# Interference with a minus sign

In Lectures on Physics, by Richard Feynman, pg 3-11, I found the following:

In case of electrons, the interfering amplitudes for exchange interfere with a negative sign.

I was unable to understand this. On reading further, it seems that electrons have spins unlike alpha particles and that causes them them to interfere in this manner.

I am unable to understand the connection between the statement and the reason given. I would be grateful for any clarity on this matter.

I didn't truly understand the + sign either, but it made more sense when I thought about it like either this or that happens to reach the final state, there are two ways to get there, so I need to add the probability amplitudes of both to get the final amplitude. (We always use + to indicate or in probability.)

Now to your question. My explanation below is not discussing the same experiment as on pg 3-11 of Feynman. But if you think about it carefully, you will see the connection. The interference occurs when we are talking about identical particles. Consider two identical particles. We describe their state by some probability amplitude. E.g., the square of this amplitude gives you probability of finding one electron at point A and another electron at point B. Imagine that you exchanged the two particles. In this process, something can happen to the amplitude, but only in such a way as to keep the square the same (if we exchange two identical particles, the probability of finding these two particles at points A and B will not change). Mathematically, this means that exchange operation can only affect the complex phase of the original probability amplitude (probability amplitude gets multiplied by $e^{i\varphi}$ with some $\varphi$). Now imagine, you exchange the two particles again. The amplitude again changes (by the same factor $e^{i\varphi}$). However, two exchanges is the same thing as not exchanging at all! So after two exchanges, you must have the original probability amplitude. In the end, this means that there could be two possibilities. The exchange operation keeps probability amplitude the same (trivial case) and the exchange operation multiplies the amplitude by a -1 (non-trivial). Now, this thinking will not lead any further, in a sense it cannot explain why electrons behave the way they do. Experimental evidence suggests that Nature realizes both cases, those particles that do not change the sign of amplitude upon exchange (bosons) and those that do change the sign of amplitude (fermions). Electrons belong to the second category. That's it. This is experimental fact, which we have to accept. At least this is the spirit, in which Feynman explains Quantum Mechanics. We cannot always understand all laws of nature, but we can still describe them (at least we hope to).