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Gauge Theory
for Beginners
Muhammad Hassaan Saleem
(PhD student, SUNY Albany, USA)
Electromagnetism
Example
๐ต = โˆ‡ ร— ๐ด
where ๐‘จ is vector potential and ๐‘ฉ is magnetic field.
Define another potential ๐ดโ€ฒ i.e.
๐ดโ€ฒ = ๐ด + ๐›๐›ผ (1)
where ๐›ผ is an arbitrary differentiable function.
โ‡’ ๐ตโ€ฒ = โˆ‡ ร— ๐ด + ๐›๐›ผ = โˆ‡ ร— ๐ด = ๐ต
So, ๐ดโ€ฒ and ๐ด are physically equivalent and thus, ๐ด is not
measurable (i.e. it is unphysical).
Transformation (1) is known as gauge transformation and
thus, ๐ด and ๐ดโ€ฒ are called gauge equivalent.
Gauge
Choices
So, โˆ‡. ๐ด doesnโ€™t matter. We can choose for value for it. These
choices are called gauge choices and this procedure is called
gauge fixing.
Examples
โˆ‡. ๐ด = 0 (Coulomb Gauge)
โˆ‡. ๐ด = โˆ’
1
๐‘2
๐œ•๐œ™
๐œ•๐‘ก
(Lorenz Gauge)
Quantum
Mechanics
Example
and local U(1)
gauge
symmetry.
Take a quantum particle of mass ๐‘š and charge ๐‘ž and let it
interact it with a background field (๐œ™, ๐ด). The Hamiltonian is;
๐ป =
1
2๐‘š
๐‘ โˆ’
๐‘ž๐ด
๐‘
2
+ ๐‘ž๐œ™
The Schrodinger equation is thus,
๐‘–โ„
๐œ•๐œ“ ๐‘ฅ, ๐‘ก
๐œ•๐‘ก
=
1
2๐‘š
โˆ’๐‘–โ„โˆ‡ โˆ’
๐‘ž๐ด
๐‘
2
๐œ“ ๐‘ฅ, ๐‘ก + ๐‘ž๐œ™ ๐œ“(๐‘ฅ, ๐‘ก)
and is invariant under following transformations
๐œ“ ๐‘ฅ, ๐‘ก โ†’ ๐‘’๐‘–๐œ† ๐‘ฅ,๐‘ก ๐œ“(๐‘ฅ, ๐‘ก)
๐ด โ†’ ๐ด โˆ’
โ„๐‘
๐‘ž
โˆ‡๐œ† ๐‘ฅ, ๐‘ก = ๐ด โˆ’
๐‘–โ„๐‘
๐‘ž
ei๐œ† ๐‘ฅ,๐‘ก โˆ‡ ei๐œ† ๐‘ฅ,๐‘ก
โ€ 
๐œ™ โ†’ ๐œ™ +
โ„
๐‘ž
๐œ•๐œ† ๐‘ฅ, ๐‘ก
๐œ•๐‘ก
= ๐œ™ +
โ„
๐‘ž
ei๐œ† ๐‘ฅ,๐‘ก
๐œ•
๐œ•๐‘ก
ei๐œ† ๐‘ฅ,๐‘ก โ€ 
๐‘’๐‘–๐œ† ๐‘ฅ,๐‘ก โˆˆ ๐‘ˆ(1) group. This is an example of local ๐‘ˆ(1) gauge
transformation.
Bring in QFT
QFT is a theory of quantized fields.
Different fields have different transformation properties and
correspond to different mathematical quantities.
Different fields have different spin particles.
Examples
For electrodynamics, we need photons and electrons i.e. spin
one and spin half particles. So, we use ๐ด๐œ‡ and ๐œ“ fields.
Field Spin
Scalars ๐œ™ 0
Spinors ๐œ“ 1/2
Vector ๐ด๐œ‡ 1
Second Rank Tensor ๐‘‡๐œ‡๐œˆ 2
Making of
QED
(Simplest, real
world gauge
theory)
Q. What is the free electron lagrangian (with mass ๐‘š and charge ๐‘’)?
Ans. It is the Dirac lagrangian i.e.
โ„’๐ท๐‘–๐‘Ÿ๐‘Ž๐‘ = ๐œ“ ๐‘–โ„๐›พ๐œ‡
๐œ•๐œ‡ โˆ’ ๐‘š ๐œ“
Q. How to get Maxwell lagrangian?
Ans. Write Maxwell equations in covariant form and then find the
lagrangian. It turns out to be
โ„’๐‘€๐‘Ž๐‘ฅ๐‘ค๐‘’๐‘™๐‘™ = โˆ’
1
4
๐น
๐œ‡๐œˆ๐น๐œ‡๐œˆ
where ๐น
๐œ‡๐œˆ = ๐œ•๐œ‡๐ด๐œˆ โˆ’ ๐œ•๐œˆ๐ด๐œ‡
๐‘’โˆ’, ๐›พ donโ€™t interact โ‡’ total lagrangian is just the sum.
Q. What is the interaction term?
Ans. It can only be one term i.e. โ„’๐ผ๐‘›๐‘ก = โˆ’๐‘’๐œ“๐›พ๐œ‡
๐ด๐œ‡๐œ“ (Lorentz invariant,
renormalizable).
Q. Why not ๐œ–๐›ผ๐›ฝ๐œ‡๐œˆ๐น๐œ‡๐œˆ
๐น๐›ผ๐›ฝ
?
Ans. CP violating, just a surface term.
So, QED lagrangian is
โ„’๐‘„๐ธ๐ท = โ„’๐ท๐‘–๐‘Ÿ๐‘Ž๐‘ + โ„’๐‘€๐‘Ž๐‘ฅ๐‘ค๐‘’๐‘™๐‘™ + โ„’๐ผ๐‘›๐‘ก
= ๐œ“ ๐‘–โ„๐›พ๐œ‡
๐œ•๐œ‡ โˆ’ ๐‘š ๐œ“ โˆ’
1
4
๐น
๐œ‡๐œˆ๐น๐œ‡๐œˆ
โˆ’ ๐‘’๐œ“๐›พ๐œ‡
๐ด๐œ‡๐œ“
Predictions
and Successes
of QED
Using Feynman rules for QED, we can do important calculations.
