Module 10: Euclidean Path Integrals
Unveiling the Deep Connection Between Quantum Mechanics and Statistical Physics
← Back to Course | Next: Module 11 →
1 Introduction
In our previous module, we introduced the concept of Wick rotation—the analytical continuation of time to purely imaginary values, \(t = -i\tau\). Now, it is time to fully exploit this transformation.
By defining the Euclidean path integral, we will discover a remarkable mapping between quantum mechanics and classical statistical mechanics. In Minkowski space (real time), paths interfere with oscillating phases, \(e^{iS/\hbar}\). In Euclidean space (imaginary time), paths are suppressed by decaying exponential weights, \(e^{-S_E/\hbar}\). This lack of oscillation might seem like a loss of quantum “magic”, but it provides us with an unparalleled tool to extract the hardest-to-reach information about a quantum system: its ground state and lowest-lying excitations.
Let’s embark on this journey into the profound interplay of imaginary time, ground states, and thermodynamics!
2 10.1 The Euclidean Propagator
Recall the standard propagator in real time \(t\), which is the coordinate representation of the time-evolution operator: \[ K(x_b, t_b; x_a, t_a) = \langle x_b | e^{-i\hat{H}(t_b - t_a)/\hbar} | x_a \rangle \]
If we perform a Wick rotation to imaginary time \(t = -i\tau\), the evolution operator becomes \(e^{-\hat{H}(\tau_b - \tau_a)/\hbar}\). This leads us to define the Euclidean propagator: \[ K_E(x_b, \tau_b; x_a, \tau_a) = \langle x_b | e^{-\hat{H}(\tau_b - \tau_a)/\hbar} | x_a \rangle \]
Notice the dramatic change! The real-time propagator is generally complex and describes oscillatory quantum interference. In stark contrast, the Euclidean propagator is strictly real and positive (assuming real boundary conditions \(x_a, x_b\)). It represents the matrix elements of the imaginary-time evolution operator, which is purely a damping factor rather than a unitary rotation.
Because the weight of each path is exactly \(e^{-S_E/\hbar}\), and since the Euclidean action \(S_E\) is real and bounded from below, the Euclidean propagator acts like a diffusion process. Instead of waves traveling and interfering, “probability” smoothly diffuses and settles into the lowest energy configuration.
3 10.2 Spectral Representation
To see exactly why the Euclidean propagator is so useful, let’s express it in terms of the energy eigenstates of the system. Let the states \(|n\rangle\) be the eigenstates of the Hamiltonian with eigenvalues \(E_n\): \[ \hat{H}|n\rangle = E_n|n\rangle \] We assume the spectrum is bounded from below, with \(E_0 \le E_1 \le E_2 \le \dots\).
We can insert a complete set of these energy eigenstates, \(\sum_n |n\rangle \langle n| = \mathbb{I}\), into the definition of the Euclidean propagator. Let \(\tau = \tau_b - \tau_a\). Then: \[ K_E(x_b, \tau; x_a, 0) = \langle x_b | e^{-\hat{H}\tau/\hbar} \left( \sum_n |n\rangle\langle n| \right) | x_a \rangle \] \[ K_E(x_b, \tau; x_a, 0) = \sum_n \langle x_b | n \rangle \langle n | x_a \rangle e^{-E_n\tau/\hbar} \] Since \(\langle x_b | n \rangle = \psi_n(x_b)\) and \(\langle n | x_a \rangle = \psi_n^*(x_a)\), we obtain the spectral representation: \[ \boxed{ K_E(x_b, \tau; x_a, 0) = \sum_{n=0}^{\infty} \psi_n(x_b)\psi_n^*(x_a) e^{-E_n\tau/\hbar} } \]
Textbook reference: Shankar, Principles of Quantum Mechanics, Ch. 8.
Notice that these are real exponentials, not oscillating phases! In real time, the terms would be \(e^{-iE_nt/\hbar}\), keeping all states “alive” forever. In imaginary time, each term decays exponentially at a rate determined by its energy \(E_n\). The higher the energy, the faster it vanishes.
