1. Introduction

The "many worlds" interpretation of the universe, although an ancient concept, is still a disturbing and confusing idea to many people. The confusion may reside in what it is meant by the word "world", or the more recently coined term "universe" which for many people implies or is equivalent to the "whole" or a single unity consisting of individual parts, e.g. galaxies, stars, planets, moons, life. Yet, instead of a single "universe" there may be "universes" such that the "universe" is not the entirety of existence, but a subset of many possible universes, i.e. the "many worlds." The same could be said of the more ancient term, "cosmos."

The ancient Greek philosophers, such as Democritus, considered the "cosmos" to consist of cycles alternated by states of order and disorder among the elements of nature. Each cycle, therefore, represents a different world. However, this concept was challenged by Aristotle who taught that the stars were fixed and unmoving and the universe was a finite sphere. The fixed stars were part of this celestial sphere. Aristotle's views were accepted by the Catholic church which viewed the universe as perfect and unchanging and which circled around Earth at the center. Although the religious Earth-centered universe was toppled by the heliocentric theory of Copernicus, the belief remained that our solar system (and therefore Earth) was at the center of a single galaxy with fixed stars devoid of other planets.

In the 16th century, Giordano Bruno (Singer, 1950) realized that the newly developed heliocentric theory of Copernicus could be applied to the other stars, which, he believed, were just like our sun. Since planets were orbiting around the sun, then it seemed logical that other stars were also ringed with orbiting planets. If there were other planets, Bruno argued, then other living beings might live on those planets. If there were an infinite number of stars, then there would be an infinite number of worlds each populated by living beings just like those on Earth. Bruno believed his views were supported by Biblical teaching. Since God is infinite and eternal the universe must also be infinite and eternal. An infinite God must have created an infinite universe and an infinite number of solar systems. Since the infinite God was everywhere, no planet or star had more importance than any other as God's presence was equally everywhere. Earth, therefore was neither at the center of the physical or biological universe.

Bruno's teachings were an anathema to the Church and he was arrested by the Roman Inquisition and charged with heresy in matters of dogmatic theology, and heresy for claiming the existence of a plurality of worlds and their eternity. Bruno was burned at the stake in 1600 by the Church authorities and his written works were banned, burned, and their publication prohibited.

"Multiple worlds" did not die with Bruno. In 1755, with considerably less risk, the Prussian philosopher Immanuel Kant again advanced the idea of multiple planets each populated by different inhabitants. Kant also interpreted what had been described as nebulous stars (nebula), as systems of stars each with independent worlds. Alexander von Humbolt in 1850 (Rioja & Ordoñez, 2006) described these as "island universes."

Therefore, it could be said, if there are multiple stars, with multiple planets, and an infinity of multiple "island universes" then what has been termed the "universe" may also be a subset of "many worlds," or, one "universe" among many.

Einstein's theory of relativity is also supportive of the "many worlds" conception of the cosmos. As summed up by Einstein (1952): "When a smaller box s is situated, relatively at rest, inside the hollow space of a larger box S, then the hollow space of s is a part of the hollow space of S, and the same "space", which contains both of them, belongs to each of the boxes. When s is in motion with respect to S, however, the concept is less simple. One is then inclined to think that s encloses always the same space, but a variable part of the space S. It then becomes necessary to apportion to each box its particular space, not thought of as bounded, and to assume that these two spaces are in motion with respect to each other.

Before one has become aware of this complication, space appears as an unbounded medium or container in which material objects swim around. But it must now be remembered that there is an infinite number of spaces, which are in motion with respect to each other.

The concept of space as something existing objectively and independent of things belongs to pre-scientific thought, but not so the idea of the existence of an infinite number of spaces in motion relatively to each other. This latter idea is indeed logically unavoidable" (Einstein,1952).

