## Saturday, 7 January 2012

### 9. Symmetry is Supreme

The edifice of modern physics, which we humans can be justly proud of, would have been unthinkable, even impossible, without the help and insights the scientists got from recognizing the role played by symmetry (and broken symmetry) in shaping the fundamental laws of Nature.
Long before any science, man was fascinated by symmetry (Klaus Mainzer).
Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection (Hermann Weyl).
Fundamental symmetry principles dictate the basic laws of physics, control the structure of matter, and define the fundamental forces of Nature (Lederman & Hill).
As far as I see, all a priori statements in physics have their origin in symmetry (Hermann Weyl).

We speak of symmetry in any situation in which we can rearrange things (positions in space, values of quantities in time or direction, etc.) and still get the same answer for any physical question we may ask about the rearranged system. In the picture above, the flower has '5-fold rotational symmetry': If we rotate it by 2π/5 or 72o, we cannot tell whether the rotation was indeed performed or not. The flower is 'invariant' under this 'symmetry operation'.

Take the law of conservation of energy. It says that the total energy cannot be enhanced or diminished. What that means is that if you measure the total energy today, and then again measure it tomorrow, you get the same value. The total energy remains INVARIANT as time progresses. So this is time-symmetry of total energy. There is a deep connection between conservation laws and symmetry.

We see so much symmetry in Nature, so the question arises: Why is that so? In my recent book Latent, Manifest, and Broken Symmetry (2011) I have argued at length that the basic underlying cause of this is simply the second law of thermodynamics ('minimization of free energy'). I shall not repeat the argument here in this short post.

But we also see so much BROKEN SYMMETRY (or reduced symmetry) around us. It is the second law again. If a system can lower its free energy further by making a phase transition to a lower-symmetry phase or state, it would do so. The familiar example is liquid water freezing to crystals of ice on cooling to 0oC. Liquid water is highly symmetric: It looks the same from any direction, meaning that it is invariant to any rotation around any axis of rotation; it is isotropic. By contrast, a crystal of ice has a far lower degree of rotational symmetry; it is anisotropic.

The figure above (called the 'Mexican-hat potential') is an illustration of spontaneous breaking of symmetry. The ball could have rolled down along any direction, but once a random choice has been made spontaneously, the final state has a lower ('broken') symmetry. This figure also illustrates ‘unstable equilibrium’. The top position has zero slope, and therefore corresponds to equilibrium. But this is unstable equilibrium because it is not robust against even a minor displacement away from equilibrium. Unstable equilibrium can lead to spontaneous breaking of symmetry as the system seeks a state of stable equilibrium.

The moment of the Big Bang (cf. Part 2) was the moment of highest symmetry: No structure, no preferred direction, no inhomogeneity. Then a very brief interlude of cosmic inflation occurred (cf. Part 8),  during which even the tiniest of quantum fluctuations got amplified hugely; this created gradients. The emergence of matter from the earlier radiation field was a symmetry-breaking phase transition. The radiation field was translation-invariant, and the appearance of matter broke, among other things, the translational symmetry. A field called the Higgs field has been postulated in cosmology to understand these phenomena. This field breaks the symmetries of the interactions among the elementary particles, and gives the particles their mass.

Nature abhors gradients because they embody states away from equilibrium, meaning that there is scope for lowering the free energy. Phase transitions occur to lower the free energy, and there is usually a concomitant lowering or 'breaking' of symmetry.

THE ENTIRE HISTORY OF THE COSMOS IS ONE BIG SAGA OF HOW SYMMETRY-BREAKING TRANSITIONS HAVE OCCURRED AGAIN AND AGAIN.

[Sometimes it is not very appropriate to use the term 'phase transition'; a better description is 'bifurcation in phase space', but that is only a technicality.]

When objects interact with one another, only four types of interaction can be possibly involved: the gravitational interaction; the electromagnetic interaction; the weak nuclear interaction; and the strong nuclear interaction. But at the moment of the Big Bang there was only one 'unified' interaction. It is only because of the cooling of the universe that the four interactions known to us today emerged one by one, engendered by symmetry-breaking phase transitions.