QED gives us certain insights e.g., corrections in Coulomb potential;
Conventional
Form of
QED
lagrangian
The QED lagrangian is
โ„’๐‘„๐ธ๐ท = ๐œ“ ๐‘–โ„๐›พ๐œ‡
๐œ•๐œ‡ โˆ’ ๐‘š ๐œ“ โˆ’
1
4
๐น
๐œ‡๐œˆ๐น๐œ‡๐œˆ
โˆ’ ๐‘’๐œ“๐›พ๐œ‡
๐ด๐œ‡๐œ“
It can be written as
โ„’๐‘„๐ธ๐ท = ๐œ“ ๐‘–โ„๐›พ๐œ‡(๐œ•๐œ‡ + ๐‘–๐‘’๐ด๐œ‡) โˆ’ ๐‘š ๐œ“ โˆ’
1
4
๐น
๐œ‡๐œˆ๐น๐œ‡๐œˆ
= ๐œ“ ๐‘–โ„๐›พ๐œ‡๐ท๐œ‡ โˆ’ ๐‘š ๐œ“ โˆ’
1
4
๐น
๐œ‡๐œˆ๐น๐œ‡๐œˆ
where ๐ท๐œ‡ = ๐œ•๐œ‡ + ๐‘–๐‘’๐ด๐œ‡ is called the covariant derivative.
So, The conventional form of the QED lagrangian is given as
โ„’๐‘„๐ธ๐ท = ๐œ“ ๐‘–โ„๐›พ๐œ‡๐ท๐œ‡ โˆ’ ๐‘š ๐œ“ โˆ’
1
4
๐น
๐œ‡๐œˆ๐น๐œ‡๐œˆ
Local
symmetry
approach to
QED
We can derive the form of QED lagrangian starting from Dirac
lagrangian alone and demanding ๐‘ˆ(1) local symmetry.
Dirac lagrangian is
โ„’๐ท๐‘–๐‘Ÿ๐‘Ž๐‘ = ๐œ“ ๐‘–โ„๐›พ๐œ‡๐œ•๐œ‡ โˆ’ ๐‘š ๐œ“
Demand symmetry under ๐‘ˆ(1) transformation
๐œ“ ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ(๐‘ฅ)
๐œ“
This renders
๐œ“๐œ“ โ†’ ๐œ“๐‘’โˆ’๐‘–๐›ผ(๐‘ฅ)๐‘’๐‘–๐›ผ ๐‘ฅ ๐œ“ = ๐œ“๐œ“
So, mass term is invariant.
The derivative term (also called kinetic term) transforms as
๐œ“๐œ•๐œ‡๐œ“ โ†’ ๐œ“๐‘’โˆ’๐‘–๐›ผ ๐‘ฅ ๐œ•๐œ‡ ๐‘’๐‘–๐›ผ ๐‘ฅ ๐œ“
= ๐œ“๐œ•๐œ‡๐œ“ + ๐‘–๐œ•๐œ‡๐›ผ ๐‘ฅ ๐œ“ ๐œ“ โ‰  ๐œ“๐œ•๐œ‡๐œ“
The kinetic term isnโ€™t invariant. WHY?
Exploring the
problem
Choose a direction in spacetime specified by a vector ๐œ‚๐œ‡
. The directional
derivative of ๐œ“(๐‘ฅ) in this direction is;
๐œ‚๐œ‡
๐œ•๐œ‡๐œ“ ๐‘ฅ = lim
๐œ–โ†’0
1
๐œ–
๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ ๐œ“ ๐‘ฅ
Under gauge transformation, we have;
๐œ‚๐œ‡
๐œ•๐œ‡๐œ“ ๐‘ฅ = lim
๐œ–โ†’0
1
๐œ–
๐‘’๐‘–๐›ผ ๐‘ฅ+๐œ–๐œ‚
๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ ๐‘’๐‘–๐›ผ ๐‘ฅ
๐œ“ ๐‘ฅ
Different exponentials cause the problem.
Invent a function ๐‘ˆ(๐‘ฆ, ๐‘ฅ) such that ๐‘ˆ ๐‘ฆ, ๐‘ฅ ๐œ“(๐‘ฅ) transforms like ๐œ“ ๐‘ฆ i.e.
