## Quantum Field Theory

### 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)
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 05q shows the non-relativistic wave (field), e.g., of an electron in harmonic potential.

#### 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 (see the similarity in Figure 05q, and also 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, and 05q) 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.

#### Figure 05q Normal Modes [view large image]

See "Planck Scale".
This vacuum energy density is supposed to be applicable everywhere and at anytime. It is not subjected to the effect of cosmic expansion.
BTW, the cosmological constant = 8G/c4.

In fact the upper integration limit for is an infinity, which can be eliminated by a dubious procedure similar to the renormalization in QED.
Anyway, if the inherently infinite vacuum energy density is bare, the observed vacuum energy density can be expressed as obs = |bare - | 10-8 erg/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. The probability amplitude for its occurrence also blows up. It is represented by the Feynman diagram below :

 This vacuum energy Feynman graph has 2 fermion and 1 boson propagators, there is no external lines, i.e., 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 the 4-momentum limits running from 0 to infinity. This was the problem with loop Feynman graphs 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).

BTW, the technique of renormalization is just a disguise for our ignorance on the true nature of anything. Somehow the procedure works for some special cases, i.e., those that do not depend on the detailed knowledge of the next level. It actually produces some meaningful results.

To demonstrate the existence of virtual particles, it is noticed that virtual photons in between two parallel metal plates placed a short distance apart can exist only when they can form a standing wave, thus there are fewer photons in each cubic centimeter of vacuum between the plates than there are in the vacuum outside. So, in effect, there is an excess pressure from outside pushing the plates together. This is known as Casimir effect (see Figure 05s). The resulting force is very small, but it has been measured (for plates separated by gaps of a few nanometers), proving that quantum fluctuations of the vacuum are a real phenomenon.

#### Figure 05s Casimir Effect [view large image]

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 05t). The range of k that can produce a viable cross-over time is shown in the table below :

#### Figure 05t 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 (time in unit of 1/H0), 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).

The role of dark energy in "Cyclic Universe" becomes the interacting agent between two oscillating 3-D branes. It also evolves along with each cycle (Figure 05u,a) moving from one false vacuum to another true vacuum, which is identified as the dark energy in that cycle. For example, in
the current cycle where we happen to live in, the energy density of the false vacuum is about 10119 ergs/cm3. Such energy has been converted to various entities since then. The true vacuum has attained its current value of about 10-8 ergs/cm3 since the beginning of this cycle. Therefore, the calculated vacuum energy density of 10123 ergs/cm3 is not the dark energy, it is instead the energy released into this cycle of the universe (Figure 05u,b).

#### Figure 05u 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.
It seems that the appearance of false vacuum was rather common in the early stage of the universe. For examples, the mechanism has been applied to the Higgs, axion, inflaton, and now a new kind of field in the cyclic universe. Nobody knows how to create a false vacuum, one suggestion by Alan Guth (the father of "Inflationary Universe") is to raise the temperature to about 1029 K, and then rapidly cooling it - similar to the condition in the inflation period posited in his theory.

### Hole Theory

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

Many interesting things occur when the electron meets the hole (see Figure 06a). Over the past three decades, the use of e+e- collisions to probe the vacuum have yielded a great deal of information about the nature of the strong, weak, and electromagnetic interactions, and have played a major role in establishing the Standard Model of elementary particles. There are many special features of the e+e- experiments:

• The e+e- pair often annihilate into a vacuum state of pure energy (in the form of electromagnetic radiation or other kind of bosons). All the quantum numbers of the initial particles disappear by cancellation. The energy is then free to sample the hidden, negative-energy content of the vacuum without the inhibiting effects of conservation laws. Such energy can be in the form of photon in diagram (a) of Figure 06a boosting any particle pair from the negative-energy sea.
• When an electron and positron are collided head-on, with equal and opposite momentum, the center of mass of the reaction is absolutely stationary. In such circumstance, the angular distribution of created particles can be measured directly and significant asymmetries detected much more easily.
• Since the e+e- pair can annihilate to a vacuum state, it means that the reactions are very clean. There is no debris surviving from the initial state to mask interesting new effects or to confuse the detection.
• The range of phenomena open to study depends upon the total energy of the collision. But while the e+e- beams are kept in orbit they emit their energy in the form of synchrotron radiation. So, to achieve very high energies, it is necessary to build very large rings and to input a lot of radio frequency power to replenish the lost energy. Therefore, the search for new particles has led to the construction of increasingly more powerful and bigger accelerators.
The basic e+e- reactions:

• Electromagnetic processes -
• The elastic scattering as shown in diagram (b), Figure 06a.
• Creation of +- as shown in diagram (c), Figure 06a. The single photon must be virtual (off-shell) as it does not obey energy-momentum conservation.
• Production of two real photons as shown in diagram (d), Figure 06a.
Accurate measurement of these electromagnetic effects is used to test the validity of QED at very high energies. It confirms that the leptons are indeed fundamental point-like particles with size < 10-16 cm.
• Annihilation into Z0 boson - The e+e- can annihilate into Z0 boson involving weak interaction (diagram (f), Figure 06a). The virtual Z0 boson is free to explore the "weak" content of the vacuum. The electroweak theory predicts that slightly more +s should emerge in one hemisphere than in the other (with the reverse asymmetry for the -s). Such parity violation was duly confirmed in 1981.
• Hadron (composite particles containing quarks) productions - One of the most significant quantities in particle physics is the ratio R of the cross section for e+e- hadrons to that for e+e- +-, as a function of the colliding energy:  ---------- (45)
It compares a well-understood reaction (muon-pair production) with the class of less well-known reactions (hadron production). The ratio R is relatively straight-forward to observe experimentally. Only two charged "prongs" are seen to emerge in muon-pair production whereas, almost invariably, more emerge from a hadronic final state (digram (e), Figure 06a). So the ratio can be obtained from the numbers of these different events. Surprisingly, the ratio R is constant over large energy ranges, indicating that the complicated hadronic state is produced in much the same way as the simple muon pair. In this case, the virtual photon is probing the negative-energy sea of hadrons in the vacuum instead of electrons or muons.
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
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,
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 pair creation mentioned above requires the presence of an atomic nucleus to maintain conservation of momentum and to provide the second photon to complete the process, which is second order (meaning proportional to =e2/c and related to the # of interaction points) with cross section (~ probability) about 1 barn (~ 10-24 cm2). For pair creation truly plucks out from vacuum, the process is fourth order
with a much smaller cross section ~ 10-7 b as shown in Figure 06e, in which, diagram (a) shows the photon-photon scattering when E < 2mc2. The process that produces the pair out of the vacuum requires very high photon intensity (such as very powerful laser not yet invented) to compensate for the much lower cross section. The diagram on the right shows the various laser facilities over the years. It predicts the laser capability could reach such requirement (the "Schwinger limit") by 2030.

#### Figure 06e Schwinger Limit [view large image]

See "What's inside nothing? This laser will rip it up to find out", New Scientist, January 26-February 1, 2019 for the world's most intense laser at ELI (Extreme Light Infra-structure) in Romania, 2019.

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

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