There are two views on the god particle as expressed in the cartoons above. The left expresses the trivial way the god particle influences the ordinary man. The one on the right expresses the actual experimental situation when the finding of a particle depends on a statistically relevant blip on a screen. It is the relevance of this blip that forms the basis of the god particle. What makes it significant is the amount of money spent on it.
In philosophicsl/religious treatises/testaments on the origin of matter/life one inevitably ends up starting with a philosophy or with a bang. The Old Testament of some Hebrews would have the world being created in seven days nearly 6000 years ago, being off by a factor of a million or so. This kind of error is sometimes allowed in cosmology. That it started suddenly is the common philosophical point and therefore similar no matter what Michael Moore would say abut it. After all, there is this (theologically) much debated line
The Mexican hat potential is always cited when discussing the Higg’s particle or “god particle”. It is important nevertheless to understand some aspects of real-life chemistry and physics that has applied the Mexican hat potential to routinely observed processes before one moves towards discussions on the Higg’s boson. I struggled (remarkably unsuccessfully at that time) with the Mexican Hat potential when I was trying to understand the Jahn-Teller distortion in molecules which occupy orbital (~spatial) states which are degenerate (energetically) equivalent. For such Jahn-Teller effects the distortion is usually studied in terms of molecular normal mode (pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation) vibrational coordinates, Qq,e (see Fig 3 left bottom for, say Cu2+ atoms with one d- electron in doubly degenerate orbital states, the arrows indicate direction of vibrating motion) Landau and Teller were among the first (late 1930s) to discuss the stability of molecules with orbitally degenerate state. The first order potential energy function for linear coupling between the normal modes for simple harmonic oscillators and the occupied electronic energy levels yields a double well potential.
The Mexican Hat potential is also sometimes known as the “wine bottle” potential when there is punting at the bottom of the bottle. The lowest-energy state of the system is not at the symmetry point at the top of the punt but is down in the trough of the wine bottle. The occupation of the trough at any point will break the symmetry. Another way which we think is useful (not only because it could be new) is to the bottle is held parallel to the earths gravitational field --- say, parallel to the vertical bars of the window grill of a cheap-ish apartment builing in Pune (Fig 4, left). If one manages to tilt the apartment complex without changing the direction of the gravitational field (Fig 4, right; this usually happens when the wine in the bottle reaches the dregs stage) and the bottle is still thought to be held straight, there is more liquid on one side and the symmetry of the “wine bottle” potential is broken.
The tilting of a bottle could be taken as the beginning of a swirling motion where liquid flows in the trough because of an introduction of a swirling motion introduce by the combined action of a tilt and rotational motion. If the liquid had no resistance to flow, say, because of a continuum of minima in the Mexican Hat potential--- it was a superfluid --- the flow will be perpetual once the initial flow of liquid was started. In this case it is not straightforward to imagine a tilt if the fluid had no charge. But once the fluid is imagined to be a flow of perpetual current then the fluid may be thought to be a superconducting current and the direction of tilt would be related to the direction of the magnetic field that is expelled by a superconducting current (Meissner effect).
The physics of the swirling or rolling of a ball in the “wine bottle” or “Mexican Hat” potential is looked upon sometimes as a transformation under continuous symmetry of the way the total energy of the system is expressed (its Hamiltonian operator). There is one mode which is thought to be gapless or no energy change on traversing the degenerate levels of the minimum potential. This is called the Goldstone Theorem and the massless mode is the Goldstone mode. The Goldstone mode is described usually by referring to a ferromagnet in which all the magnetic moments (taking up a direction in a magnetic field) are aligned in one direction (Fig 5, top, magnetic moments indicated by arrow). This arrow may refer to any direction. In Fig 5, middle, the finger of god reaching out to Adam in Michael Angel’s painting in the Sistine chapel would reach out in the same spiritual direction for all aadmi (hindi for man = adam). There are some critical points that arise from the system being infinitely large because, I suppose, one cannot roll up infinity into a three-dimensional ball or even flatten it into a two-dimensional disc. In this way an infinitesimally small point and an infinity become equivalent as they cannot be divided.
