In tis slide, an introduction to string theory has been given. Apart from that, a simple proof of 26 dimensions of bosonic string theory is given (following Zwiebach's approach). I explained this presentation in two parts (on my YouTube channel). Here are the links _______________________________________________ Part 1 https://www.youtube.com/watch?v=QQA4JQ6Y-eo&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp _______________________________________________ Part 2 https://www.youtube.com/watch?v=vhLCtLn79jE&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp&index=2 _______________________________________________

1. The basics that every physics
student should know about
string theory
Muhammad Hassaan Saleem
Department of Physics
University of the Punjab

2. Contents
1) Why string theory?
2) Ideas and potential of string theory
3) Experimental support

3. Why string theory?
1) Conflicts in Physics
2) Quantum gravity

4. The conflicts in Physics: An overview
 In physics, we have seen a lot of conflicts between theories and we have
witnessed their solutions too. The solution for the conflict between general
relativity and Quantum mechanics is a mystery.

5. The conflicts in Physics: GR vs. QM
1) General Relativity: Idea
General relativity is the Einstein’s theory of gravity. It says that the gravitational
phenomena is the manifestation of the curvature of space time around a massive
body or energy.
The two equations that are fundamental to general relativity are as follows.

6. The conflicts in Physics: GR vs. QM
1) General relativity: Demonstration

7. The conflicts in Physics: GR vs. QM
2) Quantum Field theory
The quantum theory can describe the three forces of nature i.e.
electromagnetism, weak force and the strong force but it cant describe the
gravitational force as it is not renormalizable.
Our best quantum theory (called the standard model) describes the universe
(other than gravity) in terms of 18 particles as shown below.

8. The conflicts in Physics: GR vs. QM
2) Quantum field theory
The fundamental equation of the standard model is given as

9. Quantum gravity
The solution to the conflict between general relativity and quantum theory is
studied in the subject of quantum gravity. This subject is concerned to the ways
to quantize gravity. The major frameworks in quantum gravity are as follows
String theory Loop Quantum Gravity (LQG)

10. Ideas and potential
of string theory
1) Vibrating strings
2) Extra dimensions
3) Calculating the 20 numbers
4) Calculating the Hawking Beckenstein entropy
5) Solving the hierarchy problem

11. The basic idea: Enter string vibrations
The basic idea of string theory is that all the matter is made up of tiny vibrating
strings. The different types of particles that we see around us are just
manifestations of the same strings vibrating in different modes.

12. Action of free relativistic particle
 Action
𝑆 = −𝑚𝑐
𝑃
𝑑𝑠 𝑤ℎ𝑒𝑟𝑒 𝑑𝑠 = 𝑐 𝑑𝑡 1 −
𝑣2
𝑐2
 Hamiltonian (by using Legendre transform)
𝐻 =
𝑚𝑐2
1 −
𝑣2
𝑐2
 Equation of motion
𝑑𝑝 𝜇
𝑑𝜏
= 0 𝑤ℎ𝑒𝑟𝑒 𝑝 𝜇 =
𝜕𝐿
𝜕𝑥 𝜇

13. Action of free relativistic string
(Nambo-Goto Action)
𝑆 ∝ 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡𝑤𝑜 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑤𝑜𝑟𝑙𝑑 𝑠ℎ𝑒𝑒𝑡
𝑁𝑜𝑡𝑒 ∶ 𝐴 = 𝑑𝜎 𝑑𝜏
𝜕𝑋
𝜕𝜎
𝜕𝑋
𝜕𝜏
2
−
𝜕𝑋
𝜕𝜎
2
𝜕𝑋
𝜕𝜏
2
𝑤ℎ𝑒𝑟𝑒 𝑋 𝜇 𝑖𝑠 𝑠𝑝𝑎𝑐𝑒𝑡𝑖𝑚𝑒 𝑣𝑒𝑐𝑡𝑜𝑟
 Action (Nambo-Goto Action)
𝑆 = −
𝑇0
𝑐
𝜏 𝑖
𝜏 𝑓
0
𝜎1
𝑑𝜎 𝑑𝜏
𝜕𝑋
𝜕𝜎
𝜕𝑋
𝜕𝜏
2
−
𝜕𝑋
𝜕𝜎
2
𝜕𝑋
𝜕𝜏
2
𝐿 = −
𝑇0
𝑐
0
𝜎1
𝑑𝜎
𝜕𝑋
𝜕𝜎
𝜕𝑋
𝜕𝜏
2
−
𝜕𝑋
𝜕𝜎
2
𝜕𝑋
𝜕𝜏
2
=
0
𝜎1
𝑑𝜎 ℒ

