the identity element e).

*the IDENTITY and INVERSEs are unique.

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2
Q

What is an Abelian group?

A

An abelian group has a composition law that is commutative (i.e: a * b = b * a). {for all elements a and b in G}

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3
Q

Why is (all the real numbers, multiplication) not a group?

A

There is no inverse element for 0 (at least not a unique one).

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4
Q

What does the discrete group S_n refer to?

What is the order? Is it abelian?

A

The possible permutations of n objects.
{order n!, non-abelian}

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5
Q

What does the discrete group Z_n refer to?

What is the order? Is it abelian?

A

(Integers mod n, addition modulo n)

{order n, abelian}

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6
Q

What does the discrete group C_n refer to?

What is the order? Is it abelian?

A

The cyclic group of order n. {so a^n = e}

Isomorphic to Z_n : abelian.

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7
Q

What is a proper subgroup H of group G?

What is the symbol denoting this?

A

H contains only elements that are also elements of G.

H does not only contain the identity element.

H is not identical to G.

H c G

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8
Q

What is the “left coset” of a subgroup H of G, for element g of G?

What about the “right coset”?

A

left coset: gH

right coset: Hg

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9
Q

What is Lagrange’s Theorem?

How might we go about proving this?

A

For any two (left) cosets of H: g_1 H and g_2 H :
either these cosets are identical, or their intersection is the null element.

Proof by contradiction, assume that both conclusions of the theorem are not true. Propose g_3 which is in the intersection of the two cosets. We can show that g_2 is in the first coset, and so the cosets are identical.

*check notes

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10
Q

What is the symbol for “there exists”?

A

Backwards capital E.

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11
Q

What is the coset decomposition of a proper subgroup H of G?

What about the coset space G/H?

A

Coset decomposition basically says that G = the union of left cosets of H with elements g of G. In general, not all elements g will have to be used, the number of elements (nu) required to make the coset decomposition is the “index” of G/H.

G/H is the set of cosets required for a coset decomposition of G in H.

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12
Q

What is a group homomorphism?

i.e: what does it mean for group A to be homomorphic to group B?

A

All elements a in A are mapped each to a single element b of B. f(a) = b

f(a1*a2) = f(a1) $ f(a2) {where * is the composition law of A and $ of B}

f(A) is a subgroup of B in general, i.e: some elements b in B will not have a counterpart in A.

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13
Q

What is a group isomorphism?

i.e: what does it mean for groups A and B to be isomorphic?

What symbol is used to denote this?

A

Basically a 1:1 mapping, f(A) = B.

Homomorphism with all elements matched.

Symbol is equals sign with ~ on top.

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14
Q

What is a group endomorphism?

What about automorphism?

A

Endomorphism means that group A is homomorphic with itself under some operation f(A).

Automorphism means that a group A is isomorphic with itself under some operation f(A).

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15
Q

What does the label “General (G)” mean for a continuous group?

A

Determinant is not equal to 0.

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16
Q

What does the label “Linear (L)” mean for a continuous group?

A

Representable by NxN matrices.

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17
Q

What does the label “Special (S)” mean for a continuous group?

A

Determinant is equal to 1.

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18
Q

What does the label “Orthogonal (O)” mean for a continuous group?

A

Matrices (in matrix representation of the group) are orthogonal:

Transpose = inverse

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19
Q

What does the label “Unitary (U)” mean for a continuous group?

A

Matrices (in matrix representation of the group) are unitary:

Hermitian conjugate = inverse

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20
Q

What is the significance of the # independent real parameters of a group?

A

= the # of group generators
= the # of real group parameters.

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21
Q

Prove that the number of independent parameters of O(N, R) is N(N-1)/2.

A
  • check notes

-start off with N^2 parameters for real NxN matrix.

-Apply orthogonality condition, use index notation, make sure not to double count the constraints.

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22
Q

How many independent real parameters does GL(N, C) have?
What about SL(N, C)?

A

GL(N, C) has 2N^2 independent real parameters.

SL(N, C) has 2(N^2 - 1) independent real parameters.

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23
Q

How many independent real parameters does O(N, R) have?