To paraphrase Einstein, the universe can be conceived as a geometrical representation of space-time, with all the fields which would represent the matter content of the universe being defined in such a space-time geometry, albeit as causally disconnected regions within the whole of space-time. For instance, let us take a spatially flat space-time, which is therefore infinite, endorsed with a cosmological constant. That might perfectly be the state of our current universe. Then, the universe exists as an event horizon behind which no signal sent by us can travel along. Therefore, because no signal can travel faster than light and the light sent by us now would take an infinite time to reach the event horizon (and therefore no further), observers separated a long distance in such an infinite space-time may become causally disconnected with respect to each other. These causally disconnected islands could be considered as single isolated universes. More generally speaking, the physical manifold to be considered in order to represent the universe can be topologically more bizarre than that representing s at space-time endorsed with a cosmological constant. Therefore, causally disconnected regions of space and thus many worlds (the multiverse) are predicted by relativity theory. This would provide us with another example of a many-world system within the realm of the contemporary gravitational theory.

2. A SIMPLE QUANTUM MODEL OF THE MULTIVERSE

It is the aim of this paper to show a simplified model of the quantum version of such a multiverse.

As described by Hawking (Hawking, 1990), quantum mechanically disconnected regions of the whole space-time manifold can be physically described by different quantum oscillators, each one representing the simply connected region in which a non-simply connected space-time can be divided. Each quantum oscillator represents a single classically isolated universe within a multiverse scenario. The quantum state of such a multiverse is given then by the state of a set of quantum oscillators.

As conceptualized by the standard Copenhagen model, the perceptual act of knowing, or representing space-time creates a discontinuity in the continuum; what has been described as the "collapse function." However, according to Everett's formulation (Everett, 1957), there is no collapse function, as the observers are embedded observers. In Everett's many-worlds model, the subjective appearance of wavefunction collapse is explained by the mechanism of quantum decoherence; every possible outcome of every event defines or exists in its own "world". Further, these worlds are entangled, although they may be infinitely far apart, emerging and disappearing into the quantum field. Therefore, any plausible representation of space-time cannot be considered as representing the whole of space time, but instead a disconnected fragment of it and only one possibility out of an infinite number of possible outcomes. Each conception, or perception of space time has a non-zero probability of being a physical realization of the universe as a whole.

Quantum fluctuations can refer to the subatomic nature of space time, where particles of energy come into existence only to be annihilated. However, these events take place so quickly and in such minute spaces that the law of conservation is not violated. These fluctuations also give rise to the uncertainty principle as applied to existence and non-existence, or the inability to know several properties of a single element simultaneously. However, since space time can exist at increasingly small spaces, to understand the implications requires an analyses employing the 1) Planck length, which is based on the speed of light in a vacuum, 2) Planck's constant (the size of quanta), and 3) the gravitational constant (the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them).

Quantum fluctuations among the different geometrical representations of space-time become extremely important in the vacuum state of gravity at Planck length. For example, as the scale of time and space grows smaller the compression effects of the collapse function causes the amount of energy contained within virtual particles to greatly increase. According to Einstein's theory of general relativity, increases in energy increases the curvature of spacetime. Therefore, decreases in space time increase energy, such that at infinitely small scales the energy of the fluctuations would rip apart spacetime, giving it a "foamy" character. Small regions of Planck length may rip out of the parent space-time and become tiny disconnected space-time regions on the order of the Planck length. The resulting bubbling soup of space-time foam creates virtual baby universes which are suddenly and continuously popping into existence and are also just as suddenly annihilated (Hawking, 1978; Wheeler, 1957).

Once a universe nucleates in the space-time foam, it may bubble and eventually jump into an inflationary period (Vilenkin, 1982). Some believe this explains the origin of our current universe, a bit of flotsam bubbling up in the sea of space-time. Therefore, one way or another, inevitably the "many worlds" quantum multiverse has be considered in a quantum theory of the universe, at least as far as the superposition principle is assumed to be held in such a theory.

