Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part III, Chapter 2

Path Integrals in Quantum Mechanics

Harmonic oscillator, Gaussian integration, and the Euclidean path integral

Page 2 of 3

2.4 Harmonic Oscillator Path Integral

The harmonic oscillator is the most important exactly solvable system in all of physics. Its path integral provides a template for every Gaussian functional integral we will encounter in quantum field theory.

The action for a particle of mass m in a harmonic potential with frequency $\omega$ is:

$$S[x(t)] = \int_{t_a}^{t_b} dt\left[\frac{1}{2}m\dot{x}^2 - \frac{1}{2}m\omega^2 x^2\right]$$

We split the path into the classical solution $x_{\text{cl}}(t)$ plus fluctuations $y(t)$:

$$x(t) = x_{\text{cl}}(t) + y(t), \qquad y(t_a) = y(t_b) = 0$$

Because the action is quadratic, it splits cleanly:

$$S[x_{\text{cl}} + y] = S[x_{\text{cl}}] + S_2[y]$$

There is no linear term in y because $x_{\text{cl}}$ satisfies the equations of motion ($\delta S / \delta x\big|_{x_{\text{cl}}} = 0$). The fluctuation action is:

$$S_2[y] = \frac{m}{2}\int_{t_a}^{t_b} dt\left[\dot{y}^2 - \omega^2 y^2\right]$$

Classical Action

The classical path satisfying $\ddot{x}_{\text{cl}} = -\omega^2 x_{\text{cl}}$ with boundary conditions $x_{\text{cl}}(t_a) = x_a$, $x_{\text{cl}}(t_b) = x_b$ is:

$$x_{\text{cl}}(t) = \frac{x_a \sin\omega(t_b - t) + x_b \sin\omega(t - t_a)}{\sin\omega T}$$

where $T = t_b - t_a$. Evaluating the action on this path gives:

$$S_{\text{cl}} = \frac{m\omega}{2\sin\omega T}\left[(x_a^2 + x_b^2)\cos\omega T - 2x_a x_b\right]$$

Gaussian Integration over Fluctuations

The path integral factorizes:

$$K(x_b,t_b;x_a,t_a) = e^{iS_{\text{cl}}/\hbar} \int \mathcal{D}y\, e^{iS_2[y]/\hbar}$$

The fluctuation integral is a functional Gaussian. We integrate by parts to write:

$$S_2[y] = -\frac{m}{2}\int_{t_a}^{t_b} dt\; y(t)\left(\frac{d^2}{dt^2} + \omega^2\right)y(t)$$

Expand y(t) in the eigenbasis of the operator $\hat{O} = -d^2/dt^2 - \omega^2$ with Dirichlet boundary conditions. The eigenfunctions are $y_n(t) = \sin(n\pi(t-t_a)/T)$ with eigenvalues $\lambda_n = (n\pi/T)^2 - \omega^2$. The functional integral becomes an infinite product of ordinary Gaussian integrals:

$$\int \mathcal{D}y\, e^{iS_2[y]/\hbar} = \prod_{n=1}^{\infty}\sqrt{\frac{2\pi i\hbar}{m\lambda_n}} = \left[\det\left(\frac{m}{2\pi i\hbar}\hat{O}\right)\right]^{-1/2}$$

💡Functional Determinants

The determinant of a differential operator is the product of all its eigenvalues. This is infinite and requires regularization, but ratios of determinants are well-defined. We compute $\det(\hat{O}_\omega)/\det(\hat{O}_0)$ and use the known free-particle result to fix the overall normalization.

Using $\prod_{n=1}^{\infty}\left[1 - \omega^2 T^2/(n\pi)^2\right] = \sin(\omega T)/(\omega T)$, we obtain:

$$\int \mathcal{D}y\, e^{iS_2[y]/\hbar} = \sqrt{\frac{m\omega}{2\pi i\hbar \sin\omega T}}$$

The Propagator

Combining the classical factor with the fluctuation determinant gives the celebrated result:

$$\boxed{K(x_b,t_b;x_a,t_a) = \sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega T}}\exp\left\{\frac{im\omega}{2\hbar\sin\omega T}\left[(x_a^2+x_b^2)\cos\omega T - 2x_a x_b\right]\right\}}$$

Consistency check: Take $\omega \to 0$. Then $\sin\omega T \to \omega T$,$\cos\omega T \to 1$, and the propagator reduces to the free-particle result$K_{\text{free}} = \sqrt{m/(2\pi i\hbar T)}\,\exp[im(x_b - x_a)^2/(2\hbar T)]$.