4 10.3 Ground-State Projection — The Magic of Large \(\tau\)
What happens if we take the limit of very large imaginary time, \(\tau \to \infty\)? Let’s factor out the term corresponding to the ground state energy, \(E_0\): \[ K_E(x_b, \tau; x_a, 0) = e^{-E_0\tau/\hbar} \left[ \psi_0(x_b)\psi_0^*(x_a) + \sum_{n=1}^{\infty} \psi_n(x_b)\psi_n^*(x_a) e^{-(E_n-E_0)\tau/\hbar} \right] \]
As \(\tau\) grows very large, the exponential factors in the sum, \(e^{-(E_n-E_0)\tau/\hbar}\), will strongly tend to zero because \(E_n > E_0\) for all \(n \ge 1\). Consequently, all excited-state contributions are exponentially suppressed relative to the ground state.
In the strict limit \(\tau \to \infty\), only the ground state survives: \[ \boxed{ \lim_{\tau \to \infty} K_E(x_b, \tau; x_a, 0) = \psi_0(x_b)\psi_0^*(x_a) e^{-E_0\tau/\hbar} } \]
THIS IS INCREDIBLY POWERFUL. If we have a complicated interacting theory where we do not know the exact eigenstates, we can simply construct the Euclidean path integral, evaluate it for large \(\tau\), and it will automatically filter out all excited states and hand us the ground state wave function and energy! This is the primary workhorse technique in Lattice Quantum Chromodynamics (LQCD) to extract the masses of protons, neutrons, and other hadrons.
5 10.4 Extracting the Ground-State Energy
We can extract the exact numerical value of the ground-state energy directly from the asymptotic behavior of the Euclidean propagator. Rearranging the large-\(\tau\) limit: \[ \ln K_E(x_b, \tau; x_a, 0) \approx -\frac{E_0\tau}{\hbar} + \ln(\psi_0(x_b)\psi_0^*(x_a)) \] If we divide by \(\tau\) and take the limit, the second term vanishes: \[ \boxed{ E_0 = -\lim_{\tau\to\infty} \frac{\hbar}{\tau} \ln K_E(x_b, \tau; x_a, 0) } \]
An even more elegant way to do this, without depending on the boundary coordinates \(x_a\) and \(x_b\), is to set \(x_b = x_a = x\) and integrate over all space. This defines an object \(Z(\tau)\): \[ Z(\tau) = \int dx K_E(x, \tau; x, 0) = \int dx \sum_{n} |\psi_n(x)|^2 e^{-E_n\tau/\hbar} \] Using the normalization of the wavefunctions \(\int dx |\psi_n(x)|^2 = 1\), we get: \[ Z(\tau) = \sum_n e^{-E_n\tau/\hbar} \] Taking the large-\(\tau\) limit here gives \(Z(\tau) \approx e^{-E_0\tau/\hbar}\). Thus: \[ \boxed{ E_0 = -\lim_{\tau\to\infty} \frac{\hbar}{\tau} \ln Z(\tau) } \]
6 10.5 Extracting the Energy Gap
What if we want to know the first excited state, or more specifically, the energy gap \(E_1 - E_0\)? We look at the Euclidean two-point correlation function: \[ C(\tau_1 - \tau_2) = \langle x(\tau_1) x(\tau_2) \rangle_E = \frac{1}{Z} \int \mathcal{D}x \, x(\tau_1) x(\tau_2) e^{-S_E/\hbar} \] In the operator formalism, assuming \(\tau_1 > \tau_2\) and evaluating in the ground state (which we obtain implicitly by taking the total time