As we known, if there is time derivative interaction in $\mathcal L_\mathrm{int}$, then $\mathcal{H}_\mathrm{int}\neq -\mathcal{L}_\mathrm{int}$. For example, Scalar QED,
$$
\begin{aligned}
\mathcal{L}_\mathrm{int}&= -ie \phi^\dagger(\partial_\mu \phi) A^\mu+ie(\partial_\mu \phi^\dagger) \phi A^\mu +e^2\phi^\dagger \phi A_\mu A^\mu, \\
\mathcal{H}_\mathrm{int}&=-\mathcal{L}_\mathrm{int} -e^2 \phi^\dagger\phi (A^0)^2.
\end{aligned}
$$
There is the last term breaking Lorentz invariance.
Derivation:
\begin{eqnarray}
\mathcal{L}&=&(\partial_\mu+i e A_\mu)\phi (\partial^\mu-i e A^\mu)\phi^\dagger-m^2\phi^\dagger \phi\\
&=&\mathcal{L}_0^\mathrm{KG}+\mathcal{L}_\mathrm{int},
\end{eqnarray}
where
$$\mathcal{L}_0^\mathrm{KG}=\partial_\mu\phi \partial^\mu\phi^\dagger-m^2\phi^\dagger \phi,$$
$$\mathcal{L}_\mathrm{int}= -ie \phi^\dagger(\partial_\mu \phi) A^\mu+ie(\partial_\mu \phi^\dagger) \phi A^\mu +e^2\phi^\dagger \phi A_\mu A^\mu, $$
$$\pi=\frac{\partial \mathcal{ L}}{\partial(\partial_0 \phi)}=\partial^0\phi^\dagger-i e A^0 \phi^\dagger,$$
$$\pi^\dagger=\frac{\partial \mathcal{ L}}{\partial(\partial_0 \phi^\dagger)}=\partial^0\phi+i e A^0 \phi, $$
\begin{eqnarray}
\mathcal{H}&=&\pi \dot\phi+\pi^\dagger \dot\phi^\dagger-\mathcal{L} \\
&=&\pi \dot\phi+\pi^\dagger \dot\phi^\dagger-(\dot\phi^\dagger\dot\phi-\nabla\phi^\dagger \cdot\nabla\phi-m^2 \phi^\dagger\phi)-\mathcal{L}_\mathrm{int} \\
&=&\pi(\pi^\dagger-ieA^0\phi)+\pi^\dagger (\pi +ieA^0\phi^\dagger)-((\pi^\dagger-ieA^0\phi)(\pi +ieA^0\phi^\dagger) \\ &&-\nabla\phi^\dagger \cdot\nabla\phi-m^2 \phi^\dagger\phi)-\mathcal{L}_\mathrm{int}\\
&=&(\pi^\dagger \pi + \nabla\phi^\dagger \cdot\nabla\phi+m^2 \phi^\dagger\phi)-\mathcal{L}_\mathrm{int} -e^2 \phi^\dagger\phi (A^0)^2 \\
&=&\mathcal{H}_0^\mathrm{KG}+\mathcal{H}_\mathrm{int}.
\end{eqnarray}
My questions:
-
The Feynman Rules for Scalar QED is here. But we see there is an extra term in interaction Hamitonian $ -e^2 \phi^\dagger\phi (A^0)^2$, according to Wick's theorem, it should have some contribution to Feynman Rule which does not occur in this textbook. I've computed this vertex and I find it's nonzero. Why there is no Feynman rules for such Lorentz breaking term?
-
As we known, for path integral quantization, the coordinate space path integral:
$$Z_1= \int D q\ \exp\left(\int dt\ L(q,\dot q) \right).$$
And phase space path integral:
$$Z_2= \int D p\, D q\ \exp\left(\int dt\ p\dot q -H(p,q) \right).$$
Only for this type Lagrangian $L=\dot q^2-V(p)$, then $Z_1=Z_2$. (The Feynman Rules for Scalar QED in textbook is same as what is derived by coordinate space path integral. )
I consider the 2nd method of path integral quantization is always equivalent to canonical quantization. So for Scalar QED, are these two kinds of path integral quantization same? How to prove? - For non-abelian gauge theory, there is derivative interaction even in gauge field itself. It seems that all textbooks use $Z_1$ to get the Feynman rules. Are these two kinds of path integral quantization same in non-abelian gauge field? If not same, why we choose the coordinate space path integral? It's the axiom because it coincides with experiment?
Best Answer
Some general remarks:
In the operator formalism, the non-covariant extra term in the Hamiltonian is cancelled by a non-covariant term coming from the naïve time ordering symbol: $$ \mathrm T\sim\mathrm T_\mathrm{cov}-e^2\phi^2A^2_0 $$
You can find the details in ref.1, section 6-1-4.
On the other hand, the case of the path-integral formalism is covered by item 2 below.
The formal equivalence $Z_1=Z_2$ can be proven for any Hamiltonian of the form $$ H\sim A^{ij}\pi_i\pi_j+B^i(\phi)\pi_i+C(\phi) $$ of which your Hamiltonian is an example of. For the proof and relevant discussion, see ref.2, Vol.1., section 9.3.
For the discussion of the path-integral quantisation of non-abelian gauge theories, see ref.2, Vol.2, chapters 15.4 -- 15.8. Ref.1, chapter 12-2 is also worth a read. In short, "$Z_1=Z_2$" up to the subtleties introduced by gauge-invariance.
References
[1] Itzykson & Zuber, Quantum field theory.
[2] Weinberg, Quantum theory of fields.