Comparison with Operator Formalism

In the energy eigenbasis, the propagator is:

$$K(x_b,T;x_a,0) = \sum_{n=0}^{\infty} \psi_n(x_b)\psi_n^*(x_a)\,e^{-iE_n T/\hbar}$$

where $E_n = \hbar\omega(n + 1/2)$ and $\psi_n$ are Hermite-Gauss functions. One can verify that the Mehler kernel identity reproduces the path integral result exactly, confirming the equivalence of the two formalisms.

2.5 Euclidean Path Integral

The Minkowski path integral has a rapidly oscillating integrand $e^{iS/\hbar}$. We can make it convergent by performing a Wick rotation to imaginary time:

$$t \;\to\; -i\tau, \qquad \tau \in [0,\beta\hbar]$$

Under this substitution $dt = -i\,d\tau$ and $\dot{x}^2 = (dx/dt)^2 \to -(dx/d\tau)^2$, so:

$$iS = i\int dt\left[\frac{m}{2}\dot{x}^2 - V(x)\right] \;\longrightarrow\; -\int_0^{\beta\hbar} d\tau\left[\frac{m}{2}\left(\frac{dx}{d\tau}\right)^2 + V(x)\right] = -S_E$$

The Euclidean action $S_E$ is positive-definite (for reasonable potentials), so the integrand$e^{-S_E/\hbar}$ is a genuine damping factor. The Euclidean propagator is:

$$\boxed{K_E(x_b,\tau_b;x_a,\tau_a) = \int \mathcal{D}x(\tau)\, e^{-S_E[x]/\hbar}}$$

💡Why Euclidean?

After Wick rotation, the Minkowski metric $ds^2 = -dt^2 + dx^2$ becomes the Euclidean metric $ds^2 = d\tau^2 + dx^2$. The path integral now resembles a classical statistical mechanics partition function with $S_E$ playing the role of energy.

2.6 Connection to the Partition Function

A profound connection emerges when we impose periodic boundary conditions$x(0) = x(\beta\hbar) = x$ and integrate over x:

$$Z(\beta) = \int dx\, K_E(x,\beta\hbar; x, 0) = \int dx \int_{x(0)=x(\beta\hbar)=x} \mathcal{D}x(\tau)\, e^{-S_E/\hbar}$$

But we also know from quantum statistical mechanics that the partition function is:

$$Z(\beta) = \mathrm{Tr}\, e^{-\beta \hat{H}} = \sum_n e^{-\beta E_n}$$

Therefore the Euclidean path integral with periodic boundary conditions computes the thermal partition function. For the harmonic oscillator:

$$Z(\beta) = \sum_{n=0}^{\infty} e^{-\beta\hbar\omega(n+1/2)} = \frac{e^{-\beta\hbar\omega/2}}{1 - e^{-\beta\hbar\omega}} = \frac{1}{2\sinh(\beta\hbar\omega/2)}$$

This can also be obtained directly from the Euclidean path integral using the functional determinant of $-d^2/d\tau^2 + \omega^2$ with periodic boundary conditions, providing a beautiful consistency check.

Ground State Energy from Euclidean Time

Taking $\beta \to \infty$ (zero temperature) projects onto the ground state:

$$E_0 = -\lim_{\beta\to\infty} \frac{1}{\beta}\ln Z(\beta) = \frac{\hbar\omega}{2}$$

The Euclidean path integral thus provides a powerful non-perturbative method for extracting ground state properties. This idea is the foundation of lattice field theory and Monte Carlo simulations in QFT.

Key Concepts (This Page)

  • Harmonic oscillator path integral splits into classical action plus fluctuation determinant
  • Fluctuation integral is a functional Gaussian evaluated via eigenvalue product
  • Result matches operator formalism (Mehler kernel) exactly
  • Wick rotation $t \to -i\tau$ converts oscillating integrand to damped one
  • Euclidean path integral with periodic BCs gives thermal partition function $Z(\beta) = \mathrm{Tr}\,e^{-\beta H}$
  • Ground state energy extracted from $\beta \to \infty$ limit
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