What about SO(N, R)?

A

both have N(N-1)/2

*i guess the additional requirement of determinant = 0 doesn’t add anything as we already require the matrices to be hermitian?

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24
Q

What is an example of a SO(2) process?

A

Passive rotation in 2D through some angle about an axis z.

*Check notes for relevant rotation matrix.

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25
Q

What is the formula for the limit representation of e^x?

A

lim{N->inf} [ 1 + x/N ]^N = e^x

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26
Q

What is the general formula for the construction of Lie Generators?

How is an element of a group formed from its generator?

A

*check wall notes

*check wall notes

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27
Q

What is an example of an SO(3) process?

A

Passive rotations in 3D through some angle about a given vector.

*Check notes for matrix representation.

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28
Q

What are the generators of the SO(3) group? (fund. rep.)

Give the general group element.

A

(X_k)_ij = -i levi-cevita_ijk

General group element e^-i phi (n dot X)

Generators of SO(N) groups are labelled X.

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29
Q

What is an example of a SU(2) process?

A

Passive (abstract) rotation of a complex 2D vector through some angle about some unit vector.

The angle (group parameter) can take values between 0 and 4pi. *analogy with fermion spin.

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30
Q

What are the generators of SU(2)? (fund. rep.)

Therefore, what commutation relation do they satisfy?

A

Pauli matrices / 2

Commutator = i levi-cevita T
(*check notes)

Generators of SU(N) groups are labelled T.

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31
Q

Why might we expect connection between SO(3) and SU(2) groups?

What is a faithful isomorphism between these groups?

A

Both have 3 generators (3 group parameters, 3 independent real parameters).

Their generators satisfy identical commutation relations.

SO(3) is isomorphic to SO(2) / Z_2
(i.e: the coset space of SO(2) with Z_2, which is simply the 2x2 identity and its negative)
*note that in this coset rep. theta goes from 0 to 2pi only, and is therefore correspondent to phi.

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32
Q

What are the requirements for the set of generators of a group to satisfy the Lie Algebra?

A

Must be closed under commutation relation. The constants are defined to be i * the “structure constants”.

Must satisfy the Jacobi Identity (always true for matrix representable generators).

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33
Q

What is the “fundamental representation” of a Lie Algebra?

A

Matrices d(F)xd(F) where d(F) is the minimal number of dimensions required to realise the Lie Algebra for continuous group G.

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34
Q

What are the generators of the U(1) group? (fund. rep.)

A

m where m is in Z.

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35
Q

How is the adjoint representation of a Lie Algebra defined?

A

*Check wall notes, related to the structure constants. (i.e: no adjoint representation for abelian groups)

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36
Q

What is the metric of the manifold defined by the adjoint representation of a Lie Algebra?

A

The cartan metric, equal to the trace of two adjoint rep.s

*check wall notes

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37
Q

How is a real Lie Algebra defined?

A

All structure constants are real.

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38
Q

How is a compact group defined?
What are some properties?

A

Positive definite cartan metric with real Lie Algebra.

Compact groups have elements bounded from below and above. The cartan metric can be diagonalised and normalised to the dirac delta, i.e: contra- and co-variant representations of generators are equal.

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39
Q

What are Casimir operators?

A

(in some representation of a Lie Algebra) Matrix representations that commute with all generators of the Lie Algebra.

*Check notes for denotion.

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40
Q

What is an example of a Casimir operator from QM?

A

Angular momentum squared commutes with all angular momentum operators.

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41
Q

What is the Casimir operator for a compact group?

A

Simply equal to the identity matrix (for the dimensionality of the rep.) * a “Casimir eigenvalue”.
Also equal to the sum over all squared generators.

*check wall notes

*known as Schur’s lemma

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42
Q

How is the normalisation of a certain rep. of generators defined? (compact group)

A

Tr(the product of two generators) = normalisation(in that rep.) * dirac delta

*check wall notes

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43
Q

How does the “successive operation” work?
(the composition law for the discrete group S (permutation group))

A

e.g: for S_3:

(1,3,2) means 1->3, 3->2, 2->1

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44
Q

What must be true of groups A and B for the coset space A/B to be valid?

