## sabato 25 febbraio 2012

### Symmetry and gravity

The standard model is a perfectly predictive model for the electromagnetic interactions, weak and strong. However, in its "simple" version does not include the mass of the particles. Why?
In the theories of modern physics "symmetry" is fundamental. All the conservation laws (momentum, energy, spin, angular momentum, baryon number, lepton number) are a consequence of the presence of certain symmetries.
Since experimentally is observed conservation of these quantities, it is essential that the theory has its symmetries. Without mass, the theory has the symmetry required, quantities are preserved, and the theory agrees with experiment.
In the moment that you add mass to the theory, break some of the fundamental symmetries, and the theory is no longer consistent with the experiment.
So you have to think about: no mass interactions are well described. By adding the mass the theory doesen’t correctly describe the interactions.
Yet the mass exists in nature! How do you solve the problem?

It 'obvious that the mass term must be added. We must only find a way to do it without destroying the symmetry.
If the mass term added breaks the symmetry, must be added another term, a new term, which behaves in the opposite way to the mass term for the particular symmetry: the two effects are equal and opposite , cancel out, and we have a theory with the mass in which the symmetries are preserved.

In the theories currently experimentally testing, the new term is the so called “Higgs boson” and the mechanism by which is given mass to elementary particles is, in fact, the "Higgs mechanism".
However there are other ways, formally more elegant, to achieve this result through the principle of "spontaneous symmetry breaking."
The easiest way to understand what is meant is to apply the " spin chain”.

The spin chain (one-dimensional) is the model that represents a elementary magnet (unidimensional).

Imagine a set of points (nodes) on a line which corresponds to a sequence of integers each of which is associated with a "spin" worth + or -.

....
+ + - + - + - + + + --- + - + + - + - + - ....

In this model we add a dynamic or do fluctuate over time these nodes in dependence on temperature, according to the dynamics determined by a certain mathematical law.

We have a model for which, at very high temperature, the probability for each node to be + or - is 50%. In this phase, in fact, temperature (thermal agitation) wins completely on the magnetization (the interaction of the neighboring nodes is too low compared with thermal agitation).
If then mediate the spins, and we call the average magnetization M, we obtain the value M = 0.

Decreasing the temperature, the condition M = 0 is kept up since to a certain critical temperature Tc. Below the critical temperature thermal agitation is no longer sufficient to swing the spin at random and, therefore, begin to form all positive or all negative areas (the interaction between neighboring nodes start winning on thermal agitation). Thermal agitation is still acting, although less so.
Then the magnetization M is different from zero (for example, just below the critical temperature, it might be M = 0.1).

Down again with temperature, approaching at T = 0, thermal agitation is weaker and weaker, likely random fluctuations always less, and spin chain tends to the value M = 1 (or -1, depending on the case decided by the fluctuations around the transition T = Tc).

In summary:

T> Tc : M = 0 ;
T = Tc : transition state, where the spins begin to mate ;
T Where is the spontaneous breaking of symmetry? And, above all, what we're talking about symmetry?
Symmetry is always compared to a transformation.
In this case the transformation is to invert the sign of each spin:

+ ----> -
-----> +

It 's also clear that in phase T> Tc, since the + and - are randomized, will be nothing the magnetization M. Then the system is symmetrical respect to this transformation.

The same can not be said for the phase T In fact, being M different from 0, applying the same transformation

+ ----> -
-----> +

that is you will obtain :

M ----> M-

and, therefore, the transformation is no more symmetric for the system.

There was a spontaneous symmetry breaking.
This kind of mechanism is the basis of phase changes in thermodynamic systems.
In the example spins represent electrons magnetic spin moments or " quarter quantum numbers" that define the Pauli exclusion principle for which, in the stable octet configurations of chemical elements, the last two electrons of the outer orbitals must have equal and opposite spins (+ ½ and - ½) ; or, if we consider the electrons as localized particles, rotate around its own axis one in a direction and one in the other, to give two equal and opposite magnetic fields. In reality electrons occuping atomic orbitals do not appear exactly as localized particles. The Heisenberg uncertainty principle, which underlies whole Quantum Mechanics, provides for you can not determine with certainty position and momentum of an electron. These, in fact, are defined by the possible solutions (eigenvalues) of Shroedinger probability wave function or, as they say, the "collapse" of the wave function in one of the possible "eigenstates" of the quantum system under consideration (the 'orbits of the electron position and momentum in this case). In fact the atomic orbital is to be considered a probability density distribution area of finding electron at a certain distance from the nucleus. ​​The integral of this area is 1 (the "normalization” condition) and represents the "sum" of the likelihood of the electron presence around the nucleus, which is a mathematical certainty.
The "collapsed" (eigenstate) quantum state is the cause of the observed physical state of the macroscopic system.
Therefore breaking quantum symmetry determines status changes in the macroscopic physical systems and also phase transition in thermodynamic systems.
Going back to the spin moments and considering in place of a spin chain a "spin network" perfectly symmetrical, homogeneous and isotropic (many + and - according to a three-dimensional spatial lattice and neatly alternates) will get a "potential" magnetic field Substituting even one only + with a - we have a symmetry breaking and the triggering “domino” effect which results in the transition from potential to “actuated” field, that is it reveals its properties (the magnetization with the relative lines of force in this case).
Let’s see another example of transformation.
Consider a planetary system with two bodies. The potential energy which describes the system is central, that is it depends only on the distance of the two bodies and not on the absolute position.

One possible transformation is " get closer or farther away radially the two objects" or "rotate an object around another, keeping constant the distance."
Both are transformations, but only the second is a symmetry because the potential, which contains all the information needed to describe the system, does not vary unless it changes the radial distance.
We know, thanks to Noether theorem, that each symmetry corresponds to a conserved quantity.
The theorem allows to calculate this quantity starting from the symmetry. Applying the theorem to the cited example, we obtain the angular momentum as conserved quantity.
The exact same procedure also applies to the quantum level: each symmetry, corresponds to a conserved quantity.
Momentum, energy, but not only that : all the quantum numbers. lepton number, baryon number, spin, and so on are all conserved quantities as a consequence of symmetry presence in the standard model.
The symmetry previously described for the spin network can 'be extended to any distribution of pulse/energy sources and, for example, also can describe a gravitational field when this symmetry breaks. However, in this case, the sources of gravitational energy would be electromagnetic and the energy released by symmetry breaking would be greater then more dense the three-dimensional lattice. In fact thought electron as a wave and not particle, the energy associated to the electromagnetic field would be E = hv, with h Planck's constant and v frequency (Hz) of the field oscillation, greater then more close is the wave length. Symmetry of the interference between transverse waves of equal amplitude, frequency and phase would give standing waves and, therefore, would make the field only potential. The symmetry spin network breaking, caused also by a small change in frequency, phase or amplitude of oscillation, would give the releasing of potential energy bridled in the lattice/three-dimensional matrix, giving rise to the lines of force of the gravitational field which, of course, would be dynamic.
However the matrix/lattice would be only a “discrete” model of an underlying physical reality consisting of a "continuum" made up of matter/energy distributed as waves.

The spin network, then, might be useful to define a mathematical/ geometrical simplified calculation that describes formally the field.

Stefano Gusman.