Fermion Anticommutation Relations (nightmare!)

[latentcorpse]

Hi. I've been thinking about this proof for over a day now and have reached the point where I can't come up with any new approaches!

I'm trying to prove equation (5.15) in these notes:
http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf


Just above eqn (5.15) we are told that the proof should be exactly the same as the proof of (5.5) which is done on p107.

I completely understand the proof on p107 for the commutation relations.
When trying to prove the anticommutation relations, the only difference is going to be a minus sign on the second term.

In other words, the proof is the same as on p107 except we have

$$\{ \psi( \vec{x}), \psi^\dagger \vec{y} \} = \displaystyle\sum_{s} \in \frac{d^3p}{(2 \pi)^3} \frac{1}{2E_{\vec{p}}} \left( u^s( \vec{p} \bar{u}^s(\vec{p}) \gamma^0 e^{i \vec{p} \cdot ( \vec{x}-\vec{y})} + v^s(\vec{p}) \bar{v}^s(\vec{p}) \gamma^0 e^{-i \vec{p} \cdot (\vec{x} - \vec{y})} \right)$$

This means that if we follow through the next few steps on p107 we arrive at

$$\int \frac{d^3p}{(2 \pi)^3} \frac{1}{E_{\vec{p}}} ( p_i \gamma^i + m ) \gamma^0 e^{i \vec{p} \cdot ( \vec{x} - \vec{y} )}$$

and as far as I can tell there is no way to make that into a delta function!


ANY HELP IS GREATLY APPRECIATED!

 

[Bill_K]

Do you really trust this reference?? Because other places give different results!

For a scalar field the commutator is usually quoted as [..x, y are 4-vectors..]

[φ(x), φ*(y)] = iΔ(x - y)

where Δ is the "invariant delta function", defined by

iΔ(x) = (2π)^-3∫d³k/2ω [e^-ik·x - e^ik·x]

Δ(x) vanishes for equal times, and its time derivative at equal times is a 3-dimensional δ function: (∂Δ(x)/∂x⁰)|(x⁰ = 0) = - δ³(x). Notice it is not true that [φ(x), φ*(y)] at equal times is a δ-function.. you must include the time derivative.

The corresponding relation for Dirac spinor fields is

{ψ(x), ψ†(y)} = -iS(x - y)

where

S(x - y) = (iγ^μ∂_μ + m)Δ(x - y)

 

[Einj]

I think relations (5.15) are correct (although I'm not sure for the (2π)$^{3}$ that probably depends on the normalization of u and v spinors). I think you can extract an expression for c and b operators using the spinor scalar product. For example, if I'm not wrong, the following equation should be true:
$$b^{r}_{\vec{p}}=\int{d^3x U^{\dagger}(r\vec{p})\psi(x)}$$
where:

$$U(r\vec{p})=\frac{1}{(2\pi)^{3/2}}\frac{1}{\sqrt{2E_{\vec{p}}}}u^s(\vec{p})e^{-ipx}$$

When you have extract both b and c you can anticommutate them and using the anticommutation relations on ψ and ψ$^{+}$ you should reach your goal.

I hope I'm not wrong, as I'm using my old notes and there should be something different from yours but I think the main idea should be ok.

 

[Bill_K]

Tong's proof is trying to go the other way. He assumes (5.15) and tries to prove (5.14). But as I indicated above, (5.14) is wrong. {ψ(x), ψ†(y)} is not equal to a delta function.

 

[Einj]

Maybe I'm missing something. Is ψ the fermionic field? If so, I'm quite sure that 5.14 are correct. I think you got wrong when you said that:

$$[\phi(x),\phi^\dagger(y)]=i\Delta(x-y)$$

as the correct relations, for scalar fields, should be:

$$[\phi(x),\dot{\phi}^\dagger(y)]=i\delta^3(\vec{x}-\vec{y})$$
and
$$[\phi(x),\phi(y)]=i\Delta(x-y)$$.

And the same thing is valid for fermionic fields. You can see for example Mandl-Shaw "Quantum Field Theory" equations 3.25 and 3.42.
I hope I'm not wrong, if so I'm sorry

 

[Bill_K]

Einj, The commutation relations for the Klein-Gordon field are slightly different depending on whether the field is real or complex.

For a real field, [φ(x), φ(y)] = -iΔ(x - y) (Sorry, I had the sign wrong.)

Whereas for a complex scalar field, [φ(x), φ*(y)] = -iΔ(x - y)

Some references call the invariant function D(x -y) instead of Δ(x - y).

Note that your first two equations are equivalent. You can get the second one by taking the time derivative wrt y of the first one, since the equal-time time derivative of Δ(x - y) is δ³(x - y).

And the same thing is valid for fermionic fields.

No, for a Dirac field ψ(x) there's an additional factor: S(x - y) = (iγ^μ∂_μ + m)Δ(x - y). The expression derived by the OP has this factor in it.

 

[Einj]

Ok, I knew that (and of course I said the same thing because S is quite the equivalent of Δ but for fermionic field). However, what should be the fermonic equivalent of my second equation?

 

[Bill_K]

One big difference between S and Δ: S is a 4 x 4 matrix!

what should be the fermonic equivalent of my second equation?

To get a delta function out of Δ, you need a time derivative. But in the Dirac case, the time derivative is already there, thanks to the (iγ^μ∂_μ + m) in front. At equal times, Δ(x - y) is zero, so all the terms in the definition of S drop out except for the time derivative. You get

{ψ(x), ψ†(y)}|(at x⁰ = y⁰) = - γ⁰∂₀Δ(x - y) = γ⁰ δ³(x - y)

(To leave no doubt about the notation, ψ† is what appears in the bilinear covariants, e.g. j^μ = eψ†γ^μψ.) And so, if you'd rather, using ψ† = ψ*γ⁰,

{ψ(x), ψ*(y)}|(at x⁰ = y⁰) = δ³(x - y)

Maybe this is what Tong had in mind when he wrote (5.14).

 

[Einj]

Ok, understood. Sorry for the misunderstanding.

 

[nikkoo]

Ok, understood
http://www.centplay.com/affiliate/id_139/