Is it physically possible that we may one day simulate the entire universe with every single particle, field and law of physics factored in? Can n number of particles (say the number of particles that make up my computer) represent what happens with "more than n particles" without neglecting, generalizing or rounding up anything. If so, would it be possible for the beings in the simulated universe to know about it?
I am assuming that you do want a simulation of the whole universe and not just a theory of everything.
Your question should be decomposed into two questions.
The first is really a mathematical question: Can a part (the simulator) simulate the whole ?
Given a positive answer to the first question, the second is whether the mathematical structures thus identified can be used to describe the universe.
To be true, I am mostly incompetent on both accounts, and I am only trying to make sense of the question, not stating too fast that it is impossible. So please do not take this as an answer (who would have one?) but rather as speculation on how an answer the question could make sense.
A part that simulate the whole means that somehow you can define a structure preserving bijection between the part and the whole. I am not quite sure I am correct, but this reminds me of the self-similarity and fractal structures ... To be checked with someone more competent than myself in fractals. Then the question would be whether a fractal structure is compatible with what we know of the universe. Building a bijection between an infinite set and an infinite subpart of that set is quite common. Can it be done in a way that preserves the laws describing the universe?
But such a bijection is possible only if the universe is infinite, and then the simulator would have to be infinite too.
Another constraint might be that the simulator should be a localized fragment of the whole, rather than spread uniformly (as you would have with a mapping of integers on the multiples of some integer $p$, these multiples playing the role of the simulator. But then, I am not sure how "localized fragment" should be defined meaningfully. This is why I was inclined to consider fractal structures, rather than more general structures that are isomorphic to some of their subparts.
But I have to leave it to more advanced physicists than I to tell whether that can be compatible with what we know of the physics of the universe.
Of course not, you would have to also simulate the simulation, etc. ad infinitum.
To address one of the OP's comments: no, this does not mean we can never have a theory of everything. A theory of everything is a theory that can describe every type of fundamental particle and interaction; there is nothing in this definition that says you have to simulate the entire universe if you have one!
Is it concievable that we may one day simulate the entire universe with every single particle
Who should type the properties of every single particle into the computer? Even if the calculation power were available (which it aint) there is nobody who would live the time to make the input.
But more seriously, Stephen Wolfram has some good recitals on Youtube about the universe possibly beeing a cellular automaton, which means that it would take the whole universe to simulate the whole universe (because no simplifications can be made if you want to trace every particle).
The next problem would be that the quantum world is rather probabilistic than deterministic.
A computer made up of n (finite) particles will not be able to simulate all the states of a larger system. This is known as the Pigeon hole principle.
If the simulator is made up of an infinite number of particles, it may be possible. But it would have to already exist; it would be impossible to construct.
Consider: Connect the output of the simulator to an LED such that "yes" = LED ON and "no" = LED OFF. Query the simulator on whether the LED will be OFF when it renders output. What will happen? The simulator will violate its own prediction when it renders output either way.
Beings inside the simulation should be able to know that they are in a simulation. Information always leaks.
protected by Qmechanic♦ Dec 22 '15 at 13:30
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