# What is the point of path integral for boson and fermion?

I am a beginner to study QFT and confused about path integral for boson or fermion.

I have read about the path integral for single particle, and finished some problems. But I cannot understand the next chapter which is about path integral for boson and fermion.

What is the difference between the two kinds of path integrals? What is the point of path integral for boson and fermion?

I find that there are big differences in forms between single particle, boson and fermion. I do not understand why the book uses different forms to discuss them. Even the path integral for the spin system has a new form to discuss.

In my opinion, I think the biggest difference between single particle and boson (fermion) is statistics, but how to consider about the statistical properties in path integral? Just using one symbol for differential, $D$, to take the place of the original symbol, $d$, is enough?

• are you really asking about the difference between quantum mechanics and QFT? and what book? Commented Feb 6, 2014 at 9:57
• I am asking about QFT, I am confused about path integral for bosons and Fermions. I cannot understand the formula and I want some help. Btw, the book is QFT in condensed matter physics, by N. Nagaosa, a very brief book. Maybe it is not a good book for a beginner, and I also know many other books, but I just have this one book on my hand to read. Commented Feb 6, 2014 at 12:56
• The point of the path integral is to determine the probability at the particle in question will go from point A to point B. Commented Jan 30, 2017 at 9:58

The fields that enter into the bosonic path integral are just commuting $c$-number fields - in other words, the field values are just plain old complex numbers, not quantum operators. So you don't need to keep track of the ordering of the fields, which makes them much easier to work with.
By the way, you never need to use the path integral - it's mathematically equivalent to the canonically-quantized Hamiltonian formalism with operator-valued fields, which looks a lot more like what you do in nonrelativistic QM. The path integral formalism's just much more mathematically convenient to work with, although admittedly more abstract and less physically intuitive (especially in the fermionic case). (Also, the path integral formalism's manifestly Lorentz invariant, while the Lorentz invariance of the canonical Hamiltonian formalism is much harder to see, because the Hamiltonian transforms under Lorentz transformations as the $T_{00}$ component of the rank-two stress-energy tensor, rather than as a Lorentz scalar like the Lagrangian density.)