HJM’s theory started with a formulation containing 2 “free” processes — the drift (alpha) and vol (sigma) of inst fwd rate
, both functions of time and could be stochastic.
Note the vol is defined differently from the Black-Scholes vol.
Note this is under physical measure (not Q measure).
Note the fwd rate is instantaneous, not the simply compounded.
We then try to replicate one zero bond (shorter maturity) using another (longer maturity), and found that the drift process alpha(t) is constrained and restricted by the vol process sigma(t), under P measure. In other words, the 2 processes are not “up to you”. The absence of arbitrage enforces certain restrictions on the drift – see Jeff’s lecture notes.
Under Q measure, the new drift process  is completely determined by the vol process. This is a major feature of HJM framework. Hull-white focuses on this vol process and models it as an exponential function of time-to-maturity:
That “T” above is confusing. It is a constant in the “df” stochastic integral formula and refers to the forward start date of the (overnight, or even shorter) underlying forward loan, with accrual period 0.
 completely unrelated to the physical drift alpha(t)
Why bother to change to Q measure? I feel we cannot do any option pricing under P measure. P measure is subjective. Each investor could have her own P measure.
Pricing under Q is theoretically sound but mathematically clumsy due to stochastic interest rate, so we change numeraire again to the T-maturity zero bond.
Before HJM, (I believe) the earlier TS models can’t support replication between bonds of 2 maturities — bond prices are inconsistent and arbitrage-able