Path Integrals for Fermions and Bosons

According to the spin-statistics theorem, fermion systems are described by totally antisymmetric wave functions and bosonic systems by symmetric ones. In other words, the wave functions must be antisymmetric/symmetric under the exchange of two identical particles,

$$\Psi_{\rm B/F}({\bf R}) = (\pm 1)^{\mathcal{P}}\; \Psi_{\rm B/F}({\mathcal{P}}{\bf R}) \quad,$$ (53)

where ${\mathcal{P}}$ stands for any of the $N!$ permutations of the particle label in the many-body coordinate $R$. In systems with additional internal degrees of freedom such as spin, the permutation is applied to those as well. The $+$ sign corresponds bosonic systems (B) and the $-$ sign to fermionic systems (F). This symmetry property can be realized by applying an antisymmetrization/symmetrization operator to a wave function for distinguishable particles $\Psi_D({\bf R})$,

$$\Psi_{\rm B/F}({\bf R}) = \frac{1}{N!} \sum_{\mathcal{P}}\; (\pm 1)^{\mathcal{P}}\; \Psi_{\rm D}({\mathcal{P}}{\bf R}) \quad.$$ (54)

The density matrix for a fermion/bosonic system is constructed from these states and can be written as,

$$\rho_{\rm B/F}({\bf R},{\bf R}';\beta) = \frac{1}{N!} \sum_{... ...}\; \rho_{\rm D}({\bf R},{\mathcal{P}}{\bf R}' ; \beta) \quad.$$ (55)

One can (anti)symmetrize with respect to the first or second argument or both. All three ways are equivalent and lead to the same physical observables. The (anti)symmetry enters into the path integral formalism as a sum over all $N!$ permutations. In addition to the integral over all configurations of paths, one has to sum over possible permutations of final set of coordinates ${\bf R}'$. Eq. 2.19 then reads,

$$\rho_{\rm B/F}({\bf R},{\bf R}';\beta) = \frac{1}{N!} \sum_{\mathcal{P}}\; (\pm 1)^{\mathcal{P}}\; \int \!... ...\! \int {\bf d}{\bf R}_{1}\:{\bf d}{\bf R}_{2}\:\ldots\: {\bf d}{\bf R}_{M-1}\;$$

$$\quad \rho_{\rm D}({\bf R},{\bf R}_{1}; \tau ) \: \rho_{\rm D}({\... ...f R}_{2};\tau ) \ldots \rho_{\rm D}({\bf R}_{M-1},{\mathcal{P}}{\bf R}' ;\tau )$$ (56)

$$= \frac{1}{N!} \sum_{\mathcal{P}}\; (\pm 1)^{\mathcal{P}} \! \! \! ... ...athcal{P}}{\bf R}' } \! \! \! \! {\bf d}{\bf R}_t \;\; e^{-S[{\bf R}_t] }\quad.$$ (57)

The $\beta \to 0$ limit from Eq. 2.14 now becomes,

$$\rho_{\rm B/F}({\bf R},{\bf R}';0) = \frac{1}{N!} \sum_{\mat... ...)^{\mathcal{P}}\; \delta({\bf R}-{\mathcal{P}}{\bf R}') \quad.$$ (58)

In most applications, one uses the path integrals to calculate averages from Eq. 2.7. There one needs the trace of the density matrix, which means one sums up all closed paths (for sampling with open paths see chapter 5). For distinguishable particles, they start at any ${\bf R}$ and return to it. For fermions and bosons one also sums over paths that return to a permuted set of coordinates given by ${\mathcal{P}}{\bf R}$. Those contributions become relevant if the degeneracy parameter (Eq. 2.17) is the order of 1 or greater.

The path integral technique has been applied extensively to bosonic systems in particular to liquid $^4$He (Ceperley, 1995; Grüter et al., 1997). It is an exact method because all permutations carry the same sign and one does not have to deal with cancellation effects of positive and negative contributions as in fermionic systems. Those will be discussed in section 2.6. We call a method exact (see (Ceperley, 1996)), if it only contains approximations, which can be controlled by an adjustable parameter, and therefore converges to the exact numerical results with increasing accuracy.