1. Noiseless Subsystems and the Emergence of Symmetries
Tomasz Konopka
Fotini Markopoulou
Thanks to:
Loops 05, Lee Smolin, and other
members of Perimeter Institute

2. Outline Constrained dynamics Map into noiseless subsystems Examples
Interpretation
Outlook: gravity

3. Constrained Dynamics Hilbert space of unconstrained system is
Consider Constraints
Physical States
Observables

4. System and Environment
Suppose the system is coupled to an environment (bath)
Evolution Hamiltonian
Some states of the system are noisy, others are noiseless

5. Noiseless States Evolution by Noiseless states
Noisy states some other state

6. Definition of Interactions
Define a new operator
Acting on a physical state
This ‘stabilizer condition’ implies that physical states are unaffected by

7. Example: Circular Motion
Classical particle in 2D
Lagrange multiplier betrays presence of centripetal force
Think of particle and environment
Solutions noiseless states

8. Example: E&M Electromagnetism Hamiltonian
Vector potential the ‘system’
Scalar potential the ‘environment’
Noiseless states are physical states

9. Relativistic Particle
The Hamiltonian is a constraint
Solutions are time reparam. inv.
Treat as an operator in a Hilbert space of an environment
Noiseless states are physical states

10. Interpretation
Map from constrained system to another system coupled to a bath
Noiseless states exhibit symmetries of the constrained system
Gauge and reparametrization invariance emerge in the noiseless states

11. Outlook: Gravity Gravity is a constrained system
What is the ‘environment of the universe’?