In my last post I referenced Kerson Huang’s book, *Fundamental Forces of Nature: The Story of Gauge Fields*. Huang starts with ^{th} Century following the development of quantum mechanics.

What is significant is that, if one overlooks his short detour to include Relativity Theory, Huang traces the world of physics from the scale of our everyday world to a smaller and smaller scale, resulting in the ‘Standard Model’, which includes the innards of nuclear particles: quarks and gluons, amongst numerous others. The significance of scale is a particular feature of Huang’s treatise that he reveals right at the end. I said in my previous post that the book doesn’t include ‘String Theory’, but Huang does explain its origins, almost in passing.

Quantum mechanics is such a tantalising yet daunting area of the natural world for me. I’ve read Richard Feynman’s book, *QED; The Strange Theory of Light and Matter* (1985), which explains everything and nothing. Feynman, who won a Nobel Prize for his pioneering work in this area, says right at the beginning that ‘no one understands quantum mechanics’, and I think that’s a very important point. QED (quantum electrodynamics) is the most successful theory ever (both Feynman and Huang, who quotes Freeman Dyson, agree on that) yet no one really understands how it works. Feynman’s book explains brilliantly, with no equations whatsoever, how one can work something out from the summing of ‘all possible paths’ to produce the path of ‘least action’; he even uses the analogy of a stopwatch to provide analogue phase changes (for each path) otherwise described by ‘complex algebra’ differential equations (the famous Schrodinger’s equation) that are used in real quantum mechanical calculations. But he doesn’t explain why we need to allow for ‘all possible paths’ in the first place, a consequence of the well-known, but enigmatic, superposition aspect of quantum phenomena. And no one else can explain it either, despite attempts to propose ‘many worlds’ interpretations and ‘Schrodinger Cats’ in simultaneous states of life and death. This is where philosophy and science collide, and so far, philosophy is still all at sea.

Huang explains how it is the mathematical concept called the Lagrangian that defines the ‘Least Action’ or ‘Least Effort’ principle, effectively the Kinetic Energy minus the Potential Energy. But Huang filled in another piece of the puzzle for me when he explained that we go from one Lagrangian to another as we change the scale of our observations. Even now, this is something that I only vaguely understand, yet I feel it is very important, because I’ve always believed that scale plays a role in the laws of physics, and Huang has effectively confirmed that, and gives a potted history of its theoretical evolution.

In a very early post (Sep.07), **The Universe’s Interpreters**, I make the point that the natural world exists as worlds within worlds, almost ad-infinitum, and we humans have the unique ability (amongst Earth species) to conceptualise worlds within worlds, ad-infinitum, therefore giving us the privileged position of being able to comprehend the universe that actually created us.

Huang lists a host of people, including Murray Gell-Mann, Francis Low, David Gross, Frank Wilczek and David Politzer for demonstrating a logarithmic relationship between energy and the ‘coupling constant’ (charge). Energy increases for QED (electrons and photons) and decreases for QCD (quarks and gluons). Then, Nikolai Bogoliubov, Curtis Callan and Kurt Symanzik proposed the ‘Renormalisation Group Trajectory’ or RG trajectory including a mathematical equation to describe it. The RG trajectory (according to Huang) takes us from ‘Classical Physics to Quantum Mechanics to QED to Yang-Mills’ (nucleon physics) – increasing energy with decreasing scale. Kenneth Wilson realised that the so-called ‘cutoff’ in renormalisation parameters that changes with scale, and therefore changes the Lagrangian from one range of energies to another, has a physical basis. In other words, these physical laws expressed in mathematics only work within a parameter or range of scale and change when we go from one parameter of scale to another (Hang uses the term ‘crossover’). Each one, as Huang points out, initiated its own scientific revolution during our discovery process, but in reality, reveal to us different layers of nature. Huang also references Leo Kadanoff and Michael Fisher as also contributing to our understanding of RG trajectories.

