1. Gauge theory and gravity
/정호정

2. What is gauge theory?
-Type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations
-Gauge: Specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian
-Gauge transformations: Transformations between possible gauges

3. Where gauge theories are used
Quantum electrodynamics (abelian, U1)
Standard Model (non-abelian U(1)×SU(2)×SU(3))
- Gravitation (Lanczos tensor)
- Quantum gravity

4. Gauge theory in classical physics

5. Gauge symmetry in electrodynamics
Maxwell’s equation 𝐵 =∇× 𝐴 , 𝐸 =−∇𝜑− 𝜕 𝐴 𝜕𝑡 A →𝐴+∇𝑓 ,𝜑→𝜑− 𝜕𝑓 𝜕𝑡

6. Gauge symmetry in G.R Translational invariance
Coordinate change (e.x 𝑡→𝑡+ 𝑡 3 𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡 0 𝑖𝑠 𝑢𝑛𝑖𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 )
same Einstein’s equation

7. Lanczos tensor(H) Rank 3 tensor generates Weyl tensor Symmetry
-Habc+Hbac=0
-Habc+Hbca+Hcab=0
Weyl Lanczos equation

8. Weyl-Lanczos equations are invariant under the gauge transformation
Set we get
this choice simplifies Weyl-Lanczos equation

9. Quantum mechanics with gauge theory
Mathematical trick?
Any physical meaning?

10. Aharonov-Bohm effect
Potential rather than field
Gauge invariance

11. Type of gauge symmetry
Electron phases- form Abelian group (0°≤𝜃<360°) called U(1)
Non commutative gauge group (Yang Mills theory)
Describe a field that interacts with itself
e.x) Gravitational fields in G.R, Nuclear forces

12. Gauge-Bosons Gauge symmetry=> existence of interactions
Superposition of two sates A,B
𝜃 0 →A:100%, B:0%
𝜃 90 →A:0%, B:100%

13. Momentum conservation breaks?
(Energy observe in point X) (Energy observe in Y)
→Gauge function 𝜃 𝑥 ℎ𝑎𝑠 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑚𝑒𝑎𝑛𝑖𝑛𝑔!
If A,B are electrons: 𝜃-> electromagnetic fields
A:20% B:80%
A:21%
B:80%

14. 𝜃 𝑎𝑠 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒
Wave particle duality
Electro magnetic waves Photon
Generally, particle of 𝜃 is called gauge bosons

15. Electromagnetic force
Standard model
Electromagnetic force
Weak nuclear force
Strong nuclear force

16. Gravity? Einstein-Cartan theory
locally gauge symmetry, non-zero torsion
Gauge gravitation theory
extension of Yang-Mills theory
Quantum gravity
string theory, loop quantum theory