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Higgs Field, the Eternal and Ubiquitous (2021 Edition)


Contents

Spontaneous Symmetry Breaking (SSB) and Phase Transition
Inflaton and Higgs as Metastable States
A Semi-classical Model of Inflation (2022 Addition)
A Novel Higgs Field
A Novel Idea on Dark Energy (2022 Addition)
Theories of Cosmic Inflation and B-mode Polarization (2022 Addition)
(Modified) Quantum Field History
Quantum Field History à la Gauge Theory

Spontaneous Symmetry Breaking (SSB) and Phase Transition

Symmetry Breaking The existence of asymmetric solutions to a symmetric theory is common to many branches of physics. The reason lies in the fact that the symmetric state is not the state of minimum energy, i.e., the ground state, and that in the process of evolving towards the ground state, the intrinsic symmetry of the system has been broken. Figure 01 shows that the initial position of the marble on top of the bump is symmetric but not in a

Figure 01 Symmetry Breaking

state of minimum energy. A small perturbation will cause the rotational symmetry to be broken and the system to assume a stable state configuration. When the symmetry of a physical system is broken in this way, it is often referred to as "spontaneous symmetry breaking (SSB)".

As shown in Figure 01, energy is released when the system transits from the symmetric phase to the lower energy Higgs phase in the
Phase Transition Standard Model process of SSB. Such process is similar to the phase transition (see Figure 02,a) between the gas-liquid-solid phases, in which energy is released to the environment and the rotational symmetry changes to translational with the change of gas/liquid phase to solid. By such analogy, the SSB of electro-weak at 10-12 sec after the Big Bang can be considered as the transition of restmass-less particles moving at the speed of light to motion constrained by having restmass (see Figure 03 of the Standard Model in symmetric and Higgs phases).

Figure 02 Phase Transition

Figure 03 Standard Model

Figure 02,b portrays another interpretation of phase transition via lowering the potential energy V by some kind of catalyzing agent.

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Inflaton and Higgs as Metastable States

Since the confirmation of the Higgs mass at 125 Gev in 2012, there are more researches on the link between the Higgs field and cosmic inflation. It is remarkable that the Planck data on CMB are indeed compatible with treating the Higgs (in particle physics) as the inflaton (in cosmology), provided the field is non-minimally coupled to gravity (see a short discussion in Inflation and the Higgs particle, 2014).

By tweaking the energy curve as shown in Figure 04,a; a novel idea in 2017 (See "The Higgs Bang") merges the inflaton and the Higgs fields into one scalar field, the inflaton and Higgs just correspond to different phase of the universe. In addition, the scheme requires an extra scalar field to make it consistent. It is known as the dilaton (~ cosmological energy density, i.e., the dark energy). Thus, this new theory resolves three outstanding paradoxes in one shoot (1. endowing mass, 2. unifying inflaton, 3. adding drak energy). Such "Inflaton-Higgs-Dilaton" model can be tested by a particular twist in the polarization of the CMBR. The presence of a dilaton field can also be detected by certain gravitational wave imprint on CMBR. This kind of data may be available by 2020.
Higgs as Inflaton The obvious omission in this model is the absence of the reheating process which is responsible for the generation of matter at the end of inflation (see "Inflation Era"). According to another model developed in 2014 (see "Inflation, LHC and the Higgs boson"), it is indeed possible to produce a potential V with an inflationary plateau at large values of the scalar field, and two minima (Figure 04,b).

Figure 04 Higgs as Inflaton
[view large image]

A much simplified version of the mathematic derivation is sketched below to illustrate the theoretical formulation.

The Lagrangian (density) for gravity in General Relativity has the form : = (-g)1/2R ---------- (1)
where R is the scalar curvature and g is the determinant formed from the space-time metric tensor gik.