3. THE THIRD QUANTIZED STATE OF THE MULTIVERSE

Let us consider the wave function of the universe (Hartle & Hawking, 1983), the Wheeler-De Witt equation (DeWitt, 1967) and a simple model of space-time filled with a vacuum energy represented by a cosmological constant. The universal wave function is the quantum state of the totality of existence, regarded as the basic physical fundamental physical entity, obeying at all times a deterministic wave equation. The Wheeler-De Witt equation (De Witt, 1967) refers to an observer perceiving, or an operator acting on a wave field consisting of configurations on all of spacetime. That is, instead of conceptualizing the universe as defined on a 3-dimensional space-like surface, the wave function contains all of the information about the geometry and matter content of the universe; the functional field configurations of all space-time.

The wave function can be obtained via the Wheeler-De Witt equation and can be directly related to the Schrodinger equation which describes how the quantum state of a physical system changes in time at the subatomic, atomic, and macroscopic level and perhaps in relation to the whole universe.

According to the Galilean theory of relativity (also known as Galilean invariance), the fundamental laws of physics are the same in all inertial frames. For example, an observer beneath the deck of a ship at sea, would not be able to tell whether the ship were moving or stationary so long as the ship is traveling at constant speed, without rocking, on a smooth sea. Likewise, the speed of light is the same for all inertial observers regardless of the state of motion of the source. Time is related to space, time, and motion. Therefore, time invariance is a demand of the principle of relativity.

As can be seen by the following Schrodinger equation, no time derivative appears in order to satisfy the time invariance demanded by the principle of relativity, i.e. H ∅ = 0. In the model being considered, the wave function of the universe, ∅ ≡ ∅(a), is given by

where the derivatives are taken with respect to the scale factor, a ≡ a(t). The frequency w(a) is related to the potential term in the Friedmann equation and for the present case it reads (González-Díaz & Robles-Pérez, 2008), where Λ is the cosmological constant, c is the speed of light and h is the Planck constant. On the other hand, Eq. (1) can be formally seen as the classical equation of motion of a harmonic oscillator with a time-dependent frequency (see, Lewis & Riesenfeld, 1969), where it is the scale factor which can be taken to play the role of time. The quantum state of a many-universe system or multiverse, Ψ can then be defined as the solution of the third quantized Schrodinger equation (Strominger, 1990),

where the Hamiltonian H is given, in the case being considered, by (Robles-Perez et al., 2009)

being p_∅ the momentum conjugate to the second quantized wave function of a single universe, ∅. Following the usual interpretation of particle physics, the quantum number N which labels the basis states of harmonic oscillators should be thought of then as the number of universes within a multiverse scenario. Let us notice however that the interpretation of such a quantum state need not to be made in terms of a certain number of universes. For those who does not feel contented with a many world interpretation of the universe the quantum state given by Eq. (2) may be also interpreted as the quantum state of a particular eld con guration, where it is the second quantized wave function of the universe the eld which is now quantized. Then, the quantum number N might be considered just as a quantum label without any further interpretation in terms of a number of realized universes. Anyway, the quantum state of each disconnected region of the whole space-time is given in the present model by the state of a quantum harmonic oscillator. Let us also notice that other energy-matter contents of the universe can also be depicted by the state of a harmonic oscillator as well as open and close universes. In a general case, the Friedmann equation can be written as,

where, p ≡ aa, is the conjugate momentum to the scale factor, the third term in Eq. (4) accounts for the spatial geometry of the space-time, being k = -1; 0, 1 for hyperbolic, at and close universes, respectively, and p₀ is the energy density of the fluid which may fill up the universe, being w. the proportionality constant in the equation of state of such a uid, i.e., p = wp , where p and p are the pressure and the energy density of the uid, respectively. Then, when a canonical quantization procedure is taken, p → =ihϑ_a and the quantum state of the multiverse is still represented by the harmonic oscillator obtained from Eqs. (2) and (3), with a new more complicated frequency given now by,

Therefore, the general quantum state of a multiverse made up of disconnected regions, each one being dominated by a particular kind of energy-matter content is given by a set of quantum harmonic oscillators, i.e.

where Ni is the number of universes of type i, represented by a second quantized wave function φi, which are filled up with an energy-matter content described by the frequency w_i(a).

