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.

## 20 comments:

Mathematics

does"allow us to go conceptually where we can't physically." I'm not a Ph.D. physicist, nor do I play one on television, but I did marry one, so I'm absorbed a certain amount of physics osmotically, on top of what I'd acquired in the same manner as you (autodidactically), and can find not a thing in your post that you haven't presented both correctly and with extraordinary lucidity, so let me express my admiration for your exposition before going on to the one point that has always seemed (to me) not consciously to be recognized by theorists (and which tends to support both Feynman's cry of "nonsense," and your own skepticism about the ultimate realizability of a GUT). It's simply this:A mathematical theory is just a predictive isomorphism, strings of symbols that, when manipulated, yield other strings of symbols. Now all these symbols can stand for things (time and space and mass and temperature and so forth), and the system *can* be explanatory in the sense that it produces results that accord with our observations of physical reality. But they're still just mathematical systems that

representaspects of reality, not the reality itself, and historically, we've kept discovering that our systems needed tweaking to account for hitherto unobserved anomalies, hence the development of new conceptualizations embodied mathematically in the form of new systems of symbols. But nor are thesenewmathematical systems anything other than mathematical systems with a certain apparent explanatory adequacy. They're not the same thing as the reality. Even if string theory, to take an example, is shown conclusively to work at a predictive level, that doesn't mean that real strings exist, only that the math exists.Which sort of recapitulates in a different way something you've perhaps already expressed more cogently, but shorter version: I tend to agree with you about the "GUT" issues. :)

Well, but is this a feature of reality or of us? It's one thing to say that we'll never find a theory of everything and quite another to say that none exists: the former would be merely annoying whereas the latter would be sort of horrifying, in a way.

Hi Larry,

Well, I certainly don't think we've come to the end of science.

The history of science, in the last century, has revealed that as we go to smaller and smaller scales by using bigger and better particle accelerators, we uncover new mathematical relationships, or laws of physics, that we didn't even know existed.

GUT requires looking at the energy or scale at the big bang, when all the 4 gauge theories we currently know about: electromagnetic, electroweak, strong and gravity; all meet. Approximately 10exp.16 to 10exp.20 GeV. But, as Woit points out, the physics at this level may not be the same as the physics we already know, and history suggests it almost certainly won't be.

Woit argues a very strong case that 20 years of string theories hasn't even produced a proper 'scientific' theory. But, string theory aside, it may be that the extremities of scale of the laws of physics are not unified in a way we expect to find. I think it's quite possible that nature follows different laws at different levels (or scales) and the belief in finding one 'law' or GUT, that mathematically combines them all together, may be a chimera.

As I like to say: only future generations know how ignorant the current generation is. And, of course, the current generation includes me.

Regards, Paul.

Hi Larry,

Oh, I'd never asseverate that none exists. On the contrary, if there's one theory that more-or-less adequately predicts the behavior of the universe, to the extent that we're able currently to observe that behavior, then there're provably an

infinitudeof such theories. How you'd choose among them I'd guess would be on the basis of theoretical economy, if you had no other. And I'm not even saying that we'll never find one: only that, if we do, it'll be a mathematical construct, and not the same thing as reality itself. And also that there's no reasona priorito believe that onehas toexist (God seems neither inclined to "play dice," nor necessarily everywhere to adhere to principles of maximal semantic economy).I do tend instinctively to embrace Paul's view that a GUT may not be found because multiple, disparate and non-unifiable explanatory mechanisms may actually be necessary to account for the observable phenomenon, but that's just an instinct, akin to Paul's: it's really not possible to know. I was just focussing more on a pet peeve of mine: the failure to distinguish between mathematical constructs and the physical universe whose properties they seek to characterize. If you look out your window, you won't see any tensors zooming around through space. Anyway, no cause for "horror" just yet. (Though I *am* reminded of Rilke's observation: "Jeder Engel ist Schrecklich.") :) :)

Hi PK,

Thanks for your generous comments.

As I've said in previous posts, that I know you've read, mathematics is one the most efficacious mediums we have for bridging our inner and outer worlds.

To paraphrase Robyn Arionrhod from

Einstein's Heroes, mathematics is really about the relationships between numbers rather than the numbers themselves. In response to the age-old philosophical question: is mathematics invented or discovered? Well, I would say the 'symbols' (numbers) are invented, but the relationships are not.The fact that nature seems to obey some of these relationships (if not all of them) is one of the reasons that we are 'privileged' to be able to comprehend the very universe that created us, as I say in my post.

Regards, Paul.

Hi PK again,

Our comments passed in cyberspace, so my last comment is in response to your first comment.

I confess to being a bit of a Pythagorean, so I may be guilty of your accusation. The tensors exist in our head, yet, wihout Reimann's geometry, we wouldn't know that the universe obeys said tensors to an inordinately accurate degree, as people like Penrose like to remind us. Penrose, by the way, is not a fan of string theories either.

Regards, Paul.

