Old fashioned perturbation theory

Last updated Aug 5, 2023

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Old-fashioned pertubation theory (OFPT) is the pertubation theory of QM applied to a theory with second quantization. It’s not lorentz invariant (?) but still gives the same results as Feynman methods.

Like in normal QM perturbation theory we split the hamiltonian $$H = H_0 + V,$$ where $H_0$ solutions are known and $V$ is a small perturbation.

• Difference to QM: $H_0$ has continous spectrum

We have a free state which is known $$H_0\ket{\phi} = E\ket{\phi}.$$ We want to find an eigenstate $\ket{\psi}$ of the full hamiltonian $H$ with the same energy $E$ $$H\ket{\psi} = E\ket{\psi}.$$ This leads to $$\ket{\psi}=\ket{\phi}+\frac{1}{E-H_0}V\ket{\psi}.$$ The inverse of $E-H_0$ is not defined. Since $E$ is an eigenvalue of $H_0$ we have $\det(E-H_0)=0$. This is singularity regulated by adding $i\epsilon$ and leads to the Lippmann-Schwinger equation

Definition: Lippmann-Schwinger equation For a hamiltonian $H=H_0+V$ the eigenstates $\ket{\psi}$ are given in terms of the eigenstates $\ket{\phi}$ of the free hamiltonian $H_0$ by the solution of the Lippmann-Schwinger equation $$\ket{\psi} = \ket{\phi} + \frac{1}{E-H_0+i\epsilon}V\ket{\psi}$$

To solve for $\ket{\psi}$ on assumes that $V\ket{\psi}$ can be expressed in terms of $\ket{\phi}$ with some operator $T$ $V\ket{\psi}=T\ket{\phi}$. $T$ is the transfer matrix.

This leads to an operator equation $$T = V + V\frac{1}{E-H_0}T,$$ which can be solved perturbatatively in $V$.

If we have free initial and final states $\ket{\phi_i}$ and $\ket{\phi_f}$ the transition rate is the matrix element $\braket{\phi_f|T|\phi_i}$. I guess this is because $T\ket{\phi}$ is the interaction $V\ket{\psi}$ of the full theory, which describes what happens in the scattering.

The perturbative solution is some series $$T_{fi} = V_{fi} + \sum_j V_{fj}\Pi_{LS}(j)V_{ji} + \sum_{jk} V_{fj}\Pi_{LS}(j)V_{jk}\Pi_{LS}(k)V_{ki}+…,$$ with the matrix elements $T_{fi} = \braket{\phi_f|T|\phi_i}$, $V_{jk} = \braket{\phi_j|V|\phi_k}$ and the Lippmann-schwinger kernel $\Pi_{LS}(j)=\frac{1}{E-E_j}$. We sum over $j$, which represents all possible intermediate states.

The different terms correspond to the different orders of the Born approximation.

Example: Coulombs law

We Calculate $e-e$ scattering: $e(\vec{p_1})e(\vec{p_2}) \to e(\vec{p_3})e(\vec{p_4})$

• Initial and final states have the same energy $E$ due to conservation (as required by Lippmann-Schwinger)
• Intermediate states can have different energy $E_n$

Write out the states: $$\ket{i}=\ket{\psi^1_e\psi^2_e},\quad \bra{f}=\bra{\psi^3_e\psi^4_e}$$

Interaction term (photon approximated as scalar) $$V = \frac{1}{2}e \int d^3x \psi_e(x)\phi(x)\psi_e(x)$$

Transfer Matrix $T_{fi}$ is calculated according to the expansion formula term by term:

1. $V_{fi}$: single insertion of the interaction will leave a state with a photon therefore $$\braket{f|V|i}=0$$
2. $\sum_n V_{fn}\frac{1}{E-E_n}V_{ni}$: Two possible intermediate states:
   1. Retarded Case: $e_1$ emits photon first (has momentum $\vec{p}_3$ afterwards): Intermediate state: $\ket{n}=\ket{\psi^3\phi^\gamma\psi^2}$ Interaction: $V^{(R)}_{ni}= \braket{\psi^3\phi^\gamma|V|\psi^1}$
   2. Advanced Case: $e_2$ emits photon first (has momentum $\vec{p}_4$ afterwards): Intermediate state: $\ket{n}=\ket{\psi^4\phi^\gamma\psi^2}$ Interaction: $V^{(A)}_{ni}= \braket{\psi^4\phi^\gamma|V|\psi^2}$