Higgs, Time Variation
    Figure 05 portrays the time evolution of the inflaton/Higgs field with the potential given by Eq.(3b). It is computed by a constant value of the self-coupling constant . It is also possible to get some idea about each stage of the evolution by inspecting dv/d in Figure 04,b :
  1. For dv/d > 0 at large value of , it is in flation era.
  2. For dv/d = 0 at the 1st minimum, has reached a true vacuum (Figure 05).
  3. For dv/d < 0, it involves reheating and ends up in another false vacuum state.
  4. For dv/d > 0 after the 2nd false vacuum, the is making the transition to the final Higgs phase. This is the electro-weak SSB.
  5. For dv/d = 0 at the 2nd minimum, this is the SM domain as of today.

Figure 05 Higgs/Inflaton, Evolution with Time

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A Semi-classical Model of Inflation (2022 Addition)

Similar to the semi-classical Bohr model of hydrogen atom in the pre-quantum mechanical era, a semi-classical model of inflation can be formulated without any quantization scheme. A one dimensional case is considered just for illustration purpose. The derivation starts with the space-time interval in flat space, i.e., k = 0 (see "Space-time Digram" and Figure 05a) :
Semi-classical Model

Figure 05a Semi-classical Model [view large image]

According to the consensus view on cosmic inflation, the duration and expanse during that period are :
T ~ c(10-32/10-44) ~ 1022, R ~ (10-6/10-33) ~ 1027 and T = R =1, ds2 = 0 time-like in the beginning.
Thus, R >> T and ds2 < 0 thereafter, such events are qualified as "Inflation" - QED.

If this Planck size entity also contains (an energy equivalent) Planck mass MPL = (c/G)½ = 2.17x10-5gm, its corresponding Schwarzschild Radius rs is equal to twice the Planck length, i.e., rs = 2GMPL/c2 = 2LPL implying that the initial radius is inside the event horizon, i.e., such Planck scale particle is a black hole. It has been shown that such black hole would evaporate in time interval :
tev = 2x1067(MPL/Msun)3 years ~ 6x10-40 sec
indicating that the inflation occurred during the process of "Black Hole Evaporation" (see Hawking radiation).

N.B. :
In classical cosmology, the scale factor R is a time dependent variable determined by the Friedmann Equation
(dR/dt)2 - 8GR2 / 3 = - kc2.
In this semi-classical approach, it becomes a "free running" variable independent of time. As shown in the next section, R is the running variable in the "Quantized Friedmann Equation" which does not involve time.

BTW, the "LPL-tPL" is the absolute minimum space-time. Because any system below such limit violates the uncertainty principle of
t tPL (for the corresponding Planck mass/energy, i.e., EPLt = EPLtPL) and similarly for x LPL rendering it meaningless (see "What Is The Smallest Possible Distance In The Universe?").

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A Novel Higgs Field

§§§ This is just a mathematical model.

There is an alternate way to associate the Higgs before and during cosmic inflation. The formulation starts from a very small universe at Planck scale with Mass/Energy M = MPL and size r0 = LPL. Accordingly, its energy density PL ~ 10114 erg/cm3. Such entity is at the realm of quantum gravity. Unfortunately, there is no such universally accepted theory for now. One of the ad hoc solutions is to quantize the Friedmann Equation for cosmic expansion. Mathematically, the procedure transforms a non-linear equation such as Eq(5a) into a linear equation such as Eq.(6) which admits linear combination of its solutions and thus the property of "quantum" superposition.

The classical Friedmann Equation for the evolution of scale factor R is in the form : (dR/dt)2 - 2GM / [R(r0)3] = -kc2 ---------- (5a),
where k is the curvature of the universe, it is actually a form of energy curled up in space.

For the Planck scale model, it can be re-written (in term of the Planck time tPL) as : (dR/dt)2 - 2 / [R(tPL)2] = -kc2 ---------- (5b).

In first quantization to endow wave property to a particle the linear momentum px and position x has to satisfy the commutative relation :
Energy Level and Cosmic Expansion

Figure 06 H Atom Energy Level & Cosmic Radius

Now the hydrogenic wave function n(R) can be identified to the inflaton field (or the Higgs field , or whatever designation). Inflation corresponds to the transition from the level of n 1 to the continuum (n ). Energy should be infused to pump the transition as shown in Figure 07. After inflation, the
dynamic equation reverts back to Eq.(5a) with k 0. The gravity couples only to mass/energy, detached from the Higgs field.

