Multiply creation operator by a phase factor A basic question, but I'm not completely confident what I'm doing is legit.
I can multiply a creation operator by an arbitrary phase factor and it doesn't change any physics. True?
I have a Hamiltonian
$$H=\sum_\mathbf{q}(f_\mathbf{q} b_\mathbf{q}^\dagger a_\mathbf{q} + f_\mathbf{q}^* a_\mathbf{q}^\dagger b_\mathbf{q})$$
where $a_\mathbf{q}$ and $b_\mathbf{q}$ commute. If $f_\mathbf{q} = \exp(i q_y)F(q_x)$ can I redefine $b_\mathbf{q} \rightarrow \exp(i q_y) b_\mathbf{q}$ to 'get rid' of the exponential factor? My energy spectrum is translationally invariant, it does not depend on $q_y$, which is why I think I can do this.
Many thanks for any and all help!
 A: This depends on whether the corresponding quadratures have physical meaning in your specific example. This is because if $a=x+ip$, then changing $a\mapsto a'=e^{i\theta}a$ corresponds to the canonical transformation
\begin{align}
x\mapsto x'= \cos(\theta)\, x -\sin(\theta) \,p,
\\
p\mapsto p'=\sin(\theta)\, x +\cos(\theta) \,p.
\end{align}
This could be undesirable if, say, your harmonic mode is the actual harmonic motion of a physical particle, in which case $x$ and $p$ have definite physical meaning. In other cases, such as a photonic mode, this corresponds to a phase in the mode and is intrinsically a feature of the quadratures, is dealt with explicitly in quadrature measurement experiments such as homodyne and heterodyne detection, and is not a problem at all. If $a$ represents something more fancy then you need to check, but you're probably safe.
Other than this effect on the quadratures, the transformation you're doing is perfectly legitimate.
Another way to see this is to see the effect of the transformed annihilation operator on the number basis,
$$ a'|n⟩=e^{i\theta}\sqrt{n}|n-1⟩,$$
as simply indicating a change of phase in the number basis. Thus, you can achieve this equally well by setting $$|n⟩'=e^{in\theta}|n⟩.$$ If you're going to do interference experiments between different number states then you obviously need to worry about their relative phase, but other than that you're perfectly free to set and re-set the phases of this basis.
