Quantum Field Theory (III)
Feynman Diagrams
あの日見た1PIの伝播関数は僕たちはまだ知らない
Foundations of Field Theory
Heisenberg Picture and Interaction Picture
- Heisenberg picture:
\[
\phi(t,\vb*{x}) = e^{iH(t-t_0)} \phi(t_0,\vb*{x}) e^{-iH(t-t_0)}.
\]
- Interaction picture:
\[
\phi_I(t,\vb*{x}) = e^{iH_0(t-t_0)} \phi(t_0, \vb*{x}) e^{-iH_0(t-t_0)}.
\]
Scalar Field
- Expansion (Schrödinger picture):
\[
\phi(t_0, \vb*{x}) = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \qty(a_{\vb*{p}} e^{i\vb*{p}\cdot \vb*{x}} + a^\dagger_{\vb*{p}} e^{-i\vb*{p}\cdot \vb*{x}}).
\]
- Expansion (Interaction picture):
\[
\phi_I(t,\vb*{x}) = \left. \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \qty(a_{\vb*{p}} e^{-ip\cdot x} + a^\dagger_{\vb*{p}} e^{ip\cdot x}) \right\vert_{x^0 = t-t_0}.
\]
Dirac Field
- Expansion:
\[\begin{align*}
\psi(x) &= \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_s \qty(a^s_{\vb*{p}} u^s(p) e^{-ip\cdot x} + b^{s\dagger}_{\vb*{p}} v^s(p) e^{ip\cdot x}), \\
\overline{\psi}(x) &= \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_s \qty(b^s_{\vb*{p}} \overline{v}^s(p) e^{-ip\cdot x} + a^{s\dagger}_{\vb*{p}} \overline{u}^s(p) e^{ip\cdot x}).
\end{align*}{}\]
Electromagnetic Field
- Free electromagnetic field (under the Lorenz gauge):
\[
\partial^2 A_\mu = 0.
\]
- Each component of \(A\) obeys the Klein-Gordon equation with \(m=0\).
- Expansion:
\[
A_\mu(x) = \int \frac{\dd{^3} p}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} \sum_{r=0}^3 \qty(a_{\vb*{p}}^r\epsilon^r_\mu{(p)}e^{-ip\cdot x} + a^{r\dagger}_{\vb*{p}} \epsilon^{r*}_{\mu}(p) e^{ip\cdot x}).
\]
- In the integration we have \(p^2=0\), and
- \(\epsilon_\mu(p)\) is a four-vector.
Propagator
- \(\ket{\Omega}\) is usually different from \(\ket{0}\).
- Note the extra minus sign for Dirac fields.
- Feynman propagator
- Scalar field (definition):
\[
D_F = \bra{0}T\phi(x)\phi(y) \ket{0}.
\]
- Scalar field (in momentum space):
\[
D_F(x-y) = \int \frac{\dd{^4 p}}{(2\pi)^4} \frac{i}{p^2 - m^2 + i\epsilon} e^{-ip\cdot (x-y)}.
\]
- Dirac field (definition):
\[
S_F(x-y) = \bra{0}T\psi(x)\overline{\psi}(y)\ket{0}.
\]
- Dirac field (in momentum space):
\[
S_F(x-y) = \int \frac{\dd{^4 p}}{(2\pi)^4} \frac{i(\unicode{x2215}\kern-.5em {p} + m)}{p^2 - m^2 + i\epsilon} e^{-ip\cdot (x-y)}.
\]
- Two-point correlation function, or two-point Green's function:
\[
\bra{\Omega}T\phi(x)\phi(y)\ket{\Omega},
\]
where \(\ket{\Omega}\) is the ground state of the interacting theory.
- Källén-Lehmann spectral representation: analytic structure of two-point correlation,
\[
\color{red} \bra{\Omega} T\phi(x)\phi(y) \ket{\Omega} = \int_0^\infty \frac{\dd{M^2}}{2\pi} \rho(M^2) D_F(x-y, M^2).
\]
- Spectral density:
\[
\color{red} \rho(M^2) = \sum_\lambda (2\pi) \delta(M^2 - m_\lambda^2) \abs{\bra{\Omega} \phi(0) \ket{\lambda_0}}^2.
\]
- \[
\rho(M^2) = 2\pi\delta(M^2 - m^2) \cdot Z + (\text{nothing else until } M^2 \gtrapprox (2m)^2).
\]
- The leading contribution is from the single-particle state.
- There are poles from single-particle bounded states near \(M^2 = (2m)^2\).
- \(Z\) is referred to as the field-strength renormalization.
- Physical mass: \(m\) is the exact mass of a single particle, i.e. the exact energy eigenvalue at rest, different from the value of the mass parameter in the Lagrangian.
- Only the physical mass is directly observable.
- Bare mass: \(m_0\) in the Lagrangian.
- \(\ket{\lambda_{0}}\): an eigenstate of \(H\) annihilated by \(\vb*{P}\).
- Källén-Lehmann spectral representation for Scalar fields:
\[
\int \dd{^4 x} e^{ip\cdot x} \bra{\Omega} T\phi(x) \phi(0) \ket{\Omega} = \frac{iZ}{p^2 - m^2 + i\epsilon} + \int_{\sim 4m^2}^\infty \frac{\dd{M^2}}{2\pi} \rho(M^2) \frac{i}{p^2 - M^2 + i\epsilon}.
\]
- \(Z_2\): the probability for the quantum field to create or annihilate an exact one-particle eigenstate of \(H\),
\[\sqrt{Z_2} = \bra{\Omega} \phi(0) \ket{p}.\]
- Källén-Lehmann spectral representation for Dirac fields:
\[
\int \dd{^4 x} e^{ip\cdot x} \bra{\Omega} T\psi(x)\overline{\psi}(0) \ket{\Omega} = \frac{i Z_2 (\unicode{x2215}\kern-.5em {p} + m)}{p^2 - m^2 + i\epsilon} + \cdots.
\]
- \(Z_2\): the probability for the quantum field to create or annihilate an exact one-particle eigenstate of \(H\),
\[\sqrt{Z_2} u^s(p) = \bra{\Omega}\psi(0)\ket{p,s}.\]
Path Integral
- Scalar field:
\begin{align*}
\color{orange} Z_0[J] &\color{orange}= \int \mathcal{D}\varphi\, e^{i\int \dd{^4 x} [\mathcal{L_0} + J\varphi]} \\
&\color{orange}= \exp\qty[\frac{1}{2}\iint \dd{^4 x} \dd{^4 x'} J(x) S_F(x-x') J(x')].
\end{align*}
- \begin{align*}
\color{darkcyan} Z[J] &\color{darkcyan}= e^{i\int \dd{^4} \mathcal{L}_1(\frac{1}{i}\frac{\delta}{\delta J(x)})} Z_0[J].
\end{align*}
- \(\mathcal{L}_1\) is a perturbation,
\[
\mathcal{L} = \mathcal{L}_0 + \mathcal{L}_1.
\]
We solve the problem where
- the incident particles occupy a volume of length \(l_B\) with number density \(\rho_B\) and velocity (uniformly) \(v\);
- the target particles occupy a volume of length \(l_A\) with number density \(\rho_A\); and
- the area common to the two branches is \(A\).
We define a few quantities below.
- Cross section:
\[
\color{orange}\sigma = \frac{\text{Number of scattering events}}{\rho_A l_A \rho_B l_B A},
\]
- Equivalently,
\begin{align*}
&\text{Number of events} \\
&= \sigma l_A l_B \int \dd{^2 x} \rho_A(x) \rho_B(x) \\
&= \frac{\sigma N_A N_B}{A}.
\end{align*}
- Differential cross section:
\[
\color{orange}\frac{\dd{\sigma}}{\dd{^3 p_1}\cdots \dd{^3 p_n}}
\]
- \(p_1,\cdots,p_n\) denote the momenta of final states;
- integration over any small \(\dd{^3 p_1}\cdots \dd{^3 p_n}\), gives the cross section for scattering into the final-state momentum space.