14. Equations of motion for relativistic
strings
Equations of motion turn out to be
𝜕𝑃𝜇
𝜏
𝜕𝜏
+
𝜕𝑃𝜇
𝜎
𝜕𝜎
= 0
Where
𝑃𝜇
𝜎 =
𝜕ℒ
𝜕𝜎
= −
𝑇0
𝑐
𝑋. 𝑋 𝑋′
𝜇 − 𝑋′ 2 𝑋 𝜇
𝑋. 𝑋′ 2
− 𝑋
2
𝑋′ 2
𝑃𝜇
𝜏 =
𝜕ℒ
𝜕𝜏
= −
𝑇0
𝑐
𝑋. 𝑋 𝑋 𝜇 − 𝑋
2
𝑋′ 𝜇
𝑋. 𝑋′
2
− 𝑋
2
𝑋′ 2
“A GOOD PARAMETRIZATION IS NEEDED”

15. Good parameterization
The parameterization is given by
𝑛. 𝑋 𝜏, 𝜎 = 𝛽𝛼′
𝑛. 𝑝 𝜏 , 𝑛. 𝑝 =
2𝜋
𝛽
𝑛. 𝑃 𝜏
𝑤ℎ𝑒𝑟𝑒 𝛼′
=
1
2𝜋𝑇0
(𝑐 = 1)
𝛽 =
2 (𝑜𝑝𝑒𝑛 𝑠𝑡𝑟𝑖𝑛𝑔𝑠)
1 (𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑡𝑟𝑖𝑛𝑔𝑠)
This parameterization gives
𝑃𝜇
𝜏
=
1
2𝜋𝛼′ 𝑋 𝜇 𝑃𝜇
𝜎
= −
1
2𝜋𝛼′ 𝑋 𝜇
′
The free string equation becomes
𝑋 𝜇
− 𝑋′′𝜇
= 0 (𝐴 𝑤𝑎𝑣𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛)

16. Solution in light cone guage
Ordinary spacetime coordinates 𝑥0
, 𝑥1
, 𝑥2
, 𝑥3
, … = 𝑥0
, 𝑥1
, 𝑥 𝐼
𝐼 = 2,3, … , 𝐷
Light cone coordinates 𝑥+, 𝑥−, 𝑥2, 𝑥3, … = 𝑥+, 𝑥−, 𝑥 𝐼 𝐼 = 2,3, … , 𝐷
𝑥+ =
𝑥0
+ 𝑥1
2
𝑥− =
𝑥0
− 𝑥1
2
Wave equation with Lorentz indices solves to give
𝑋 𝜇
𝜏, 𝜎 = 𝑥0
𝜇
+ 2𝛼′
𝑝 𝜇
𝜏 − 𝑖 2𝛼′
𝑛=1
∞
𝑎 𝑛
𝜇∗
𝑒 𝑖𝑛𝜏
− 𝑎 𝑛
𝜇
𝑒−𝑖𝑛𝜏
cos 𝑛𝜎
𝑛
In Light cone gauge (𝑋+
= 2𝛼′
𝑝+
𝜏), we get
𝑋 𝐼 𝜏, 𝜎 = 𝑥0
𝐼
+ 2𝛼′ 𝑝 𝐼 𝜏 − 𝑖 2𝛼′
𝑛=1
∞
𝑎 𝑛
𝐼 ∗
𝑒 𝑖𝑛𝜏 − 𝑎 𝑛
𝐼 𝑒−𝑖𝑛𝜏
cos 𝑛𝜎
𝑛
𝛼 𝑛
−
=
1
𝑝+ 2𝛼′
𝐿 𝑛
⊥
𝑤ℎ𝑒𝑟𝑒 𝐿 𝑛
⊥
=
1
2
𝑝 ∈𝑍
𝛼 𝑛−𝑝
𝐼
𝛼 𝑝
𝐼

17. Quantization of relativistic strings
The operators used in the quantization
(𝑋 𝐼
𝜏, 𝜎 , 𝑥0
−
𝜏 , 𝑃 𝜏𝐼
𝜏, 𝜎 , 𝑝+
(𝜏))
Postulated commutation relations
𝑋 𝐼
𝜏, 𝜎 , 𝑃 𝜏𝐼
𝜏, 𝜎′ = 𝑖𝜂 𝐼𝐽
𝛿(𝜎 − 𝜎′)
𝑥−(𝜏), 𝑝+(𝜏) = −𝑖
This leads to the following commutation relations of 𝛼 𝑛
𝐼
𝛼 𝑛
𝐼 , 𝛼 𝑚
𝐽
= 𝑚𝜂 𝐼𝐽 𝛿 𝑚+𝑛 ,0
Reminder : The definition of Virasoro modes (Now operators)
𝐿 𝑛
⊥
=
1
2
𝑝 ∈𝑍
𝛼 𝑛−𝑝
𝐼
𝛼 𝑝
𝐼
The commutators of Virasoro operators become (Witt algebra)
𝐿 𝑚
⊥ , 𝐿 𝑛
⊥ = 𝑚 − 𝑛 𝐿 𝑚+𝑛
⊥ 𝑓𝑜𝑟 𝑚 + 𝑛 ≠ 0

