One more observational consequence of many-worlds quantum theory

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Russian State University of I. Kant

УДК 524.85

One more observational consequence of many-worlds quantum theory

A.V. Yurov

Introduction

In article [1] Don Page has presented the forcible argument that the lifetime of the de Sitter universe t_max < 10⁶⁰ yr. On the other hand, the string theory prediction grants the dS universe as much time as ф< recurrence time ~ yr [2], [3] (the matter of whether it should be seconds, years or even millenniums is really unessential for such monstrous numbers).

It is possible to lower this value to the ф ~ yr and even to the limit of ф ~ yr for models with instantons of Kachru, Pearson and Verlinde [4] and with 2 Klebanov-Strassler (see [5]) throats [6].

But, nevertheless, even with assumption that one of those models do describe our Universe, the magnitude ф will still be way too large as compared to Page's 10⁶⁰ yr.

Thus we are facing the following dilemma: either the dark energy is not pure positive cosmological constant at all (and stringscape should not have any significant long-lived positive metastable minima) or the dark energy IS the cosmological constant and our ideas about stringscape (and the strings theory in general) are absolutely false!

Actually, the nature of dark energy is one of those questions, which can be redirected to astronomers. It appears, that there exist some observational series which proved to be suciently difficult to explain with the assumption that the dark energy should be some scalar field (quintessence) rather then cosmological constant. (The phenomenon of this kind is, for example, the drift of unhomogenous local volume (1 Mpc) with the regular Hubble flow inside [7]).

Of course, those results can appear to be of statistically insignificant nature, but if not, then it would mean one strong graphic evidence of presence of the cosmological constant.

So, does there exist some kind of ”loophole” in the Page reasoning? Something, that would allow us to conclude that the lifetime of dS universe can be t_max ~ t_f with t_f ~ googleplex?

As we shall show further, such loophole indeed exists. To present it we will have to re-examine the essense of the Page's argumentation, and it will be done in the next Section. In Section III we'll consider the case of phantom cosmology to show that it surprisingly grants us t_max = t_f, and therefore, in such universes the Page reasoning doesn't lead to inevitable conflict, as differs from dS models. Next Section is essentially devoted to the universes filled with vacuum energy and to the way of prevention of Page conclusion t_max « t_f . Here we will show that it is possible to obtain t_max ~ t_f in assumption that the total number of human observations is the quantum variable. And Section V is the overall conclusion.

Page argument

Following to Page, suppose that the process of observation is described by some localized positive operator A, such that application of it to any state ш leads to positive central tendency. This implies that every possible observation has some positive probability of occurrence in the given volume (e.g., as a vacuum fluctuation). Therefore, we can treat the observers as the standart quantum objects. With this in mind, Page has calculated the action for the brain of a human observer: Sbr ~10¹⁶ J Чs, and the probability p_br ~ . Then, Page made an estimation for 4-volume for the brain (V₄(br)), taken in process of making the observation: V₄(br)

Next, let's assume that we are living in dS universe filled with vacuum, energy density с_Л ~ 10^-29 gramm /cm³ of which greatly exceeds the total density of all other energy components in the universe. Then we appears to be merely prisoners in ”cosmic prison” of a radius R = c/H, where Hubble constant H = V8рGс_Л /3. After 10¹⁷ yr each and every star in the universe will be either black hole, black dwarf or neutron star; 10¹⁰ Gyr later the temperature of neutron stars will decrease to less than 100 K. It is mildly speaking unlikely that human-observers will be able to endure in such inhospitable universe.

However, it wouldn't really matter at that point, because no life (including human-observers) will be able to exist there forever due to both proton decay (it's time life t_pr > 10³² yr) and the exponential fallo of the density of matter (information, being processed in ever-expanding universes was considered in [8].

There has been shown that an infinite amount of information can be processed via the usage of temperature gradients created by gravitational tidal energy, but only in assumption that the cosmological constant is equally zero). Therefore if t_f ~ googleplex then except for unimaginably tiny initial period from the big bang to tpr the universe will be absolutely dead. It is definitely not bright future at all!

On the other hand, taking into account the unimaginably long lifetime of such universe we shall conclude that all possible events, including those with extremely low probability, will someday occur. One of the most interesting of those unlikely events would be the spontaneous appearance from quantum fluctuations of ”observers”, surrounded by ”environment” suitable to permit the ”observation”.

