Student Seminar: Classical and Quantum Integrable Systems

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The form of the L-operator and the R-matrix is essentially the same. We check ” ” “(λ 1 − λ 2 )I ab + iP ab L ia (λ 1 )L ib (λ 2 ) = L ib (λ 2 )L ia (λ 1 ) “(λ 1 − λ 2 )I ab + iP ab , which leads to “ ” “ ” (λ 1 − λ 2 ) L ia (λ 1 )L ib (λ 2 ) − L ib (λ 2 )L ia (λ 1 ) = iP ab L ia (λ 2 )L ib (λ 1 ) − L ia (λ 1 )L ib (λ 2 ) , It is easy to see that L ia (λ 1 )L ib (λ 2 ) − L ib (λ 2 )L ia (λ 1 ) = P ib P ia − P ia P ib and ” iP ab “L ia (λ 2 )L ib (λ 1 ) − L ia (λ 1 )L ib (λ 2 ) = (λ 1 − λ 2 )P ab (P ib − P ia ) = (λ 1 − λ 2 )(P ib P ai − P ia P ib ) This proves the statement. The relation (5.3) is called the fundamental commutation relation. Yang-Baxter equation. It is convenient to suppress the index of the quantum space and write the fundamental commutation relation as R ab (λ 1 − λ 2 )L a (λ 1 )L b (λ 2 ) = L b (λ 2 )L a (λ 1 )R ab (λ 1 − λ 2 ) . We can think about L as being 2 × 2 matrix whose matrix elements are generators of a certain associative algebra (operators). Relations (5.3) define the then the commutation relations between the generators of this algebra. Substituting the indices a and b for 1 and 2 we will write the general form of the fundamental commutation relations R 12 (λ 1 , λ 2 )L 1 (λ 1 )L 2 (λ 2 ) = L 2 (λ 2 )L 1 (λ 1 )R 12 (λ 1 , λ 2 ) . What the R-matrix does is that it interchange the position of the matrices L 1 and L 2 . Consider a triple product L 1 L 2 L 3 = R12 −1 L 2 L 1 R 12 L 3 = R12 −1 L 2 L 1 L 3 R 12 = = R12 −1 R13 −1 L 2 L 3 L 1 R 13 R 12 = R12 −1 R13 −1 R23 −1 L 3 L 2 L 1 R 23 R 13 R 12 . Essentially, we brought the product L 1 L 2 L 3 to the form L 3 L 2 L 1 . However, we can reach the same effect by changing the order of permutations L 1 L 2 L 3 = R23 −1 L 1 L 3 L 2 R 23 = R23 −1 R13 −1 L 3 L 1 L 2 R 13 R 12 = = R12 −1 R13 −1 L 2 L 3 L 1 R 13 R 12 = R23 −1 R13 −1 R12 −1 L 3 L 2 L 1 R 12 R 13 R 23 . Thus, if we require that we do not generate new triple relations between the elements of L we should impose the following condition on the R-matrix: This is the quantum Yang-Baxter equation. R 12 R 13 R 23 = R 23 R 13 R 12 . (5.4) – 69 –
Semi-classical limit and quantization. Why the quantum Yang-Baxter equation is called “quantum”? Assume that the R-matrix depends on the additional parameter and when → 0 it expands into the power series starting with the unit: R 12 = I 12 + r 12 + · · · Expanding quantum Yang-Baxter equation we see that the leading terms as well as the terms proportional to cancel out. At order 2 we find ) ([r 2 12 , r 13 ] + [r 12 , r 23 ] + [r 13 , r 23 ] + O( 3 ) = 0 . At order 2 we find the classical Yang-Baxter equation. Thus, the quantum Yang- Baxter equation can be considered as the deformation (or quantization) of the classical Yang-Baxter equation. Further we recall the relation between the Poisson bracket of classical observables and the Poisson brachet of their quantum counterparts {A, B} = lim [Â, ˆB] →0 1 Now we notice that the fundamental computation relations can be written in the equivalent form [L 1 (λ 1 ), L 2 (λ 2 )] = i λ 1 − λ 2 [P 12 , L 1 (λ 1 )L 2 (λ 2 )] . This formula allows for the semi-classical limit }{{} L → }{{} L quantum classical and it defines the Poisson bracket on the space of classical L -operators 1 [ {L 1 (λ 1 ), L 2 (λ 2 )} = lim →0 [L i ] 1(λ 1 ), L 2 (λ 2 )] = P 12 , L 1 (λ 1 )L 2 (λ 2 ) . λ 1 − λ 2 We see that r = i P appears to be the classical r-matrix. Thus, the semi-classical λ limit of the fundamental commutation relations is nothing else as the Sklyanin bracket for classical Heisengerg magnetic. Inversely, we can think about the fundamental commutation relations as quantization of the Poisson algebra of the classical L-operators. Monodromy and transfer matrix. For a chain of length L define the monodromy as the ordered product of L-operators along the chain 14 T a (λ) = L L,a (λ) . . . L 1,a (λ). 14 Recall the definition of the monodromy as the path-ordered exponent in the classical case. – 70 –
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Preprint typeset in JHEP style - HY
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7. Homework exercises 95 7.1 Semina
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which can be written as follows {q
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To get more insight on the Liouvill
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4. The equations of motion with Ham
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Let us show that the variables (I i
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We have H = p2 2 p2 + V (x) = 2 + V
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We can now see how the solution can
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Thus, and, therefore, the half-peri
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Page 107 and 108: Exercise 4 Consider the non-linear
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Student Seminar: Classical and Quantum Integrable Systems

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