QFT and coordinate transformations

Question

I was going through this lecture, where Leonard Susskind explains that these two Feynman diagrams are considered to be equivalent in QFT:

One diagram explains the same phenomena in one possible way and the other describes the same thing in a different way. But, when I look at it, it seems an awful lot that if you take one of the diagrams, and rotate the entire space by $90$ degrees, then you get the second diagram.

I know that one of the axis represents time, so it is not as easy as I described, but imagine drawing one of these on a paper and then rotating the whole paper.

So, of course rotating the entire co-ordinate system is a coordinate transformation, but I don't know whether any coordinate transformations apply in QFT.

So, my question is, are these two situations equivalent i.e. if I make a $90$ degree rotation of the coordinate system, is the same as the second possible process for the phenomena? And if they are similar, would this mean that you would have to count the second possibility in the path integral, as it is the same first possibility turned over?

Answer 1

I haven't seen the lecture, but if you take the time axis to be the same in each diagram, the processes are not equivalent, and you cannot "rotate" one diagram into the other. The key is that although space and time both are a part of Minkowski space and "mix" under Lorentz transformations, they are not entirely equivalent: there is no coordinate transformation that rotates the t axis into the x axis, for example. The reason is that the Minkowski metric $\eta_{\mu\nu}$ gives a minus sign to space and a plus sign to time. If a spacetime vector $x^\mu$ points "mostly" in the direction of time, it will have a positive Minkowski norm $x^\mu x_\mu$; since Lorentz transformations preserve Minkowski norm, the spacetime vector will have a positive norm in every frame, indicating it is points "mostly" in the direction of time in every coordinate system. This means that there is no valid transformation that swaps space and time axes: as a result, the two diagrams you gave are not related by a coordinate transformation, and therefore represent distinct physical processes.

There are, however, two senses in which the diagrams could be said to be "equivalent".

1. There exist physical processes in which both diagrams contribute to the path integral. In positron-electron scattering, for example, both of the diagrams you listed contribute to the overall probability for the process (along with a third diagram with equal contribution, and infinite others with smaller contributions). In the sense of "corresponding to the same process" or "never occurring without the other", the two diagrams are therefore equivalent.
2. There are distinct physical processes corresponding to diagram 1 and diagram 2 that are equivalent on the level of math. Instead of both diagrams representing positron-electron scattering, we can take the first diagram to represent positron-electron annihilation/pair production and the second diagram to represent electron-electron scattering. Then, when you write down the mathematical expression corresponding to the two processes, it turns out that they're equal!

In short: there's no sense in which flipping time and space axes is a valid coordinate transformation, so diagrams related by this transformation are not physically equivalent. The diagrams are, however, related by two weaker senses of equivalence: any physical process involving diagram 1 will necessarily involve diagram 2, and there exist distinct physical processes for which diagram 1 and diagram 2 yield the same mathematical expression.