u/golgho__

Hi !

Disclaimer : although the title seems quite philosophical, I wish to discuss what the equations of standard QM tell us about reality, and only that. Despite the aversion of this channel for crackpot philosophical digression (which I respect and support) it still seemed the more appropriate to me, due to the higher average level of posts and answers in this channel wrt to others. I hope this is fine.

1. Context :

A few days ago, I was chatting with a colleague from the philosophy department. She asked me "as a physicist, do you think QM challenges Aristotle's principle of non contradiction" ?

The Stanford encyclopedia for philosophy says that Aristotle's principle of non contradiction means that "It is impossible for the same thing to belong and not to belong at the same time to the same thing and in the same respect".

I believe what triggered my colleague's question is that she probably has read stuff like "in QM, an electron can be in several places at the same time", which is quite common in more or less pop science articles.

2. My attempt to construct a rigorous answer to her question

I believe QM does not contradict the principle of non contradiction, but it forces us to distinguish between what the system is (ie its state vector) vs what we can say about it (ie what we can measure, aka the spectrum of the operator).

Let's take the example of "can an electron be at different places at the same time" ? I believe the answer is no, because being in a superposition state does not mean being at different places at the same time.

Disclaimer : for simplification, I will assume the position basis is a 1D discrete basis. I will always assume that the state vector is normalized.

What the system is = the vector state. What I can say about the system = the outcome of a measurement (ie the eigenvalue that pops out if I apply the operator to the state vector).

a) Special case : the vector state is an eigenstate of the position operator.
This is the only case when I can fairly say "the system is at position x_i". In this special case, I recover the good old Newtonian perspective where 1) I can measure the position of the system without altering its state and 2) the position is both a fundamental aspect of what the system is (it's the fact that the ith coordinate of the state vector equals 1 in the position basis while all others equal 0) and it's what I can say about the system (it's the eigenvalue that pops out if I apply the position operator to the vector state).

b) General case : the vector state is in a superposition state. The question "what is the position of the system" is ill defined. The reason is that, unlike in the Newtonian perspective where the position is both a fundamental aspect of what the system is and what I can say about it, in QM position is only a fundamental aspect of what I can say about it.
It is true that there are several non-zero coordinates of the state vector in the position basis. But that does not mean that the system is at different positions at the same time. Special case a) has defined what it means to be at position x_i : it means the i_th coordinate of the state vector in the position basis has value 1. If the state is in a superposition state, there is no such coordinate : they all have values < 1.

Therefore QM is not in conflict with Aristotle's principle of non contradiction : either the system has a well defined single position, or it has no position at all.

What do you think of my attempt to answer this question ? Do I miss something ? Did I make any conceptual and/or physical mistakes ?

nb : I willingly let aside the delicate question of the collapse (or branching or whatever your favorite interpretation) of the state vector on one eigenstate during the measurement process. There, the question becomes really tricky of what really happens but I guess the honest answer is that current QM is floppy about it. I guess one could say that right at the "moment of the collapse" (whatever that means), the principle of non contradiction is somehow challenged because the system switches more or less instantaneously from a superposition state to an eigenstate for no obvious reason (in Copenhagen) or for an unknown hypothetical external reason (in objective collapse). Many world somehow manages to escape this contradiction, but the costs are quite high.