Btw this is me, Server. I just decided to change name b4 it is used

I thought they were using cognition to an extent. Firstly their appearance is of rebellion, which is why they have personas. Secondly fake gun looks real so it is real. Thirdly all the main boss are also rebels to a certain degree, that's why they have palaces. If it's the metaverse, they can't complete until the public trusts phantom Thevies. If it doesn't help, then please elaborate a little more, okay.

Can they use illusion as a power? Like if theyre to cut their body in half, will it really cut into half?

Well seeing how if you kill a shadow of someone they die in the real world, i would guess not. Also seeing how Ryuji could have died in the first palace by a bunch of guards, they would just die. The gun are the closet thing to it, also They could make a cog. clone so using that maybe. Don't know if it helps or not. If not i can try to clarify.

In the palaces, they can use cognition a bit (like with the guns), but mainly they can't. The creator/owner of the palace can use cognition, hence what their palace is. To explain whether or not they can die in the metaverse, they can. When you kill someone's shadow, they die. Rather than that the PTs are in the metaverse themselves, in place of their shadows, which means they are easily susceptible to death.

Is the Metaverse related to other world like the TV world?

probably not at all

Nah, they're a little related in the fact that they're both shadow worlds but the main diffrence would be the rules that apply to each world which is effected by what form of divinity made the shadow world. An important thing to note with the TV world is that a lot of what formed there was based on people perspective of who they wanted to know more about. In a way you could argue that, that was a form of cognition but that's more of the speculative side.

Well first of all, Morgana specifically mentions MEMENTOS as the Collective Unconscious. Palaces are more individual to that specific person's cognition, while Mementos is everyone's palace. In Jung's work, the Collective Unconscious is where all humans draw their root symbols and meanings from, and they bring it to their personal unconscious. The only way the Phantom Thieves can affect the Collective is by affecting EVERYONE, which they do, as each of their acts brings more awareness to the population of the world, and expands the scope of what the Phantom Thieves can explore in the Collective Unconscious. To put it simply, palaces are more influenced by their owners, so the Phan Thieves would have to affect the owner's cognition, but in Mementos, they have to affect EVERYONE'S cognition

I just have another question, its kinda related to the theme but maybe not. After the calling card has been sent, the ruler of the palace wiil be on their guard and the security will be tight but there 's still a chance for the phantom theives to slip through right? but based on the whole cognition thing, wouldnt the defence supposed to be completely impenetratable?

When the ruler realizes their Treasure is at risk, rather than go full defense mode, they basically say "Come and take it!" and up the security to 99%. They want you to fight them, because they believe they can defeat you. That's why they don't lock everything down.

Yeah, in every single case, they shadows are basically taunting the PTs to TRY and steal it