N.B. The scale factor R in classical cosmology has changed from a function of time to just a running variable upon quantization serving a different role as running variable in cosmic inflation.

Cosmic Energy Density Evolution of Fields Effort to formulate theory in time or size smaller than the Planck scale (~10-44 sec, Figure 08) is doomed to fail as any system below such limit violates the uncertainty principle of t tPL (for the corresponding Planck mass/energy, i.e., EPLt = EPLtPL) and x LPL rendering it meaningless (see "What Is The Smallest Possible Distance In The Universe?").

Figure 07 Cosmic Energy Density [view large image]

Figure 08 Cosmic Evolution

BTW, the cosmic evolution in Figure 08 shows that the structures in quantum domain had completed within a very short time interval about 10-32 sec. Then it went through a period of making the atoms for building the large structures,
Eventually, gravity took over the process of assembling the large structures such as stars, galaxies, ... The living world is the latest to emerge (see "Origin of Life"). Awareness of Reality comes even later.

Meanwhile, the energy density at the ground state of this "Novel Higgs Field" is ~ 10114 ergs/cm3 (see "Planck Scale"), which is a value often quoted mistakingly as the vacuum energy (dark energy). In fact, it has a value of ~ 0.75x10-8 ergs/cm3 ~ 0.75x10-29 g/cm3 (see "A Novel Idea on Dark Energy" in next section).

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A Novel Idea on Dark Energy (2022 Addition)

A crucial information about the dark energy density is the realization that it is constant over the entire eon. The now discredited "Steady State Theory" in cosmology has a similar aspect in such observation. This theory also subscribes to the "Cosmological Principle" over the entire eon (while its Big Bang rivalry adopts the same principle but the cosmic aspect changes over time). Thus, the Steady State Theory is applicable to a "small" region, e.g., the void in "supercluster" at a given time, e.g., in particular, a very tiny volume of Planck length LPL. It can be shown (see mathematics below) that upon quantization, as the space opens up in cosmic expansion, the newly created Planck size volume is replenished with a fixed amount of energy (density) which is called "Dark Energy (density)" (see figure 08l).

In a way, this scenario is similar to the "Loop Quantum Gravity" with the Planck scale volume as the basic unit of space. In addition, it is posited that cosmic expansion corresponds to adding such unit which also receives a fixed amount of energy E or in term of energy density (the so-called Dark Energy).

Here's the math. -- In Standard (Classical) Cosmology, the Friedmann Equation for vacuum energy only universe with k =0 is :

(dR/dt)2 = R2 c2 / 3 ---------- (8a)
where is the cosmological constant. It can be reduced to a form adopted by the Steady State Theory as :

(dR/dt) = HR ---------- (8b)
where H = (c2/3)1/2 is the Hubble constant in the Steady State Theory.

Quantization is required, if we consider a very small spatial volume of the Planck size, i.e., VPL = (LPL)3. According to the quantization procedure in "Novel Higgs Field", (dR/dt) = (-i/tPL)(d/dR) operating on the wave function , i.e., Eq.(8b) becomes :

Dark Energy

Figure 08b Dark Energy [view large image]

The scale factor R in quantum domain can run from 1 to a huge magnitude of about 1063 comparing to the range of 0 - 1 in classical cosmology. It can be estimated by the case of "novel inflation model" which produces such humongous value at the end of inflation when R/n ~ 1 for n ~ 1063 with k ~ 10-62 cm-2 (see Eqs.(7),(8) and "Flat Spatial Curvature").

N.B. The scale factor R in classical cosmology has changed from a function of time to just a running variable upon quantization serving a different role as running variable in cosmic inflation.