๐‘ˆ ๐‘ฆ, ๐‘ฅ ๐œ“ ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ ๐‘ฆ ๐‘ˆ ๐‘ฆ, ๐‘ฅ ๐œ“(๐‘ฅ) โ‡’ ๐‘ˆ ๐‘ฆ, ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ ๐‘ฆ ๐‘ˆ(๐‘ฆ, ๐‘ฅ)๐‘’โˆ’๐‘–๐›ผ ๐‘ฅ
Moreover, it is plausible to set ๐‘ˆ ๐‘ฅ, ๐‘ฅ = 1. Now, let first argument differ
slightly from ๐‘ฅ and do a Taylorโ€™s expansion;
๐‘ˆ ๐‘ฅ + ๐œ–๐œ‚, ๐‘ฅ = 1 +
๐œ•๐‘ˆ ๐‘ฆ, ๐‘ฅ
๐œ•๐‘ฆ๐œ‡
๐‘ฆ=๐‘ฅ
๐œ–๐œ‚๐œ‡
+ ๐’ช ๐œ–2
Set
๐œ•๐‘ˆ ๐‘ฆ, ๐‘ฅ
๐œ•๐‘ฆ๐œ‡
๐‘ฆ=๐‘ฅ
= โˆ’๐‘–๐‘’๐ด๐œ‡ ๐‘ฅ โ‡’ ๐‘ˆ(๐‘ฅ + ๐œ–๐œ‚, ๐‘ฅ) = 1 โˆ’ ๐‘–๐‘’๐œ–๐œ‚๐œ‡
๐ด๐œ‡ ๐‘ฅ + ๐’ช ๐œ–2
Exploring the
problem
Define a new kind of derivative called ๐ท๐œ‡๐œ“ as
๐œ‚๐œ‡
๐ท๐œ‡๐œ“ ๐‘ฅ = lim
๐œ–โ†’0
1
๐œ–
๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ ๐‘ˆ ๐‘ฅ + ๐œ–๐œ‚, ๐‘ฅ ๐œ“ ๐‘ฅ
It turns out to be
๐œ‚๐œ‡
๐ท๐œ‡๐œ“ ๐‘ฅ = lim
๐œ–โ†’0
1
๐œ–
(๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ 1 โˆ’ ๐‘–๐‘’๐œ–๐œ‚๐œ‡
๐ด๐œ‡ ๐œ“(๐‘ฅ))
= lim
๐œ–โ†’0
1
๐œ–
๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ ๐œ“ ๐‘ฅ + ๐‘–๐‘’๐œ‚๐œ‡
๐ด๐œ‡๐œ“(๐‘ฅ)
= ๐œ‚๐œ‡
๐œ•๐œ‡๐œ“ ๐‘ฅ + ๐‘–๐‘’๐ด๐œ‡๐œ“ ๐‘ฅ โ‡’ ๐ท๐œ‡๐œ“ ๐‘ฅ = ๐œ•๐œ‡๐œ“ ๐‘ฅ + ๐‘–๐‘’๐ด๐œ‡๐œ“ ๐‘ฅ
We recover the same covariant derivative !!!
We check that ๐ท๐œ‡๐œ“ ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ ๐‘ฅ
๐ท๐œ‡๐œ“(๐‘ฅ) and thus, the following
lagrangian is invariant under local ๐‘ˆ(1) symmetry;
๐œ“ ๐‘–โ„๐›พ๐œ‡๐ท๐œ‡ โˆ’ ๐‘š ๐œ“
WHAT ABOUT MAXWELL LAGRANGIAN?
What about
Maxwellโ€™s
lagrangian?
First ask that how does ๐ด๐œ‡ transform?
We know that
๐‘ˆ ๐‘ฆ, ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ ๐‘ฆ
๐‘ˆ(๐‘ฆ, ๐‘ฅ)๐‘’โˆ’๐‘–๐›ผ ๐‘ฅ
Set ๐‘ฆ = ๐‘ฅ + ๐œ–๐œ‚ and we have;
1 โˆ’ ๐‘–๐‘’๐œ–๐œ‚๐œ‡๐ด๐œ‡(๐‘ฅ) โ†’ ๐‘’๐‘–๐›ผ ๐‘ฅ+๐œ–๐œ‚ 1 โˆ’ ๐‘–๐‘’๐œ–๐œ‚๐œ‡๐ด๐œ‡(๐‘ฅ) ๐‘’โˆ’๐‘–๐›ผ ๐‘ฅ
โ‡’ ๐ด๐œ‡ ๐‘ฅ โ†’ ๐ด๐œ‡(๐‘ฅ) โˆ’
๐‘–
๐‘’
๐œ•๐œ‡๐›ผ(๐‘ฅ)
We recover the same gauge transformation for ๐ด๐œ‡ as before!
๐น
๐œ‡๐œˆ is gauge invariant and thus Maxwell lagrangian can be
โˆ’
1
4
๐น
๐œ‡๐œˆ๐น๐œ‡๐œˆ
as it is Lorentz invariant and renormalizable.
Moral: QED can be deduced by demanding local ๐‘ˆ(1) gauge
symmetry.
Crash Course
on generators
and
parameters of
groups
๐‘ˆ(1) is the group is 1 ร— 1 matrices which are unitary (i.e. complex numbers). So, one
parameter needed to specify.
๐‘†๐‘ˆ ๐‘ is the group of complex matrices which are unitary and have determinant 1.
An ๐‘ ร— ๐‘ complex matrix needs 2๐‘2
parameters to be specified. However, ๐‘†๐‘ˆ(๐‘)
matrices need ๐‘2
โˆ’ 1 parameters to be specified (think why).
Every group has an identity element ๐•€.
Elements near identity can be specified as follows (for ๐‘†๐‘ˆ(๐‘) groups);
๐•€๐‘ โˆ’ ๐‘–
๐‘—=1
๐ท
๐›ผ๐‘—๐‘‡๐‘— = ๐•€๐‘ โˆ’ ๐‘– ๐›ผ๐‘—๐‘‡๐‘—
Where ๐ท is the number of parameters (which is ๐‘2
โˆ’ 1 in case of ๐‘†๐‘ˆ(๐‘)), ๐›ผ๐‘— are
parameters and ๐‘‡๐‘— are matrices called generators.
An arbitrary element of the ๐‘†๐‘ˆ(๐‘) group denoted as ๐’ช(๐›ผ1, โ€ฆ , ๐›ผ๐ท)can be written by
exponentiation
๐’ช ๐›ผ1, โ€ฆ , ๐›ผ๐ท = exp(๐‘– ๐›ผ๐‘—๐‘‡๐‘—)
The element of ๐‘ˆ(1) is ๐‘’๐‘–๐›ผ
and thus, the generator of ๐‘ˆ(1) is just 1.
The generators satisfy ๐‘‡๐‘Ž, ๐‘‡๐‘ = ๐‘–๐‘“๐‘Ž๐‘๐‘๐‘‡๐‘ where ๐‘“๐‘Ž๐‘๐‘ are called structure constants.