- Decay rate:
\[
\color{orange}\Gamma = \frac{\text{Number of decays per unit time}}{\text{Number of } A \text{ particles present}}.
\]
- Breit-Wigner formula: near the reasonance energy, the scattering amplitude is given by
\begin{align*}
& f(E) \propto \frac{1}{E - E_0 + i\Gamma/2}\\
&\xlongequal{\text{relativistic}} \frac{1}{p^2 - m^2 + im\Gamma} \\
&= \frac{1}{2E_{\vb*{p}}(p^0 - E_{\vb*{p}} + i(m/E_{\vb*{p}})\Gamma/2)},
\end{align*}
- and the cross section by
\[
\sigma \propto \frac{1}{(E-E_0)^2 + \Gamma^2/4}.
\]
Cross Section and S-Matrix
We use Heisenberg picture in this subsection.
We reduce the cross section to the \(\mathcal{M}\) matrix.
- \(B\) denotes the incident particles.
- Normalization: \[\ket{\vb*{k}} = \sqrt{2E_{\vb*{k}}}a_{\vb*{k}}^\dagger\ket{0}.\]
- \(\bra{\phi}\ket{\phi} = 1\) if
\[
\int \frac{\dd{^3 k}}{(2\pi)^3} \abs{\phi(\vb*{k})}^2 = 1.
\]
- Expansion of in states: plane wave to in state,
\[
\ket{\phi_A \phi_B}_{\text{in}} = \int \frac{\dd{^3 k_A}}{(2\pi)^3} \int \frac{\dd{^3 k_B}}{(2\pi)^3} \frac{\phi_A(\vb*{k}_A) \phi_B(\vb*{k}_B) e^{-i\vb*{b}\cdot \vb*{k}_B}}{\sqrt{(2E_A) (2E_B)}}\ket{\vb*{k}_A \vb*{k}_B}_{\text{in}}.
\]
- Constructed in the far past.
- \(\phi_{B}(\vb*{k}_B)\) is the state of the incident particle, with \(\vb*{b} = 0\).
- Expansion of out states: plane wave to out state,
\[
_{\text{out}}\bra{\phi_1\phi_2\cdots} = \qty(\prod_f \int \frac{\dd{^3 p_f}}{(2\pi)^3} \frac{\phi_f (\vb*{p}_f)}{\sqrt{2E_f}}) {_{\text{out}}\bra{\vb*{p}_1 \vb*{p}_2 \cdots}}.
\]
- Constructed in the far future.
- \(S\)-matrix:
\begin{align*}
&\color{orange}\phantom{{}={}} _{\text{out}}\bra{\vb*{p}_1\vb*{p}_2\cdots}\ket{\vb*{k}_A \vb*{k}_B}_{\text{in}} \\
&\color{orange}= \bra{\vb*{p}_1\vb*{p}_2\cdots} e^{-iH(2T)} \ket{\vb*{k}_A \vb*{k}_B} \\
&\color{orange}= \bra{\vb*{p}_1\vb*{p}_2\cdots} S \ket{\vb*{k}_A \vb*{k}_B}.
\end{align*}
- \(T\)-matrix:
\[
\color{orange} S = \mathbb{1} + iT
\]
- Invariant matrix element \(\mathcal{M}\): brakets between in and out plane waves,
\[
\color{orange} \bra{\vb*{p}_1\vb*{p}_2\cdots} iT \ket{\vb*{k}_A \vb*{k}_B} = (2\pi)^4 \delta^{(4)}\qty(k_A + k_B - \sum p_f)\cdot i\mathcal{M}\qty(k_A,k_B \rightarrow p_f),
\]
- The \(\delta^{(4)}\) is imposed by the conservation of momentum.
- The probability of getting a certain out state (depending on the impact parameter): brakets of in and out states to probability,
\[
\mathcal{P}(AB\rightarrow 1,2,\cdots,n) = \qty(\prod_f \frac{\dd{^3 p_f}}{(2\pi)^3}\frac{1}{2E_f})\abs{{_{\text{out}}\bra{\vb*{p}_1\cdots \vb*{p}_n}\ket{\phi_A \phi_B}_{\text{in}} }}^2.
\]
- Cross section: probability to cross section, summing over all impact parameters,
\[
\sigma = \int \dd{^2 b} \mathcal{P}(\vb*{b}).
\]
- Putting together:
\[\begin{align*}
\dd{\sigma} &= \qty(\prod_f \frac{\dd{^3 p_f}}{(2\pi)^3}\frac{1}{2E_f}) \frac{\abs{\mathcal{M}(p_A,p_B\rightarrow \qty{p_f})}^2}{2E_A 2E_B \abs{v_A - v_B}} \int \frac{\dd{^3 k_A}}{(2\pi)^3} \int \frac{\dd{^3 p_B}}{(2\pi)^3} \\
&{\phantom{{}={}}} \times \abs{\phi_A \qty(\vb*{k}_A)}^2 \abs{\phi_B\qty(\vb*{k}_B)}^2 (2\pi)^4 \delta^{(4)}\qty(k_A + k_B - \sum p_f).
\end{align*}{}\]
- Conclusion: \(\mathcal{M}\) matrix to cross section,
\[\begin{align*}
\color{red}\dd{\sigma} &= \color{red}\frac{1}{2E_A 2E_B \abs{v_A - v_B}} \qty(\prod_f \frac{\dd{^3 p_f}}{(2\pi)^3}\frac{1}{2E_f}) \\
&{\phantom{{}={}}} \color{red}\times \abs{\mathcal{M}(p_A,p_B\rightarrow \qty{p_f})}^2 (2\pi)^4 \delta^{(4)}\qty(p_A+p_B - \sum p_f).
\end{align*}{}\]
- We demanded the wave packet be smooth (not so sharp).
When doing integration to get the total cross section or decay rate for a final state of \(n\) particles, one should either restrict the integration to inequivalent configurations or divide the result by \(n!\).
- Two-body scattering:
- In the center-of-mass frame, \(\vb*{p_1} = -\vb*{p}_2\).
- \(E_{\mathrm{cm}}\) denote the total energy.
- Cross section of two-particles:
\[
\color{red} \qty(\dv{\sigma}{\Omega})_{\text{CM}} = \frac{1}{2E_A 2E_B \abs{v_A - v_B}} \frac{\abs{\vb*{p}_1}}{(2\pi)^2 4E_{\text{cm}}}\abs{\mathcal{M}(p_A,p_B \rightarrow p_1,p_2)}^2.
\]
- Identical mass:
\[
\color{red} \qty(\dv{\sigma}{\Omega})_{\text{CM}} = \frac{\abs{\mathcal{M}}^2}{64\pi^2 E^2_{\text{cm}}}.
\]
- Decay rate:
\[
\color{red} \dd{\Gamma} = \frac{1}{2m_A}\qty(\prod_f \frac{\dd{^3 p_f}}{(2\pi)^3} \frac{1}{2E_f}) \abs{\mathcal{M}(m_A \rightarrow \qty{p_f})}^2 (2\pi)^4 \delta^{(4)}\qty(p_A - \sum p_f).
\]
- \(M^2(p^2)\) denotes the sum of all 1PI insertions into the \(\phi^4\) propagator. Let \(m\) be defined by
\[m^2 - m_0^2 - \Re M^2(m^2) = 0.\]
- The two-point correlation is found to be
\[\frac{i Z}{p^2 - m^2 - iZ \Im M^2(p^2)}.\]
- Decay rate identified to be
\[\Gamma = -\frac{Z}{m} \Im M^2(m^2).\]
- cf. Breit-Wigner formula
\[\sigma \propto \abs{\frac{1}{p^2 - m^2 + im\Gamma}}^2.\]
Born Approximation
Born approximation:
\[
\bra{p'}iT\ket{p} = -i\tilde{V}\qty(\vb*{q})(2\pi)\delta \qty(E_{\vb*{p}'} - E_{\vb*{p}})
\]
where \(\vb*{q} = \vb*{p}' - \vb*{p}\).