Quantum Fields Meanwhile the quantum fields are also homogeneous, isotropic, and constant over the eon; it is tempting to identify each of the 24 quantum fields (Figure 08b2) to the eigen-function with different value of H (labeled by n, Hn and n for n = 1, 2, 3, ... 24). It is estimated that they would have energy density 10-8 erg/cm3, with the corresponding vacuum (zero point) energy En = hn/2, where n = (k/m)1/2, m is the mass, k the spring constant for the nth particle. The elementary particles in this world are just in the excited state of the quantum fields. As the Universe expands, the quantum fields have to be replenished similarly to the dark energy.

Figure 08b2 Quantum Fields

3-D U in 4-D Bulk As for where does the infusing energy come from ? The quick answer would be from the 4th spatial dimension (Figure 08b3). The corollary for contraction to the beginning is to remove such energy back to the bulk. Thus, such reversal process would not end up with infinite matter-energy as oft-quoted in the past. Unfortunately, the "Superstring Theory" is not viable anymore due to failure of finding any experimental evidence.
Or, is the bulk just another 3-D environment beyond this Universe ?

Figure 08b3

This is part of the progress in science -
more problems are created following the resolution of an old one.

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Theories of Cosmic Inflation and B-mode Polarization (2022 Addition)

Primordial Fluctuations The pattern on Cosmic Microwave Background (CMB) has shown conclusively that it is the relic of density fluctuation. The theory of cosmic inflation is rather convincingly (not 100%) ascribed as the origin. Mathematically, the CMB (the earliest observable) is expressed by the Power Spectrum :
Ps(k) = Ask(ns-1) ---------- (9),
where k is the Fourier transform of the spatial coordinate x, ns 1 the scalar index, As the scalar amplitude, the subscript "s" denotes scalar type perturbation (quantum fluctuation), Ps(k) represents the density fluctuation in k-space corresponding to random fluctuation in x-space (see Figure 08c).

Figure 08c Primordial Fluctuations

Some versions of the "Inflation Theory" also include the fluctuation in gravity (or metric tensor in General Relativity paradigm). It is often referred as primordial gravitational wave which is expressed as
Pt(k) = Atknt ---------- (10),
where t denotes tensor type. The amplitude At should be very small comparing to As since gravity is much weaker than the inflaton field responsible for the scalar (density) fluctuation. The ratio r = Pt/Ps is crucial to constrain various types of inflation theory.

Scalar fluctuation originated from random quantum fluctuation of the inflaton field , which in turn induced the density fluctuation .
acoustic oscillations acoustic oscillations The gravitational attraction in the density enhanced regions, and the radiation repulsion acted together to produce the incoherent acoustic oscillations (noise). Compressing a gas heats it up; letting it expand cools it down - this is the origin of the temperature variation which eventually shows up as CMB. As shown in Figures 08d and 08e, the main peak is the oscillatory mode that went through 1/4 of a period (reaching maximal compression) at the time of recombination (between electrons and protons to form neutral atoms). The lower peaks correspond to the harmonic series of the main peak frequency.

Figure 08d acoustic oscillations [view large image]

Figure 08e Power Spectrum

acoustic oscillations In angular coordinates, the temperature fluctuation is expressed mathematically by :



where Ym(,), is the Spherical Harmonics, and = 0 denotes the monopole, = 1 the dipole, = 2 the quadrupole, ..., and m can be any integer between - and . Each alm constitutes a multipole mode. The lowest multipoles are the largest-area, continent- and ocean-size undulations on the temperature map. Higher multipoles are like successively smaller-area plateaus, mountains and hills (trenches and valleys) inserted on top of the larger features. The entire complicated topography is the sum of the individual multipoles (Figure 08f).

Figure 08f Multipole

Eq.(11) is the Inverse Fourier Transform back to the x-space with integration replaced by summation over the multipole moment , Ylm playing the role of eikx, and alm the power spectrum Ps(k) in Eq.(9).

B-mode Polarization There are two distinct gravitational fluctuations corresponding to two kinds of polarizations as shown in Figure 08g. The E-mode is similar to the scalar fluctuation with g for = = 1,2,3 producing compression/expansion perturbation. While the B-mode is for producing shear perturbation.

