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I've always wondered (and was re-inspired to explore further from these two videos) that if at a single point of time we know about the complete state (position, momentum, spins, everything.) of every particle in an isolated system (e.g., the whole Universe), can we perfectly determine its future? And by same analogy, also the past?

I understand that according to the uncertainty principle, it is impossible to determine all state properties of a particle accurately at the same time.

So my question is, is it like future of an isolated system is already determined but is just not perfectly predictable by an observer because of limitations in observability?

UPDATE: To be more clear, the question is not about whether we can or can not determine the future but about whether or not the future of an isolated system is already fixed. I.e., given exactly same initial conditions, will two isolated systems always be in same state with time? Is our future already fixed even though we'll never know it in advance?

UPDATE: Would appreciate less technical and more accessible explanations. I'm not from Physics background but am interested in it nonetheless.

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It depends. Do you mean by "determine its future" that the future states are determined, or that measurement results are determined? You might also enjoy this or this question. – ACuriousMind Jul 27 '14 at 3:00
@ACuriousMind future states. – Vikrant Chaudhary Jul 27 '14 at 3:03
FYI, "hidden variable" is a term used to talk about theories in which quantum uncertainties merely represent limits of our measurement/knowledge, but that in reality there's an "actual value". Mathematics and experiment have ruled out various entire kinds of hidden variable theory as unworkable or inconsistent with observation, and I think that area of research is worth your time. It speaks to your intuition that things might be more predetermined than we're able to measure, and shows that some of the ways we might like this to happen actually aren't the case. – Steve Jessop Jul 27 '14 at 9:42
@bechira: well, Einstein though it was relevant to the matter of determinism (and said so), but then Einstein was wrong about hidden variables anyway. In any case I think the questioner probably doesn't know enough to ask his question precisely (especially around exactly what a "state" is that he's asking about), and that the answer "the Schroedinger equation evolves predictably" doesn't satisfy the request. But hey, if it's the straightforward answer then post it, and if the questioner accepts it then I'm flat wrong and it does satisfy :-) – Steve Jessop Jul 27 '14 at 15:11
@SteveJessop "questioner probably doesn't know enough to ask his question precisely". Yes, I agree. :-) – Vikrant Chaudhary Jul 27 '14 at 15:14

There are a few ways to answer your question, and I will try to list some of them.

  1. According to Quantum Mechanics, and due to the Heisenberg Uncertainty Principle, we cannot predict the future state (position and momentum) of any system. Given the state of a system in classical phase space $(\textbf{r}(t_0), \textbf{p}(t_0))$, we cannot determine the state at some later time $t$. However, given a quantum state $ | \Psi (\textbf{r}, t_0)\rangle$, we can use Schrodinger's equation $i \hbar \partial_t | \Psi \rangle = H | \Psi \rangle$ to predict the state's evolution. The difference here is we are tracking the probability of the system being in some classical state$^1$, not which state it will be in.

  2. Unfortunately, we do not have a "theory of everything" at the moment, only effective theories that cover certain domains (specifically, certain energy domains and length scales). Even if we were given the quantum state of the Universe, we wouldn't have the physics to determine its time-evolution. In some sense, we may never arrive at such a theory, and only have better and better effective theories that cover a wider range of natural phenomena.

Edit: It's important to note that quantum mechanics doesn't make any philosophical arguments about the observer's role in nature. It isn't that we don't have enough information to know exactly where the particle is or what its momentum is at some future time. It's that the particle doesn't even have a well defined position or momentum until we measure it. What happens when we interact with the particle is currently up for interpretation by different interpretations, and there is no definitive answer at the moment.

Edit 2 (Less technical explanation): It's difficult to answer your question in a nontechnical way because we need to define what you mean by "perfectly determine [the Universe's] future." Remember, by the uncertainty principle, it is impossible to determine the exact position and momentum of a particle. So, I'm assuming you mean that we begin with the complete quantum state of the system in question. All we can do we this state is determine the probabilities of each particle having a position/momentum within some range of values. Theoretically, yes, we can determine the future quantum state of the system (and thus the future probabilities). This is my answer in part 1. In part 2, I explain that our current understanding of the Universe is incomplete. At the moment, we use our best estimates of a Theory of Everything would look like. However, these estimates only cover certain areas of physics, and some are incompatible at the moment (e.g., General Relativity and Quantum Field Theory). In this sense, we cannot determine the future, even if we had access to the current state of the Universe.