- Assuming nonrelativistic normalization:
\[
\bra{p'}\ket{p} = \delta(p'-p).
\]
LSZ Reduction Formula
\[\color{red}
\begin{align*}
& \prod_1^n \int \dd{^4 x_i} e^{ip_i\cdot x_i} \prod_1^m \int \dd{^4 y_j} e^{-ik_i \cdot y_j} \bra{\Omega} T\qty{\phi(x_1) \cdots \phi(x_n) \phi(y_1) \cdots \phi(y_m)} \ket{\Omega} \\
& \underset{\substack{p_i^0 \rightarrow E_{\vb*{p}_i} \\ k_j^0 \rightarrow E_{\vb*{p}_j}}}{\sim} \qty( \prod_1^n \frac{\sqrt{Z} i}{p_i^2 - m^2 + i\epsilon} )\qty( \prod_1^m \frac{\sqrt{Z} i}{k_j^2 - m^2 + i\epsilon} ) \bra{\vb*{p}_1 \cdots \vb*{p}_n} S \ket{\vb*{k_1} \cdots \vb*{k}_m}.
\end{align*}
\]
The Optical Theorem
- From the unitarity of \(S\):
\[\begin{align*}
&\phantom{{}={}}-i[\mathcal{M}(k_1k_2 \rightarrow p_1p_2) - \mathcal{M}^*(p_1p_2 \rightarrow k_1k_2)] \\
&= \sum_n \qty(\prod_{i=1}^n \int \frac{\dd{^3 q_i}}{(2\pi)^3} \frac{1}{2E_i}) \mathcal{M}^*(p_1p_2 \rightarrow \qty{q_i}) \mathcal{M}(k_1k_2 \rightarrow \qty{q_i}) \times (2\pi)^4 \delta^{(4)}(k_1 + k_2 - \sum_i q_i).
\end{align*}{}\]
- Abbreviated as
\[-i[\mathcal{M}(a\rightarrow b) - \mathcal{M}^*(b\rightarrow a)] = \sum_f \int \dd{\Pi_f} \mathcal{M}^*(b\rightarrow f) \mathcal{M}(a\rightarrow f).\]
- With the factors in the formula for cross section, we obtain the optical theorem,
\[\color{red} \Im \mathcal{M}(k_1, k_2 \rightarrow k_1, k_2) = 2 E_{\mathrm{cm}} p_{\mathrm{cm}} \sigma_{\mathrm{tot}}(k_1,k_2 \rightarrow \mathrm{anything}).\]
- Feynman diagrams yields imaginary part only when the virtual particles in the diagram go on-shell.
- Discontinuity: above the energy threshold,
\[\operatorname{Disc} \mathcal{M}(s) = 2i \Im \mathcal{M}(s+i\epsilon).\]
Perturbation Expansion of Correlation Functions
The perturbed Hamiltonian has the form
\[
H = H_0 + H_{\text{int}} = H_0 + H_{\text{Klein-Gordon}}.
\]
In the \(\phi^4\) theory
\[
H = H_0 + H_{\text{int}} = H_{\text{Klein-Gordon}} + \int \dd{^3 x} \frac{\lambda}{4!} \phi^4(x).
\]
Obtaining the Field Operator%%% Obtaining \(\phi\)
- \(\phi\) under Schrödinger picture:
\[
\color{darkcyan}\phi(t,\vb*{x}) = U^\dagger(t,t_0)\phi_I(t,\vb*{x})U(t,t_0)
\]
- Time evolution operator under the interaction picture:
\[
\color{orange}U(t,t_0) = e^{iH_0(t-t_0)}e^{-iH(t-t_0)}
\]
- Equation of motion:
\[
i\pdv{}{t}U(t,t_0) = H_I(t) U(t,t_0).
\]
- Interaction Hamiltonian:
\[
\color{orange}H_I(t) = e^{iH_0(t-t_0)} (H_{\mathrm{int}}) e^{-iH_0 (t-t_0)}.
\]
- Dyson series:
\begin{align*}
U(t,t_0) &= \mathbb{1} + (-i) \int_{t_0}^t \dd{t_1} H_I(t_1) + (-i)^2 \int_{t_0}^{t} \dd{t_1} \int_{t_0}^{t_1} \dd{t_2} H_I(t_1) H_I(t_2) + (-i)^3 \int_{t_0}^{t} \dd{t_1} \int_{t_0}^{t_1} \dd{t_2} \int_{t_0}^{t_2} \dd{t_3} H_I(t_1) H_I(t_2) H_I(t_3) + \dots \\
&= 1 + (-i) \int_{t_0}^t \dd{t_1} H_I(t_1) + \frac{(-i)^2}{2!} \int_{t_0}^t \dd{t_1} \dd{t_2} T\qty{H_I(t_1)H_I(t_2)} + \cdots \\
\color{red}U(t,t') &\color{red}= T\qty{\exp\qty[-i\int_{t'}^t\dd{t''}H_I(t'')]}.
\end{align*}
- \(U(t,t')\) is only defined for \(t>t'\).
- For \(t_1 \ge t_2 \ge t_3\),
- \[
U(t_1,t_2) U(t_2,t_3) = U(t_1,t_3),
\]
- \[
U(t_1,t_3)[U(t_2,t_3)]^\dagger = U(t_1,t_2).
\]
Obtaining the Vacuum State%%% Obtaining \(\ket{\Omega}{}\)
- We follow this recipe:
- \[
\ket{\Omega} = e^{-iHT} \ket{0}.
\]
- \[
\color{red} T \rightarrow \infty(1-i\epsilon).
\]
- \[E_0 = \bra{\Omega}H\ket{\Omega}.\]
- \[H_0\ket{0} = 0.\]
- \[
U(t,t') = e^{iH_0(t-t_0)} e^{-iH(t-t')}e^{-iH_0(t'-t_0)}.
\]
- \(\Omega\) from vacuum:
- \[
\ket{\Omega} = \qty(e^{-iE_0(t_0 - (-T))}\bra{\Omega}\ket{0})^{-1} U(t_0,-T)\ket{0}.
\]
- \[
\bra{\Omega} = \bra{0}U(T,t_0) \qty(e^{-iE_0(T-t_0)}\bra{0}\ket{\Omega})^{-1}.
\]
- Normalization:
\[
\bra{\Omega}\ket{\Omega} = 1.
\]
Derivation
With
\[
e^{-iHT}\ket{0} = e^{-iE_0 T} \ket{\Omega}\bra{\Omega}\ket{0} + \sum_{n\neq 0} e^{-iE_n T}\ket{n}\bra{n} \ket{0},
\]
we find that
\[
\ket{\Omega} = \qty(e^{-iE_0 T}\bra{\Omega}\ket{0})^{-1} e^{-iHT}\ket{0},
\]
which simplifies to
\[
\ket{\Omega} = \qty(e^{-iE_0(t_0 - (-T))}\bra{\Omega}\ket{0}) U(t_0, -T)\ket{0}.
\]
What are the Consequences of the Imaginary Part of \(T\)?
After applying the momentum space Feynman rules, we are facing with
\[
\int_{-T}^T \dd{z^0} \int \dd{^3 z} e^{-i(p_1 + p_2 + p_3 - p_4)\cdot z}.
\]
To prevent the integral from blowing up, we demand
\[
p^0 \propto (1+i\epsilon),
\]
which is equivalent to the Feynman prescription of propagator.
Obtaining the Correlation Function (Canonical Quantization)
- Two-point correlation function:
\[
\color{red} \bra{\Omega}T\qty{\phi(x)\phi(y)}\ket{\Omega}= \dfrac{\bra{0}T\qty{\phi_I(x)\phi_I(y)\exp\qty[-i\int_{-T}^{T}\dd{t}H_I(t)]}\ket{0}}{\bra{0}T\qty{\exp\qty[-i\int_{-T}^T \dd{t} H_I(t)]}\ket{0}}.
