Quantum Groups and curved space-time
Today I will present the last post on Hopf algebras and I will talk about quantum groups (which are a special kind of Hopf algebras) and their amazing application to curves space-time. Alfredo Iorio pointed me his paper (http://arxiv.org/pdf/hep-th/0104162v1.pdf) which I studied in detail.
In second quantization he starts with the Weyl-Heisenberg algebra \((a, a^{\dagger}, N, c)\):
\([a, a^{\dagger}] = 2c\)
\([N, a] = −a \)
\([N, a^{\dagger}] = a^{\dagger} \)
\([c, \cdot] = 0\)
and then the Hopf algebra coproduct is introduced:
\(\Delta a = a\otimes I + I \otimes a\)
So far nothing special but then the coproduct can be deformed forming a quantum group:
\(\Delta a_q = a_q \otimes q^c + q^{-c}\otimes a_q\)
where \(q\) is the deformation parameter related to a geometric-like series:
\({[x]}_q = \frac{q^x - q^{-x}}{q - q^-1}\)
As a nice math puzzle, what is the series which gives rise to \({[n]}_q\)? It is obviously related to \( 1+q+q^2+\cdots + q^n\) but it is not quite that.
So what is this to do with anything?
So what is this to do with anything?
Quantum group deformations can be induced by gravitational fields. For technical details please read Iorio's paper from above.
I am still studying quantum groups and I do not want to venture stating more because my intuition in this area is work in progress, but clearly they have a real physical interpretation.
The deformation Δa(q) = a(q)⊗q^c + q^{−c}⊗a(q) is related to Bogoliubov operators. We may write
ReplyDeleteq^c = Ie^{2πiφ} = I(cos(2πφ) + sin(2πφ)),
for φ a parameter related to acceleration. Of course in addition
(q^c)^{-1} = Ie^{-2πiφ} = I(cos(2πφ) - sin(2πφ)).
This gives
Δa(q) = a(q)⊗I(cos(2πφ) + sin(2πφ)) + (cos(2πφ) - sin(2πφ))I⊗a(q)
= (a(q)⊗I + I⊗a(q))cos(2πφ) + (a(q)⊗I - I⊗a(q))sin(2πφ)
We may of course form the Hermitian conjugate
Δa^†(q) = a^†(q)⊗q^{-c} + q⊗a^†(q)
= (a^†(q)⊗I + I⊗a^†(q))cos(2πφ) + (a^†(q)⊗I - I⊗a^†(q))sinh(2πφ).
The commutator [Δa(q), Δa^†(q)] is
[Δa(q), Δa^†(q)] = ([a(q), a^†(q)] ⊗I + I⊗[a(q), a^†(q)])(cos^2(2πφ) + sin^2(2πφ)),
which gives a unit product.
The hyperbolic trigonometric functions obtained from φ --- > iφ are squeeze parameters and we have a C* form of the Bogoliubov operators
b = au + a^†v, b^† = a^†u + av
for u = cosh(2πφ) and v = sinh(2πφ). The commutator of b and b^† is
[b, b^†] = u^2 – v^2 = 1.
The q-deformed algebra has the same u^2 – v^2 = 1. This then means we have the operators employed in computing Hawking and Unruh radiation.
LC