Gosh, where do I begin? It’s going to be a hugely busy day on Monday. I’m excited about Monday first and foremost because of the colloquium. Charles (Chuck) Stevens from the Salk Institute generously accepted my invitation to come and visit USC and give a special colloquium to an audience made up of people from both the Physics and Astronomy Department and the Neuroscience Research Institute. This is because his talk is going to be about thing of interest to both camps! He’s applying physicist approaches -the sort that I do- to biological problems. This is a big deal to me because it is very hard to know when problems in biology actually really need or care about the sorts of things that physicists care about. The last thing you need as a biologist… when you’re happily gathering all those juicy data on various systems and finding out all sorts of highly specialized context-dependent mechanisms… the last thing you need is some arrogant physicist coming in an babbling stuff like “symmetry dictates that…”, “universal features…”, and “…single underlying principle”, and stuff like that. But sometimes -just sometimes- it might be possible to bring that sort of thinking to some problems and actually learn something. This is what Chuck is going to tell us a bit about, with some examples. See his abstract here. Actually, here’s a bit of it:
Physics-style theory – for example, variational principles and symmetry arguments – is not frequently used in biology, but can be a very powerful way to understand biological systems. I describe the unique characteristics of physics-style theory in the biological context, and illustrate this approach to determine the value of an exponent in a scaling law that governs aspects of brain architecture.
I heard this talk in Aspen last Summer (or a version of it) and never got around to reporting on it. This time I am going to go one better. I’ve arranged for it to be recorded and put on the web. I will give you the link when it is done.
I’ll be running around all day tomorrow during his visit, since I’ll be hosting him (jointly with neurobiology). This will involve making sure that he gets from meeting to meeting with various people who want to talk to him, lunch, dinner, etc. It’s going to be rough…
And then, I’ll be dashing off to a concert. The legendary composer Steve Reich is coming to campus to introduce a concert in his honour, where various groups of musicians perform several of his pieces!! Actually, I should put that in a separate post, so I’ll stop here.
-cvj
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I should write down here the operator solution for the general electron and or positron state. I will use the “hat” language used in Clifford algebra where the Dirac vectors are replaced by [tex]\gamma_0 \to \hat{t},\;\; \gamma_1 \to \hat{x},\;\; \gamma_2 \to \hat{y},\;\; \gamma_3 \to \hat{z}. [/tex] The reason for the substitution is that the hat notation is obviously geometric, and requires less space.
Okay, an electron or positron has spin in some direction defined by a unit vector. The operator for spin 1/2 in the Dirac algebra in the [tex](a_x,a_y,a_z)[/tex] direction is:
[tex]\hat{a} = i\hat{x}\hat{y}\hat{z}(a_x\hat{x}+a_y\hat{y} + a_z\hat{z})[/tex]
The projection operator for the state with spin +1/2 in the [tex]\vec{a}[/tex] direction is therefore:
[tex](1+\hat{a})/2[/tex]
To distinguish between electron and positron I need a “charge” operator. Peskin & Schroeder use [tex]Q = \hat{t}[/tex], and the projection operator for this is [tex](1\pm \hat{t})/2[/tex]
Therefore, the complete solution for the density operator (qubit) representation of an electron / positron is the product of these two (commuting) projection operators:
[tex](1 + i\hat{x}\hat{y}\hat{z}(a_x\hat{x}+a_y\hat{y} + a_z\hat{z}))(1\pm \hat{t})/4[/tex]
where the P&S convention is that the + sign gives the electron and – sign gives the positron. To translate this into the usual (inelegant) spinor notation, one chooses a representation for the operators, and computes the matrix equivalent:
[tex]\frac{1}{4}\left(\begin{array}{cccc}
1+a_z&a_x-ia_y&\pm(1+a_z)&\pm(a_x-ia_y)\\
a_x+ia_y&1-a_z&\pm(a_x+ia_y)&\pm(1-a_z)\\
\pm(1+a_z)&\pm(a_x-ia_y)&1+a_z&a_x-ia_y\\
\pm(a_x+ia_y)&\pm(1-a_z)&a_x+ia_y&1-a_z\end{array}\right)[/tex]
The spinor is any non zero column of the above matrix. For the particular example that P&S slaves over, [tex]\vec{a} = (0,0,1)[/tex] and the resulting spinor is (1,0,1,0), which is what one obtains as a special case above (times 1/2).
Here, let me better explain how physics cripples itself with symmetry when there are other, far more practical and easy to use methods of achieving the same thing.
Pick up your copy of Peskin and Schroeder and look up the section early in the book where they derive the constant spinors in the chiral representation. The derivation takes a half dozen pages and involves assumptions about spinors as representatives of the Lorentz or Poincare symmetry. At the end, a spinor pops out, but it was a hell of a lot of work. This is the way particle theory is done because it is based on the assumption that symmetry is fundamental.
The way I do particle physics is from the operator point of view. For the Dirac algebra, it is the operators themselves, as elements of a Clifford algebra that are foundational, and so the theory is directly related to the vectors of the Clifford algebra, [tex]\gamma_0, \gamma_1, \gamma_2, \gamma_3[/tex].
In the operator language, symmetry, and spinors, are just mathematical conveniences, it is the operators that are fundamental. And these operators are geometric (i.e. Hestenes’ Geometric Algebra) in nature. In this language, the problems which are so messy in the symmetry language become quite trivial in the operator language. For example, the wikipedia entry for spinors uses operator theory. In one page it derives the spinor for an electron or positron in an arbitrary orientation. About 1/6th of P&S’s efforts for more results.
Standard physics makes the assumption that relativity is perfect (a symmetry principle) and derives everything from this, at great effort and with oodles of undetermined constants (as arise in any attempt to use symmetry to solve a math problem). Operator physics makes the assumption that the geometry of spacetime is the Dirac algebra and gets the same results trivially.
Now let’s try and unify the particles. From the symmetry point of view, you have a damned near infinite number of possible choices of symmetry, which particles are present, and how they interact. From the operator point of view, the only thing you can do is generalize the Dirac algebra to a more general Clifford algebra. Which one of these techniques makes sense to you? How many vacua can 5000 particlel theorists explore in 30 years given that there are [tex]10^{500}[/tex] to look through? Operator theory is very restrictive, and that means that you have a hope in Hell of solving the problem.
The reason that obsolete techniques become obsolete in physics, in the face of the very strong sociological pressures towards conservatism, is because graduate students don’t like to solve problems the hard way. Symmetry is the hard way and is doomed.
Just between you and me, the whole problem with physics nowadays is precisely the reliance on symmetry arguments. Yes, it is possible to make great strides knowing symmetry, but the primary use of it is when you are clueless about what is going on at the fundamental level. This is the case with physics now, but the next advance will be to give physics a foundation in real things.
The foundations of physics are shrouded in mystery, symmetry is a useful tool for this. Biology is based on chemistry and all that, the foundations are very practical and are well understood at an intuitive level. If anything, the biologists should be over giving assistance to the physicists.
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