Research Paper Notes on Particle Correlation in Wigner Function Approach

Research Paper Notes on Particle Correlation in Wigner Function Approach

nucl-th/9909018 Boson spectra and correlations for thermal locally equilibrium systems
Yuri M. Sinyukov Preprint ITP-93-8E, Heavy Ion Phys.10:113-136,1999

P.4 (4) Firstly, $$p^{\mu}e^{ik\cdot x}$$ can be seen as a conserved current, when $$p^{\mu}$$ is a given (constant) four-vector satisfying $$p\cdot k=0$$ (see below)


 * $$\begin{align}

\partial_{\mu}(p^{\mu}e^{ik\cdot x})=ip^{\mu}k_{\mu}e^{ik\cdot x}=i(p\cdot k)e^{ik\cdot x}=0 \end{align}$$

One may apply Gauss's law to such conserved current, so that the surface integral of $$p^{\mu}e^{ik\cdot x}$$ on any given surface can be replaced by an integral sharing the same boundary. In particular, we will replace the integral on surface $$\sigma_{\mu}$$ by an integral on the surface $\Sigma_{\mu}$ with constant time.


 * $$\begin{align}

\int d\Sigma_{\mu}p^{\mu}e^{ik\cdot x}=\int d\sigma^t_{\mu}p^{\mu}e^{ik\cdot x} \end{align}$$

Therefore some terms can be simplified as follows


 * $$\begin{align}

&d\sigma^t=d^3x \hat{e}_t\\ &\int d\sigma^t_{\mu}p^{\mu}e^{ik\cdot x}=\int d\sigma^t_{\mu}p^{\mu}e^{ik^0t}e^{-i\mathbf{k\cdot x}}=p^0e^{ik^0t}(\int d^3x e^{-i\mathbf{k\cdot x}})=p^0e^{ik^0t}(2\pi)^3\delta(\mathbf{k}) \end{align}$$

We note that the above argument maybe seems too clumsy and not necessary, one may simply argue that if a quantity is invariant, one can evaluate it in any frame of reference.

P.4 (5) We will first introduce some notations and some identities to be used later
 * $$p=\frac{p_1+p_2}{2},q=p_2-p_1$$

therefore
 * $$p_1=p-\frac{q}{2},p_2=p+\frac{q}{2}$$

Note that $$p_1,p_2$$ are two 4-momenta of detected hadrons (on the mass shell), one immediate consequence is


 * $$p\cdot q = \frac{1}{2}(p_2^2-p_1^2)=\frac{1}{2}(m^2-m^2)=0 $$

At a certain moment, we will want to write down $$\delta(p\cdot u) $$ in terms of $$\delta^4$$ function. To this end, we define $$u'$$ as $$u=q+u'$$. For any function $$f(u)$$ one has,


 * $$\begin{align}

&f(u) \delta(p\cdot u)\delta(\mathbf{q}-\mathbf{u}) \\ &=f(u) \delta(p\cdot (q+u'))\delta(\mathbf{q}-\mathbf{u}) \\ &=f(u) \delta(p^0{u^0}')\delta(\mathbf{q}-\mathbf{u}) \\ &=f(u) \delta({u^0}')\delta(\mathbf{q}-\mathbf{u})/p^0 \\ &=f(u) \delta(q^0-u^0)\delta(\mathbf{q}-\mathbf{u})/p^0 \\ &=f(u) \delta(q-u)/p^0 \end{align}$$

Both $$p,q$$ are seen as constant in the above expression, where $$u$$ is the variable. We have made use of $$\delta^3$$ function of 3-momentum $$\delta(\mathbf{q}-\mathbf{u}) $$, where $$\mathbf{u}$$ is not necessarily on the mass shell, which implies $$u'=({u^0}',0,0,0) $$. We also note that


 * $$\begin{align}

d^4u=du^0d\mathbf{u}=du'^0d\mathbf{u},d\mathbf{u}\equiv d^3u \end{align}$$


 * $$\begin{align}

&d^4u p^0 e^{i(q^0-u^0)t}\delta(p\cdot u)\delta(\mathbf{q}-\mathbf{u}) =d^4u p^0 e^{i(q^0-u^0)t} \delta(q-u)/p^0 =d^4u e^{i(q^0-u^0)t}\delta(q-u) =d^4u \delta(q-u) \end{align}$$

The above expressions will be used below. Now we are in the position to derive (5)


 * $$\begin{align}

&\langle a^+(p_1)a(p_2)\rangle\\ &=\langle a^+(p-\frac{q}{2})a(p+\frac{q}{2})\rangle \\ &=\int d^4u \delta(q-u)\langle a^+(p-\frac{u}{2})a(p+\frac{u}{2})\rangle \\ &=\int d^4u(2\pi)^{-3}\delta(p\cdot u)\langle a^+(p-\frac{u}{2})a(p+\frac{u}{2})\rangle\times[(2\pi)^3p^0e^{i(q^0-u^0)t}\delta(\mathbf{q}-\mathbf{u})] \\ &= \int d^4u\int d\Sigma_{\mu} p^{\mu}e^{i(q-u)\cdot x}(2\pi)^{-3}\delta(p\cdot u)\langle a^+(p-\frac{u}{2})a(p+\frac{u}{2})\rangle \\ &=\int d\Sigma_{\mu} p^{\mu}e^{iq\cdot x} \times\int d^4u (2\pi)^{-3}e^{-iu\cdot x}\delta(p\cdot u)\langle a^+(p-\frac{u}{2})a(p+\frac{u}{2})\rangle \\ &=\int d\Sigma_{\mu} p^{\mu}e^{iq\cdot x} f_W(x,p) =\int d\sigma_{\mu} p^{\mu}e^{iq\cdot x} f_W(x,p) \end{align}$$

We would like to give a few comments on (5). The identity (5) involves an invariant form of $$\langle a^+(p_1)a(p_2)\rangle$$, it helps to express things in terms of hydrodynamic variables. However, by itself, (5) is irrelevant to physical quantities such as fluid velocity. In this respect, it merely transfers the problem into the calculation of Wigner function. In practice, for systems almost in equilibrium the Wigner function can be further approximated by the equilibrium distribution at local rest frame $$f_W\to f(p\cdot U,T,\mu)$$ (see [6] of the paper). This approximation is only valid when $$\vec{p}_1 \sim \vec{p}_2$$, which is due to the fact that the density matrix is almost diagonal when the system is not far away from equilibrium. The later implies, one has to do the Bogoliubov transformation first, then apply (5). In particular, if the two 3-momenta are back to back ($$\vec{k}_1,\vec{k}_2$$), they will be parallel after the Bogoliubov transformation (consider, e.g., $$\langle a^+(k_1)a^+(k_2)\rangle \to \langle  b^+(k_1)b(-k_2)\rangle$$), this is when $$f_W=f$$ is a good approximation. It is worth pointing out that the two corresponding 4-momenta in this case are $$p_1=(k_1^0,\vec{k}_1),p_2=(k_2^0,-\vec{k}_2)$$ before substituting them into (5).