  We present Herschel SPIRE-FTS observations of the Antennae (NGC 4038/39), a
well studied, nearby ($22$ Mpc) ongoing merger between two gas rich spiral
galaxies. We detect 5 CO transitions ($J=4-3$ to $J=8-7$), both [CI]
transitions and the [NII]$205\mu m$ transition across the entire system, which
we supplement with ground based observations of the CO $J=1-0$, $J=2-1$ and
$J=3-2$ transitions, and Herschel PACS observations of [CII] and [OI]$63\mu m$.
Using the CO and [CI] transitions, we perform both a LTE analysis of [CI], and
a non-LTE radiative transfer analysis of CO and [CI] using the radiative
transfer code RADEX along with a Bayesian likelihood analysis. We find that
there are two components to the molecular gas: a cold ($T_{kin}\sim 10-30$ K)
and a warm ($T_{kin} \gtrsim 100$ K) component. By comparing the warm gas mass
to previously observed values, we determine a CO abundance in the warm gas of
$x_{CO} \sim 5\times 10^{-5}$. If the CO abundance is the same in the warm and
cold gas phases, this abundance corresponds to a CO $J=1-0$ luminosity-to-mass
conversion factor of $\alpha_{CO} \sim 7 \ M_{\odot}{pc^{-2} \ (K \ km \
s^{-1})^{-1}}$ in the cold component, similar to the value for normal spiral
galaxies. We estimate the cooling from H$_2$, [CII], CO and [OI]$63\mu m$ to be
$\sim 0.01 L_{\odot}/M_{\odot}$. We compare PDR models to the ratio of the flux
of various CO transitions, along with the ratio of the CO flux to the
far-infrared flux in NGC 4038, NGC 4039 and the overlap region. We find that
the densities recovered from our non-LTE analysis are consistent with a
background far-ultraviolet field of strength $G_0\sim 1000$. Finally, we find
that a combination of turbulent heating, due to the ongoing merger, and
supernova and stellar winds are sufficient to heat the molecular gas.