Peter Higgs gave an illuminating talk at Kings College, London, on 24th November 2010 (my wife completed sixty that day) which he titled “My Life as a boson”.
t is perhaps best to get Higgs’ viewpoint from this talk:-
The feature of spontaneous symmetry breaking is that you have some continuous symmetry that is broken by the ground state of the condensed-matter system. If it is of infinite volume, which of course a ferromagnet never is, then the ground state is degenerate. But if you were in an infinite ferromagnet then there would be a spontaneous magnetization, which could point in any direction, and you would think that the system was no longer rotationally invariant.
The Mass of particles
One of the more discussed topics in modern particle physics seems to be the origin of mass. This should be strange to us since we have been brought up in the belief that the central feature of Newton’s view is the conservation of mass so that mass is the crucial defining property of matter. There should be no origin because all of us, including elementary particles are born with a mass. Mass can be redistributed, but neither created nor destroyed. In modern physics there is no equivalent fundamental concept for mass.It turns out that in special relativity it is energy that is conserved, and in general relativity gravity responds to energy through space-time curvature and not mass.
There seems to be a new way to look at properties of physics, which is the property of emergence. It emerges from the interaction of smaller entities which themselves do not show it. For instance, the P,V,T of a gas of atoms or molecules are not anticipated from the property of a single atom or molecule though they may depend on the properties of those individuals. When the physicists consider emergence, mas appears as an approximate, emergent property of matter so that the understanding of the origins of mass of standard matter (colour gluons, up- and down- quarks, electrons and photons) depends on understanding the properties of the relevant elementary quanta.
“I would like to emphasize that this theory of the origin of the mass of standard matter makes no reference to symmetry breaking in the electroweak sector, nor to the Higgs condensate, nor of course to the Higgs particle.” (Frank Wilczek, in Origins of Mass, 2012
It is useful to familiarize one-self with mass being expressed in terms of energy as electron-volts. From the well known expression E = mc2, mass is expressed in units of eV/c2, where eV is an electron volt, c is the speed of light in a vacuum. Most times mass is simply expressed terms of 1/c2 units as electron volts, eV. 0.51 MeV is the rest mass of an electron or positron and an electron-positron annihilation yield nearly 1 MeV of energy; 0.94 GeV is the mass of a proton, 200 MeV energy released in fission of 239Pu or 235U atoms.
The next term one need to be (as usual vaguely) familiar with is The Standard Model (SM). comprises a set of fermionic fields whose quanta of excitation are fermions with spin ½ such as electrons, protons, quarks, etc. The Fermions obey Pauli exclusion principle by which two fermionic particles cannot occupy the same state at the same time and which leads to the Fermi-Dirac statistics. There is also a bosonic field whose quanta of excitations are bosons following the Bose-Einstein statistics in which two bosons can occupy the same state at the same time. I am not clear why Fermi, an Italian, is associated with statistics of fermions that gives others their space and Bose, a Bengali developed a statistics where one is used to one’s space being overcrowded. Actually a Bengali is unaware of the existence of others as long his wisdom can flow uninterrupted, non-contradicted (resistance less) preferably at the same time and space in perfectly coherent incoherence. That is why the Bengalis are so loved
A little about gauge and gauge theory
In simplest terms, I think, a change in gauge may be looked upon as the way we convert our measurement systems on going from America to Europe (e.g., feet to metres, gallons to litres, pounds to kilograms, dollars to euros). This is one example of a gauge transformation in which we have changed our measurement coordinates (gauge) with changes in our location. We do not change the absolute (a more difficult concept to define, perhaps, especially when we are talking of god-like terms) value (gauge invariance, --- “What’s in a name? that which we call a rose/By any other name would smell as sweet;/ So Romeo would, were he not romeo call’d”) of the property being measured. Put this way, gauge theory is straightforward. Gauge invariance must be maintained to ensure that the results are not dependent on the choice of gauge.