Example:
๐œŽ๐‘Ž
2
,
๐œŽ๐‘
2
= ๐‘–๐œ–๐‘Ž๐‘๐‘
๐œŽ๐‘
2
for ๐‘†๐‘ˆ 2 and hence, ๐‘“๐‘Ž๐‘๐‘ = ๐œ–๐‘Ž๐‘๐‘
Generalizing
QED
Let ๐œ“(๐‘ฅ) be a multiplet now (donโ€™t use the word vector) with ๐‘ components as follows;
๐œ“1
๐‘ฅ
โ‹ฎ
๐œ“๐‘
๐‘ฅ
So, we have an index now (๐œ“๐‘—
(๐‘ฅ)) called the gauge index.
Now, demand symmetry under the transformation
๐œ“๐‘—
๐‘ฅ โ†’ [exp(๐‘–๐›ผ๐‘˜
๐‘‡๐‘˜
)]๐‘—๐‘™
๐œ“๐‘™
(๐‘ฅ)
This is a non-Abelian gauge transformation (in this case, ๐‘†๐‘ˆ(๐‘)).
Process similar to QED gives;
๐‘ˆ ๐‘ฅ + ๐œ–๐œ‚, ๐‘ฅ = 1 + ๐‘–๐‘”๐œ–๐œ‚๐œ‡
๐ด๐œ‡
๐‘—
๐‘‡๐‘—
+ ๐’ช(๐œ–2
)
๐ท๐œ‡๐œ“๐‘—
= ๐•€ ๐œ•๐œ‡ โˆ’ ๐‘–๐‘”๐ด๐œ‡
๐‘˜
๐‘‡๐‘˜ ๐‘—๐‘™
๐œ“๐‘™
= ๐œ•๐œ‡๐œ“๐‘—
โˆ’ ๐‘–๐‘”๐ด๐œ‡
๐‘˜
๐‘‡๐‘˜
๐œ“
๐‘—
๐‘จ๐
๐’‹
๐‘ป๐’‹
โ†’ ๐ž๐ฑ๐ฉ ๐’Š๐œถ๐’
๐‘ป๐’
๐‘จ๐
๐’‹
๐‘ป๐’‹
+
๐’Š
๐’ˆ
๐๐ ๐ž๐ฑ๐ฉ ๐’Š๐œถ๐’‘
๐‘ป๐’‘ โ€ 
๐น
๐œ‡๐œˆ
๐‘—
๐‘‡๐‘—
= ๐œ•๐œ‡๐ด๐œˆ
๐‘—
๐‘‡๐‘—
โˆ’ ๐œ•๐œˆ ๐ด๐œ‡
๐‘™
๐‘‡๐‘™
+ ๐‘”๐‘“๐‘–๐‘—๐‘˜
๐ด๐œ‡
๐‘–
๐ด๐œˆ
๐‘—
๐‘‡๐‘˜
Not gauge inv.
Gauge Invariant quantity is
โˆ’
1
2
tr F๐œ‡๐œˆ
j
๐‘‡๐‘—
2
= โˆ’
1
4
F๐œ‡๐œˆ
j 2
The final lagrangian is (called Yang Mills lagrangian)
โ„’๐‘Œ๐‘€ = โˆ’
1
4
๐น
๐œ‡๐œˆ
๐‘— 2
+ ๐œ“๐‘—
๐‘–๐›พ๐œ‡
๐ท๐œ‡ โˆ’ ๐•€ ๐‘š
๐‘—๐‘˜
๐œ“๐‘˜
= โˆ’
1
4
๐น
๐œ‡๐œˆ
๐‘— 2
+ ๐œ“ ๐‘–๐›พ๐œ‡
๐ท๐œ‡ โˆ’ ๐‘š ๐œ“
Peculiarity of
Non-Abelian
gauge theories
1) The extra terms in ๐น
๐œ‡๐œˆ
๐‘—
๐‘‡๐‘— i.e. ๐‘”๐‘“๐‘–๐‘—๐‘˜๐ด๐œ‡
๐‘– ๐ด๐œˆ
๐‘—
๐‘‡๐‘˜ give extra vertices in
Feynman diagrams i.e.
i.e. it allows the force carriers to interact with one another.
2) Non Abelian ๐‘†๐‘ˆ(๐‘) gauge theories are asymptotically free (๐›ฝ < 0)
๐›ฝ ๐‘” = โˆ’
๐‘”3
4๐œ‹ 2
11
3
๐ถ2 ๐บ โˆ’
4
3
๐‘›๐‘“๐ถ ๐‘Ÿ
No other renormalizable theory in four spacetime dimension is
asymptotically free (proved by Coleman and Gross Phys. Rev. Lett. 31,
851 (1973)).
Some famous
Non-Abelian
gauge theories
1) Quantum Chromodynamics
๏ƒ˜ ๐‘†๐‘ˆ(3) gauge group.
๏ƒ˜ The gauge index can be ๐‘– = 1,2,3 โ‡’ three colors.
2) Electroweak Theory
๏ƒ˜ ๐‘†๐‘ˆ 2 ๐ฟ ร— ๐‘ˆ 1 ๐‘Œ gauge group.
๏ƒ˜ It is a chiral gauge theory (violates parity).
3) Standard Model
๏ƒ˜ ๐‘†๐‘ˆ 3 ร— ๐‘†๐‘ˆ 2 ๐ฟ ร— ๐‘ˆ 1 ๐‘Œ gauge group.
4) Glashow Georgi Model (Candidate GUT)
๏ƒ˜ ๐‘†๐‘ˆ(5) gauge group.
๏ƒ˜ Contains ๐‘‹, ๐‘Œ bosons with mass โˆผ 1016 GeV
๏ƒ˜ Implies proton decay.
Super Yang
Mills theories
โ€ข Yang Mills theories in the presence of supersymmetry are
called Super Yang Mills theories.
โ€ข Supersymmetry is NOT a local gauge symmetry. It is an
extension of spacetime symmetry (Poincare symmetry).
โ€ข ๐›ฝ function for ๐‘ = 4 Super Yang Mills theories is zero
(proved by N. Seiberg in 1988).
Thanks
for
listening Questions?

