Particle Physics lecture 11 Gauge theories in particle physics and the standard model Yazid Delenda D´ epartement des Sciences de la mati` ere Facult´ e des Sciences - UHLB http://delenda.wordpress.com/teaching/particlephysics/ Batna, 04 January 2015 (http://delenda.wordpress.com) Particle Physics - lecture 11 1 / 35
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Particle Physicslecture 11
Gauge theories in particle physics and thestandard model
Yazid Delenda
Departement des Sciences de la matiereFaculte des Sciences - UHLB
where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:
∂L∂ψ
= (iγµ∂µ −m)ψ,
∂L∂(∂µψ)
= 0. (2)
The Euler-Lagrange equation gives the Dirac equation:
(iγµ∂µ −m)ψ = 0. (3)
so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.
where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:
∂L∂ψ
= (iγµ∂µ −m)ψ,
∂L∂(∂µψ)
= 0. (2)
The Euler-Lagrange equation gives the Dirac equation:
(iγµ∂µ −m)ψ = 0. (3)
so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.
where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:
∂L∂ψ
= (iγµ∂µ −m)ψ,
∂L∂(∂µψ)
= 0. (2)
The Euler-Lagrange equation gives the Dirac equation:
(iγµ∂µ −m)ψ = 0. (3)
so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.
where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:
∂L∂ψ
= (iγµ∂µ −m)ψ,
∂L∂(∂µψ)
= 0. (2)
The Euler-Lagrange equation gives the Dirac equation:
(iγµ∂µ −m)ψ = 0. (3)
so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.
where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:
∂L∂ψ
= (iγµ∂µ −m)ψ,
∂L∂(∂µψ)
= 0. (2)
The Euler-Lagrange equation gives the Dirac equation:
(iγµ∂µ −m)ψ = 0. (3)
so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.
where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:
∂L∂ψ
= (iγµ∂µ −m)ψ,
∂L∂(∂µψ)
= 0. (2)
The Euler-Lagrange equation gives the Dirac equation:
(iγµ∂µ −m)ψ = 0. (3)
so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.
In order to make the Lagrangian invariant under such a transformation weshall need to introduce a new field Aµ which, under a certaintransformation, is capable of absorbing the extra term.We do this bymaking the replacement ∂µ → Dµ, such that Dµ transforms like:
Dµψ → Dµ′ψ′ = eieΩ(x)Dµψ, (7)
then:ψγµD
µψ → ψ′γµD
µ′ψ′ = ψγµDµψ. (8)
and this way the Dirac Lagrangian remains invariant under local phasetransformations.
In order to make the Lagrangian invariant under such a transformation weshall need to introduce a new field Aµ which, under a certaintransformation, is capable of absorbing the extra term.We do this bymaking the replacement ∂µ → Dµ, such that Dµ transforms like:
Dµψ → Dµ′ψ′ = eieΩ(x)Dµψ, (7)
then:ψγµD
µψ → ψ′γµD
µ′ψ′ = ψγµDµψ. (8)
and this way the Dirac Lagrangian remains invariant under local phasetransformations.
In order to make the Lagrangian invariant under such a transformation weshall need to introduce a new field Aµ which, under a certaintransformation, is capable of absorbing the extra term.We do this bymaking the replacement ∂µ → Dµ, such that Dµ transforms like:
Dµψ → Dµ′ψ′ = eieΩ(x)Dµψ, (7)
then:ψγµD
µψ → ψ′γµD
µ′ψ′ = ψγµDµψ. (8)
and this way the Dirac Lagrangian remains invariant under local phasetransformations.
In order to make the Lagrangian invariant under such a transformation weshall need to introduce a new field Aµ which, under a certaintransformation, is capable of absorbing the extra term.We do this bymaking the replacement ∂µ → Dµ, such that Dµ transforms like:
Dµψ → Dµ′ψ′ = eieΩ(x)Dµψ, (7)
then:ψγµD
µψ → ψ′γµD
µ′ψ′ = ψγµDµψ. (8)
and this way the Dirac Lagrangian remains invariant under local phasetransformations.
