Flashcards in Gauge Theory, Non-Abelian Gauge Theory and the Weak Interaction Deck (36)

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1

##
Global Transformation

Definition

### -a transformation that is the same everywhere in spacetime

2

## How do we show that the Dirac equation is invariant under a global transformation?

###
-the most general transformation that can be applied to the spinor in the Dirac equation is:

Ψ -> e^(iα)Ψ

-where α is a constant

-sub in and show invariance

3

## Symmetry Group

###
-a set of transformations under which an equation remains invariant

-several such transformations in sequence still retains invariance

4

##
Symmetry of the Dirac Equation

The U(1) Group

###
-the Dirace equation is invariant under the transformation:

Ψ -> e^(iα)Ψ

-here α can be any real number and the exponential can be any complex number of modulus 1

-several such transformations in sequence still retains invariance

-since this group has a continuous parameter labelling its elements, in addition to a group structure it also has the structure of a smooth manifold, a Lie group

-in particular, since the transformations are unitary and only have one degree of freedom, this group is known as the U(1) group

5

##
Symmetry of the Dirac Equation

Global Symmetry

###
-since α is constant, we are applying the same transformation to Ψ at all spacetime points, the transformation is global

-we can say that the Dirac equation has global U(1) symmetry

6

##
Local Transformation

Definition

###
-a transformation is local if it is a function of spacetime coordinates

-i.e. if the transformation applied is different at different points in spacetime

7

## Does the Dirac equation have local U(1) symmetry?

###
-apply transformation:

Ψ -> e^(iα(x))Ψ

-find that the U(1) symmetry in the Dirac equation IS NOT local

8

## Can the Dirac equation be modified to accommodate local U(1) symmetry?

###
-the reason that the Dirac equation is not locally U(1) symmetric is that the derivative ∂μΨ does not transform the same way as Ψ itself under a U(1) transformation

-if we replace this derivative with a derivative that does transform the same way, a 'gauge-covariant' derivative

-then we do have local U(1) symmetry is then:

Aμ' = Aμ - 1/g ∂μ α

9

## What are the conditions for local U(1) symmetry of the Dirac equation?

###
-once the derivative has been switched for a gauge-covariant derivative, we do have local U(1) symmetry as long as:

Aμ' = Aμ - 1/g ∂μ α

-that is, we can accommodate local U(1) symmetry in the Dirac equation as long as the fermion described couples to some other vector quantity with gauge symmetry

-e.g. coupled to photon

10

##
Gauge-Covariant Derivative

Definition

###
Dμ = ∂μ + igAμ(x)

-here Aμ(x) is some vector and g is a constant

-we require:

DμΨ -> e^(iα(x))DμΨ

when

Ψ -> e^(iα(x))Ψ

11

## How can we produce gauge-covariant quantities?

###
-acting twice with a gauge-covariant derivative still produces something gauge-covariant:

DμDνΨ -> e^(iα(x))DμDνΨ

-any linear combination of such terms is also gauge-covariant, in particular the commutator [Dμ,Dν]

12

## Is the field-strength tensor gauge-covariant?

###
-yes, it can be shown that:

[Dμ,Dν] = igFμν

-this in turn means that the Maxwell equation can be included as an equation of motion for Aμ without breaking U(1) symmetry

13

## Does Aμ have mass?

###
-from the Proca equation we know that massive vector particles do not have gauge symmetry

-so Aμ is a massless vector field that is invariant under gauge transformations and obeys the Maxwell equation

-a photon matches this description, so we can accommodate local U(1) symmetry into the Dirac equation if and only if the fermion we wish to describe is coupled to an electromagnetic field

14

## Conserved Current for a Dirac Particle

### j^μ = Ψ_ γμ Ψ

15

## Write down a fully local U(1) symmetric theory of a fermion and a vector particle

###
(iD/ - m)Ψ = 0

∂μ F^μν = qj^ν = q Ψ_ γμ Ψ

-these are (once again) the equations for quantum electrodynamics, this time arrived at using only the requirement of local U(1) symmetry

16

## Global Colour Symmetry

###
-if we were to replace all the red quarks with blue and vice versa, there would be no effect on the universe

-there is a global symmetry in how the colours are assigned to quarks

-we can take this further, replacing all blue quarks with a superposition as long as a corresponding transformation is made to the other colours to retain orthonormality