Hi Paul,

The comment about tensors wasn't intended at all as an "accusation;" just a whimsical way of flailing my arms to differentiate between the mathematical representation and the thing represented. You're exactly right that the tensors *do* exist in our heads (or, at least, internal cognitive representations isomorphic to the external mathematical symbology that's isomorphic to the physical phenomena observable in the universe reside in our heads, insofar as we're capable of constructing yet some other neuronally-based model-theoretic interpretation thereof! :)) Actually, the more I think about it, the tensor I get! :)

Regards, Peter

Thanks Peter,

You're entitled to accuse me of many things. It's an area where we don't quite meet, along with a lot of others I expect - nothing wrong with that.

I am a Platonist, at least mathematically speaking. In fact, the Platonic realm has a lot in common with the Chinese Tao (ref. prev. post).

Having said that, I agree with Penrose, possibly the best known living Platonist, that you have to 'find' the 'right' mathematics (ref:

The Road to Reality). I think he sympathises with Peter Woit's views on 'String Theory'; he's just more circumspect in expressing his opinions (he gives his endorsement on the cover).I admit to being an admirer of Penrose, as you would when you find someone who philosophically agrees with you in a number of areas, though not all, but then no one does that I believe. We don't live in the world of Gurus anymore - or, at least, I hope not.

Or is that my 'Aussieness' showing through - we are traditionally a bit wary of Gurus of any persuasion.

Regards, Paul.

Hi Paul,

Well, we do disagree on one point, which is that I'm *entitled* to "accuse" you of *anything* ("accuse" having, for me, the pejorative connotation of a sanctimonious, judgmental attitude, which I'd like to hope I mostly avoid; I try to, anyway). We can, on the other hand, amicably agree to disagree, which is fine. I guess at one point I rather blithely and offhandedly expressed my disdain for Plato's writing (though not for *all* of his ideas), but there's a story behind that, which has partly to do with my having been exposed to The Republic at an extremely early age (12), when I regrettably lacked tolerance of anything that didn't seem to me rigorously logical. And then I used to get into public arguments with a colleague from another department (philosophy) who was an enthusiastic Nietzschean as well as a devout Platonist, and I may have let my distaste for the one slop over onto the other, whereas there wasn't really any a priori connection. Also, I had reread Plato's one work on linguistics, The Cratylus, back when I was studying that field, and I think even the most ardent of Platonic apologists have had a hard time maintaining that that particular essay can be interpreted to have any coherent meaning whatsoever. Howsoever, I harbor no animus whatsoever towards Platonists, per se, and wouldn't "accuse" you of Platonism, either, using that verb. I sense we probably have different semantic fields for what "Platonist" means, in any case.

And then, of course, I remember having (I thought, innocently) used the word "rebarbative" in a whimsical vein in describing Penrose's historical attack on classical AI, of which I was one of the early practitioners. Anyway, I certainly hadn't meant to convey either grumpiness or accusatoriness in either case (not, at the time, realizing that you were a particular admirer of Penrose and Plato, both).

Bottom line: I also think it's perfectly fine (and conducive to interesting discussions) that we have somewhat disparate views. No "accusations" ever intended. :)

Regards, Peter

This is where your education shines through. I've actually never read Plato's Republic - I've only read some of his Socratic dialogues. In my curtailed course on philosophy (I went overseas and wrote

Elvene, and never returned to studying it officially) which pretty well sums up my whole education - never finished anything. I never finished that last sentence - in my curtailed course, I studied Aristotle, in particular hisNicomachean Ethics, and some of the stoics, both Greek and Roman.So my only knowledge of Plato centres around the influence of Socrates and Pythagoras. I do know (according to Kitty Ferguson,

The Music of Pythagoras) that Plato adopted Pythagoras's 'quadrivium' of mathematics, geometry, astronomy and music, for his famous 'Academy', from Pythagoras's most successful student, Archytas, whom Plato actively sought out, or so Ms Ferguson claims, and she is a scholar.Regards, Paul.

On that subject: I actually wrote a short essay, and won a modest prize when I submitted it to

Philosophy Now, a UK magazine.You can read the essay here.

Regards, Paul.

Hi Paul,

I think you make a good case, predicated on his contributions to the form (and even the inception) of the academy as we know it, which stands irrespective of whether I concur with all of his philosophical opinions. Managing (especially as a non-academician) to get an article published in a philosophy magazine -- let alone winning a prize for it! -- is a d*mned impressive achievement, so my sombrero is off to you, in any case! Kudos!

Regards, Peter

Hi PK,

To be fair and honest, I won a prize along with about 12 others I expect. Anyone whose submission was published received a prize. I never saw the result because getting issues here seems to be a hit and miss affair, but I received a prize, which was a book by Raymond Tallis, so that's how I know.

I added an addendum to the post if you haven't noticed.

Regards, Paul.

Hi Paul,

Saw your addendum, and found the referenced article lucid and interesting (but then, stuff in the New Scientist usually is). And here I thought "string theory" was only good for tying up Schroedinger cats! :) :)

Hello Paul,

QED (quantum electrodynamics) is the most successful theory ever... yet no one really understands how it works.Feynman is fond of saying that nobody understands quantum mechanics. There is a sense in which that statement is false, a sense in which it is controversial, and a sense it which it is true.