\]
For each factor inserted to the LHS, we inserted an identical one on the RHS.
Obtaining the Correlation Function (Path Integral)
- Path integral formulation:
\[
\color{red} \bra{\Omega} T\phi_H(x_1) \phi_H(x_2) \ket{\Omega} = \dfrac{\int \mathcal{D}\phi\, \phi(x_1) \phi(x_2) \exp\qty[i\int_{-T}^T \dd{^4 x}\mathcal{L}]}{\int \mathcal{D}\phi\, \exp\qty[i\int_{-T}^T \dd{^4 x}\mathcal{L}]}.
\]
- From the generating functional:
\[
\color{red} \bra{\Omega} T\phi(x_1)\phi(x_2) \ket{\Omega} = \frac{1}{Z_0}\left.\qty(-i\frac{\delta}{\delta J(x_1)})\qty(-i\frac{\delta}{\delta J(x_2)}) Z[J]\right|_{J=0}.
\]
Due to an unfortunate collision of notation, \(Z_0\) may stands for both the generating functional \(Z_0[J_A,J_B,\cdots]\) or the constant \[Z_0 = Z_0[0,0,\cdots].\]
- Write down the perturbation \(\mathcal{L}_1\), which may contain various kinds of fields.
- It's important that the fields occur in product form, e.g.
\[
\mathcal{L}_1 = g\phi_A \phi_B,
\]
so that we may write the perturbation as products of variations of the generating functional.
- Obtain the unperturbed generating functional \(Z_0[J_A, J_B, \cdots]\). This should admit the form
\[
\begin{align*}
Z_0[J_A,J_B,\cdots] &= \qty(\int \mathcal{D}\varphi_A \int \mathcal{D} \varphi_B \cdots) \exp\qty[i\int \dd{^4 x} \qty[\mathcal{L}_A + J_A \phi_A]] \exp\qty[i\int \dd{^4 x} \qty[\mathcal{L}_B + J_B \phi_B]] \\
&= Z_0 \cdot \exp\qty[-\frac{1}{2}\iint \dd{^4 x} \dd{^4 x'} J_A(x) D^A_F(x-x') J_A(x')] \exp\qty[-\frac{1}{2}\iint \dd{^4 x \dd{^4 x'}} J_B(x) D^B_F(x-x') J_B(x')]\cdots.
\end{align*}
\]
- To obtain the perturbed generating functional, we
- replace each item in the perturbation as a functional derivative with respect to the corresponding source:
\[
g\phi_A \phi_B \rightarrow g\qty(-i\frac{\delta}{\delta J_A})\qty(-i\frac{\delta}{\delta J_B}).
\]
- Extra care should be taken for the possible minus sign for Dirac fields.
- then exponentiate it and let it act on the unperturbed generating functional:
\[
\begin{align*}
Z[J_A, J_B,\cdots] &= Z_0 \cdot \exp\qty[\int \dd{^4 x} ig\qty(-i\frac{\delta}{\delta J_A(x)})\qty(-i\frac{\delta}{\delta J_B(x)})]Z_0[J_A,J_B,\cdots] \\
&= Z_0 e^{iW[J_A, J_B, \cdots]}.
\end{align*}{}
\]
- To obtain the correlation amplitude, we apply functional derivatives to the perturbed generating functional:
\[
\begin{align*}
\color{red} \bra{\Omega} T\phi_A(x_1) \phi_A(x_2) \ket{\Omega} &\color{red} = \frac{Z_0}{Z[0,0,\cdots]} \left.\qty(-i \frac{\delta}{\delta J_A(x_1)})\qty(-i \frac{\delta}{\delta J_A(x_2)}) \cdot \exp\qty[\int \dd{^4 x} ig\qty(-i\frac{\delta}{\delta J_A(x)})\qty(-i\frac{\delta}{\delta J_B(x)})]Z_0[J_A,J_B,\cdots]\right\vert_{J_A = J_B = \cdots = 0} \\
&\color{red} = \left.\qty(-i \frac{\delta}{\delta J_A(x_1)})\qty(-i \frac{\delta}{\delta J_A(x_2)}) \cdot i W[J_A, J_B, \cdots]\right\vert_{J_A = J_B = \cdots = 0}.
\end{align*}{}
\]
Obtaining the T-Matrix
- T-matrix:
\[\begin{align*}
&\bra{\vb*{p}_1 \cdots \vb*{p}_n} iT \ket{\vb*{p}_A \vb*{p}_B} \\
&= \qty({_0\bra{\vb*{p}_1 \cdots \vb*{p}_1}T\qty(\exp\qty[-i\int_{-T}^T \dd{t} H_I(t)])\ket{\vb*{p}_A \vb*{p}_B}_0})_{\substack{\text{connected,}\\\text{amputated}}}.
\end{align*}\]
Wick's Theorem: Correlation Amplitude to Feynman Propagators
In contractions, \(\phi\) is always under the interaction picture even without the subscript \(I\).
- \(\phi^+\) and \(\phi^-\) denote the annihilation and creation parts of \(\phi\) respectively, i.e.
- \[\color{warning} \phi^+_I(x) = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} a_{\vb*{p}} e^{-ip\cdot x}; \]
- \[\color{warning} \phi^-_I(x) = \int \frac{\dd{^3 p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\vb*{p}}}} a^\dagger_{\vb*{p}} e^{+ip\cdot x}. \]
The time-ordered product of a Dirac field contains a factor of the sign of the permutation, e.g.
\[
T\qty(\psi_1 \psi_2 \psi_3 \psi_4) = (-1)^3 \psi_3 \psi_1 \psi_4 \psi_2
\]
if \[x_3^0 > x_1^0 > x_4^0 > x_2^0.\]
- \(N(XYZ)\) that turns \(XYZ\) into normal order,
- Scalar field:\[
\color{warning} N(a_{\vb*{p}}a_{\vb*{k}}^\dagger a_{\vb*{q}}) = a^\dagger_{\vb*{k}}a_{\vb*{p}}a_{\vb*{q}}.
\]
- Dirac field:\[
\color{warning}N(a_{\vb*{p}} a_{\vb*{q}} a_{\vb*{r}}^\dagger) = (-1)^2 a_{\vb*{r}}^\dagger a_{\vb*{p}} a_{\vb*{q}} = (-1)^3 a_{\vb*{r}}^\dagger a_{\vb*{q}} a_{\vb*{p}}.
\]
- Wick contraction: different from the convention in most literatures, here we use color to denote contraction,
\[
{\color{blue}\phi}(x){\color{blue}\phi}(y) \color{orange} = T\qty{\phi(x)\phi(y)} - N\qty{\phi(x)\phi(y)}.
\]
More generally,
\[
{\color{blue} A}{\color{blue} B} = T\qty{AB} - N\qty{AB}.
\]
- For a scalar field:\[
{\color{blue}\phi}(x){\color{blue}\phi}(y) \color{orange} = \begin{cases}
\qty[\phi^+(x), \phi^-(y)] & \text{for } x^0 > y^0; \\
\qty[\phi^+(y), \phi^-(x)] & \text{for } y^0 > x^0. \\
\end{cases}
\]
- The contraction yields the Feynman propagator,
\[
{\color{blue}\phi}(x){\color{blue}\phi}(y) = D_F(x-y).
\]
- Surprisingly, the contraction yields a number, without bra-keting with \(\ket{0}\).
- For a Dirac field:\[
{\color{blue}\psi}(x) {\color{blue} \overline{\psi}}(y) = \begin{cases}
\phantom{-}\qty{\psi^+(x), \overline{\psi}^-(y)}, & \text{for } x^0 > y^0; \\
-\qty{\overline{\psi}^+(y),\psi^-(x)}, & \text{for } x^0 < y^0.
