Conformal weights in Laurent expansions

Let me use the book by di Francesco et al. Including the conformal weight in the power of $z$ in the Laurent expansion, as in equation (6.8), allows you to have the nice property (6.10), namely $phi_{-m} = phi^{dagger}_m$. For the energy momentum tensor, it gives $L_{-m} = L^{dagger}_m$.

It is just a nice convention.

The conformal weights are included because they come from the conformal transformation which maps the cylinder to the radial plane. Start with the cylinder where $x sim x + 2pi$. Define complex coordinates $w=t+ix$ so $w sim w + 2pi i$. Then, all operators admit the expansion $$ Phi(w,{bar w}) = sum_{m,n} phi_{m,n} e^{-m w} e^{-n {bar w} } , qquad m,n in {mathbb Z} $$

Now, we perform the conformal transformation which moves us to the radial plane. In particular, we define $$ z = e^{ w} = e^{ i x} e^{t} , qquad {bar z} = e^{ {bar w}}. $$ Then, note that $t to -infty$ maps to the origin and $t to infty$ maps to infinity. Now, under conformal transformations a primary field transforms as begin{align} Phi(z,{bar z}) &= (partial_z w)^h(partial_{bar z} {bar w})^{{bar h}}Phi (w,{bar w}) = sum_{m,n} phi_{m,n} z^{-m-h} {bar z}^{-n-{bar h}} end{align}

ASIDE

In Euclidean signature, the adjoint property is $$ Phi(t,x)^dagger = Phi(-t,x) ~~implies~~ Phi(w,{bar w})^dagger = Phi(-{bar w},-w) ~~implies~~ phi_{m,n}^dagger = phi_{-m,-n} . $$ This is because time evolution is defined by $Phi(t,x) = e^{H t} Phi(0,x) e^{-H t}$. We assume here that $Phi(0,x)$ is Hermitian. On the radial plane, the adjoint property then reads begin{align} Phi(z,{bar z})^dagger &= sum_{m,n} phi^dagger_{m,n}{bar z}^{-m-h} z^{-n-{bar h}} &= {bar z}^{-2h} z^{ - 2 {bar h} }sum_{m,n} phi_{m,n} (1/{bar z})^{-m-h} (1/z)^{-n-{bar h}} &= {bar z}^{-2h} z^{ - 2 {bar h} } Phi(1/{bar z},1/z). end{align}

This is simply a convenience, and in the end you should be able to work with any convention. In fact, in the mathematical literature on vertex operator algebras you will often encounter,

$$ Y(omega,z) = sum omega_n z^{-n-1} = sum L_n z^{-n-2}$$

where $Y$ maps a state to a field, i.e. the stress-energy tensor has modes expanded as if $h=1$ in your notation. This might seem odd but it is actually useful to have a consistent convention to use for any state, regardless of conformal dimension, when you have operators in your algebra of varying conformal dimension.