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  1. Quantum mechanics says that the entropy of an unobserved system remains constant. As such, the apparent growth of entropy is a subjective illusion. If we consider the wave function of the universe, its entropy remains constant at the value of the Big Bang, that is zero or one basic unit.

  2. Holographic principle says the surface of the cosmic event horizon is proportional to the universe's entropy.

Given these two principles one should conclude that the area of the boundary of the universe remains always zero or at Planck scale, while what we see as the big universe is illusion.

Is this true? If not, what is explanation?

I have one conjecture solving the paradox, please tell me if it is plausible. My conjecture is that our universe has no event horizon, and instead it is bounded by the particle horizon which is the Big Bang. Since the surface area of this horizon exactly coincides with the surface area of the initial universe, the paradox is resolved: our universe always had and has zero (or basic unit) of entropy.

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  • $\begingroup$ can you give a link for your statement 1. ? seems to contradict the definition of quantum entropy physics.stackexchange.com/q/43816 $\endgroup$ – anna v Feb 8 '14 at 5:12
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    $\begingroup$ @annav See physics.stackexchange.com/a/63420/19976 $\endgroup$ – joshphysics Feb 8 '14 at 5:15
  • $\begingroup$ When one reaches paradoxes in combining two different physics frameworks, as here GR and QM one has to look at the assumptions used. Big Bangs black holes and horizons are the framework of GR. We do not have a consistent framework of quantized GR except maybe string models that have not been validated with data. QM goes to classical mechanics through decoherence where unitarity is lost and the argument of constant entropy with it. This has to be examined in the context of a unified QM GR theory. Lets hope some expert in strings picks this up. $\endgroup$ – anna v Feb 8 '14 at 6:08
  • $\begingroup$ Look up the definition of "entanglement entropy": dividing space into different sections (as eg a horizon does) creates an entropy that is proportional to the surface area, even though the total entropy is zero. $\endgroup$ – WIMP Feb 8 '14 at 9:23
  • $\begingroup$ The fact that the entropy of a microstate is zero and invariant under time translation is not solely a quantum property. In classical mechanics this is expressed via the Liouville theorem and in the quantum version by the quantum Liouville theorem. Now, if one looks at macroscopic variables or state/emergent variables, their effective evolution is not unitary anymore and their associated entropy can grow even in an isolated system. $\endgroup$ – gatsu Feb 8 '14 at 11:01
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I will wade in where angels fear to tread.

1> Quantum mechanics says that the entropy of an unobserved system remains constant. As such, the apparent growth of entropy is a subjective illusion.

Quantum mechanics states this for a system that has a wavefunction and all the incoming outgoing lines of the integral which describes a physical measurement are coherent. Coherent means all the phases are known and their functional dependence in time too.

If we consider the wave function of the universe,

Here is a big hole in the argument. Quantum mechanics is a great theoretical model of the microcosm, order of fermi or angstrom and currently with nano studies,nanos of the centimeter. That is the region it has been validated in. There do exist superconducting wires of order of kilometer which also can be modeled by quantum mechanics. When we come to orders of magnitude as the universe great care has to be taken.

The universe, physicists agree, has been described well by General relativity . The validation of the theory has happened using cosmological observations of order of light years, billions. We have interpreted the time projection on the heavenly sphere and a bit further with our satellites, and GR is by consensus accepted as a good model. The extrapolation of this model leads to the singularity called the Big Bang.

Now we have to decide what we mean by entropy in GR. Classical entropy, counting microstates, might be assumed to be 0 since there was only one microstate at the singularity. Classically there is no reason to assume that the entropy stays constant. Thermodynamics and logic tell us that classically it is increasing. Certainly by the stage we are now it is enormous, classically, since the number of microstates is close to infinity.

To assume that :

its entropy remains constant at the value of the Big Bang, that is zero or one basic unit.

Is a logical fallacy, in my opinion.

We have good theoretical proofs that classical mechanics emerges smoothly from quantum mechanics, with no contradictions, including the law of entropy. This happens because of decoherence. If we assume a density matrix describing the universe, the off diagonal elements of this matrix , i.e. the phases, are lost by the time we see the universe that exists around us. This is because the macrocosm is modeled beautifully by classical mechanics and the way the two frameworks join is through decoherence as the system becomes macroscopic.

There exists no well accepted quantized model of gravity that includes the standard model of elementary particles. String theories have candidates but no model has been validated as yet. When one extrapolates to the early universe when the approach to the singularity reaches sizes compatible with the microcosm of elementary particles a semi-classical model explains the first time slices of the big bang:

Approximately 10^−37 seconds into the expansion, a phase transition caused a cosmic inflation, during which the Universe grew exponentially. After inflation stopped, the Universe consisted of a quark–gluon plasma, as well as all other elementary particles.[24] Temperatures were so high that the random motions of particles were at relativistic speeds, and particle–antiparticle pairs of all kinds were being continuously created and destroyed in collisions.

Already a quark gluon plasma treated semiclassicaly has an enormous number of microstates as there is no reason to suppose that the density matrix describing it has not decohered, i.e.lost the phase information. Otherwise the term "plasma" should not be used .

At about 10−6 seconds, quarks and gluons combined to form baryons such as protons and neutrons.

Now what about before 10^-37 seconds. Can we assume that the density matrix is still coherent and there exists one wave function for the universe at that time? It is a moot point, to be examined once we have a Theory Of Everything, i.e. particles and gravity in a single quantized model. It might be that it is constant at those very early segments of time.

your second assumption:

Holographic principle says the surface of the cosmic event horizon is proportional to the universe's entropy.

used to conclude :

Given these two principles one should conclude that the area of the boundary of the universe remains always zero or at Planck scale, while what we see as the big universe is illusion.

may be a valid question before 10^-37 seconds and the horizon of the universe then, but not to the present state of the universe.

There is no logical contradiction. We can estimate an entropy for the present state of the universe and it is not zero because the present state is described adequately by classical mechanics.

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