\end{cases}
\]
- The contraction yields the Feynman propagator:\[
{\color{blue}\psi}(x) {\color{blue} \overline{\psi}}(y) = S_F(x-y),
\]
- or vanishes:
\[
{\color{blue}\psi}{\color{blue}\psi} = {\color{blue}\overline{\psi}}{\color{blue}\overline{\psi}} = 0.
\]
- The following notation is used, although it seems that contraction should not be put in \(N\qty{\cdots}\):
- Scalar field: \[N\qty{{\color{blue}\phi_1}\phi_2{\color{blue}\phi_3}\phi_4} = D_F(x_1 - x_3)\cdot N\qty{\phi_2\phi_4}.\]
- Dirac field: \[
N({\color{blue}\psi_1} \psi_2 {\color{blue}\overline{\psi}_3} \overline{\psi}_4) = -{\color{blue}\psi_1}{\color{blue}\overline{\psi}_3} N(\psi_2 \overline{\psi}_4) = -S_F(x_1 - x_3)N(\psi_2 \overline{\psi}_4).
\]
With the normal ordering we consider only either the \(a_{\vb*{p}}\) or \(a_{\vb*{p}}^\dagger\) part in the expansion of \(\color{blue}\phi_I\).
- Contracting field with creation and annihilation operators:
- \[
{\color{blue}\phi_I}(x)\ket{\color{blue}\vb*{p}} = {\color{blue}\phi_I}(x){\color{blue}a^\dagger_{\vb*{p}}}\ket{0} = e^{-ip\cdot x}\ket{0}{ }.
\]
- \[
\bra{\color{blue}\vb*{p}}{\color{blue}\phi_I}(x) = \bra{0}{\color{blue} a_{\vb*{p}}}{\color{blue}\phi_I}(x) = \bra{0} e^{+ip\cdot x}{ }.
\]
- Mixing different kinds of fields: Do I contract a Klein-Gordon field with a Dirac field?
- I don't know. I am fabricating shit here. No. If the two fields decouple in the unperturbed Hamiltonian,
then their creation and annihilation operators commutes, and therefore we don't give a fuck to their relative ordering. See also
Wick's theorem:
\begin{align*}
& \color{red} T\qty{\phi(x_1)\phi(x_2)\cdots \phi(x_m)} \\
& \color{red} = N\qty{\phi(x_1)\phi(x_2)\cdots \phi(x_m) + \text{all possible contractions}}.
\end{align*}
For \(m=4\) we have
\begin{align*}
& T\qty{\phi_1\phi_2\phi_3\phi_4} \\
&= N\big\{ \phi_1\phi_2\phi_3\phi_4 \\
& + {\color{blue}\phi_1}{\color{blue}\phi_2}\phi_3\phi_4 + {\color{blue}\phi_1}\phi_2{\color{blue}\phi_3}\phi_4 + {\color{blue}\phi_1}\phi_2\phi_3{\color{blue}\phi_4} \\
& + \phi_1{\color{blue}\phi_2}{\color{blue}\phi_3}\phi_4 + \phi_1{\color{blue}\phi_2}\phi_3{\color{blue}\phi_4} + \phi_1\phi_2{\color{blue}\phi_3}{\color{blue}\phi_4} \\
& + {\color{blue}\phi_1}{\color{blue}\phi_2}{\color{magenta}\phi_3}{\color{magenta}\phi_4} + {\color{blue}\phi_1}{\color{magenta}\phi_2}{\color{blue}\phi_3}{\color{magenta}\phi_4} + {\color{blue}\phi_1}{\color{magenta}\phi_2}{\color{magenta}\phi_3}{\color{blue}\phi_4}\big\}.
\end{align*}
Since the normal order put annihilation on the right, we have
\begin{align*}
&\bra{0}T\qty{\phi_1\phi_2\phi_3\phi_4}\ket{0} \\
&= D_F(x_1 - x_2) D_F(x_3 - x_4) \\
&+ D_F(x_1 - x_3) D_F(x_2 - x_4) \\
&+ D_F(x_1 - x_4) D_F(x_2 - x_3).
\end{align*}
Only fully paired contractions are present in\[\color{red}\bra{0}T\qty{\cdots}\ket{0}.\]
Feynman Diagrams: Wick Contractions / Functional Derivatives Visualized
The \(m=4\) example may be rewritten using Feynman diagrams as
- Line \(=\) propagator \(=\) contraction \(=\) whatever you multiply.
- Vertex:
- In position space: vertex \(=\) integration, \(\displaystyle \int \dd{^4 x}\).
- In momentum space: vertex \(=\) intergration \(\displaystyle \int \dd{^4 p}\) and \(\delta(\sum p)\).
Correlation Amplitude to Diagram (Canonical Quantization)
How do we convert the expression for \(\bra{\Omega}T\qty{\phi(x)\phi(y)}\ket{\Omega}{}\) into diagram?
- Converting the numerator to diagram:
- The numerator is given by
\[
\bra{0}T\qty{\phi_I(x)\phi_I(y)\exp\qty[-i\int_{-T}^{T}\dd{t}H_I(t)]}\ket{0}.
\]
- Expand the \(\exp\) via Taylor series. Note that \(H_I\) consists of fields (\(\varphi, \psi\), etc.):
\[
\bra{0} T\qty{ \phi(x)\phi(y) + \phi(x)\phi(y)\qty[-i\int \dd{t} H_I(t)] + \cdots } \ket{0}.
\]
- Apply Wick's theorem to each item after doing the Taylor expansion.
- Retain only full contractions.
- Convert contractions into Feynman propagators.
- Conclusion:
\[
\color{red} \bra{0}T\qty[\phi(x_1)\phi(x_2)\cdots \phi(x_n)]\ket{0} = \qty(\sum \text{ all possible diagrams with } n \text{ external points}).
\]
- Taking the denominator into account:
- It can be shown that the disconnected diagrams (disconnected from the two external points) contribute a factor that cancels the denominator in the equation for \(\bra{\Omega}T\qty{\phi(x)\phi(y)}\ket{\Omega}{}\).
Conclusion.
\[
\begin{align*}
&\phantom{{}={}} \color{red}\bra{\Omega}T\qty[\phi(x_1)\phi(x_2)\cdots \phi(x_n)]\ket{\Omega} \\ &\color{red}= \qty(\sum \text{ connected diagrams with } n \text{ external points}).
\end{align*}
\]
For the \(\phi^4\) perturbation to the first order we find
\begin{align*}
& \bra{0}T\qty{\phi(x)\phi(y)\qty(\frac{-i\lambda}{4!})\int \dd{^4 z}\phi(z)\phi(z)\phi(z)\phi(z)}\ket{0} \\
& = 3\cdot \qty(\frac{-i\lambda}{4!})D_F(x-y) \int \dd{^4 z} D_F(z-z)D_F(z-z) \\
& + 12 \cdot \qty(\frac{-i\lambda}{4!}) \int \dd{^4 z} D_F(x-z)D_F(y-z)D_F(z-z).
\end{align*}
This may be rewritten using Feynman diagrams as
\(+\)
.
One of the contractions in the \(\lambda^3\) term is given by
\begin{align*}
& \bra{0}{\color{blue}\phi}(x){\color{magenta}\phi}(y) \frac{1}{3!} \qty(\frac{-i\lambda}{4!})^3 \int \dd{^4 z} {\color{blue}\phi}{\color{cyan}\phi}{\color{cyan}\phi}{\color{green}\phi} \int \dd{^4 w} {\color{green}\phi}{\color{orange}\phi}{\color{red}\phi}{\color{magenta}\phi} \int \dd{^4 u} {\color{red}\phi}{\color{orange}\phi}{\color{purple}\phi}{\color{purple}\phi} \ket{0} \\
&= \frac{1}{3!}\qty(\frac{-i\lambda}{4!})^3 \int \dd{^4 z} \dd{^4 w} \dd{^4 u} D_F(x-z) D_F(z-z) D_F (z-w) D_F(w-y) D_F^2(w-u) D_F(u-u),
\end{align*}
which accounts for
\[
10368 = 3! \times 4\cdot 3 \times 4\cdot 3\cdot 2 \times 4\cdot 3 \times 1/2
\]
contractions. The Feynman diagram is
.