Broken-symmetry-considerations occupy centre-stage in theoretical physics. Particularly in particle physics, progress often means postulating, and then discovering, a new broken symmetry. Why only a broken (or ‘hidden’) new symmetry? Because an unbroken symmetry would have been manifest already, and would thus be old symmetry rather than new symmetry. Broken symmetries in Nature are not always very obvious; one has to go looking for them.
Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop (C. N. Yang, Nobel laureate).

Thou hast no faults, or I no faults can spy;
Thou art all beauty, or all blindness I (Christopher Codrington).

'Diatoms are single-celled organisms found in oceans all over the world. There are estimated to be 100,000 species of these micron-sized creatures in existence, and they play a crucial role as one of the main food sources for marine organisms, including fish, molluscs and tunicates, such as sea squirts' http://www.sciencealert.com.au/features/20141809-26195.html.

Again a very useful and fascinating blog post. Incidentally , I was reading physicist Victor Stenger's "God: The failed Hypothesis" few days ago . In this book , Stenger discusses about 'Spontaneous Symmetry Breaking' to explain the apparent double spiral helix pattern visible in Sun Flower head. He demonstrated this process using a simple computer simulation. Quoting from the book :

"With this simple computer program, I have demonstrated the process called spontaneous symmetry breaking, whereby the symmetry of a system is broken naturally, that is, without being forced on the system by some asymmetric mechanism."

He tells us that this symmetry is broken naturally. Could you please explain a bit how this process occurs naturally ?

Thanks

2. Thank you, Priyabrata. Symmetry is an organizing principle, but only a secondary organizing principle. The mother of all organizing principles is the second law of thermodynamics for open systems. Perhaps I can state the following law: Any open system tends to acquire the highest symmetry compatible with the requirement of lowest free energy. Subject to this law, the symmetry of an open system can break if the external conditions change. Another important idea is that unstable equilibrium leads to breaking of symmetry as the system tries to reach a state of different but stable equilibrium.

Spontaneous symmetry breaking (SSB) occurs in any system that is not stable to even tiny random fluctuations. The importance of SSB is that it leads to self-organization and build-up of complexity. As I shall explain in future posts, the emergence of life out of non-life is simply an example of self-organization, unaided by any designer or creator.

By the way, if anybody wants a soft copy of my book on symmetry, I can send the 5 MB pdf file. Just send me an email at vkw1412@gmail.com

The series is very exciting, I learnt how order comes into existence via second law of thermodynamics in the previous posts. Now it is engrossing to learn that symmetry is inevitably broken down as the interactions cause a lowering of free energy.

It is incredible how Nature's property to abhor gradients, can be used to explain observable phenomenon and processes.

I have a question regarding creation of matter from the primordial radiation field. I suppose the universe was in the "inflation" stage, how did theoretical physicists figure out that the radiation field was translation symmetric ? If so, I imagine it to be a sphere of radiation growing exponentially. Now if the creation of matter caused Spontaneous (tranlsational) symmetry breaking (SSB), it must have caused matter to be created in a non homogenous manner in the primordial radiation field. As the universe continues to expand, we should find a map of regions where matter is more concentrated than others. To paraphrase it, the density of matter in space won't be uniform in all directions. Does it align with our cosmological observations ?

4. Your questions will get answered in future posts. Still, if you are comfortable with the jargon, here are some quick replies:

1. A radiation field has so much homogeneity that we can assume translational invariance.

2. It is difficult to visualize, but the expansion of the universe, during and after inflation, is in an n-dimensional hyperspace, although most of these dimensions have got 'curled up', as modelled by the M-theory. The exponentially growing 'sphere of radiation' you mention was in this hyperspace, with the proviso that there was no 'inside', and everything happened on the (n-1)-dimensional 'surface'. Because of the phenomenal amount of inflation, this surface became flat. That is why our universe obeys the 'flat' Euclidean geometry.

3. The Higgs field emerged and broke the translational symmetry by imparting mass to many types of fundamental particles. Inhomogeneities grew when matter underwent clumping. Moreover, inflation had anyway caused a huge amplification of some very tiny quantum fluctuations. The CMB (Cosmic Microwave Background) map is a graphic record of these inhomogeneities, and also a resounding proof for the validity of the Big Bang model.