A

B must be a proper subgroup of A.

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45
Q

How can we show that a the generators of a unitary group are hermitian?

A

Consider the representation of group elements in terms of generators.

We know that the group elements are unitary, applying this relation results in the generators being hermitian.

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46
Q

What is an abstract way to show that some space does not form a group?

A

To form a group, the generators of the space must form a Lie Algebra.
If the generators do not satisfy the two conditions for a Lie Algebra, the space does not form a group.

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47
Q

What does it mean for a matrix to be hermitian?

A

Equal to its hermitian conjugate.

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48
Q

What is the general definition of the casimir operator?

A

*Check wall notes

49
Q

What is the definition of the normalisation of the adjoint representation?

A

Trace(T_a * T_b) = T diracdelta_ab

*adjoint labels not given here

50
Q

What is the transpose of (ABC)?

A

C^T B^T A^T

51
Q

What is the cyclic trace property?

A

Tr(ABC) = Tr(BCA) = Tr(CAB)

52
Q

What is the commutator [AB, C]?

A

[AB, C] = A[B, C] + [A, C]B

53
Q

How many independent real parameters does SU(N) have?

A

N^2 - 1

2N^2 - N^2 (due to unitarity) - 1 (due to det = 1)

54
Q

Justify the fact that the O(N, R) group and the SO(N, R) groups have dimensionality N(N-1)/2

A

N^2 free parameters for NxN matrix.

Orthogonality imposes constraints, O_ba O_bc = delta_ac

The number of these constraints is the sum of c and a from 1 to N. However, we must avoid double-counting, so divide by two. i.e: the sum from 1 to N, which is N(N+1)/2. Subtract this from N^2 to get N(N-1)/2.

55
Q

What is the dimensionality of the fundamental representation of the generators of a group?

A

Equal to the dimensionality of the group (N).

56
Q

What is the dimensionality of the adjoint representation?

A

Equal to d(G) = the # of independent real parameters = the number of generators.

57
Q

Give two examples of compact Lie groups.

A

SO(N)
SU(N)

58
Q

What is the normalisation of fund. and adj. rep’s for SU(N) theories?

A

T_fun. = 1/2

T_adj. = N

59
Q

What is a one-particle-irreducible (1PI) diagram?

A

If a diagram is not 1PI then an internal line can be cut resulting in two separate diagrams (that both make sense individually).

59
Q

What is a one-particle-irreducible (1PI) diagram?

A

If a diagram is not 1PI then an internal line can be cut resulting in two separate diagrams (that both make sense individually).

60
Q

What are Yang-Mills theories?

A

Gauge theories based on non-Abelian compact groups.

(so cartan metric is ~ dirac delta)

e.g: SU(N), SO(N>2)

61
Q

How many gauge fields do we get in the YM field multiplet for a SU(N) YM theory?

A

N^2 - 1

62
Q

How does the YM field multiplet transform under SU(N) gauge transformation?

What about a global SU(N) gauge transformation?

A

*Check wall notes

Global –> group parameter is constant. Therefore the term in the derivative of the group element goes to zero and the field multiplet transforms as a rank-2 SU(N) tensor.

63
Q

What is the field strength tensor multiplet in a YM theory?

How does it transform under a SU(N) gauge transformation?

A

*Check wall notes

Transforms as a rank-2 SU(N) tensor.

64
Q

How is the general YM lagrangian defined for SU(N) theories?

How does it transform under a SU(N) gauge transformation?

A

-1/2 * trace[ lorentz-covariant product of field strength multiplets]

*check wall notes for form without trace.

Invariant under SU(N), easy to show using the transformation of the field strength tensor and the cyclic trace property.

65
Q

How many gauge bosons do SU(N) YM theories predict?

What symmetry group is QCD based on? How many gauge bosons do we get?

A

N^2 - 1 gauge bosons.

QCD is based on the pure YM SU(3) theory, resulting in 8 gauge bosons (the gluons).

66
Q

Why do YM theory gauge bosons self-interact with triple and quartic vertices?