So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′
(indicated above) and Aµ → Aµ′
(which is yet to be determined to be ableto absorb the extra term) we have:
So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′
(indicated above) and Aµ → Aµ′
(which is yet to be determined to be ableto absorb the extra term) we have:
So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′
(indicated above) and Aµ → Aµ′
(which is yet to be determined to be ableto absorb the extra term) we have:
So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′
(indicated above) and Aµ → Aµ′
(which is yet to be determined to be ableto absorb the extra term) we have:
So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′
(indicated above) and Aµ → Aµ′
(which is yet to be determined to be ableto absorb the extra term) we have:
So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′
(indicated above) and Aµ → Aµ′
(which is yet to be determined to be ableto absorb the extra term) we have:
Hence we choose the transformation of Aµ as follows:
Aµ → A′µ = Aµ + ∂µΩ(x) (12)
We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):
Hence we choose the transformation of Aµ as follows:
Aµ → A′µ = Aµ + ∂µΩ(x) (12)
We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):
Hence we choose the transformation of Aµ as follows:
Aµ → A′µ = Aµ + ∂µΩ(x) (12)
We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):
Hence we choose the transformation of Aµ as follows:
Aµ → A′µ = Aµ + ∂µΩ(x) (12)
We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):
Hence we choose the transformation of Aµ as follows:
Aµ → A′µ = Aµ + ∂µΩ(x) (12)
We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):
Hence we choose the transformation of Aµ as follows:
Aµ → A′µ = Aµ + ∂µΩ(x) (12)
We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):
We can interpret −eψγνψ ≡ jν as a current density.So:
∂µFµν = jν (18)
which is equivalent to:
~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E
∂t, (19)
where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.
We can interpret −eψγνψ ≡ jν as a current density.So:
∂µFµν = jν (18)
which is equivalent to:
~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E
∂t, (19)
where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.
We can interpret −eψγνψ ≡ jν as a current density.So:
∂µFµν = jν (18)
which is equivalent to:
~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E
∂t, (19)
where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.
We can interpret −eψγνψ ≡ jν as a current density.So:
∂µFµν = jν (18)
which is equivalent to:
~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E
∂t, (19)
where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.
We can interpret −eψγνψ ≡ jν as a current density.So:
∂µFµν = jν (18)
which is equivalent to:
~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E
∂t, (19)
where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.
The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).
The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).
The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).
The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).
The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).
The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).
The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).
The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).
A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .
G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c
G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a
G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e
An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a
A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .
G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c
G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a
G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e
An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a
A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .
G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c
G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a
G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e
An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a
A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .
G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c
G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a
G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e
An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a
A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .
G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c
G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a
G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e
An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:
D(g1.g2) = D(g1).D(g2)
Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:
D(g1.g2) = D(g1).D(g2)
Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:
D(g1.g2) = D(g1).D(g2)
Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:
D(g1.g2) = D(g1).D(g2)
Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:
D(g1.g2) = D(g1).D(g2)
Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:
D(g1.g2) = D(g1).D(g2)
Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Alternatively given the generator of the group X we can write:
R(φ) = e−iXφ
which means that all elements of the group can be built from thegenerator.Alternatively:
−iX =dR
dφ
∣∣∣∣φ=0
where we note that X is hermitian (because R is unitary) and thatdetR = 1⇒ Tr(X) = 0.A more interesting group is SO(3) which describes rotations in 3D spaceand ultimately angular momentum.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Alternatively given the generator of the group X we can write:
R(φ) = e−iXφ
which means that all elements of the group can be built from thegenerator.Alternatively:
−iX =dR
dφ
∣∣∣∣φ=0
where we note that X is hermitian (because R is unitary) and thatdetR = 1⇒ Tr(X) = 0.A more interesting group is SO(3) which describes rotations in 3D spaceand ultimately angular momentum.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Alternatively given the generator of the group X we can write:
R(φ) = e−iXφ
which means that all elements of the group can be built from thegenerator.Alternatively:
−iX =dR
dφ
∣∣∣∣φ=0
where we note that X is hermitian (because R is unitary) and thatdetR = 1⇒ Tr(X) = 0.A more interesting group is SO(3) which describes rotations in 3D spaceand ultimately angular momentum.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
Alternatively given the generator of the group X we can write:
R(φ) = e−iXφ
which means that all elements of the group can be built from thegenerator.Alternatively:
−iX =dR
dφ
∣∣∣∣φ=0
where we note that X is hermitian (because R is unitary) and thatdetR = 1⇒ Tr(X) = 0.A more interesting group is SO(3) which describes rotations in 3D spaceand ultimately angular momentum.