With this conclusion, it would be only natural to ask: can we in principle be one of those ”vacuum observers”? And, more generally: under what circumstances will the ordered (i.e. classical) observations dominate over vacuum ones? Page gives the following answer: if t< t_max = 10⁶⁰ yr, and only then will the human observations be with high probability ordered.

Otherwise, almost all observations in the universe will have its root in vacuum fluctuations. As the result, in universe with t_max ~ tf ~ googolplex our, obviously ordered, observations are to be considered as something embarrassingly atypical.

Page concludes that ”This extreme atypicality is like an extremely low likelihood, counting as very strong observational evidence against any theory predicting such a long-lived universe with a quantum state that can allow localized observations”, and makes the prediction that the universe just will not last long enough to give 4-volume > .

To show this in work, let's consider the total 4-volume of universe:

(1)

The probability of vacuum fluctuations p_vac < p_br whereas the probability of ordered occurrences p_ord > p_br. Multiplying V₄(t) by p_ord results in the volume of the part of total V₄(t) where ordered occurrences are dominating ones.

Now let N be the number of observations, made during the past human history. The product N·V₄(br) will mark the part of total V₄(t) where ordered human observations all take place. If humans are the typical observers (anthropic principle!) then one can expect that

V₄(t) p_ord ~ V₄(br) N. (2)

Substituting a(t)= a₀e^Ht into the (1) one get V₄(t). Following Page we can evaluate N ~ e⁴⁸. Substituting N and V₄(t) into the (2) allows us to express p_ord. Finally, using the inequality p_ord > p_br one comes to conclusion that, under those circumstances, the timelife of the dS universe is indeed t < t_max = 10⁶⁰ yr.

Phantom energy

Let's see, what will happen with Page results in the universe filled with phantom energy. It appears, that in contrast to dS models, for such universes we get a remarkable concordance: t_f = t_max up to very high degree of accuracy.

Before we start, we should mention, that the particular interest to the models with phantom fields arises from their prediction of so-called ”Cosmic Doomsday” alias big rip [9]. Since for the phantom energy we have w = p/(c²с)= ?1?? with ? > 0, the integration of the Einstein-Friedmann equations for the flat universe results in

(3)

Where . Choosing t = 0 as the present time, a⁰ ~ 10²⁸ cm and с⁰ =1.4с_c/(2 + 3?) as the present values of the scale factor and the density (If ? « 1 then с₀ ~ 0.7 Ч 10^-29 g/ cm³), at time t = t_f = 1/о, we automatically get the big rip.

Now, let's return to our question. Equations (1) and (3), taken together, lead to

where V₄(0) = a₀ = 10¹¹² cm⁴ = e²⁵⁸ cm⁴. Using Page approach we have

The second member of the equation is

therefore

(4)

In the case ? « 1 we get

(5)

Now, we have to consider 3 different cases.

a. t_f ~ 1.685 Ч 10⁶⁷ s = 5.3 Ч 10⁵⁹ yr. In this case the power of exponent in (5) is small enough to use the expansion in Taylor's series. It's application results in inequality t < t_max =5.3 Ч 10⁵⁹ yr.

This leaves us with the same problem as in dS situation: the end of the world will take place at t = t_f but ordered observation will be dominating ones while t «t_f only.

b. t_f ~1.685 Ч 10⁶⁷ s.

In this case

Here we come to remarkable difference between phantom and dS cosmologies. While in the last case we have t_max = 10⁶⁰ yr ~ t_f > googolplex, where t_max follows from Page's reasoning and t_f is the string theory prediction, in the former case the situation can be much more agreeable: in fact, the validity of the b condition ensures that t_max ~ t_f. It can be shown that t_max > t_f very fast when t_f decreases. If t_f =5.3 Ч 10⁵⁰ yr then and t_f = 22 Gyr stands for, thus actually erasing the very di?erence between t_max and t_f .

c. t_f ~ 1.685 Ч 10⁶⁷ s. This case implies

Therefore, in such Universe only about half of all observers can assuredly consider themselves classical and having the naturally ordered observations, which is su?ciently better then what we had in dS universe, yet still being far from perfect.

Summarizing all of the above, we can conclude that the one and essentially the only convenient case is b. After all, for t_f < 10⁵⁹ yr it gives us t_f ~ t_max for granted!

