Imaginary Time Path Integrals and the Partition Function

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the sixth lesson in our module on the path integral formulation of quantum mechanics.

In our last lesson, we saw how the classical world, governed by the Principle of Least Action, emerges from the quantum path integral. The key was the destructive interference between the phases, $e^{i S /ℏ}$ , of all non-stationary paths, leaving only the classical trajectory to contribute constructively. This analysis was all done in real time, $t$ .

Today, we will explore a powerful and widely used technique: performing the path integral in imaginary time. By making the substitution $t \to - i τ$ , a procedure known as a Wick rotation, we will see that the oscillating phase factor transforms into a real, decaying exponential. This seemingly abstract mathematical trick forges a profound and practical connection between quantum mechanics and statistical mechanics.

Our learning goal is to evaluate path integrals in imaginary time and relate them to the partition function from statistical mechanics. This connection is not just a theoretical curiosity; it forms the foundation of powerful numerical methods like Quantum Monte Carlo, which are indispensable in your field of interest, condensed matter physics, for studying many-body systems.

From Real-Time Oscillations to Imaginary-Time Damping

Let's begin by recalling the propagator, or Green's function, expressed as a sum over the energy eigenstates of the system:

K (x_{f}, t; x_{i}, 0) = ⟨ x_{f} ∣ e^{- i H t /ℏ} ∣ x_{i} ⟩ = n \sum ψ_{n} (x_{f}) ψ_{n}^{*} (x_{i}) e^{- i E_{n} t /ℏ}

The complex exponential $e^{- i E_{n} t /ℏ}$ describes the oscillatory time evolution of each eigenstate. As we've seen, summing these oscillating terms is what leads to interference phenomena.

Now, let's see what happens when we analytically continue time to the imaginary axis.

17.4 Feynman Quantum Path Integrals I

The following video explains the concept and consequences of performing a Wick rotation. It clearly demonstrates how this transforms the propagator and provides a method for isolating the ground state of a quantum system.

Please watch the segment from 28:39 to 33:06. Focus on how the substitution t = -iτ changes the time-evolution term and what this implies for the sum over eigenstates, especially in the limit of large imaginary time, τ.

As the video explained, the Wick rotation $t \to - i τ$ transforms the time-evolution operator:

e^{- i H t /ℏ} t \to - i τ e^{- i H (- i τ) /ℏ} = e^{- H τ /ℏ}

The propagator in imaginary time, often called the Euclidean propagator, becomes:

K_{E} (x_{f}, τ; x_{i}, 0) = n \sum ψ_{n} (x_{f}) ψ_{n}^{*} (x_{i}) e^{- E_{n} τ /ℏ}

The crucial change is that the oscillatory phase factors have become real, decaying exponentials.

For a large imaginary time evolution $τ \to \infty$ , the exponential terms $e^{- E_{n} τ /ℏ}$ decay at different rates. The term corresponding to the highest energy will decay the fastest, while the term for the ground state energy $E_{0}$ will decay the slowest. Consequently, for large $τ$ , the sum is completely dominated by the ground state:

K_{E} (x_{f}, τ; x_{i}, 0) \approx ψ_{0} (x_{f}) ψ_{0}^{*} (x_{i}) e^{- E_{0} τ /ℏ} (for large τ)

This provides a powerful method for extracting ground state properties—a central task in many condensed matter problems.

The Bridge to Statistical Mechanics

This decaying exponential form, $e^{- E τ /ℏ}$ , is highly suggestive. It looks remarkably similar to the Boltzmann factor, $e^{- E / k_{B} T}$ , from statistical mechanics. Let's make this connection explicit.

In statistical mechanics, the central object for a system in thermal equilibrium at temperature $T$ is the canonical partition function, $Z$ , defined as the trace of the Boltzmann operator:

Z = Tr (e^{- β H}) = n \sum ⟨ n ∣ e^{- β H} ∣ n ⟩ = n \sum e^{- β E_{n}}

where $β = 1/ (k_{B} T)$ .

Now, let's write out the trace in the position basis:

Z = \int d x ⟨ x ∣ e^{- β H} ∣ x ⟩

The term inside the integral, $⟨ x ∣ e^{- β H} ∣ x ⟩$ , is precisely the Euclidean propagator from a point $x$ back to itself, $K_{E} (x, ℏ β; x, 0)$ , if we make the identification:

τ = ℏ β = \frac{ℏ}{k _{B} T}

This is the fundamental connection: Quantum mechanical evolution through an imaginary time interval $τ$ is mathematically equivalent to the statistical mechanics of a system in thermal equilibrium at a temperature $T = ℏ/ (k_{B} τ)$ .

The following notes provide a more formal and detailed derivation of this relationship.

Introduction to path integrals

These lecture notes provide a concise but comprehensive derivation of the imaginary time path integral and its relation to the equilibrium density matrix and partition function. Your background in statistical mechanics will be very useful here.

Please read Section 2.9, 'Imaginary Time Path Integral' (pp. 24-26). Pay close attention to how the equilibrium density matrix ρ_β(x, x') is identified with the propagator in imaginary time. Follow the derivation of the Euclidean action S_E and see how the partition function Z is calculated for the harmonic oscillator as an example.

The Path Integral for the Partition Function

Having established the formal connection, we can now write down the path integral representation of the partition function.

