Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.naic.edu/~blewis/aaa.pdf Дата изменения: Wed Jan 14 21:33:55 2015 Дата индексирования: Sat Apr 9 23:29:42 2016 Кодировка: Поисковые слова: dark energy

Towards understanding dark energy
B. M. Lewis emeritus, Arecibo Observatory 120 L Street, Aguadil la PR00603
(Dated: January 13, 2015)

Abstract
That space is p ervaded by quantum fields is a basic tenet of QFT. It is accordingly usual to characterize every p ossible oscillation of every field mode as a simple harmonic oscillator (SHO). Summing the zero p oint energies of the resulting ensemble of SHOs leads to an average density for the Universe 120 orders of magnitude larger than observations p ermit: this celebrated mismatch is known as the Cosmological Constant Problem. The present approach to resolving this problem is to make a first principles analysis of the nexus b etween space and the putative existence of a quantum field. As a result the attribution of prop erties to space can only proceed logically from a completely time-indep endent p oint of view. This prescription resolves the Cosmological Constant Problem. Further, in an expanding space, it also provides an explanation for b oth the magnitude and the inherent nature of dark energy.
PACS numbers: 95.36.+x, 98.80.Es, 98.80.-k, 04.20.Cvy Keywords: dark energy, Cosmological Constant Problem



bmllewis@gmail.com

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I.

INTRODUCTION

Observations of supernovae [1, 2], the microwave background from WMAP [3], and the Planck satellite [4], as well as other diverse investigations [5­10] show that the expansion of the Universe is accelerating. These studies are well fitted by CDM mo dels that assign just 4% of the mass content of the Universe to ordinary matter as we know it, with 28% assigned to so called dark matter, and the rest to dark energy. While the intrinsic nature of both dark matter and dark energy has still to be established, every mo del, as a consequence of a non-zero cosmological constant attributes dark energy to some form of vacuum energy [11]. And every mo del finds a very small average density univ for the Universe. However this observational result disagrees strongly with present expectations of quantum field theory (QFT).

QFT represents every possible mo de of oscillation of a field at every point in space as a simple harmonic oscillator (SHO). A summation over phase space of the positive zero-point energies of every mo de of a single scalar field down to the Planck scale then leads to 120 orders of magnitude larger a value for univ than observations permit [11­14]. This mismatch is known as the Cosmological Constant Problem (aka, the Cosmological Problem). Moreover several quantum fields contribute mass to the vacuum: these include the quark-antiquark condensate, the Higgs condensate, and the weak superconducting condensate to name a few. Additional mass is also added by each field via its random quantum fluctuations. So the Cosmological Problem is a ma jor concern, together with the nature of dark energy, and the small observed value of univ . A resolution for these three interlinked problems is posed here.

The present investigation began with the realization that the standard treatment for quantizing a SHO tacitly assumes that the mass term in its Hamiltonian is positive. That dictates a direction for its arrow of time and restricts a SHO to positive eigenvalues, thus engendering the Cosmological Problem. This approach to the SHO therefore specifies that quantum fields as entities are inherently time-dependent. To make any progress it follows that the mass term in the Hamiltonian must not be restricted to positive masses. Our starting point in §II begins by lo oking at the logical connection between physical space and a putative quantum field, which is thereby shown to be inherently time-independent. To 2


satisfy this requirement one must attribute both ±energy levels to physical space, which solves the Cosmological Problem. But this resolution do es require one to accept that the nature of reality allows wavefunctions to propagate with both directions for the arrow of time. Section III reconciles this outcome with experience.

Dark energy is the topic of §IV. Strictly speaking every mo de in a quantum field is eternal given a constant volume of physical space, whereon the net vacuum energy is zero. However, observations demonstrate that physical space is expanding and has a small vacuum energy density. Consequently this energy must be contributed by the new field mo des initiated by the expansion of the Universe. This context is analyzed while assuming that new space is added to the Universe via an accretion of small spatial cells: a tiny surplus of negative vacuum energy proportional to the change in volume is a consequence. Section V summarizes the main results.

II.