Calculate the matrix elements by inserting $V$ and the field operators $$ \begin{align*} &V^{R}_{ni}=\frac{e}{2}\int d^3x \int \frac{d^3p_a}{(2\pi)^3} \frac{d^3p_\gamma}{(2\pi)^3} \frac{d^3p_b}{(2\pi)^3} \frac{1}{\sqrt{2w_{p_a}}}\frac{1}{\sqrt{2w_{p_\gamma}}}\frac{1}{\sqrt{2w_{p_b}}} \\ &\braket{\psi^3\phi^\gamma| (a_{p_a}e^{i\vec{p_a}\vec{x}} + a^\dagger_{p_a}e^{-i\vec{p_a}\vec{x}})(a_{p_\gamma}e^{i\vec{p_\gamma}\vec{x}} + a^\dagger_{p_\gamma}e^{-i\vec{p_\gamma}\vec{x}})(a_{p_b}e^{i\vec{p_b}\vec{x}} + a^\dagger_{p_b}e^{-i\vec{p_b}\vec{x}}) |\psi^1} \end{align*} $$

Collecting the nonzero terms we find one term which contributes with $\braket{\psi^3\phi^\gamma|a_{p_a}a^\dagger_{p_\phi}a^\dagger_{p_b}|\psi^1}$ and one with $\braket{\psi^3\phi^\gamma|a^\dagger_{p_a}a^\dagger_{p_\phi}a_{p_b}|\psi^1}$, so that the electron creation and annihalation operators switch the electron $\psi^3$ and $\psi^1$ they create. We therefore get a factor $2$. All other terms don’t contribute since the creation and annihalation operators lead to different states on the left and the right.

With $\braket{\phi_p|\phi(x)|0}=e^{-i\vec{p}\vec{x}}$ and $\braket{0|\phi(x)|\phi_p}=e^{i\vec{p}\vec{x}}$ we get $$V^{(R)}=e(2\pi)^2\delta^3(\vec{p}1-\vec{p}3-\vec{p}\gamma),$$ with the expression for the delta distribution $\delta^3(\vec{x}-\vec{a}) = \frac{1}{2\pi}\int^{\infty}{-\infty}e^{i\vec{p}(\vec{x}-\vec{a})}d^3p$.

The advanced interaction term $V^{(A)}{ni}$ can be calculated similarly. We can plug both cases into the expansion formular for $T{fi}$ and calculate $E_n$ from kinematics. With the two different intermediate energies $E^{(R)}n = E_3 + E_2 + E\gamma$ and $E^{(A)}n = E_4 + E_1 + E\gamma$ we get the full $T$ matrix $$T_{fi} = \frac{2e^2E_\gamma}{(\Delta E)^2 - (E_\gamma)^2}$$ with $\Delta E = E_1 - E_3 = E_4 - E_2$.

Note that in OFPT:

• All states are on-shell all the time
• 3-momentum is conserved at vertices
• Energy is not conserved at vertices

In some naive quantum filed theoretical calculations divergent integrals occur, which are suspicious since they are unphysical. These divergences can eventually be fixed by Renormalization.

A simple theory with a photon in second quantization $$H = \int \frac{d^3k}{(2\pi)^3} w_k \left(a^\dagger_k a_k + \frac{1}{2}\right)$$ leads to the energy of the vakkum $E_0$ $$E_0 = \braket{0|H|0} = \frac{1}{2}\int \frac{d^3k}{(2\pi)^3}|\vec{k}| = \infty,$$ which diverges. This divergence can be disregarded since the zero point energy is not physical in a sense that it cant be measured. Divergences only have to cancel in the calculation of final physical observables but not in intermediate results.

In quantum field theory an electron can emmit and annihalate a photon resulting in the self energy of the electron. At second order only a single photon is created and annihilated again, resulting in a loop.

The loop integral in OFPT diverges.