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A Lagrangian path integral approach to the qubit

A. Ibort, M. Jiménez-Vázquez

Abstract

A Lagrangian description of the qubit based on a generalization of Schwinger's picture of Quantum Mechanics using the notion of groupoids is presented. In this formalism a Feynman-like computation of its probability amplitudes is done. The Lagrangian is interpreted as a function on the groupoid describing the quantum system. Such Lagrangian determines a self-adjoint element on its associated algebra. Feynman's paths are replaced by histories on the groupoid which form themselves a groupoid. A simple method to compute the sum over all histories is discussed. The unitarity of the propagator obtained in this way imposes quantization conditions on the Lagrangian of the theory. Some particular instances of them are discussed in detail.

A Lagrangian path integral approach to the qubit

Abstract

A Lagrangian description of the qubit based on a generalization of Schwinger's picture of Quantum Mechanics using the notion of groupoids is presented. In this formalism a Feynman-like computation of its probability amplitudes is done. The Lagrangian is interpreted as a function on the groupoid describing the quantum system. Such Lagrangian determines a self-adjoint element on its associated algebra. Feynman's paths are replaced by histories on the groupoid which form themselves a groupoid. A simple method to compute the sum over all histories is discussed. The unitarity of the propagator obtained in this way imposes quantization conditions on the Lagrangian of the theory. Some particular instances of them are discussed in detail.
Paper Structure (14 sections, 73 equations, 3 figures, 1 table)

This paper contains 14 sections, 73 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction: The Lagrangian and Quantum Mechanics
  2. Quantum systems and groupoids: the qubit
  3. The kinematical description of quantum systems: outcomes and transitions: the qubit and other simple examples
  4. The Qubit
  5. Observables: the algebra of a groupoid
  6. States and dynamics
  7. The path integral description of quantum systems in the groupoids picture
  8. From Feynman's paths to histories: the qubit again
  9. Feynman's path integral picture of quantum mechanics
  10. Beyond Feynman's paths: histories
  11. The Lagrangian and the qubit
  12. Computing amplitudes in the groupoid of histories
  13. A path integral computation of the qubit probability amplitudes
  14. Discussion and conclusions

Figures (3)

  • Figure 1: Left: Diagrammatic representation of the qubit groupoid. Right: the $A_2$ Dynking diagram.
  • Figure 2: Diagrams representing histories on a groupoid. The vertical axis is the time axis and the horizontal axis represents the space of outcomes $\Omega$ of the system. On the left two composable histories $w_1, w_2$ are displayed (orange line). In blue a discrete approximation to $w_1, w_2$ are shown as well as their steps $w_1: \alpha_1, \alpha_2,\alpha_3, \alpha_4$, and $w_2 : \beta_1, \beta_2, \beta_3, \beta_4$. On the right hand side a future-oriented discrete history $\tilde{w}_1$ (in blue), and a past-oriented history $\tilde{w}_2$ (in red) that can be composed with $\tilde{w}_1$ are shown.
  • Figure 3: Abstract representation of how a history $w$ (in blue) on the qubit groupoid $A_2$ can be decomposed as another history $\tilde{w}$ (in red) with the same origin $(+, \tau )$ and end $(-, 8\tau )$, and a loop $\sigma$ (in green).