THE ATTRIBUTES OF A QUANTUM FIELD

The Cosmological Problem arises from attributions accorded physical space via the QFT tenet that physical space is pervaded by quantum fields. Our first step towards resolving this 120 orders of magnitude problem begins by parsing the link between space and the existence of a quantum field. Let S be the set of items and understandings that are inherently time-less, that in a sense exist "outside of time." S then contains those verities and concepts that do not change with the passage of time, such as the integer-number verity 1+1=2, as well as mathematical truths, such as Pythagoras's Theorem. It also contains the conceptualization of space described by Euclidean geometry. For simplicity lets call this space Euclidean. It supplies the uniform, passive, background domain within which items may be placed at will, although any subsequent accounting for systematic changes to their relative positions and characteristics do es require the addition of a time co ordinate. But in using Euclidean space as the mo del for the flat physical space of our reality, we tacitly acknowledge its time-less aspect as a member of S, even if it is simultaneously used as a "primitive" or component part of space-time. This is seen for instance in establishing a mass density, for which time is usually irrelevant. So the strictly lo cal physical space, within 3


which QFT is deployed, can be considered to be Euclidean.

QFT posits that given space there is always an accompanying quota of quantum fields [13, 14]. Whereon quantum fields are inherently attributes of space per se, rather than of space-time. Logically, therefore, this conceptualization of quantum fields requires them to be also time-less and members of S. One cannot predicate the existence of a time-dependent entity solely on the existence of an inherently time-less one.

How do es this work in practice? If, for example, an electron is regarded as the excitation of a particular field mo de at a specific position and time, it constitutes an identifiable space-time point that evolves in time. Clearly the moment to moment intensity of a field mo de is time-dependent, and traces a world line in space-time. Nevertheless, the fluctuating intensity of a mo de is conceptually distinct from the underlying existence of the electron field as an entity, which is time-less. Consider as an analogy to the electron field a pond, which is a small bo dy of water. The pond exists in space, and could be in principle independent of time, and thus a member of S. But any and all ripples on its surface are time dependent. So to o are the disturbances inherent in the existence of a quantum field. The field itself, however, remains an intrinsically time-less entity, as must be the attributes it confers on space.

The attributes of a quantum field are usually derived by representing every possible mo de of its oscillation at every point in space by a SHO. These are quantized via the `timeindependent' SchrЁ odinger Equation [15]. However every textbo ok tacitly assumes that the mass term in its Hamiltonian is positive, which on its own mandates a set of positive energy eigenvalues and a positive zero point energy. That engenders the Cosmological Problem. Moreover, the solution has an explicit time dependence, e-Et , with time progressing towards the future. This `time-independent' SchrЁ odinger Equation is not, therefore, completely time-independent, as the initial restriction to positive mass smuggles the direction for the arrow of time into the solution. Merely regarding the resulting set of energy levels as the sole attributes of a quantum field then fails on two counts, both in predicting an impossibly large univ , as well as in importing an explicit time-dependence. The proper solution must include a matching set of negative energy levels. 4


Following in Dirac's fo otsteps, we obtain a negative mass by adopting the negative ro ot of the relativistic expression for energy E = - m2 c4 + p2 c2 and dividing by c2 . On substitution into the Hamiltonian H for the SHO, one gets a solution with a mirror-image set of negative eigenvalues, En = -(n + 1/2) h for n 0, and a negative zero point energy -h /2. This solution has the same evolution as the usual treatment when -t replaces t, and so is linked to time chronologically progressing towards the past. Time is thus inescapably incorporated into a solution via the assumption made about the nature of the mass term: positive mass/energy equates to a normal time flow, negative mass/energy to a reversed flow of time. Indeed Weinberg [16] notes that the transformation operator T for the chronological reversal in the flow of time is both linear and unitary, so that T H T
-1

= -E . He
-1

further notes that "for every state of energy E there is then another state T -E". This subsumes and generalizes the SHO case.

of energy

To obtain a fully time-independent solution for the SHO therefore requires a superposition of wavefunctions, with one following normal time, the other chronologically reversed time. The result is then indeterminate with respect to time until a field mo de is excited. Consequently the attributes of a time-less quantum field embrace both the positive and negative sets of energy levels. Contextuality offers a yet more explicit interpretation. Experimental tests of Bell's Theorem [17, 18] demonstrate that where a binary choice between two states exists, such as in the spin orientation of an electron, neither is the actual physical case until the experiment forces a result: it is as if the possibility of the electron's spin as a physical quantity is suppressed until prompted. The same circumstances apply to the binary choice operating with respect to the direction of the arrow of time for an excitation in a quantum field right up to the moment when a specific mo de is excited. Every field mo de is in consequence time-less until excited, so summing over the ensemble of positive and negative zero point energies necessarily yields 0: as zero point energies are the default, unexcited states of a quantum field, they can never be excited. That abolishes the Cosmological Problem.

