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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 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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    $\begingroup$ This is very interesting question, but it has two problems: 1) can you please edit it a bit to ease reading? 2) It feels to me to be bit too broad. Can you please pinpoint one question you really need to get answered $\endgroup$ – Pavel Janicek Jun 15 '16 at 9:17
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    $\begingroup$ This question doesn't appear to have a point - what is it you are asking? Your first few sentences don't make sense. $\endgroup$ – Rory Alsop Jun 15 '16 at 9:30
  • $\begingroup$ Two electrons cannot share all their properties, two electrons doing the same thing in a different position is fine. Photons can be in the same place with the same properties. $\endgroup$ – Donald Hobson Jun 15 '16 at 10:41
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    $\begingroup$ There's a question here of how deep do you go. To me, all Emperor Penguins are identical, to a geneticist all Emperor Penguins are unique. Do you want your world to have something that looks like an Emperor Penguin or do you want it to have the same Emperor Penguins. $\endgroup$ – Separatrix Jun 15 '16 at 10:58
  • $\begingroup$ @DonaldHobson I wonder if the two photons with the same properties are somehow distinguishable? To refer to Separatrix's example: Will I be able to find a differences between this two photons if I go deeper? $\endgroup$ – BobbyPi Jun 15 '16 at 12:48
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The existence of a second, identical earth crucially depends on the question whether the universe is finite (closed) or not. In an infinite (open) universe, there are infinitely many twin earths, and there are earths with all possible small deviations of history as well. The reason is, that quantum mechanics only allow for a finite number (although immensely huge) of arrangements of matter. Given an infinite universe .... gotcha!

The only remaining problem is to find and reach a twin earth: This is impossible because the twin earths are (with almost certain probability) beyond the horizon of the observable universe.

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  • $\begingroup$ Afaik, open universe just expands infinitely, not "contains any combination of atoms you could imagine". Could you please clarify why would the earth have a twin in an open universe? $\endgroup$ – user8808 Jun 15 '16 at 11:01
  • $\begingroup$ @Roux: Some years ago, there was an article in "Spektrum der Wissenschaft" (the German edition of "Scientific American") explaining different types of infinity in the universe. Unfortunately, I don't have it at hand to be more specific. $\endgroup$ – jk - Reinstate Monica Jun 15 '16 at 12:15
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I'm considering only the chance of something happening in an infinite universe, where anything statistically possible, becomes highly probable, simply because of the sheer volume of opportunity.

There are two key factors at play here.

One is related to the law of large numbers. Given enough opportunity something very similar will arise, no matter how small the odds of this happening, in a truly infinite universe it's odds on to happen. There will also be a considerable number of quite similar though not actually identical worlds.

The other is convergent evolution. Given a very similar world, in a very similar star system, very similar lifeforms are likely to arise to fill the very similar ecological niches. There are going to be black and white seabirds (we have a lot of black and white seabirds), of which at least one is liable to end up flightless, and sooner or later could well end up looking a lot like an Emperor Penguin. To an expert in Emperor Penguins it'll be very different, but not to your average punter.

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One relevant concept is that of a phase space. The basic idea is that if one entity can exist in two states, then two of them can define four states, three can define eight, and N can have 2 to the power N states. By the time N is large, the number of possible ensemble states or combinations gets so large that there is not enough time in a universe for any state to repeat, or for more than an infinitesimal fraction of possible states ever to exist.

For example in biology: no creature that has ever existed as a consequence of sexual reproduction will ever exist again as a consequence of a future sexual act. Indeed, only an infinitesimal fraction of the possible individual instances of a species will ever actually be realized.

So uniqueness follows from simply from the number of possible combinations of entities compared to the number of entities being permuted and the finite size of the universe within which they are being combined. In practical terms the more important question is about when two similar permutations might seem to be identical, although we know that at a fundamental level they cannot be so.

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If the universe is big enough it will repeat in detail over extended regions.

I've seen calculations going over the number of possible quantum states in various sized regions from the planet up to the entire observable universe (the hubble volume). If the universe is larger than that volume multiplied by the number of states, than there must be duplicates.

In particular, the universe might be infinite in spacial extent. That's an open question as yet. If so, then there are an infinite number of duplicate Earths down to the last atom.

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It has been stated (although not proven) that "if anything is not expressly forbidden, it is inevitably compulsory". Therefore the question would become one of scale: Is it possible for me to observe truly identical existence? Consider the state of matter when involved in quantum entanglement, that the two deviated elements are identical is assumed to be true, but they can be said to 'suffer' from uncertainty, because of the same mechanism. From a scaled perspective, two entangled particles, two (seemingly) identical penguins and two spiral galaxies present a threshold of scale that would conceivably prevent verification within the remaining life of the observable universe.

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