4. ENTANGLED STATES IN THE MULTIVERSE

The quantum state given by Eq. (6) may describe both a multiverse of macroscopic parent universes as well as one of tiny baby universes. Formally the quantum state of both systems is given by the same wave function (6), and the di erence just resides in the relative value of the scale factor: large scales of order of our Hubble length for parent universes and the smallest scales of Planck length for baby universes. In both cases, annihilation and creation operators for such universes, b and b⁺, can be defined (Robles-Pérez, et al., 2009). The Hamiltonian given by Eq. (3) can be written then as,

where β+ and β₀ are given functions which depend on the scale factor (Robles-Pérez, et al., 2009). For baby universes of order of the Planck length, β+ → v_o/4 with v_o being a constant given by, v_o ≈ 0.3 (c²Λ/h²) ^1/6 , for a at spacetime endorsed with a cosmological constant Λ. The non-zero value of the parameters β+ introduces then high order correlations among the number states of the quantum multiverse and it implies that the vacuum state of gravity is represented rather by a squeezed state (Grishchuk & Sidorov, 1990; Robles-Pérez, et al., 2009), being therefore a highly non-classical state (Reid & Walls, 1986; González-Díaz 1992). On the other hand, for parent universes with a relatively large value of the scale factor the quantum correlations among the number states asymptotically disappear. In that case, β+ → 0 and β₀ → w(a) the Hamiltonian (3) turns out to be then the usual Hamiltonian of an uncorrelated harmonic oscillator, i.e. H = w(a)(N + 1/2), being N ≡ b⁺b, the number operator of universes in the whole multiverse. Let us notice that the number of universes in the multiverse does not depend then on the value of the scale factor, i.e. it is scale factor invariant,

with N ≠ N(a) (Lewis & Riesenfeld, 1969; Robles-Pérez, et al., 2009). That result was expected because to take the scale factor as a time variable is nothing more than to apply a time transformation given by, a ≡ a (t), and it was not expected that the number of universes in the whole multiverse would depend on a particular coordinate choice, or equivalently on a particular choice of the reference system, within a single universe.

However, generally speaking, the quantum states of the universes which form up the whole multiverse are not uncorrelated. In fact, from the Hamiltonian (7) it can be clearly inferred that universes should be created (and annihilated) in pairs, and their state corresponds then to that of a squeezed state or two-universe coherent state. In the realm of quantum optics the name of two-photon coherent state for the squeezed states is introduced because such states are generated in non-linear processes of light emission which involve two photons with the same frequency (Yuen, 1975; Walls, 1983; Walls & Milburn, 2008). In the space-time foam of baby universes it would mean that two universes with the same energy-matter content are created at a given point of the space-time of the parent universe in which they are embedded. Let us notice however that in the case of a multiverse made up of parent universes, in which no common space-time among universes can generally be posed, the pair of correlated universes has to be taken rather as being created in a Hilbert space which is not described in terms of space-time coordinates but of statistical ones. Nevertheless, unlike in quantum optics, the meaning of the number states in the multiverse is not that clear. In quantum optics, the states N_wo(to), t_o can be thought of as representing the number N of photons at a given time t_o with frequency w(t_o). Such a number of photons can be collected in a pulse counter and therefore, at another given time t₁the number state Nw1(t1), t₁ can represent either the number of photons with frequency t₁ at time t₁ or the number of pulses with frequency w₁ collected in the counter at time t₁. It eventually depends on the experimental setting, i.e. on whether the photon counter can register pulses of frequency w₀ or those of frequency w₁.

In the case of a multiverse made up of parent universes is not that clear the interpretation of a number N of universes with a given frequency w(a). Notwithstanding, if universes are created in correlated pairs two interesting features can at least be inferred: rst, an entanglement energy might appear in each single universe due to the entangled state with their partners which would provide us with a vacuum energy of the universe that could account both for a high value of the cosmological constant in their initial stages, when the entanglement between the pair of universes is higher, and for its small value nowadays, when the correlations are asymptotically disappearing. Secondly, it would imply therefore that single universes could not be considered that isolated systems if we could in principle sense the e ects of other universes in ours. That, on the other hand, would open a possibility of being able to test the whole multiverse proposal.

