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Boydstun

Group Theory and Physics

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These look like good references for readers with a background in undergraduate physics (through intermediate classical and quantum mechanics):

Group Theory

A Physicist’s Survey

Pierre Ramond (2010)

Group Theory for the Standard Model of Particle Physics and Beyond

Ken J. Barnes (2010)

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“I think it will be much more exciting if we don’t find the Higgs. That will show something is wrong, and we need to think again. I have a bet of $100 that we won’t find the Higgs.” – Stephen Hawking (The Times 9/9/08; Far Reach)

From an interview* with Steven Weinberg on 6/28/2011 by Zinta Lundborg:

Lundborg – What about the particle everyone’s looking for—the Higgs boson?

Weinberg – Because the Higgs boson is really required by the simplest version of the theory that unifies the weak and electromagnetic forces,* it’s very likely to be discovered. The theory has other versions which would lead to the discovery of other kinds of particles, the so-called technicolor particles. We have a fair degree of certainty that one or the other of those, and very likely the Higgs boson, will be discovered. In fact, it’s so likely that we already anticipate it, so it probably won’t get us anything new. What we really need is something that we don’t anticipate.

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Another quest at the LHC is for particles that constitute dark matter.* ** That would be cold dark matter, for which a light neutralino might be the particle.

Recent modeling of dwarf galaxies suggests dark matter emerged later after the initial singularity and is of higher energy than the “cold” dark matter sought at LHC.* “Warm” dark matter would be out of the range of the LHC. The sterile neutrino is a plausible candidate for warm dark matter, and if that is right, warm dark matter could be detected in the future by telltale X-rays.*

Neutrinos

Neutralinos

Sterile Neutrinos as a Dark Matter Candidate

Some evidence for sterile neutrinos as dark matter is reported here.

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It will be interesting to see how it plays out. It sounds like no one--including the experimenters--actually believes it; they are looking for the mistake they think they must have made, somewhere, somehow. (As they should be--physicists must be certain of something that overturns well-established theory.)

Edited by Steve D'Ippolito

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The 60 nsec differential multiplied by the speed of light in vacuum is (rounded to appropriate significant figures) 18 meters.

If the transmitting and receiving locations are 18 meters closer together than they think, the result is explained. That is 18 meters out of the approximately 730 kilometers of the direct path through the Earth. They claim to know the distance accurately to within 20cm, but leave all of the geodesy discussion in the references 24, 25, and 26 in the linked paper. That would be the first place to look for an error in my opinion. Performing the experiment on a different site would require entirely replacing this data.

We also know reference frames with a stronger gravity field appear to be slower than frames with less gravity. At the deepest point in the Earth of the 730 km path of the neutrinos the gravity will be slightly less than at the surface, and the neutrinos will appear to move faster. If this was considered, it is buried somewhere in one of the references.

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Neutrino speed anomaly may signal some new physics of elementary particles and spacetime.

. . .

(Related thread)

Superluminal neutrino possibility analyzed with respect to energy transfer and particle transformations when speed limits are different for different elementary particles: Sci. Am. – 10/2/11

Paper – Andrew Cohen and Sheldon Glashow

Edited by Boydstun

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Not five sigma, but something is up at Atlas and CMS on the Higgs boson.

Existence of the Higgs boson “is a prediction that stems from a very mathematical approach to understanding the Universe, which is guided by the idea that it is simple at heart.” —

.

Higgs Missing Report (Fermilab page)

See especially:

“Is this all just a theory?”

“What if no one ever finds the Higgs Boson?”

“Are there alternate theories?”

.

. . .“I think it will be much more exciting if we don’t find the Higgs. That will show something is wrong, and we need to think again. I have a bet of $100 that we won’t find the Higgs.” – Stephen Hawking (The Times 9/9/08; Far Reach)

From an interview* with Steven Weinberg on 6/28/2011 by Zinta Lundborg:

Lundborg – What about the particle everyone’s looking for—the Higgs boson?

Weinberg – Because the Higgs boson is really required by the simplest version of the theory that unifies the weak and electromagnetic forces,* it’s very likely to be discovered. The theory has other versions which would lead to the discovery of other kinds of particles, the so-called technicolor particles. We have a fair degree of certainty that one or the other of those, and very likely the Higgs boson, will be discovered. In fact, it’s so likely that we already anticipate it, so it probably won’t get us anything new. What we really need is something that we don’t anticipate.

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Happy Birthday, Electron

Frank Wilczek (author of Lightness of Being)

“One could say that the electron was conceived in 1892 and delivered in 1897.”

“Although the Higgs particle is sometimes credited with giving matter mass, its contribution to the mass of ordinary matter is actually quite small. Lorentz’s beautiful idea, in modern form accounts for most of it.”

