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Path Integral Formulation

Field: Quantum Theory

A description of quantum theory that generalizes the action principle of classical mechanics.

Sequence of Expressions

The probability density of finding a particle at position xbx_b at time tbt_b, given it started at xax_a at tat_a, is given by the square modulus of the propagator, which is calculated via the path integral: \nP(xb,tb∣xa,ta)=∣⟨xb,tb∣xa,ta⟩∣2=∣∫Dx(t)eiℏS[x(t)]∣2P(x_b, t_b | x_a, t_a) = |\langle x_b, t_b | x_a, t_a \rangle|^2 = \left| \int \mathcal{D}x(t) e^{\frac{i}{\hbar} S[x(t)]} \right|^2
Definition

Phase Space

The phase space is defined by the coordinates (q,p)(q, p), where qq is the position and pp is the canonical momentum. The path integral formulation can be extended to include the evolution in phase space, often involving the Wigner function or the Moyal bracket, where the measure is dPhase=dqdpd\text{Phase} = dq dp. The propagator in phase space is related to the Wigner transform of the density matrix ρ\rho.
Define the Lagrangian L(x,x˙,t)L(x, \dot{x}, t). The Action Integral SS is the time integral of the Lagrangian along a path x(t)x(t): \nS[x(t)]=∫t1t2L(x(t),x˙(t),t)dtS[x(t)] = \int_{t_1}^{t_2} L(x(t), \dot{x}(t), t) dt
The functional integral ∫Dx(t)\int \mathcal{D}x(t) represents the summation over all possible paths x(t)x(t) in the configuration space. The weighted amplitude is defined by the exponential of the action S[x(t)]S[x(t)]: \n⟨xb,tb∣xa,ta⟩=∫Dx(t)eiℏS[x(t)]\langle x_b, t_b | x_a, t_a \rangle = \int \mathcal{D}x(t) e^{\frac{i}{\hbar} S[x(t)]}
The Wigner function W(q,p,t)W(q, p, t) provides a quasi-probability distribution in phase space, defined via the Fourier transform of the density matrix ρ(q,q′,t)\rho(q, q', t): \nW(q,p,t)=12πℏ∫−∞∞eipτ/ℏ⟨q−τ/2∣ρ(t)∣q+τ/2⟩dτW(q, p, t) = \frac{1}{2\pi \hbar} \int_{-\infty}^{\infty} e^{i p \tau / \hbar} \langle q - \tau/2 | \rho(t) | q + \tau/2 \rangle d\tau
The path integral formulation relates the Lagrangian LL to the Hamiltonian HH via the Legendre transform: H(q,p,t)=p⋅q˙−L(q,q˙,t)H(q, p, t) = p \cdot \dot{q} - L(q, \dot{q}, t). The path integral incorporates this by defining the action SS using the canonical variables (q,p)(q, p) and the Hamiltonian HH: \nS=∫t1t2(p⋅q˙−H(q,p,t))dtS = \int_{t_1}^{t_2} \big( p \cdot \dot{q} - H(q, p, t) \big) dt
Let ⟨xb,tb∣xa,ta⟩\langle x_b, t_b | x_a, t_a \rangle be the quantum mechanical propagator (amplitude). The Path Integral Formulation defines this amplitude as the functional integral over all possible paths x(t)x(t) connecting xax_a at tat_a to xbx_b at tbt_b: \n⟨xb,tb∣xa,ta⟩=∫Dx(t)eiℏS[x(t)]\langle x_b, t_b | x_a, t_a \rangle = \int \mathcal{D}x(t) e^{\frac{i}{\hbar} S[x(t)]}
The classical path xcl(t)x_{cl}(t) that minimizes the action integral S[x(t)]S[x(t)] is determined by the Euler-Lagrange equations, derived from the variation δS=0\delta S = 0: \n∂L∂x−ddt(∂L∂x˙)=0\frac{\partial L}{\partial x} - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}} \right) = 0
The Path Integral Formulation provides an alternative representation for the quantum mechanical propagator K(xb,tb;xa,ta)K(x_b, t_b; x_a, t_a), which satisfies the Schrödinger equation: \n∂K∂tb=iℏH^K\frac{\partial K}{\partial t_b} = \frac{i}{\hbar} \hat{H} K \nIn the path integral approach, this evolution is given by the functional integral over the Hamiltonian HH: \nK(xb,tb;xa,ta)=∫Dx(t)eiℏS[x(t)]K(x_b, t_b; x_a, t_a) = \int \mathcal{D}x(t) e^{\frac{i}{\hbar} S[x(t)]}
A Feynman path is any continuous, differentiable trajectory x(t)x(t) connecting the initial point xax_a at tat_a to the final point xbx_b at tbt_b. The path integral sums the contributions of all such paths, weighted by the phase factor eiS[x(t)]/ℏe^{iS[x(t)]/\hbar}: \n⟨xb,tb∣xa,ta⟩=∑all paths x(t)A[x(t)]eiℏS[x(t)]\langle x_b, t_b | x_a, t_a \rangle = \sum_{\text{all paths } x(t)} A[x(t)] e^{\frac{i}{\hbar} S[x(t)]}