5. MANY WORLDS AND QUANTUM ENTANGLEMENT

The quantum mechanics of "many-worlds" rests upon the objective reality of the quantum field and the wave function. However, in the "many worlds" quantum model proposed by Everett, the observer does not trigger a collapse of the wave function through the process of observing. Observers are not external, they are internal and embedded, and this means there is no observation-triggered wavefunction collapse; a view which is contrary to the standard Copenhagen interpretation. However, if there is no wave form collapse, how can there by "many worlds" or multiple universes? This can be reconciled by quantum mechanisms leading to the creation of "quantum foam" where universes bubble up, but which remain independent, yet part of the quantum field, linked non-local connections made possible via "Quantum Entanglement."

Quantum entanglement is a property of a quantum mechanical state where two or more objects, or worlds (that is, the quantum states of the many objects) are linked together even though the individual objects are spatially separated in space-time. They are linked but not linked, as each object is entangled in the other. Naturally such a conception violates all the classical concepts of space and time. However, this is where Heisenberg's uncertainty principle can be applied. If the many worlds are conceptualized as individual particles emerging in and out of space time, then as each particle appears in one location the other is now in an ambiguous state. That is, as one of the particles is observed, its entangled pair collapses into the very same state, resulting in an "entangled creation" such that one cannot be described without reference to the other. Although this may seem to imply that one universe instantaneously influence other universes, when conceived in terms of infinity, and as these systems may be infinitely far apart, then they do not necessary affect one another as information cannot be transmitted faster than the speed of light and it would take an infinite amount of time.

It is for this reason that Einstein famously derided entanglement as "spukhafte Fernwirkung" ("spooky action at a distance").

Therefore, seemingly we are presented with non-deterministic events occurring within a quantum world governed (or explained by) the deterministic equations of quantum physics. However, in the quantum world an infinite number of quantum outcomes are possible. As each world emerges from the foam of the quantum sea, they are linked, like the infinite branches and roots of a cosmic tree where every possible quantum outcome is realised. Some might described this as order emerging from disorder, or conversely, disorder from order, and this has led to the idea of quantum decoherence.

According to the mechanism of quantum decoherence every possible outcome of every event has its own unique history and is defined by its history. Therefore, every possible world is its history. Since there are an infinite number of possibilities, there are an infinite number of worlds. If event A does not occur in this universe, it has occurred in some other universe or universes which are entangled and which emerge and are then submerged like flotsam in the quantum sea.

6. CONCLUSIONS AND FURTHER COMMENTS

The many-world interpretation of the cosmos is an ancient idea which is consistent with and supported by current cosmological theories. Likewise, causally disconnected regions in the whole space-time are supported by Einstein's theory of relativity. In simplified models with high symmetry, these "many worlds" can be described quantum mechanically in a third quantization formalism by quantum harmonic oscillators, being the frequency of such harmonic oscillators related to the potential term in the Friedmann equation.

The third quantization formalism can be applied both to the space-time foam of baby universes of Planck length and to a set of parent universes with a Hubble length on the order of our universe. The creation of universes in correlated pairs has unavoidably to be assumed in both cases. The smaller in the length scale of the universes the higher are the quantum correlations in their entangled state and therefore, it might supply us with a mechanism which would account for both a high value of the cosmological constant in the flationary period and a smaller value now, provided that one of the consequences of their entangled state is the appearance of an entanglement energy for the vacuum of each single universe.

It is important from our point of view that in general the effects of quantum correlations between otherwise disconnected universes might be detected in our single universe, allowing therefore the possibility of testing the multiverse proposal. Furthermore, quantum correlations in the state of the whole multiverse would also permit the possibility to open quantum communication channels among different disconnected regions or universes. Classical communication channels of the kind of either warp-drive machines or wormholes connecting different universes, for example, might allow us to communicate with innumerable "many worlds: and eventually journey across the quantum sea of the multiverse.