As an aside, there is one mystery arising from quantum field theory, highlighted by Huang, that I had never heard of before: when time becomes purely imaginary it reduces quantum theory to statistical mechanics, so that time relates to absolute temperature. Actually, a very simple mathematical relationship involving *t* (time), T (Temperature), *i* (square route of -1), and h (Planck’s constant). It is tempting to think that this mathematical relation arises from the fact that entropy is the only physical law we know of that gives a direction to time, with entropy being related to temperature, but Huang doesn’t make this connection, so there probably isn’t one. (Entropy, or the second law of thermodynamics, is the only law in physics that insists on a direction for time; relativity theory and quantum mechanics both allow for time reversal – so that bit is true. Reference: Roger* *Penrose’s* The Emperor’s New Mind*.)

Finally, noticeable by its absence in all this, is gravity, described brilliantly by Einstein’s General Theory of Relativity. Gravity and general relativity is effectively the Lagrangian for cosmological scales, but, as everyone knows, there is no place for gravity in the Standard Model – Einstein’s General Theory of Relativity stands alone. The best exposition on relativity theory, that I’ve read (both the special and general theories) is by Richard Feynman in *Six Not-So-Easy Pieces*, where he describes the ‘Least Action’ principle in terms of relativistic energy or ‘maximum relativistic time’. This is intuitively opposite to the ‘principle of least time’, as postulated by Pierre de Fermat (in the 17^{th} Century) found in the optical phenomenon of refraction and now accepted as scientific fact, yet it is the same principle. Feynman demonstrates mathematically that the principle of maximum relativistic time (therefore ‘Least Action’) gives the correct trajectory of a projectile in flight in a gravitational field. As I describe in an earlier post (Mar.08) **The Laws of Nature**, Fermat’s principle in refraction and Feynman’s mathematical description of ‘Least Action’ in relativistic physics both relate to how the light or the projectile finds the ‘right’ path – the path that requires minimum effort, satisfying the Lagrangian: Kinetic Energy minus Potential Energy as a minimum. Feynman also demonstrates how quantum mechanics gives the answer that light follows the ‘least time’ principle using his analogue version of QED, in his book titled, *QED* (as I described above). So Feynman effectively demonstrates that the ‘Least Action’ principle applies consistently in relativity theory, classical optics and QED.

Huang gives very little space to ‘Grand Unifying Theories of Everything’ (known generically as GUT), but, of course, String Theory is the great contender. One of the best books I’ve read on String Theory is Peter Woit’s *Not Even Wrong; The Failure of String Theory and the Continuing Challenge to Unify the Laws of Physics*. Woit covers much of the same territory as Huang in his explanation of gauge theories, quantum field theory and the Standard Model, but then continues onto String Theory, explaining how it became the latest paradigm in our search for theoretical answers (if not experimental ones) and, specifically, the role of Edward Witten in its evolvement. In fact, reading Huang’s book, and writing this post, has forced me to re-read Woit’s book. Woit, like Huang, is a physicist and a mathematician, and I am humbled when I read these guys. Unlike me, they actually know what they're talking about.

Whilst Woit is highly critical of String Theory (or string theories to be more accurate), he is deeply respectful of

One of the points that Woit makes is that String Theory evolved out of a ‘Bootstrap’ theory (also mentioned by Huang) developed by Geoffrey Chew in opposition to QCD and the highly successful ‘Standard Model’. This theory developed from an ‘S matrix theory’ that Woit is almost contemptuous of, because some of its followers, including Fritjof Capra, refused to admit its demise, even after the Standard Model became one of the great success stories in recent theoretical physics. Woit is particularly scathing of Capra’s *The Tao of Physics*. (Capra’s ideas, by the way, are not to be confused with Huang’s poetical allusion to Taoism, nor mine, that I discussed in the previous post.)

But ‘Bootstrap’ theory aside, Woit has other issues with String Theory and its derivatives, for which he provides an exhaustive and illuminating history. Woit readily admits, by the way, that if you want a more positive picture of String Theory there are other books available, by authors like Brian Greene and Michio Kaku, and he generously lists them (some of which I’ve read).