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Gauge Theory for Beginners.pptx

  • 1. Gauge Theory for Beginners Muhammad Hassaan Saleem (PhD student, SUNY Albany, USA)
  • 2. Electromagnetism Example ๐ต = โˆ‡ ร— ๐ด where ๐‘จ is vector potential and ๐‘ฉ is magnetic field. Define another potential ๐ดโ€ฒ i.e. ๐ดโ€ฒ = ๐ด + ๐›๐›ผ (1) where ๐›ผ is an arbitrary differentiable function. โ‡’ ๐ตโ€ฒ = โˆ‡ ร— ๐ด + ๐›๐›ผ = โˆ‡ ร— ๐ด = ๐ต So, ๐ดโ€ฒ and ๐ด are physically equivalent and thus, ๐ด is not measurable (i.e. it is unphysical). Transformation (1) is known as gauge transformation and thus, ๐ด and ๐ดโ€ฒ are called gauge equivalent.
  • 3. Gauge Choices So, โˆ‡. ๐ด doesnโ€™t matter. We can choose for value for it. These choices are called gauge choices and this procedure is called gauge fixing. Examples โˆ‡. ๐ด = 0 (Coulomb Gauge) โˆ‡. ๐ด = โˆ’ 1 ๐‘2 ๐œ•๐œ™ ๐œ•๐‘ก (Lorenz Gauge)
  • 4. Quantum Mechanics Example and local U(1) gauge symmetry. Take a quantum particle of mass ๐‘š and charge ๐‘ž and let it interact it with a background field (๐œ™, ๐ด). The Hamiltonian is; ๐ป = 1 2๐‘š ๐‘ โˆ’ ๐‘ž๐ด ๐‘ 2 + ๐‘ž๐œ™ The Schrodinger equation is thus, ๐‘–โ„ ๐œ•๐œ“ ๐‘ฅ, ๐‘ก ๐œ•๐‘ก = 1 2๐‘š โˆ’๐‘–โ„โˆ‡ โˆ’ ๐‘ž๐ด ๐‘ 2 ๐œ“ ๐‘ฅ, ๐‘ก + ๐‘ž๐œ™ ๐œ“(๐‘ฅ, ๐‘ก) and is invariant under following transformations ๐œ“ ๐‘ฅ, ๐‘ก โ†’ ๐‘’๐‘–๐œ† ๐‘ฅ,๐‘ก ๐œ“(๐‘ฅ, ๐‘ก) ๐ด โ†’ ๐ด โˆ’ โ„๐‘ ๐‘ž โˆ‡๐œ† ๐‘ฅ, ๐‘ก = ๐ด โˆ’ ๐‘–โ„๐‘ ๐‘ž ei๐œ† ๐‘ฅ,๐‘ก โˆ‡ ei๐œ† ๐‘ฅ,๐‘ก โ€  ๐œ™ โ†’ ๐œ™ + โ„ ๐‘ž ๐œ•๐œ† ๐‘ฅ, ๐‘ก ๐œ•๐‘ก = ๐œ™ + โ„ ๐‘ž ei๐œ† ๐‘ฅ,๐‘ก ๐œ• ๐œ•๐‘ก ei๐œ† ๐‘ฅ,๐‘ก โ€  ๐‘’๐‘–๐œ† ๐‘ฅ,๐‘ก โˆˆ ๐‘ˆ(1) group. This is an example of local ๐‘ˆ(1) gauge transformation.
  • 5. Bring in QFT QFT is a theory of quantized fields. Different fields have different transformation properties and correspond to different mathematical quantities. Different fields have different spin particles. Examples For electrodynamics, we need photons and electrons i.e. spin one and spin half particles. So, we use ๐ด๐œ‡ and ๐œ“ fields. Field Spin Scalars ๐œ™ 0 Spinors ๐œ“ 1/2 Vector ๐ด๐œ‡ 1 Second Rank Tensor ๐‘‡๐œ‡๐œˆ 2
  • 6. Making of QED (Simplest, real world gauge theory) Q. What is the free electron lagrangian (with mass ๐‘š and charge ๐‘’)? Ans. It is the Dirac lagrangian i.e. โ„’๐ท๐‘–๐‘Ÿ๐‘Ž๐‘ = ๐œ“ ๐‘–โ„๐›พ๐œ‡ ๐œ•๐œ‡ โˆ’ ๐‘š ๐œ“ Q. How to get Maxwell lagrangian? Ans. Write Maxwell equations in covariant form and then find the lagrangian. It turns out to be โ„’๐‘€๐‘Ž๐‘ฅ๐‘ค๐‘’๐‘™๐‘™ = โˆ’ 1 4 ๐น ๐œ‡๐œˆ๐น๐œ‡๐œˆ where ๐น ๐œ‡๐œˆ = ๐œ•๐œ‡๐ด๐œˆ โˆ’ ๐œ•๐œˆ๐ด๐œ‡ ๐‘’โˆ’, ๐›พ donโ€™t interact โ‡’ total lagrangian is just the sum. Q. What is the interaction term? Ans. It can only be one term i.e. โ„’๐ผ๐‘›๐‘ก = โˆ’๐‘’๐œ“๐›พ๐œ‡ ๐ด๐œ‡๐œ“ (Lorentz invariant, renormalizable). Q. Why not ๐œ–๐›ผ๐›ฝ๐œ‡๐œˆ๐น๐œ‡๐œˆ ๐น๐›ผ๐›ฝ ? Ans. CP violating, just a surface term. So, QED lagrangian is โ„’๐‘„๐ธ๐ท = โ„’๐ท๐‘–๐‘Ÿ๐‘Ž๐‘ + โ„’๐‘€๐‘Ž๐‘ฅ๐‘ค๐‘’๐‘™๐‘™ + โ„’๐ผ๐‘›๐‘ก = ๐œ“ ๐‘–โ„๐›พ๐œ‡ ๐œ•๐œ‡ โˆ’ ๐‘š ๐œ“ โˆ’ 1 4 ๐น ๐œ‡๐œˆ๐น๐œ‡๐œˆ โˆ’ ๐‘’๐œ“๐›พ๐œ‡ ๐ด๐œ‡๐œ“