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
In fact SO(3) generators satisfy commutation relations for angularmomentum in quantum mechanics:
[Xi, Xj ] = iεijkXj
The same algebra is also satisfied by 2× 2 hermitian matrices Xi = σi/2,with σi the Pauli matrices,which act on 2-component spinors,e.g theelectron spin-up and spin down-states :
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
In fact SO(3) generators satisfy commutation relations for angularmomentum in quantum mechanics:
[Xi, Xj ] = iεijkXj
The same algebra is also satisfied by 2× 2 hermitian matrices Xi = σi/2,with σi the Pauli matrices,which act on 2-component spinors,e.g theelectron spin-up and spin down-states :
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
In fact SO(3) generators satisfy commutation relations for angularmomentum in quantum mechanics:
[Xi, Xj ] = iεijkXj
The same algebra is also satisfied by 2× 2 hermitian matrices Xi = σi/2,with σi the Pauli matrices,which act on 2-component spinors,e.g theelectron spin-up and spin down-states :
Non-Abelian gauge invariance Group representations and continuous or Lie groups
Non-Abelian gauge invarianceGroup representations and continuous or Lie groups
In fact SO(3) generators satisfy commutation relations for angularmomentum in quantum mechanics:
[Xi, Xj ] = iεijkXj
The same algebra is also satisfied by 2× 2 hermitian matrices Xi = σi/2,with σi the Pauli matrices,which act on 2-component spinors,e.g theelectron spin-up and spin down-states :
SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:
U = eiωata
where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).
SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:
U = eiωata
where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).
SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:
U = eiωata
where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).
SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:
U = eiωata
where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).
SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:
U = eiωata
where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).
SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:
U = eiωata
where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).
SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:
U = eiωata
where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).
SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:
U = eiωata
where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).
SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:
U = eiωata
where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).
where fabc are the structure constants of the group.The generators of thegroup are also normalised as:
tr[tatb] = TF δab
with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:
L(x) =∑
f
qif (x)(iγµ∂µ −mf )qjf (x)δij
where the sum runs over flavours and i and j are colour indices which runover colours.
where fabc are the structure constants of the group.The generators of thegroup are also normalised as:
tr[tatb] = TF δab
with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:
L(x) =∑
f
qif (x)(iγµ∂µ −mf )qjf (x)δij
where the sum runs over flavours and i and j are colour indices which runover colours.
where fabc are the structure constants of the group.The generators of thegroup are also normalised as:
tr[tatb] = TF δab
with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:
L(x) =∑
f
qif (x)(iγµ∂µ −mf )qjf (x)δij
where the sum runs over flavours and i and j are colour indices which runover colours.
where fabc are the structure constants of the group.The generators of thegroup are also normalised as:
tr[tatb] = TF δab
with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:
L(x) =∑
f
qif (x)(iγµ∂µ −mf )qjf (x)δij
where the sum runs over flavours and i and j are colour indices which runover colours.
where fabc are the structure constants of the group.The generators of thegroup are also normalised as:
tr[tatb] = TF δab
with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:
L(x) =∑
f
qif (x)(iγµ∂µ −mf )qjf (x)δij
where the sum runs over flavours and i and j are colour indices which runover colours.
where fabc are the structure constants of the group.The generators of thegroup are also normalised as:
tr[tatb] = TF δab
with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:
L(x) =∑
f
qif (x)(iγµ∂µ −mf )qjf (x)δij
where the sum runs over flavours and i and j are colour indices which runover colours.
where fabc are the structure constants of the group.The generators of thegroup are also normalised as:
tr[tatb] = TF δab
with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:
L(x) =∑
f
qif (x)(iγµ∂µ −mf )qjf (x)δij
where the sum runs over flavours and i and j are colour indices which runover colours.
Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:
qi(x)→ Uijqj(x)
where U = eigωata and g and ωa are constants.Note that the sum over a is
in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:
Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:
qi(x)→ Uijqj(x)
where U = eigωata and g and ωa are constants.Note that the sum over a is
in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:
Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:
qi(x)→ Uijqj(x)
where U = eigωata and g and ωa are constants.Note that the sum over a is
in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:
Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:
qi(x)→ Uijqj(x)
where U = eigωata and g and ωa are constants.Note that the sum over a is
in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:
Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:
qi(x)→ Uijqj(x)
where U = eigωata and g and ωa are constants.Note that the sum over a is
in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:
Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:
qi(x)→ Uijqj(x)
where U = eigωata and g and ωa are constants.Note that the sum over a is
in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:
(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).