The number of coarse-grained histories

Let's return back to the case of dS universe and seeming inconsistency between t_max and t_f (t_max ~ t_f ), that has been found in it. The core of Page's argumentation is the equation (2).

But let's inspect carefully the quantities, forming it. It is clear that, by complete analogy with p_br, quantity p_ord should be calculated by quantum laws. As a matter of fact, p_ord = Therefore, l.h.s of equation (2) is dependant on But the equivalence will hold only if the same will be true for the r.h.s.! If the value V₄(br) is purely classical, then N is the only remaining candidate for the dependency on

At a first glance this conclusion seems absolutely grotesque, but it appears to be right in touch with Page reasoning. As a matter of fact, in his article Page deals with quantum (or semi-classical) observers.

The number of quantum observers N is the quantum quantity and hence, must be calculated by the quantum laws. From this point of view, it is no wonder that N will depend on

But if this is correct, then one can't use Page estimate (N ~ e⁴⁸) anymore. Unfortunately, we can't calculate N explicitly, but we can evaluate it upon usage of very simple quantum-based reasoning. It is already clear that ”new” N should be much greater then e⁴⁸. As we shall see, this number can exceed even googolplex, thus totally refuting Page argument.

One can roughly evaluate the number N as the number of coarse-grained histories: where Nc is the number of spacetime cells and Nb is the number of relevant bins in field space. In the article [10] Garriga and Vilenkin have made this for the spacetime volume with the size R = ct₀ where t₀ = 10¹⁰ yr. As a result they got N ~ e. Substituting this value into the (2) one get t_max = 10²⁶¹ yr. This number is by many orders greater than Page's 10⁶⁰ yr but is still too small in comparison with . However, the number N easily allows for additional increase up to the point, where t_max will be comparable with strings predictions.

Indeed, remaining in framework of quantum theory we should consider all possible observers, including those who are living in much older universes where vacuum energy already exceeds the total density of all the other energy components in the universe. In such universes

For example, if t = 10¹⁷ yr (the era of black holes) one have V₄ = and if t = 10³² yr (the low bound of proton lifetime) then

The number of spacetime cells of size L will be in first case and in the second one. But in all cases the values of N are given by googolplex-like numbers:

Substituting them in (2) we finally get t_max =yr or t_max =yr.

The interesting fact here is that both these numbers lies in remarkable agreement with the results of [6]. In particular, for the case of 3 -branes with some parameters there have been obtained theoretical value t_f ~ (lifetime on the NS5-brane).

Decays due to decompactification are much faster: t_f ~. Those are results of the models with the single KS throat. Consideration of 2 KS throats (such models are more interesting since they result in positive cosmological constant whereas the models with single KS throat result in Л < 0) in case of KPV instantons leads to such value as t_f ~ -very good agreement with the previously obtained t_max =.

In the case of general position one can conclude that

yr,

cosmological quantum mechanics phantom

where is the maximal possible lifetime of ”human-observers”. Thus, if t_max =googolplex one get = 10¹¹⁷ yr while t_max = yr implies yr. Of course, it can be difficult to imagine that 10¹¹⁷ yr later the universe will be filled by ”human-observers”. Besides, it can be argued whether such ”observers” fits into the set being reviewed or not. But the answer is very simple: whenever the probability of finding ourselves in such universe has the nonzero value, we have to take it into account.

Finally, we should answer the following question: are those ”auxiliary” observers real, or not, i.e. can we ascribe all of them to some really existing Universes, or are they nothing more then ”vacuum probabilities”? The answer is: yes, they have to be real; otherwise, we are facing the situation, where the quantum objects are required to explain the existence of e⁴⁸ (real) objects. Here is the same Page's paradox, only in other form and aggravated by much worsen numbers!

Conclusions

As we have seen, the assumption that N is the total number of quantum observers results in such lifetime of universe which is comparable with strings predictions. This creates the very strong grounds for serious consideration of such strange possibility. After all, the quantum nature of N seems to be absolutely inevitable in quantum cosmology.

Of course, such state of affairs is something highly unusual in ”everyday” quantum mechanics. It has already become a common fact, that in laboratory research with neutron interferometer the neutron passing through a beam splitter will split into ”two neutrons”. But in lab we don't expect that the same will be true for us. Observers are classical objects ”ad definition”.