  • 7. Predictions and Successes of QED Using Feynman rules for QED, we can do important calculations. QED gives us certain insights e.g., corrections in Coulomb potential;
  • 8. Conventional Form of QED lagrangian The QED lagrangian is โ„’๐‘„๐ธ๐ท = ๐œ“ ๐‘–โ„๐›พ๐œ‡ ๐œ•๐œ‡ โˆ’ ๐‘š ๐œ“ โˆ’ 1 4 ๐น ๐œ‡๐œˆ๐น๐œ‡๐œˆ โˆ’ ๐‘’๐œ“๐›พ๐œ‡ ๐ด๐œ‡๐œ“ It can be written as โ„’๐‘„๐ธ๐ท = ๐œ“ ๐‘–โ„๐›พ๐œ‡(๐œ•๐œ‡ + ๐‘–๐‘’๐ด๐œ‡) โˆ’ ๐‘š ๐œ“ โˆ’ 1 4 ๐น ๐œ‡๐œˆ๐น๐œ‡๐œˆ = ๐œ“ ๐‘–โ„๐›พ๐œ‡๐ท๐œ‡ โˆ’ ๐‘š ๐œ“ โˆ’ 1 4 ๐น ๐œ‡๐œˆ๐น๐œ‡๐œˆ where ๐ท๐œ‡ = ๐œ•๐œ‡ + ๐‘–๐‘’๐ด๐œ‡ is called the covariant derivative. So, The conventional form of the QED lagrangian is given as โ„’๐‘„๐ธ๐ท = ๐œ“ ๐‘–โ„๐›พ๐œ‡๐ท๐œ‡ โˆ’ ๐‘š ๐œ“ โˆ’ 1 4 ๐น ๐œ‡๐œˆ๐น๐œ‡๐œˆ
  • 9. Local symmetry approach to QED We can derive the form of QED lagrangian starting from Dirac lagrangian alone and demanding ๐‘ˆ(1) local symmetry. Dirac lagrangian is โ„’๐ท๐‘–๐‘Ÿ๐‘Ž๐‘ = ๐œ“ ๐‘–โ„๐›พ๐œ‡๐œ•๐œ‡ โˆ’ ๐‘š ๐œ“ Demand symmetry under ๐‘ˆ(1) transformation ๐œ“ ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ(๐‘ฅ) ๐œ“ This renders ๐œ“๐œ“ โ†’ ๐œ“๐‘’โˆ’๐‘–๐›ผ(๐‘ฅ)๐‘’๐‘–๐›ผ ๐‘ฅ ๐œ“ = ๐œ“๐œ“ So, mass term is invariant. The derivative term (also called kinetic term) transforms as ๐œ“๐œ•๐œ‡๐œ“ โ†’ ๐œ“๐‘’โˆ’๐‘–๐›ผ ๐‘ฅ ๐œ•๐œ‡ ๐‘’๐‘–๐›ผ ๐‘ฅ ๐œ“ = ๐œ“๐œ•๐œ‡๐œ“ + ๐‘–๐œ•๐œ‡๐›ผ ๐‘ฅ ๐œ“ ๐œ“ โ‰  ๐œ“๐œ•๐œ‡๐œ“ The kinetic term isnโ€™t invariant. WHY?
  • 10. Exploring the problem Choose a direction in spacetime specified by a vector ๐œ‚๐œ‡ . The directional derivative of ๐œ“(๐‘ฅ) in this direction is; ๐œ‚๐œ‡ ๐œ•๐œ‡๐œ“ ๐‘ฅ = lim ๐œ–โ†’0 1 ๐œ– ๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ ๐œ“ ๐‘ฅ Under gauge transformation, we have; ๐œ‚๐œ‡ ๐œ•๐œ‡๐œ“ ๐‘ฅ = lim ๐œ–โ†’0 1 ๐œ– ๐‘’๐‘–๐›ผ ๐‘ฅ+๐œ–๐œ‚ ๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ ๐‘’๐‘–๐›ผ ๐‘ฅ ๐œ“ ๐‘ฅ Different exponentials cause the problem. Invent a function ๐‘ˆ(๐‘ฆ, ๐‘ฅ) such that ๐‘ˆ ๐‘ฆ, ๐‘ฅ ๐œ“(๐‘ฅ) transforms like ๐œ“ ๐‘ฆ i.e. ๐‘ˆ ๐‘ฆ, ๐‘ฅ ๐œ“ ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ ๐‘ฆ ๐‘ˆ ๐‘ฆ, ๐‘ฅ ๐œ“(๐‘ฅ) โ‡’ ๐‘ˆ ๐‘ฆ, ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ ๐‘ฆ ๐‘ˆ(๐‘ฆ, ๐‘ฅ)๐‘’โˆ’๐‘–๐›ผ ๐‘ฅ Moreover, it is plausible to set ๐‘ˆ ๐‘ฅ, ๐‘ฅ = 1. Now, let first argument differ slightly from ๐‘ฅ and do a Taylorโ€™s expansion; ๐‘ˆ ๐‘ฅ + ๐œ–๐œ‚, ๐‘ฅ = 1 + ๐œ•๐‘ˆ ๐‘ฆ, ๐‘ฅ ๐œ•๐‘ฆ๐œ‡ ๐‘ฆ=๐‘ฅ ๐œ–๐œ‚๐œ‡ + ๐’ช ๐œ–2 Set ๐œ•๐‘ˆ ๐‘ฆ, ๐‘ฅ ๐œ•๐‘ฆ๐œ‡ ๐‘ฆ=๐‘ฅ = โˆ’๐‘–๐‘’๐ด๐œ‡ ๐‘ฅ โ‡’ ๐‘ˆ(๐‘ฅ + ๐œ–๐œ‚, ๐‘ฅ) = 1 โˆ’ ๐‘–๐‘’๐œ–๐œ‚๐œ‡ ๐ด๐œ‡ ๐‘ฅ + ๐’ช ๐œ–2
