Friday, January 23, 2015

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 ( 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? 

The q-deformation o reproduces the typical structures of a quantized field in a space–time with horizon!!! 

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.

1 comment:

  1. The deformation Δa(q) = a(q)⊗q^c + q^{−c}⊗a(q) is related to Bogoliubov operators. We may write

    q^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.