Old fashioned perturbation theory

Last updated Dec 29, 2024

• Onepass

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.

We have a free state which is known and has energy $E$ $$H_0\ket{\phi} = E\ket{\phi}.$$ We want to find an eigenstate $\ket{\psi}$ of the full hamiltonian $H$ with the same energy $E$ (because energy is conserved ?) $$H\ket{\psi} = E\ket{\psi}.$$ 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 $$\ket{\psi} = \ket{\phi} + \frac{1}{E-H_0+i\epsilon}V\ket{\psi}$$ The full wavefunction of the interacting theory $\ket{\psi}$ is given by the free wavefunction $\ket{\phi}$ and a scattering term.

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

Definition: Transfer matrix

The transfer matrix is an operator connecting the effect of the potential on the interacting eigenstates $V\ket{\psi}$ with the free eigenstates $\ket{\phi}$ $$V\ket{\psi}=T\ket{\phi}$$

This leads to an operator equation for $T$, which is the operator of interest (if we know it we know the effects of the scattering)

$$T = V + V\frac{1}{E-H_0}T,$$

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 $\braket{\phi_f|T|\phi_i}$. This is because $T\ket{\phi}$ is the interaction $V\ket{\psi}$ of the full theory, which describes what happens in the scattering.

Old-fashioned perturbation theory

The perturbative solution for $T$ 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 perturbative soultion $T_{fi}$ can be represented in terms of Feynman diagrams. 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 and electrons alls approximated as scalar) $$V = \frac{1}{2}e \int d^3x \psi_e(x)\phi(x)\psi_e(x)$$ This corresponds to vertices of e.g. one electron turning into one photon and a new electron with different momentum. 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 (fourier rep. in terms of creation and annihilation operators $a$ for the scalar field) $$ \begin{align*} &V^{R}_{ni}=\frac{e}{2}\int d^3x \int \frac{d^3p_a}{(2\pi)^3} \frac{d^3p_\gamma}{(2\pi)^3} \frac{d^3p_b}{(2\pi)^3} \frac{1}{\sqrt{2w_{p_a}}}\frac{1}{\sqrt{2w_{p_\gamma}}}\frac{1}{\sqrt{2w_{p_b}}} \\ &\braket{\psi^3\phi^\gamma| (a_{p_a}e^{i\vec{p_a}\vec{x}} + a^\dagger_{p_a}e^{-i\vec{p_a}\vec{x}})(a_{p_\gamma}e^{i\vec{p_\gamma}\vec{x}} + a^\dagger_{p_\gamma}e^{-i\vec{p_\gamma}\vec{x}})(a_{p_b}e^{i\vec{p_b}\vec{x}} + a^\dagger_{p_b}e^{-i\vec{p_b}\vec{x}}) |\psi^1} \end{align*} $$

Collecting the nonzero terms we find one term which contributes with $\braket{\psi^3\phi^\gamma|a_{p_a}a^\dagger_{p_\gamma}a^\dagger_{p_b}|\psi^1}$ and one with $\braket{\psi^3\phi^\gamma|a^\dagger_{p_a}a^\dagger_{p_\gamma}a_{p_b}|\psi^1}$, 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. You could calculate this with create and annihilation operators, but it is simpler to use the field operator representation and the fact that $$\braket{\phi^\gamma|\phi(x)|0}=e^{-\vec{p}_\gamma\cdot\vec{x}}$$ We get $$V^{(R)}=\frac{e}{2}\int d^3x\braket{\psi^3\phi^\gamma|\psi(x)\phi(x)\psi(x)|\psi^1} = e\int d^3x=e(2\pi)^2\delta^3(\vec{p}_1-\vec{p}_3-\vec{p}_\gamma),$$ with the expression for the delta distribution $\delta^3(\vec{x}-\vec{a}) = \frac{1}{2\pi}\int^{\infty}_{-\infty}e^{i\vec{p}(\vec{x}-\vec{a})}d^3p$. We used that the factor $2$ arises again because we can pair each of the electron field operators $\phi(x)$ with each electron in the state. Then we use the identity for the field operators given above. This is possible because the states factorize into single states with different momentum.

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 Old-fashioned perturbation theory (OFPT):

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

In the future we will use Feynman diagrams, which trade the on-shell’ness agains energy conservation at vertices (and thus four momentum conservation).

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 vacuum $E_0$ $$E_0 = \braket{0|H|0} = \frac{1}{2}\int \frac{d^3k}{(2\pi)^3}|\vec{k}| = \infty,$$ 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. Things can be fixed by Renormalization