2026-10-01 · chapter Yang–Mills and the mass gapPart 2 Physical preparationpath integrals

The idea of the path integral — add up every path, and making time imaginary turns it into probability

The path integral rewrites quantum mechanics as the sum of contributions from every path the particle could take. Feynman (1948) wrote the amplitude (the propagator) for a particle leaving $a$ at time 0 to be found at $b$ at time $t$ in the following form.

$S$ is the action of 2-01, and the paths are all those satisfying $x(0)=a$, $x(t)=b$. Every path brings a complex number of absolute value 1, differing only in phase, and the probability is the square of the absolute value of the sum Known. The idea is said to go back to Dirac (1933). As a metaphor: just as light passes through both slits and interferes, the particle passes along every path and interferes. Precisely, the "sum" in (1) is not defined as it stands, and is read as the limit of the discretised integrals of section 03.

Many computable path integrals reduce to Gaussian integrals. Let $A$ be a positive-definite symmetric matrix; diagonalising it and using the one-variable $\int e^{-ax^2/2+Jx}dx=\sqrt{2\pi/a}\,e^{J^2/2a}$ for each eigenvalue,

Normalised, this is the Gaussian distribution with mean 0 and covariance matrix $A^{-1}$ (1-08). Differentiating with respect to $J$ and putting $J=0$,

The expectation of a product of $2k$ factors is the sum over the $(2k-1)!!$ ways of pairing them off (Isserlis 1918) Known. In physics the same combinatorics appears in operator form in Wick (1950), and goes by the name of Wick's theorem. Drawing these pairings as lines gives Feynman diagrams (4-03). Example. For $A=\begin{pmatrix}2&-1\\-1&2\end{pmatrix}$ the integral is $2\pi/\sqrt3$, $\langle x_1^2\rangle=\frac23$, $\langle x_1x_2\rangle=\frac13$, $\langle x_1^2x_2^2\rangle=\frac49+2\cdot\frac19=\frac23$ Computed. The value is set by $\det A$, and the correlations by $A^{-1}$.

There are three ingredients. Composition: from $e^{-i(t_1+t_2)H/\hbar}=e^{-it_2H/\hbar}e^{-it_1H/\hbar}$ comes the formula $K(b,t_1+t_2;a,0)=\int K(b,t_1+t_2;c,t_1)K(c,t_1;a,0)\,dc$ that sums over all intermediate places $c$. The short-time kernel: for a free particle, exactly

and the exponent is the action of the uniform-velocity path. The product formula: split the Hamiltonian as $H=T+V$. $T=P^2/2m$ is the operator of kinetic energy ($P=-i\hbar\,d/dx$ is the momentum of 2-05), and $V$ is the operator of multiplication by the potential energy $V(x)$. $T$ and $V$ do not commute, but if, for example, $T+V$ is self-adjoint on the intersection of the domains of $T$ and $V$, then $e^{-it(T+V)/\hbar}=\lim_N\bigl(e^{-i\varepsilon T/\hbar}e^{-i\varepsilon V/\hbar}\bigr)^N$, $\varepsilon=t/N$ holds Known. It is a special case of Trotter's (1959) product theorem for contraction semigroups, and this form is in chapter VIII of Reed–Simon I. Putting these together, with $x_0=a$, $x_N=b$,

The sum in the exponent is a Riemann sum for the action of a broken-line path, that is, the discretised action of 2-01. (1) is shorthand for this limit, also written $\int\mathcal Dx\,e^{iS/\hbar}$.

The integrand of (5) has absolute value 1 and oscillates, and does not converge absolutely. Figure 1 shows the model case $\int e^{ix^2/2}dx=\sqrt{2\pi}\,e^{i\pi/4}$ (the Fresnel integral). On the right is the trace of the partial integral (the integral with a moving upper limit) $F(x)=\int_{-6}^{x}w\,dx'$; for $\theta=0$ both ends spiral and cancel, and the net progress happens only near the stationary point $x=0$ where the slope of the phase is 0. For $\hbar\to0$ (the sign is the sign of $S''(x_c)$) Known

In path space too, the contribution comes from near the classical path with $\delta S=0$ (2-01). If the action is quadratic, (6) is exact, and (4) has that form. Example. A 1 g ball moving at 1 cm/s for 1 second gives $S/\hbar\approx4.7\times10^{26}$ Computed. For displaced paths the phase turns round many times, and they cancel.

ComputedThe numbers in the figure were computed on this machine (the trapezoidal rule on the spot). Table 1 gives values from the complex error function in python3.

ComputedTable 1 The absolute value of the difference between $\int_{-X}^{X}w\,dx$ and the exact value $\sqrt{2\pi}\,e^{i(\pi/2-\theta)/2}$.

In real time the error decreases only like $2/X$; rotating even slightly makes it vanish at the rate $e^{-\sin\theta\,X^2/2}$.

