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continually materialize out of the vacuum for a short time and then vanish (with energy allowable by the uncertainty principle). Such movement means that there is a lowest energy level for the virtual particles. The lowest part of the quantum fields at vacuum state can be approximated by a harmonic oscillator as shown in Figure 05p. The lowest energy level has an energy E_{0} = /2 for n = 0 corresponding to no real particle according to "Quantum Field Theory" (QFT). Since there are an infinite number of harmonic oscillators per unit volume, the total zero-point energy density is, in fact, infinite. The process of renormalization is to move the zero point energy up to E_{0} so that E_{n} = n with n = 0, 1, 2, ... ; the infinity is thus removed. Such manipulation is justified by the fact that only the difference in energy is measurable. Now, the quantum vacuum is defined as no excitation of field quanta, i.e., no real particles are present. In other word, it is at a state of minimum (and zero) energy. The classical analogy would be a collection of motionless harmonic oscillators. | |
Figure 05p Zero Point Energy [view large image] |
BTW, the excitation of virtual particle is often explained as an embezzling bank teller who takes money (virtual particle with certain energy) from the till and has it returned before the auditor comes over to check. The visit would be more often (shorter interval) with more money entrusted/misappropriated (more energy). |
Figure 05q Normal Modes |
This vacuum energy density is supposed to be applicable everywhere and at anytime. It is not subjected to the effect of cosmic expansion. |
Anyway, if the inherently infinite vacuum energy density is _{bare}, the observed vacuum energy density can be expressed as _{obs} = |_{bare} - _{}| 10^{-29} gm/cm^{3}, where _{} is another infinite entity calculated from virtual particles agitation to modify _{bare}. Now the dark energy can be identified with the vacuum energy density as represented by the Feynman diagram below (ignored by QED) :
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Figure 05r Virtual Particle |
The vacuum energy diagram shown above is a very special case in QED, where it doesn't contribute to any verifiable process and thus has been ignored all these years until it emerges as a possible explanation for dark energy in the last few years (see its difference from Vacuum Polarization). |
The dominance of dark energy over matter occurred at about 7 billion years since the Big Bang according to some astronomical observations. It is found that only constant dark energy density or a very slowly varying one in standard cosmology can reproduce a rough match for this value of transitional time. All other forms proportional to 1/R^{k} (R is the scale factor) would fail the test (Figure 05s). The range of k that can produce a viable cross-over time is shown in the table below : | |
Figure 05s Dark Energy Era [view large image] |
We are going to use the standard cosmological equation with the additional 1/R^{k} dependence on the cosmological "constant" term (time in unit of 1/H_{0}), i.e., |
(dR/dt)^{2}/R^{2} = _{m}/R^{3} + _{}/R^{k}, which can be recast to express the cosmic acceleration explicitly (d^{2}R/dt^{2}) = R[- _{m}/(2R^{3}) + (1-k/2)_{}/R^{k}] = 0 at cross-over time; then the scale factor R = e^{q/(3-k)}, where q = ln(_{m}) - ln[(2-k)_{}], with _{m} = 0.26, _{} = 0.74; giving the red shift z = (1-R)/R, while the cross-over time since Big Bang in Gyr (10^{9} years) T is computed by a Cosmic Calculator with the above parameters and 1/H_{0} = 13.7 Gyr (or H_{0} = 71.4 km/s-Mpc). |
the current cycle where we happen to live in, the energy density of the false vacuum is about 10^{119} ergs/cm^{3}. Such energy has been converted to various entities since then. The true vacuum has attained its current value of about 10^{-8} ergs/cm^{3} since the beginning of this cycle. Therefore, the calculated vacuum energy density of 10^{123} ergs/cm^{3} is not the dark energy, it is instead the energy released into this cycle of the universe (Figure 05t,b). | |
Figure 05t Dark Energy in Cyclic Universe [view large image] |
This kind of consideration can be applied to other cosmological theory in general (such as the inflation theory) with the false vacuum energy density coming from some source. |
Another kind of vacuum structure is prescribed by the Dirac theory of spin-1/2 particles in 1930. It admits the existence of both positive- and negative-energy particles: E = _{}(m_{0}^{2}c^{4} + p^{2}c^{2})^{1/2}. The concept of negative-energy entities is wholly alien to our knowledge of the universe. All things of physical significance are associated with varying amounts of positive energy. To get around the problem, Dirac proposed an energy spectrum containing all electrons in the universe (see Figure 06a). In addition to the normal positive-energy spectrum, it also contains the negative-energy variety, which spans the spectrum from -m_{0}c^{2} down to negative infinity. All the negative-energy levels are filled, thus the positive-energy particle is inhibited from transition into these lower energy states. Thus, there is no observable effect in the real world. Only when there is enough energy available, e.g., E _{} 2m_{0}c^{2}, a real particle and anti-particle pair with positive-energy can be created from this unseen sea of negative-energy particles. The particle is the electron originally resided in the negative energy | |
Figure 06a Negative Energy [view large image] |
region, while the anti-particle (positron) can be interpreted as the hole in the vacated energy level acquiring a mass m_{0}. The law of charge conservation demands that this anti-particle carries a positive charge. |
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Closer examination of diagram (e), Figure 06a reveals that the intermediate product is a pair of quarks as shown in the insert within Figure 06b. The quark confinement process creates the observed hadrons in the final state. As the quarks are observed to be point-like (in deep inelastic scattering) and spin 1/2, the intermediate process e^{+}e^{-} q_{} is very similar to the process e^{+}e^{-} ^{+}^{-}, the only difference being that the charges on the quarks are only some fractions of that on the muons. This explains the constancy of the | |
Figure 06b Reaction Ratio [view large image] |
ratio R mentioned earlier and displayed in Figure 06b. As for the pronounced spikes which punctuate the curve in Figure 06b. These shapes are formed at certain energies of the e^{+}e^{-} collision, when the q_{} pair have just the correct mass to appear as a single-meson resonance. They are the SU(3) flavour symmetry mesons with mass ~ 1 Gev. |
spectrum from the different values for the spin and the orbital angular momentum of the constituent quarks as shown in Figure 06c. In the same notation for the atomic spectra, S, P and D in the diagram refer respectively to orbital angular momentum 0, , and 2. The resemblance to the atomic spectrum is understandable because of asymptotic freedom as the c and _{} bound themselves together loosely. The force between the quarks can be formulated as a potential acting in the vicinity of a colour charge. Thus instead of the Coulomb potential in the case of the atom, the quarks interact via the potential: | |
Figure 06c Meson Spectrum [view large image] |
the quark and antiquark are produced with very large momenta, moving in opposite directions known as two-jet event. The fragmentation into hadrons then takes place, preferentially along the direction of the motion the quark and antiquark, resulting in jets of hadrons which become more and more collimated as energy is increased (see left diagram of Figure 06d). The measurement of the angular distribution of jet axes confirms that the spin of quarks is indeed 1/2. At even higher energy the quark or antiquark radiating a gluon, | |
Figure 06d Two- and Three-Jet Event [view large image] |
which forms a separate jet of its own. This three-jet event is shown in the right diagram of Fgiure 06d. |