Now you must realize that from a scientific perspective, this kind of time travel is near impossible. Whenever the object reappears, if it materialises, it will invariably take the space of whatever was there in that volume already. And probably cause some intense action at the molecular level. The only way such a device can work is if it creates an absolute vaccuum at the place where the object is supposed to teletemport (I just made that word up). For this, the device should know well in hand when an incoming object is expected, and as such, shouldn't allow outside interaction during that period. If the said device is what is the object of this question, then there are two cases possible: 1. Causality is maintained, and when you open the microwave for the first time, you already find a box inside. You take it out, put the new box in, and resend it for your past self to find. The box undergoes a linear timeline containing a loop. There are no other timelines or world branchings involved. 2. 781
Causality is not maintained, and when you open the microwave for the first time, it is empty. You proceed to send it back in time. This creates a branching timeline, where the box is already in there, and when you open the microwave, you find the box already inside. You take it out, and may or may not decide to put the box in again.
In the second case, the box completely disappeared from the first worldline, and there are two boxes in the second. If the you from the second world also decides to send the box back, you might end up in a pseudo causal system, where the box is sent to another parallel worldline, where it again comes out from the microwave, and a different you again puts his box back in. So while to each of you it would appear that you are in a causal system and indeed getting your box back, you are in fact getting the box from the previous worldline, and yours is getting sent to the next. It "appears" to be a causal system for everyone but the initial you. One way to check which one you are in is, to not completely observe the box that came out of the microwave, and make a tiny mark somewhere on the underside of the box you are going to send into the microwave. If the box that came out of the microwave doesn't have this mark, you can be sure of multiple worldlines. If the mark is there, it still mightn't be a causal system, as the previous worldline's you could also have decided to mark the box. My 2 cents 783