Student Seminar: Classical and Quantum Integrable Systems

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.
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