The biggest problem, according to Woit, is with ‘supersymmetry’, the ‘Holy Grail’ of String theory and its derivatives. To quote his concluding paragraph on its incompatibility with the Standard Model:

‘*As far as anyone can tell, the idea of super-symmetry contains a fundamental incompatibility between observations of particle masses, which require spontaneous super-symmetry breaking to be large, and observations of gravity, which require it to be small or non-existent.*’

Feynman, in a 1987 interview, the year before his death, was even more damning:

‘*Now I know that other old men have been very foolish in saying things like this, and, therefore, I would be very foolish to say this is nonsense. I am going to be very foolish, because I do feel strongly that this is nonsense! I can’t help it, even though I know the danger in such a point of view.*’

Woit does elaborate on one of the benefits of String Theory, which is the cross-fertilisation, for want of a better term, between physics and mathematics, that he believes was badly needed. In fact, he devotes considerable space to the interaction between mathematics and physics, both historically and philosophically.

One of the truly extraordinary features of mathematics is that it allows us to go intellectually and conceptually where we can’t go physically. One can’t help but wonder if **The Unreasonable Effectiveness of Mathematics** (a quote from Eugene Wigner) I referenced Penrose’s 3 perspectives of reality: physical, mental and Platonic, where the Platonic realm is mathematical, therefore abstract. The mental (consciousness) arises from the physical, the Platonic from the mental, and the physical from the Platonic (not unlike a self-perpetuating Escher graphic). In other words, not everything Platonic relates to the physical, although if there are an infinite number of universes (the multiverse) then perhaps it does. But my point is that

Leaving aside, for the moment, the idea of a multiverse (very popular, I might add, and discussed by Woit) mathematics is comfortable with dealing with infinities and multiple dimensions in a way that we are not. The current version of String Theory (Superstring Theory or M Theory) requires 10 dimensions, which means that 6 spatial dimensions need to effectively disappear, or be so physically insignificant as to be invisible, even at the sub-nuclear level.

I, for one, am a little sceptical of a ‘grand unified theory of everything’ because history has shown that the resolution of one set of mysteries always uncovers others. We always think that we are at the final limit of nature’s secrets, yet we never are, and, obviously, never have been.

Huang’s exposition has highlighted the apparent reality that the laws of physics, therefore nature, are scale dependent. Many people overlook this, and talk about quantum physics as if it really works at all scales, including the one we are familiar with, and the mathematics doesn’t contradict this, just the reality we observe (refer Addendum 2 below, and Timmo's comments in the thread for a more knowledgable perspective). Penrose has argued that there is something missing in our knowledge to explain how classical physics ‘emerges’ from quantum mechanics, in a similar way that consciousness apparently ‘emerges’ from neuron activity. But the fact that physics has different laws at different levels reflects what we observe and is consistent with nature at all levels, including biology (refer my post in Feb.09 on Hofstadter’s book, *Godel, Escher, Bach*: **Artificial Intelligence and Consciousness**).

Therefore, I don’t expect we’ll find a ‘Theory of Everything’ that encompasses all levels of nature in one mathematical expression, but a lot of people, including many who work in the field, seem to think we will. The fact that we need to go to 10 or more dimensions to achieve this, makes it more speculative than physically probable, in my view. When I think of the 10 dimensions required, I’m reminded of all the epicycles that were needed to make Ptolemy’s model of the solar system compatible with observations.

I’m not saying we already know all the answers because we obviously don’t, but I am saying that maybe we never will. Every time we’ve uncovered one layer of reality we’ve found another layer underneath, or beyond. The Standard Model suggests we have finally reached rock bottom, but even if we have, the fact that there are mysteries still unsolved suggests to me that there are still further mysteries yet to be uncovered, because that’s the one consistency that the history of science has revealed thus far.

Addendum: There is an article in this week's *New Scientist* (30 May 2009) on how String Theory, or a variant of it has been useful, not in cosmology, but in condensed matter physics and high temperature superconductivity What string theory is really good for

Addendum 2: I want to thank Timmo for his valuable and knowledgable contribution that you can view in the thread of comments below. He provides more detailed information and analysis on Feynman's publications in particular.