  • 11. Exploring the problem Define a new kind of derivative called ๐ท๐œ‡๐œ“ as ๐œ‚๐œ‡ ๐ท๐œ‡๐œ“ ๐‘ฅ = lim ๐œ–โ†’0 1 ๐œ– ๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ ๐‘ˆ ๐‘ฅ + ๐œ–๐œ‚, ๐‘ฅ ๐œ“ ๐‘ฅ It turns out to be ๐œ‚๐œ‡ ๐ท๐œ‡๐œ“ ๐‘ฅ = lim ๐œ–โ†’0 1 ๐œ– (๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ 1 โˆ’ ๐‘–๐‘’๐œ–๐œ‚๐œ‡ ๐ด๐œ‡ ๐œ“(๐‘ฅ)) = lim ๐œ–โ†’0 1 ๐œ– ๐œ“ ๐‘ฅ + ๐œ–๐œ‚ โˆ’ ๐œ“ ๐‘ฅ + ๐‘–๐‘’๐œ‚๐œ‡ ๐ด๐œ‡๐œ“(๐‘ฅ) = ๐œ‚๐œ‡ ๐œ•๐œ‡๐œ“ ๐‘ฅ + ๐‘–๐‘’๐ด๐œ‡๐œ“ ๐‘ฅ โ‡’ ๐ท๐œ‡๐œ“ ๐‘ฅ = ๐œ•๐œ‡๐œ“ ๐‘ฅ + ๐‘–๐‘’๐ด๐œ‡๐œ“ ๐‘ฅ We recover the same covariant derivative !!! We check that ๐ท๐œ‡๐œ“ ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ ๐‘ฅ ๐ท๐œ‡๐œ“(๐‘ฅ) and thus, the following lagrangian is invariant under local ๐‘ˆ(1) symmetry; ๐œ“ ๐‘–โ„๐›พ๐œ‡๐ท๐œ‡ โˆ’ ๐‘š ๐œ“ WHAT ABOUT MAXWELL LAGRANGIAN?
  • 12. What about Maxwellโ€™s lagrangian? First ask that how does ๐ด๐œ‡ transform? We know that ๐‘ˆ ๐‘ฆ, ๐‘ฅ โ†’ ๐‘’๐‘–๐›ผ ๐‘ฆ ๐‘ˆ(๐‘ฆ, ๐‘ฅ)๐‘’โˆ’๐‘–๐›ผ ๐‘ฅ Set ๐‘ฆ = ๐‘ฅ + ๐œ–๐œ‚ and we have; 1 โˆ’ ๐‘–๐‘’๐œ–๐œ‚๐œ‡๐ด๐œ‡(๐‘ฅ) โ†’ ๐‘’๐‘–๐›ผ ๐‘ฅ+๐œ–๐œ‚ 1 โˆ’ ๐‘–๐‘’๐œ–๐œ‚๐œ‡๐ด๐œ‡(๐‘ฅ) ๐‘’โˆ’๐‘–๐›ผ ๐‘ฅ โ‡’ ๐ด๐œ‡ ๐‘ฅ โ†’ ๐ด๐œ‡(๐‘ฅ) โˆ’ ๐‘– ๐‘’ ๐œ•๐œ‡๐›ผ(๐‘ฅ) We recover the same gauge transformation for ๐ด๐œ‡ as before! ๐น ๐œ‡๐œˆ is gauge invariant and thus Maxwell lagrangian can be โˆ’ 1 4 ๐น ๐œ‡๐œˆ๐น๐œ‡๐œˆ as it is Lorentz invariant and renormalizable. Moral: QED can be deduced by demanding local ๐‘ˆ(1) gauge symmetry.
  • 13. Crash Course on generators and parameters of groups ๐‘ˆ(1) is the group is 1 ร— 1 matrices which are unitary (i.e. complex numbers). So, one parameter needed to specify. ๐‘†๐‘ˆ ๐‘ is the group of complex matrices which are unitary and have determinant 1. An ๐‘ ร— ๐‘ complex matrix needs 2๐‘2 parameters to be specified. However, ๐‘†๐‘ˆ(๐‘) matrices need ๐‘2 โˆ’ 1 parameters to be specified (think why). Every group has an identity element ๐•€. Elements near identity can be specified as follows (for ๐‘†๐‘ˆ(๐‘) groups); ๐•€๐‘ โˆ’ ๐‘– ๐‘—=1 ๐ท ๐›ผ๐‘—๐‘‡๐‘— = ๐•€๐‘ โˆ’ ๐‘– ๐›ผ๐‘—๐‘‡๐‘— Where ๐ท is the number of parameters (which is ๐‘2 โˆ’ 1 in case of ๐‘†๐‘ˆ(๐‘)), ๐›ผ๐‘— are parameters and ๐‘‡๐‘— are matrices called generators. An arbitrary element of the ๐‘†๐‘ˆ(๐‘) group denoted as ๐’ช(๐›ผ1, โ€ฆ , ๐›ผ๐ท)can be written by exponentiation ๐’ช ๐›ผ1, โ€ฆ , ๐›ผ๐ท = exp(๐‘– ๐›ผ๐‘—๐‘‡๐‘—) The element of ๐‘ˆ(1) is ๐‘’๐‘–๐›ผ and thus, the generator of ๐‘ˆ(1) is just 1. The generators satisfy ๐‘‡๐‘Ž, ๐‘‡๐‘ = ๐‘–๐‘“๐‘Ž๐‘๐‘๐‘‡๐‘ where ๐‘“๐‘Ž๐‘๐‘ are called structure constants. Example: ๐œŽ๐‘Ž 2 , ๐œŽ๐‘ 2 = ๐‘–๐œ–๐‘Ž๐‘๐‘ ๐œŽ๐‘ 2 for ๐‘†๐‘ˆ 2 and hence, ๐‘“๐‘Ž๐‘๐‘ = ๐œ–๐‘Ž๐‘๐‘
