The underlying principle for the introduction of path integrals in
imaginary time is the product property of the density matrix stating
that the low temperature density matrix can be expressed as a product
of high temperature matrices. In operator notation this reads,

(27) |

This is called a path integral in imaginary time. The expression is exact for any . In the limit , it becomes a continuous paths beginning at and ending at .

The reason for using a path integral is in the limit of high
temperature, the density matrix can be calculated. Usually the
Hamiltonian can be split in a kinetic and in a potential part,
and the density matrix can expressed using the following
operator identity (Raedt and Raedt, 1983),

(29) |

(30) |

(31) |

(32) |

The density matrix for a system of particles in the primitive approximation is given by,

In the path integral formalism this is written as,

(35) |

The free particle terms act like a weight over all Brownian random walks (BRW) in imaginary time starting at and ending at . In the limit of , this leads to the Feynman-Kac relation

In the

(37) |

Already, the primitive approximation is a sufficient basis for a path integral Monte Carlo simulation. However, the required number of time slices to reach accurate results would be enormous. The following sections, we discuss methods to derive a more accurate high temperature density matrix in order to reduce the number of slices to a computationally feasible level. A pair action will be derived, which contains the exact solution of the two particle problem. This means only one time slice is required for a simulation of two particles at any temperature. However, in many particle systems, a path integral is needed because of many-particle effects and the fermion nodes, which will be discussed in section 2.6.