Figure 08g Tensor Polarization [view large image]

Thus, the scalar fluctuation generates only E-mode polarization; while the tensor fluctuation (gravitational wave) generates both E-mode and B-mode polarizations.

B-mode Polarization Polarization in CMB can be detected by using a polarizer (by the antenna in astronomical instrument). The corresponding formulas to Eq.(11) for CMB in E-mode and B-mode are :

where E(n) and B(n) are CMB for the E-mode, and B-mode respectively, and n denotes the direction on the sky (see "Probing Inflation with CMB Polarization").

Figure 08h CMB Spectra, and Polarization Patterns

Figure 08h,a shows the un-polarized, E-mode and B-mode CMB from both scalar and tensor perturbations (dotted curves). The B-mode by lensing is detected from only a small patch of the sky via gravitational lensing. Figure 08h,b shows the patterns around hot and cold spots.

By the 2010's, advance in technology has allowed observation of the CMB to the very low intensity of the B-mode polarization. The
BICEP2 Experiment r-Ratio BICEP2's (Figure 08i) first attempt was a failure because it has not taken into account the inter-galactic dust (see "B-mode Detection, 2010's"). Since then it has created a precise inter-galactic model to subtract out the masking. The 2021 BICEP2 measurement provides the tightest constraint to gravitational perurbation with r < 0.036. The BICEP2 team have arrived at this limit by combining data at three frequencies (96 GHz, 150 GHz, 220 GHz) from their own experiment, complemented by archival data from the WMAP and Planck.

Figure 08i BICEP2 Experiment [view large image]

Figure 08j r-Ratio [view large image]

Figure 08j plots the r-ratio r = Pt/Ps vs the scalar index ns. It encompasses the prediction of some Inflation Models, and experimental measurement (current and future). It is related to the e-fold expansion of space during inflation and is
designed by Lf/Li = eN or N = n(Lf/Li), where L is the size of the universe for the initial (i) and final (f) state of the inflation. To be consistent with the theory of inflation, the value of N has to be between 40 and 60 (to solve the cosmological problems and to be in agreement with the measurement of ns respectively).

The latest measured value of r would rule out the monomial models (the yellow patch in Figure 08j); while the Starobinsky model (in purple) is within the new bounds from BICEP2 (in blue). On the other hand, the models related to cyclic universe would not produce gravitational wave, i.e., r = 0. However, this kind of models is based on the now defunct "Theory of Super-string".
In conclusion, the novel model satisfies most of the cosmological criterions providing yet another description for the beginning of the UNIVERSE.

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(Modified) Quantum Field History

According to the the above scenario of "quantum cosmology", the quantum field history has to be modified such that the Big Bang occurred amid quantum uncertainty with no definite time and size. It is only at the Planck scale that a concrete event occurred.
Quantum Field History The universe starts with gravity (which goes on by itself) and a quantum wave function that is identified as the Higgs field. This Higgs field went on to undergo many phase changes and finally settled down with 3 distinct quantum fields plus the Higgs field itself as shown in Figure 09. Such phenomena is similar to the phase transition of water (see Figure 02), which can exist in the form of gas, liquid, and solid in macroscopic states (similar to the strong, weak, and em quantum fields in Figure 09).

Figure 09 Quantum Field History (modified) [view large image]


NB : The gravitational field is a classical field that does not follow the rules for the quantum fields. It only coupled to the Higgs field at the very early cosmic time in the form of quantum gravity. It detaches from the Higgs after inflation when the size of the universe is beyond the quantum domain.
    The role of the Higgs field in the different phases is discussed briefly in the followings :