Time-independent solutions for the SHO also eliminate the ultraviolet catastrophe faced in summing their zero-point energy contributions down to the Planck scale and beyond, 5


as both ± energy fields are similarly affected and exactly cancel. Moreover, the physical reality of quantum fields is demonstrated by the Casimir effect [19]. This is a consequence of the Uncertainty Principle, as the intensity of a field mo de can never be precisely known, so it must constantly fluctuate. That in turn results in the virtual presence of positive mass entities exerting a measurable physical pressure on positive masses. Nevertheless, since fluctuations o ccur equally often in both ± energy fields, the net mass/energy density of the vacuum is always zero in a constant volume Universe.

III.

RECONCILING EXPERIENCE

The Cosmological Problem disappears once quantum fields are treated as time-less entities. Yet the historical development of QFT has required every interaction to be strictly asso ciated with positive mass/energy interchanges. Weinberg [16] accordingly defines a second version of a time-reversal transformation T such that T H T
-1

= E , where E is

positive. This definition, by requiring T to be anti-linear and anti-unitary, agrees with the applicability of physical laws that allow the evolution of a system to be recapped by inserting -t in place of t in all equations. QFT uses the definition to enable interactions between positive mass/energy components A + B ... C + D ... to be completely reversible within the framework of CPT symmetry. Were this the only operable definition for T , however, every quantum field would be restricted to contributing just positive mass to univ . Whence an insoluble Cosmological Problem.

Nor is the usual applicability of QFT affected by the intro duction of sets of negative energy levels, as QFT has always been concerned with positive mass / energy entities, which implicitly share our sense of time. Likewise, there is no difficulty in applying the usual laws of physics, since the success of QFT serves to demonstrate that positive mass/energy entities only interact directly with similar entities. Just gravity senses both, as it audits the arithmetic net mass/energy content of space.

Are there any valid ob jections to this solution? The most often voiced is that should negative energy levels hypothetically exist, extra energy could be extracted from a positive 6


energy system by interactions setting up a cascade to more negative energy states, thus providing a perpetual source of free energy. That is rightly ridiculed. But it ignores the limit to such cascades set by the positive zero point energy governing every entity characterized by positive energy. The logic is also bypassed by the QBist philosophy underwriting recent approaches to quantum paradoxes [20], which requires the observer to be included in all considerations. The salient point then is that a SHO is limited to positive energy states whenever the mass term in its Hamiltonian is positive: since it then has no other states, every possible change between its energy levels is mediated by a positive energy entity. Indeed no positive mass entity (the observer/photon in this case) interacting with it has access to any negative energy states. And vice versa for the case of a negative energy SHO. Hence there is no possibility of a cascade down an infinite negative-energy well and no additional source of realizable energy. Consequently this ob jection is far from persuasive.

There is a natural reluctance, however, to invoking a reversal in the arrow of time. Our conception of time is formulated as a way of making sense of data reaching us through our senses, which literally involves interactions between parts of ourselves and the world around us. This necessarily pro ceeds in our own frame of reference. As our world is constituted of positive mass / energy, that apprehension dictates a single direction for our arrow of time. Thus all of our sense experience provides us with just one sense to the arrow of time, which is in consequence firmly ro oted in our conception of time. Weyl [21] begins his treatise Space - Time - Matter with a philosophical intro duction. In this he points to insights from Galileo, Descartes, and Hobbes on the sub jectivity implicit in the qualities we infer from our senses, when, for instance, they recognize a green colour. Weyl go es on to point out that Kant extended this mo de of introspection to the point of view that "space and spatial characteristics likewise have no objective significance in the absolute sense; in other words that space, to o, is only a form of our perception". As to o is time. While general relativity refines our understanding of both space and time, our apprehensions about the intrinsic nature of time still retain the perceptual notion that time necessarily flows in a single direction, that it is solely characterized by a steady forward progress. We need to abandon this perception, to allow the essential nature of reality to permit wavefunctions to simultaneously propagate in both time directions.