  • 14. Generalizing QED Let ๐œ“(๐‘ฅ) be a multiplet now (donโ€™t use the word vector) with ๐‘ components as follows; ๐œ“1 ๐‘ฅ โ‹ฎ ๐œ“๐‘ ๐‘ฅ So, we have an index now (๐œ“๐‘— (๐‘ฅ)) called the gauge index. Now, demand symmetry under the transformation ๐œ“๐‘— ๐‘ฅ โ†’ [exp(๐‘–๐›ผ๐‘˜ ๐‘‡๐‘˜ )]๐‘—๐‘™ ๐œ“๐‘™ (๐‘ฅ) This is a non-Abelian gauge transformation (in this case, ๐‘†๐‘ˆ(๐‘)). Process similar to QED gives; ๐‘ˆ ๐‘ฅ + ๐œ–๐œ‚, ๐‘ฅ = 1 + ๐‘–๐‘”๐œ–๐œ‚๐œ‡ ๐ด๐œ‡ ๐‘— ๐‘‡๐‘— + ๐’ช(๐œ–2 ) ๐ท๐œ‡๐œ“๐‘— = ๐•€ ๐œ•๐œ‡ โˆ’ ๐‘–๐‘”๐ด๐œ‡ ๐‘˜ ๐‘‡๐‘˜ ๐‘—๐‘™ ๐œ“๐‘™ = ๐œ•๐œ‡๐œ“๐‘— โˆ’ ๐‘–๐‘”๐ด๐œ‡ ๐‘˜ ๐‘‡๐‘˜ ๐œ“ ๐‘— ๐‘จ๐ ๐’‹ ๐‘ป๐’‹ โ†’ ๐ž๐ฑ๐ฉ ๐’Š๐œถ๐’ ๐‘ป๐’ ๐‘จ๐ ๐’‹ ๐‘ป๐’‹ + ๐’Š ๐’ˆ ๐๐ ๐ž๐ฑ๐ฉ ๐’Š๐œถ๐’‘ ๐‘ป๐’‘ โ€  ๐น ๐œ‡๐œˆ ๐‘— ๐‘‡๐‘— = ๐œ•๐œ‡๐ด๐œˆ ๐‘— ๐‘‡๐‘— โˆ’ ๐œ•๐œˆ ๐ด๐œ‡ ๐‘™ ๐‘‡๐‘™ + ๐‘”๐‘“๐‘–๐‘—๐‘˜ ๐ด๐œ‡ ๐‘– ๐ด๐œˆ ๐‘— ๐‘‡๐‘˜ Not gauge inv. Gauge Invariant quantity is โˆ’ 1 2 tr F๐œ‡๐œˆ j ๐‘‡๐‘— 2 = โˆ’ 1 4 F๐œ‡๐œˆ j 2 The final lagrangian is (called Yang Mills lagrangian) โ„’๐‘Œ๐‘€ = โˆ’ 1 4 ๐น ๐œ‡๐œˆ ๐‘— 2 + ๐œ“๐‘— ๐‘–๐›พ๐œ‡ ๐ท๐œ‡ โˆ’ ๐•€ ๐‘š ๐‘—๐‘˜ ๐œ“๐‘˜ = โˆ’ 1 4 ๐น ๐œ‡๐œˆ ๐‘— 2 + ๐œ“ ๐‘–๐›พ๐œ‡ ๐ท๐œ‡ โˆ’ ๐‘š ๐œ“
  • 15. Peculiarity of Non-Abelian gauge theories 1) The extra terms in ๐น ๐œ‡๐œˆ ๐‘— ๐‘‡๐‘— i.e. ๐‘”๐‘“๐‘–๐‘—๐‘˜๐ด๐œ‡ ๐‘– ๐ด๐œˆ ๐‘— ๐‘‡๐‘˜ give extra vertices in Feynman diagrams i.e. i.e. it allows the force carriers to interact with one another. 2) Non Abelian ๐‘†๐‘ˆ(๐‘) gauge theories are asymptotically free (๐›ฝ < 0) ๐›ฝ ๐‘” = โˆ’ ๐‘”3 4๐œ‹ 2 11 3 ๐ถ2 ๐บ โˆ’ 4 3 ๐‘›๐‘“๐ถ ๐‘Ÿ No other renormalizable theory in four spacetime dimension is asymptotically free (proved by Coleman and Gross Phys. Rev. Lett. 31, 851 (1973)).
  • 16. Some famous Non-Abelian gauge theories 1) Quantum Chromodynamics ๏ƒ˜ ๐‘†๐‘ˆ(3) gauge group. ๏ƒ˜ The gauge index can be ๐‘– = 1,2,3 โ‡’ three colors. 2) Electroweak Theory ๏ƒ˜ ๐‘†๐‘ˆ 2 ๐ฟ ร— ๐‘ˆ 1 ๐‘Œ gauge group. ๏ƒ˜ It is a chiral gauge theory (violates parity). 3) Standard Model ๏ƒ˜ ๐‘†๐‘ˆ 3 ร— ๐‘†๐‘ˆ 2 ๐ฟ ร— ๐‘ˆ 1 ๐‘Œ gauge group. 4) Glashow Georgi Model (Candidate GUT) ๏ƒ˜ ๐‘†๐‘ˆ(5) gauge group. ๏ƒ˜ Contains ๐‘‹, ๐‘Œ bosons with mass โˆผ 1016 GeV ๏ƒ˜ Implies proton decay.
  • 17. Super Yang Mills theories โ€ข Yang Mills theories in the presence of supersymmetry are called Super Yang Mills theories. โ€ข Supersymmetry is NOT a local gauge symmetry. It is an extension of spacetime symmetry (Poincare symmetry). โ€ข ๐›ฝ function for ๐‘ = 4 Super Yang Mills theories is zero (proved by N. Seiberg in 1988).