# In QFT, does the “sum over paths” of the Feynman formulation requires that all the possible world-lines share the same proper time?

QUESTION:

1. In QFT, does the “sum over paths” of the Feynman formulation requires that all the possible world lines share the same proper time for a given process?

RELATED QUESTIONS:

1. Since one acknowledges the fact that this integration is performed in accordance to the underlying principle of extremal action, is there any constraint(s) for the allowed proper time interval(s) spawn by this sum?

2. How the ensemble of encompassed world lines is related to the uncertainty (position-momentum or time-energy) principles (for instance, considering virtual exchange quanta being on or off mass shell)?

• Do you set up your experiment in such a way that only paths with the same proper time count? In that case, yes. Imagine what happens to a classical wavefront when you add that requirement. Whether quantum or classical domain, it's a different experiment. – CuriousOne Aug 14 '16 at 4:55
• @CuriousOne : Thanks for your comment. As it may appear, I am new to QFT. What do you mean by " Whether quantum or classical domain, it's a different experiment."? I am considering the quantum domain for a given experiment performed in a fixed stationnary lab. – Wilj Aug 14 '16 at 5:16
• You have probably seen these videos: youtube.com/watch?v=1zmRulUaDk8, and then there is the issue that fixing the time will lead to an energy uncertainty. Whether we do this in QFT or classical theory, the effects are similar. – CuriousOne Aug 14 '16 at 5:25

In QFT, it isn't the paths of particles that were summed. Rather, it is the field configurations. So proper time is not relevant here. What happened for the usual amplitude computation in QFT is that one specifies field configuration at $t=-\infty$ and $t=\infty$ in a specific coordinate, and 'path integral' considers all possible fields with the specified boundary condition weighted by $e^{iS[\phi,J]}$.