Replacing the step $\varepsilon$ in (5) by $-i\delta$ ($\delta\gt0$), the exponent becomes $-\frac\delta\hbar\bigl[\frac m2(\Delta x/\delta)^2+V\bigr]$, and the sign of the potential energy changes. With $\tau=N\delta$,

We call $S_E$ the Euclidean action. The weight is positive, and if $V$ is bounded below the integral converges absolutely. The $V=0$ part is a random walk whose increments independently follow normal distributions with variance $\hbar\delta/m$; for $N\to\infty$ it becomes a probability measure on continuous paths, the Wiener measure (the distribution of Brownian motion), and $V$ enters as a weight.

$X_s$ is Brownian motion leaving $x$ and spreading with variance $\hbar s/m$. (8) is called the Feynman–Kac formula (Kac 1949). It is a theorem when, for example, $V$ is bounded and continuous, and it holds for quite general $V$ bounded below Known. In real time there is no measure; in imaginary time there is. This is one reason to construct field theories on the Euclidean side (the rotation for fields is in 4-02 §02 and 4-10). Example. The stationary point $\delta S_E=0$ is motion in $-V$, and the $S_0$ of the double well of 2-05 was the action of that path (the instanton, 3-05).

The roughness of paths and the measure

The paths themselves are nowhere differentiable with probability 1 Known. Since $\Delta x\sim\sqrt\delta$, the expectation $\langle S_{\rm kin}\rangle$ of the kinetic term $S_{\rm kin}=\sum(\Delta x)^2/2\delta$ ($\hbar=m=1$) of $S_E$ grows in proportion to the number of steps Computed (for the ring of section 07, 6.97, 126.96, 2046.96 for $\beta=4$, $N=16,256,4096$; almost $N/2$). The potential-energy part $\sum\delta\,V(x_j)$ on its own approaches $\int_0^\tau V(X_s)\,ds$, but the weight of the kinetic term $e^{-S_{\rm kin}/\hbar}$ and $\mathcal Dx$ have no limits separately; only the product survives, as the Wiener measure (both in 1-09).

In thermal equilibrium a state $n$ appears with weight $e^{-\beta E_n}$ ($\beta=1/(k_B\times\text{temperature})$, $k_B$ is Boltzmann's constant), and their sum is the partition function (2-08). Taking the trace means putting $a=b$ and integrating over $a$, so by (7) it becomes an integral over closed paths on a ring of length $\beta\hbar$.

Read in discretised form, it is the partition function of a classical chain in which $x_0,\dots,x_{N-1}$ arranged on a ring are joined by springs $\frac{m}{2\delta}(x_{j+1}-x_j)^2$ and each point carries $V$. Quantum mechanics takes the same form as one-dimensional classical statistical mechanics, with $\hbar$ playing the role of temperature and $\beta\hbar$ that of the length of the chain. The exposition of the problem attributes to Symanzik the view of reading the Euclidean expectations of a field theory as a classical statistical-mechanical ensemble with density proportional to $e^{-S}$ (Jaffe–Witten §3) Known. The chain of this section is its one-dimensional version.

Example: the oscillator ring

For the oscillator, $Z=1/(2\sinh\frac{\beta\hbar\omega}{2})$; the distribution of one point $x(0)$ of the ring is the distribution of position in thermal equilibrium, with variance

and at low temperature it approaches the $|\psi_0|^2$ of the ground state (2-06), at high temperature the classical Boltzmann distribution.

ComputedThe numbers in the figure were computed on this machine. The paths are exact samples of (7) with the ring cut into 256 pieces (the real Fourier components are independent normal variables, drawn with random numbers from a fixed seed). Table 2 gives python3 values from the same procedure.

ComputedTable 2 $\langle x^2\rangle$ at one point of the ring (unit $\hbar/m\omega$; the ground state gives 0.5).

At high temperature, on a short ring, the path shrinks almost to a point (a classical particle); at low temperature, quantum fluctuations are seen as fluctuations of a path stretched in the imaginary-time direction.

$\hbar=m=\omega=1$: cut a ring of length $\beta$ into $N$ pieces. From here on $a=\beta/N$ is the step width (the lattice spacing), a different symbol from the starting point $a$ of sections 01–06. The $A$ of $S_E=\frac12x\cdot Ax$ is $(Ax)_j=\frac1a(2x_j-x_{j+1}-x_{j-1})+ax_j$ (indices counted modulo $N$), a circulant matrix connected only to neighbours. Its eigenvectors are $v_j=e^{2\pi ijk/N}$ (discrete Fourier, $k=0,\dots,N-1$), and substituting $v_{j\pm1}=e^{\pm2\pi ik/N}v_j$ gives the eigenvalues $\frac1a(2-2\cos\frac{2\pi k}{N})+a$. From (2) and the normalisation $(2\pi a)^{-N/2}$,

Compared with the continuous $1/(2\sinh\frac\beta2)$, the discretised system is the same as an oscillator with frequency $\Omega_a=\kappa/a=1-a^2/24+\cdots$. The correlations are set by $A^{-1}$, and for $N\to\infty$,

As in (8) of 2-06, correlations decrease at the rate of the gap. The correlation length counted in lattice units is $\xi=1/\kappa=1/(a\Omega_a)$.

