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We know that the lowering and raising operators in quantum mechanics are defined as

\begin{array}{l} a =\frac{1}{\sqrt{2}}(X+i P) \\ a^{\dagger} =\frac{1}{\sqrt{2}}(X-i P), \end{array}

respectively.

I was reading in this book page 257 about the different quantization schemes and he mentioned that the Wick-ordered quantization scheme is useful for quantum field theory and they are defined as:

\begin{aligned} a &=X+i \alpha P \\ a^{\dagger} &=X-i \alpha P \end{aligned}

He also mentioned that they differ by a constant from the first two raising and lowering operators and the commutator of $a$ and $a^\dagger$ is not $I$ but rather $2\alpha \hbar I$.

Does all of this mean that the raising and lowering operators of quantum mechanics are different than those of quantum field theory?

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Yes, normally one uses a convenient normalization for creation and annihilation operators. In QM there is usually only one frequency, ω, in your problem, so you incorporate it into the normalization of $a$ and $a^\dagger\equiv a^*$, (11.10), and it virtually disappears from the problem, $$ D^2={\hbar \over m\omega}, \\ a=\frac{1}{\sqrt 2} \left ({X\over\sqrt{D}} +i {\sqrt{D}\over \hbar}P\right ),\\ a^*=\frac{1}{\sqrt 2}\left ({X\over\sqrt{D}} -i {\sqrt{D}\over \hbar}P\right ),\leadsto \\ [a,a^*]=I, $$ given $[X,P]=i\hbar I$.

In the case you are asking about, the many frequencies are not irrelevant, and become part of the normalization of each oscillator, so , as stated, \begin{aligned} a &=X+i \alpha P \\ a^{\dagger} &=X-i \alpha P \end{aligned} and, by a similar calculation, $[a,a^*]=2\alpha \hbar I$, now. Presumably, he would proceed later down to fix $\alpha$, depending on the QFT oscillators he might be dealing with.

N.B. Some QFT books, but not all!, dealing with the infinity of commuting oscillators comprising QFT, normalize them as $$[a_k,a^\dagger _p]=(2\pi)^3 \delta^3(\vec p -\vec k). $$ But some also incorporate the energy of each in the normalization. Caveat lector!

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  • $\begingroup$ Thank you. Can you please suggest me a good source on Weyl Quantization that might help me fully understand Stone-Von Neumann's Theorem? $\endgroup$ – Sally Jun 28 at 13:49
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    $\begingroup$ Have you tried [Wikipedia](Stone-Von Neumann's Theorem) and phase space formulation, and references therein? Our booklet covers the waterfront pretty well, we thought. $\endgroup$ – Cosmas Zachos Jun 28 at 15:33

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