Explicit form of annihilation and creation operators for Dirac field

[QFT1995]

I'm unclear on what exactly an annihilation or creation operator looks like in QFT. In QM these operators for the simple harmonic oscillator had an explicit form in terms of

$$
\hat{a}^\dagger = \frac{1}{\sqrt{2}}\left(- \frac{\mathrm{d}}{\mathrm{d}q} + q \right),\;\;\;\hat{a} = \frac{1}{\sqrt{2}}\left(\frac{\mathrm{d}}{\mathrm{d}q} + q \right)
$$
however I cannot find any explicit terms for these in QFT. My question is, is it possible to formulate an expression for them in terms of a differential operator or do we just assume that they exist in QFT? I am particularly interested in the Massive Thirring Model (Dirac field in 1+1D with self interactions).

 

[DarMM]

They act on quite an structured set of functions. It is possible to write them explicitly but consider that a creation operator $a_{p}^{\dagger}$ acting on a one particle state $f\left(q\right)$ maps it into a two particle state $f\left(p,q\right)$ also they need to be smeared or else the result won't be a state. The creation operator for example acts as:
$$a_{p}^{\dagger} f = \delta\left(q-p\right)\otimes f, \quad f \in \mathcal{H}_{-\frac{1}{2}}\left(\mathbb{R}^{3}\right)^{\otimes n}$$

This isn't really usable in a analytic sense like the ones from QM since they don't map between functions of the same fixed number of variables. One just needs their algebraic properties.

 

[Demystifier]

My question is, is it possible to formulate an expression for them in terms of a differential operator

It is possible, in terms of functional derivatives. See e.g. https://www.amazon.com/dp/0201360799/?tag=pfamazon01-20 Eq. (10.40).

 

[QFT1995]

Thank you both.

It is possible, in terms of functional derivatives. See e.g. https://www.amazon.com/dp/0201360799/?tag=pfamazon01-20 Eq. (10.40).

I checked the reference you provided and it helped so thank you. Do you know what the creation and annihilation operators would look like for a fermionic field? I don't have much experience with the path integral formalism and I'm struggling with the manipulations.

 

[vanhees71]

Well, the path integral is well worth learning when dealing with QFT. It makes some issues much more simple (though it's still complicated enough). Particularly quantizing local gauge symmetric theories (among them the Standard Model of HEPs) is much more complicated in the (covariant) operator formalism.

As it turns out the path integral for fermions needs the introduction of "Grassmann numbers" rather than usual complex numbers to describe the fields integrated over in the path integral. A good textbook introducing QFT in the "path-integral-first" way is

D. Bailin, A. Love, Introduction to Gauge Field Theory, Adam Hilger, Bristol and Boston (1986).

 

[Demystifier]

I checked the reference you provided and it helped so thank you. Do you know what the creation and annihilation operators would look like for a fermionic field? I don't have much experience with the path integral formalism and I'm struggling with the manipulations.

For fermions you need functional derivatives with respect to Grassmann valued fields. See e.g. my http://de.arxiv.org/abs/quant-ph/0302152 Eqs. (9) and (11).

 

[vanhees71]

Of course you can also work in the operator formalism. For gauge theories it's a pretty complicated eneavor though; I'd recommend to learn only QED in the operator formalism, then learn path-integral methods and then go to the non-Abelian case.

For free fields, for which a mode decomposition, usually in terms of the momentum-spin single-particle eigenbasis, makes sense. The fermionic case is not much different from the bosonic one. The only difference is that you have anti-commutators instead of commutators, i.e.,
$$\{ \hat{a}(\vec{p},\sigma),\hat{a}(\vec{p}',\sigma') \} =0, \quad \{ \hat{a}(\vec{p},\sigma),\hat{a}^{\dagger}(\vec{p}',\sigma') \} = N(\vec{p}) \delta^{(3)}(\vec{p}-\vec{p}') \delta_{\sigma \sigma'}.$$
The normalization factor is a matter of convention. Some textbooks use the simple but non-covariant one, $N(\vec{p})=1$, others use the covariant one with $N(\vec{p})=(2 \pi)^3 2 E_{\vec{p}}$.