- #1

hgandh

- 27

- 2

*Quantum Theory of Fields Vol. 1,*Chapter 4. Under interchange of two identical species of particles we have:

\begin{equation}

\Phi_...p,\sigma,n...p',\sigma',n...= \pm \Phi_...p',\sigma',n...p,\sigma,n...

\end{equation}

Plus sign for bosons and minus for fermions. As far as I understand, this is just a statement on labeling identical particles. The labeling of particle 1 with ##p,\sigma## and particle 2 with ##p',\sigma'## is the same physical state as vice versa because we cannot distinguish the two particles. Now this is where I get confused.

I am not sure if I understand this correctly but this is my interpretation: In the first case, we can treat all particles of different species as distinguishable and therefore fix them in some order. In the second, some particles of different species are indistinguishable and we define the state-vector in accordance with some standard state-vector. Lastly, we treat all bosons as indistinguishable from each other as well as all fermions. If somebody could help explain this quote it would be much appreciated."What about interchanges of particles belonging to different species? If we like, we can avoid this question by simply agreeing from the beginning to label the state-vector by listing all photon momenta and helicities first, then all electron momenta and spin z-components, and so on through the table of elementary particle types. Alternatively, we can allow the particle labels to appear in any order anddefinethe state-vectors with particle labels in an arbitrary order as equal to to the state-vector with particle labels in some standard order times phase factors, whose dependence on the interchange of particles of different species can be anything we like. In order to deal with symmetries like isospin invariance that relate particles of different species, it is convenient to adopt a convention that generalizes (1): the state-vector will be taken to be symmetric under interchange of any bosons with each other, or any bosons with any fermions, and antisymmetric with respect to interchange of any fermions with each other, in all cases, whether the particles are of the same species or not."