$^1$ Specifically, by "tracking the probability of the $\ldots$ classical state," I mean that given the quantum state $ | \Psi (x, t_0)\rangle$ at some time $t_0$, we can use Schrodinger's equation to determine the probability of the particle being located between some $x$ and $dx$ (or between $p$ and $dp$ in momentum space) at $t_0 + \Delta t$. The probability is given by

$$ P = \langle \Psi (x, t_0 + \Delta t)| \Psi (x, t_0 + \Delta t)\rangle dx. $$

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More or less, even if we get a "lot" of different quantum states, most of them can be cancelled out cus they are unlikely to happen. Say, one of the result simplifies to a world where students get marks for sleeping more! But, that is plainly not possible, so we reject it. Similarly, most of the quantum states could be rejected, and if we look at the ones which are possible, they would be few. So, yes, future is more or less predictable. – Awal Garg Jul 27 '14 at 17:09

A deterministic universe need not be predictable. And even a deterministic universe not hampered by any limits to observability need not be predictable.

As an example take a toy universe consisting of an infinite chain of $0$'s and $1$'s. This 1D cellular universe evolves according to cellular automata rule-110: the state of a cell becomes $1$, unless the current value of the cell and its right neighbor are both equal to $0$, or if the current value of the cell and both its neighbors are all $1$s. It goes without saying that this 'universe' is fully deterministic with each of its discrete states being defined with zero uncertainty.

It has been proven that such a 'rule-110 universe' is capable of universal computation. Now we can ask the question: can this universal computation capability be deployed to predict the future states of this cellular universe? In other words: is there a shortcut within the rule-110 universe that allows it to anticipate its own future state?

The answer is 'no'. The universal computation capability does not provide a shortcut. The fastest route to get to generic future states is to 'play out the full evolution' and predictions are not possible within this rule-110 universe.

Added following the update to the question: a question like "Is our future already fixed even though we'll never know it in advance?" can be meaningful only when operationalized. This means that the term "fixed" needs to be defined in a way that allows us to test if our future is "fixed". The only viable way to do this is to interpret the question "is our future fixed?" as being synonymous to "do the laws of physics allow us - at least in principle - to predict our future?". The above 'rule-110 automata reasoning' indicates the answer to that question to be "no".

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A shortcut shouldn't be needed, it is deterministic when every time the simulation is run from the same start state ('play out the evolution') the result is the same. – API-Beast Jul 28 '14 at 10:02
@API-Beast: Exactly this, the question wasn't asking whether the future could be determined by us, but whether it 'is' determined. – David Mulder Jul 28 '14 at 10:34
@DavidMulder -- the question posed: "So my question is, is it like future of an isolated system is already determined but is just not perfectly predictable by an observer because of limitations in observability?" – Johannes Jul 28 '14 at 13:21

Even in a quantum universe, all evolution is deterministic if interpreted under Many-Worlds interpretation. So all possible futures could be "already determined", but you would still be unable to know which of those futures will be directly experienced by your qualia, since qualia experiences are always described by non-unitary probabilistic projection operators.

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Which is one of the reasons why "many worlds" doesn't really solve any problems. – Brandon Enright Jul 27 '14 at 3:43
MWI is not meant to solve any particular problem: It is just the natural, observer-invariant description of quantum mechanics that preserves unitarity at all times. Nothing more, nothing less – diffeomorphism Jul 27 '14 at 5:42
@diffeomorphism Indeed. It is funny how quickly people seize on an opportunity to throw away well estabilished theories, such as the finite speed information can propagate at. Copenhagen interpretation really breaks a lot of those, and MWI doesn't. So really, it does solve a problem - a problem that the (still) mainstream quantum physics interpretation flatly breaks well estabilished physical laws and theories, not to mention the absurd weight it gives to something like human-level observation. – Luaan Jul 28 '14 at 7:36

So my question is, is it like future of an isolated system is already determined but is just not perfectly predictable by an observer because of limitations in observability?

In quantum mechanics the outcome of measurements is in general not determined when given the exact physical state of the system in general. But apart from the uncertainty arising from this, the future is as determined as is the present. If the entire solar system were to be put in a big perfectly isolated box, then you could in principle perform a measurement to determine who is the US president in the year 2017. This measurement would collapse the wavefunction of the box such that it would evolve exactly toward the measured outcome.

If $O$ is an observable, then to measure this observable in the state that lies a distance t into the future, you would need to measure the observable $\exp\left(\frac{i H t}{\hbar}\right)O\exp\left(-\frac{i H t}{\hbar}\right)$ in the current state.

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The universe cannot be predicted from a single data point about a moment in time because inertia does not exist in any one moment but is critical to how a system would develop.

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protected by Qmechanic Jul 27 '14 at 16:21

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