interval \(T \to \infty\)), this correlation function is: \[ C(\tau_1 - \tau_2) = \langle 0 | \hat{x}(\tau_1) \hat{x}(\tau_2) | 0 \rangle \] We use the imaginary-time Heisenberg picture: \(\hat{x}(\tau) = e^{\hat{H}\tau/\hbar} \hat{x}(0) e^{-\hat{H}\tau/\hbar}\). \[ C(\tau_1 - \tau_2) = \langle 0 | e^{\hat{H}\tau_1/\hbar} \hat{x} e^{-\hat{H}\tau_1/\hbar} e^{\hat{H}\tau_2/\hbar} \hat{x} e^{-\hat{H}\tau_2/\hbar} | 0 \rangle \] Since \(\hat{H}|0\rangle = E_0|0\rangle\), the outer operators give \(e^{E_0\tau_1/\hbar}\) and \(e^{-E_0\tau_2/\hbar}\): \[ C(\tau_1 - \tau_2) = \langle 0 | e^{E_0\tau_1/\hbar} \hat{x} e^{-\hat{H}(\tau_1-\tau_2)/\hbar} \hat{x} e^{-E_0\tau_2/\hbar} | 0 \rangle \] Now, we boldly insert a complete set of energy eigenstates \(\sum_n |n\rangle \langle n| = \mathbb{I}\) between the two \(\hat{x}\) operators: \[ C(\tau_1 - \tau_2) = \sum_n e^{E_0\tau_1/\hbar} \langle 0 | \hat{x} | n \rangle e^{-E_n(\tau_1-\tau_2)/\hbar} \langle n | \hat{x} | 0 \rangle e^{-E_0\tau_2/\hbar} \] \[ C(\tau_1 - \tau_2) = \sum_n |\langle 0|\hat{x}|n\rangle|^2 e^{-(E_n - E_0)(\tau_1 - \tau_2)/\hbar} \] Let \(\Delta\tau = \tau_1 - \tau_2\). For large separations \(|\Delta\tau| \to \infty\), the dominant term comes from the lowest energy level for which the matrix element \(\langle 0|\hat{x}|n\rangle\) is non-zero. For a symmetric potential, \(\langle 0|\hat{x}|0\rangle = 0\). Thus, the lowest surviving state is \(n=1\). \[ \boxed{ C(\Delta\tau) \xrightarrow{|\Delta\tau| \to \infty} |\langle 0|\hat{x}|1\rangle|^2 e^{-(E_1 - E_0)|\Delta\tau|/\hbar} } \] The exponential decay rate of the correlation function explicitly gives the energy gap \(E_1 - E_0\)!
7 10.6 The Euclidean Generating Functional
To systematically compute these correlation functions, we construct the Euclidean Generating Functional: \[ \boxed{ Z_E[J] = \int \mathcal{D}x \, \exp\left[ -\frac{S_E[x]}{\hbar} + \int J(\tau)x(\tau)d\tau \right] } \]
Compare this to the Minkowski version, where we had \(e^{iS/\hbar + i \int Jx dt}\). In Euclidean space, the \(i\)’s vanish, making this purely a statistical weighting function. We can generate arbitrary \(n\)-point Euclidean correlation functions through functional differentiation: \[ \langle x(\tau_1) \dots x(\tau_n) \rangle_E = \frac{1}{Z_E[0]} \frac{\delta^n Z_E[J]}{\delta J(\tau_1) \dots \delta J(\tau_n)} \Bigg|_{J=0} \]
8 10.7 The Partition Function & Periodic Paths
Now we reach THE BIG RESULT of this module. What happens if we take the integral we defined earlier, \(Z(\beta\hbar) = \int dx K_E(x, \beta\hbar; x, 0)\)? We have already shown that: \[ \int dx K_E(x, \beta\hbar; x, 0) = \sum_n e^{-\beta E_n} \] But wait… \(\sum_n e^{-\beta E_n}\) is exactly the canonical partition function from statistical mechanics, provided we identify \(\beta\) with \(1 / (k_B T)\)!
