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The Laws of nature are universally applicable and at every point in force. Together they shape our universe but are all "shapes" unique?

For example, is it possible that there is a second identical earth somewhere in the universe?

Or is it possible that there exists an exact copy of a tree somewhere sometimes on earth? And what about a grain of sand; can one find two identical grains of sand?

Or going to even smaller scales; can it be that two free electron are identical? I know that two bounded electrons cannot occupy exactly the same state because of Pauli's principle and thus are not identical. I also know that I can replace an electron with another electron without causing any change; an electron is an electron. But are they actually indistinguishable? As an analogy:

Imagine there are two workers with the same skills. Worker 1 is controlling device X. If worker 1 is replaced with one of his colleagues, the device will work as before. If a third person A is observing the device, he will notice no change. He is unable to say who controls the device right now (worker 1 or 2). For A it makes no difference what worker is controlling the device; the device keeps its functionality. For A it is as if there is just one worker. Now imagine you are one of the workers. Again each worker is interchangeable without altering the functionality of the device. But from your perspective the workers are unique. They just perform the same task. For A all workers are identical and he is not able to distinguish the workers from each other because the replacement of one worker with another is not changing the function of the device. However, a worker is able to distinguish the workers and thus for him they are not identical.

I want to know: If I have an object A, will a second object exists that is an exact copy of A and cannot be distinguished from the original?

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closed as unclear what you're asking by knzhou, CuriousOne, John Rennie, ACuriousMind, user36790 Jun 26 '16 at 3:13

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ All electrons, protons, atoms etc. are exactly the same. They are so much the same that we have to symmetrize/anti-symmetrize our equations to get the correct answers. As you go up in scale the number of possible combinations of these identical objects increases exponentially and the probability of finding the same "thing" twice becomes vanishingly small. There is, if you like, "no chance in hell", that there is a second earth or even anything remotely close. $\endgroup$ – CuriousOne Jun 24 '16 at 19:53
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    $\begingroup$ You have Pauli's exclusion principle backwards: it is because electrons are all identical that they cannot have the same state. $\endgroup$ – Stéphane Rollandin Jun 24 '16 at 20:07
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If I have an object A, will a second object exists that is an exact copy of A and cannot be distinguished from the original?

If A is a very simple object such as an electron, a nucleon, or even an atom or a small molecule, yes. It is actually an important notion in quantum mechanics that indistinguishable objects (fermions, bosons) collectively behave differently from distinguishable ones.

What if A is a tree, or Earth? When we look at the structure of such an object it appears as a complex, open system. From a reductionist point of view, it can be considered as composed by smaller objects, ultimately atoms. Atoms are themselves composed but in the context of your question we can stop there because as pointed out above similar atoms are indistinguishable, so in principle we could build an exact copy of a tree from the same atoms it is made of. But even if we managed to do that, or if it happened naturally by chance, it would be an exact copy for a fleeting instant only. An open system is constantly interacting with its environment (a tree feeds, breathes, grows and decays, Earth sends and receives EM radiation, is subject to tidal forces, is impacted by different sort of objects). So for a copy to stay a faithful copy, we should also copy the environment. Even that would probably not be enough though, because the dynamics of a complex object is so sensitive to initial conditions that the minutest difference will develop and soon lead to a clearly different state. Ultimately it is safe to say that any open system with a sufficiently large number of degrees of freedom can only be unique.

We are left with the intermediary isolated mesoscopic systems. Isolated means negligible interaction with environment, mesoscopic means not too many degrees of freedom. These objects are more likely to appear in a similar state and may at times be indistinguishable.

Note that a grain of sand is a macroscopic object. It contains in the order of $10^{19}$ atoms, which means a staggeringly high number of possible ways to assemble them. There are no two identical grains of sand on Earth.

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The answer to your question is we do not know, and will never know. As you stated it, the uniqueness property depends on our ability to perceive a difference. But that ability is always limited to some level of detail, beyond which differences, if any, are no longer perceived. For instance some people has the idea that each elementary particle contains inside is a universe in itself. If such were the case then the answer would be yes, uniqueness is a property of nature, just we do not know it yet. Current theories do assume though that all elementary particles of the same kind are identical, but as I said, this might be a consequence of our limitation to find a difference.

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