IThere is also Wigner’s statement that the invariance of observables under certain transformations implies the existence of an unitary operator on the various states. If the unitary operator commutes (Abelian group operation does not depend on the order they are applied as in the addition of integers, with the unity operation being addition with 0 as an integer) with the operator (Hamiltonian) that describes the dynamics of the system, it gives to characteristics (spectrum or observables) that describe the system. However, there are systems (such as those with long-range coherence including superconductors, superfluids, iferromagnets) in which the physical observable (local symmetry) does not reflect the invariant global symmetry of the system. This is the spontaneous symmetry breaking, as noticed by Higgs’ observer inside a ferromagnet (see above).
n the more complicated world of particle physicists, gauge theory is applied to areas (I don’t venture to understand) such as string theory and loop quantum gravity where, essentially, point particles are replaced by one-dimensional string-like (sutra?) objects and their path in space trace surfaces instead of lines and are described by appropriate languages from topology and geometry.
A gauge boson is a particle that carries an interaction between elementary particles by the exchange of these gauge bosons between them. Thus photons (mass ~ 0 GeV) carry electromagnetic interactions, the two W± (mass 80.4 GeV) and Z0 bosons (91.2 GeV) carry weak interactions and the gluons (mass 0 GeV) carry the strong interaction. All of these have spin = 1; except for the W± boson which have charge ± 1, the others are charge neutral.
The Lagrangian equations (that describes the state of a dynamic system in terms of position coordinates and their time derivatives that summarises the dynamics of the system) express a gauge invariance that cannot have a mass term so that it implies that gauge fields are massless and a mass cannot change as Newton said they won’t. Gauge symmetry offers a way to describe interactions due to invariance
properties of Lagrangians. The gauge principle provides a method to transform a Lagrangian that is invariant with respect to global Abelian symmetry from the U(1) group (unitary transforming) into a Lagrangian that is invariant with respect to local symmetry, or gauge-invariant.
By having a spontaneous symmetry breaking such that new background forces can be introduced, one may overcome the concepts of gauge invariance and introduce/add a mass because of a change in the background. Such a change cannot be a consequence of the initial starting equations. It is put in by hand due to the spontaneous changes in symmetry (as we did, we think, by the “drunken” building in Fig 4). This is a key step in the Salam-Weinberg electroweak theory.
Curvature and Mass
The mass of a particle (say, the earth) placed in the wine-less end of the bottle (see inset of Fig 4, left) would be different from that when immersed in the wine-full end (see inset of Fig 4, right) of the bottle, for example. If we take the tilt (director) angle as a measure or gauge then the extent of tilt may be taken as a measure of the change in the gauge. The director characterizing the local direction of tilt is a two-dimensional vector sicne it depends on the position in the trough. The continuous rotation symmetry along the bottle axis is broken in the director. The symmetry breaking occurs spontaneously in the tilted bottle. All directions are equivalent long the central symmetry axis (there is no change in energy) as long as the tilt angle is the same.
The curvature up the side of the wine bottle is often called a “mass". In the big bang theory this curvature has a relationship with real physical mass, The relationship between curvature and mass is reminiscent of the mass of a charge carrier in an energy vs wave-vector (E vs k) plot in band theory of solids Fig 6, left). The energy of a free electron, Efree, has a parabolic dependence on Its wave vector, k, such that Efree µ k2/m0, m0 being the mass of a free electron. In the presence of an attractive potential such as in a periodic lattice of a crystal, the energy, Ecrystal µ k2/m*, where m* is an effective mass. In the periodic potential of the lattice, the E vs k diagram shows allowed and forbidden region. In an extended zone scheme shows an energy band which are separated by band gap. Going by the curvature, it is apparent that the opening of a gap would lead to changes in curvature and therefore to changes in mass.