First, let's see how the action transforms under a Wick rotation. The real-time action is $S = \int L d t = \int (\frac{1}{2} m x ˙^{2} - V (x)) d t$ . Under the substitution $t \to - i τ$ , we have $d t \to - i d τ$ and $x ˙ = d x / d t \to i (d x / d τ)$ . The phase in the path integral becomes:

\frac{i S}{ℏ} = \frac{i}{ℏ} \int (\frac{1}{2} m (i \frac{d x}{d τ})^{2} - V (x)) (- i d τ) = - \frac{1}{ℏ} \int_{0}^{ℏ β} (\frac{1}{2} m (\frac{d x}{d τ})^{2} + V (x)) d τ

We define the quantity in the integral as the Euclidean Action, $S_{E}$ :

S_{E} [x (τ)] = \int_{0}^{ℏ β} (\frac{1}{2} m (\frac{d x}{d τ})^{2} + V (x)) d τ

Notice that the Euclidean Lagrangian is $L_{E} = T + V$ , akin to the total energy.

The partition function, being a trace, requires summing over paths that are periodic in imaginary time, meaning they start and end at the same position, $x (0) = x (ℏ β)$ . The path integral for $Z$ is then a sum over all such closed loops:

Z = \oint D [x (τ)] e^{- S_{E} [x (τ)] /ℏ}

This is a remarkable result. The quantum partition function is expressed as a sum over all possible classical paths (loops) in imaginary time. Each path is weighted by a real, positive definite factor $e^{- S_{E} /ℏ}$ , which can be interpreted as a probability distribution. This makes the problem amenable to statistical sampling methods.

Application: Path Integral Monte Carlo (PIMC)

This formulation is the basis for the Path Integral Monte Carlo (PIMC) method, a powerful numerical technique you'll often encounter in computational condensed matter physics. Since you enjoy writing simulations, it's worth seeing how this theoretical framework translates directly into a computational algorithm.

The core idea of PIMC is to discretize the imaginary time "path" $x (τ)$ into a finite number of "beads" or time slices. The Euclidean action then becomes a sum over the interactions between adjacent beads.

pimc-beads

CAPTION: A schematic representation of the PIMC method. The continuous quantum path of a particle in imaginary time (left) is discretized into a "necklace" of M classical beads (right). The interactions between these beads are governed by an effective classical potential derived from the Euclidean action. This transforms the quantum problem into a classical statistical mechanics problem that can be simulated.

The quantum particle is mapped onto a classical "ring polymer." The partition function of this N-particle quantum system in $d$ dimensions becomes equivalent to the partition function of a classical system of $N \times M$ particles in $d + 1$ dimensions. This classical system can then be simulated using standard Monte Carlo techniques, like the Metropolis algorithm, to calculate thermodynamic properties ( $E$ , $C_{V}$ , etc.) of the original quantum system.

The Path Integral Monte Carlo Method

The following resource provides an excellent overview of the PIMC method, making the connection between the imaginary time path integral and a practical simulation algorithm very clear. It's an excellent example of how the abstract concepts we're learning are put to work.

Please read the first six pages of this document (up to the start of the code listing). Pay attention to: (1) how the continuous time interval is discretized; (2) how the final expression for the partition function Z is written in terms of classical variables; and (3) the interpretation of the discretized path as a classical system of interacting particles (the 'ring polymer').

Test your understanding!

In the PIMC method, the quantum path is mapped onto a classical ring polymer. The "beads" of this polymer are coupled by harmonic springs, representing the kinetic energy term in the Euclidean action.

If you increase the number of time slices (beads) $M$ while keeping the total imaginary time $τ = ℏ β$ fixed, would you expect the effective spring constant connecting the beads to get stronger or weaker? Why?

Show answer

The effective spring constant would get stronger. The kinetic energy term in the discretized Euclidean action for two adjacent beads $j$ and $j + 1$ is proportional to $\frac{m}{2} (\frac{x _{j + 1} - x _{j}}{Δ τ})^{2}$ . This is the potential energy of a harmonic spring with spring constant $k \propto m / (Δ τ)^{2}$ .

Since $Δ τ = τ / M$ , increasing the number of beads $M$ makes the imaginary time step $Δ τ$ smaller. This causes the effective spring constant $k$ to increase, coupling the beads more tightly. In the limit $M \to \infty$ , $Δ τ \to 0$ , and the polymer becomes a continuous, rigid path, which is the classical limit.

Conclusion

Today, we've made one of the most important conceptual leaps in the study of path integrals. By rotating time into the imaginary plane, we transformed a problem of quantum interference into one of statistical probabilities.

Key Takeaways:

Wick Rotation ( $t \to - i τ$ ) changes the oscillating propagator $e^{- i H t /ℏ}$ into a decaying operator $e^{- H τ /ℏ}$ .
In the limit of large imaginary time, the Euclidean propagator projects out the ground state of the system.
There is a formal equivalence between quantum mechanics in imaginary time $τ$ and statistical mechanics at inverse temperature $β = τ /ℏ$ .
The partition function $Z$ can be written as a path integral over all paths that are periodic in imaginary time, $x (0) = x (ℏ β)$ .
The weighting factor in this path integral is $e^{- S_{E} /ℏ}$ , where the Euclidean Action is $S_{E} = \int (T + V) d τ$ . This allows for probabilistic sampling methods like PIMC.

Preview of the Next Lesson:
We have now developed a significant portion of the path integral formalism. We've seen how to calculate it for simple systems, how the classical limit emerges, and how it connects to statistical mechanics. One powerful application of this machinery is in calculating tunneling phenomena. In our next lesson, we will apply the imaginary time path integral to a double-well potential and derive the instanton contribution to the tunneling amplitude, providing a beautiful semi-classical picture of this quintessentially quantum effect.