The correlation length and the continuum limit

ComputedTable 3 The ring with $\beta=4$ ($Z_N$: the numerical determinant and (11) agree to 10 digits).

The discrepancy vanishes like $a^2$, and the correlation length grows like $1/a$ (Figure 3). Approaching the continuum means bringing the lattice chain towards a state in which "the correlation length is many times the lattice spacing", in the language of statistical mechanics towards a critical point. We hold the physical gap $1/(\xi a)$ fixed and let $\xi\to\infty$. In lattice gauge theory too, the continuum limit is thought to be taken in this form Physics. That the limit exists in four dimensions has not been shown (5-05, 5-09, 6-07).

ComputedThe numbers in the figure were computed on this machine (the closed formula (12) computed on the spot; $\kappa$, $\xi$, $\Omega_a$ at $a=1,\frac12,\dots,\frac1{16}$ agree with the python3 values to the digits shown). On the left, the smaller $a$, the further the dots continue; on the right, the dots settle onto the same dashed line.

Even without Gaussians, discretised integrals can be computed as products of matrices. Make one step of (7) ($\hbar=m=1$, step $a$) into the integral kernel of $e^{-aV/2}e^{-aP^2/2}e^{-aV/2}$

then $Z_N=\operatorname{Tr}T_a^N$; this is the transfer matrix of statistical mechanics. Writing the eigenvalues as $e^{-aE_n(a)}$, $E_n(a)\to E_n$ as $a\to0$. The kernel is positive at every point ($T_a(x,y)\gt0$). Moreover $V=x^4$ grows at infinity, so $\iint T_a(x,y)^2\,dx\,dy\le\frac{1}{2\pi a}\bigl(\int e^{-aV}dx\bigr)^2\lt\infty$, and $T_a$ is compact (an operator that can be approximated as closely as we like by finite-dimensional matrices, and whose eigenvalues accumulate nowhere but 0). With these two together, the integral-operator version of the Perron–Frobenius theorem (said to begin with Jentzsch 1912; the form for $L^2$ is in Reed–Simon IV) shows that the largest eigenvalue is simple and its eigenfunction can be taken positive Known. Hence the ground state is non-degenerate (2-05, 5-03).

ComputedTable 4 The levels of the transfer matrix for $H=-\frac12\frac{d^2}{dx^2}+x^4$ (lattice width for $x$ ≤ 0.02, $|x|\le3.2$) and their differences from the values of Table 1 of 2-05 ($E_0=0.667986$, $E_1-E_0=1.725658$). Halving the lattice width changes nothing to 9 digits.

The difference is roughly 1/4 each time the step is halved. The discretised path integral and operator quantum mechanics give the same levels and gap.

We extend the discretisation of section 07 directly to fields. In field theory the path $x(\tau)$ is replaced by a field $\phi(x)$ ($d$ is the dimension of spacetime including imaginary time), and $S_E=\int(\frac12|\nabla\phi|^2+\frac12m^2\phi^2+\cdots)d^dx$ ($m$ is here the mass of the field; 4-02). Discretising on a lattice of spacing $a$, a field without interactions is again the Gaussian integral of (2), and the fluctuation at a point is a diagonal entry of $A^{-1}$.

$d=1$ is the chain of section 07 itself; (14) equals the value $a/(2\sinh\kappa)$ of (12) at $j=0$, and converges to $1/2m$. For $d\ge2$ it diverges. The divergence of the variance at a point is also treated in 1-09, from the side of Gaussian measures in infinite dimensions. Here we look, as a continuation of discretising the path integral, at what happens each time the step is halved.

ComputedEach time the step is halved, $d=1$ converges to $1/2$, $d=2$ grows by about $\ln2/(2\pi)\approx0.110$ each time ($G_a(0)\approx\frac1{2\pi}\log\frac1a+0.276$), $d=3$ doubles ($\approx0.2527/a$), and $d=4$ quadruples ($\approx0.1549/a^2$) ($m=1$; (14) is turned into a one-dimensional integral of the Bessel function $I_0$ and integrated numerically). The table of values for each step and the figure for choosing the dimension are in 1-09 Table 3, Figure 3, which this book takes as authoritative. A figure setting it beside other quantities that switch at four is the dimension dial of 1-12 §10.

Fields in the continuum limit and Yang–Mills

A field in the continuum limit is a distribution without values at points (1-11). Products such as $\phi(x)^2$ need subtractions (renormalisation) (4-04, 6-03).

In four-dimensional Yang–Mills this divergence, the gauge freedom (4-08) and the interaction are handled at once. The exposition of the problem says that for a compact group and a finite (periodic) spacetime, Wilson's lattice makes the functional integral a finite-dimensional integral, and that one must then show that the limit lattice spacing → 0, volume → ∞ exists for the expectations of gauge-invariant observables (Jaffe–Witten §6.5) Known. For what is special about four dimensions, see the four-dimensions article.

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