A

In the YM field strength tensor multiplet there is a quadratic term in the gauge field multiplet.

As the field strength tensor appears twice in the lagrangian, this leads to both triple and quartic vertices between the gauge bosons.

67
Q

What are the generators in the fund. rep. for SU(3) (QCD)?

A

Gell-mann matrices (denoted lambda) divided by 2.

(there are 8 of them)

68
Q

How are quarks represented in QCD?

A

As a vector of three colours in the fund. rep.

They are Dirac fermions, and are expressed in the Chiral (Weyl) basis of LH and RH spinors.

69
Q

How is qbar defined?
How does it transform under SU(3)?
(quark in QCD)

A

q(hermitian conjugate) gamma^0

transforms to qbar U(hermitian conjugate)
{should be easy to show, remembering that gamma and U commute as they act on different group spaces}

*check notes

70
Q

What is the lagrangian used to describe quark-gluon interactions?

A

*check notes

YM gauge field term + term in the SU(3) covariant derivative - quark mass term

71
Q

How does q (quark in QCD) transform under SU(3)?

What about the SU(3) covariant derivative?

A

q -> Uq

D_mu q -> U D_mu q
{should be easy to show, remembering the gauge transformation of the gauge field multiplet}

72
Q

Why must we use a gauge fixing term in YM theories?

What is the “unitary gauge”?

A

We require the field propagator in order to calculate observables. If we do not fix the gauge, this propagator will be singular.
A gauge fixing term violates local gauge symmetry, with a term epsilon.

As epsilon -> 0, the gauge fixing term in the lagrangian -> 0. Applying this limit is called the unitary gauge.

73
Q

What is the gauge fixing term used for QCD?

A

*Check notes

1/epsilon * Trace[(covariant derivative of gauge field multiplet)^2]

74
Q

What is the E-L equation of motion?

A

*check wall notes.

75
Q

Why are Fadeev-Popov ghosts introduced to the QCD lagrangian?

A

These grassman-valued complex fields are introduced to restore LOCAL SU(N) symmetry to the QCD lagrangian after the gauge-fixing term is added.

i.e: invariant under the BRS transformation, which ensures unitarity and renormalisability of YM gauge theories.

76
Q

How do we define renormalisations/”counter-terms” for wavefunctions?

What about general parameters?

A

Bare wavefunction = Z^1/2 * renormalised wavefunction = renormalised wavefunction + delta(renormalised wavefunction) {CT}

Bare parameter = Z * renormalised parameter = renormalised parameter + delta(renormalised parameter) {CT}

77
Q

What are the on-shell renormalisation conditions?
(used for self-energy)

A

The effective action{2} = 0, when p^2 = m^2.

1/i * d{effective action}/d{p^2} = 1, evaluated at p^2 = m^2.

78
Q

What does a Wick rotation correspond to?

A

Using a sophisticated contour integral to rotate the integral over the real axis to an integral over the complex axis.

k^o –> i k_E

79
Q

What is a UV-cutoff and why is it applied?

What are some drawbacks to this approach?

A

Loop integrals are divergent due to their integrating over infinite limits.

Applying a UV-cutoff shortens these limits with some parameter Lambda.

Applying a UV cutoff explicitly breaks gauge (and other) symmetries.

80
Q

What is Dimensional Regularisation?

A

After Wick rotating, transform from a 4D space to a (4 - 2epsilon)D space. Where epsilon is a small positive factor. Apply a “t’ Hooft mass” to preserve dimensionality.

81
Q

What is the relationship between UV-cutoff and Dimensional Regularisation?

Specifically, what is the equivalent of:
Lambda^2 :
ln(Lambda^2 / m^2) :
in DR.

A

Lambda^2 –> 0 + O(epsilon)

ln(Lambda^2 / m^2) –> 1/epsilon + 1 -gamma_E + ln(4pi) + ln(mu^2 / m^2)

*probably don’t worry about this too much

82
Q

What is Weinberg’s theorem?

A

To all orders in perturbation theory (loop expansion), the renormalised 1PI effective action (Gamma) does not depend on the renormalisation scheme:
-independent of Lambda in UV-cutoff.
-independent of mu in DR.