Symmetry Factor
To obtain the overall constant of a diagram in the \(\phi^4\) theory, we apply the following recipe:
- We drop the factor \(\displaystyle \frac{1}{k!}\qty(\frac{1}{4!})^k\) in front of each item, i.e. multiply by \(k! (4!)^k\), which
- copies this diagram for \(k!\) times: taking into accound the exchange of vertices;
- and then copies for \((4!)^k\) times: on each vertex we arbitrarily permute the four incoming lines.
- Then we divide the item by the symmetry factor:
- The permutation above may not result in a new diagram, and therefore we should divide by a symmtry factor to prevent these diagrams being counted multiple times.
- In the above example, the following permutations does not yield a new diagram, which contribute a symmetry factor \(S = 2\cdot 2\cdot 2 = 8\).
- exchanging the two ends of the loop on \(z\),
- exchanging the two ends of the loop on \(u\),
- exchanging the two propagators connecting \(w\) and \(u\).
- Lines in the diagrams are refered to as propagators.
- Internal points (\(z\), \(u\), \(w\), etc.) are called vertices.
Correlation Amplitude to Diagram (Path Integral)
- We apply the Taylor expansion to
\[
\begin{align*}
Z[J_A, J_B,\cdots] &= Z_0 \cdot \exp\qty[\int \dd{^4 x} ig\qty(-i\frac{\delta}{\delta J_A(x)})\qty(-i\frac{\delta}{\delta J_B(x)})]Z_0[J_A,J_B,\cdots] \\
&= Z_0 \cdot \sum_{V=0}^\infty \qty(\int \dd{^4 x} ig\qty(-i\frac{\delta}{\delta J_A(x)})\qty(-i\frac{\delta}{\delta J_B(x)}))^V \sum_{P=0}^\infty \qty( {-\frac{1}{2}\iint \dd{^4 x} \dd{^4 x'} J_A(x) D^A_F(x-x') J_A(x') -\frac{1}{2}\iint \dd{^4 x \dd{^4 x'}} J_B(x) D^B_F(x-x') J_B(x')} )^P.
\end{align*}
\]
- For each integral in \(\sum_P\), we associate with it a propagator (line).
- For each integral in \(\sum_V\), we associate with it an internal vertex.
- For each source not killed, we assocaite with it an external vertex.
- We are not interested in the full form of \(Z\). With \(Z=e^{iW}\) we identify \(iW\) with the contribution from all connected diagrams.
- We count only connected diagrams.
- To obtain the correlation amplitude, we apply
\[
\qty(-i \frac{\delta}{\delta J})
\]
on \(iW\). Every instance of such functional derivative kills a source, giving rise to an external vertex.
T-Matrix to Diagram
In the equation for the \(T\)-matrix
\[\begin{align*}
&\bra{\vb*{p}_1 \cdots \vb*{p}_n} iT \ket{\vb*{p}_A \vb*{p}_B} \\
&= \qty({_0\bra{\vb*{p}_1 \cdots \vb*{p}_1}T\qty(\exp\qty[-i\int_{-T}^T \dd{t} H_I(t)])\ket{\vb*{p}_A \vb*{p}_B}_0})_{\substack{\text{connected,}\\\text{amputated}}},
\end{align*}{ }\]
the terminologies are explained below.
- Connected: only the fully connected (i.e. all vertices are connected to each other) diagrams are counted.
- Amputated: performed an operation that starts from the tip of each external leg and cut at the last point at which removing a single propagator would separates the leg from the rest of the diagram.
Conclusion.
\begin{align*}
&\phantom{{}={}}\color{red} i\mathcal{M} \cdot (2\pi)^4 \delta^{(4)}\qty(p_A + p_B - \sum p_f) \\
&\color{red}= \begin{pmatrix}
\text{sum of all connected, amputated Feynman} \\
\text{diagrams with } p_A, p_B \text{ incoming, } p_f \text{ outgoing}
\end{pmatrix}.
\end{align*}
Conclusion.
\[
\color{red} i\mathcal{M} = \begin{pmatrix}\text{sum of all connected, amputated diagrams}\\ \text{in the momentum space}\end{pmatrix}.
\]
The two rules above should be revised for loop diagrams.
- \(S\)-matrix and Feynman diagram.
\[\bra{\vb*{p}_1 \cdots \vb*{p}_n} S \ket{\vb*{k}_1 \vb*{k}_2} = \qty(\sqrt{Z})^{n+2} \qty[\text{Amputated}].\]
Determination of Symmetry Factor
The symmetry factor counts the number of ways of interchanging components without changing the diagram.
The symmetry factor of
is \(2\cdot 2\cdot 2 = 8\).
Disconnected Diagrams
- A diagram is disconnected if it is disconnected from all external points.
- A diagram connected to some of the external points is deemed as connected.
- Disconnected diagrams are called vacuum bubbles.
- Let \(V_i\) denote the disconnected diagrams as well as their values.
- The \(V_i\)'s are not isomorphic to each other.
- The diagrams representing the two-point correlation function yield the sum
\begin{align*}
&\phantom{{}={}}\sum_{\text{connected}} \sum_{\qty{n_i}} \qty(\begin{array}{c}
\text{value of} \\
\text{connected piece}
\end{array}) \times \qty(\prod_i \frac{1}{n_i!} {(V_i)}^{n_i}) \\
&= \qty{\sum \text{connected}} \times \sum_{\qty{n_i}} \qty(\prod_i \frac{1}{n_i!}{(V_i)^{n_i}}) \\
&= \qty{\sum \text{connected}} \times \exp\qty{\sum_i V_i}.
\end{align*}
- The contribution to the denominator in the two-point correlation function is exactly
\[
\exp\qty{\sum_i V_i}.
\]
- Therefore, we may take only connected diagrams in our calculation.
- Each disconnected diagram contains a factor
\[
(2\pi)^4 \delta^{(4)}(0) = 2T\cdot V.
\]
Constructing Feynmann Rules (Position Space)
- Write down the perturbation \(H_I\), which may contain various kinds of fields \(\phi\), \(\psi\), etc.
- Obtain the Feynman propagators: \(D_F\), \(S_F\), etc.
- Ideally, this is equivalent to asking (from the recipe via canonical quantization): what are \({\color{blue}\phi}(x){\color{blue}\phi}(y)\), \({\color{blue}\psi}(x){\color{blue}\overline{\psi}}(y)\), etc.?
- The contractions should be functions of \((x-y)\) only thanks to the homogeneity of spacetime.
- Follow the procedure specified by Correlation Amplitude to Diagram to write down the diagrams.
- For each propagator, associate with it a factor equal to the contraction it represents, \({\color{blue}\phi}(x){\color{blue}\phi}(y)\), \({\color{blue}\psi}(x){\color{blue}\overline{\psi}}(y)\), etc.
- For each vertex, associate with it an integration
\[
(-i\lambda)\int \dd{^4 z}.
\]
- We should have associated
\[
-i\frac{\lambda}{4!} \int \dd{^4 z}
\]
for the \(\displaystyle \frac{\lambda}{4!} \phi^4\) perturbation. However, to better take symmetries of diagrams into account,
we kill the \(1/4!\) factor to make copies of the diagram for all permutations of lines.
- For each external point, associate a factor \(1\).
- Divide by the symmetry factor.
Constructing Feynmann Rules (Momentum Space)
- Contruct the position space Feynman rule first.
- Rewrite the contractions in the momentum space.
- \[
{\color{blue}\phi}(x){\color{blue}\phi}(y) = {\color{magenta}\int \frac{\dd{^4 p}}{(2\pi)^4}} e^{-ip\cdot({\color{darkcyan}x}-{\color{darkgreen}y})} {\color{orange}\tilde{\Delta}(p)}.