It is true that in the late 1950s and early 1960s the exclusive particle physics community was frustrated in their progress with particles and their origin. In particular the problem seemed to be to find a mechanism that gives a photon a mass. It was at this point that Nambu listened to Schrieffer (both graduate students at that time) and speculated on the way the broken symmetry gives mass to fermions. Again referring to Higgett’s 2010 talk at King’s college he writes *referring to Fig 6, right)
“the analogy that I think Nambu noticed was with the Dirac theory of a spin-half fermion. If you have a massless Dirac particle there are excited states of positive energy and there’s the sea, and there is no gap between them, but if you have a massive particle then those are split apart. There is an energy difference between the sea and the states of the fermions, so there is a BCS-like gap of 2mc2. This is basically the way in which Nambu’s BCS model involving spontaneous symmetry breaking generates fermion masses.”
In the Nambu-Goldstone mode, the ground state is not invariant under the symmetry. Instead, there is a continuous family of exactly degenerate, not invariant, ground states that are related by the generator of the broken symmetry. For every spontaneously broken symmetry there is a particle species with zero mass which are called Goldstone particle or Goldstone bosons when they are spin-less. The currents of broken symmetries create these particels from the vacuum. One may have the ball in the bottle of Fig 6 to be a ball infinite dimension
The “wine-in-the-tilted-bottle” model requires to be modified to make a proper connection with the Higgs’ model. In the first place the tilt in the wine bottle need not be so exaggerated. The tilt may be treated as a perturbation around the ring wirh a wave number, k. We may may make the tilt infinitesimally small or k tending to zero..When the relaxation time of the perturbation becomes long compared to the relevant time scale there is perpetual flow... as in a superconductor. The wine at the bottom of the “wine bottle” potential will then be seen as that shown in Fig 7 with a phase mode and an amplitude mode. It is the amplotude mode from the curvature that gives mass to the boson.
On the Mass of a Photon
The massiveness of a photon goes against some of our most cherished lessons from what we think is fundamental physics. A consequence of Maxwell’s description of electromagnetism is the constant speed of all electromagnetic radiation in vacuum. However, many of the properties of light is obtained from universal constants such as the Planck’s constant, h, and the velocity of light, c, and a spin angular momentum h/2p, for a frequency, v. As it turns out, the idea of a finite photon mass does not contradict developments in classical electromagnetism or quantum electro dynamics (QED). Experiments to find limits to the rest mass of a photon shows it to be less than 10-50 9.
With a nonzero photon mass, the usual Maxwell equations transform into the so-called Proca equations which are valid for the charge carriers in superconductors and for, the basis for understanding massive photons and the London penetration depth, lL ((the distance magnetic flux can penetrate into a superconductor). It turns out that the damping of a magnetic field inside a superconductor is totally caused by the photon having a finite mass. The propagator (the probability amplitude for a particle to travel from one place to another in a given time) for a massless photon with distance r fall off as 1/r. When a photon obtains a finite mass with a Compton wavelength 1/mA, the propagator drops as exp(-mAr) /r. The London penetration depth, lL ~ 1/mA. As early as 1992 in a paper titled “On the London equations” (PNAS, 1992, v. 89,no. 22, 10673–10675.] Sternberg had shown how the classical London equations for superconductors can be written in a form that yield the Proca equations. The abstract of this paper has the sentence: In particular, the field itself acts as its own charge carrier.
The Higgs field has non-zero strength everywhere and the particles such as W±, Z0 and photons aquire mass by interacting wit the Higgs field. Since the field carries its own charge carrier, the Higgs field should have a Higgs particle. The Higgs amplitude of the vibrational amplitude up- and down- the sides of the Mexican Hat potential (or wine-bottle potential, Fig 7) has a rest mass that the Large Hadron Collider experiments were looking for. The mass of the Higgs particle as determined by experiment (126 GeV ~ 133 mass of protons) would correspond to an atomic nucleus of atomic number 133. This value is close to the value of the fine structure constant of ~ 137. This value gives the relativistic limits of stability for the mass of an atom.
