Quantum Theory – A view from the inside Part II
Let us make a few formal assumptions about the virtual universe and the observer to make the following exploration easier:
- The virtual universe has a concept of space and locality. All interactions only act locally and propagate with some kind of speed limit.
- Time is imposed from the outside as part of the simulation.
- The observer is a mechanism that is contained within a finite region of space for all time
What does the observer know about his universe in the best case? And maybe even more importantly, what does he not know?
First, everything he can possibly know must be encoded somehow in the sequence of states that the universe passes through, because that’s all there is.
Then, he cannot know about certain symmetries of the formulation of the simulation. For example, if the virtual universe is a cellular automaton with an integer number in each cell, then surely the observer cannot decide which numbers are in the cells. At best he can figure out only their relationship. For example, he could add 5 to all numbers and still get a cellular automation law that generates the exact sequence of states but with an offset of 5 in each cell. However, there might be some choice of numbers that makes the laws of his observed physics very simple or beautiful. And even if the original simulation used entirely different numbers, the observer would surely prefer the more elegant representation.
Another example of an unobservable symmetry is the state space of quantum theory. If we simulate the unitary evolution of a state vector, then it is impossible to decide from any mechanism that is itself simulated, what the magnitude or phase of the state vector is. That is because quantum theory is linear and the state evolution commutes with scalar multiplication. Also, there is no most elegant choice for a complex factor and the observer would decide that all work equally well. Remember, we are not assuming any kind of statistical interpretation of the state vector, just unitary dynamics, so normalization is not an obvious elegant choice. But we may assume that he want his state descriptions to be unique and so he identifies all vectors that are related by a nonzero complex factor. The resulting projective space containing the rays of the original state space is then his new state space.
As simple as it may seem, this is an important point. In fact, we will find a symmetry that will allow us to make a statement about reconstructing the quantum state of the universe similar to this one, with surprising consequences.