# Dual Resonance Model: Fermions

I am going through Ramond's 1971 paper Dual Theory for Free Fermions Phys Rev D3 10, 2415 where he first attempts to introduce fermions into the conventional dual resonance model.

I get the 'gist' of what he's doing: he draws an analogy of the bosonic oscillators satisfying the Klein-Gordon equation, and extends it to incorporate some version of the Dirac equation. Great.

Now without resorting to string theory (since I know nothing about it) and perhaps minimally resorting to field theory (after all, this is still S-matrix theory, right?), how can I understand his correspondence principle (eqn 3)?

$$p^2-m^2=\langle P\rangle\!\cdot\!\langle P\rangle-m^2\rightarrow\langle P\!\cdot\! P\rangle-m^2$$

(the same correspondence principle appears in Frampton's 1986 book "Dual Resonance Models" equation 5.63). Is this a special property of harmonic oscillators?

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It seems that Ramond is appealing to the fact that in the classical limit average of the square is equal to square of average. But, is that justifiable in the quantum theory? Why is he allowed to use correspondence principle? – QuantumDot Jul 29 '12 at 23:54
Ramond is doing string theory (he's founding it), so it is impossible to answer this question without resorting to string theory. Just a nitpick. Also, you should be aware that what Ramond is doing is a string picture, which he modifies by adding fermions, but for some reason known only to him, he has chosen the string coordinate to be timelike, so that the oscillators are temporal. – Ron Maimon Jul 30 '12 at 21:12
@Maimon Oh man; I was hoping you (Ron Maimon) would provide a nice and lengthy answer involving Dual Resonance Formalism as your post history suggests you have studied it in great detail. Sigh, oh well; what's the answer in terms of string theory? – QuantumDot Jul 31 '12 at 2:41
@Maimon If I sent you his paper electronically, would you be able to decipher it? – QuantumDot Jul 31 '12 at 18:57
Yes, it was pretty clear--- I am likebox at g mail dot com. but your question is really about the preamble not the main thing, and this part is just a review of oscillator representation of strings in light cone gauge (in Ramond's ideosyncratic string picture which has a time oscillators, but mathematically it's the same). This is covered well in the first chapter of Polchinsky, and in Green Schwarz and Witten sections on light-cone gauge quantization. If you just look at the mathematics of the mode expansion you'll get it. The main things in Ramond are the F's, the SUSY. – Ron Maimon Jul 31 '12 at 23:30