Let’s look at this from the path integral side. The propagator \(K_E(x, \beta\hbar; x, 0)\) is the sum over all paths that start at \(x\) at \(\tau=0\) and end at the same \(x\) at \(\tau=\beta\hbar\). Integrating over all starting/ending points \(x\) means we are summing over all possible periodic paths. \[ \boxed{ Z(\beta) = \text{Tr}(e^{-\beta\hat{H}}) = \oint \mathcal{D}x \, e^{-S_E/\hbar} } \] Here, the symbol \(\oint \mathcal{D}x\) specifically denotes integration over paths that satisfy the periodic boundary condition: \[ x(0) = x(\beta\hbar) \]
This is one of the most beautiful connections in theoretical physics. A quantum mechanical system in thermal equilibrium at temperature \(T\) is mathematically completely equivalent to the same quantum system evolving in imaginary time over a bounded interval of length \(\beta\hbar = \hbar/(k_B T)\) with periodic boundary conditions!
9 10.8 The Thermal Circle
Because of the periodic boundary condition \(x(0) = x(\beta\hbar)\), the imaginary time coordinate \(\tau\) ceases to be a straight infinite line. Instead, it is topologically compactified into a circle of circumference \(\beta\hbar\).
Imagine a cylinder. The spatial coordinates run along the length of the cylinder, while the imaginary time coordinate wraps around the circumference. - At \(T \to 0\) (\(\beta \to \infty\)), the circumference becomes infinitely large, and the cylinder unrolls into an infinite flat sheet (recovering zero-temperature quantum mechanics). - At high temperatures \(T \to \infty\) (\(\beta \to 0\)), the cylinder squashes down into an infinitesimally thin loop. The paths have no “room” to fluctuate in imaginary time, and the system collapses entirely into classical mechanics!
10 📝 Worked Examples
10.0.1 Worked Example 1: Harmonic Oscillator Partition Function
Problem: Compute the canonical partition function \(Z(\beta)\) for the quantum harmonic oscillator directly from the Euclidean path integral.
Solution: From Module 9, the Euclidean propagator for the harmonic oscillator is: \[ K_E(x_b, \tau; x_a, 0) = \sqrt{\frac{m\omega}{2\pi\hbar\sinh(\omega\tau)}} \exp\left[ -\frac{m\omega}{2\hbar\sinh(\omega\tau)} \left( (x_a^2+x_b^2)\cosh(\omega\tau) - 2x_a x_b \right) \right] \]
To find the partition function, we set \(x_a = x_b = x\), \(\tau = \beta\hbar\), and integrate over \(x\): \[ Z(\beta) = \int_{-\infty}^{\infty} dx K_E(x, \beta\hbar; x, 0) \] \[ Z(\beta) = \int_{-\infty}^{\infty} dx \sqrt{\frac{m\omega}{2\pi\hbar\sinh(\beta\hbar\omega)}} \exp\left[ -\frac{m\omega}{2\hbar\sinh(\beta\hbar\omega)} x^2 \left( 2\cosh(\beta\hbar\omega) - 2 \right) \right] \]
We use the half-angle identity \(\cosh(\theta) - 1 = 2\sinh^2(\theta/2)\) to simplify the term in the parentheses: \[ 2\cosh(\beta\hbar\omega) - 2 = 4\sinh^2\left(\frac{\beta\hbar\omega}{2}\right) \] And the double-angle identity for the denominator: \(\sinh(\beta\hbar\omega) = 2\sinh(\frac{\beta\hbar\omega}{2})\cosh(\frac{\beta\hbar\omega}{2})\). Substituting these into the exponent: \[ -\frac{m\omega}{2\hbar \left( 2\sinh(\frac{\beta\hbar\omega}{2})\cosh(\frac{\beta\hbar\omega}{2}) \right)} \left[ 4\sinh^2\left(\frac{\beta\hbar\omega}{2}\right) x^2 \right] = -\frac{m\omega}{\hbar} \frac{\sinh(\frac{\beta\hbar\omega}{2})}{\cosh(\frac{\beta\hbar\omega}{2})} x^2 = -\frac{m\omega}{\hbar} \tanh\left(\frac{\beta\hbar\omega}{2}\right) x^2 \]
