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Feynman checkers: lattice quantum field theory with real time

Mikhail Skopenkov, Alexey Ustinov

Abstract

We present a new completely elementary model that describes the creation, annihilation, and motion of non-interacting electrons and positrons along a line. It is a modification of the model known under the names Feynman checkers or one-dimensional quantum walk. It can be viewed as a six-vertex model with certain complex weights of the vertices. The discrete model is consistent with the continuum quantum field theory, namely, reproduces the known expected charge density as the lattice step tends to zero. It is exactly solvable in terms of hypergeometric functions. We introduce interaction resembling Fermi's theory and establish perturbation expansion.

Feynman checkers: lattice quantum field theory with real time

Abstract

We present a new completely elementary model that describes the creation, annihilation, and motion of non-interacting electrons and positrons along a line. It is a modification of the model known under the names Feynman checkers or one-dimensional quantum walk. It can be viewed as a six-vertex model with certain complex weights of the vertices. The discrete model is consistent with the continuum quantum field theory, namely, reproduces the known expected charge density as the lattice step tends to zero. It is exactly solvable in terms of hypergeometric functions. We introduce interaction resembling Fermi's theory and establish perturbation expansion.
Paper Structure (23 sections, 56 theorems, 146 equations, 7 figures, 2 tables)

This paper contains 23 sections, 56 theorems, 146 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Organization of the paper
  4. Feynman anticheckers: statements
  5. Construction outline
  6. Axiomatic definition
  7. Exact solution
  8. Asymptotic formulae
  9. Identities
  10. Combinatorial definition
  11. Generalizations to several particles
  12. Identical particles in Feynman checkers
  13. Identical particles in Feynman anticheckers
  14. Fermi theory and Feynman diagrams
  15. Open problems
  16. ...and 8 more sections

Key Result

Theorem 1

The functions $A_k(x,t,m,\varepsilon,\delta)$ and the lattice propagator $\widetilde{A}_k(x,t,m,\varepsilon)$ are well-defined, that is, there exists a unique pair of functions satisfying axioms 1--3, and limit eq-def-anti-alg exists for each $(x,t)\in\varepsilon \mathbb{Z}^2$ and $k\in\{1,2\}$. For where the minus signs are taken when $t<0$. For $(x+t)/\varepsilon+k$ odd, limit eq-def-anti-alg is

Figures (7)

  • Figure 1: Feynman (anti)checkers as a six-vertex model. Weights of odd (top) and even (bottom) vertices depend on mass $m$, lattice step $\varepsilon$, and small imaginary mass $\delta$. Each configuration has also an overall sign defined globally in terms of a loop decomposition. See Definition \ref{['def-anti-subgraphs']}.
  • Figure 2: Normalized expected charge density in the discrete model (dots) and continuum quantum field theory (curves). Plots depict the left (dots) and the right side (curves) of \ref{['eq-cor-anti-uniform']} for mass $m=4$, lattice step $\varepsilon=0.03$, time $t=6$, and position $x$ being an even multiple of $\varepsilon$.
  • Figure 3: A checker path (left). A generalized checker path (right). See Definitions \ref{['def-mass']} and \ref{['def-anti-combi']}.
  • Figure 4: Plots of (the normalized imaginary part of) the lattice propagators $\mathrm{Im}\,\widetilde{A}_1(x,6,4,0.03)/0.12$ (top, dots) and $\mathrm{Im}\,\widetilde{A}_2(x,6,4,0.03)/0.12$ (bottom, dots) for $x/0.06\in\mathbb{Z}$ and $x/0.06+1/2\in\mathbb{Z}$ respectively, their analytic approximations from Theorems \ref{['th-continuum-limit']} (dark curve) and \ref{['th-anti-ergenium']} (light curve). The former approximation is the imaginary part of the spin-$1/2$ Feynman propagator $\mathrm{Im}\,G^F_{11}(x,6,4)$ (top, dark) and $\mathrm{Im}\,G^F_{12}(x,6,4)$ (bottom, dark) given by \ref{['eq-feynman-propagator']}.
  • Figure 5: Lattices of sizes $1$ and $2$ (left); see Example \ref{['ex-1x1']}. Notation for edges (right).
  • ...and 2 more figures

Theorems & Definitions (131)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1: Consistency of the axioms and concordance to Feynman's model
  • Proposition 1: Fourier integral
  • Example 1: Massless and heavy particles
  • Example 2: Unit mass and lattice step
  • Proposition 2: Rational basis
  • Proposition 3: Exact solution
  • Remark 2
  • ...and 121 more