We have a very strong

technicalunderstanding of quantum mechanics: one can learn the rules and play according to them. We know what the mathematical character of the theory is; we can use it to create models of phenomena we are interested in and compare theoretical predictions to experimental data; we can teach the theory to students and explain it collegues. There is a crystal clear grasp of the functional, operational workings of quantum mechanics.It is controversial whether there is a

conceptual, or philosophical, understanding of quantum mechanics: there are rival accounts of what the world is like according to quantum mechanics. My own sympathies lie with Niels Bohr, Aage Petersen, and Leon Rosenfeld who championed the complementarity interpretation of the formalism. But, I notice that there are deep connections between the various interpretations (Bohr and Bohm both emphasizewholeness). Sometimes people say that Bohmian mechanics is just a many-worlds theory in disguise!It is certainly true that we do not have a

intuitivegrasp of quantum mechanics: the results are shocking, confusing, and leave one asking 'But... how could it be that way?' Though, it is also worth noting that Newtonian mechanics also sometimes defies intuitive expectations, as anyone who has tried to teach it finds out! Even very simple ideas, like the motion of the Earth, were bizarre to the people who were first wrestling with it. Nowadays everyone hears this since they are young and don't see what an incredible claim it is!Hello Paul,

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 actionPerhaps I can clarify what he is saying. In the first two lectures he lays out the quantum-mechanical laws of motion:

(1) The probability P for an event is given by the absolute square of a complex number called the probability amplitude ψ. P = |ψ|^2

(2) If an event can happen in alternative ways, 1 and 2, then the amplitude ψ is given by summing the amplitudes for the event to occur via 1 and via 2: ψ = ψ1 + ψ2.

(3) If an event occurs in temporally contiguous stages A and B, the amplitude for the event is the product to the amplitudes for each stage: ψ = ψA × ψB

Feynman does not state the rules in this way. Complex numbers can be represented as arrows, or vectors on the complex plane, so the rules (1)-(3) I just wrote down can be given in terms of adding and multiplying arrows in terms of the prescription that he gives.

What you produce in a given problem is not necessary the path of 'lease action'. He gives the example of a mirror and shows how to regain the classical answer that the beam hits, say, the middle of the mirror and bounces off at the same angle it hits the mirrror. There is an amplitude ψ for the light to hit all of the various parts of the mirror, but, away from the classical path, the phases of all those ψ's, of all those arrows, basically vary at random and cancel. So, you recover the classical answer. And, if you cut away every part of the mirror but the middle you get the same result.

But, if we cut away parts of the mirror at regular intervals to prevent the random cancellation away from the path of least time -- thereby making a diffraction grating -- we don't get the 'classical' answer! Whether or not the dominant contribution to the final arrow is the classical path of least time depends on the details of the situation!

Hello Paul,

I’ve always believed that scale plays a role in the laws of physics...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 observeQuantum mechanics

doeswork at all scales. Through Ehrenfest's theorem, you can show that F = ma comes out as a special case of quantum mechanics. Moreoever, I just gave you an example of classical results coming out of Feynman's approach. If you stray too far from the path of least action -- the classical trajectory -- the phases can vary at random and make the arrows cancel.There is something else here though. The laws of physics are not scale-invariant in the following way: I cannot just double the size of everything in the universe and get the same results I had before. Feynman talks about this in Six-Not-So-Easy pieces.

when time becomes purely imaginary it reduces quantum theory to statistical mechanics, so that time relates to absolute temperature...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.This is just a formal thing. In statistical mechanics, one tries to calculate what is called the partition function Z. If you can calculate Z, then you can derive all of the thermodynamic properties you are interested in (e.g. the chemical potential, heat capacity, phase transition temperatures). So, various schemes for calculating Z have been developed. The 'imaginary-time formalism' allows you to import all of the techniques of Feynman's path integral formulation of quantum mechanics to the project of calculating Z. It's just a mathematical or formal thing.

Hi Paul,

...String Theory, explaining how it became the latest paradigm in our search for theoretical answers... the idea of a multiverse (very popular, I might add...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.Did string Theory ever become a paradigm? The development of String Theory had promised to provide a quantum theory of gravitation. But, and I sense this is part of its current decline and demise, no feasible experimental checks have been offered by the theory. The hypotheses of String Theory, to the best of my knowledge, remain beyond experimental probing -- it's just speculation until we can go make some measurements. It's not a theory of everything so much as a theory of nothing.

Multiverses are popular? It's too bad if that's the case. The claim that there are is a multi-verse is pseudo-scientific bunk -- it's neither here nor there when it comes to developing explanatory theories or giving experimentalists something to look for. Maybe it's true, but this is just wild speculation.

Personally, I think it is irresponsible, unprofessional, unethical for scientists to publically engage in these kinds of speculations and mislead people into thinking this is part of the science.

Hope that helps! :-)

Hi Timmo,

Yes, it helps a great deal. I appreciate the time and effort you've given to your 'appraisal'.

I agree with your comments about both string theory and the so-called multiverse.

You make me realise how much I don't know and that is a very good thing.

I'll give you an acknowledgement in the original post.

Best regards, Paul.

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