This leaves us with a standard Gaussian integral \(\int e^{-ax^2} dx = \sqrt{\pi/a}\): \[ Z(\beta) = \sqrt{\frac{m\omega}{4\pi\hbar\sinh(\frac{\beta\hbar\omega}{2})\cosh(\frac{\beta\hbar\omega}{2})}} \int_{-\infty}^{\infty} dx \exp\left[ -\frac{m\omega}{\hbar} \tanh\left(\frac{\beta\hbar\omega}{2}\right) x^2 \right] \] \[ Z(\beta) = \sqrt{\frac{m\omega}{4\pi\hbar\sinh(\frac{\beta\hbar\omega}{2})\cosh(\frac{\beta\hbar\omega}{2})}} \sqrt{\frac{\pi\hbar}{m\omega \tanh(\frac{\beta\hbar\omega}{2})}} \] Notice how the \(m, \omega, \pi\), and \(\hbar\) neatly cancel out inside the combined square root: \[ Z(\beta) = \sqrt{\frac{1}{4\sinh(\frac{\beta\hbar\omega}{2})\cosh(\frac{\beta\hbar\omega}{2})} \times \frac{\cosh(\frac{\beta\hbar\omega}{2})}{\sinh(\frac{\beta\hbar\omega}{2})}} = \sqrt{\frac{1}{4\sinh^2(\frac{\beta\hbar\omega}{2})}} = \frac{1}{2\sinh(\frac{\beta\hbar\omega}{2})} \]
To verify this, we expand using \(2\sinh(y) = e^y - e^{-y}\): \[ Z(\beta) = \frac{1}{e^{\beta\hbar\omega/2} - e^{-\beta\hbar\omega/2}} = \frac{e^{-\beta\hbar\omega/2}}{1 - e^{-\beta\hbar\omega}} = e^{-\beta\hbar\omega/2} \sum_{n=0}^{\infty} (e^{-\beta\hbar\omega})^n = \sum_{n=0}^{\infty} e^{-\beta\hbar\omega(n+1/2)} \] This precisely matches the known partition function \(\sum_n e^{-\beta E_n}\) with \(E_n = \hbar\omega(n+1/2)\).
10.0.2 Worked Example 2: Extracting Energy Gaps
Problem: Use the results of the harmonic oscillator to extract \(E_0\) and the gap \(E_1 - E_0\).
Solution: From Section 10.4, \(E_0 = -\lim_{\tau\to\infty} \frac{\hbar}{\tau} \ln Z(\tau)\). Using our result from above with \(\beta\hbar \to \tau\): \[ Z(\tau) = \frac{e^{-\omega\tau/2}}{1 - e^{-\omega\tau}} \] \[ \ln Z(\tau) = -\frac{\omega\tau}{2} - \ln(1 - e^{-\omega\tau}) \] In the limit \(\tau \to \infty\), \(e^{-\omega\tau} \to 0\), so the logarithmic term vanishes. \[ E_0 = -\lim_{\tau\to\infty} \frac{\hbar}{\tau} \left( -\frac{\omega\tau}{2} \right) = \frac{\hbar\omega}{2} \] This is the correct zero-point energy!
For the gap, recall the continuous Euclidean correlation function for the HO (which we derive from \(Z_E[J]\)): \[ C(\tau) = \frac{\hbar}{2m\omega} e^{-\omega|\tau|} \] Comparing this to the general asymptotic form \(C(\tau) \propto e^{-(E_1 - E_0)|\tau|/\hbar}\), we immediately see that: \[ \frac{E_1 - E_0}{\hbar} = \omega \implies E_1 - E_0 = \hbar\omega \] The energy gap is exactly \(\hbar\omega\), as expected.
10.1 🤔 Quick Question 1
We forced the paths to end exactly where they started, \(x(\beta\hbar) = x(0)\). Why does this particular geometry have anything to do with temperature?
Setting the endpoints to be equal (\(x_a = x_b = x\)) and integrating over all \(x\) is the continuous coordinate representation of the Trace operation in linear algebra: \(\int \langle x | \hat{O} | x \rangle dx = \text{Tr}(\hat{O})\). The canonical partition function from statistical mechanics is fundamentally defined as the trace of the Boltzmann weight: \(Z = \text{Tr}(e^{-\beta\hat{H}})\). Thus, making the paths periodic and integrating is the path integral’s way of executing the trace operation required by thermal physics.