\]
- The diagram is left intact. However, each propagator should be assigned a momentum (with direction).
\[
x - y \rightarrow p.
\]
- We associate each propagator with the factor \(\color{orange}\tilde{\Delta}(p)\):
- \[
{\color{blue}\phi}(x){\color{blue}\phi}(y) \rightarrow {\color{orange}\tilde{\Delta}(p)}.
\]
- For each vertex, kill the \(\displaystyle \int \dd{^4 z}\) in the position space Feynmann rules:
\[
(-i\lambda) \int \dd{^4 z} \rightarrow -i\lambda \cdots.
\]
- The integral in \(z\) should be replaced by a \(\delta\)-function representing the conservation of momentum:
\[
(-i\lambda) \int \dd{^4 z} \rightarrow (-i\lambda) {\color{darkgreen}(2\pi)^4 \delta\qty(\sum p)}.
\]
- For each external points, associate a factor \(\color{darkcyan} e^{-ip\cdot x}\) if the momentum points towards \(x\), and \(e^{+ip\cdot x}\) if pointing away from \(x\).
- Integrate over each momentum:
\[
\color{magenta} \int \frac{\dd{^4 p}}{(2\pi)^4}.
\]
- Divide by the symmetry factor.
Feynman Rules for Scattering (Position Space)
- The procedure for position space Feynman rules applies when the contraction does not involve in and out states.
- In addition to contraction between fields, we have to obtain the factor for contraction between field and in and out states.
\[
{\color{blue} \phi_I}(x)\ket{\color{blue} \vb*{p}} = {\color{orange} \tilde{\phi}_I(p) e^{-ip\cdot x}}\ket{0},
\]
- Contractions between in and out states themselves, i.e.
\[
{\color{blue}a_{\vb*{q}}a^\dagger_{\vb*{p}}} = {\color{orange} \delta^{(4)}}(p-q),
\]
does not occur in (fully) connected diagrams.
- Always a line there when we are doing contraction and obtaining something as a factor!
Feynman Rules for Scattering (Momentum Space)
- Construct the position space Feynman rule for scattering first.
- The procedure for momentum space Feynman rules applies when the contraction does not involve in and out states.
- Replace \({\color{orange} \tilde{\phi}_I(p) e^{-ip\cdot x}}\) by its Fourier transform:
\[
{\color{orange} \tilde{\phi}_I(p) e^{-ip\cdot x}} \rightarrow {\color{orange} \tilde{\phi}_I(p)}.
\]
- External \(p\)'s should not be integrated.
Rules for scattering in the position space is for \[
i\mathcal{M} \cdot (2\pi)^4 \delta^{(4)}\qty(p_A + p_B - \sum p_f),
\] while in the momentum space for \[
i\mathcal{M}.
\]
Fermion Trivia
- We denote
- scalar particles by dashed lines,
- fermions by solid lines, and
- photons by dashed lines.
- Direction of momentum:
- On internal fermion lines, the momentum must be assigned in the direction of particle-number flow.
- On external fermion lines, the momentum is assigned in the direction of particle number for fermions while oppsite for antifermions.
- Symmetry of diagrams:
- We always drop the \(\displaystyle \frac{1}{k!}\) factor from \(\exp\), since it cancels out the permutation of vertices.
- The diagrams of Yukawa theory has no symmetry factors, because the three fields cannot replace one for another.
- Determining the overall sign of a diagram:
- We should not allow any intruder operators between a Dirac field operator and a creation or annihilation operator.
- However, pure complex numbers (\(D_F\), \(S_F\), etc.) are ok.
- For a specific contraction, we permute the field operators to eliminate the intruders.
- Fully contracted operators between are ok.
- Tricks:
- \({\color{blue} \overline{\psi}\psi}\) commutes with any operator.
- Closed loop:
\[
{\color{darkcyan} \overline{\psi}}{\color{blue}\psi}{\color{blue} \overline{\psi}}{\color{darkcyan}\psi} = - \operatorname{tr}{[S_F S_F]}.
\]
Example: Quartic Interaction
\[
H_{\mathrm{int}} = \int \dd{^3 x} \frac{\lambda}{4!} \phi^4(\vb*{x}).
\]
Position Space Feynman Rules (Quartic Interaction)
- For each propagator,
\(=D_F(x-y)\);
- For each vertex,
\(\displaystyle =(-i\lambda) \int \dd{^4 z}\);
- For each external point,
\(=1\);
- Divide by the symmetry factor.
Momentum Space Feynman Rules (Quartic Interaction)
- For each propagator,
\(\displaystyle = \frac{i}{p^2 - m^2 + i\epsilon}\);
- For each vertex,
\(\displaystyle = -i\lambda\);
- For each external point,
\(\displaystyle = e^{-ip\cdot x}\);
- Impose momentum conservation at each vertex;
- Integrate over each undetermined momentum: \(\displaystyle \int \frac{\dd{^4 p}}{(2\pi)^4}\);
- Divide by the symmetry factor.
In the integration on \(p\), we should take \(p^0\) as \(p^0\propto (1+i\epsilon)\), i.e. using the Ferynman prescription.
Position Space Feynman Rules for Scattering (Quartic Interaction)
- \[{\color{blue}\phi_I}(x)\ket{\color{blue}\vb*{p}} = e^{-ip\cdot x}\ket{0}{ };\]

- \[\bra{\color{blue}\vb*{p}}{\color{blue}\phi_I}(x) = \bra{0} e^{+ip\cdot x}{ }.\]

- For each propagator,
\(=D_F(x-y)\);
- For each vertex,
\(\displaystyle =(-i\lambda) \int \dd{^4 z}\);
- For each external line,
\(=e^{-ip\cdot x}\);
- Divide by the symmetry factor.
We evalute
\[\begin{align*}
&\bra{\vb*{p}_1 \cdots \vb*{p}_n} iT \ket{\vb*{p}_A \vb*{p}_B} \\
&= \qty({_0\bra{\vb*{p}_1 \cdots \vb*{p}_1}T\qty(\exp\qty[-i\int_{-T}^T \dd{t} H_I(t)])\ket{\vb*{p}_A \vb*{p}_B}_0})_{\substack{\text{connected,}\\\text{amputated}}},
\end{align*}{ }\]
to the zeroth and first order below, ignoring the prescription connected and amputated first.
Zeroth order: trivial because of the conservation of momentum.
\begin{align*}
_0\bra{\vb*{p}_1 \vb*{p}_2}\ket{\vb*{p}_A \vb*{p}_B}_0 &= \sqrt{2E_1 2E_2 2E_A 2E_B} \bra{0} a_1 a_2 a_A^\dagger a_B^\dagger \ket{0} \\
&= 2E_A 2E_B (2\pi)^6 \qty(\delta^{(3)}\qty(\vb*{p}_A - \vb*{p}_1) \delta^{(3)}\qty(\vb*{p}_B - \vb*{p}_2) + \delta^{(3)}\qty(\vb*{p}_A - \vb*{p}_2)\delta^{(3)}\qty(\vb*{p}_B - \vb*{p}_1)).
\end{align*}
\(+\) 
First order: using the Wick's theorem we find
\begin{align*}
& _0\bra{\vb*{p}_1 \vb*{p}_2} T\qty({-i\frac{\lambda}{4!}\int \dd{^4 x}\phi_I^4 (x)}) \ket{\vb*{p}_A \vb*{p}_B}_0 \\
&= {_0 \bra{\vb*{p}_1 \vb*{p}_2} N\qty({-i\frac{\lambda}{4!}\int \dd{^4 x}\phi_I^4(x) + \text{contractions}}) \ket{\vb*{p_A}\vb*{p}_B}_0}.
\end{align*}
- Contraction of type \({\color{blue}\phi}{\color{blue}\phi}{\color{magenta}\phi}{\color{magenta}\phi}\) (trivial again):
- Contraction of type \({\color{blue}\phi}{\color{blue}\phi}{\phi}{\phi}\): the two uncontracted \(\phi\)'s should have one contracted to the bra and one to the ket. Trivial again.