Anderson found in 1963 in a paper entitled Plasmons, Gauge Invariance, and Mass that when a superconducting Fermi liquid is charged, the scalar zero-mass excitations become longitudinal plasmon modes of finite mass of a superconductor. It has been pointed out (including by Higgs) that Anderson’s proposal was a speculation since he did not discuss any relativistic model so that he could not demonstrate (through Lorentz invariance) how the Goldstone theorem could be avoided.
A critical question would be whether such a Higgs boson mass value allows to extrapolate the standard model to very large energies witout leading to vacuum instabilities (which is an end-of-the-world scenario, in my understanding, at least for some particles). The most critical ingredient in evaluating this instability is the top quark with a mass of ~ 173.2 GeV. Whether the electroweak vacuum is stable or not up to the largest possible high-energy scale, would depend critically on a precise determination of the Higgs boson and top quark masses. An estimate by Alekhin and co workers show that the mass of the Higgs boson is at the brink of a stability (see Fig 8, click to expand for reading the legend).,
This must be exciting from a Russian roulette point of view since the stability of our universe would depend on the theoretical models for the “god particle”.
The Higgs field that permeates all space and gives mass to some bosons including photons now becomes conceptually similar to the luminferous ether , that is a material substance that causes light to have an unique velocity as it passes through it. The famous experiments of Michelson and Morley disproved this concept when they showed that the measured velocity of light did not depend on the direction of the Earth’s motion seemingly demolished this theory and established the special theory of relativity and Einstein’s story became a folklore. Higgs electro-weak field is a modern relativistic, quantum mechanical, non-Newtonian resurrection of the ether.
When Einstein applied his general theory of relativity to the structure of the cosmos, Newtonian laws still guided scientific efforts and to be consistent with Newton his model, Einstein had to have in his model an infinite Euclidean three-dimensional space with the mass (or density, r) being distributed uniformly. It seems that Einstein realized that for such a model the universe would be unstable to collapse or expansion. For Newtonian gravity, in which the gravitational field is conservative, one uses the Poisson equation (Ñ2F = 4pkF) for a scalar gravitational potential, F. Einstein found it necessary to add a cosmological constant, l, that modifies the Poisson equation (to Ñ2F + lF)= 4pkF) to his field equation. By doing so, it seems, Einstein had introduced a new repulsive force on every particle that balanced the catastrophic consequence of an attractive forces due to the distribution of mass, Einstein himself required a justification of the cosmological constant, l, by empirical observation. The cosmological constant was required by Einstein to get a quasi-static distribution so as to obtain “ ... the fact of the small velocities of the stars.”.
Beginning of Time: Setting up a Supercurrent
In superconducting coils a supercurrent can be set up by having the rate of change of an external magnetic field with time being not equal to zero. Taking this up in some way may mean that the fact that a supercurrent is important in setting up the Higgs field or generating the Higgs particle can only be done at the beginning of time itself ... or at the origin of the time coordinate itself when the Minkowski spacetime is three dimensional space alone. I don’t know if this makes some sense to the special relativity people as they always have the past and future light cones meeting at the origin.
It is perhaps because of this that in real time time, when time exists, mass is given to gauge bosons and the realisation of existence begins. Causality or Adristam comes into being.
I hope it makes some sense in the context of the beginning of the day in the life of Brahma.
The cosmological constant was thought by Einstein to be the residual space-time curvature when matter was created/removed. It would therefore seem that at time t = 0, some supercurrent contributing to the relativistic gauge field still existed. So whether Brahma’s days and night may continue in cycles there is the eternal Brahman overseeing this cycle.
There is no Creatio ex Nihilo?
In the meanwhile Shiva continues with his dance of creation and destruction as Brahman said he should (I think). In the meanwhile, we may add our "aum" to te sparklers from the experiments on the Higgs particle in the spirit of Diwali --- our own Festival of Lights