10.2 🤔 Quick Question 2
We saw how \(E_0\) dominates at very large \(\tau\), and \(E_1\) dominates the decay of the correlation function. Are higher excited states like \(E_2, E_3\) lost forever?
Yes, they can be extracted! While \(E_0\) and \(E_1\) dominate the asymptotic (\(\tau \to \infty\)) limits, the full correlation function contains a sum over all states: \(\sum_n |c_n|^2 e^{-(E_n-E_0)\tau/\hbar}\). By carefully measuring (or simulating) the correlation function at intermediate values of \(\tau\), and fitting it to a sum of multiple decaying exponentials, one can systematically extract \(E_2, E_3\), etc. This requires high precision, but it is routinely done in lattice QCD.
11 Summary Table
| Concept | Minkowski (Real Time, \(t\)) | Euclidean (Imaginary Time, \(\tau\)) |
|---|---|---|
| Time Variable | \(t\) | \(\tau = i t\) |
| Action | \(S = \int (\frac{1}{2}m\dot{x}^2 - V(x))dt\) | \(S_E = \int (\frac{1}{2}m\dot{x}^2 + V(x))d\tau\) |
| Path Weight | \(e^{iS/\hbar}\) (Oscillating phase) | \(e^{-S_E/\hbar}\) (Exponential damping) |
| Evolution Op. | \(e^{-i\hat{H}t/\hbar}\) (Unitary rotation) | \(e^{-\hat{H}\tau/\hbar}\) (Hermitian decay) |
| Asymptotic (\(\infty\)) | Highly oscillatory, hard to define | Projects completely onto the Ground State \(E_0\) |
| Trace \(\to\) Physics | — | Canonical Partition Function at \(T = \frac{\hbar}{k_B \beta}\) |
11.0.0.1 🎯 Checkpoint
Before moving on, verify that you understand: 1. Why the limit \(\tau \to \infty\) filters out all excited states and leaves only the ground state. 2. The mathematical steps showing that inserting complete sets of states into \(\langle x(\tau_1)x(\tau_2)\rangle_E\) leads to an exponential decay controlled by \(E_1 - E_0\). 3. The topological distinction between zero-temperature QFT (an infinite imaginary time axis) and finite-temperature QFT (a circular imaginary time axis).
12 Practice Problems
- The Double Well Potential: Suppose a particle is in a double well potential. At large \(\tau\), the correlation function \(C(\tau)\) is measured to decay as \(A e^{-\Delta E \tau/\hbar}\). Physically, what does \(\Delta E\) represent in this specific system, and why is it very small?
- Euclidean Propagator of a Free Particle: Perform a Wick rotation on the real-time free particle propagator. Verify that it satisfies the diffusion equation rather than the Schrödinger equation.
- High Temperature Limit: Start with the path integral expression for \(Z(\beta)\). Take the limit \(\beta \to 0\) (high temperature). Argue that the kinetic energy term strictly enforces that the only contributing paths are constants \(x(\tau) = x_0\). Show that this recovers the classical partition function \(Z \propto \int dx_0 e^{-\beta V(x_0)}\).
- Parity and Energy Gaps: In Section 10.5, we assumed \(\langle 0 | \hat{x} | 0 \rangle = 0\) because of symmetry, which led to the leading term being \(n=1\). If we had an asymmetric potential where \(\langle 0 | \hat{x} | 0 \rangle \ne 0\), how would you modify the observable to correctly extract the gap \(E_1 - E_0\)? (Hint: Consider the connected correlation function \(\langle x(\tau_1)x(\tau_2)\rangle - \langle x \rangle^2\)).
- Finite Temperature Correlation: At finite temperature \(T = 1/(k_B\beta)\), the time direction is a circle. Re-derive the two point correlation function expansion, but this time trace over all states instead of just taking the ground state expectation. Show that the correlation function \(C(\tau)\) must be periodic in \(\tau\) with period \(\beta\hbar\).