*bare quantities are dependent on epsilon only, whereas normalised quantities are dependent on mu only.

83
Q

What are the renormalisation group equations (RGE’s)?

A

*check notes, not sure if required to memorise.

Equations defining dimensionless parameters that describe the mu-dependence of renormalised quantities. Here mu is referred to as the RG-scale.

84
Q

How is the ratio R defined in renormalisation group theory?

A

the ratio of the wavefunction between two values of the RG-scale mu.

*can be shown to be equal to an integral - check notes

Using Weinberg’s theorem, these ratios can be shown to form a group: the Renormalisation Group.

85
Q

When considering the running of kinematic parameters, what does the factor t refer to?

How does t relate to the IR and UV limits?

A

t = ln(mu/mu_0)

in the IR limit (mu/mu_0 –> 0), t –> -inf

in the UV limit (mu/mu_0 –> +inf), t –> +inf

86
Q

How does the function beta from the RGE’s tell us about the behaviour of a coupling in the UV or IR limits?

A

The behaviour is dependent on the sign of the first derivative of beta w.r.t. the coupling, evaluated at the roots.

If this parameter is > 0, this corresponds to an IR stable fixed point.

If this parameter is < 0, this corresponds to a UV stable fixed point.

87
Q

What is the behaviour of a pure YM theory in the UV/IR limits?

I.e: what does asymptotic freedom mean?

A

Pure YM theories such as QCD are ASYMPTOTICALLY FREE: at very high energy scales (UV) the coupling –> 0.

In the IR limit, (low energy scales), the coupling gets very large and the theory becomes non-perturbative. This is referred to as the CONFINEMENT SCALE.

88
Q

What is the confinement scale, how would we go about finding an estimate for this?

A

The scale at which the coupling diverges to infinity.

Use the RGE for the coupling to get an integral in the coupling = an integral in t. The limit of the integral in the coupling will be +inf. Use this to work out the t value for the confinement scale, relative to some measured value.

This can then be inverted to get the confinement scale.

89
Q

Relate the derivative appearing in the RGE beta factor for a coupling to the derivative of that coupling w.r.t. t.

A

*check notes

90
Q

What does a “free” lagrangian correspond to?

A

The coupling –> 0.

91
Q

What are the three dirac matrices?

What is the law describing their products?

A

*Check wall notes

92
Q

What is the matrix commutator [AB, C]?

A

A[B,C] + [A,C]B

93
Q

When considering a ground state of a lagrangian, what are the vacuum equations?

Where do these come from?

A

We are looking for solutions where phi = const.
i.e: we can neglect the first term in the lagrangian.
The second term will only involve the potential since the kinetic term involves only derivatives of the field.

Therefore the vacuum equations are given by d(V)/d(phi) = 0.

94
Q

For a SO(2) model, what do we require in order to have spontaneous symmetry breaking?

A

m^2 < 0

in order to have a non-zero ground state for the field(s).

95
Q

How can we work out the mass spectrum for a SSB theory?

A

Consider one possible vacuum solution and expand to first order (creating reparametrised fields).

The mass terms for these fields (quadratic term) can be found and the mass inferred.

*be careful about factors of 1/2.

96
Q

What is the Goldstone theorem?

A

If a theory possesses a global symmetry group G, which spontaneously breaks to a smaller group H c G, there exists a massless goldstone boson for each BROKEN GENERATOR X of G.

The broken generators form a vacuum manifold given by coset space M = G/H.

*only applicable to global symmetries, and only to theories with more than (1+1)D

97
Q

Why does the Goldstone theorem not apply to locally gauge symmetric theories?

A

The goldstone boson in this case can be gauged away via the Higgs Mechanism, and so removed from the physical spectrum.

98
Q

What are the key steps in the Goldstone theorem proof?

A

Global symmetry –> kinetic term invariant –> potential term must be invariant –> d(V)/d(Phi) = 0 –> key equation (*check notes).

