Gauge theory and gravity 20160781/정호정

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

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

Gauge theory in classical physics

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

Gauge symmetry in G.R Translational invariance Coordinate change (e.x 𝑡→𝑡+ 𝑡 3 𝑡 0 2 𝑤ℎ𝑒𝑟𝑒 𝑡 0 𝑖𝑠 𝑢𝑛𝑖𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 ) same Einstein’s equation

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

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

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

Aharonov-Bohm effect Potential rather than field Gauge invariance

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

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%

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%

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

Electromagnetic force Standard model Electromagnetic force Weak nuclear force Strong nuclear force

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