Of related interest: Representing Electrons

A Biographical Approach to Theoretical Entities

Theodore Arabatzis (Chicago 2006)

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Concerning the monumental proof, in my lifetime, of the exhaustive classification of finite simple groups, Stephen Ornes, notes in Scientific American (July 2015):

The work [the classification theorem] brings order to group theory, which is the mathematical study of symmetry. Research on symmetry, in turn, is critical to scientific areas such as modern particle physics. The Standard Model—the cornerstone theory that lays out all known particles in existence, found and yet to be found—depends on the tools of symmetry provided by group theory. . . .

 

Group theory also led physicists to the unsettling idea that mass itself . . . formed because symmetry broke down at some fundamental level. Moreover, that idea pointed the way to the discovery of the most celebrated particle in recent years, the Higgs boson, which can exist only if symmetry falters at the quantum scale. . . .

 

Symmetry is the concept that something can undergo a series of transformations—spinning, folding, reflecting, moving through time—and, at the end of all those changes, appear unchanged. . . .

 

[The classification theorem] demonstrates with mathematical precision that any kind of symmetry can be . . . grouped into one of four families . . . . In the future, it could lead to other profound discoveries about the fabric of the universe and the nature of reality. . . .

 

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Boydstun, I'm curious. 

 

I think you may be most qualified to answer :  What would an Objectivist's integration and understanding of the applicability of mathematical abstractions such as group theory and symmetry to reality and identity, look like?  I'd like a taste of what the explanation would be of A. this is what is in reality, B. these are the abstractions and why they are coherent with reality.

 

I have a background in physics and my colleagues at the time seemed perplexed and often confused reality with abstraction...  

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SL,

 

I’m not yet to the bottom of those issues and the issues of the nature of truth in abstract algebraic and topological concepts and facts to our existential classifications and measurement. In general strokes, I expect that although our method of arriving at truth in pure mathematics, though different than our empirical experimental ways of gaining knowledge of existence, is ultimately based on our experience of fundamentals of the physical world. And similarly, I expect it goes for our methods of logic, such as which inferences do not derail truth.

 

Groups are a kind of mathematical category, as you may know. Not every structure of interest in pure mathematics is a category, but considering the elements of what constitutes a category—class of objects, set of morphisms on them, rule of associative composition of morphisms, and an identity morphism for all those morphisms—would I think be a good bunch to consider in thinking about what surface connections these elements have to our everyday experience and manipulations in the world. Beyond being a bare category, various specific further conditions are added to get the various kinds of categories we have found productive in physics, such as groups, algebras, vector spaces, topological spaces, and Hilbert spaces.

 

The ways in which Rand's somewhat expansive notion of identity is related to such abstract structures as mathematical category in its bare bones does not look offhand too different from what we would say about her identity concept in relation to logical inference and standard logical identity, to nature of concepts of existential things, and to theory of predication and definitions. Historical look of how we got our scientific concepts (e.g. energy) and how we got our mathematical concepts (e.g. derivative of a function) and the similarity and difference in how changes in those two classes of concepts have come about would likely help us in pursuit of specifying how mathematical concepts such as groups stand in relation to the concrete physical world.

 

Another window on groups would be in the mathematical symmetry aspect of it (which I presume depends crucially on the addition of an inverse function to mappings of sets into sets [where we have taken the objects for groups to be sets and the morphisms to be mappings]). I’d like to think about symmetry in groups in relation to our more usual symmetries.

 

I do not know, at least not yet, if Rand’s identity conception of all existents and causation, together with her measurement-values-suspended way of understanding concepts lead to any special insight into the degree to which mathematical symmetry has come to such salience as we get deeper into the fundamental physics of existence. The importance of invariant quantities under certain classes of transformations seems not to have been foretold by any philosophers, rather to have been arrived at by scientists, who, to be sure, get to think philosophically as they forge new tools. I do not know whether Rand’s teachings on identity and on measurement analysis of concepts has anything to bring to that grand tool or to understanding its success in fullest context.

 

The physicists I have known thought of mathematical structures as potential models for physical applications. Whether such abstractions are instantiated in physical reality is lead by observation and experiment. I imagine in doing theoretical physics, one may think mostly of implications in the theoretical structures, but everyone I knew understood that those were abstract structures and that physics is about that which is physically realized, that physics is profoundly experimental.

 

Thanks for the questions. Hope this ramble may have touched on your interests at least a bit. I’ll mention a couple of books of interest for some of this philosophical reflection, especially future:

 

Category Theory for the Sciences

D. I. Spivak

 

Symmetries in Physics – Philosophical Reflections

K. Brading and E. Castellani, editors (2003)

 

Robert Knapp’s book Mathematics is about the World has a final chapter “Abstract Groups and the Measurement of Symmetry.” It seems to introduce the mathematical area, but have little specific to Rand’s innovations by way of philosophical assimilation.

