Paul P. Mealing

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Thursday, 28 May 2009

Nature’s Layers of Reality: from Cosmology to QED to The Standard Model

In my last post I referenced Kerson Huang’s book, Fundamental Forces of Nature: The Story of Gauge Fields. Huang starts with Newton’s equation, F=ma, and works through the history of physics right up to the so-called ‘Standard Model’. The theory of ‘gauge fields’ is effectively the theme of his book, which means that the best part of it is spent in the 20th 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 17th 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 Witten, who was at Princeton at the same time as Woit.

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 Witten’s genius, along with others, hasn’t gone somewhere that the physical universe can’t follow. In a previous post (Mar.09), 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 Witten and his colleagues may well be exploring a part of the Platonic realm that doesn’t relate specifically to ‘our’ universe.

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.


Thursday, 14 May 2009

Socrates, Russell, Sartre, God and Taoism

An unlikely congregation, but bear with me and it will all become clear. Earlier this week I received 2 new books from Amazon UK: The Mind’s I, by Douglas R. Hofstadter and Daniel C. Dennett; and Fundamental Forces of Nature; The Story of Gauge Fields, by Kerson Huang.

Huang is a Chinese born American, now Professor of Physics, Emeritus, at MIT, and 79 years old when he published this book in 2007. The book covers all of physics, in a historical, therefore evolutionary, context, from Newtonian physics (F= ma) up to QED (quantum electrodynamics) and beyond, though it doesn’t include String Theory. The presentation is very unusual, with equations kept deliberately minimalist, yet he manages to explain, for example, the subtle difference between Faraday’s equations and Maxwell’s (an extra term effectively) that led to the prediction of electromagnetic waves propagating at the speed of light. He also introduces mathematical concepts like Lagrangians and Hamiltonians early in his treatise; an unusual approach.

Its relevance to the title of this post is at the end, where he quotes a Taoist poet, Qu Yuan (340-278 BC) who wrote a series of questions called Tian Wen (Ask Heaven):

At the primordial beginning

Who was the Reporter?

Before the universe took shape.

How could one measure it?

(Huang also provides the original Mandarin.)

Then he quotes Russell on mathematical beauty:

A beauty so cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without gorgeous trappings or painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

He follows this quote with the following rumination of his own:

Physics is truth. It sails down a trajectory in the space of Lagrangians, when the energy scale shrinks from that set by the Big Bang.

I sometimes think that God is in the mathematics; I’ll explain myself at the end.

But the subject of this post really comes from an essay written by Raymond M. Smullyan (in Dennett’s and Hofstadter’s book) titled, Is God a Taoist?. It’s very cleverly written in the style of a Socratic dialogue between God and a mortal, who wants God to relieve him of free will. It reminds me of Sartre’s seminal essay, Existentialism is a humanism, with its famous quote: ‘man is condemned to be free’. I once wrote an entire essay founded on that quote alone, but that’s not the subject of this post.

Smullyan manages to cover an array of topics, including free will and morality, in which, via a lengthy Socratic dialogue, he concludes that the real virtue of free will is that it mandates responsibility for the infliction of suffering on others. In other words, you know when you’ve done it, and you will feel guilt and remorse as a consequence. This is not a verbatim interpretation, just my own summary of it. The dialogue effectively gets the mortal to admit this when God offers to free him of all guilt associated with his ‘free will’. So the choice then of allowing God to rid him of free will, and its consequences, becomes a moral choice in itself, therefore turning the moral dilemma back on itself.

But it’s the particular Eastern references in this essay that appealed to me, in which Smullyan incorporates the idea of God as a process. (A concept I’ve flirted with myself, though Smullyan’s concept is more Eastern in influence.)

To quote Smullyan’s God character in the dialogue:

My role in the scheme of things... is neither to punish nor reward, but to aid the process by which all sentient beings achieve ultimate perfection.

Then to elaborate:

…it is inaccurate to speak of my role in the scheme of things. I am the scheme of things. Secondly, it is equally misleading to speak of my aiding the process of sentient beings attaining enlightenment. I am the process. The ancient Taoists were quite close when they said of me (whom they called “Tao”) that I do not do things, yet through me all things get done. In more modern terms, I am not the cause of Cosmic Process. I am the Cosmic Process itself.

Smullyan, then (as God) quotes the Mahayana Buddhists:

The best way of helping others is by first seeing the light [in]oneself.

He also addresses the issue of personality (of God)

But the so-called “personality” of a being is really more in the eyes of the beholder than in the being itself.

I hope I haven’t been too disparate in this rendition of someone else’s essay. Hofstadter provides his own commentary at the end, with particular reference to the role of free will which he describes thus: ‘a person is an amalgamation of many subpersons, all with wills of their own.’ He says: ‘It’s a common myth that each person is a unity.’ I assume he’s talking about split brains, but I won’t explore that issue here, as Smullyan’s essay has other resonances for me. (I admit I'm not doing justice to Hofstadter, but I don't want to get distracted; maybe another post.)

I’ve said in previous posts that God is an experience, which is one reason I claim religion is totally subjective, because it’s an experience that can’t be shared – it’s unique to the person who has it and only they can interpret it. The essay by Smullyan makes only passing reference to this idea of God (when he discusses personality). I believe he’s referring to a more universal concept, but in an Eastern context rather than a Western one.

I can’t help but make a connection between Huang’s book and Smullyan’s essay, because they both relate to 2 of my lifelong passions: science and religion. Mathematics has given us such extraordinary insights into the physical processes of the universe, at every level, and the idea of God as the process itself, in which we play a very small part is an appealing one. And calling it the Tao, effectively rids it of human personality.

Most people would make no connection between these 2 ideas, but I sometimes think I am a Pythagorean at heart. Mathematics is such a magical medium that one cannot dissociate it from God, especially if God is the Tao, and Tao is ‘the scheme of things’.