18. The problems with 𝑚 + 𝑛 = 0
The full case with 𝑚 + 𝑛 = 0 case included gives us the following commutation
relation (Virasoro algebra)
𝐿 𝑚
⊥
, 𝐿 𝑛
⊥
= 𝑚 − 𝑛 𝐿 𝑚+𝑛
⊥
+
𝐷 − 2
12
𝑚3
− 𝑚 𝛿 𝑚+𝑛,0
The ordering in zeroth Virasoro mode becomes ambiguous
𝐿0
⊥
=
1
2
𝑝 ∈𝑍
𝛼−𝑝
𝐼
𝛼 𝑝
𝐼
(𝑅𝑒𝑚𝑒𝑚𝑏𝑒𝑟 𝑡ℎ𝑎𝑡 𝛼 𝑛
𝐼
, 𝛼 𝑚
𝐽
= 𝑚𝜂 𝐼𝐽
𝛿 𝑚+𝑛 ,0)
We add a constant (𝑎) in the definition of 𝑝−
to get
2𝛼′𝑝−
=
1
𝑝+ (𝐿0
⊥
+ 𝑎)

19. The Lorentz generators in light cone
gauge
The Lorentz generators are given in light cone coordinates as (with their
antisymmetric counterparts)
(𝑀+ −, 𝑀+𝐼, 𝑀−𝐼, 𝑀 𝐼𝐽)
They need to be Hermitian and this sets the 𝑀−𝐼
generator to be of the form
𝑀−𝐼 = 𝑥0
−
𝑝 𝐼 −
1
4𝛼′ 𝑝+
𝑥0
𝐼
, 𝐿0
⊥
+ 𝑎 −
𝑖
𝑝+ 2𝛼′
𝑛=1
∞
1
𝑛
(𝐿−𝑛
⊥ 𝛼 𝑛
𝐼 − 𝛼−𝑛
𝐼 𝐿 𝑛
⊥)
The commutator [𝑀−𝐼, 𝑀−𝐽] should be zero. This commutators turns out to be
𝑀−𝐼
, 𝑀−𝐽
= −
1
𝛼′ 𝑝+2
𝑚=1
∞
(𝛼−𝑚
𝐼
𝛼 𝑚
𝐽
− 𝛼−𝑚
𝐽
𝛼 𝑚
𝐼
) m 1 −
D − 2
24
+
1
m
D − 2
12
+ a
⇒ m 1 −
D − 2
24
+
1
m
D − 2
12
+ a = 0 ∀ 𝑚 ∈ 𝑍 ⟹ 𝐷 = 26 & 𝑎 = −1

20. A small description of superstrings
Inclusion of fermions ⟹ inclusion of new, anticommuting world sheet variables
𝜓1 𝜏, 𝜎 and 𝜓2(𝜏, 𝜎).
Then, the theory is called superstring theory.
The action is given by
𝑆 𝜓 =
1
2𝜋
𝑑𝜏
0
𝜋
𝑑𝜎 [𝜓1
𝐼
𝜕𝜏 + 𝜕 𝜎 𝜓2
𝐼
+ 𝜓2
𝐼
𝜕𝜏 − 𝜕 𝜎 𝜓1
𝐼
]
No. of Dimensions =10
In M-Theory , No. of dimensions =11

21. The surprise: Extra dimensions
The theory works only in 26 dimensions (bosonic strings), 10 (superstrings) or 11 (in M
theory) present on Calabi-Yau manifolds (of plank scale).

22. A mystery of physics: The 20 numbers
A big mystery of physics has been the very finely adjusted values of the 20
numbers. The answer is still unfound using the standard model and general
relativity.

23. The potential of string theory
1) Calculating the 20 numbers
 Calabi yau manifolds have almost 10500
shapes.
 Maths gives no clue.
 This idea gives rise to the idea of multiverse.

24. The potential of string theory
2) Calculating the Black hole entropy
The black hole entropy can be calculated by the string theory models an they
have been found in agreement with the derivation of Stephen hawking and
Beckenstein using general relativity and quantum mechanics.

25. The potential of string theory
3) Solving the hierarchy problem
String theory can actually explain the fact that gravity is very weak as compared
to the other forces (i.e. the hierarchy problem). It says that gravity isn’t that
weak, it just appears to be weak as its strength gets distributed in other
dimensions because it is made of closed strings.

26. Experimental support
1) Lack of experimental support
2) The search for evidence

27. Lack of experimental proof
Energies that we have to probe the distances of the order of the plank’s scale or
to see the extra dimensions are enormously high.
Simple example is the particle in a box problem with a compactified extra
dimension.
𝐸 𝑘,𝑙 =
ℎ2
8𝜋2 𝑚
𝑘𝜋
𝑎
2
+
𝑙
𝑅
2
𝑅 is the radius of the small extra dimension

28. The search for evidence
The modern particle accelerators are searching for extra dimensions and
supersymmetry (it is one of the predictions of string theory) that can help in
validating the string theory.

29. Questions?