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I do not know, at least not yet, if Rand’s identity conception of all existents and causation, together with her measurement-values-suspended way of understanding concepts lead to any special insight into the degree to which mathematical symmetry has come to such salience as we get deeper into the fundamental physics of existence.

 From the AR Lexicon:

"If nothing exists, there can be no consciousness: a consciousness with nothing to be conscious of is a contradiction in terms. A consciousness conscious of nothing but itself is a contradiction in terms: before it could identify itself as consciousness, it had to be conscious of something. If that which you claim to perceive does not exist, what you possess is not consciousness. "

 

This statement of Rand's (no matter how hard it tries) cannot escape the self-reference problem - a la: Liar's Paradox, Russell's Paradox, Turing Halt problem or Godel's Incompleteness theorem.  

This statement cannot be formally proven (as we require in mathematics).

 

All of our models - Group Theory, Symmetry, Newtonian Mechanics, General Relativity , Quantum Mechanics, etc., etc., etc. – cannot escape EPISTEMOLOGICAL PARADOXES that arise from our METHOD of abstraction. Our models (abstractions) are not a substitute for reality. The map is not the territory....

 

The boundaries between Higgs Bosons, electrons, quarks, neutrinos, etc., are epistemic, not existential.

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Budd,

 

There is no primacy of the formal over the existential. The boundaries of formal systems, such as incompleteness at some orders of logic (or completeness at other orders) has little-to-nothing to do with the existential structures and boundaries we have discovered. Arm-chairing does not cut it in physics. The math must be learned; the experiments and observations must be designed and analyzed.

 

The only parts of logic have I seen have possible special bearing in physics is over alternatives in De Morgan type rules in attempts to understand aspects of quantum mechanics. And that has nothing to do with semantic or syntactic paradoxes or with limiting-theorems on deduction in formal logical systems. The other is in work on quantum-assisted computability, which incorporates results on limitation theorems on computability. By far the greater implications for physics from formal structures and their internal constraints are from mathematics, not logic and not even, for example, the formalization of group theory in suitable mathematical predicate logic (chap. XII of Richard Epstein's Classical Mathematical Logic). We have found that what is germane from formal disciplines to our ever more penetrating physics is theory of partial differential equations and geometrical spaces and various mathematical categories such as groups, topological vector spaces, and Lie algebras.

Edited by Boydstun

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Here's what I'm trying to suggest.   Correct me if I'm wrong or if it's not relevant to this post.

 

Suppose we created a new type of metal and shaped it into a wide-flange beam.  We would use one type of instrument/test to determine it's tensile capacity, another type of instrument/test to determine it's compresssive capacity and so on too for modulus of elasticity, moment, shear, deflection, thermal expansion, buckling, etc.

 

Each test and, and our mathematical formulation of each property, essentially stands alone from all others.  There is no one, unified formulation for every possible property of the metal.

 

Would this (or does this) not also apply to our description of the different parts of an atom?

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Budd, I think it is quite relevant to the thread and a good thought. There is a history of success in physics, and in chemistry, geology, and biology too, of uncovering underlying unifying explanations of diverse phenomena. Sometimes, as with chaos theory and thermodynamics, we have found phenomena to which the theory with its equations apply that are different from the phenomena for which we first developed the theory. We get some explanation and some new unity in our understanding of the world in those successes.

 

We strive in addition for another, stronger sort of unity, and this one is the main arena under the tent so to speak. The standard model of the elementary particles, including their quantum mechanics and including the unified theory of the electric, magnetic, and weak and strong nuclear forces (such unity of those forces as we have so far) is a case of bringing a diverse range of phenomena under a single dynamical picture. This is not only a satisfaction of the intellect; this kind of unification increases what materials and devices we are able to invent. The strength-of-solids family of properties you listed is also a case in which we have achieved some amount of unified underlying explanation by atomic, solid-state physics. Similarly, with properties of gases and liquids: we have had some success with molecular underlying explanation of their properties through statistical mechanics. We have also unified various features of classical thermodynamics in its generality by underlying statistical mechanics.

 

However, those various properties of solids you mentioned remain phenomena just as different (and similar) as ever they were. Such phenomena are existential, and their existential differences do not get attenuated by our learning of underlying common causes of them. Two more “howevers” are these: I don’t see any advance assurance that we can always find such unifying dynamics as we pursue, for I don’t see any advance assurance that such a thing is there. We are gambling. Secondly, I don’t want to leave an impression that it is only through grasp of such underlying unities of dynamics that we are able to make any technological advances. People who first came up with ways of forming a bearing for a wheel on an axel, thereby inventing load-bearing wheels, would have had no such sophisticated understanding. 

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