  1. Planck Scale (the Constants) - Soon after Max Planck introduced the Planck constant h in 1899 to account for the spectrum of blackbody radiation, he realized that the only way to construct a unit of length out of h = 6.625x10-27 erg-sec, the velocity of light c = 3x1010 cm/sec, and the gravitational constant G = 6.67x10-8 cm3/sec2-gm, is LPL = (Gh/c3)1/2. This is the Planck length.
    Subsequently, it transpires that the realm of Planck scale would involve entities with size of the order 10-33 cm and associated with high energy phenomena at 1019 Gev (see list below). Such environment is considered to occur at or near the moment of Big Bang - a most favorable theory for the creation of this universe. Here's the Planck scale :
    Quantum Field History
    • Planck Length - LPL = GMPL/c2 = (G/c3)1/2 = 1.62x10-33 cm.
    • Planck Time - tPL = LPL/c = (G/c5)1/2 = 5.39x10-44 sec.
    • Planck Mass - MPL = (c/G)1/2 = 2.17x10-5 gm.
    • Planck Energy - EPL = MPLc2 = (c5/G)1/2 = 1.22x1019 Gev = 1.95x1016 erg.
    • Planck Temperature - TPL = EPL/kB = (c5/GkB2)1/2 = 1.42x1032 oK.
    • Planck Energy Density - PL = (c7/G2) ~ 10114 ergs/cm3.
      Note that at speed of light c, there is enough time tPL for signal to travel the length of LPL, i.e., no problem with homogeneous/isotropic of the universe; and EPLtPL = satisfies the Uncertainty Principle.

    Figure 09b Convergence of Quantum Gravity [view large image]

    Figure 09b shows all kinds of objects in this world with different size and mass. Ultimately, they all emerge from a tiny speck at the Planck scale via the action of Quantum Gravity (see the corresponding Table).

  2. Planck Era (the Theory) - Theoretically, the moment of Big Bang should be related to General Relativity, specifically, the Friedmann Equation in Standard Cosmology. This equation has to be quantized if the initial entity is in Planck scale as discussed in "Novel Higgs Field". It turns out that the wave function is similar to the Hydrogen atom's with the scale factor R as independent variable. This wave function is identified to the Higgs field . The transition of from the state of huge curvature k to k = 0 represents the period of inflation at the end of which it approaches an oscillatory form ~ Ae-i(t-kx), which is the solution for the "Klein-Gordon Equation of the Scalar Field". At this stage, the Higgs field could be the only quantum field around. It used to be called "Grand Unification" by assuming that all the quantum fields are in existence all the time but merged together at this point.
    BTW, the "flatness" problem is resolved by the transition of curvature k ~ 1/LPL to k 0 in the quantum cosmological model.

  3. Quark-Gluon Plasma
  4. Reheating and "Electroweak Unification" - During inflation, the environment lost a lot of energy in the process. Reheating is another process whereby the energy of the environment is replenished by energy from the Higgs field in the form of quarks and gluons (the Quark-Gluon Plasma QGP, see Figure 10). Separation of the quark fields (of strong interaction) at this period represents a form of phase transition. The re-shuffling also moves the Higgs field to a false vacuum state (mathematically by adding the potential V() to the Klein-Gordon Equation, see Eq.(3b)).
  5. Figure 10 Quark-Gluon Plasma [view large image]

  6. Electroweak Spontaneous Symmetry Breaking (SSB) - At 10-12 sec after BB (Figure 09), the Higgs field decays from the false vacuum to the true vacuum precipitating another phase transition. According to the Standard Model (SM, Figure 11), all the quantum fields already were in existence at the state of false vacuum. However, the quantum fields were not interacting with the
    SM, Parameters Mass, Origin of Higgs and hence the quantum particles (excited from the fields) are restmass-less moving at the velocity of light (Figure 12). Most of them acquired restmass only after SSB. The event separated the electroweak field into the electro-magnetic and weak; it also made the left/right-handed versions of the quarks and leptons acting differently in weak interaction.

    Figure 11 SM, Parameters

    Figure 12 Mass, Origin of
    [view large image]

    Another phase change occurs about 10-5 sec after BB. This one transforms the Quark-Gluon Plasma to the bound state of Hadons (see "Chiral Symmetry Breaking").
    BTW, BB here denotes "Big Bang" not "Best Before" (no pun intended).

