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Quantum Field Theory

Quantum Vacuum

(Zero Point Energy, Vacuum Energy Density, Hole Theory, Pair Creation and Annihilation)

Zero Point Energy

In classical physics, empty space is called the vacuum. The classical vacuum is utterly featureless. However, in quantum theory, the vacuum is a much more complex entity. The uncertainty principle t E > allows virtual particles (each kind corresponding to a specific quantum field)
Zero Point Energy 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 E0 = /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 E0 so that En = 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).

Vacuum Energy Density

The virtual particles popping up briefly from the zero point energy is one of the possible explanations for the enigmatic dark energy driving the acceleration of cosmic expansion. To simplify the calculation of the energy density of such activity, the dynamics of the harmonic oscillator is replaced by normal modes in a cube with length L (Figure 05q, also see the similarity with the Square Well Potential). This is similar to the Standing wave produced upon reflection when the incident wave encounters another medium with different property. In this case, the potential barriers (as represented by the potential curve of the harmonic oscillator in Figure 05p) is responsible for confining the wave within a spatial volumn. The simplified derivation in the following shows the huge difference between observation and such explanation.
Normal Modes

Figure 05q Normal Modes
[view large image]

This vacuum energy density is supposed to be applicable everywhere and at anytime. It is not subjected to the effect of cosmic expansion.

In fact the upper integration limit for is an infinity, which can be eliminated by a dubious procedure similar to the renormalization in QED.
Virtual Particle Anyway, if the inherently infinite vacuum energy density is bare, the observed vacuum energy density can be expressed as obs = |bare - | 10-29 gm/cm3, 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) :
(it has 2 fermion and 1 boson propagators, there is no vertice - meaning no interaction except gravitationally as dark energy to accelerate the cosmic expansion).
Mathematically, the real particles are derived from the quantized solutions of the wave equation, while virtual particles are related to the Green's function (a.k.a. propagator) associated with the formulation. Figure 05r portrays these two kinds pictorially. The lower diagram illustrates the relationship between virtual particles and propagator, which involves divergent integration with limits running from 0 to infinity. This was the problem with electron self-energy plaguing QED for a while.

Figure 05r Virtual Particle
[view large image]

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).

Dark Energy Era 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/Rk (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/Rk dependence on the cosmological "constant" term, i.e.,

(dR/dt)2/R2 = m/R3 + /Rk, which can be recast to express the cosmic acceleration explicitly

(d2R/dt2) = R[- m/(2R3) + (1-k/2)/Rk] = 0 at cross-over time; then the scale factor

R = eq/(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 (109 years)

T is computed by a Cosmic Calculator with the above parameters and 1/H0 = 13.7 Gyr (or H0 = 71.4 km/s-Mpc).

Hole Theory

Negative Energy States 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 = (m02c4 + p2c2)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 -m0c2 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 2m0c2, 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 m0. The law of charge conservation demands that this anti-particle carries a positive charge.

The development of quantum field theory in the 1930's made it possible to treat the positron as a "real" particle rather than the absence of a particle, and makes the vacuum the state in which no real particles exist instead of an infinite sea of particles. It recaptures all the valid predictions of the Dirac sea, such as electron-positron annihilation.

Pair Creation and Annihilation

Reaction Ratio 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.
As the collision energies increase, new resonance spikes occur at 3.096 Gev and 3.687 Gev. It was discovered subsequently that they are the products of a new type of quark called "charm", which together with the "strange" quark form the second generation of quark. With this discovery the number of mesons is expanded into the 16-plet of spin-0 mesons generated by SU(4) flavour symmetry. Similarly, the spikes at 10 Gev and 10.40 Gev in Figure 06b lead to the discovery of the bottom quark for the third generation. The SU(4) flavour symmetry has to be enlarged to SU(5) so that the basic multiplet of spin-0 mesons is now expanded to a 25-plet. It is further realized that the c mesons form a
Meson Spectrum 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]

V = - 4s / (3r) + ar ,

with which the Schrodinger equation in non-relativistic quantum mechanics yields a most satisfactory match to the observed meson mass level (see Figure 06c). This formula combines a Coulomb potential at short ranges with an attractive potential rising linearly at longer distance, giving rise to the ever-increasing forces of quark confinement. Thus, the masses of the c mesons provide direct support for the QCD picture of inter-quark forces containing both asymptotic freedom at short ranges and confining forces at longer distance. As the beam energy is increased,
Jets 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.

The next topic is about the Higgs field, which prescribes yet another kind of structure in the quantum vacuum.

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