(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).
(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).
(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).
(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).
The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.
The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.
The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.
The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.
The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.
From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:
F aµν = ∂µAaν − ∂νAaµ
However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:
From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:
F aµν = ∂µAaν − ∂νAaµ
However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:
From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:
F aµν = ∂µAaν − ∂νAaµ
However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:
From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:
F aµν = ∂µAaν − ∂νAaµ
However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:
From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:
F aµν = ∂µAaν − ∂νAaµ
However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:
From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:
F aµν = ∂µAaν − ∂νAaµ
However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:
and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).
We can thus express the full QCD Lagrangian as:
LQCD =∑
f
qif (δijiγµ∂µ+gsγ
µAaµtaij−mfδij)q
jf−
1
4F aµνF
µν,a+Lgauge+Lghost
and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.
and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).
We can thus express the full QCD Lagrangian as:
LQCD =∑
f
qif (δijiγµ∂µ+gsγ
µAaµtaij−mfδij)q
jf−
1
4F aµνF
µν,a+Lgauge+Lghost
and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.
and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).
We can thus express the full QCD Lagrangian as:
LQCD =∑
f
qif (δijiγµ∂µ+gsγ
µAaµtaij−mfδij)q
jf−
1
4F aµνF
µν,a+Lgauge+Lghost
and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.
and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).
We can thus express the full QCD Lagrangian as:
LQCD =∑
f
qif (δijiγµ∂µ+gsγ
µAaµtaij−mfδij)q
jf−
1
4F aµνF
µν,a+Lgauge+Lghost
and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.
and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).
We can thus express the full QCD Lagrangian as:
LQCD =∑
f
qif (δijiγµ∂µ+gsγ
µAaµtaij−mfδij)q
jf−
1
4F aµνF
µν,a+Lgauge+Lghost
and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.
and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).
We can thus express the full QCD Lagrangian as:
LQCD =∑
f
qif (δijiγµ∂µ+gsγ
µAaµtaij−mfδij)q
jf−
1
4F aµνF
µν,a+Lgauge+Lghost
and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.
and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).
We can thus express the full QCD Lagrangian as:
LQCD =∑
f
qif (δijiγµ∂µ+gsγ
µAaµtaij−mfδij)q
jf−
1
4F aµνF
µν,a+Lgauge+Lghost
and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.
If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA
cν as well as AaµA
bνA
µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.
If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA
cν as well as AaµA
bνA
µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.
If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA
cν as well as AaµA
bνA
µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.
If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA
cν as well as AaµA
bνA
µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.
If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA
cν as well as AaµA
bνA
µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.
If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA
cν as well as AaµA
bνA
µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.
If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA
cν as well as AaµA
bνA
µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.
Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.
Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.
Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.
Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.
Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.
Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.
Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.
Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.
Finally we note that the generators satisfy the following algebra:
∑
a
taijtajk = δikCF
tr(T aT b) = CAδab
where CF = (N2c − 1)/(2Nc) = 4/3 and CA = Nc = 3 are the casimir
operators in the fundamental and adjoint representations respectively,theyrepresent the colour factors for the emission of a gluon from a quark, andof a gluon from a gluon.
Finally we note that the generators satisfy the following algebra:
∑
a
taijtajk = δikCF
tr(T aT b) = CAδab
where CF = (N2c − 1)/(2Nc) = 4/3 and CA = Nc = 3 are the casimir
operators in the fundamental and adjoint representations respectively,theyrepresent the colour factors for the emission of a gluon from a quark, andof a gluon from a gluon.
Finally we note that the generators satisfy the following algebra:
∑
a
taijtajk = δikCF
tr(T aT b) = CAδab
where CF = (N2c − 1)/(2Nc) = 4/3 and CA = Nc = 3 are the casimir
operators in the fundamental and adjoint representations respectively,theyrepresent the colour factors for the emission of a gluon from a quark, andof a gluon from a gluon.
Finally we note that the generators satisfy the following algebra:
∑
a
taijtajk = δikCF
tr(T aT b) = CAδab
where CF = (N2c − 1)/(2Nc) = 4/3 and CA = Nc = 3 are the casimir
operators in the fundamental and adjoint representations respectively,theyrepresent the colour factors for the emission of a gluon from a quark, andof a gluon from a gluon.