However, in quantum cosmology this situation changes drastically. Since we are nothing but the part of the universe we have no choice but to consider ourselves as quantum objects. Page has shown in [11] that quantum cosmology can give observational consequences of many-worlds quantum theory. We think that our results can be consider as one more observational evidence of validity of many-worlds quantum theory.

References

1. Page Don N., The Lifetime of the Universe // ArXiv, 2003, hep-th/0510003.

2. Goheer N., Kleban M., and Susskind L., The Trouble With De Sitter Space // J. High Energy Phys, 07, 2003, p.056, hep-th/0212209

3. Kachru S., Kallosh R., Linde A., Trivedi S. P., De Sitter Vacua in String Theory // Phys.Rev. D, 2003, 68, p.046005, hep-th/0301240.

4. Kachru S., Pearson J., and Verlinde J., Brane/Flux Annihilation And the String Dual Of a Non-Supersymmetric Field Theory // JHEP, 2002, 06, p.021, hep-th/0112197.

5. Klebanov R. and Strassler M. J., Supergravity And a Confining Gauge Theory: Duality Cascades and chiSB-resolution of Naked Singularities // JHEP, 2000, 08, p.052, hep-th/0007191.

6. Frey R., Lippert M., and Williams B., The Fall of Stringy De Sitter // Phys. Rev. D, 2003, 68, p.046008, hep-th/0305018.

7. Chernin D., Cosmic Vacuum // PHYS-USP, 2001, 44 (11), p.1099-1118.

8. Barrow J.D. and Hervik S., Information Processing in Ever-expanding Universes // Phys.Lett. B, 2003, 566 p.1-7, gr-qc/0302076.

9. Caldwell R.R., Kamionkowski M. and Weinberg N.N., Phantom Energy and the Cosmic Doomsday // Phys. Rev. Lett., 2003, 91, p.071301, [astro-ph/0302506].

10. Garriga J. and Vilenkin A., Many Worlds in One // Phys.Rev. D, 2001, 64, p.043511, grqc/0102010.

11. Page Don N., Observational Consequences of Many-Worlds Quantum Theory // ArXiv, 2004, quantph/9904004.

Annotation

УДК 524.85

One more observational consequence of many-worlds quantum theory. A.V. Yurov. Russian State University of I. Kant, 236014, Kaliningrad, Al. Nevsky str., 14, Electronic address: artyom_yurov@mail.ru

Using new cosmological doomsday argument Page predicts that the maximal lifetime of de Sitter universe should be t_max = 10⁶⁰ yr which is way too small in comparison with strings predictions (ф>googolplex). However, since this prediction is dependant on the total number of human observations, we show that Page arguments result instead in astounding conclusion that this number is the quantum variable and is therefore much greater then Page's estimation. Identifying it with the number of coarse-grained histories in de Sitter universe we get the lifetime of the universe comparable with strings predictions. Moreover, it seems that this result can be considered as another one of the observational evidences of validity of the many-worlds quantum theory. Finally, we show that for the universe filled with phantom energy t_max ~ t_f up to very high precision.

Keywords: de Sitter solutions, cosmological measure, “freak” observers, Everett interpretation, branes

Аннотация

Еще одно наблюдаемое свидетельство в пользу «многомировой» интерпретации квантовой механики. А.В. Юров

Дон Пэйдж предсказывает, что максимальное время жизни де ситтеровской вселенной должно составлять число порядка tmax=10⁶⁰ лет, что намного меньше оценок, сделанных в рамках струнного пейзажа. Однако это предсказание зависит от полного числа наблюдений, которое следует считать квантовой, а не классической переменной. Мы отождествляем число наблюдений с числом крупногранулированных историй, вычисленных с использованием голографического принципа. В этом случае оценки, полученные методом Пэйджа, и оценки, сделанные в рамках теории струн, становятся сравнимы. Интересно то, что сам метод вычислений основан на использовании эвереттовской интерпретации квантовой механики. В заключение мы показываем, что при наличии фантомной компоненты ожидаемое, по Пэйджу, время жизни Вселенной автоматически оказывается сравнимо с временем до финальной сингулярности «большого разрыва».

Ключевые слова: вселенная де Ситтера, вакуумные флуктуации, струнный пейзаж, больцмановские наблюдатели, интерпретация Эверетта

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