What is the meaning of $PV^\gamma=$ constant in an adiabatic process? In an adiabatic process, heat transfer doesn't occur and hence $\Delta U=-W$ (an increase in internal energy due to work done on the system or a decrease in internal energy due to work done on the surrounding).
My textbook then says (for monatomic gas) adiabatic process takes the form: $PV^{5/3} =$ constant
What does this equation means? From my understanding, "$PV^{5/3} =$ constant" pinpoints to a specific Pressure and Volume of the system and implies a specific Temperature of the system. This is pretty much equivalent to $PV=nRT$ where a specific Pressure and Volume implies a specific Temperature of the system.
But in an Adiabatic process the Temperature, Pressure and Volume can change (otherwise if nothing changes, what's the point of there being a process? and from $\Delta U=-W$ there should be something going on, the temperature should change which will cause a change in the volume of the system [but not a change in pressure since we're assuming the process is quasi-static and thus pressure is constant]) and thus  what is the meaning of the equation? Why is this equation specific to adiabatic process?
 A: The equation for adiabtic process you wrote does not fix both volume and pressure...you can pick any pressure and then and only then, the volume is fixed, and it is calculated via formula you wrote. I can give you this example: lets say that we observe isotermic process, that is, with constant temperature, and we know that for this process there is a formula PV=const. That does not mean that both P and V are fixed. Lets say, PV=2.
Then, we can write: P=2/V. So, if V=8, for example, P= 2/8=0.25. Same thing for adiabtic process with the difference that the dependence goes with -5/3 and not -1 like here. 
A: It means that during an adiabatic process if you monitor both the pressure and the volume then when you calculate the quantity $PV^\gamma$ every point you will get the same result. For example, $PV=nRT$ can be stated as $PV=$ constant for an isothermal process. You are correct that temperature is also changing for an adiabatic process, so you can combine the ideal gas law with $PV^\gamma=$ constant to get $PT^{\gamma/(1-\gamma)}=$ constant or $TV^{\gamma-1}=$ constant.
A: You are correct in saying that, in an adiabatic reversible process, the pressure, volume, and temperature are changing in tandem.  However, the temperature is changing in such a way that, along the process path, the equation $PV^{\gamma}=const$ is always satisfied.  And, in an adiabatic process, the parameter $\gamma$ is equal to the ratio of the specific heat at constant pressure to the specific heat at constant volume.
When you mentioned a quasi-static process path, I think you were thinking of a stair-step arrangement in which, over each step, the incremental change takes place at constant pressure.  This is only an approximation to a perfectly reversible process in which the applied pressure is changing very gradually (with time).  However, it is close enough to say that, if you determine the pressure and volume at the beginning and end of each incremental step, to an excellent approximation $P_iV_i^{\gamma}=P_fV_f^{\gamma}$.