- Contraction of type \(\phi\phi\phi\phi\), i.e. all \(\phi\)'s are contracted to the bra and ket:
\begin{align*}
&(4!)\cdot \qty(-i\frac{\lambda}{4!})\int \dd{^4 x} e^{-i(p_A + p_B - p_1 - p_2)\cdot x} \\
&= -i\lambda (2\pi)^4 \delta^{(4)}\qty(p_A + p_B - p_1 - p_2).
\end{align*}
With the definition of \(\mathcal{M}\) we find that \(\mathcal{M} = -\lambda\).
Momentum Space Feynman Rules for Scattering (Quartic Interaction)
- For each propagator,
\(\displaystyle = \frac{i}{p^2 - m^2 + i\epsilon}\);
- For each vertex,
\(\displaystyle = -i\lambda\);
- For each external point,
\(\displaystyle = 1\);
- Impose momentum conservation at each vertex;
- Integrate over each undetermined momentum: \(\displaystyle \int \frac{\dd{^4 p}}{(2\pi)^4}\);
- Divide by the symmetry factor.
Example: Yukawa Theory
\[
\begin{align*}
H &= H_{\text{Dirac}} + H_{\text{Klein-Gordon}} + H_{\text{int}} \\
&= H_{\text{Dirac}} + H_{\text{Klein-Gordon}} + \int \dd{^3 x} g\overline{\psi}\psi\phi.
\end{align*}
\]
Momentum Space Feynman Rules for Scattering (Yukawa Theory)
- Propagators:
\({\color{blue}\phi}(x){\color{blue}\phi}(y) =\)
\(\displaystyle = \frac{i}{q^2 - m_\phi^2 + i\epsilon}{};\)
\({\color{blue}\psi}(x){\color{blue}\overline{\psi}}(y) =\)
\(\displaystyle = \frac{i(\unicode{x2215}\kern-.5em {p} + m)}{p^2 - m^2 + i\epsilon}{}.\)
- Vertices:
\(=-ig\).
- External leg contractions:
- \({\color{blue}\phi}\ket{\color{blue}\vb*{q}} =\)
\(=1\);
- \(\bra{\color{blue}\vb*{q}}{\color{blue}\phi} =\)
\(=1\);
- \({\color{blue}\psi}\underbrace{\ket{\color{blue}\vb*{p},s}}_{\text{fermion}} =\)
\(=u^s(p)\);
- \(\underbrace{\bra{\color{blue}\vb*{p},s}}_{\text{fermion}}{\color{blue}\overline{\psi}} =\)
\(=\overline{u}^s(p)\);
- \({\color{blue}\overline{\psi}}\underbrace{\ket{\color{blue} \vb*{k},s}}_{\text{antifermion}} =\)
\(= \overline{v}^s(k)\);
- \(\underbrace{\bra{\color{blue}\vb*{k},s}}_{\text{antifermion}} {\color{blue}\psi} =\)
\(=v^s(k)\).
- Impose momentum conservation at each vertex.
- Integrate over undetermined loop momentum.
- Figure out the overall sign of the diagram.
We consider
\[
\mathrm{fermion}(p) + \mathrm{fermion}(k) \longrightarrow \mathrm{fermion}(p') + \mathrm{fermion}(k').
\]
- Lowest order:
\[
_0\bra{\vb*{p}',\vb*{k}'} T\qty({\frac{1}{2!}(-ig) \int \dd{^4 x} \overline{\psi}_I \psi_I \phi_I (-ig) \int \dd{^4 y} \overline{\psi}_I \psi_I \phi_I}) \ket{\vb*{p},\vb*{k}}_0.
\]
- Implicitly we assocaite spins \(s, r, s', r'\) to the momenta.
- We drop the factor \(1/2!\) for any diagrams since we could interchange \(x\) and \(y\).
- Contractions:
- \[
\bra{0}{\color{blue} a_{\vb*{k}'}}{\color{darkcyan} a_{\vb*{p}'}} {\color{blue} \overline{\psi}_x} {\color{magenta} \psi_x} {\color{darkcyan} \overline{\psi}_y} {\color{orange} \psi_y} {\color{orange}a_{\vb*{p}}^\dagger} {\color{magenta} a_{\vb*{k}}^\dagger}\ket{0}.
\]
- Move \({\color{darkcyan} \psi_y}\) two spaces to the left.
- Factor: \((-1)^2 = +1\).
- \[
\bra{0}{\color{darkcyan} a_{\vb*{k}'}}{\color{blue} a_{\vb*{p}'}} {\color{blue} \overline{\psi}_x} {\color{magenta} \psi_x} {\color{darkcyan} \overline{\psi}_y} {\color{orange} \psi_y} {\color{orange}a_{\vb*{p}}^\dagger} {\color{magenta} a_{\vb*{k}}^\dagger}\ket{0}.
\]
- Move \({\color{darkcyan} \psi_y}\) one spaces to the left.
- Factor: \(-1\).
Conclusion:
\(i\mathcal{M} =\)
\(+\)
\begin{align*}
&= (-ig^2) \qty( \overline{u}\qty(p') u\qty(p) \frac{1}{(p'-p)^2 - m_\phi^2} \overline{u}\qty(k') u\qty(k) - \overline{u}\qty(p') u\qty(k) \frac{1}{(p'-k)^2 - m_\phi^2} \overline{u}\qty(k') u\qty(p) ).
\end{align*}
Typo: wavy lines should be replaced with dashed lines here.
The Yukawa Potential
- For distinguishable particles, only the first diagram contributes.
- \[
i\mathcal{M} \approx \frac{ig^2}{\abs{\vb*{p}' - \vb*{p}}^2 + m_\phi^2} 2m\delta^{ss'} 2m\delta^{rr'}.
\]
- With the Born approximation we find
\[
V(r) = -\frac{g^2}{4\pi}\frac{1}{r}e^{-m_\phi r}.
\]
Example: Quantum Electrodynamics
\[
H_{\text{int}} = \int \dd{^3 x} e\overline{\psi}\gamma^\mu \psi A_\mu.
\]
Momentum Space Feynman Rules for Scattering (Quantum Electrodynamics)
- Fermion rules from the Yukawa theory apply here.
- New vertex:
\(= -ie\gamma^\mu\).
- Photon propagator:
\(\displaystyle = \frac{-ig_{\mu\nu}}{q^2 + i\epsilon}\).
- External photon lines:
- \({\color{blue}A_\mu}\ket{\color{blue}\vb*{p}}=\)
\(=\epsilon_\mu(p)\).
- \(\bra{\color{blue}\vb*{p}}{\color{blue}A_\mu}=\)
\(=\epsilon^*_\mu(p)\).
- Initial and final state photons should be transversely polarized.
- \[
\epsilon^\mu = (0, \vb*{\epsilon}).
\]
- \[
\vb*{p}\cdot \vb*{\epsilon} = 0.
\]
We consider
\[
\mathrm{fermion}(p) + \mathrm{fermion}(k) \longrightarrow \mathrm{fermion}(p') + \mathrm{fermion}(k').
\]
\(i\mathcal{M} =\)
\begin{align*}
&= (-ie)^2 \overline{u}\qty(p') \gamma^\mu u\qty(p) \frac{-ig_{\mu\nu}}{\qty(p'-p)^2}\overline{u}\qty(k')\gamma^\nu u\qty(k).
\end{align*}
The Coulomb Potential
- \[
i\mathcal{M} \approx \frac{-ie^2}{\abs{\vb*{p}' - \vb*{p}}^2}(2m\xi'^\dagger \xi)_p (2m\xi'^\dagger \xi)_k.
\]
- Comparing this to the Yukawa case we obtain a repulsive Coulomb potential
\[
V(r) = \frac{e^2}{4\pi r} = \frac{\alpha}{r}.
\]
- For scattering between an electron and an anti-electron, we get an attractive potential.