-in fact we can transform the 'vector' of quark colours (r,b,g) with some 3x3 matric M as long as it retains orthonormality, i.e. as long as M is unitary

-if we also have det(M)=1, these transformations form a group, SU(3)

17

##
Unitary Matrix

Definition

### M† = M^(-1)

18

## How many degrees of freedom are there in choosing M?

###
-3x3 matrix with complex entries => 18 real numbers

-BUT

-unitary => 9 entries fixed

-det(M)=1 => 1 entry fixed

=> 8 degrees of freedom

19

## Infinitessimal Transformations

###
-since the 8 degrees of freedom for M are continuous parameters αi, we can find transformatino matrices that are only infinitesimally removed from the identity

-in fact, we can find 8 such independent matrices, Ti

-a general infinitesimal transformation can then be constructed as a linear combination of these:

𝛿M = 𝛿 αi Ti

20

##
Constructing Matrices from the Infinitesimal Transformations

Description

###
-to construct any matrix in the group, choose a 'direction' in the space group of transformations and a 'distance' from the identity

-we can construct a finite transformation from repeated application of infinitesimal transformations

21

##
Constructing Matrices from the Infinitesimal Transformations

M(α1)

###
-assuming αi=0 for i=2,...,8 we can see the transformation M(α1) can be written:

M = lim ( 1 + α1/N)^N

= e^(α1T1)

(where the limit is taken as N->∞)

-since each factor represents an infinitesimal change in the T1 direction

-this limit gives the exponential function

22

##
Constructing Matrices from the Infinitesimal Transformations

any SU(3) matrix

###
-any member of SU(3)can be written as:

M = e^(i αiTi)

-where the repeated indices indicate summation

-the 8 independent infinitesimal transformation matrices are called generators for the group, and must by linearly independent, Hermitian and traceless

23

## Generators and Group Structure

###
-since the entire symmetry group is generated by the generator matrices, most of the structure of the group can be determined by looking at how they interact with each other

-looking at the commutators of the generators we define a set of structure constants, fijk, for the group such that:

[Ti,Tj] = 2i fijk Tk

24

##
Albeian and Non-Albeian

Definition

###
-if the structure constant of a group are non-zero, the group elements will not generally commute with each other, in this case the group is said to be non-Abelian

-Abelian groups are, by definition, groups in which group operations commute

25

## Choice of Generators

###
-as long as the matrices meet the criteria ( linear independence, Hermitian, traceless) they may be chosen arbitrarily

-each choice of generators is a representation e.g. the Gell-Mann matrices can be used to represent the generators of the SU(3) group

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## Pauli Matrices and SU(2)

###
-the Pauli matrices are possible set of generators for the SU(2) group

-restricting only to T1, T2 and T3 of the Gell-Mann matrices would give SU(2) transformations in the first two components of the vector

27

## Dirac equation for non-interacting quarks?

###
-if quarks did not interact, we could write down the Dirac equation for each colour as:

(i∂/ - m)𝜱 = 0

-where 𝜱 is a 3x1 vector with entries ψr, ψg, ψb

-this is invariant under transformation:

𝜱 -> m𝜱 = e^(iαiTi)𝜱

-hence we have invariance under global SU(3) symmetry

-but NOT local SU(3) symmetry

28

## Introducing local SU(3) symmetry to the Dirac equation for non-interacting quarks

###
-introduce a covariant derivative Dμ such that:

Dμ 𝜱 -> e^(iαiTi) Dμ 𝜱

-so let:

Dμ = ∂μ + i gs Aμ^i Ti

-where gs is the strong force coupling constant and repeated i index indicates summation over i=1,...,8

-construct the commutator [Dμ,Dν]

-for the earlier case of the photon, [Aμ,Aν] vanished, but in this case it is non-zero and we have to define Gμν^i

-we now have a theory of 3 quark colours that interact with 8 independent massless vector particles with gauge invariance, these are the gluons and we have produced the strong interaction

29

## Equation of Motion for Gluons

###
Dμ G^iμν = j^νi

-here j^νi is the source term or current for the ith gluon just as the electric current j^μ was the source for photons in the EM case

-and

G^iμν = ∂μAν^i - ∂νAμ^i - 2*gs*f^ijk*Aμ^j*Aν^k

-this equation of motion for gluons tells us that gluons interact among themselves

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