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It is easy to see on a physiological basis where our perceptions about time come from. Our senses transmit information via nerve cells that communicate with other cells via synapses. These macroscopic one-way units in themselves are not time-reversible. Thus the mo del of time we assemble from our perceptions derives from data filtered through a set of dio des. Entropy enters to o, as every macroscopic change in any system we directly perceive is insulated by entropy from exact time-reversal, so even its possibility is masked from our direct perception. Yet most micro-physical pro cesses, and all of those concerned with interactions with charged particles, are time-reversible.

Special relativity showed us that the apparent rate of efflux of time in different frames of reference is determined by their relative velo cities: the rate is frame dependent. So to o is the direction of their arrows of time. The mere existence of a quantum field on the other hand is neither time-dependent nor frame-dependent; it is simply a common presence for observers in every possible frame of reference. This is crucial. A quantum field is inherently an environmental feature within space-time, so its properties must be set from a completely time-independent standpoint. That is not done when its properties are assigned from a frame of reference with a preferred direction for the arrow of time. This realization mandates a Copernican-level paradigm shift in asserting that the nature of reality simultaneously embraces both directions for the arrow of time. Is this conclusion supported by anything else? Yes, it also provides an explanation for the nature of dark energy and the finite small value of univ . This follows.

IV.

DARK ENERGY

Observations have shown for 15 years with ever greater precision that the expansion of the Universe is accelerating. An essential result of CDM mo dels is that dark energy is a form of vacuum energy that retains its density irrespective of the expansion. It is thus a property of space. Yet CDM mo dels always use the simplifying assumption that space and all its contents can be treated as absolutely homogeneous. This do es not allow for lo cal mass concentrations of any kind, whether these be stars, galaxies, or clusters of clusters of galaxies. Consequently these divergence-free cold-dust mo dels cannot tell us whether 8


dark energy is intrinsically positive or negative. Ma & Wang [22] resolve the ambiguity by mo delling space differently: they generalize the Einstein functional by including a potential function. Since the resulting mo del is not divergence-free, it is a better representation of the actual nature of a cosmos containing lo cal mass concentrations. Dark energy is then found to be intrinsically negative energy. This is in accord with expectations going back to Bondi [23] that negative mass acts to repel positive mass: hence the cosmic acceleration.

Quantum fields, in a constant volume Universe that embraces both directions for the arrow of time, contribute nothing to univ . Since the cosmological redshift tells us that space is expanding, this feature must be central to an explanation for dark energy. Let us assume that the expansion is generated via an eruption of discrete spatial cells into pre-existing space [24]. Whereon the generation of new space implies a corresponding initiation of new field mo des to the Universe. So `individual' mo des are not eternal: new mo des must arise when space expands and disappear if it contracts. Changes in extant space compel changes to the complement of field mo des, which may in turn trace spatial changes to some extent. Hence energy is added/subtracted from the Universe as a consequence of changes to extant space, which o ccur within spacetime. The eruption of a new spatial cell begins from a point, since the continuity of space-time, together with the finite velo city-of-light limit for spatial changes to be effected, prevents a spatial cell from erupting as a completed entity into pre-existing space. One can therefore figuratively draw a sphere at an infinitesimal radius around a point, prior to a new spatial cell erupting from it. As the eruption pro ceeds, the radius of the sphere in the pre-existing space-time expands at the velo city of light until the cell reaches its full size. This sequence is seen identically by every possible observer in normal space-time.

Figure 1 shows the space-time diagram for a cell erupting into the flat space of a reference frame S The world line ABC in S is displaced infinitesimally from the adjacent world line of S. the central point as far as B, whereon the new spatial cell begins to expand at the moment linked to B. Its center moves from B to D at the velo city of light during the expansion in S while time is incremented by t. At the cessation of expansion, the world line DE of S, S' the center of the cell is again parallel to ABC. But time is stationary in the S' frame of the center of the cel l along BD for observers lo cated beyond its periphery. Further, FIG. 1 is 9


