Feynman Diagrams in Many-Body Perturbation Theory

Hello! Welcome back to our study of diagrammatic many-body perturbation theory.

In our last lesson, we defined the single-particle Green's function, $G$, and understood its physical meaning as a propagator for a particle or hole moving through an interacting system. We saw that the spectral function, which is directly related to $G$, contains rich information about the system's elementary excitations. The crucial question we left unanswered was: how do we actually calculate the Green's function for a complex interacting system?

Today, we will answer that question. We are going to develop a systematic method to approximate the Green's function using perturbation theory. This will lead us directly to Feynman diagrams, the graphical language of modern many-body physics. By the end of this lesson, you will be able to construct these diagrams and, more importantly, interpret them as representations of physical interaction processes. You will also learn the "Feynman rules" for translating these pictures into precise mathematical expressions.

1. The Perturbation Expansion for the Green's Function

Our starting point is the formal expression for the time-ordered Green's function, which we'll write in position-time representation for clarity:
$G(x,t;x^{\prime },t^{\prime })=-i⟨{\Psi }_0|T[{\hat{\psi }}_H(x,t){\hat{\psi }}_H^{†}(x^{\prime },t^{\prime })]|{\Psi }_0⟩$
Here, $|{\Psi }_0⟩$ is the exact, interacting ground state of the N-particle system, and the field operators are in the Heisenberg picture, evolving under the full Hamiltonian $H=H_0+H_{int}$.

The central problem is that we don't know $|{\Psi }_0⟩$. The core idea of perturbation theory is to systematically relate everything back to the non-interacting system, whose ground state $|0⟩$ and time evolution (under $H_0$) we do know. This is achieved using a formalism called the interaction picture. The derivation is somewhat involved, but it rests on a cornerstone result known as the Gell-Mann and Low theorem.

The key result from your reading is the celebrated formula for the interacting Green's function:

where the S-matrix is given by

Let's dissect this powerful expression:

1. Expectation values are over the non-interacting ground state $|0⟩$. This is a huge simplification.
2. Operators are in the interaction picture (${\hat{\psi }}_I$, $H_{int}(t)$). This means their time evolution is governed only by the non-interacting Hamiltonian, $H_0$.
3. The entire effect of the interactions is packaged into the S-matrix. By expanding the exponential in the S-matrix, we generate a power series in the interaction Hamiltonian $H_{int}$. This is our perturbation expansion.

2. Evaluating the Terms: Wick's Theorem and the Birth of Diagrams

Expanding the S-matrix gives us a series of terms to calculate. For example, the first-order term will involve an expression like:

Since $H_{int}$ is typically composed of four field operators (for a two-body interaction like the Coulomb force), we are left with the daunting task of calculating the expectation value of a long string of creation and annihilation operators.

This is where Wick's theorem comes to the rescue. It provides a simple, powerful rule for breaking down these complex expectation values.

In short, Wick's theorem states that the time-ordered ground-state expectation value of any string of creation and annihilation operators is equal to the sum of all possible "full contractions." A contraction (or pairing) of two operators is simply their time-ordered expectation value, which is nothing but the non-interacting Green's function, $i{G}_0$.

So, the procedure is:

1. Expand the S-matrix to a certain order in $H_{int}$.
2. For each term, you get a long string of operators inside a time-ordered expectation value.
3. Apply Wick's theorem to break this down into a sum of products of non-interacting propagators, $G_0$.

Each term in this final sum corresponds to a specific Feynman diagram. A diagram is simply a picture that represents one specific way the particles can propagate and interact.

3. The Anatomy of a Diagram

Every Feynman diagram, regardless of the specific theory, is built from the same fundamental components. Understanding these components is key to their interpretation.

As the video explained, we can identify three key elements in the context of our many-body problem:

A collection of Feynman diagrams illustrating various interaction processes. Each diagram represents a term in a perturbative expansion.

1. Propagator (Solid Line with Arrow): Represents a non-interacting Green's function, $G_0(x,t;x^{\prime },t^{\prime })$. It describes a particle propagating freely from $(x^{\prime },t^{\prime })$ to $(x,t)$. The arrow indicates the direction of particle flow (forward in time).
2. Interaction (Wiggly Line): Represents the two-body interaction potential, e.g., the Coulomb interaction $v(x-y)$. It connects two points in spacetime and is the mechanism through which particles exchange momentum and energy.
3. Vertex: A point where two solid lines and one wiggly line meet. This represents an interaction event. At a vertex, a particle is scattered, absorbing or emitting a "virtual" quantum of momentum and energy from another particle via the interaction. Crucially, energy and momentum are conserved at each vertex.

The full, interacting Green's function $G$ (often drawn as a double line) is the sum of all possible ways a particle can get from its starting point to its end point. This includes free propagation (zero interactions), scattering once, scattering twice, and so on, ad infinitum.

A diagrammatic representation of the perturbation series for the full Green's function, G. The full propagator (double line) is the sum of the free propagator (single line) plus all possible connected diagrams representing interaction events.

