Theory

Theory

In this section, a brief but (mostly) complete introduction to quantum mechanics and its mathematical description is presented. This is intended to serve as a reference to the quantum mechanics implemented within the Qhronology package, in addition to providing a pedagogical summary of the basic physics relevant to this project. Included are discussions on antichronological time travel from the perspectives of both classical and quantum physics. The exposition on these topics is designed to be sufficiently comprehensive so as to provide a pathway for the determined reader to make a foray into the fascinating area of research that is the quantum mechanics of time travel.

Quantum mechanics

• Mathematical foundations of quantum mechanics
  • Vector spaces
  • Linear operators and operations
  • Composite systems
  • Superoperators
  • Concavity and convexity
• Quantum mechanics on discrete Hilbert spaces
  • Quantum states
  • Quantum quantities
  • Separability and entanglement
  • Quantum operations
  • Quantum evolutions
  • Quantum measurements
  • Integrals over quantum states
• Quantum mechanics on continuous Hilbert spaces
  • Probabilities and the Born rule
  • Time evolution
  • The path-integral formulation
• Quantum circuitry
  • Anatomy of a quantum circuit
  • List of special gates
  • Examples
  • Remarks

Time travel

• Classical theories of time travel
  • Closed timelike curves and time machines
    • Wormhole-based time machines
  • Time-travel paradoxes
    • The principle of self-consistency
  • The Cauchy problem on spacetimes containing CTCs
  • Time-travelling billiard balls
    • Indeterministic dynamics and evolution multiplicity
    • Quantum resolutions to classical indeterminism
• Quantum theories of time travel
  • The Deutsch model (D-CTCs)
    • Deutsch’s picture: The rule of maximum entropy
    • Iterative picture: The equivalent circuit
    • Allen’s picture: Noise incorporation
  • Postselected teleportation (P-CTCs)
    • The path-integral formulation in the presence of CTCs
    • Equivalence of CTC path integrals and postselected teleportation
    • General form of P-CTC system evolution
    • Determination of the P-CTC CV state
  • Comparison of D-CTCs and P-CTCs
  • Alternative formulations of quantum time travel
    • Weighted D-CTCs (WD-CTCs)
    • Transition probabilities (T-CTCs)
    • CTC as a ring resonator
    • Indefinite causal structure and the process matrix formalism