common to every observer, and it is a tenet of QFT that every possible mo de already exists in S up to the moment linked to B. So the singular moment marking the initiation of a mo de spanning the whole of the new cel l is deferred to D. This o ccurs despite spatial change being initiated at the moment linked to B, since a new mo de cannot arise within the cell prior to the existence of a volume increase about S' Consequently the eigenfunction + that spans S'. the cell and propagates forwards in time is only initiated at D: it is not present within the volume of the growing spatial cell during the t interval following B. On the other hand its complementary eigenfunction - propagates backwards in time through the previously generated volume of the new cell as so on as - is initiated at D. The solitary, uncompensated - mo de therefore briefly injects negative energy into the Universe for the interval t. S' Time is only stationary in the S' frame of the center of the cel l, while it is viewed from beyond the periphery of the expanding cell. Indeed this is the only configuration where every possible S-frame observer is moving away from S' at the velo city of light. But the ensemble of shorter wavelength mo des engendered within a growing cell do es not expand at the velo city of light, and so is not in a developmental stasis. While the periphery of the new cell expands at the velo city of light, every other sphere contained within the cell expands S' at a slower rate. The key feature, that time is stationary in the S' frame of the center of the cell along BD, is the sole property of mo des that are simultaneously centered within the cell and constrained by its periphery. Every other mo de enabled during a cell's expansion is initiated simultaneously with paired + & - eigenfunctions, so none of them contributes any mass to the Universe. The zero point energy of a SHO has a mass of m = (h / 2) /c2 = h / 2 c for a duration t = 2 / c seconds, where h is Planck's constant. Hence the pro duct m . t = h / c2 is a scale-free invariant. Dividing this into univ gives the number of mo des generated by the expanding Universe per unit volume per second N = univ / (m . t) = univ c2 / h 8.5 1018 cm3 s-1 . While N is well determined, every cell briefly hosts n mass-contributing mo des from the various quantum fields, so the number of new spatial cells generated per second is N / n. The expansion rate of the Universe gives its fractional volume change 10


= dV/V = dR/R 7 10

-18

.

Assuming every erupting cell of final radius rc contributes to the expansion of the Universe sets rc = ( h n/(4 univ c2 ))
1/3

.

The smallest possible radius for a new cell is given by letting n = 1, so rc 10 fm and t = rc /c 3.6 10
-23

seconds. At the other extreme, if the whole of is assigned to a single
-6

new spatial cell, rc 1.2 10

cm and t = 4 10

-17

seconds.

Another consideration arises at this juncture. Thus far the expansion of the Universe is attributable to random fluctuations intro ducing new spatial cells. Yet micro-physical processes are in general time-reversible, so the inverse pro cess, wherein spatial cells are removed from space-time, may also o ccur. The expansion of the Universe implies a `Big Bang' event some time in the past. So were the Universe instead to be contracting under gravity to an eventual `Big Crunch', this would be deduced from a cosmological blueshift, which would in turn be synonymous with an ongoing contraction of space. But this expectation lo oks exactly like a gravitational blueshift, where gravitational attraction is simply regarded as a consequence of the abstraction or removal of space. Whereon, since a loss of space implies a loss of field mo des, one needs to consider their possible demise.

Figure 2 is the space-time diagram arising when a spatial cell contracts out of existence. It follows from FIG. 1 after it is flipped about the moment B at which the cell begins to evolve in time. In this case both the + & - eigenfunctions for the central mo des spanning A-D already exist at A and D, and therefore at B, before the periphery of the cell begins at E to contract towards a point. So the freeze in their evolution as a cell contracts at the velo city of light along EB leaves the presence of + & - unchanged for both external-to-the-cell observers, and for a putative observer lo cated at the center of the cell, until + & - are extinguished simultaneously with the extinction of the cell at B. Since + has no space to propagate into beyond B, the mo de is never unpaired and therefore never provides a mass contribution to a Universe operating with our sense of time.

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The asymmetry of outcomes between the genesis and extinction of a spatial cell ensures that the negative mass intro duced by a new cell is never negated by the inverse pro cess that would extinguish a cell. Consequently the observed expansion of intergalactic space is not due to a net difference in almost comparable rates of initiation and extinction of spatial cells. Thus every spatial eruption event momentarily provides a negative energy contribution to univ at a rate determined by the net expansion rate. Dark energy is in consequence negative vacuum energy.

The initiation of new spatial cells is required by the observed expansion of the Universe: apart from the expected time reversibility of micro-physical pro cesses, their extinction is only relevant if one invokes them to explain the usual effect of gravity. Extinction events about mass concentrations would then be systematically enhanced in a fashion akin to the shielding of electric charge by virtual electron/positron pairs. So the surrounds of a galaxy would appear to be a zone of contracting space that must be compensated for eventually by a zone characterized by the enhanced initiation of new spatial cells, and thus by an above-average negative mass density. This approach to the nature of gravity has a bearing to o on the debate about the actual existence of black holes. As the event horizon of a black hole is approached, space is extinguished at a rate where the n