  7. Stability of the Universe
  8. By taking into account the higher term of the SM (Standard Model) perturbation series, the vacuum potential of the Higgs field V is in the form :

  9. Figure 13 Stability of the Universe [view large image]

    and the sum is over all SM particles acquiring a Higgs-dependent mass Mi. The precise form of V1 is not important in the present context, it just shows that the Higgs potential also depends on the particles it acts upon. Furthermore, only the heaviest top quark in the sum is retained in the following consideration.
    As shown in Figure 13,a and the above formula, the vacuum potential V depends on the mass of the Higgs m and also the mass of the top quark mt. Computation of V with the various values of m and mt yields different forms, which can be stable, meta-stable, or runaway (see Figure 13,a). Figure 13,b plots the result revealing our universe is possibly in an meta-stable state sitting on the verge of catastrophic decay into the true vacuum. The revelation produced sensational news headline around the globe one morning in 2013 (soon after the discovery of Higgs in July, 2012). A second wave of doomsday prophecy occurred in 2014 with Hawking's blessing on the revelation.
See a slightly different version in "Evolution of Quantum Fields" and one more variation below.

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Quantum Field History à la Gauge Theory

Gauge Transformation In Gauge theory, the Largangian or field equation is invariant under the gauge transformation :
' = ei(at) ,
where a is the phase angle associates with the "axis", t the generator in the form of nxn matrix, n the dimension of the Gauge space, the number of phase angles is determined by the formula Na = n2 - 1 (except for n = 1, Na = 1, see a little bit more in "Unitary Groups").
Figure 14 is an example of simplest Gauge transformation with n = 1. It shows that the SO(2) rotational transformation of 2-component field (interpreted as two +/- charge states) is equivalent to a single complex field in U(1)= eiI (I = 1) Gauge group. In general SO(n+1) ~ SU(n), where "S" means that the transformation is "Special" with |det(U)| = +1, which implies continuous rotations without reflection.

Figure 14 [view large image]
Gauge Transformation

This is "Global Gauge Transformation" applicable to all space. It is the spatial dependence "Local Gauge Transformation" that generate the Gauge bosons responsible for interaction of particles.

Formulation of the Standard Model of Elementary Particle (SM) is entirely based on the concept of Gauge invariance of the theory both global and local with the U(1) gauge transformation for electromagnetism, SU(2) for leptons, and SU(3) for quarks.
    The followings trace the variations of the Gauge environment at different phase of the Quantum Field Evolution :

  1. Planck Scale Era (novel) :
    In the "Novel Quantum Cosmologic Model", the cosmic wave equation Eq.(6) is global Gauge invariant with ' = ei , where is a constant, while the wave function is identified to the Higgs field up till the end of inflation.

  2. Grand Unified Theory (conventional) :
  3. Spontaneous + Electroweak, Symmetry Breaking (SSB + ESB) :
  4. Spontaneous + Chiral, Symmetry Breaking (SSB + CSB) :
Finally, here's an analogy with the Crude Oil Distillation which may provide a better understanding of the quantum fields evolution :
Crude Oil Distillation The main way of processing crude oil is to separate it using fractional distillation. Oil is made up of many different hydrocarbons of different boiling points (Quantum Fields). The first step in distillation is to boil the oil to a very high temperature (Big Bang), usually around 400oC. When the solution boils, it forms vapors entering the fractional distillation column (GUT), which has many trays to collect the liquid (Quantum Fields). As the height in the column increases, the temperature gets cooler, so as the vapors rise, they cool. Since every hydrocarbon chain has a different boiling point, as a chain reaches a height where the temperature is lower than the boiling point, it will cool into a liquid and be collected by a tray (Symmetry Breaking Processes). This method allows for all parts of the oil to be separated and collected so that they can be further processed and refined (Standard Model). A very good animation of the distillation process can be found in Figure 20.

Figure 20 Q Fields Evolve and Crude Oil Distillation


According to this scenario, all the quantum fields were unified (mixed together) in the beginning. Each one emerges from the whole and manifests its characteristics only at suitable temperature by the process of Symmetry Breaking. Ultimately, it is the cosmic expansion (which keeps lowering the temperature) that causes all these cosmic phase changes (Figure 09).