Expand around a vacuum solution v, (there will be no linear term in the new field phi due to the vacuum eqn. This leads to the definition of a mass matrix squared. (*check notes)

Combining these two key equations leads to a final equation. The interpretation of this leads to two solutions: the broken generators corresponding the massless Goldstone bosons and the unbroken generators.

99
Q

In the U(1)_Y model with SSB, what are the two choices of expansion about the vacuum solution?

Which do we use?

A

Linear:
1/sqrt(2) [v + H + iG]

Non-linear:
1/sqrt(2) [v + H’]e^{i G’ / v}

Use the non-linear as it becomes very clear how we can choose the unitary U(1) gauge to eliminate the Goldstone boson.

*in the SM we are using a SU(2)_L model for the higgs, so there will be extra term in the SU(2) generator accompanying G’.

100
Q

How does the Higgs mechanism give mass to gauge bosons?

Where does the Higgs boson mass come from?

A

The term in the ground state is quadratic in the gauge field when the kinetic term is expanded.

In the unitary gauge, the goldstone boson term does not end up cancelling this term, so the gauge field mass term survives.

The Higgs boson is the oscillations (H) from the VEV of the scalar (higgs) field, expanding the kinetic term, we see there is also a quadratic term in H.

101
Q

What does the term unitary gauge refer to?

A

The gauge chosen such that the goldstone boson(s) are absorbed.

102
Q

What is the scalar-kinetic lagrangian term?
What about the potential for the scalar field?

A

*check wall notes

103
Q

What is the symmetry group of the SM?

How is it’s symmetry spontaneously broken?

A

G = SU(3)_C * SU(2)_L * U(1)_Y

–> SU(3)_C * U(1)_em = H

104
Q

What symmetry group is the scalar field expressed in?

What is the vacuum solution, and expansion of this field doublet in the unitary gauge?

A

SU(2)_L

Vacuum solution : 1/sqrt(2) [0 , v]

Expansion (UG) : 1/sqrt(2) [0 , v + H]

105
Q

How can the U(1)_em generator be expressed in terms of the unbroken generators?

A

Q = T^3 + Y

106
Q

What is the covariant derivative in the scalar-kinetic term (EW sector of the SM)?

A

*check notes

107
Q

Which generators correspond to the W bosons?

A

The off-diagonal generators of SU(2)_L

*the mass eigenstates are in fact a superposition of the gauge fields.

108
Q

How are the W boson mass eigenstates defined in terms of the off-diagonal SU(2)_L gauge fields?

A

*check notes

109
Q

Which generators correspond to the photon and Z bosons?

A

The diagonal generators (third component of SU(2)_L and the U(1)_Y component)

*the eigenstates are a superposition of the relevant fields.

110
Q

What are the left and right chiral projection matrices for dirac fermions in the chiral representation?

A

1/2 * (I_4 -/+ gamma_5)

gamma_5 = *check notes

111
Q

Which fermion fields are not singlets in SU(2)_L basis?

Which fermion fields are not singlets in SU(3)_C

A

Only LH fields are doublets in SU(2)_L : the LH lepton and quark doubletes.

Only quarks are triplet states of colour in SU(3)_C.

112
Q

What are the hypercharges of the different fermion fields in the SM?

A

LH leptons : -1
RH massive leptons : -2
{RH neutrinos : 0}?

LH quarks : 1/3
RH up quarks : 4/3
RH down quarks : -2/3

113
Q

What is the gauge-kinetic lagrangian for SM fermions?

A

*check wall notes

Don’t forget separate LH and RH terms.

114
Q

How are the fields for Z bosons and photons related to the gauge fields corresponding to generators?

A

*check wall notes

115
Q

What is the key result of the Goldstone theorem in equation form?

A

*check notes

116
Q

What must be applied when working out the feynman rule for a fermion loop?

A
  • (-1) factor

Take the trace (acting on gamma matrices only) - due to the dirac delta’s so remove these.

117
Q

What is a “broken” generator?

A

In SSB, a broken generator is one that does not leave the ground state invariant.

Therefore the broken generator acting on the ground state vector is non-zero.

EXAMS Summer 2023 (6 decks)

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