4. The Feynman Rules: From Pictures to Math

Feynman diagrams are more than just intuitive pictures; they are a rigorous computational tool. A set of Feynman rules allows us to unambiguously translate any diagram into its corresponding mathematical expression. The derivation of these rules involves carefully tracking all the factors of $i$, $-1$, and combinatoric numbers that arise from the S-matrix expansion and Wick's theorem.

Fortunately, several remarkable cancellations simplify the final rules immensely. The two most important are:

• Cancellation of disconnected diagrams: The denominator $⟨0|\hat{S}|0⟩$ in our master formula exactly cancels the contribution of all diagrams that consist of a connected part and one or more disconnected "vacuum bubbles". This is the Linked Cluster Theorem. We only need to sum the connected diagrams.
• Cancellation of combinatoric factors: The factor of $1/n!$ from the Taylor expansion of the S-matrix at order $n$ is cancelled by the $n!$ ways of permuting the $n$ identical interaction vertices.

This leads to a practical set of rules. While calculations in real space-time are possible, they involve complicated convolutions. It is almost always more convenient to work in frequency-momentum space, where convolutions become simple products.

Here are the Feynman rules for calculating the n-th order contribution to $G(k,\omega )$ for a system with a static two-body interaction $v(q)$:

1. Draw all topologically distinct, connected diagrams with two external lines (one incoming, one outgoing) and $n$ interaction lines.
2. Assign a momentum $k$ and frequency $\omega$ to the incoming external line.
3. For each internal line (both propagator and interaction), assign a momentum and frequency variable.
4. At each vertex, apply momentum and frequency conservation.
5. For each fermion propagator line with momentum $p$ and frequency $\Omega$, write a factor of $G_0(p,\Omega )=\frac{1}{\Omega -{ϵ}_p+i\delta \text{sign}(|p|-k_F)}$.
6. For each interaction line with momentum transfer $q$, write a factor of $v(q)$.
7. Integrate over all undetermined internal loop momenta and frequencies: for each loop, include a factor of $\int \frac{d^{d}p}{(2\pi {)}^d}\int \frac{d\Omega }{2\pi }$.
8. Multiply the entire expression by a factor of $(i{)}^n$ and a factor of $(-1{)}^F$, where $F$ is the number of closed fermion loops in the diagram.

5. An Example: First-Order Corrections

Let's apply these rules to the first-order ($n=1$) diagrams. There are two connected diagrams that correct the free propagator.

The two first-order Feynman diagrams. The left diagram represents the Hartree (or direct) interaction, and the right diagram represents the Fock (or exchange) interaction.

1. The "Direct" or "Hartree" Diagram (Left):

• Interpretation: A particle with momentum $k$ propagates along. It emits a virtual interaction quantum with zero momentum transfer ($q=0$), which goes off and interacts with the "sea" of other electrons (the closed loop), and is then reabsorbed. This represents the interaction of the electron with the average static charge density of the entire system.
• Calculation:
  • There is one interaction line ($n=1$).
  • There is one closed fermion loop ($F=1$).
  • The prefactor is $(i{)}^1(-1{)}^1=-1$.
  • The loop represents a sum over all possible particle states in the Fermi sea.

2. The "Exchange" or "Fock" Diagram (Right):

• Interpretation: This is a more subtle quantum effect. The particle with momentum $k$ interacts with another particle from the Fermi sea (momentum $k^{\prime }$). They exchange an interaction quantum $v(q)$, and in the process, they "swap places." The original particle continues with momentum $k$, but its journey has been affected by this exchange process. This diagram is a direct consequence of the Pauli exclusion principle and the antisymmetry of the fermionic wavefunction.
• Calculation:
  • There is one interaction line ($n=1$).
  • There are no closed fermion loops ($F=0$). An open line goes in one side of the interaction and out the other.
  • The prefactor is $(i{)}^1(-1{)}^0=+i$.

These two first-order diagrams together constitute the Hartree-Fock approximation within the Green's function formalism. They represent the mean-field potential experienced by a particle in the many-body system.

Conclusion

In this lesson, we have built the foundations of diagrammatic perturbation theory. We have seen how the abstract problem of calculating an interacting Green's function can be transformed into the concrete and intuitive task of drawing and evaluating pictures.

Key Takeaways:

• The interacting Green's function can be expressed as a perturbation series in the interaction strength, where all calculations are performed relative to the non-interacting system.
• Feynman diagrams are a graphical representation of the terms in this series. Each diagram corresponds to a specific physical process of propagation and scattering.
• Diagrams are built from propagators ($G_0$), interaction lines ($v$), and vertices where momentum and energy are conserved.
• A systematic set of Feynman rules allows us to translate any connected diagram into a precise mathematical expression for its contribution to the full Green's function $G$.
• The sum of all connected diagrams represents the full physics of the propagating particle.

Preview of the Next Lesson:

While we can now calculate diagrams order-by-order, this can quickly become intractable. A far more powerful approach is to recognize that we can sum infinite classes of diagrams. This leads to the central concept of the self-energy, $\Sigma$, which encapsulates all the interaction effects. We will introduce Dyson's equation, a compact and profound equation that relates the full Green's function $G$ to the free Green's function $G_0$ and the self-energy $\Sigma$. This will be our primary tool for developing sophisticated approximations in many-body theory.